Issue 
EPJ Appl. Metamat.
Volume 6, 2019
Metamaterials'2018 – Microwave, mechanical, and acoustic metamaterials



Article Number  14  
Number of page(s)  6  
DOI  https://doi.org/10.1051/epjam/2019012  
Published online  28 March 2019 
https://doi.org/10.1051/epjam/2019012
Research Article
Propagation characteristics of periodic structures possessing twist and polar glide symmetries
^{1}
Department of Electromagnetic Engineering, KTH Royal Institute of Technology, 10044 Stockholm, Sweden
^{2}
Laboratoire d'Électronique et Électromagnétisme, Sorbonne Université, 75005 Paris, France
^{*} Corresponding author: oskdah@kth.se
Received:
21
November
2018
Accepted:
4
March
2019
Published online: 28 March 2019
In this article, we provide an overview of the current state of the research in the area of twist symmetry. This symmetry is obtained by introducing multiple periods into the unit cell of a periodic structure through a rotation of consecutive periodic deformations around a symmetry axis. Attractive properties such as significantly reduced frequency dispersion and increased optical density, compared to purely periodic structures, are observed. The direct link between the symmetry order and these properties is illustrated through numerical simulations. Moreover, polar glide symmetry is introduced, and is shown to provide even further control of the dispersion properties of periodic structures, especially when combined with twist symmetry. Twist symmetries can, with benefit, be employed in the development of devices for future communication networks and space applications, where fully metallic structures with accurate control of the dispersion properties are desired.
Key words: higher symmetry / twist symmetry / polar glide symmetry / broken symmetry
© O. Dahlberg et al., published by EDP Sciences, 2019
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
A periodic structure possesses a higher symmetry if its unit cell is invariant under the composition of a translation and another geometrical operation [1–4]. The unit cell of a highersymmetric periodic structure is composed of m subunit cells, where m is the degree of symmetry to which the compound symmetry operation, , is applied. Therefore, a highersymmetric structure with periodicity p possesses both the conventional translational symmetry, , and the higher symmetry, . The generalized Floquet theorem presented by Oliner [4] proves that the Bloch modes in highersymmetric structures are eigenmodes to both the translation operation, , and the compound operation, . The two operators are linked through the relationship . From this relationship, it is deduced that the dispersion diagram for the full unit cell consists of m subsets: the spectrum of the compound operator, , and its m − 1 space harmonic branches [5]. The presence of such harmonics in the Brillouin diagram for the full unit cell results in a closing of band gaps for the first m − 1 edge points.
Twist symmetry is one type of higher symmetry, where the additional geometrical operation is a rotation around a twist axis, called here a “twist operation” [6–9]. More specifically, a structure possesses an mfold twist symmetry if the compound geometrical operation consists of a translation of p/m along a given axis (the periodicity axis) and a rotation of 2π/m around the same axis, where p is the periodicity of the unit cell, and m is an integer number. Purely periodic structures, where the periodic operation simply consists of a translation of the unit cell, can also be named 1fold twist.
The concept of polar glide symmetry was introduced in [6], where a structure is said to possess polar glide symmetry if its flat approximation of the structure possesses Cartesian glide symmetry, i.e., the additional geometric operation is a mirroring in a plane. However, it has been noted that this definition can be improved upon, and in [7], the true definition of polar glide symmetry is proposed. Here, the authors state that a structure possesses a perfect polar glide symmetry if the subunit cell is translated along a symmetry axis, followed by a mirroring with respect to a cylindrical surface, coaxial with the symmetry axis [7,8].
The electromagnetic properties of structures possessing higher symmetries, and in particular twist and glide symmetries, were extensively studied in the 60s and 70s [1–4]. These initial works focused on the study of onedimensional structures, and the results have been applied in the design of forward and backward scanning leakywave antennas [10–12]. More recently, glide symmetry has received renewed attention and it has been observed that glidesymmetric structures are optically denser and less dispersive than their purely periodic counterparts [13–16]. Moreover, purely periodic structures (i.e. nonglide) produce a stopband at selected frequencies between the first and second modes [13,17–19]. This stopband is closed by applying glide symmetry, permitting the design of wideband flat metasurface lenses [20,21]. Moreover, while the stopband between the first and second modes is suppressed in glidesymmetric structures, a wide stopband can be created between the second and third modes. This stopband can be used to design lowcost gap waveguides [22–24] and lowcost contactless waveguide flanges for millimeter wave measurements [25]. Due to the beneficial properties of glide symmetry, several methods for fast calculation of the dispersion characteristics of glidesymmetric structures (Cartesian and polar) have been presented [8,26–28]. In these methods, by using the generalized Floquet theorem [4], the computational time is reduced and a valuable physical insight to the phenomenon of glide symmetry is provided [27].
Similar properties have been demonstrated in twistsymmetric structures [6–9]. Moreover, the possibility of reaching mfold symmetry, where m is any integer (in contrast to glide symmetry which is inherently of order 2 [4]), enables further control of the dispersion characteristics in twistsymmetric structures [6–9]. Such increased control has found applications in filters [7,8] and phase shifters [9], and has been proposed to be of use in fully metallic lowdispersive leakywave antenna designs [6–8]. This article provides an overview of the recent results in twist symmetries; three studies are conducted, simulated with the Eigenmode Solver of CST Microwave Studio [29], in which the properties of twist symmetry are highlighted. Firstly, in Section 2, a study of perfect twist symmetry is conducted and the refractive properties in structures possessing such symmetry are discussed. Secondly, in Section 3, polar glide symmetry is introduced and shown to provide further control of the propagation characteristics. Thirdly, in Section 4, the twist symmetry is broken, and the link between the symmetry and the opening and closing of stopbands is illustrated. Finally, in Section 5, conclusions are drawn.
2 Perfect twist symmetry
Two different twistsymmetric configurations are considered: coaxial cables periodically loaded with either radially protruding square pins or coaxially aligned semicircular holes. Both types of discontinuities (pins and holes) are placed on the inner conductor of the coaxial cable, and there is no electrical connection between the inner and outer conductors in any of the studied scenarios. Eight different configurations, i.e. purely periodic and 2, 3 and 4fold twistsymmetric structures, for the two types of deformations, are studied and they are illustrated in Figures 1a–d and 2a–d, for pin and holetype loading, respectively. The total periodicity, p, for all pinloaded structures is 10 mm. This means that the distance between two subsequent deformations is 10 mm in the purely periodic structure and 5 mm, 3.33 mm and 2.5 mm in the 2, 3 and 4fold twistsymmetric structures, respectively. The radii of the inner, r _{ i }, and outer, r _{ o }, conductors are 1 mm and 2.5 mm, respectively. The width of the square pins, w _{ p }, is 1 mm and the pins are extruding with a height, p _{ h }, of 1.2 mm from the inner conductor. Similarly, the total periodicity, p, for the holey structures is 10 mm, resulting in similar deformation separation as in the pinloaded structures. The radii of the inner conductor, r _{ i }, is 2.3 mm. The air gap, g, between the inner and outer conductor is 0.2 mm. The holes are airfilled and have a semicircular shape with an opening angle of 180° and a radius of r _{ i }. The holes have a length, l _{ h }, of 2.2 mm, extending along the coaxial transmission line.
The simulated dispersion diagrams for the four pin and holeloaded twistsymmetric structures are presented in Figures 1e and 2e, respectively. Similar behaviour is observed using both types of discontinuities. Notably, the stopbands between the first m modes are suppressed. This behaviour arises from the fact that the modes are eigenmodes of both the translational operation, applied to the full unit cell, and the twist symmetry operation, applied to the subunit cell, as is described by the generalized Floquet theorem. More specifically, this theorem states that, in a twistsymmetric structure, the field is periodic after each twist operation, aside from a multiplication with an exponential factor e ^{ jβp/m } [4]. This leads to a degeneracy at the m − 1 first edge points of the Brillouin diagram, and hence, there are no stopbands at these points. Furthermore, only in the purely periodic structures does the first mode show significant dispersion. In all other cases, the suppression of stopbands allows for a nonzero group velocity at the boundary, resulting in a less dispersive response. As the order of the symmetry is increased, an optically denser material is realized. This property can be used to produce broadband phase delays. For instance, in [9], a reconfigurable phase shifter, in which variable phase delays are achieved through reconfiguring the order of symmetry, has been demonstrated.
In order to provide a fair comparison between purely periodic structures and mfold twistsymmetric structures (m ≥ 2), a purely periodic structure and a 4fold twistsymmetric structure, with the same distance between the deformations, are simulated. It means that the periodicity in the purely periodic structure is four times the periodicity in the 4fold structure (see insets in Figs. 1f and 2f). Again, both pin and holeloading are considered. The total periodicity, p, in the pintype structure is 6 mm, and in the holey structure the periodicity is 10 mm. All other dimensions are the same as in the previous study. Although, the true periodicity in the purely periodic structures is p/4, for the sake of clear representation, 4 periods are employed in the simulations, so that the total length of the coaxial transmission lines is kept constant.
The effective refractive index of the the pin and holey structures are presented in Figures 1f and 2f, respectively. By applying twist symmetry to a periodic structure, increased density is obtained for structures with the same distance between two consecutive deformations. This is due to the fact that the wave is forced to propagate revolving around the twist axis rather than in a straight line, which leads to an increased phase delay per unit length [6]. Moreover, as the deformations may overlap in the longitudinal direction in twistsymmetric structures, which is not possible in purely periodic structures, even further increased density is possible. If the dimensions are tuned in the 4fold structures, so that the effective refractive index of the twistsymmetric and purely periodic structures matches in the long wavelength limit, it is clear that the frequency dispersion is lower in the twistsymmetric structures. This is illustrated with the dashed lines in Figures 1f and 2f. Here, we tune the pin height, h _{ p }, to 1.075 mm and the opening angle of the hole, α, to 85^{∘} in the 4fold structures to obtain the same effective refractive index as in the purely periodic structures. In this example, the upper bound frequency, if 2% deviation from the effective refractive index in the long wavelength limit is allowed, increases from 8.8 GHz to 11.1 GHz in the pinloaded structure, and from 8.9 GHz to 13.5 GHz in the holeloaded structure.
Fig. 1 Simulated perfect twistsymmetric structures with pintype deformations. Studied structures: (a) purely periodic, (b) 2fold, (c) 3fold, and (d) 4fold. (e) Dispersion diagram using the dimensions: h _{ p } = 1.2 mm, w _{ p } = 1 mm, r _{ i } = 1 mm, r _{ o } = 2.5 mm, and p = 10 mm; (f) Effective refractive index comparison of the purely periodic and 4fold twistsymmetric structures with dimensions: h _{ p } = 1.2 mm (tuned: h _{ p } = 1.075 mm), w _{ p } = 1 mm, r _{ i } = 1 mm, r _{ o } = 2.5 mm, and p = 6 mm. 
Fig. 2 Simulated perfect twistsymmetric structures with holetype deformations: Studied structures: (a) purely periodic, (b) 2fold, (c) 3fold, and (d) 4fold. (e) Dispersion diagram using the dimensions: α = 180^{∘}, l _{ h } = 2.2 mm, r _{ i } = 2.3 mm, g = 0.2 mm, and p = 10 mm; (f) Effective refractive index comparison of the purely periodic and 4fold twistsymmetric structures with dimensions: α = 180^{∘} (tuned: α = 85^{∘}), l _{ h } = 2.2 mm, r _{ i } = 2.3 mm, g = 0.2 mm, and p = 10 mm. 
3 Polar glide symmetry
Similar to Cartesian glide symmetry, the operations describing polar glide symmetry, which are applied to the subunit cell, are: a translation of p/2 along the symmetry axis, followed by a mirroring. However, instead of mirroring the subunit cell with respect to a plane, the mirroring is done with respect to a cylindrical surface, coaxial with the periodicity axis and located at some radius, R.
To illustrate the properties of polar glidesymmetric structures, we analyze four coaxial transmission lines, loaded with coaxial halfcylindrical metallic extrusions, in which various degrees of symmetry are introduced. In such structures, the radius of the mirroring surface, R, is given by the geometrical mean radius of the two conductors of the coaxial cable, i.e. . The four studied configurations are the: purely periodic, 2fold twist, perfect polar glide and 2fold twisted polar glide structures, illustrated in Figure 3a–d, respectively. In the figures, the green metallic halfrings protrude from the inner conductor and red metallic halfrings protrude from the outer conductor of the coaxial cables, with no electrical connection between the two conductors. In the purely periodic structure (Fig. 3a), the minimal translational period is half of the period of the other configurations. The total periodicity, p, is 6 mm in all studied configurations. The radii of the inner, r _{ i }, and outer, r _{ o }, conductors are 1 mm and 2.5 mm, respectively. The opening angle of the halfcylindrical pins, γ, is 180°, the longitudinal length of the half cylinders, l _{ p }, is 1 mm, the radius of the half cylinders attached to the inner conductor (in green), r _{ pi }, is 2.3 mm, and the radius to the half cylinders attached to the outer conductor (in red), r _{ po }, is 1.08 mm (obtained from performing the mirroring in the cylindrical surface at radius ).
The dispersion curves for all cases are presented in Figure 3e. The dispersion curve of the purely periodic structure is normalized with respect to p, which is the double of the minimal period of the structure, for coherence with the other cases. We first compare the periodic structure 3a (red line) with the polar glide one 3c (violet line), i.e. we study the effect of adding one of the two halfrings on the outer conductor instead of putting them both on the inner conductor. Similarly to 2fold twist symmetry, polar glide symmetry results in a closing of the stopband at the first Brillouin zone boundary. For the first two modes, the effect of adding polar glide symmetry to a structure is small, compared to the purely periodic structure, i.e. the effect of adding a pin on the outer or inner conductor is the same. However, for the third mode, we see a significant upshift of the cutoff frequency in the polar glide structures, which can be used to widen stopbands in between the second and third modes.
Now we discuss the 2fold twist in Figure 3b, i.e. we study the effect of rotating one of the halfrings by 180°. In this case the optical density is higher, for the first two modes of propagation, compared to both the purely periodic and polar glide structures. However, the third mode is largely unaffected by the rotation of the second halfring. This results in a widening of the stopband between the second and third modes, but, in contrast to the effects of adding polar glide symmetry, this widening is caused by the downshift of the first and second mode, rather than the upshift of the third mode.
By combining 2fold twist symmetry with polar glide, i.e. applying a translation, cylindrical mirroring and a 180° rotation between each consecutive deformation, both the upshift of the third mode due the polar glide symmetry and the downshift of the first and second mode due to the 2fold twist symmetry are obtained, and hence a large stopband is opened in between the modes.
Fig. 3 Simulated structures in the study of polar glide symmetry. Studied structures: (a) purely periodic, (b) 2fold twist, (c) polar glide, and (d) twisted polar glide symmetry; (e) Dispersion diagram using the dimensions: β = 180^{∘}, r _{ pi } = 2.3 mm, r _{ po } = 1.08 mm, l _{ p } = 1 mm, r _{ i } = 1 mm, r _{ o } = 2.5 mm, and p = 6 mm. 
4 Broken twist symmetry
As explained in [4], an mfold symmetry is required to close the stopbands between the m first modes. If this symmetry is broken, in one or several ways, stopbands can be introduced in a controlled manner [7,8]. To illustrate this, we study a 4fold twistsymmetric structure, in which the symmetry is broken in two different ways.
The reference structure is a perfect 4fold, as illustrated in Figure 4a. Similar to the structure discussed in Section 2, radially protruding pins, with square crosssections, are attached to the inner conductor. These pins are denoted as 1, 2, 3 and 4 in the figure, and they are highlighted with the colours green, black, violet and red, respectively. In the initial structure, the total periodicity, p, is 10 mm, the radii of the inner, r _{ i }, and outer, r _{ o }, conductors are 1 mm and 2.5 mm, respectively, the width of the square pins, w _{ p }, is 1 mm and the pins are extruding with a height, p _{ h }, of 1.2 mm from the inner conductor. The dispersion curve, for the perfect 4fold structure, is shown in blue in Figure 4d. Similar to the perfect 4fold structures discussed above, there are no stopbands between the first 4 modes.
A first breaking of the 4fold symmetry is achieved by shifting pin 1 (green) and pin 3 (violet), 1 mm in the longitudinal direction, with respect to the other two pins, as is illustrated by the green arrows in Figure 4b. The resulting dispersion curve is shown in green in Figure 4d. Again, no stopband is observed between modes 1 and 2, and modes 3 and 4, since the structure still possesses 2fold twist symmetry. However, between modes 2 and 3, this structure exhibits a stopband. The width of the stopband can be tuned by varying the magnitude of the symmetry breaking [7,8].
Next, the 2fold symmetry of the structure in Figure 4b is broken by reducing the height of pin 1 (green) and pin 4 (red) by 0.6 mm, as illustrated by the red arrows in Figure 4c. This structure has no higher symmetry and, consequently, exhibits stopbands in between all of the first 4 modes, as shown in the red curve in Figure 4d.
The breaking of symmetry, in combination with twisted polar glide symmetry, was used in [7] to design a fully metallic reconfigurable filter.
Fig. 4 Simulated structures for the observation of the symmetry breaking. Studied structures: (a) perfect 4fold twist symmetry, (b) broken symmetry in 1 way, and (c) broken symmetry in two ways. (d) Dispersion diagram using the dimensions: h _{ p } = 1.2 mm, w _{ p } = 1 mm, r _{ i } = 1 mm, r _{ o } = 2.5 mm, and p = 10 mm. The first symmetry breaking is obtained by shifting pin 1 (green) and pin 3 (violet) by 1 mm according to green arrows. The second breaking is obtained by reducing heights of pin 1 (green) and 4 (red) by 0.6 mm according to red arrows. 
5 Conclusions
In this article, we provide an overview of the recent advances in the research regarding twistsymmetric structures. We illustrate the enhanced control of the dispersion properties, enabled by the twist symmetry operation, in periodically loaded coaxial transmission lines. More specifically, the reduced frequency dependence is highlighted and the impact of symmetry order on the phase constant is discussed. The direct relation between the higher symmetry and the absence of stopbands between the m first modes is demonstrated, together with an illustration of how full control of the pass and stopbands can be obtained in twistsymmetric structures.
Additionally, it has been demonstrated that polar glide symmetry provides similar benefits as other types of higher symmetries (Cartesian glide and twist). In particular, polar glide is shown to create large stopbands at high frequencies, and if combined with twist symmetries, the stopband can be even further increased.
Twist symmetries find applications in fully metallic phase shifters and filters. Moreover, the reduced frequency dispersion may be employed in leakywave antennas with reduced beamsquint, appropriate for future millimeter wave communication networks and space applications [30].
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Cite this article as: Oskar Dahlberg, Fatemeh Ghasemifard, Guido Valerio, Oscar QuevedoTeruel, Propagation characteristics of periodic structures possessing twist and polar glide symmetries, EPJ Appl. Metamat. 6, 14 (2019)
All Figures
Fig. 1 Simulated perfect twistsymmetric structures with pintype deformations. Studied structures: (a) purely periodic, (b) 2fold, (c) 3fold, and (d) 4fold. (e) Dispersion diagram using the dimensions: h _{ p } = 1.2 mm, w _{ p } = 1 mm, r _{ i } = 1 mm, r _{ o } = 2.5 mm, and p = 10 mm; (f) Effective refractive index comparison of the purely periodic and 4fold twistsymmetric structures with dimensions: h _{ p } = 1.2 mm (tuned: h _{ p } = 1.075 mm), w _{ p } = 1 mm, r _{ i } = 1 mm, r _{ o } = 2.5 mm, and p = 6 mm. 

In the text 
Fig. 2 Simulated perfect twistsymmetric structures with holetype deformations: Studied structures: (a) purely periodic, (b) 2fold, (c) 3fold, and (d) 4fold. (e) Dispersion diagram using the dimensions: α = 180^{∘}, l _{ h } = 2.2 mm, r _{ i } = 2.3 mm, g = 0.2 mm, and p = 10 mm; (f) Effective refractive index comparison of the purely periodic and 4fold twistsymmetric structures with dimensions: α = 180^{∘} (tuned: α = 85^{∘}), l _{ h } = 2.2 mm, r _{ i } = 2.3 mm, g = 0.2 mm, and p = 10 mm. 

In the text 
Fig. 3 Simulated structures in the study of polar glide symmetry. Studied structures: (a) purely periodic, (b) 2fold twist, (c) polar glide, and (d) twisted polar glide symmetry; (e) Dispersion diagram using the dimensions: β = 180^{∘}, r _{ pi } = 2.3 mm, r _{ po } = 1.08 mm, l _{ p } = 1 mm, r _{ i } = 1 mm, r _{ o } = 2.5 mm, and p = 6 mm. 

In the text 
Fig. 4 Simulated structures for the observation of the symmetry breaking. Studied structures: (a) perfect 4fold twist symmetry, (b) broken symmetry in 1 way, and (c) broken symmetry in two ways. (d) Dispersion diagram using the dimensions: h _{ p } = 1.2 mm, w _{ p } = 1 mm, r _{ i } = 1 mm, r _{ o } = 2.5 mm, and p = 10 mm. The first symmetry breaking is obtained by shifting pin 1 (green) and pin 3 (violet) by 1 mm according to green arrows. The second breaking is obtained by reducing heights of pin 1 (green) and 4 (red) by 0.6 mm according to red arrows. 

In the text 
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