Investigation of the electromagnetic properties of a transverse insert based on a planar layer of a chiral metamaterial in a rectangular waveguide

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Introduction

The electromagnetic properties of various artificial structures, known as metamaterials are currently under active study [1–5]. This is due to the fact that the use of composite media allows for the achievement of new electromagnetic properties that are unattainable with traditional materials. A metamaterial is a spatial composition of a medium, acting as a container, within which areas or volumes are replaced with a material of a different type. For the microwave range, the containers are usually dielectric and the embedded areas are conductive. At the development stage, it is possible to design the metamaterial structure to achieve the desired properties of electromagnetic field interaction. In most cases, resonant conductive microelements serve as the implanted areas. When these microelements have a mirror-like asymmetric shape, the metamaterial is usually referred to as chiral (from the Greek $\text{χiρo}$ – “hand”) [6–10]. In the classical sense, a chiral metamaterial is a composite medium consisting of a dielectric container in which uniformly placed and randomly oriented conductive microelements of mirror asymmetric shape are embedded. Several different types of chiral elements have been considered in the scientific literature, namely three-dimensional (such as Tellegen elements, single- and multiturn spirals, and mutually orthogonal spirals) and two-dimensional (such as S-elements, gammadions, open rings, and Archimedes spirals). Chiral materials are characterized by the propagation of two waves with right- and left-handed circular polarizations, as well as the cross-polarization of the field. These chiral metamaterials (CMM) are currently being actively used in microwave and antenna technologies. Their main include circulators, phase shifters, filters, antennas on CMM substrates, and chiral transmission lines, among others. [11–13]. It is important to note here that because chiral microelements are resonant particles, microwave devices based on CMM will exhibit frequency-selective properties.

The use of CMMs within microwave transmission line structures has garnered significant interest. Initial research on this topic was published in 1988 [14]. This study investigated the natural waves of a plane chiral waveguide bounded by ideally conducting planes. Further detailed studies have since been conducted on wave propagation in both open and closed circular uniformly filled chiral waveguides [15–17]. In [18], the natural waves of a plane two-layer chiral dielectric waveguide were studied, without restrictions on the structure thickness. A detailed theory of natural wave propagation in microwave guides is presented in [19]. Additionally, waves in chiral waveguides with impedance walls have been analyzed [20]. The analysis of rectangular cross-section waveguides requires the use of numerical methods [21]. In [22], an analysis of the natural waves of a planar chiral waveguide was performed. Technologies for creating chiral and bi-anisotropic waveguides are discussed in [23].

Interestingly, most of the aforementioned studies employ a standard mathematical model of a chiral medium, based on the Lindell–Sivola formalism [6]. This model uses material equations in the harmonic signal mode with vectors dependent on time in form $\exp \left(i\omega t\right))\text{:}$

$\overrightarrow{\mathbf{D}}=\mathrm{\epsilon }\overrightarrow{\mathbf{E}}\mp i\mathrm{\chi }\overrightarrow{\mathbf{H}},\overrightarrow{\mathbf{B}}=\mathrm{\mu }\overrightarrow{\mathbf{H}}\pm i\mathrm{\chi }\overrightarrow{\mathbf{E}},$ (1)

where $\overrightarrow{\mathbf{E}}\mathbf{,}\mathbf{ }\overrightarrow{\mathbf{H}}\mathbf{,}\mathbf{ }\overrightarrow{\mathbf{D}}\mathbf{,}\mathbf{ }\overrightarrow{\mathbf{B}}$ are the complex amplitudes of the vectors of intensity and induction of electric and magnetic fields, i is an imaginary unit; $\mathrm{\epsilon }$ is the relative dielectric permeability of the CMM; $\mathrm{\mu }$ is the relative magnetic permeability of the CMM; $\mathrm{\chi }$ is the relative chirality parameter. In Eq. (1), the upper signs correspond to the CMM based on mirror asymmetric microelements with a right twist, while the lower signs correspond to the CMM based on mirror asymmetric microelements with a left twist. The relationships of Eq. (1) are presented in the Gaussian system of units.

It is worth noting that most research assumes that the material parameters are constant, unaffected by the frequency of the incident electromagnetic field, and that the CMM is homogeneous. In other words, these studies do not consider the difference in the parameter values for the container and regions occupied by chiral microelements when calculating the effective dielectric permeability.

However, some studies have proposed generalized and specific mathematical models of chiral metamaterials that do not consider dispersion and heterogeneity [24–26]. Heterogeneity arises owing to the CMM composition: a container environment combined with areas possessing different material parameters, where mirror-like asymmetric microelements are located.

Some mathematical models of CMM are described in [34; 35].

In this study, we present a unique mathematical model for a metamaterial that factors in chirality, dispersion, and heterogeneity to accurately depict the electromagnetic properties of the chiral layer. This metamaterial is created based on a uniform assembly of thin-wire conductive spiral elements.

The primary objective of this research was to address the diffraction problem of the fundamental wave of a rectangular waveguide H₁₀ on a transverse insert made of CMM. In this case, a model featuring frequency-dependent parameters $\mathrm{\epsilon }\left(\mathrm{\omega }\right); \mathrm{\chi }\left(\mathrm{\omega }\right)$ is employed to describe the chiral layer.

1. Development of a private mathematical model of CMM based on thin-wire spiral microelements

Let us delve into the structure of a metamaterial consisting of a dielectric container with relative dielectric and magnetic permeabilities ${\mathrm{\epsilon }}_{\mathrm{c}}, {\mathrm{\mu }}_{\mathrm{c}}$, where mirror-like asymmetric conductive microelements are placed. The regions inhabited by these mirror-like asymmetric elements possess relative dielectric and magnetic permeabilities ${\mathrm{\epsilon }}_{\mathrm{s}}, {\mathrm{\mu }}_{\mathrm{s}}$. The linear dimensions of these regions are denoted by d and l represents the distance between adjacent elements. A general block diagram of the CMM is presented in Fig. 1.

 

Fig. 1. Structural general scheme of the metamaterial

Рис. 1. Структурная общая схема метаматериала

 

The effective dielectric and magnetic permeabilities of the metamaterial functionally depend on the permeabilities of both the container and chiral regions, namely $\mathrm{\epsilon }=\mathrm{\epsilon }({\mathrm{\epsilon }}_{\text{c}},{\mathrm{\epsilon }}_{\text{s}}); \mathrm{\mu }=\mathrm{\mu }({\mathrm{\mu }}_{\text{c}},{\mathrm{\mu }}_{\text{s}}).$

These functional dependencies for heterogeneous media are determined by various models, such as the Maxwell Garnett model and the Bruggeman model, among others [27–29].

The Maxwell Garnett model uses the following functional relationships:

$\mathrm{\epsilon }={\mathrm{\epsilon }}_{\text{c}}\frac{1+2{\mathrm{\alpha \epsilon }}_{\text{x}}}{1-{\mathrm{\alpha \epsilon }}_{\text{x}}};{\mathrm{\epsilon }}_{\text{x}}=\frac{{\mathrm{\epsilon }}_{\text{s}}-{\mathrm{\epsilon }}_{\text{c}}}{{\mathrm{\epsilon }}_{\text{s}}+2{\mathrm{\epsilon }}_{\text{c}}},$ (2)

where $\mathrm{\epsilon }$ represents the relative effective dielectric permeability of the metamaterial; ${\mathrm{\epsilon }}_{\mathrm{c}}$ stands for the relative dielectric permeability of the container; ${\mathrm{\epsilon }}_{\mathrm{s}}$ indicates the relative dielectric permeability of chiral regions; and $\mathrm{\alpha }$ is volume concentration of the chiral regions.

To accommodate the dispersion of dielectric permeability of chiral regions, we use the Drude–Lorentz equation:

${\mathrm{\epsilon }}_{\text{s}}\left(\mathrm{\omega }\right)={\mathrm{\epsilon }}_{\infty }+\frac{\left({\mathrm{\epsilon }}_{\text{c}}-{\mathrm{\epsilon }}_{\infty }\right){\mathrm{\omega }}_{\text{p}}^2}{{\mathrm{\omega }}_0^2+2\mathrm{i}{\mathrm{\delta }}_{\text{e}}\mathrm{\omega }-{\mathrm{\omega }}^2},$ (3)

where ${\mathrm{\epsilon }}_{\infty }$ is the asymptotic value of the dielectric permeability at $\mathrm{\omega }\to \infty ; {\mathrm{\delta }}_{\mathrm{e}}$ is the damping coefficient; ${\mathrm{\omega }}_{\mathrm{p}}^2$ represents resonant frequency of absorption; and ${\mathrm{\omega }}_0^2$ is the resonant frequency of the microelement, which is then calculated for a specific chiral microelement in a quasistationary approximation.

To consider the dispersion of the chirality parameter, we use the Condon equation [30–31]:

$\mathrm{\chi }\left(\mathrm{\omega }\right)=\frac{{\mathrm{\omega }}_0^2{\mathrm{\beta }}_0\mathrm{\omega }}{{\mathrm{\omega }}_0^2+2i{\mathrm{\delta }}_{\mathrm{x}}{\mathrm{\omega }}_0\mathrm{\omega }-{\mathrm{\omega }}^2},$ (4)

where ${\mathrm{\beta }}_0$ is a constant with an inverse time dimension and describes the degree of mirror asymmetry of the microelement; and ${\mathrm{\delta }}_{\mathrm{x}}$ is the damping coefficient of the chirality parameter.

Considering Eqs. (2) and (3), we obtain the following equation for the frequency-dependent effective dielectric permeability of the CMM:

$\mathrm{\epsilon }\left(\mathrm{\omega }\right)={\mathrm{\epsilon }}_{\text{c}}\frac{1+2{\mathrm{\alpha \epsilon }}_{\text{x}}\left(\mathrm{\omega }\right)}{1-{\mathrm{\alpha \epsilon }}_{\text{x}}\left(\mathrm{\omega }\right)};$ (5)

${\mathrm{\epsilon }}_{\text{x}}=\frac{{\mathrm{\epsilon }}_{\infty }+\frac{\left({\mathrm{\epsilon }}_{\text{c}}-{\mathrm{\epsilon }}_{\infty }\right){\mathrm{\omega }}_{\text{p}}^2}{{\mathrm{\omega }}_0^2+2i{\mathrm{\delta }}_{\text{e}}\mathrm{\omega }-{\mathrm{\omega }}^2}-{\mathrm{\epsilon }}_{\text{c}}}{{\mathrm{\epsilon }}_{\infty }+\frac{\left({\mathrm{\epsilon }}_{\text{c}}-{\mathrm{\epsilon }}_{\infty }\right){\mathrm{\omega }}_{\text{p}}^2}{{\mathrm{\omega }}_0^2+2i{\mathrm{\delta }}_{\text{e}}\mathrm{\omega }-{\mathrm{\omega }}^2}+2{\mathrm{\epsilon }}_{\text{c}}},$

The material equations for a chiral metamaterial (without considering the type of microelement) considering Eqs. (1), (4), and (5) have the following form:

$\overrightarrow{\mathbf{D}}=\mathrm{\epsilon }\left(\mathrm{\omega }\right)\overrightarrow{\mathbf{E}}\mp i\mathrm{\chi }\left(\mathrm{\omega }\right)\overrightarrow{\mathbf{H}},\overrightarrow{\mathbf{B}}=\mathrm{\mu }\overrightarrow{\mathbf{H}}\pm \mathrm{i}\mathrm{\chi }\left(\mathrm{\omega }\right)\overrightarrow{\mathbf{E}};$ (6)

$\mathrm{\epsilon }\left(\mathrm{\omega }\right)={\mathrm{\epsilon }}_{\text{c}}\frac{1+2{\mathrm{\alpha \epsilon }}_{\text{x}}\left(\mathrm{\omega }\right)}{1-{\mathrm{\alpha \epsilon }}_{\text{x}}\left(\mathrm{\omega }\right)};\mathrm{\chi }\left(\mathrm{\omega }\right)=\frac{{\mathrm{\omega }}_{\text{0}}^2{\mathrm{\beta }}_0\mathrm{\omega }}{{\mathrm{\omega }}_0^2+2i{\mathrm{\delta }}_{\text{x}}{\mathrm{\omega }}_0\mathrm{\omega }-{\mathrm{\omega }}^2};$

${\mathrm{\epsilon }}_{\text{x}}=\frac{{\mathrm{\epsilon }}_{\text{s}}\left(\mathrm{\omega }\right)-{\mathrm{\epsilon }}_{\text{c}}}{{\mathrm{\epsilon }}_{\text{s}}\left(\mathrm{\omega }\right)+2{\mathrm{\epsilon }}_{\text{c}}};{\mathrm{\epsilon }}_{\text{s}}\left(\mathrm{\omega }\right)={\mathrm{\epsilon }}_{\infty }+\frac{\left({\mathrm{\epsilon }}_{\text{c}}-{\mathrm{\epsilon }}_{\infty }\right){\mathrm{\omega }}_{\text{p}}^2}{{\mathrm{\omega }}_0^2+2i{\mathrm{\delta }}_{\text{e}}\mathrm{\omega }-{\mathrm{\omega }}^2}.$

The mathematical model of Eq. (6) is applicable when all chiral microelements have identical shapes and linear dimensions, are equidistantly located, chaotically oriented, and the magnetic permeability of the CMM remains frequency-independent.

Based on Eq. (6), we construct a particular mathematical model for the CMM based on a specific type of a mirror asymmetric element.

Next, we consider the calculation of the resonant frequency of a thin-wire conducting element in a quasistationary approximation [32].

The structure of a CMM cell based on a thin-wire spiral element is shown in Fig. 2. Figure 3 presents a cross-section of a spiral microelement.

 

Fig. 2. Cell structure of a chiral metamaterial based on a thin-wire helix

Рис. 2. Структура ячейки кирального метаматериала на основе тонкопроволочной спирали

 

Fig. 3. Cross section of a cylindrical mandrel, on which a thin-wire spiral element is wound

Рис. 3. Поперечный разрез цилиндрической оправки, на которую накручен тонкопроволочный спиральный элемент

 

In Fig. 3, the following designations are introduced: H is the container height; d is the distance between the spiral turns; R is the internal radius of the spiral; r is the wire radius; α refers to the spiral winding angle; and N indicates the number of spiral turns.

To calculate the resonant frequency in a quasistatic approximation, we use Thomson’s equation :

${\mathrm{\omega }}_0=\frac{1}{\sqrt{LC}}$ (5)

where L is the inductance of the spiral; and C is the spiral capacity.

The inductance of the spiral is calculated by the following equation:

$\mathrm{L}={\mathrm{\mu }}_{\text{с}}\frac{{\mathrm{N}}^{2}S}{g}={\mathrm{\mu }}_{\text{с}}\frac{\mathrm{\pi }{N}^2{R}_{}^2}{g},$ (6)

where S is the area of the spiral turn; g is the length of the unrolled wire; and R is the internal radius of the spiral.

The capacitance of the spiral element is determined by the wire capacitance, the interturn capacitance, and the interelement capacitance:

$C=C_{\mathrm{w}}+C_{\mathrm{it}}+C_{\mathrm{ie}}$ (7)

The capacitance of the conductive wire itself is determined using the following equation for the capacitance of a straight conductor:

$C_{\mathrm{w}}={\mathrm{\epsilon }}_{\mathrm{c}}\frac{g}{18\ln \left({\displaystyle \frac{2g}{r}}\right)-1}$, (8)

where r is the wire radius.

The interturn capacitance of the spiral is determined as follows:

$C_{\text{it}}=\frac{{\epsilon }_{\text{c}}{S}_{\text{it}}\left(N-1\right)}{d},$ (9)

where

$S_{\text{it}}=\pi \left[{(R+2r)}^2-R^2\right]$

– area of the ring formed by the wire element; d is the spiral pitch; and N is the number of turns.

The distance between the spiral turns can be expressed in terms of container height h and number of spiral turns N as follows:

$d=\frac{h}{N+1}$, (10)

Substituting Eq. (10) into Eq. (9), we obtain:

$C_{\text{it}}=\frac{{\epsilon }_{\text{c}}\pi \left[{(R+2r)}^2-R^2\right]\left(N^2-1\right)}{h}.$ (11)

The interelement capacitance of the spiral is determined as follows:

$C_{\text{ie}}=\frac{{\epsilon }_{\text{c}}{S}_{\text{мэ}}}{4l},$ (12)

where

$S_{\text{ie}}=\frac{4Nr\left(R+r\right)}{\cos \alpha }$

– area of space filled with spirals; r is the wire radius; l is the distance between chiral elements; $\mathrm{\alpha }=\pi /\left[2\left(N+1\right)\right]$ is spiral winding angle. The coefficient 1 / 4 is related to the spatial arrangement of the chiral elements in the container.

Substituting the expressions for the twist angle of the helix and the area occupied by the chiral element into Eq. (12), we obtain:

$C_{\text{ie}}=\frac{{\epsilon }_{\text{с}}Nr(R+r)}{l\cos \left[\frac{\pi }{2(N+1)}\right]},$ (13)

where R is the inner radius of the spiral; α is spiral winding angle; r is the wire radius; A is the distance between the chiral elements; N is the number of turns; and ${\mathrm{\epsilon }}_{\mathrm{c}}$ is the dielectric permeability of the container.

Substituting Eqs. (9), (11), and (13) into the Eq. (8) representing the total capacity, we obtain:

$C={\epsilon }_{\text{c}}\frac{g}{18\ln \left(\frac{2g}{r}\right)-1}+\frac{{\epsilon }_{\text{c}}rN(R+r)}{l\cos \left[\frac{\pi }{2(N+1)}\right]}+$

$+\frac{{\epsilon }_{\text{c}}\pi \left[{(R+2r)}^2-R^2\right]\left(N^2-1\right)}{h}.$ (14)

Considering Eqs. (5), (6), and (14) for the resonant frequency of a single-start spiral element, we obtain:

${\omega }_0=\frac{c\sqrt{g}}{\sqrt{\pi {\epsilon }_{\text{c}}{\mu }_{\text{c}}}NR{K}_{\text{x}}};$ (15)

$K_{\text{x}}=\sqrt{\frac{g}{18\ln \left(\frac{2g}{r}\right)-1}+}$

$\overline{+\frac{\pi \left[{(R+2r)}^2-R^2\right]\left(N^2-1\right)}{h}+\frac{rN(R+r)}{l\cos \left[\frac{\pi }{2(N+1)}\right]}}.$

Eq. (15) was obtained in a quasistatic approximation, and its application is only viable within the range $\mathrm{\omega }\in \left(0; {\mathrm{\omega }}_{\max }\right)$ where ${\mathrm{\omega }}_{\max }$ represents the maximum frequency at which the elements can be considered quasistationary $cT\gg 1$ (where c is the speed of light; and T is the period of oscillation of the electromagnetic field).

Thus, a particular mathematical model of a chiral metamaterial, based on a uniform set of thin-wire spiral elements, considers Eqs. (1), (6), and (15) has and takes the following form:

$\overrightarrow{\mathbf{D}}=\mathrm{\epsilon }\left(\mathrm{\omega }\right)\overrightarrow{\mathbf{E}}\mp i\mathrm{\chi }\left(\mathrm{\omega }\right)\overrightarrow{\mathbf{H}},\overrightarrow{\mathbf{B}}=\mathrm{\mu }\overrightarrow{\mathbf{H}}\pm i\mathrm{\chi }\left(\mathrm{\omega }\right)\overrightarrow{\mathbf{E}};$ (16)

$\mathrm{\epsilon }\left(\mathrm{\omega }\right)={\mathrm{\epsilon }}_{\text{c}}\frac{1+2{\mathrm{\alpha \epsilon }}_{\text{x}}\left(\mathrm{\omega }\right)}{1-{\mathrm{\alpha \epsilon }}_{\text{x}}\left(\mathrm{\omega }\right)};\mathrm{\chi }\left(\mathrm{\omega }\right)=\frac{{\mathrm{\omega }}_{\text{0}}^2{\mathrm{\beta }}_0\mathrm{\omega }}{{\mathrm{\omega }}_0^2+2i{\mathrm{\delta }}_{\text{x}}{\mathrm{\omega }}_0\mathrm{\omega }-{\mathrm{\omega }}^2};$

${\mathrm{\epsilon }}_{\text{x}}=\frac{{\mathrm{\epsilon }}_{\text{s}}\left(\mathrm{\omega }\right)-{\mathrm{\epsilon }}_{\text{c}}}{{\mathrm{\epsilon }}_{\text{s}}\left(\mathrm{\omega }\right)+2{\mathrm{\epsilon }}_{\text{c}}};{\mathrm{\epsilon }}_{\text{s}}\left(\mathrm{\omega }\right)={\mathrm{\epsilon }}_{\infty }+\frac{\left({\mathrm{\epsilon }}_{\text{c}}-{\mathrm{\epsilon }}_{\infty }\right){\mathrm{\omega }}_{\text{p}}^2}{{\mathrm{\omega }}_0^2+2i{\mathrm{\delta }}_{\text{e}}\mathrm{\omega }-{\mathrm{\omega }}^2};$

${\mathrm{\omega }}_0=\frac{c\sqrt{\mathit{g}}}{\sqrt{{\mathrm{\pi \epsilon }}_{\text{c}}{\mathrm{\mu }}_{\text{c}}}NR{K}_{\text{x}}},$

where

$K_{\text{x}}=\sqrt{\frac{g}{18\ln \left(\frac{2g}{r}\right)-1}+}$

$\overline{+\frac{\pi \left[{(R+2r)}^2-R^2\right]\left(N^2-1\right)}{h}+\frac{rN(R+r)}{l\cos \left[\frac{\pi }{2(N+1)}\right]}}.$

2. Problem of diffraction of the fundamental wave of a rectangular waveguide H₁₀ on a planar transverse insert made of a chiral metamaterial created using thin-wire conducting spiral microelements

This study addresses the issue of H₁₀ wave diffraction in a rectangular waveguide on a thin chiral layer, located perpendicular to the power transmission direction. The geometry of the problem is presented in Fig. 4.

 

Fig. 4. Geometry of the problem

Рис. 4. Геометрия задачи

 

At $z=0$ a thin chiral layer exists with material parameters $\mathrm{\epsilon }, \mathrm{\mu }$ and $\mathrm{\chi }$. The thickness of this chiral layer is less than the wavelength $k_{0}h\ll 1$ (where h is the layer thickness; and $k_0=\mathrm{\omega }/c$ is the wave number for a plane homogeneous electromagnetic wave in vacuum). The walls limiting the waveguide at $x=0; a$ and $y=0; b$ are assumed to be perfectly conducting $\left(\mathrm{\sigma }=\infty \right)$.

The problem of wave diffraction by a transverse chiral layer in a rectangular waveguide was solved using the method of double-sided boundary conditions (DSBCs) for a thin chiral layer [33].

We assume that a H₁₀ wave with components of complex amplitudes of the electromagnetic field E_y, H_x, H_z is incident on the chiral layer from the region $z<0$. For this geometry, we establish two-sided approximate boundary conditions for a thin chiral layer as follows [33]:

$E_y^{\left(1\right)}-E_y^{\left(2\right)}=\frac{\chi h}{2k_0{n}_{\text{c}}^2}\frac{{\partial }^2}{\partial y^2}\left(E_x^{\left(1\right)}+E_x^{\left(2\right)}\right)+$

$+\frac{i{k}_{0}h}{2}\left\{\mu \left(H_x^{\left(1\right)}+H_x^{\left(2\right)}\right)+i\chi \left(E_x^{\left(1\right)}+E_x^{\left(2\right)}\right)\right\},$ (17)

$H_y^{\left(1\right)}-H_y^{\left(2\right)}=\frac{i{\epsilon }^{\text{'}}h}{2k_0{n}_{\text{c}}^2}\frac{{\partial }^2}{\partial y^2}\left(E_x^{\left(1\right)}+E_x^{\left(2\right)}\right)-$

$-\frac{i{k}_{0}h}{2}\left\{\epsilon \left(E_x^{\left(1\right)}+E_x^{\left(2\right)}\right)-i\chi \left(H_x^{\left(1\right)}+H_x^{\left(2\right)}\right)\right\},$

$E_x^{\left(1\right)}-E_x^{\left(2\right)}=-\frac{\chi h}{2k_0{n}_{\text{c}}^2}\frac{{\partial }^2}{\partial x^2}\left(E_y^{\left(1\right)}+E_y^{\left(2\right)}\right)-$

$-\frac{i{k}_{0}h}{2}\left\{\mu \left(H_y^{\left(1\right)}+H_y^{\left(2\right)}\right)+i\chi \left(E_y^{\left(1\right)}+E_y^{\left(2\right)}\right)\right\},$

$H_x^{\left(1\right)}-H_x^{\left(2\right)}=\frac{i{\epsilon }^{\text{'}}h}{2k_0{n}_{\text{c}}^2}\frac{{\partial }^2}{\partial x^2}\left(E_y^{\left(1\right)}+E_y^{\left(2\right)}\right)+$

$+\frac{i{k}_{0}h}{2}\left\{\epsilon \left(E_y^{\left(1\right)}+E_y^{\left(2\right)}\right)-i\chi \left(H_y^{\left(1\right)}+H_y^{\left(2\right)}\right)\right\},$

where $n_{\text{c}}^2\left(\mathrm{\omega }\right)=\mathrm{\epsilon }\left(\mathrm{\omega }\right)\mathrm{\mu }-{\mathrm{\chi }}^2\left(\mathrm{\omega }\right);$ the indices “1” and “2” correspond to the areas of the waveguide located at $z<0$ and $z>h$.

Let us assume that the incident wave H₁₀ is incident on the chiral layer from $z=-\infty$ and the waveguide is matched at $z=+\infty$. The field of the incident wave at any inhomogeneity gives rise to the reflection and transmission of the fundamental wave H₁₀. From the solution of the homogeneous Helmholtz equations and considering the boundary conditions at $x=0;a$ and Maxwell’s equations, we formulate equations for the tangential components $E_y^{\left(j\right)}$ and $H_x^{\left(j\right)}$ $\left(j=1, 2\right)$ of the wave field H₁₀ in the isotropic regions 1 and 2 [33]:

$E_y^{\left(1\right)}=\left(e^{-i{\gamma }_{10}z}+R_{10}{e}^{i{\gamma }_{10}z}\right)\sin \left(\frac{\pi x}{a}\right);$ (18)

$E_y^{\left(2\right)}=T_{10}{e}^{-i{\gamma }_{10}z}\sin \left(\frac{\pi x}{a}\right),$

$\begin{array}{l}{H}_x^{\left(1\right)}=-\frac{{\gamma }_{10}}{k_0}\left(e^{-i{\gamma }_{10}z}-R_{10}{e}^{i{\gamma }_{10}z}\right)\sin \left(\frac{\pi x}{a}\right);\\ H_x^{\left(2\right)}=-\frac{{\gamma }_{10}}{k_0}{T}_{10}{e}^{-i{\gamma }_{10}z}\sin \left(\frac{\pi x}{a}\right),\end{array}$ (19)

where ${\gamma }_{10}=\sqrt{k_0^2-{\left(\pi /a\right)}^2}$ is the wave H₁₀ propagation constant in a rectangular waveguide with vacuum filling; and $R_{10}, T_{10}$ are unknown wave H₁₀ reflection and transmission coefficients, respectively. The amplitude of the incident wave H₁₀ was assumed to be 1.

Owing to the cross-polarization of the field, when wave H₁₀ is incident on the chiral layer in regions 1 and 2 of the waveguide, tangential components $E_x^{\left(j\right)}$ and $H_y^{\left(j\right)}$ $\left(j=1, 2\right)$ will also appear, and a cross-polarized wave H₀₁ with components will arise (when $a\ge 2b$):

$E_x^{\left(1\right)}=R_{01}{e}^{i{\gamma }_{01}z}\sin \left(\frac{\pi y}{b}\right);$ (20)

$E_x^{\left(2\right)}=T_{01}{e}^{-i{\gamma }_{01}z}\sin \left(\frac{\pi y}{b}\right);$

$H_y^{\left(1\right)}=-\frac{{\gamma }_{01}}{k_0}{R}_{01}{e}^{i{\gamma }_{01}z}\sin \left(\frac{\pi y}{b}\right);$

$H_y^{\left(2\right)}=\frac{{\gamma }_{01}}{k_0}{T}_{01}{e}^{-i{\gamma }_{01}z}\sin \left(\frac{\pi y}{b}\right),$

where ${\gamma }_{01}=\sqrt{k_0^2-{\left(\pi /b\right)}^2}$ is the wave H₁₀ propagation constant in a rectangular waveguide with vacuum filling; and $R_{01}, T_{01}$ are the reflection and transmission coefficients of the wave H₁₀, respectively.

Consequently, the equations for the component vectors of the electromagnetic field tangential to the layer have the following form:

$E_y^{\left(1\right)}=\left(e^{-i{\gamma }_{10}z}+R_{10}{e}^{i{\gamma }_{10}z}\right)\sin \left(\frac{\pi x}{a}\right);$ (21)

$H_y^{\left(1\right)}=-\frac{{\gamma }_{01}}{k_0}{R}_{01}{e}^{i{\gamma }_{01}z}\sin \left(\frac{\pi y}{b}\right);$

$E_x^{\left(1\right)}=R_{01}{e}^{i{\gamma }_{01}z}\sin \left(\frac{\pi y}{b}\right);$

$H_x^{\left(1\right)}=-\frac{{\gamma }_{10}}{k_0}\left(e^{-i{\gamma }_{10}z}-R_{10}{e}^{i{\gamma }_{10}z}\right)\sin \left(\frac{\pi x}{a}\right);$

in region 1;

$E_y^{\left(2\right)}=T_{10}{e}^{-i{\gamma }_{10}z}\sin \left(\frac{\pi x}{a}\right);$ (22)

$H_y^{\left(2\right)}=\frac{{\gamma }_{01}}{k_0}{T}_{01}{e}^{-i{\gamma }_{01}z}\sin \left(\frac{\pi y}{b}\right);$

$E_x^{\left(2\right)}=T_{01}{e}^{-i{\gamma }_{01}z}\sin \left(\frac{\pi y}{b}\right);$

$H_x^{\left(2\right)}=-\frac{{\gamma }_{10}}{k_0}{T}_{10}{e}^{-i{\gamma }_{10}z}\sin \left(\frac{\pi x}{a}\right);$

in region 2.

Substituting Eqs. (21) and (22) into the DSBC of Eq. (17), we obtain a system of four algebraic equations for unknown coefficients $R_{10}, R_{01}, T_{10}, T_{01}$:

$\overleftrightarrow{\mathbf{A}}\overrightarrow{\mathbf{R}}\mathbf{=}\overrightarrow{\mathbf{F}}$ (23)

where $\overrightarrow{\mathbf{R}}=\left\{R_{10},R_{01},T_{10},T_{01}\right\};$

$\overrightarrow{\mathbf{F}}=\left\{\left[-1-\frac{i{\mathrm{\mu \gamma }}_{10}h}{2}\right],\left[\frac{\chi {\gamma }_{10}h}{2}\right],\right.$

${\left.\left[-\frac{\mathrm{\chi }{k}_{0}h}{2}\left(1+{\mathrm{\alpha }}_{10}^2\right)\right],\left[-\frac{{\gamma }_{10}}{k_0}-\frac{\text{i\hspace{0.17em}}{\mathrm{\epsilon }}^{\text{'}}{k}_{0}h}{2}\left(1+{\mathrm{\alpha }}_{10}^2\right)\right]\right\}}^{\text{T}}.$

The matrix $\overleftrightarrow{\mathbf{A}}$ elements have the form:

$\begin{array}{l}{A}_{11}=1-\frac{\text{i\hspace{0.17em}}{\mathrm{\mu \gamma }}_{10}h}{2};A_{12}=\frac{\mathrm{\chi }{k}_{0}h}{2}\left(1+{\mathrm{\alpha }}_{01}^2\right);\\ A_{13}=\left[-1+\frac{\text{i\hspace{0.17em}}{\mathrm{\mu \gamma }}_{10}h}{2}\right]e^{-\text{i\hspace{0.17em}}{\gamma }_{10}h};\\ A_{14}=\frac{\mathrm{\chi }{k}_{0}h}{2}\left(1+{\mathrm{\alpha }}_{01}^2\right)e^{-\text{i\hspace{0.17em}}{\gamma }_{01}h};A_{21}=\frac{{\mathrm{\chi \gamma }}_{10}h}{2};\end{array}$

$\begin{array}{l}{A}_{22}=-\frac{{\mathrm{\gamma }}_{01}}{k_0}+\frac{\text{i\hspace{0.17em}}\epsilon k_{0}h}{2}\left(1+{\mathrm{\alpha }}_{01}^2\right);A_{23}=-\frac{{\mathrm{\chi \gamma }}_{10}h}{2}{e}^{-\text{i\hspace{0.17em}}{\gamma }_{10}h};\\ A_{24}=\left[-\frac{{\mathrm{\gamma }}_{01}}{k_0}+\frac{\text{i\hspace{0.17em}}\epsilon k_{0}h}{2}\left(1+{\mathrm{\alpha }}_{01}^2\right)\right]e^{-\text{i\hspace{0.17em}}{\gamma }_{01}h};\\ A_{31}=\frac{\mathrm{\chi }{k}_{0}h}{2}\left(1+{\mathrm{\alpha }}_{10}^2\right);A_{32}=\left[-1+\frac{\text{i\hspace{0.17em}}{\mathrm{\mu \gamma }}_{01}h}{2}\right];\end{array}$

$\begin{array}{l}{A}_{33}=\frac{\mathrm{\chi }{k}_{0}h}{2}\left(1+{\mathrm{\alpha }}_{10}^2\right)e^{-\text{i\hspace{0.17em}}{\gamma }_{10}h};\\ A_{34}=\left[1-\frac{i{\mathrm{\mu \gamma }}_{01}h}{2}\right]e^{-\text{i\hspace{0.17em}}{\gamma }_{01}h};\\ A_{41}=\left[-\frac{{\mathrm{\gamma }}_{10}}{k_0}+\frac{\text{i\hspace{0.17em}}\mathrm{\epsilon }{k}_{0}h}{2}\left(1-{\mathrm{\alpha }}_{10}^2\right)\right];A_{42}=-\frac{{\mathrm{\chi \gamma }}_{01}h}{2};\end{array}$

$\begin{array}{l}{A}_{43}=\left[-\frac{{\mathrm{\gamma }}_{10}}{k_0}+\frac{\text{i\hspace{0.17em}}\mathrm{\epsilon }{k}_{0}h}{2}\left(1-{\mathrm{\alpha }}_{10}^2\right)\right]e^{-\text{i\hspace{0.17em}}{\gamma }_{10}h};\\ A_{44}=\frac{{\mathrm{\chi \gamma }}_{01}h}{2}{e}^{-\text{i\hspace{0.17em}}{\gamma }_{01}h},\end{array}$

where ${\alpha }_{10}^2={\pi }^2/\left(k_0^2{a}^2{n}_c^2\right);$ ${\alpha }_{01}^2={\pi }^2/\left(k_0^2{b}^2{n}_c^2\right).$ The remaining parameters are determined by the system of equations (16).

Solving the system of equations (23) in the first approximation with respect to the small parameter $k_{0}h$ in an analytical form, we derive equations for the reflection and transmission coefficients of the fundamental and cross-polarized waves H₁₀ and H₀₁:

$R_{10}=\frac{k_{0}h\mathrm{\epsilon }\left[\frac{{\mathrm{\beta }}_{01}}{{\mathrm{\beta }}_{10}}\right]\left[1+{\mathrm{\alpha }}_{10}+{\mathrm{\eta }}^2{\mathrm{\beta }}_{10}^2\right]}{2\text{i\hspace{0.17em}}{\mathrm{\beta }}_{10}+k_{0}h\mathrm{\epsilon }\left(1+{\mathrm{\alpha }}_{01}\right)\left\{1+{\mathrm{\eta }}^2{\mathrm{\beta }}_{01}^2\left({\mathrm{\beta }}_{10}-{\mathrm{\beta }}_{01}\right)\right\}};$ (24)

$T_{10}=\frac{k_{0}h\epsilon \left(1+{\alpha }_{01}+{\eta }^2{\beta }_{01}^2\right)e^{-\text{i\hspace{0.17em}}{k}_{0}h{\beta }_{10}}}{2\text{i\hspace{0.17em}}{\beta }_{01}+k_{0}h\epsilon \left(1+{\alpha }_{01}\right)\left\{1+{\eta }^2{\beta }_{01}^2\left({\beta }_{10}-{\beta }_{01}\right)\right\}};$

$R_{01}=\frac{\text{i\hspace{0.17em}}\chi k_{0}h\left[-{\beta }_{10}+{\beta }_{01}\left(1-{\alpha }_{10}\right)\right]}{2\text{i\hspace{0.17em}}{\beta }_{01}+k_{0}h\epsilon \left(1+{\alpha }_{01}\right)\left\{1+{\eta }^2{\beta }_{01}^2\left({\beta }_{10}-{\beta }_{01}\right)\right\}};$

$T_{01}=\frac{\text{i\hspace{0.17em}}\chi k_{0}h\left[{\beta }_{10}+{\beta }_{01}\left(1-{\alpha }_{10}\right)\right]e^{-\text{i\hspace{0.17em}}{k}_{0}h{\beta }_{01}}}{2\text{i\hspace{0.17em}}{\beta }_{01}+k_{0}h\epsilon \left(1+{\alpha }_{01}\right)\left\{1+{\eta }^2{\beta }_{01}^2\left({\beta }_{10}-{\beta }_{01}\right)\right\}},$

where

$\mathrm{\eta }=\sqrt{\mathrm{\mu }/\mathrm{\epsilon }};{\mathrm{\beta }}_{10}=\sqrt{1-{\left[\mathrm{\pi }/\left(k_{0}a\right)\right]}^2};$

${\mathrm{\beta }}_{01}=\sqrt{1-{\left[\pi /\left(k_{0}b\right)\right]}^2}.$

3. Numerical modeling

In the numerical modeling, we considered a rectangular waveguide, in which the transverse plane comprised a predetermined thickness layer of chiral metamaterial. The metamaterial container was made of polystyrene foam with a relative dielectric permeability of 1,5. The waveguide was filled with vacuum, which has a relative dielectric permeability of 1. In this study, we calculated the frequency dependencies of the transmitted and reflected powers of the fundamental and cross-polarized waves H₁₀ and H₀₁ were calculated when the fundamental wave was incident on the chiral layer.

Let us consider the case when the metamaterial is formed by spirals with one twist $\left(N=1\right)$.

The initial values of metamaterial parameters are the following:

${\mathrm{\epsilon }}_1={\mathrm{\epsilon }}_3=1;{\mathrm{\epsilon }}_2=1,5-{10}^{-5}i\mathit{;}N=1;$

$R=0,0025$ m; $r=0,001$ m;

$d=0,0015$ m; $h=0,005$ m; $l=0,0015$ m.

Figure 5 presents the dependencies of the transmitted $20\lg \left|T_{10}\right|$ and reflected $20\lg \left|R_{10}\right|$ wave H₁₀ powers, as well as the transmitted $20\lg \left|T_{01}\right|$ and reflected $20\lg \left|R_{01}\right|$ wave H₀₁ powers, on frequency in the operating mode of a rectangular waveguide at N = 1.

 

Fig. 5. Frequency dependences of the transmitted and reflected powers for the case of single-turn spirals

Рис. 5. Частотные зависимости прошедшей и отраженной мощностей для случая одновитковых спиралей

 

As inferred from Fig. 5, for a CMM based on single-turn spirals, the fundamental wave H₁₀ traverses the chiral layer with minimal attenuation. This is because the attenuation of the transmitted power across all frequencies of the operating range is close to 0 dB. The reflection of the fundamental wave is minimal, reaching a peak at –10 dB around the frequency of 10,5 GHz. There is also a resonant minimum in the reflection of the fundamental wave H₁₀ near the frequency of 8,5 GHz. Meanwhile, the cross-polarized wave H₀₁ is reflected near the same frequency, but its maximum reflection level is −37,5 dB.

Let us consider the case in which the metamaterial is formed by spirals with two twists $\left(N=2\right)$.

The metamaterial parameter values are the following:

${\mathrm{\epsilon }}_1={\mathrm{\epsilon }}_3=1;{\mathrm{\epsilon }}_2=1,5-{10}^{-5}i;N=2;$

$R=0,0025$ m; $r=0,001$ m;

$d=0,0015$ m; $h=0,005$ m; $l=0,0015$ m.

Figure 6 presents the dependencies of the transmitted $20\lg \left|T_{10}\right|$ and reflected $20\lg \left|R_{10}\right|$ wave H₁₀ powers, as well as the transmitted $20\lg \left|T_{01}\right|$ and reflected $20\lg \left|R_{01}\right|$ wave H₀₁ powers, on frequency in the operating mode of a rectangular waveguide at $N=2$.

 

Fig. 6. Frequency dependences of the transmitted and reflected powers for the case of two-turn spirals

Рис. 6. Частотные зависимости прошедшей и отраженной мощностей для случая двухвитковых спиралей

 

As is obvious from Fig. 6, with two-turn spirals, frequency selectivity arises. Specifically, at a frequency of 10,9 GHz, there is a sharp dip in the transmission of the fundamental wave through the chiral layer in the waveguide. Near this frequency in a rectangular waveguide, wave H₀₁ becomes the fundamental wave because the transmitted and reflected power reaches maxima. The presence of a chiral layer based on two-turn spirals results in the waveguide initially not transmitting the fundamental wave H₁₀ near the resonant frequency, and a shift to the operating mode on the cross-polarized wave H₀₁, which becomes the fundamental one, occurs. In addition, Fig. 6 shows both waves over the entire operating frequency range, although the amplitude of the cross-polarized wave is extremely small (except in the region near the resonant frequency).

Let us now consider the case in which the metamaterial is formed by spirals with three twists. The parameter values for the metamaterial are the following:

${\mathrm{\epsilon }}_1={\mathrm{\epsilon }}_3=1;{\mathrm{\epsilon }}_2=\mathrm{1,5}-{10}^{-5}i;N=3;$

$R=0,0025$ m; $r=0,001$ m;

$d=0,0015$ m; $h=0,005$ m; $l=0,0015$ m.

Figure 7 presents the dependencies of the transmitted $20\lg \left|T_{10}\right|$ and reflected $20\lg \left|R_{10}\right|$ wave H₁₀ powers, as well as the transmitted $20\lg \left|T_{01}\right|$ and reflected $20\lg \left|R_{01}\right|$ wave H₀₁ powers, on frequency in the operating mode of a rectangular waveguide at N = 3.

 

Fig. 7. Frequency dependences of transmitted and reflected powers for the case of three-turn spirals

Рис. 7. Частотные зависимости прошедшей и отраженной мощностей для случая трехвитковых спиралей

 

As evident from Fig. 7, pronounced frequency selectivity is also present in this case. Near the resonant frequency of 10,1 GHz, the fundamental H₁₀ wave ceases to propagate along the waveguide and is partially reflected from the chiral layer, while the cross-polarized wave passes into the region behind the chiral layer with a larger amplitude than that of the fundamental wave. At a frequency of 11,4 GHz, there is a minimum in the transmission of the fundamental wave through the chiral layer in the waveguide. The presence of a chiral layer based on three-turn spirals results in the waveguide initially not transmitting the fundamental wave H₁₀ near the resonant frequency, leading to a shift to the operating mode of the cross-polarized wave H₀₁, which becomes the fundamental one.

Throughout our analysis, we conclude that to obtain a strong effect of frequency selectivity, it is preferable to use two- and three-turn spirals as chiral microelements. These enable the mode transition from the fundamental wave type from H₁₀ to H₀₁ near resonant frequencies. This phenomenon is not associated with waveguide dispersion but arises owing to the insertion of a heterogeneous chiral metamaterial into the waveguide.

Conclusion

This study constructs a mathematical model of achiral metamaterial based on thin-wire conducting spirals, considering the chirality, heterogeneity, and dispersion of material parameters. The work proved the frequency selectivity of wave propagation through a chiral layer located in the transverse plane of a rectangular waveguide. Furthermore, it establishes that a chiral metamaterial based on two-turn thin-wire spirals exhibits the highest degree of frequency selectivity. Our work also revealed that when a chiral metamaterial is inserted into a rectangular waveguide, a cross-polarized wave H₀₁ inevitably arises in addition to the wave of the fundamental type H₁₀.

An analysis of the frequency dependencies of the reflection and transmission coefficient modules of the fundamental H₁₀ and cross-polarized H₀₁ showed that in some narrow frequency intervals in single-wave mode, situations arise when the fundamental wave type transitions from H₁₀ to H₀₁ near resonant frequencies.

The transmission line under study has potential applications in the creation of frequency-selective filters and polarization converters in the microwave range.

About the authors

Ivan Yu. Buchnev

Povolzhskiy State University of Telecommunications and Informatics

Email: v.buchnev@psuti.ru

post-graduate student of the Department of Higher Mathematics, Povolzhskiy State University of Telecommunications and Informatics, Samara, Russia.

Research interests: electrodynamics of metamaterials.

Russian Federation, 23, L. Tolstoy Street, Samara, 443010, Russia

Oleg V. Osipov

Povolzhskiy State University of Telecommunications and Informatics

Author for correspondence.
Email: o.osipov@psuti.ru

Doctor of Physical and Mathematical Sciences, acting head of the Department of Higher Mathematics, Povolzhskiy State University of Telecommunications and Informatics, Samara, Russia.

Research interests: electrodynamics of metamaterials, microwave devices and antennas, nonlinear optics.

Russian Federation, 23, L. Tolstoy Street, Samara, 443010, Russia

References

Supplementary files