Keywords

1 Introduction

Currently, the development of metamaterial technology is experiencing rapid growth. At the same time, despite the successful studies, the broadband polarization converter in the THz range has not been implemented yet. So the idea of creating such a polarizer is fairly new and relevant. The devices for producing circularly polarized THz waves are of great interest. The electromagnetic waves with this polarization type can be used to study and diagnose various objects, including biological ones, in the THz frequency range. Chiral structures, which can exhibit polarization-selective properties, stand out among these objects. The study of such properties requires THz waves with right and left circular polarization. These waves can interact differently with chiral objects and thereby reveal their structure and distinctive features.

The polarizers of electromagnetic waves in different spectral bands can have elements of various shapes, e.g. helices with different numbers of turns, open rings directed in different positions, as well as classical or rectangular omega elements [1,2,3,4,5,6,7,8,9]. The main range of these polarizers is microwave, and, to a lesser extent, the THz band of the spectrum. Paper [1] presents numerical studies on a multi-functional polarization convertor for terahertz light, composed of a bilayer wire-split-ring structure chiral metamaterial. Paper [2] studies a highly sensitive refractive index sensor based on the conjugated bilayer Г-shaped chiral metamaterial. Paper [3] provides the studies on the bilayer structure with microstrip connecting line to achieve giant circular dichroism. Paper [4] presents a bi-layered chiral metamaterial to realize high–efficiency broadband asymmetric transmission of linearly polarized electromagnetic wave in the microwave region. An asymmetric chiral metamaterial circular polarizer based on bilayer twisted split-ring resonator structure is proposed and investigated in [5]. The plasmonic version of a 3D chiral meta-atom which consists of a loop-wire structure, namely the so-called twisted omega particle, is implemented in [6]. Paper [7] presents systematical studies of the Ω elements of classical shape as well as considers the impact of geometric parameters on chiral effects, which occur in the metamaterial. Paper [8] studies the chiral metamaterial consisting of double L resonators on two sides of the dielectric substrate. Paper [9] considers the multi-layered structure comprising two sub-wavelength gratings and split-ring resonator, printed on two sides of dielectric substrate.

This paper presents the study of the properties of a rectangular omega element for potential use of such elements in the polarizers of THz waves. Since the element being examined is flat, this facilitates the production of metamaterials and metasurfaces by using planar technologies. The second advantage of the omega element is the fact that the electric current, arising in it, induces simultaneously an electric dipole moment and a magnetic moment. This contrasts with the coupled elements of metamaterials, e.g. straight wires and split rings. The first element in this couple is necessary for producing the dielectric properties of the metamaterial, and the other one is for producing the magnetic properties. Compared to the omega element, such coupled elements are more difficult to tune, as different modes of electromagnetic waves are activated in each of them. The third advantage of the rectangular omega-element is that the conversion of the THz wave polarization will occur in the designed metamaterial only when the wave is reflected, but not when it passes through the metamaterial. This will reduce the losses of wave intensity for the absorbing metamaterial.

The implementation of the elements to convert wave polarization in two-dimensional and three-dimensional arrays of metamaterials is associated with certain difficulties. For example, helix elements have quite a complicated spatial form, and their production has a number of limitations in the size and material of the substrate. Omega elements and split rings have a planar geometry and can be easily manufactured using modern photolithography techniques.

The production of a new type of the omega-element for the THz range is due to the use of the balance property of this element. The optimality condition for the omega-element as part of a two-dimensional array was previously formulated and experimentally investigated. This was done primarily to produce an absorber of microwave and THz waves. The optimality condition determines the equality of the induced magnetic and electric dipole moments. They are equal in absolute value in the optimal element. The length of the metallized strip, forming the polarizing element, is approximately equal to half the wavelength of the incident radiation, which corresponds to the main condition of frequency resonance [10,11,12,13].

The aim of our research paper is to obtain and study a balanced rectangular omega-element and eventually produce two-dimensional and three-dimensional arrays on its basis in the THz range for frequency filtration and wave polarization conversion. In the future, a reflective polarizer of the THz wave is expected to be produced. This polarizer will allow changing the wave polarization from linear to circular when the wave is reflected from the metamaterial.

2 Simulation

The research methodology consists in numerical simulation of the electric current, arising in a rectangular omega-element under the influence of an incident electromagnetic wave. The electric dipole moment and the magnetic moment of the omega element, which have a mutually perpendicular direction, are calculated. Then the ellipticity coefficient of the reflected wave is determined, which must be close to 1 under the balance of moments condition. The influence of the omega-element geometric parameters on the possibility of obtaining a circular polarization of the reflected wave is taken into account. The change in the frequency of the incident wave near the resonance and two different directions of its electric vector are considered as well.

The study uses a traditional bianisotropic particle with an omega coupling, which is a metal Ω shape strip. It is possible to find a balance condition for the polarizability for a single uniaxial omega particle, made from a conducting wire. It is also known that some sort of restriction on the electromagnetic properties of the wire omega particle can be found in the approximation of electrically small dimensions.

All the polarizabilities of the omega particle are determined on the basis of the finite element method (FEM). Computer simulation and an analytical approach are used as well to determine the polarizability of the rectangular omega element.

The parameters of the element can be calculated approximately considering the model of a quasi-stationary current. It is expected that the current intensity does not change depending on the coordinate measured along the omega element. The design of the rectangular shape of the omega element is selected on the basis of the balance condition, since the induced electric and magnetic dipole moments are equally significant. It should be noted that electromagnetic and magnetoelectric polarizabilities are equal as well.

The polarizability balance is achieved by changing the geometric parameters of the omega element. The polarization properties of the overall metamaterial can be enhanced using the array of balanced micro-resonators on the metasurface.

Figure 1(a) and (b) shows Ω-resonator design with new structural parameters and its polarizability. As can be seen, this omega resonator is balanced, since the induced electrical and magnetic dipole moments are equally significant. This property is reflected in the formula in Fig. 1(b). It should be noted that electromagnetic and magnetoelectric polarizabilities are also equal. Consequently, this Ω-resonator fully satisfies the polarizability balance condition. Using the array of these resonators, it is possible to achieve complete polarization of electromagnetic waves. At the same time the polarizer will be inactive away from resonance. However, additional optimization of the particles position in the array is required, since their interaction in the array also significantly affects the properties of the polarizer.

Fig. 1.
figure 1

(a) The Ω-resonator design with structural parameters; (b) calculated polarizability of a balanced Ω-resonator

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The main initial parameters are shown in Fig. 1. The width of the element is approximately equal to c = 10 µm, the width of the metallized strip is t = 1 µm, the length of the element is d = 70 µm, the thickness of the metallized strip is h = 1 µm, the arm length is a = 0 µm. Figure 1 shows the arms for illustrative purposes, since in parametric simulation the parameter a can change upwards, but it is 0 (zero) when calculating a balanced rectangular Ω-element. The metallization material is copper; the element is in a vacuum in the initial conditions.

2.1 Boundary Conditions

The Ω-elements in the microwave range (2.55–3.8 GHz) were previously calculated and investigated in [10, 11]. These particles were also considered for the terahertz range as elements of effective polarizers or absorbers of electromagnetic waves [12,13,14]. At this stage of research, a rectangular Ω-element is being developed, the parameters of which are optimized for the terahertz radiation. It is shown that the metamaterial, based on an array of omega elements, can perform the functions of an effective polarization transducer of an electromagnetic wave.

Designing a balanced rectangular omega element for the THz range is associated with a change in the parameters of the element in accordance with a change in the wavelength of the incident radiation. Under the main resonance condition, the length of the optimal omega-element in the rectified state is approximately equal to half the wavelength of the incident radiation. In particular, the wavelength for the 1 THz frequency is 0.3 mm, and the length of the metallized strip, forming the omega element, will be about 0.15 mm.

The optimality condition for the omega-element means that equally significant electric dipole moment and magnetic moment are induced in the element [10]. Considering this condition, the following parameters and design of the element are obtained (Fig. 1).

The balance of the rectangular omega element can be obtained, for example, by optimizing the geometrical parameters of this omega element, designed with the use of the quasi-stationary current model.

The polarization properties of the rectangular omega element in the terahertz range are simulated. The parameters of the metasurface, consisting of omega elements, are determined. The omega-elements are necessary for the effective conversion of the incident linearly polarized wave into the reflected wave with circular polarization in the THz frequency range. The studies are carried out with different directions of the incident wave vector \( \vec{E} \). In the first case, the vector of the electric field of the incident wave oscillates in the plane of incidence, i.e. in the plane that passes through the ends of the omega element. In this case, an electric dipole moment is induced by the electric field, and a magnetic moment arises as a result of the electric current in the omega element. In the second case, the magnetic field vector of the incident wave oscillates in the plane of incidence and has a component which is perpendicular to the plane of the omega element. In this case, a magnetic moment is induced by the magnetic field, and an electric dipole moment arises as a result of the electric current in the omega-element. The moments moduli are equal in both cases, which leads to the appearance of a reflected wave with a circular polarization.

On constructing the simulation object, the boundary conditions were introduced, and the parameters of the incident electromagnetic wave were set. When addressing the current task, an incident plane wave is used. The vector of the incident wave \( \vec{E} \) is oriented either horizontally in the XY plane, i.e. in the plane parallel to the arms of the element, or vertically, along the Z axis, i.e. orthogonal to the arms of the element. The wave vector \( \vec{k} \) of the incident wave is directed along the X axis, i.e., at an angle of 45 degrees to the omega-element plane (Fig. 2).

Fig. 2.
figure 2

The direction of the vector \( \vec{k} \) of the incident EM wave relative to the structural elements of the omega cell, (a) \( \vec{E} \) vector oscillates in the XY plane, (b) \( \vec{E} \) vector is directed along the Z axis

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3 Results of the Simulation and Their Analysis

The current paper considers the case with the oblique incidence of an electromagnetic wave; \( \vec{k} \) vector of the incident wave is directed at 45° to the axis of the rectangular omega element. The calculated wavelength is 0.3 mm, and the frequency is 1 THz for the optimal balanced omega element.

The oblique incidence of the EM wave (when the vector is directed at an angle of 45 degrees to the plane of the omega element turn) allows to activate both the electric dipole moment and the magnetic moment of the omega elements, so the ellipticity coefficient of the reflected EM wave will reach maximum values ​​at the calculated frequency. In this case, the condition of half-wave resonance is satisfied, therefore the reflection coefficient takes the maximum value as well.

Consider the first case with the direction of \( \vec{E} \) vector of the incident wave in the plane parallel to the arms of the rectangular Ω-element. Parametric simulation is carried out using mainly three parameters of the element: the length, the thickness of the metallized strip and the length of the arm. The Ω-element with the parameters a = 0.3 μm, h = 1.5 μm, d = 69 μm, c = 10 μm, t = 1 μm prove to be the best polarizer of all the parameters under study. The Ω-element with such parameters shows the ellipticity coefficient of the reflected wave Кмax = 0.997 at the calculated frequency equal to 1 THz (Fig. 3). In this case, the Ω-element converts well enough the polarization of an electromagnetic wave, when it is reflected from a plane-polarized incident wave, to a circularly polarized reflected wave in a wide frequency interval. The peak of the electric field intensity of the reflected wave is observed at a frequency of 1 THz, which indicates a good excitation of the Ω-element by the electromagnetic wave at the calculated frequency. Figure 5 shows the electromagnetic wave activation of the rectangular omega element for this case.

Fig. 3.
figure 3

The graph of the frequency dependence of the ellipticity coefficient of the reflected wave for the element with the parameters: a = 0.3 μm, h = 1.5 μm, d = 69 μm, \( \vec{E} \) vector oscillates in the plane parallel to the arms of the Ω-element

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Fig. 4.
figure 4

The graph of the frequency dependence of the ellipticity coefficient of the reflected wave for the element with the parameters: a = 0.1 μm, h = 1.5 μm, d = 69 μm, \( \vec{E} \) vector oscillates in the plane orthogonal to the arms of the Ω-element

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Fig. 5.
figure 5

Field overlay: a – \( \vec{E} \) vector; b – \( \vec{H} \) vector, the Ω-element with the parameters: a = 0.3 μm, h = 1.5 μm, d = 69 μm, \( \vec{E} \) vector oscillates in the plane parallel to the arms of the Ω-element

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While analyzing the other case, when \( \vec{E} \) vector is orthogonal to the arms of the rectangular omega element, it can be concluded that the Ω element with the following parameters has the best polarization properties for this case: a = 0.1 μm, h = 1.5 μm, d = 69 μm, s = 10 μm, t = 1 μm. The Ω-element with such parameters showed the ellipticity coefficient of the reflected wave Кмax = 0.299 at the calculated frequency of 1 THz, which corresponds to the elliptically polarized reflected wave (Fig. 4). In this case, the “turn” of the omega element is penetrated by the magnetic field vector of the incident wave, i.e. the element is activated by \( \vec{H} \) vector. A small peak of the electric field intensity of the reflected wave is recorded at a frequency of 0.95 THz, which generally corresponds to theoretical calculations. A small ellipticity coefficient is explained by the insufficiently effective electric component of the incident wave, since the wave mainly activates the front leg of the rectangular element (Fig. 6).

Fig. 6.
figure 6

Field overlay: a – \( \vec{E} \) vector; b – \( \vec{H} \) vector, Ω-element with the parameters: a = 0.1 μm, h = 1.5 μm, d = 69 μm, \( \vec{E} \) vector oscillates in the plane orthogonal to the arms of the Ω-element

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This paper provides a numerical simulation of the electric current arising in a rectangular omega-element under the influence of an incident electromagnetic wave. The excited mode of electromagnetic oscillations produces both an electric dipole moment and a magnetic moment in the omega-element. These moments have a mutually perpendicular direction. The arising moments and polarizabilities of the omega-element as a bianisotropic particle have been calculated. The ellipticity coefficient of the reflected wave has been determined. This coefficient must be close to 1 under the balance of moments condition. Consequently, it becomes possible to convert the incident linearly polarized wave into a reflected wave with circular polarization. The optimal geometric parameters of the omega element have been found to obtain circular polarization of the reflected wave under the half-wave resonance condition. There has been the study of polarization conversion of the reflected wave at the frequency change of the incident wave near the resonance and for two different directions of the electric vector of the incident wave.

4 Conclusions

According to the simulation results of a single rectangular omega element, it is concluded that a balanced rectangular omega element has high polarizing properties calculated for the THz range. This element is an effective polarizer of electromagnetic waves with the ellipticity coefficient of the reflected wave close to 1, with an oblique incidence of the electromagnetic wave at an angle of 45 degrees to the element plane.

One of the advantages of the rectangular omega element under consideration is that the conversion of the THz wave polarization in the designed metamaterial will become possible for the reflected wave, and not for the wave passing through the metamaterial. This will allow using the absorbing metamaterials and reducing losses in the wave intensity with converted polarization.

The maximum value of the ellipticity coefficient, close to 1, is observed for a rectangular omega element with the following parameters: a = 0.3 μm, h = 1.5 μm, d = 69 μm, c = 10 μm, t = 1 μm. The omega element with such parameters shows the ellipticity coefficient of the reflected wave K = 0.997 at a calculated frequency of 1 THz, when the vector of the incident wave \( \vec{E} \) is directed in the plane parallel to the arms of the rectangular omega element.

The rectangular balanced omega particle is a promising element for creating a metamaterial with high polarization properties in the THz range.

Since the omega elements under study have a rectangular shape, the printed-circuit board techniques can be used to create metamaterials and metasurfaces on the basis of these elements. Vacuum-plasma technologies can also be applied to produce the omega-structured metamaterials and metasurfaces containing rectangular elements.

The research has been performed as a part of the government research program “Photonics, opto- and microelectronics” of the “Photonics” subprogram.