# Modeling of Glide-Symmetric Dielectric Structures

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Analysis of Waveguides Containing Periodic Dielectric Structures

_{x}, a glide operator can be written as:

_{x}, the height of corrugations by h

_{corr}, the width and permittivity of periodic dielectric inclusions by W

_{x1}, W

_{x2}and ε

_{1}, ε

_{2}, and the height and permittivity of parallel-plate waveguide by h

_{ppw}and ε

_{ppw}. Note that the plane z = 0 is located in the middle of the parallel-plate region. We will first analyze the simple periodic structure (i.e., the one with classical translation/mirroring symmetry, see Figure 1a) and then we will modify the analysis procedure for structures with higher symmetry in which the corrugations are shifted with respect to each other by P

_{x}/2, i.e., the structure will be glide-symmetric, as in Figure 1b.

_{PPW}is the highest-order considered Floquet mode (in total 2N

_{PPW}+ 1 modes are taken into account), and k

_{0}and η

_{0}are the wave number and the wave impedance of free space.

_{m}and the propagation constant k

_{x,}

_{0}of the propagating wave inside the parallel-plate waveguide we need to match the tangential EM components with the ones in the corrugated walls. The wave propagating in the corrugated region can be modeled as a wave propagating along a periodic array of dielectric slabs [18,20] with the following EM field distribution (N

_{corr}denotes the highest-order considered mode).

_{i}are determined by considering the wave propagation along the periodic array with an assumed progressive phase delay per unit cell. In more detail, for each possible propagation constant k

_{x,}

_{0}, i.e., for each considered progressive phase delay we need to solve the secondary (local) mode-matching problem. The problem is described with a linear system of four equations representing the continuity of the E

_{y}and H

_{z}field components at two boundaries x = ±W

_{1}/2. Since for each considered case we have six unknown coefficients as seen in Equations (5)–(7), the Floquet theorem is used to express ${B}_{i}^{5}$ and ${B}_{i}^{6}$ using ${B}_{i}^{1}$, ${B}_{i}^{2}$ and assuming progressive phase delay. The determinantal equation resulting from the linear system gives the value of the propagation constant along the interfaces γ

_{i}and then it is possible to determine the field distribution of the considered ith mode. The details of the formulation are given in Reference [18].

_{ppw}/2) and testing it with $(1/{P}_{x})\mathrm{exp}(+j{k}_{x}x)$, one equation per each Floquet harmonic is obtained (as a consequence of orthogonality of Floquet harmonics):

_{m}with the Fourier transformation of the E-field distribution at the corrugation boundary. Thereby, only the coefficients ${C}_{i}^{\pm}$ (among all coefficients A

_{m}, B

_{i}, C

_{i}and D

_{m}that describe the field distribution) are the unknowns.

_{x,}

_{0}of the propagating wave inside the parallel-plate waveguide. Therefore, we should also match the tangential magnetic field H

_{x}at the boundary between parallel-plate waveguide and corrugated region.

_{m}and ${C}_{i}^{\pm}$ (given by Equations (8) and (9)). By multiplying Equation (10) with ${({E}_{0,l}^{corr}(x))}^{*}$, where * refers to the complex conjugate, and integrating over the period we obtain the following linear system of equations whose determinant is the characteristic equation for the propagation constant k

_{x,}

_{0}

## 3. Results

_{y}field component of the TM wave for different values of assumed progressive phase delay per unit cell. The parameters of the corrugated structure are the following: ε

_{r1}= 2.56, ε

_{r2}= 1.0, P

_{x}= 0.6 λ

_{0}, W

_{x1}= 0.26 λ

_{0}and W

_{x2}= 0.34 λ

_{0}. Excellent matching of the obtained results can be noticed in Figure 2. We also tested the accuracy of the calculated propagation constant of the first two modes travelling along the periodic array of dielectric slabs (Figure 3). Again we got excellent agreement with results given in [18]. It is interesting to notice that for values k

_{x}

_{,0}less than 0.5·π/P

_{x}the first high-order mode is evanescent, i.e., the values of γ

_{2}are imaginary (k

_{x}

_{,0}is the transverse propagation constant in the corrugated region). For values k

_{x}

_{,0}larger than 0.5·π/P

_{x}the wave propagates along the dielectric slabs and consequently γ

_{2}is a real number. The dominant mode is of a propagating type for all values of k

_{x}

_{,0}.

_{PPW}= 2.56, h

_{PPW}= 10 mm, P

_{x}= 35 mm, W

_{x1}= 15 mm, W

_{x2}= 20 mm, ε

_{r1}= 10.0, ε

_{r2}= 1.0, and h

_{corr}= 10 mm. First, we analyzed the simple periodic structure (Figure 1a). In Figure 4 the dispersion diagram of the first three propagating modes are given. Note that only the values of the propagation constant larger than the free-space propagation constant k

_{0}are given, otherwise the excited mode is a fast wave and the structure is radiating part of the EM energy (i.e., we have a leaky-wave antenna). The obtained results were compared with the ones obtained using a general electromagnetic solver, CST Microwave Studio in our case, and the agreement is very good. Specifically, the Eigenmode Solver in the CST Microwave Studio package was used to find the dispersion characteristics of the considered structures, thus it was necessary to analyze only a unit cell. In x- and y- directions the periodic boundary conditions were applied, while at the top and bottom of the periodic cell the PMC boundary conditions were used to simulate infinitely long symmetric and glide-symmetric structures. Note that the unit cell was quite long in the z-direction since it was needed to ensure that the amplitude of the evanescent fields was negligible at the top and bottom boundaries.

_{x}

_{,0}= π/P

_{x}. Like in the case of the simple periodic structure (Figure 4) a much larger percentage of EM power is propagating outside the dielectric structure for the second mode. The selected time moment gives the best illustration of the E-field distribution for the depicted modes. Other time moments could be selected, as illustrated in Figure 6 where the E-field distribution of the dominant mode is given for relative phase shifts 45°, 90°, 135° and 180°, respectively.

_{x}

_{,0}= π/P

_{x}), while the upper part is mostly affected by the odd mode.

_{r1}= ε

_{PPW}= 2.56 and ε

_{r2}= 1.0; other parameters are the same as for the structure in Figure 5). In other words, such structures are made from a homogeneous media but the boundaries have a periodic variation. Results of the same type were obtained (see Figure 8), however the glide-symmetric properties were not so pronounced. It can be concluded that contrast between the permittivity of PPW and corrugated regions can also be used for tuning the dispersion properties.

_{x}/2). The obtained dispersion diagrams are located between ones for the symmetric and glide-symmetric cases, which was demonstrated in Reference [11] for metallic structures. Therefore, we have focused our investigation on the two most interesting cases, symmetric and glide-symmetric.

_{PPW}= 5 (in total 11 modes) and N

_{corr}= 2. We tested the numerical convergence of the solution, and enlarging the number of modes only slightly improved the accuracy of the solution. The needed computer time was less than a second per point in the dispersion diagram, while the CST typically needed around 1 hour for calculating the whole dispersion diagram (depending on the accuracy of the meshing). All the calculations were made on a standard PC.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Sketch of the parallel-plate dielectric waveguide with periodic structure; (

**a**) simple periodic structure with translation and mirroring symmetry, and (

**b**) periodic structure with glide symmetry. Direction of wave propagation is sketched using a red arrow.

**Figure 2.**Normalized value of the H field component of TM wave for different values of assumed progressive phase delay per unit cell; solid line—results calculated using the developed program, diamonds—calculated results from [18]; (

**a**) first mode, (

**b**) second mode.

**Figure 3.**Normalized value of the propagation constant of the first two modes travelling along the periodic array of dielectric slabs; solid line—results calculated using the developed program, diamonds—calculated results from Reference [18]. γ

_{2}= 0 is the cut-off condition of the 2

^{nd}mode, thus for the values k

_{x,0}smaller than the cut-off value (0.5·π/P

_{x}) γ

_{2}is imaginary (the mode is evanescent).

**Figure 4.**Propagation constant of the first three modes travelling along the dielectric waveguide with corrugations from Figure 1a; solid line—results calculated using the developed program (blue line: 1st even mode, red line: 1st odd mode, green line: 2nd even mode), diamonds—results calculated using CST Microwave Studio (dashed black line represents the light-line). The E-field distribution in the unit cell of the first three modes for k

_{x}

_{,0}= π/P

_{x}is also shown (the structure is infinite in the y-direction).

**Figure 5.**Propagation constant of the first two modes travelling along the glide-symmetric dielectric waveguide with corrugations (Figure 1b); solid line—results calculated using the developed program, diamonds—results calculated using CST Microwave Studio (dashed black line represents the light-line); (

**a**) glide-symmetric structure only, (

**b**) comparison of glide-symmetric and simple-symmetric structures (blue dashed line). The E-field distribution in the unit cell of the first two modes for k

_{x}

_{,0}= π/P

_{x}is also shown (the structure is infinite in the y-direction).

**Figure 6.**The E-field distribution in the unit cell of the first mode for k

_{x}

_{,0}= π/P

_{x}at different time moments where T denotes the period of oscillations (T = 1/f).

**Figure 7.**Propagation constant of the first mode travelling along the glide-symmetric dielectric waveguide with corrugations (case of thinner parallel-plate dielectric slab); solid line—results calculated using the developed program, diamonds—results calculated using CST Microwave Studio (dashed black line represents the light-line); (

**a**) case of thinner parallel-plate dielectric slab (h

_{ppw}= 2 mm), (

**b**) case of thinner corrugated region (h

_{corr}= 2 mm).

**Figure 8.**Propagation constant of the first mode travelling along the glide-symmetric dielectric waveguide with corrugations made from homogeneous material; solid line—results calculated using the developed program, diamonds—results calculated using CST Microwave Studio (dashed black line represents the light-line).

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**MDPI and ACS Style**

Sipus, Z.; Bosiljevac, M.
Modeling of Glide-Symmetric Dielectric Structures. *Symmetry* **2019**, *11*, 805.
https://doi.org/10.3390/sym11060805

**AMA Style**

Sipus Z, Bosiljevac M.
Modeling of Glide-Symmetric Dielectric Structures. *Symmetry*. 2019; 11(6):805.
https://doi.org/10.3390/sym11060805

**Chicago/Turabian Style**

Sipus, Zvonimir, and Marko Bosiljevac.
2019. "Modeling of Glide-Symmetric Dielectric Structures" *Symmetry* 11, no. 6: 805.
https://doi.org/10.3390/sym11060805