Twist and Glide Symmetries for Helix Antenna Design and Miniaturization
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S S symmetry Article Twist and Glide Symmetries for Helix Antenna Design and Miniaturization Ángel Palomares-Caballero 1,*, Pablo Padilla 2, Antonio Alex-Amor 1, Juan Valenzuela-Valdés 2 and Oscar Quevedo-Teruel 3 1 Department of Languages and Computer Science, Universidad de Málaga, 29071 Málaga, Spain; [email protected] 2 Department of Signal Theory, Telematics and Communications, Universidad de Granada, 18071 Granada, Spain; [email protected] (P.P.); [email protected] (J.V.-V.) 3 Department of Electromagnetic Engineering, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden; [email protected] * Correspondence: [email protected]; Tel.: +34-958248899 Received: 26 January 2019; Accepted: 4 March 2019; Published: 8 March 2019 Abstract: Here we propose the use of twist and glide symmetries to increase the equivalent refractive index in a helical guiding structure. Twist- and glide-symmetrical distributions are created with corrugations placed at both sides of a helical strip. Combined twist-and glide-symmetrical helical unit cells are studied in terms of their constituent parameters. The increase of the propagation constant is mainly controlled by the length of the corrugations. In our proposed helix antenna, twist and glide symmetry cells are used to reduce significantly the operational frequency compared with conventional helix antenna. Equivalently, for a given frequency of operation, the dimensions of helix are reduced with the use of higher symmetries. The theoretical results obtained for our proposed helical structure based on higher symmetries show a reduction of 42.2% in the antenna size maintaining a similar antenna performance when compared to conventional helix antennas. Keywords: higher symmetries; periodic structures; glide symmetry; twist symmetry; dispersion diagram; microwave printed circuits; helix antennas 1. Introduction Symmetrical structures are present in a huge number of natural phenomena. On occasion, symmetries have a positive impact in the physical response and properties of materials. Therefore, when symmetries are not spontaneously found in nature, engineers have found a manner to tailor them in an artificial manner [1]. The use of symmetrical geometries modifies the physical properties of materials, such as their mechanical, thermal or electromagnetic responses [2]. Among these properties, those related to electromagnetic behaviour are of great importance for conductive materials and dielectric substrates. The electromagnetic properties resulting from configurations with one-dimensional higher symmetries were initially studied in the 1960s and 1970s [3,4]. However, it has been in recent years, with the new developments on computational electromagnetics, that more complex structures, including two-dimensional and three-dimensional higher-symmetric structures, have been studied. These structures have demonstrated new possibilities for the design of microwave and millimeter-wave circuits and antennas [5]. Some of the key advantages of employing higher symmetries in the design of radiofrequency devices are: significant reduction of the frequency dispersion [6], accurate control of the equivalent refractive index [7], control and elimination of stop-bands in periodic structures [8]. Symmetry 2019, 11, 349; doi:10.3390/sym11030349 www.mdpi.com/journal/symmetry Symmetry 2019, 11, 349 2 of 14 Symmetry 2018, 10, x FOR PEER REVIEW 2 of 14 A periodic structure possesses a higher symmetry when it is invariant under a translation and another spatial spatial operator operator such such as as a arotation rotation or or mirroring. mirroring. Two Two commonly commonly used used higher higher symmetries symmetries are glideare glide and and twist twist symmetries symmetries as asillustrated illustrated in inFigure Figure 1.1 .While While the the extra extra spatial spatial operator forfor glideglide symmetries is a mirroring with respect to aa glideglide line/surfaceline/surface [9,10], [9,10], for for the twist symmetries the operator is a rotation along a twist axis [10–12]. [10–12]. Twist Twist symmetry is a more general concept than glide ◦ symmetry. For For example, example, when when the the unit unit cell cell is sy symmetricalmmetrical along the transversal direction, a 180 o twist-symmetrical structure turns outout to be glide-symmetrical too.too. (a) (b) Figure 1. Examples of higher symmetries: (a) glide-symmetrical corrugations. The corrugations Figureare periodic 1. Examples along x axis, of and higher the mirroring symmetries: plane is(za)= 0;glide-symmetrical (b) Twist-symmetrical corrugations. metallic rod withThe corrugationsinclusions rotated arej =periodic 90◦ along thealong twist x axis. axis, and the mirroring plane is z = 0; (b) Twist-symmetrical metallic rod with inclusions rotated φ = 90 o along the twist axis. Twist and glide symmetries were demonstrated to reduce the dispersion of metasurfaces, providingTwist unitand cellsglide with symmetries a flattened were frequency demonstrated dependence. to reduce This the is advantageousdispersion of formetasurfaces, the design providingof wideband unit microwave cells with a devices, flattened such frequency as planar dependence. lenses [3]. This Also, is theadvantageous use of glide for symmetries the design of in widebandmetallic holey microwave structures, devices, as reported such as in planar [4], increases lenses [3]. the Also, bandwidth the use of of the glide electromagnetic symmetries in bandgaps. metallic holeyThese structures, glide-symmetrical as reported holey in [4], structures increases have the bandwidth been proposed of the for electromagnetic cost-effective bandgaps. gap waveguide These glide-symmetricaltechnology at the holey millimeter-wave structures have frequency been prop range.osed Additionally, for cost-effective the gap combination waveguide of technology twist- and atglide-symmetrical the millimeter-wave configurations freque increasesncy range. the equivalentAdditionally, refractive the indexcombination of periodic of structurestwist- and [5], glide-symmetricalenhances the linearity configurations of the modes, increases and creates the additionalequivalent stop-bands refractive index at given of frequenciesperiodic structures [10]. In [[5],13], enhancesthe use of the higher linearity symmetries of the modes, has been and applied creates to ad produceditional a stop-bands multibeam Luneburgat given frequencies lens antenna [10]. with In [13],low scan-lossesthe use of higher and wide symmetries bandwidth. has Gapbeen waveguides applied to produce with glide-symmetrical a multibeam Luneburg holey EBG lens structures antenna withwere low proposed scan-losses in [14 ],and creating wide bandwidth. cost-effective Gap guiding waveguides structures with at glide-symmetrical high frequencies. holey Using EBG this structurestechnology, were a wideband proposed phase in [14], shifter creating was proposedcost-effective in [ 15guiding]. Finally, structures low-loss at waveguide high frequencies. flanges Usingwere proposedthis technology, in [16], a withwideband the use phase of glide-symmetricshifter was proposed holes in around [15]. Finally, the waveguide low-loss waveguide apertures. flangesIn this manner,were proposed the leakage in [16], at the with waveguide the use of joint glide-symmetric is avoided. Lastly, holes aaround recent workthe waveguide about the apertures.application In of this glide-symmetry manner, the leakage in printed at the double-sided waveguide parallel-strip joint is avoided. lines Lastly, demonstrated a recent the work potential about the application of glide-symmetry in printed double-sided parallel-strip lines demonstrated the potential of glide symmetry to produce low dispersion transmission lines and filter behavior by breaking the symmetry [9]. Symmetry 2019, 11, 349 3 of 14 Symmetry 2018, 10, x FOR PEER REVIEW 3 of 14 of glide symmetry to produce low dispersion transmission lines and filter behavior by breaking the symmetry Here, [9]. we proposed the combination of glide and twist symmetries to reduce the size of helix antennasHere, webased proposed on the increment the combination of the equivalent of glide and refractive twist symmetries that the higher to reduce symmetries the size produce. of helix antennas based on the increment of the equivalent refractive that the higher symmetries produce. 2. Materials and Methods 2. Materials and Methods For antenna designs that are based on field guidance and progressive matching towards free space,For antennaas is the designs case in that helix-based are based antennas, on field guidance the use and of higher-symmetrical progressive matching configurations towards free space, can be asbeneficial. is the case inIn helix-basedthis section, antennas, we analyze the use the of elec higher-symmetricaltromagnetic effects configurations of including can twist be beneficial. and glide Incorrugations this section, weof a analyze metallic the flat electromagnetic strip in a helix structure, effects of includingdefining the twist guidelines and glide for corrugations antenna designs. of a metallic flat strip in a helix structure, defining the guidelines for antenna designs. 2.1. Baseline Helix Antenna Designs 2.1. Baseline Helix Antenna Designs The general structure of a helix antenna is shown in Figure 2a, and it is formed by a ground planeThe and general a conducting structure of helix a helix structure. antenna The is shown basic inparameters Figure2a, andthat it define is formed a helix by a antenna ground planeare: the andradius a conducting (r), the pitch