Analysis of gyrobianisotropic media effect on the input impedance, field distribution and mutual coupling of a printed dipole antenna

In this paper, we present an analytical study for the investigation of the effects of the magnetoelectric elements of a reciprocal and nonreciprocal bianisotropic grounded substrate on the input impedance, resonant length of a dipole antenna as well as on the mutual coupling between two element printed dipole array in three configuration geometries: broadside, collinear and echelon printed on the same material. This study examines also the effect of the considered bianisotropic medium on the electric and magnetic field distributions that has been less addressed in the literature for antenna structures. Computations are based on the numerical resolution, using the spectral method of moments, of the integral equation developed through the mathematical derivation of the appropriate spectral Green’s functions of the studied dipole configuration. Original results, for chiral, achiral, Tellegen and general bi-anisotropic media cases, are obtained and discussed with the electric and magnetic field distributions for a better understanding and interpretation. These interesting results can serve as a stepping stone for further works to attract more attention to the reciprocal and non-reciprocal Tellgen media in-depth studies.


Analytical formulation
The magnetoelectric tensors of the bianisotropic layer material express the coupling between the magnetic and electric fields. The corresponding constitutive relations of a general complex bianisotropic medium are expressed, in their general form, by 7,8,28,29 : where ε is the permittivity, μ is the permeability, ξ and η are the magnetoelectric parameters expressing the coupling between the magnetic and electric fields. In general, for bianisotropic mediums, all the parameters are 3 × 3 tensors. Chiral materials can be realized using a bi-isotropic medium based on randomly distributed right-and lefthanded helices (Fig. 1) or bianisotropic based on orderly distributed right-and left-handed helices. Several configurations of these structures were presented in 20 , in particular, the structures with vertically or horizontally oriented helices (spirals) as illustrated in Fig. 2a,b, respectively.
In 28 , another type of a bianisotropic metamaterial is designed using Split Ring Resonators (SRRs), schematically shown by its unit cell depicted in Fig. 3. It consists of a pair of orthogonal SRRs. The split ring acts as a LC circuit, where the loop and gap are equivalent to an inductor and a capacitor, respectively. The bianisotropic property can be described by the fields and current distribution for a z-polarized incident wave, as reported in the literature 28 .
In 21 , the authors reported that the Tellegen material response exhibits non-reciprocal possibilities in gainless or lossless media. Suitable numerical methods were developed for computing photonic eigen-modes for designing and characterizing topological order based on Tellegen metacrystals. Strong Tellegen responses can be stimulated in metamaterials composed of magnetic ferrites associated with bianisotropic elements, and it would be simple www.nature.com/scientificreports/ to implement this proposal experimentally in the microwave domain 21 . Artificial media, such as chiral and SRRs, can also be used to design such systems. In 30 , the authors investigated the scattering of electromagnetic waves from bounded moving media. They explained the phenomenon of the intraluminal regime of double transmission of waves downstream through a stationary interface between regular medium and moving media. The refractive index and impedance relations for bulk moving media were derived in the subluminal and intraluminal regimes.
In the present work, we consider for study a complex bianisotropic medium characterized by 3 × 3 permittivity, permeability and magnetoelectric tensors. Assuming that the coupling between the magnetic and electric fields only exists in the x-y plane. These tensors have the following forms:   Unit cell of the bi-anisotropic metamaterial medium described by Eq. 2. The copper is inserted in a dielectric medium with a relative permittivity ε r = 2.2. 28 .
Generally, the magnetoelectric coupling effects can be classified as reciprocal or non-reciprocal. In reciprocal media, as can be easily derived from the Lorentz reciprocity theorem, the permittivity and permeability dyadics are symmetric, and the two magnetoelectric dyadic coefficients are related 31 as [η] = − [ξ] T , only one magnetoelectric dyadic is sufficient to describe the nonreciprocal coupling. Indeed, one of the two dyadics is the transpose of the other The non-reciprocity parameters χ xy and ς xy are needed to model natural magnetoelectric effect which occurs, for example, in some ferromagnetic and anti-ferromagnetic crystals. Recently, it has been suggested how such media can be realized as artificial composites for microwave applications 32,33 . Because these tensors are real valued, the Tellegen response breaks time reversal symmetry, distinguishing it from other bianisotropic electromagnetic responses (such as SRRs and chiral meta-molecules), which do not 21,34,35 .
The jξ xy and jη xy in some conditions are responsible for chiral effects in isotropic Pasteur media. As is commonly accepted, we use the name chirality parameter for the coupling jξ xy and jη xy for anisotropic reciprocal media, too 32,33 .
In what follows, we investigate the following five cases according to the conditions of consideration [28][29][30][31][32][33] , which are: The general planar dipole antenna geometry and the associated coordinate system, with the optical axis oz as direction of propagation, are illustrated in Fig. 4. The herein studied planar structure is based on a complex bianisotropic grounded substrate. The presented configuration is used to investigate the effect of bianisotropy on the input impedance of the printed dipole (Fig. 4a) and to evaluate the mutual coupling between two-dipole antenna array (Fig. 4b).
In this case study, a realistic bianisotropic material design is considered that supports both TE and TM surface modes. This material can be realized with complex imaginary-valued elements of the constitutive parameters. This is done by inserting a periodic set of inclusions in a dielectric substrate 9 . A realistic experimental model of this bianisotropic metamaterial is presented in 9,28 . The simplest cases of this medium have been studied as a substrate of a dipole antenna by Sayad et al. 26 , where the magnetoelectric elements were equal and purely imaginary. The results show that the chirality parameter serves as an additional parameter that could be utilized to control or adjust the input impedance for bandwidth improvement and miniaturization of the antenna size. The case of non-equal magnetoelectric elements was studied by Zebiri et al. in 12 .
In 12 , the results showed that the contribution of the magnetoelectric elements ξ and η in the calculation of the input impedance takes the average form of 1 2 (ξ + η) in the dyadic Green's function described by Eq. 20 in 12 . On the other hand, the gyro-chiral parameter contribution in the electromagnetic field components is denoted by the factor e −κ 0 1 2 (ξ −η)·z , in Eqs. 9 and 10 12 , that expresses gain or loss, depending on the choice of the two magnetoelectric parameters ξ and η.
In 8,[11][12][13][14]26 , only cases of media with imaginary valued magnetoelectric elements have been investigated, i.e., the case of a reciprocal chiral media [ξ] = −[η] T . In this work, we will examine the more complex issue, taking into consideration reciprocity and non-reciprocity for complex valued magnetoeclectic element (chiral, achiral, Tellegen and general bi-anisotropic cases). The present work will carry out this novelty of cases that have never been investigated before. The first result which should be taken into consideration is that these media have a significant potential which must be thoroughly studied.
The expected waves propagating in a grounded dielectric slab are surface wave modes which are either TE or TM with respect to the interface normal 36 . We assume that the propagation is along the + z direction with e −jβz as a factor of propagation. The longitudinal electromagnetic field components E z and H z are found to satisfy two decoupled homogeneous second-degree differential wave equations: The particularity of this medium is described by the extra term Ŵ 1 ∂Ẽ z ∂z in the second-order differential wave equation. This additional term can be interpreted by a loss or a gain in the amplitude of the electromagnetic fields 12 , which is reminiscent of the Schrodinger equation for an electron in presence of a magnetic potential 21 .
For [ξ] = [η] , Ŵ 1 = 0 , we meet the case of reciprocal bianisotropic media. It is confirmed that the non-reciprocity contributes to the appearance of the Ŵ 1 = 0 and which can be interpreted by the presence of a magnetic moment 21 .

Method of solution
Because there are two distinct regions: dielectric (region 1) and air (region 2), we must separately define the fields components in these two regions and then match the tangential fields on the interface air-dielectric. Solving the two differential Eqs. (2a) and (2b) for Ẽ z and H z in region 1, gives: where A e , B e , A h and B h are complex constants and (4a) www.nature.com/scientificreports/ In the absence of the Tellegen parameter ( ς xy = χ xy = 0 ) the expressions above are the same as those found in 12 .Thus we find that the solution for a plane EM wave in a bianisotropic (lossy or lossless) medium consists of a nonreciprocal z + -directed propagating wave and a z --directed one with e κ 0 κ c z e jκ t κ 0 z . For the chiral, this term will be e κ 0 κ c z .There may be a gain in one direction and a loss in the other. The non-reciprocal medium (Tellegen) contributes by a phase e jκ t κ 0 z in the solution which must be deeply examined in future works. However, these two traveling waves no longer have a constant amplitude as they move, but rather they decay exponentially with the traveled distance, as indicated by the e −αz term in the z + -directed wave and the e αz term for the z-directed one. The solution resembles that of a lossy dielectric medium only in one direction and just for one medium ( η xy = −ξ xy ς xy = −χ xy ). This is a very interesting feature that has to be well considered. Under these conditions, the medium behaves like an isotropic dielectric with the presence of the term e κ 0 κ c z e jκ t κ 0 z .
In the air region, the field components decay with respect to z, for which, we assume the following expressions: where C e and C h are complex constants. The application of the appropriate boundary conditions at z = 0 and z = d allows the determination of the complex constants A e , B e , C e , A h , B h and C h that appear in the electromagnetic field component expressions in both regions 26 .
Algebraic development of the derived mathematical expressions leads to the formulation of the estimated electric field at the interface air-dielectric between the two regions with respect to the current densities J x and J y . The spectral Green's tensor is derived satisfying the following system of equations 12,26 . where J x and J y are the Fourier transforms of the current densities on the conducting strips.
In the analysis of narrow dipoles configurations, the herein considered structure, the cross-current density in the y-direction is commonly ignored, as it is assumed that the width of the dipole is negligible 26 . Consequently, G xx is the only presented Green's function, since the others are not involved in the calculations. For this bianisotropic medium, G xx is derived and it is given by: with The magnetoelectric-depending sub-cases of this general bianisotropic medium can be verified. For χ xy = ς xy = 0 , the Green's tensor expression is the same as that found in 12 . For [ξ] = [η] � = 0 , we find the expression derived in 26 and for a dielectric with a uniaxial anisotropy [ξ] = [η] = 0 , we obtain the same medium and expressions as those treated in 24,25 .
The mathematical manipulation of the resulting equations gives the formulation of the electric field evaluated at the interface air-dielectric in terms of the current densities J x and J y . By applying the boundary conditions, the expressions of the x, y and z components of the electric and magnetic fields in the dielectric and air regions can be formulated as follows: 1st region (dielectric): where J x and J y are the Fourier transforms of the current densities, and In the present analysis, we also aimed to examine the effect of the gyro-bianisotropic medium on the electric and magnetic field distributions that has been less investigated in previous works in the literature. To evaluate the (10a) E x1 (α, β, z) = j e 1 2 κ 0 κ c (z−d)

Numerical results
In this work, we are interested in the investigation of the effect of the gyro-bianisotropic substrate on the input impedance, the resonant length of the dipole and the mutual coupling between two-element printed dipole array along three configurations. The five cases: chiral, achiral, Tellegen and complex bianisotropic mediums, are investigated and the related original results are discussed and commented.
Validation. Before discussing the results of this study, a validation of the method and the solution technique is undertaken by comparing with studies published in literature.
In order to test the efficiency of the employed method and the accuracy of the solution technique, an initial comparative study is carried out. We have initially considered the isotropic case (ε t = ε z = 3.25 and μ r = 1). Figure 5 shows the plot of the complex input impedance (real and imaginary parts) of a printed planar dipole of width W = 0.0004λ 0 with respect to the normalized length L/λ 0 . The dipole is printed on an anisotropic grounded dielectric slab with a thickness of d = 0.1060λ 0 . Figure 6a-c present the calculated mutual coupling between the printed dipoles for collinear, echelon and broadside configurations, respectively, for L = 150 mm, W = 0.5 mm, f = 500 MHz and d = 1.58 mm.
A comparison representation of the input impedance and mutual coupling configurations for the same configuration parameters used above is given in Figs. 5 and 6. The representation shows a good agreement compared to available data reported in literature 22-25 . Effect of the magneto-electric parameters on the input impedance. Non-reciprocal achiral ( ξ xy = −η xy ). Figure 7 shows the input impedance for a non-reciprocal achiral medium case ( ξ xy = −η xy = 1 ), compared with the isotropic case. No effect is noticed on the input impedance. This can be justified by the electromagnetic fields and Green's function expressions. In this case ( ξ xy = −η xy ), it is noticed that no contribution of the achirality is observed in the expressions of Eqs. 6 and 10.
Reciprocal chiral ( ξ xy = η xy ). Figure 8a,b illustrate the variation of the input impedance of a reciprocal chiral dipole antenna for different positive and negative values of ξ xy and η xy with a permittivity of ε r = 3.25 and a permeability of µ r = 1 , compared to the isotropic case medium. In this case, the effect of the parameters ξ xy and η xy is reciprocal, either on the shape of the input impedance (Fig. 8a,b) or on the resonance frequency (Fig. 9).
The maximum (peak) of the input impedance and the resonance frequency increases with the increasing positive values of ξ xy and inversely for negative values. For ξ xy = η xy = 0.5 ( ξ xy = η xy = -0.5), an increase (a decrease) of 50% of the input impedance peak is observed.
Reciprocal Tellegen medium ( χ xy = ς xy ). Figure 11a,b show an increase in the input impedance amplitude for positive values of χ xy and a decrease for negative ones. However, the resonance points shifted to the left compared with the isotropic case with for positive values (Fig. l1a). As for the negative values, the resonance points moved right (Fig. 11b). We can notice that the Tellegen case affects significantly the input impedance. Figures 11 and 12 illustrate the effect of the bianisotropic Tellegen element on the input impedance and the resonant frequency, respectively. The values of this latter are determined from the input impedance shown in www.nature.com/scientificreports/ Fig. 11, from the zero crossing of the reactance curve (imaginary part) 37,38 . Figure 12 shows that the resonance frequency decreases significantly with increasing reciprocal Tellegen medium, unlike the case where the elements are purely imaginary.     Figure 13 shows, the mutual coupling for a non-reciprocal achiral medium case ( ξ xy = −η xy = 1 ), compared with the isotropic case. No effect is noticed on the mutual coupling.   Non-reciprocal Tellegen medium ( χ xy = −ς xy ). In this case ( χ xy = −ς xy ), similarly to the case of non-reciprocal achiral ( ξ xy = −η xy = 1 ), it is noticed that no effect of the non-reciprocal Tellegen medium is absolutely observed on the input impedance (Fig. 10) and the mutual coupling (Fig. 15).
Reciprocal Tellegen medium ( χ xy = ς xy ). Figure 16a-c show the effect of the magnetoelectric element with real, imaginary, positive and negative values on the mutual coupling of the three cases of two dipole configurations, respectively. From Fig. 16a-c, the Tellegen case ( χ xy = ς xy = 0.5, ξ xy = η xy = 0) has a reciprocal behavior whereas in the case where ( χ xy = ς xy = 0, ξ xy = η xy = 0.5) is imaginary, a slight difference between the effect of the positive and negative elements is noticed. This can be justified by exploring the equations and developed expressions elaborated by Zebiri in 8 .
A shift of the mutual coupling curves, of the three configurations, with respect to the isotropic cases, in the direction of increasing G and S is noticed for higher values of ξ xy in the first studied case ( ξ xy is imaginary) and it is inversely in the second case ( ξ xy real).
The quasi-periodicity caused by surface waves increases with increasing magnetoelectric elements of the chiral medium and inversely for the Tellegen case. As an example, in the case of echelon configuration, we observe that the quasi-period equals 200 mm for the chiral element ( ξ xy = η xy = 1) and 150 mm for the Tellegen element ( χ xy = ς xy ).
Effect of the substrate thickness on the mutual coupling. This sub-section deals with the effect of different substrates for different values of χ xy , ς xy , ξ xy , η xy and selected substrate thicknesses: 0.8 mm and 4.25 mm. Figure 17a-c illustrate the mutual coupling between the printed dipoles for these different configurations. The increase in coupling for greater substrate thicknesses is due to the increase in spatial and surface wave modes 39 . As the dipoles are spaced from the ground due to the thickness of the substrates, stronger space and www.nature.com/scientificreports/ surface wave modes are generated, resulting in further radiated power and more effect on parasitic elements in the vicinity of the dipoles 25 .
The beneficial choice of the medium, that must be taken into account, is that of the chiral one. As it has been shown, this latter presents a decrease in coupling of more than 30% for certain values of S or G, on the other  Effect of the magneto-electric parameters on the field's distributions. The electric field distributions in the XY, XZ and YZ planes are shown in Fig. 18a-c. The field distributions illustrate a conventional isotropic dipole with the XZ plane as the E plane. Similarly, the distributions of the magnetic field in the XY, XZ and YZ planes are presented in Fig. 18d,e. We observe rotating lines of the magnetic field around the dipole in the YZ plane (the H plane) according to Fig. 18e. The electric and magnetic field distributions in the XY, XZ and YZ planes are shown in Fig. 19a-f, respectively. The lines of the electromagnetic field for this case have not changed and have kept the shape of the conventional isotropic substrate dipole (Fig. 18a-f). A slight increase in the components of the electromagnetic field is noticed.
The sign of the elements ( ξ xy = 0.5 and ξ xy = −0.5 ) influences the distribution of the fields differently in this case. From Fig. 20a-f, for ξ xy = 0.5 , we notice that the amplitude of the field components increases by 14% in the E plane with a slight decrease in the other planes. A slight decrease in the components of the magnetic field is noticed also for the three planes. The z-directed wavelength λz in this case has undergone an increase.
According to Fig. 21a-f, for ξ xy = −0.5 , we notice that the effect in this case is reversed compared to the previous case ( ξ xy = 0.5 ). The amplitude of the electric field components decreases by 24% in the E plane with a slight increase in the other planes. λz in this case experienced a decrease. However, for this case a more important decrease in the magnetic field components is noticed for the three planes.
From Fig. 22a-f, the electric field distribution in the XZ plane (Fig. 22c) is completely different from that of the isotropic case and from the previous cases, and remains unrelated because the electric field in this case is weaker compared to that in the XZ plane (E plane).
From Fig. 23a-f, the distribution of the electromagnetic field has kept the same shape, except that the components of the electric field have almost doubled in the E plane. Similarly, the components of the magnetic field are multiplied by about 1.7 in plane H.    www.nature.com/scientificreports/ For χ xy = ς xy = −0.5 case (Fig. 24a-f), the field amplitudes are almost the same with a particular change in the distribution of the electric field in the YZ plane (Fig. 24c). The distribution has completely changed compared to the other cases and even for the isotropic case. Chiral and Tellegen elements advantageous combined effect on the input impedance and mutual coupling. The advantageous combined effect cases of Tellegen and chiral mediums, represented by complex values of the magnetoelectric element (real and imaginary parts, respectively), on the input impedance and coupling are illustrated by Figs. 25 and 26. The selection of the elements values is based on reducing the input impedance amplitude, for matching purposes, and an improving decoupling between the antenna system elements.
The first element selection is based on the above-mentioned case ( χ xy = ς xy = ξ xy = η xy = − 1), where a significant decrease in the peak of the input impedance accompanied with an increase in the resonant length is observed. The second selection is considered to obtain a total peak decrease of more than 75%, compared with the isotropic case, to achieve 1.02KΩ for χ xy = ς xy = − 0.8 and ξ xy = η xy = − 2.1, all without altering the resonant frequency value.    Fig. 26, it is observed that the effect of the bianisotropic medium reciprocal (chiral and Tellegen) is different depending on the configuration as well as the distance between dipoles. For the collinear configuration, the chosen medium has a decoupled structure of more than 25% for the entire range between λ/4 to λ/2 (150-300 mm). For the broadside configuration, the medium shows a significant decreased mutual coupling (almost the half) for two closer dipoles (160-210 mm). For the echelon configuration, the chiral combined with the Tellegen presents an advantageous effect in the same region (160-210 mm) and above λ/2 as well. A summary of the obtained results is well described in Table 1.

Conclusions
In this paper, a rigorous mathematical formulation describing the bianisotropic medium electromagnetic behavior is presented and the input impedance, the electromagnetic field distributions and the mutual coupling in two-element printed dipole array are investigated. The field distributions in such a medium will be the start of an in-depth study of the bianisotropic medium behavior to better understand the effects of this medium on different parameters of the dipole antenna, for which some have been shown here. The medium bianisotropy effect on the input impedance of a single dipole configuration is evaluated. It is shown that, with increasing magnetoelectric elements of the chiral medium, the quasi-periodicity caused by surface waves increases and inversely for the Tellegen medium. An appropriate selection of the magnetoelectric elements, combined effect of chiral and Tellegen mediums, leads to a significant decrease in the input impedance. A peak decrease of more than 75% is achieved without altering the resonance frequency, which is advantageous for matching possibilities. It is also concluded that this choice leads to a better decoupling compared with the isotropic case. An improvement