Implementing a transmission line electrical circuit model in a complementary metamolecule waveguide

Initially, we construct the metamolecule from its constituent meta-atoms, namely, a split ring resonator (SRR) and a complementary SRR (CSRR) which have characteristic oscillator frequencies in the microwave frequency band (especially in the X-band). Next, an one-dimensional metamolecule waveguide is created from an array of the CSRR-SRR metamolecules. The CSRR is the negative image of the SRR therefore this waveguide is called Complementary Metamolecule Waveguide (CM-WG). Using electromagnetic (EM) simulations, we study the operation frequency bands of the CM-WG changing the polarization of the incident EM wave as well as its unit cell dimension. In continue, we develop an electrical transmission line model, we precisely calculate the electric and magnetic coupling coefficients between the metamolecules and verify the results of the EM simulations. We conclude, from the electromagnetic simulations and the analytical considerations, that the polarization of the incident EM wave and the distance between the metamolecules affect the bandwidth of the frequency bands as well as the properties of the propagation.


Introduction
Metamaterials are artificial, effectively homogeneous structures, featuring negative refractive index at specific frequency bands, where the effective permittivity and permeability are simultaneously negative [1][2][3]. In contrast to ordinary materials which are composed by atoms, metamaterials are constructed by structural components, called 'meta-atoms' [4][5][6]. Metamaterials may be divided in magnetic (MMs) and electric (EMs) ones, consisting of resonant elements with strong magnetic/electric response. The most common MMs/EMs element is the Split Ring Resonator (SRR) and Complementary SRR (CSRR). In microwave regime the SRR is constructed from two concentric metallic split rings which are printed on a dielectric circuit board. The CSRR is the negative footprint of the SRR, namely, the metallic parts of the SRR have been replaced from dielectric and vice versa [4,7]. The SRR and CSRR are used for the realization of various microwave devises, including filters [8], diplexers [9], antennas [10] and so on.
Traditional electromagnetic theory is usually used for studying metamaterials structures [11]. However, a convinient framework for the studying of the metamaterials, especially in the microwave regime, is the transmission line (TL) theory [12]. In this case the properties of medium ò and μ are connected with serial and shunt impendances of the TL model. Thus, the periodic structures of SRRs and CSRRs are described by the equivalent transmission line models [13][14][15].
Our objective is to study a one-dimensional complementary metamolecule waveguide (CM-WG), using a equivalent TL model, analytically and numerically. The CM-WG is a subwavelength waveguide that consists of an array of meta-atoms with complementary characteristics, SRR and CSRR, harmonically coupled with their nearest neighbors. In recent years, self-complementary structures [16,17] are used for the design of metasurfaces mainly for the manipulation of the polarization of incident EM waves [18]. In the case of the proposed metamolecule, the orientations of meta-atoms and the polarization of the incident EM field determine the induced electric/magnetic dipoles. Thus, the CSRR (SRR) can be excited either by a normal component of Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. the electric (magnetic) incident field, or by a component of the magnetic (electric) field along the slit of the CSRR (SRR); consequently cross polarization effects appear in the CSRR (SRR) [19,20] structures. Recently, optical excitations were used in the nanoscale metamolecule structures in order to achieve tunability through polarization [21,22]. In our study, we have chosen the square geometry for the meta-atoms because both their geometry and the orientation of the polarization of the incident electromagnetic field define their electromagnetic response. Initially, in order to theoretically and numerically explore the electromagnetic (EM) properties of the proposed structures we use 3D EM analysis software package CST Studio Suite R [23] for the electromagnetic simulations. The simulation results for these structures are obtained, using boundary conditions (electric and magnetic walls) on the transverse directions (x and y axes) and open boundary conditions along the wave propagation (z-axis). Thus, the SRR meta-atom appears a transmission minimum at 8.56GHz and the CSRR meta-atom appears a transmission maximum at 12.35 GHz, as depicted in figure 1. The size of the meta-atoms are much smaller than the free space wavelength of the propagating EM field at resonance (in our case each meta-atoms size is about one-tenth of that wavelength). Next, we design and study a structure composed by meta-atoms (SRR/CSRR) in axial configuration, as depicted in figure 1 (bottom panel). This hybrid structure of meta-atoms (SRR and CSRR), we call CSRR-SRR Metamolecule, has transverse dimensions 2.5mm × 2.5 mm and deep 0.534 mm. In the case of the proposed CSRR-SRR Metamolecule structure, the orientations of meta- atoms and the polarization of the incident EM field determine the induced electric/magnetic dipoles. Thus, the CSRR (SRR) can be excited by a component of the magnetic (electric) field along the slit of the CSRR (SRR).

The results of the simulations
The results of the simulations of the transmission coefficient for the horizontal and vertical polarization of incident EM wave of the CSRR-SRR metamolecule structure, are depicted in figure 1 (bottom right panel). The SRR meta-atom (CSRR meta-atom) coupled directly with the incident EM wave via the electric (magnetic) field. Also, it is fairly obvious that in this (dimer) configuration the SRR meta-atom and CSRR meta-atom are coupled together. Thus, the CSRR-SRR Metamolecule structure is excited and two frequency transmission bands appear. The first frequency band is a narrow band and a maximum appears about of 8.5 GHz near the resonance of SRR meta-atom. The second frequency transmission band is a wideband and appears at a maximum about 10.5 GHz. Now, we analyze the structure of Complementary Metamolecule Waveguide (CM-WG) made up by CSRR-SRR metamolecules as depicted in figure 2. By using the eigenmode solver of 3D EM analysis software package CST Studio Suite R [23], with boundary conditions (electric and magnetic walls) on the transverse directions (x and y axes) and periodic boundary conditions along the wave propagation (z-axis) , the dispersion diagram is obtained. Each resonator (SRR and CSRR) is coupled electrically and magnetically to its nearest neighbor as shown from our simulations which are well supported by theoretical considerations. Initially, we consider the chain configuration where the smaller distance between nearest meta-atoms is R 1 = 0.267 mm and the larger one is R 2 = 1.733 mm, namely the length of the unit cell is d = R 1 + R 2 = 2 mm. Then, from simulations results, the dispersion diagram is illustrated in figure 2, where we plot the frequency f as a function of the dimensionless wave number k , for the incident horizontal and vertical polarized wave (note that here we consider a semi-  [25,26]. Also, in the case of the vertical polarization (bottom panel), there exists two frequency bands where EM wave propagation is possible: the LF band is a RH frequency band [cf. solid (blue) line], for 9.08 GHz < f < 9.14 GHz, and HF band is a RH frequency band [cf dashed (red) line], for 10.31 GHz < f < 11.6 GHz. In the same case there exists a gap for 9.14 GHz < f < 10.31 GHz, where EM wave propagation is not possible. The width of the LF, HF and gap is about 0.06GHz, 1.29 GHz and 1.17 GHz, respectively. It is important to notice that the polarization of the incident EM wave changes the behavior of the upper branch (HF band) of the dispersion diagram from LH frequency band to RH frequency band and vice versa. The lower branch (LF band) of the dispersion diagram remains a RH frequency band.
Next, we repeat the above simulations increasing the length d of the unit cell, at the values 2, 2.5, 3, 3.5, 4 and 5 in mm. Then, we notice that these changes of the unit cell length don't affect the LH (RH) attitudes of the LF and HF bands in the changes of the polarization of the incident EM wave, namely, for the different values of d the dispersion relation diagrams are similarly as in the case of d = 2 mm (see figure 2). However, the widths of the LF, HF and gap are varied, as depicted in the figure 3.
As seen in the figure 3 for the horizontal polarization of incident EM wave as d is increased, the width of gap (for k = π) initially is increased and next it tends in a constant value, while the width of HF and LF bands are decreased. It is worth noting that in this case the HF band is decreased by exponential rate and the structure of CM-WG behaves like a SRR structure. In contrary with the previous case, as seen in the figure 3 for the vertical polarization of incident EM wave, as d is increased the width of gap (for k = 0) is decreased while the width of HF and LF bands are decreased. In this case the LF band is decreased by exponential rate and the structure of CM-WG behaves like a CSRR structure.

The analytical model
Both meta-atoms (SRR and CSRR), can be represented by a simple LC parallel resonator circuit (see figure 4), but nevertheless the significance of parameters L and C is different for the SRR and CSRR [19]. It means that our electric circuit model satisfies both horizontal and vertical polarization and consequently two different transmission bands appear , in contrary to other similar circuit models [24]. We start with an equivalent electrical lattice model of the CM-WG where the different resonators, namely the SRRs and CSRRs meta-atoms, are coupled magnetically and electrically to their nearest neighbors. In this case, we shall consider the configuration where the neighboring meta-atoms appear to have different magnetic and electric coupling depending on their relative distance. Thus, L c and C c are the self-inductance and self-capacitance of the CSRR, L s and C s are the self-inductance and self-capacitance of the SRR, L M and and C M is the mutual inductance and the mutual capacitance between the meta atoms. Let us now consider Kirchhoff's voltage and current laws for this (see the L s C s and L c C c combination in figure 4) equivalent circuit, namely,  lead to the following system At this point, in order to further simplify equation (7), we define the equivalent capacitance C 0 and inductance L 0 as . Then, we obtain: where the coefficients are given by: The coefficient λ (μ) is the magnetic (electric) coupling coefficient from the nearest neighbour (which is at the smallest distance).

Linear analysis
We now assume plane wave solutions of equation (10), of the form , where k and ω denote the wave number and angular frequency, respectively, while the amplitude of the wave is V 0 = 1. Substituting the above ansatz into equation (10), and keeping only the linear terms in V 0 , we obtain the following linear dispersion relation: Now, it is useful to calculate the values of parameters λ and μ in order to plot the dispersion relation using the equation (12). Thus, from equation (12) it follows that (1) for k = π/2, Next, we substitute the above values of λ and μ in the equation (12) and we plot the dispersion relation, namely, the frequency f in as function of wavenumber k, as depicted in figure 5.
Similarly, we consider the case of the vertical polarization (E y ) of the incident EM wave where we obtain the simulation results for the (k, f (GHz)), namely, (0, 9.08), (π/2, 9.11) and (π, 9.14) for LF band as well as (0, 10.31), (π/2, 11.3) and (π, 11.61) for HF band. Following the same procedure like the case of horizontal polarization for the values of λ and μ, we calculate 0.2867 and 0.2443 in the HF band and 0.105 and 0.0071 in the LF band, respectively. Also, we substitute the above values of λ and μ in the equation (12) and we plot the dispersion relation, as depicted in figure 5.
As shown in figure 5 there is a very good agreement between the analytical considerations and the results from the EM simulations. This agreement indicates that the equivalent electric circuit model is a very good approximation for the study of the CM-WG. It has been mentioned (see previous section) that in the case of horizontal polarization the HF (LF) band is a LH (RH) frequency band for all the values of the length of unit cell  (d). We notice that in the HF (LF) band the difference between the coefficient λ and the coefficient μ, namely λ − μ, is negative (positive) , for all the values of d. Similarly, in the case of vertical polarization both the HF and LF bands are RH frequency bands, for all the values of the length of unit cell (d), and the difference λ − μ is positive. All these results are depicted in figure 6.
Finally, according to the above notifications we can use the difference λ − μ as a criterion to characterize the behaviour of a frequency band as LH (λ − μ < 0 ) or RH (λ − μ > 0 ). Consequently, this criterion can be used to identify the frequency regime where the phase velocity is parallel (RH region) or antiparallel (LH region) with the group velocity.

Conclusion
In conclusion, we have performed an analytical and electromagnetic simulation study of a complementary metamolecule waveguide (CM-WG). The unit-cell circuit of the latter was constructed thus resembling the unit cell circuit of a one-dimensional transmission line. In this framework, we derived the dispersion relation for small-amplitude linear plane waves. We have found that two frequency transmission bands appear and the polarization of the incident EM wave defines if the upper band will have LH or RH behavior. We have also observed that as the length of the unit cell is increased the CM-WG behaves like a SRR (CSRR) structure, in the case of horizontal (vertical) polarization. Also, we showed that there is a very good agreement between the analytical considerations and the results from the EM simulations for the dispersion relation. Thus, the equivalent electric circuit model is a very good approximation for the study of the CM-WG. Finally, we showed that the difference between the coupling coefficients (λ − μ) is a criterion for the LH or RH behavior of the CM-WG. These results will be use in the studies for linear and nonlinear CM-WG structures which are currently in progress and will be reported in future publications.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).