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## Dispersion Relation and Brillouin Zone

Propagation properties of a structure are obtained by studying its dispersion relation. The dispersion relation is a relation between the frequency and propagation constants of an electromagnetic mode. It is commonly visualized in a plot showing the frequency vs. the phase constant of the Floquet mode: the dispersion diagram. Substitution of (2.13) in (2.4) provides the dispersion relation for the 1-D periodic structure along the x direction. As it was mentioned before, changing kx by integral multiples of q = 2p/p does not change the corresponding frequency. This means that the dispersion relation w (kx) is periodic with period q which means that it is enough to find the dispersion relation for p/p < kx < p/p and the rest of the dispersion diagram is known from the periodicity of the result. This region of kx values is called the Brillouin zone.

When a periodic structure has a rotation, mirror reflection or inversion symmetry, its dispersion diagram (w (k) where k is the wave vector) is also symmetric with the same symmetry [1]. In these cases, there are further redundancies within the Brillouin zone and the smallest region for which the frequency band is not repeating itself is called the irreducible Brillouin zone [1].

The Brillouin zone for a periodic structure is defined by its lattice vectors.

For instance, a periodic structure with a rectangular lattice is shown in Fig. 2.2 (a). The lattice vectors are px = pxxˆ and py = pyyˆ. The reciprocal lattice vectors in kxky plane are qx = (2p/px) ˆ x and qy = 2p/py yˆ which are depicted in Fig. 2.2 (b). The Brillouin zone is shown in Fig. 2.2 (c) with orange rectangular area around the center which is bounded by red lines kx = p/px and ky = p/py.

### Higher Symmetries in Periodic Structures

Periodic structures with higher symmetries are structures that can be described by additional geometrical operations beyond the usual periodic condition. Periodic structures are invariant under translations of period length.

In periodic structures with higher symmetries, in addition to the periodic condition, the unit cell is invariant under a composite transformation consisting of either translation and rotation or translation and mirroring. The higher symmetry that is constructed by translation and rotation is called twist symmetry (also called screw symmetry) and the one constructed by translation and mirroring is called glide symmetry.

Fig. 2.4 shows two structures with glide symmetry. A translation of half a period and a mirroring operation with respect to the mirroring plane (also called the glide plane) ensures a periodic structure with a glide symmetry.

Fig. 2.4 (a) shows a 1-D glide symmetry which is achieved by translation and mirroring of the PEC slabs in the line. Fig. 2.4 (b) shows glide symmetry in a 2-D periodic structure. To better understand the glide symmetry in periodic structures, we define the glide symmetry operator in one and two dimensions. The 1-D glide symmetry operator Gp ˆ x can be considered as a composition of an x-translation of length p/2 and a z-reflection around the glide plane (here we choose z = 0 as the glide plane) [2]: Gp ˆ x : (x, z) ! (x + p/2,z).

#### Applications of Higher Symmetries

Introducing a spatial higher symmetry within a unit cell of a periodic structure provides interesting dispersive properties that make these structures proper candidates for numerous electromagnetic applications. In the 1960’s and 70’s, one dimensional periodic structures with twist and glide symmetries were studied using a generalized Floquet theorem to provide initial insights on the characteristics of these symmetries [2, 3, 4, 5]. In these works, the influence of higher symmetries on the propagation characteristics of the guided and radiating fields were studied. [2] introduces a generalized Floquet theorem for periodically loaded closed waveguides possessing twist and glide symmetries. The theorem states that the Bloch modes are eigenvectors of the glide or twist symmetry operator characterizing the structure.

Based upon this theorem, it also presents a method to construct the Brillouin diagrams of higher symmetric structures qualitatively. This method predicts which stop bands will be present or absent in the dispersion diagram by considering the mode couplings between modes of the unloaded structure and their space harmonics. [3] discusses the higher symmetry operators as well.

By means of mode-coupling theory it discusses how higher symmetries can uncouple some space harmonics of the unloaded guide and then suppress some stopbands. The results of these classical theoretical studies, such asthe lack of a stopband and a larger passband will be discussed in the next paragraphs.

**Reduced frequency dispersion and lens antennas**

Recently, a surge of interest in studying 2-D periodic structures with glide symmetry has started in the framework of metamaterial research [6, 7, 8, 9, 10]. It was demonstrated in [11] that application of higher symmetries to two dimensional periodic structures reduces their frequency dispersion. To understand this phenomenon, let’s consider the periodic bed of nails structure with a top plate whose unit cell is shown in Fig. 2.6 (a). Fig. 2.6 (b) shows the glide-symmetric version of this structure which is achieved by mirroring the central pin to the top plate and moving half a period P/2 in both the lateral directions. Fig. 2.7 depicts the normalized phase constant versus the frequency for the two aforementioned unit cells. The parameters were chosen as p = 4 mm, d = 0.5 mm, h = 1 mm and g = 0.1 mm. Firstly, the stopband has moved to higher frequencies for the glide-symmetric structure. Secondly, the curve of the first Bloch mode has become almost straight for the glide-symmetric cell. This linear dependency over a wide range of frequency means that this mode is essentially nondispersive. [11] also investigates a glide-symmetric holey structure rather than the bed of nails configuration. Glide-symmetric holey metasurfaces show similar characteristics to their bed of nails counterparts. Fig. 2.8 (a) shows the unit cell of a holey periodic structure with a top plate cover. Fig. 2.8 (b) shows the glide-symmetric counterpart where the glide symmetry was achieved by moving the mirrored holes half a unit cell P/2 in both lateral directions. The dispersion diagram of these two unit cells are compared in Fig. 2.9. This figure shows that the glide-symmetric unit cell has a higher refractive index compared to the non-glide symmetric cell, realizing a denser material. This is very useful for lens design. We also notice that similar to the bed of nails configuration, the glide-symmetric holey structure is again nondispersive.

This can be observed by the linear dependence of the phase constant on the frequency for the first Bloch mode.

This is very important for the design of planar lens antennas since the bandwidth can be noticeably increased. [11] discusses this characteristic and suggests that the less dispersive behavior of metasurfaces with higher symmetries can be utilized in the design of ultrawideband lenses. The paper furthermore discusses how the glide-symmetric holey surfaces can be used to achieve the refractive index profile of a two-dimensional Luneburg lens.

Different hole depths h could be used to achieve different refractive indices required to realize a Luneburg lens.

**Table of contents :**

Declaration of Authorship

Abstract

**1 Introduction **

1.1 Organization of the Document

**2 Periodic and Higher Symmetric Structures **

2.1 Wave Propagation in Periodic Structures

2.1.1 Periodic Structures

2.1.2 Floquet Theorem

2.1.3 Dispersion Relation and Brillouin Zone

2.2 Higher Symmetries in Periodic Structures

2.3 Applications of Higher Symmetries

2.3.1 Early studies

2.3.2 Reduced frequency dispersion and lens antennas

2.3.3 Backward radiation and leaky-wave antennas

2.3.4 Enhanced stopband and microwave components

2.3.5 Breaking glide symmetry

2.3.6 Glide-symmetric flanges

2.3.7 Matching improvement of dielectric discontinuities and printed circuits

2.3.8 Properties of twist symmetric lines

2.4 Existing models for higher symmetries

2.5 Content of the thesis

**3 Multimodal T Matrix Analyses for Cells with Higher Symmetries **

3.1 Transmission-Matrix Analysis

3.2 Extension of the T Matrix Analysis

3.2.1 Multimodal T Matrix

3.2.2 Analysis of 1-D Periodic Structures

3.2.3 Analysis of 2-D Periodic Structures

3.3 Glide Symmetric Structures

3.3.1 Glide Symmetry Along One Direction

Periodic-structure associated to a glide structure

Reducible and irreducible glide-symmetric structures .

3.3.2 1-D Glide-Symmetric Boundary Conditions Along One

Direction

3.3.3 Glide Symmetry Along Two Directions

3.3.4 2-D Glide Conditions on the Boundaries of a Quarter of a Non-Minimal Unit Cell

3.3.5 Parameter Study on the Reducible Condition

3.3.6 Computation Time

3.4 Twist-Symmetric Structures

3.4.1 Reducible and Irreducible twist structures

3.4.2 Twist Symmetry Conditions on a Sub-unit Cell

3.5 Conclusions

**4 Reconfigurable Waveguide Technology Based on Glide Symmetry **

4.1 Introduction on MM-Wave Switches

4.2 ReconfigurableWaveguide Design

4.2.1 Design of Guiding Medium and EBG

Parametric Study of the EBG Unit Cell

The “on” State

Two-State Reconfigurability

4.2.2 Feeding Mechanism and Impedance Matching

4.2.3 Simulation Results

4.3 The Improved ReconfigurableWaveguide

4.3.1 Modifications of the guiding medium

4.3.2 Modifications of the EBG medium

4.3.3 The Final Structure

4.3.4 Simulation Results

SingleWaveguide

AdjacentWaveguides

Comparisons with Non-Glide Waveguides

4.3.5 Measurements

4.4 Conclusions

**5 Conclusion **

5.1 Contributions

5.2 Future Work

**Bibliography **