Glide-Symmetric Metallic Structures with Elliptical Holes for Lens Compression
Antonio Alex-Amor, Fatemeh Ghasemifard, Guido Valerio, Mahsa Ebrahimpouri, Pablo Padilla, José M. Fernández-González, Oscar Quevedo-Teruel
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IEEE Transactions on Microwave Theory and Techniques .Citation information: DOI 10.1109/TMTT.2020.3011004 a r X i v : . [ phy s i c s . a pp - ph ] J a n EEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. X, NO. X, 2020 2
Glide-Symmetric Metallic Structures with EllipticalHoles for Lens Compression
Antonio Alex-Amor, Fatemeh Ghasemifard,
Graduate Student Member, IEEE,
Guido Valerio,
SeniorMember, IEEE , Mahsa Ebrahimpouri,
Student Member, IEEE,
Pablo Padilla, Jos´e M. Fern´andez-Gonz´alez,
SeniorMember, IEEE , and Oscar Quevedo-Teruel,
Senior Member, IEEE
Abstract —In this paper, we study the wave propagation in ametallic parallel-plate structure with glide-symmetric ellipticalholes. To perform this study, we derived a mode-matchingtechnique based on the generalized Floquet theorem for glide-symmetric structures. This mode-matching technique benefitsfrom a lower computational cost since it takes advantage ofthe glide symmetry in the structure. It also provides physicalinsight on the specific properties of Floquet modes propagatingin these specific structures. With our analysis, we demonstratethat glide-symmetric structures with periodic elliptical holesexhibit an anisotropic refractive index over a wide range offrequencies. The equivalent refractive index can be controlledby tuning the dimensions of the holes. Finally, by combiningthe anisotropy related to the elliptical holes and transformationoptics, a Maxwell fish-eye lens with a 33.33% size compressionis designed. This lens operates in a wideband frequency rangefrom 2.5 GHz to 10 GHz.
Index Terms —Dispersion analysis, elliptical holes, anisotropy,mode-matching, metasurfaces, periodic structures, glide symme-try, generalized Floquet theorem, Mathieu functions, PPW.
I. I
NTRODUCTION M ETAMATERIALS are artificial materials formed bysubwavelength periodic structures that exhibit extraor-dinary macroscopic properties [1]–[4]. Metasurfaces [5]–[7],the two-dimensional version of metamaterials, opened the pos-sibility of manufacturing low-profile and cost-effective devices[8]. Metasurfaces have been used for controlling (allowing orsuppressing) the propagation of electromagnetic waves in a
Manuscript received X, 2019; revised X, 2019. This work was partiallyfunded by the Ministerio de Ciencia Innovaci´on y Universidades under theproject TIN2016-75097-P, with European Union FEDER funds under theproject FUTURERADIO “Radio systems and technologies for high capacityterrestrial and satellite communications in an hyperconnected world” (projectnumber TEC2017-85529-C3-1-R), by the French National Research AgencyGrant Number ANR-16-CE24-0030, by the Vinnova project High-5 (2018-01522) under the Strategic Programme on Smart Electronic Systems, and bythe Stiftelsen ˚Aforsk project H-Materials (18-302).A. Alex-Amor is with the Departamento de Lenguajes y Ciencias de laComputaci´on, Universidad de M´alaga, 29071 M´alaga, Spain.A. Alex-Amor and J. M. Fern´andez-Gonz´alez are with the Information Pro-cessing and Telecommunications Center, Universidad Polit´ecnica de Madrid,28040 Madrid, Spain (e-mail: [email protected]; [email protected])F. Ghasemifard, M. Ebrahimpouri and O. Quevedo-Teruel are with theDepartment of Electromagnetic Engineering, School of Electrical Engineeringand Computer Science, KTH Royal Institute of Technology, SE-100 44 Stock-holm, Sweden (e-mail: [email protected]; [email protected]; [email protected])G. Valerio is with UR2, Laboratoire d’ ´Electronique and ´Electromagn´etisme,Sorbonne Universit´e, F-75005 Paris, France (e-mail: [email protected])P. Padilla is with the Departamento de Teor´ıa de la Se˜nal, Telem´aticay Comunicaciones, Universidad de Granada, 18071 Granada, Spain (email:[email protected]) given direction [9], [10], to produce electromagnetic cloaking[11], [12], wavefront transformation [13], [14], and graded-index lenses [15]–[17].Recent studies have demonstrated the distinctive dispersiveproperties related with the presence of higher symmetries inperiodic structures. For example, introducing glide and twistsymmetries in periodic structures can increase their equiva-lent refractive index [18], [19]. This feature has been usedfor antenna miniaturization [20], mutual coupling reduction[21] or to produce a compact and low-loss phase shifter atmillimeter waves [22]. Furthermore, with the use of highersymmetries, the stopband between the first and the secondmodes of conventional periodic structures is suppressed [23]–[25] and the frequency dispersion of the first propagatingmodes is reduced [18], [26]–[28]. The latter can be usedfor realizing fully-metallic low-loss wideband lenses for 5Gcommunication systems [17], [29]. Additionally, it has beenrecently proved that breaking symmetries can be used forfiltering purposes [30].Previous works have studied in detail the implementation of isotropic holey metasurfaces [31]–[33], composed of squaredand circular holes. In [33], a comparison between holeymetasurfaces composed of square, circular and triangular holesis presented. It is shown that square and triangular holespossess higher equivalent refractive indexes and wider stop-bands compared to the configuration that uses circular holes.On the other hand, the frequency dispersion of the structurewith circular holes is normally lower compared to triangularand square holes. However, some specific lenses require acertain level of anisotropy , for example, optically transformedlenses [34]–[36]. In this case, rectangular holes can provide thenecessary anisotropy [37], [38]. However, rectangular holes aredifficult to manufacture with milling techniques. Alternatively,holey structures with elliptical holes are a practical solution toimplement anisotropy in parallel-plate structures. In order totune the equivalent refractive index, a change of the ellipticityand depth of the holes can be applied, similarly to circularholes in isotropic implementations [39].Glide-symmetric periodic structures require specific mod-elling tools due to the strong interaction between the layers[40], [41]. Even though circuit approaches [28] and homoge-nized impedance models [42] are fast tools to characterize thedispersion properties of a great variety of periodic structures,these methods are difficult to implement for a glide-symmetricconfiguration due to the strong coupling between the metalliclayers. Furthermore, while integral-equation formulations can
EEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. X, NO. X, 2020 3 handle multiple types of geometries [43]–[45], they do not givephysical insight on the operation of the structure due to theirpurely numerical approach. In this paper, we apply a generalmode-matching technique to study the dispersion features andthe anisotropy related to glide-symmetric metallic structureswith periodic elliptic holes. Our results confirm the widebandnature of glide-symmetric structures.This paper is organized as follows. Section II presents thegeneral mode-matching formulation applied to the ellipticalholey structure. The fields inside the holes are described bymeans of Mathieu functions, and they are subsequently usedto obtain the dispersion equation. Section III presents thedispersion analysis for all the geometric parameters of thestructure. Furthermore, a study on the anisotropic behaviorof the equivalent refractive index is also carried out. SectionIV presents the design of a compressed Maxwell fish-eyelens made of glide-symmetric periodic elliptical holes. Finally,conclusions are drawn in Section V.II. F
ORMULATION
In this section, we propose a mode-matching technique toanalyze the 2D periodic holey structure composed of ellipticalholes shown in Fig. 1(a). Fig. 1(b) shows a top view of thestructure when the metallic plate is removed. Mat and brightred colors in Fig. 1(b) relate the elliptical holes of the bottomand top layers. Fig. 1(c) depicts the glide-symmetric version ofthe holey metasurface, and Fig. 1(d) the non-glide-symmetricversion. Both structures are periodic, with period d , along the x - and y -directions and are bounded along the z -direction. Thegap region is filled with a dielectric of relative permittivity ε r and the elliptical holes are filled with a dielectric of relativepermittivity ε r hole . The z = 0 plane is chosen to be on thetop of the non-glide-symmetric unit cell, and in the center ofthe gap between plates in the glide-symmetric case. The gapbetween the plates is g and g/ in the glide-symmetric andnon-glide-symmetric structures, respectively. A. Mode-matching Formulation
In this section, first, we briefly present the mode-matchingformulation for this problem in a suitable coordinate system.An elliptic cylindrical coordinate system is used due to thegeometry of the hole. As stated in Appendix A, the ellipticcylindrical coordinate system is defined by three coordinates:radial ξ , angular η , and longitudinal z . Using this coordinatesystem, the tangential fields in the surface of the hole can beexpressed as E WG t ( ξ, η, z = − g/
2) = N (cid:88) i =1 r − i C i Φ i ( ξ, η ) (1) H WG t ( ξ, η, z = − g/
2) = N (cid:88) i =1 r + i Y i C i [ ˆz × Φ i ( ξ, η )] (2)where C i is the coefficient of the i -th mode, Φ i ( ξ, η ) is themodal function that represents the i -th mode inside the hole,and Y i is the wave admitance of the i -th mode. The elliptical (a) (b)(c) (d)(e) Fig. 1 : (a) 2D periodic holey structure and (b) a top view of it.Unit cells of the (c) glide-symmetric ( s = d/ ) and (d) non-glide-symmetric configurations. (e) Cross section of an elliptical hole . hole can be regarded as a short-circuited elliptical waveguide.Thus, the coefficients r ± i in (1) and (2) are calculated as r ± i = 1 ± e − j k zi h hole . (3)The term k zi = (cid:112) ε r hole k − k ti is the wavenumber in thelongitudinal direction of the elliptical waveguide, ε r hole is therelative dielectric constant inside the hole, k ti is the transversalwavenumber and h hole is the hole depth in Fig. 1.On the other hand, the tangential fields in the gap regioncan be expressed as a series of Floquet harmonics: E GAP t = 1 d (cid:88) p (cid:88) s e − j ( k x,p x + k y,s y ) ˜e GAP t,ps ( z ) (4) H GAP t = 1 d (cid:88) p (cid:88) s e − j ( k x,p x + k y,s y ) ˜h GAP t,ps ( z ) (5)where k x,p = k x, + πpd , k y,s = k y, + πsd , p and s are the integer values specifying the order of the Floquetharmonics, and ˜e GAP t,ps ( z ) , ˜h GAP t,ps ( z ) are the field amplitudesof each transversal electric and magnetic Floquet harmonic.Both field-amplitude terms are expressed as a sum of sine andcosine functions [39]. EEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. X, NO. X, 2020 4
By applying the boundary conditions (continuity of electricand magnetic tangential fields at z = − g/ and z = g/ forthe glide case, and at z = − g/ and z = 0 for the non-glidecase), the following linear system of equations is obtained[39], [46]: N (cid:88) i =1 C i α ri = 0 , r = 1 , ..., N (6)where N is the number of modal functions considered in theelliptical hole and α ri = jk η d Y i I ri + r − i r + i (cid:88) p (cid:88) s (cid:101) f ps ( k z,ps ) β ri ( k ps ) . (7)By setting the determinant of the coefficient matrix α ri tozero, the dispersion equation can be obtained. In (7), I ri isthe inner product of the r -th and i -th modal functions. Theparameter β ri ( k ps ) is expressed as β ri ( k ps ) = β ri ( k x,p , k y,s , k z,ps )= k − k y,s k z,ps (cid:101) φ xi (cid:101) φ x ∗ r + k x,p k y,s k z,ps (cid:101) φ yi (cid:101) φ x ∗ r + k − k x,p k z,ps (cid:101) φ yi (cid:101) φ y ∗ r + k x,p k y,s k z,ps (cid:101) φ xi (cid:101) φ y ∗ r (8)where k z,ps = (cid:113) ε r k − k x,p − k y,s is the vertical wavenum-ber of the ( p, s ) -th harmonic, ε r is the relative dielectric con-stant in the gap region, (cid:101) φ xi and (cid:101) φ yi are the x and y -componentsof the Fourier transform of the i -th modal function, and thesymbol (*) denotes the complex conjugate. Finally, the verticalspectral function (cid:101) f ps in (7) distinguishes between the glide-symmetric and non-glide-symmetric cases. In the non-glide-symmetric case, it is (cid:101) f ps = cot ( k z,ps g/ (9)and in the glide-symmetric case, it is (cid:101) f ps = (cid:40) − tan ( k z,ps g/ p + s even cot ( k z,ps g/ p + s odd (10) B. Numerical Implementation
Elliptical holes can be seen as short-circuited ellipticalwaveguides. The foundations of elliptical waveguides werefirst established in [47] and subsequently extended in [48].Additionally, some mode-matching proposals can be found inthe literature [49]–[51], normally oriented to analyze discon-tinuities in an elliptical waveguide. As detailed in AppendixB, the modal functions Φ i involved in the mode-matching areexpressed in terms of Mathieu functions [52]. They are solu-tions of Mathieu’s differential equation, the governing waveequation of a homogeneous, lossless elliptical waveguide.After obtaining the modal functions, the next step is tocalculate the coefficients α ri , as expressed in (7). For thispurpose, we need to find I ri and calculate the Fourier trans-forms of the modal functions (cid:101) Φ i = (cid:101) φ xi ˆx + (cid:101) φ yi ˆy to computeequation (8). The two-dimensional Fourier transform of the modal func-tions is defined as (cid:101) Φ i ( k x,p , k y,s ) = (cid:90) S hole Φ i e j ( k x,p x + k y,s y ) dS. (11)The differential surface dS in an elliptic coordinate systemcan be expressed as dS = h ξ dξdη , since the scale factorof the ξ and η coordinates is the same (see Appendix A).From Appendices A and B, we know how both φ xi and φ yi components of the modal functions can be derived from the φ ξi and φ ηi components. Thus, the components of the Fouriertransform are written as (cid:101) φ x i = h (cid:90) ξ ξ =0 (cid:90) πη =0 ( φ ξi sinh ξ cos η − φ ηi cosh ξ sin η ) × e j ( k x,p x + k y,s y ) h ξ dξdη (12) (cid:101) φ y i = h (cid:90) ξ ξ =0 (cid:90) πη =0 ( φ ξi cosh ξ sin η + φ ηi sinh ξ cos η ) × e j ( k x,p x + k y,s y ) h ξ dξdη. (13)where h = √ a − b . In addition, the inner product of the r -th and i -th modal functions is expressed as I ri = (cid:90) S hole Φ r · Φ i dS = (cid:90) ξ (cid:90) π (cid:16) φ ξr φ ξi + φ ηr φ ηi (cid:17) h ξ dξdη. (14)For the case of perfect metallic waveguides, the modal func-tions are orthogonal regardless of the geometry of the hole[59]. Thus, the inner product I ri is zero when r (cid:54) = i and non-zero when r = i . This implies that I ri is a diagonal matrix.The integrals (12)–(14) are computed with Gaussian quadra-ture sums [53] with thirty terms. We have considered fifteenmodes inside the hole and nine Floquet modes in the gapregion. These values guarantee the convergence of the mode-matching method, since the ratio / . is in concor-dance with the estimation discussed in [39]. However, moreFloquet modes were used to accurately compute the resultsfor small values of the ratio a/d [38].As a difference with respect to [38] and [39], this codehas been partially optimized when finding the zeros of thedeterminant of the coefficient matrix in (6) by taking advantageof the physical properties of the modes. As seen in Figs. 2–10, the dispersion diagrams are always continuous functionswithout an abrupt change. Additionally, the dispersion diagramof the first mode is always below the light line. Thus, bysimply searching the frequency range between the frequencyassociated with the previous point and the one associated withthe light line at a particular wavenumber, the computation timeof the code is significantly reduced.III. R ESULTS
In this section, we study the anisotropic behavior of theunit cell for different values of the parameters of the ellipticalhole. In order to check the validity of the results, all dispersionanalyses are compared with the commercial software
CSTMicrowave Studio . We will use a reference unit cell to studythe individual effect of each parameter in the equivalentrefractive index. The dimensions of the reference unit cell are:
EEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. X, NO. X, 2020 5
Fig. 2 : Dispersion diagram of the irreducible Brilloin zone for theglide-symmetric ( red lines ) and non-glide-symmetric ( blue lines ) unitcells, obtained with the proposed mode-matching ( solid lines ) and
CST ( dashed lines ). The gray line represents the light line.Fig. 3 : Equivalent refractive index for wave propagation in x - and y -directions in the reference glide-symmetric ( red lines ) and non-glide-symmetric unit cells ( blue lines ). (a) (b) Fig. 4: Constitutive parameters in the reference glide-symmetric ( redlines ) and non-glide-symmetric ( blue lines ) unit cells : (a) Real partof the permeability. (b) Real part of the permittivity. period d = 4 . mm, eccentricity e = 0 . , gap g = 0 . mm,semi-major axis a = 1 . mm, and hole depth h hole = 1 . mm. In addition, the gap region and the holes are filled withair ( ε r = ε r hole = 1 ).Fig. 2 shows the dispersion diagram over the irreducibleBrillouin zone for the glide-symmetric (red) and non-glide-symmetric (blue) reference unit cells. There an excellentagreement between the proposed mode-matching (solid lines)and the commercial software CST (dashed lines). These resultsdemonstrate the advantages of using glide symmetry. Due tothe vanishing of the stopband between the first and the secondmodes, the frequency dispersion of the first mode is stronglyreduced. The higher linearity of the red curves in the Γ -X andY- Γ intervals, compared to the blue curves, demonstrates thatthe equivalent refractive index is almost invariant over a widerbandwidth in the case of the glide-symmetric configuration(Fig. 3). This fact is highly desired for the design of widebandlenses. The third mode of the glide-symmetric configurationhas also been plotted in Fig. 2. This mode is the continuationof the second mode at the ending point of the Y- Γ interval (50GHz), as the yellow circle indicates. Due to the anisotropicbehavior of the unit cell, the third mode crosses under thesecond mode in the Γ -X interval. For the sake of conciseness,only the first and second modes will be plotted in the nextfigures.Fig. 3 represents the variation of the equivalent refractiveindex of both glide-symmetric (red curves) and non-glide-symmetric (blue curves) structures. As previously discussed,the operational bandwidth of the non-glide-symmetric case isnarrower compared with the glide-symmetric case. Further-more, the anisotropy of this structure can be visualized inthe figure ( n y (cid:54) = n x ). The wave that propagates along the x -direction ( k y, = 0 ) is less disturbed by the presence of thehole. Thus, the equivalent refractive index is smaller in the x -direction than in the y -direction. Although it is not plottedin Fig. 3, our code is able to calculate the equivalent refractiveindex in other propagation directions different from x and y .For instance, the equivalent refractive index associated to themain diagonal (45 ◦ ) of the Brillouin zone is calculated bysetting ≤ k x. = k y, ≤ π/d .Fig. 4 illustrates the constitutive parameters of the glide-symmetric (red) and non-glide-symmetric (blue) unit cells.Following [54], the scattering parameters can be used toextract the constitutive parameters of the structures understudy. A single row of unit cells (periodic along y direction)is utilized for this purpose. A plane wave that propagatesin the parallel-plate along x direction and impinges the rowof unit cells is used to excite the structure and extract theS-parameters. For 2-D periodic structures, the permeabilitycomponents along x and y directions ( µ xx and µ yy ) mustbe extracted separately. Depending on the orientation of theelliptical holes, we derive from the S-parameters the sets { µ xx ( f ) , ε z ( f ) } and { µ yy ( f ) , ε z ( f ) } . The first set is obtainedif the semi-major axis of all elliptical holes that conformthe mentioned row is oriented along y direction. The latteris obtained if the semi-major axis of the elliptical holes isoriented along x direction. Note that ε z is obtained at thesame time from both configurations, since the electric field EEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. X, NO. X, 2020 6
Fig. 5 : Dispersion diagram of the glide-symmetric unit cell fordifferent values of the relative permittivity in the gap region, obtainedwith the proposed mode-matching ( solid lines ) and
CST ( dashedlines ). The other geometrical parameters of the unit cell are d = 4 . mm, e = 0 . , g = 0 . mm, a = 1 . mm, and h hole = 1 . mm. Thegray line represents the light line. is oriented along z direction in the parallel-plate waveguide.When simulating the structures in CST , two parallel-platesections are added at both sides of the row (in x direction)in order to deembed the ports and extract the S-parametersthat lead to the constitutive parameters.The constitutive parameters represent the transverse com-ponents of the permeability and the normal component of thepermittivity for a wave propagation in x and y -directions.It is important to remark that the refractive indexes arecross-related with the constitutive parameters ( n x ∝ √ µ yy , n y ∝ √ µ xx ), as later depicted in Section IV. As observed inFig. 4, the permeability is higher than the permittivity, and thelatter remains invariant for both propagating directions. Thisfact indicates that the unit cell has a strong magnetic behavior.Therefore, the anisotropy and the variations in the equivalentrefractive index (Fig. 3) are mainly due to the variations inthe effective permeability.In Fig. 5, the variation of the propagation constant as aconsequence of filling the gap region with a material of relativepermittivity ε r is studied over the Γ -X (propagation in the x -direction, k y, = 0 ) and Γ -Y (propagation in the y -direction, k x, = 0 ) intervals. Since it is only necessary to representthe Y- Γ -X interval to qualitatively observe the presence of theanisotropy, for the sake of conciseness, the interval X-M-Yhas not been plotted. As expected, the higher the permittivity,the denser the structure is, so the modes move away from thelight line. Beyond that, the frequency dispersion does not seemto increase as the permittivity increases.Fig. 6 presents the influence of filling the elliptical hole witha dielectric of relative permittivity ε r hole on the propagationconstant. Similar to Fig. 5, the equivalent refractive index in-creases as the hole is filled with denser materials. Additionally,the structure is also more frequency dispersive above 30 GHz Fig. 6: Dispersion diagram of the glide-symmetric unit cell fordifferent dielectrics inside the hole, obtained with the proposed mode-matching ( solid lines ) and
CST ( dashed lines ). The other geometricalparameters of the unit cell are d = 4 . mm, e = 0 . , g = 0 . mm, a = 1 . mm, and h hole = 1 . mm. The gray line represents the lightline.Fig. 7 : Dispersion diagram of the glide-symmetric unit cell fordifferent values of the hole depth, obtained with the proposed mode-matching ( solid lines ) and
CST ( dashed lines ). The other geometricalparameters of the unit cell are d = 4 . mm, e = 0 . , g = 0 . mm, a = 1 . mm. The gray line represents the light line. as the relative permittivity of the dielectric is higher.In Fig. 7, the variation of the hole depth in the glide-symmetric unit cell is studied. For the three different casesconsidered here ( h hole = 0 . mm, h hole = 1 . mm and h hole = 2 . mm), the agreement between our formulationand CST is noticeable. Two phenomena are highlighted in thefigure. The results demonstrate that the equivalent refractiveindex increases as the hole depth increases too. This facthas been recently exploited in the work of [29] to designa fully-metallic lens for 5G applications. However, there isa limit where the hole depth has no longer influence overthe dispersion diagram (blue curve), as also pointed out in
EEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. X, NO. X, 2020 7
Fig. 8 : Dispersion diagram of the glide-symmetric unit cell fordifferent values of the semi-major axis of the hole, obtained withthe proposed mode-matching ( solid lines ) and
CST ( dashed lines ).The other geometrical parameters of the unit cell are d = 4 . mm, e = 0 . , g = 0 . mm, and h hole = 1 . mm. The gray line representsthe light line.Fig. 9 : Dispersion diagram of the glide-symmetric unit cell fordifferent values of the gap, obtained with the proposed mode-matching ( solid lines ) and
CST ( dashed lines ). The other geometricalparameters of the unit cell are d = 4 . mm, e = 0 . , a = 1 . mm,and h hole = 1 . mm. The gray line represents the light line. [39], since the modes exponentially attenuate inside the hole.Furthermore, the unit cell presents an anisotropic behaviour.Note that by changing the hole depth, the equivalent refractiveindex has a small variation in the Γ -X interval. However,the equivalent refractive index varies remarkably in the Y- Γ interval. This anisotropy can be used to produce the opticallytransformed lens presented in Section IV.Fig. 8 represents the variation of the propagation constantfor different values of the semi-major axis a of the ellipse.Three cases are plotted: a = 1 . mm, a = 1 . mm and a =1 mm. The equivalent refractive index increases in the Y- Γ interval as the semi-major axis of the ellipse increases too. This Fig. 10 : Dispersion diagram of the glide-symmetric unit cell fordifferent eccentricities of the hole, obtained with the proposed mode-matching ( solid lines ) and
CST ( dashed lines ). The other geometricalparameters of the unit cell are d = 4 . mm, g = 0 . mm, a = 1 . mm, and h hole = 1 . mm. The gray line represents the light line.Fig. 11 : Equivalent refractive index of the glide-symmetric unitcell for wave propagation in x - and y -directions. The geometricalparameters of the unit cell are d = 4 . mm, g = 0 . mm, and a = 1 . mm. is also related with the cross section of the hole: the bigger thehole is, the greater the interaction of the propagating wave withthe hole. Thus, the equivalent refractive index approaches theunity as the hole shrinks and the structure becomes a commonparallel-plate waveguide.In Fig. 9, the influence of the gap height on the propagationconstant is studied. In this case, three different values areconsidered: g = 1 mm, g = 0 . mm and g = 0 . mm.Even though the gap height is not a parameter to be typicallytuned for lens designs, its influence on the equivalent refractiveindex cannot be neglected. Some discrepancies between ourformulation and CST appear for really small values of g such as g = 0 . mm, as more than fifteen modes need EEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. X, NO. X, 2020 8 to be considered here for accurate results. As expected, theequivalent refractive index becomes higher as the gap heightis smaller, since the interaction between the top and bottommetallic layers is stronger.The variation of the propagation constant as a consequenceof modifying the eccentricity of the hole is studied in Fig. 10.The design of anisotropic lenses may require very flattenedellipses, so the effect of the anisotropy is highly marked in thestructure. Thus, two high-eccentricities values are consideredhere: e = 0 . and e = 0 . . As the eccentricity of the holeincreases, the modes in the dispersion diagram approach thelight line.Finally, Fig. 11 presents a parametric study of the variationin the equivalent refractive index of the glide-symmetric unitcell when modifying the hole depth and the eccentricity ofthe ellipse. The potential of glide-symmetric structures withelliptical holes is reflected in the figure. Firstly, flattenedellipses easily control the anisotropy of the structure in aneasy manner, since n x is near to one and n y varies. Secondly,the reduction of the frequency dispersion in the first mode isappreciable in Fig. 11.IV. C OMPRESSED M AXWELL F ISH -E YE L ENS
In this section, we take advantage of both the broadbandresponse due to glide symmetry and the anisotropy related toelliptical holes to produce a broadband compressed Maxwellfish-eye lens (MFE). Thus, we present a proof-of-conceptdesign with a compression factor of 33.33% and widebandcapabilities.The Maxwell fish-eye lens is a rotationally-symmetricgraded-index lens that focuses any point-source excitationto the opposite side of the circle [55], [56]. The requiredrefractive index for this lens is given by: n ( r ) = 2 n r/R ) (15)where r = (cid:112) x + y , R is the radius of the lens and n isthe background refractive index, assumed to be one here.Here, we attempt to compress the lens by a factor β whilemaintaining the same aperture in the other direction. Trans-formation optics (TO) and the anisotropy related to ellipticalholes can be used for our purpose. Essentially, TO relates theelectromagnetic fields and constitutive parameters of the orig-inal space, namely virtual space ( x, y, z coordinates), to thedesired transformed space, known as physical space ( x (cid:48) , y (cid:48) , z (cid:48) coordinates). Since we are compressing the lens along the y direction, the coordinates of the transformation are: x (cid:48) = x, y (cid:48) = yβ , z (cid:48) = z. (16)In the physical space, we are interested in having isotropicmaterials outside the lens, similar to the virtual space (air: ε , µ ). Thus, the formulation of [36] particularized to theMFE lens leads us to the constitutive parameters: ε (cid:48) = ε /β
00 0 1 (17a) (a)(b) Fig. 12: (a) Top view of the compressed Maxwell fish-eye lens with ametasurface detail and (b) glide-symmetric unit cell. Elliptical holesare filled with a dielectric of relative permittivity ε r hole . µ (cid:48) = µ n ( r (cid:48) ) /β
00 0 1 (17b)where r (cid:48) = (cid:112) x (cid:48) + ( βy (cid:48) ) . From the material properties ofthe physical space, the refractive index can be calculated: n (cid:48) x = (cid:113) ε (cid:48) zz µ (cid:48) yy (18a) n (cid:48) y = (cid:112) ε (cid:48) zz µ (cid:48) xx . (18b)Note that, since the refractive index is cross-related with theconstitutive parameters ( n (cid:48) x ∝ (cid:112) µ (cid:48) yy , n (cid:48) y ∝ (cid:112) µ (cid:48) xx ) and β > ,it follows from equations (17) and (18) that n (cid:48) x < n (cid:48) y . This is,the refractive index is higher along the compressed direction.Therefore, the elliptical holes must be oriented as shown inFig. 12.The schematic for the compressed MFE lens is shown inFig. 12. Bright and mat red colors relate the holes of thebottom and top layers, respectively. In order to compare ourresults with [57], the compression factor is chosen to be β = 4 / (33.33 %). The radius of the lens is set to R = 9 cm. As illustrated in Fig. 12(b), we have placed a dielectric ofrelative permittivity ε r hole inside the elliptical holes to increasethe equivalent refractive index in both x and y directions andreach the required values.The map of the required refractive indexes is displayed inFig. 13. Some of the values extracted from equation (18) areless than one. In particular, the minimum index associated tothe propagation in the x -direction is n (cid:48) x, min = 1 /β = 0 . .Since the proposed unit cell cannot reach these values, themap plot of Fig. 13 was normalized by /β so the minimumrequired index is one.Fig. 14 shows the equivalent refractive index obtained at10 GHz with the unit cell of Fig. 12(b) for different hole EEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. X, NO. X, 2020 9 (a) (b)
Fig. 13: Map of refractive indexes after normalization for a 33.33%compression in the MFE lens: (a) n (cid:48) x , (b) n (cid:48) y . (a) (b) Fig. 14: Equivalent refractive index extracted at 10 GHz from theglide-symmetric unit cell when varying the eccentricity and themajor-axis of the ellipse: (a) n (cid:48) x , (b) n (cid:48) y . The geometrical parametersof the unit cell are: d = 4 . mm, g = 0 . mm, and h hole = 2 . mm. sizes when filled with Rogers-3010 dielectric ( ε r hole = 10 . ).In order to implement the compressed MFE lens, for eachspatial point, we must select from Fig. 14 the values of theeccentricity and major-axis that match the equivalent refractiveindexes closest to the map of Fig. 13. In the center of the lens,the required refractive indexes are higher. As a consequence,the holes are bigger there.Subsequently, the compressed MFE lens is realized andsimulated in CST Microwave Studio . The normalized absolutevalue of the electric field is displayed in Fig. 15 from 2.5GHz to 10 GHz for three excitation points at 0 o , 45 o and90 o . As observed, the compressed lens shows an ultrawidebandperformance, enhanced by the use of glide symmetry. Sincethe maximum equivalent refractive index calculated in Fig. 14is n (cid:48) y, max = 2 . , the maximum achievable compression factorin the glide case will be β G = 43 . %. On the other hand, afurther simulation in CST reveals that the non-glide-symmetricversion of the unit cell of Fig. 12(b) only offers a maximumcompression factor of β NG = 3 . %.Finally, a performance comparison is made with respect[57], where a MFE lens was compressed with a conformaltransformation and fully isotropic dielectric materials. Theapplied compression factor in [57] is identical to our work,a 33.33 %. However, their design needs a higher variationof the refractive index ( . / .
68 = 3 . ) compared to ours( . ) in order to fulfill the same compression requirements.As a conclusion, the use of anisotropic materials such as theelliptical holes reduces the required variation of the refractiveindex when compared to an isotropic design. (a)(b)(c) Fig. 15: Normalized electric field distribution (absolute value) of thecompressed MFE lens at different frequencies: (a) 10 GHz, (b) 5GHz, (c) 2.5 GHz. The point source is located at 0 o , 45 o and 90 o .The black line indicates the contour of the lens. V. C
ONCLUSION
In this paper, we analyzed the wave propagation in a glide-symmetric metallic structure composed of periodic ellipticalholes. For this purpose, a mode-matching based on the gen-eralized Floquet theorem has been applied. The dispersiondiagrams of the unit cell have been obtained for the differentvalues of all the geometric parameters. It has been demon-strated that glide-symmetric structures with periodic ellipticalholes exhibit an anisotropic refractive index over a wide rangeof frequencies. Furthermore, it has been confirmed that therefractive index can be tuned in a wider bandwidth in theglide-symmetric case.In order to validate the capabilities of the presented structurefor practical scenarios, a compressed lens has been designed.With the use of periodic elliptical holes in a glide-symmetricconfiguration, the profile of a MFE lens has been compressedof 33.33 %. The simulated MFE lens is wideband, operatingfrom 2.5 GHz to 10 GHz. It is reported how the use ofglide symmetry increases the equivalent refractive index, sohigher compression factors can be achieved. For the samedimensions and materials in the unit cell, the non-glide versiononly offers a maximum compression factor of 3.05 %, whilewith glide symmetry, a 43% is achieved. Finally, a comparisonbetween isotropic and anisotropic materials is done. The use ofanisotropic materials substantially reduces the required rangeof refractive index when compressing the lens. Thus, periodicelliptical holes arranged in a glide-symmetric configuration area promising candidate for the manufacturing of reduced-sizelow-loss low-cost wideband planar lenses.
EEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. X, NO. X, 2020 10 A PPENDIX AE LLIPTIC C YLINDRICAL C OORDINATES
Elliptic cylindrical coordinates, depicted in Fig. 16, aredefined by a radial coordinate ξ , which describes a set ofconfocal ellipses, an angular coordinate η , which describes aset of hyperbolae sharing the same foci, and a longitudinalcoordinate z . Their relation with the cartesian coordinates( x, y, z ) is [58]: x = h cosh ξ cos ηy = h sinh ξ sin ηz = z (19)where h = ae is the focal distance of the ellipse, a , b are the semi-major and semi-minor axes of the ellipse, and e = √ a − b /a is the eccentricity of the ellipse. From thedefinition of a vector r = x ˆx + y ˆy + z ˆz and the use of (19),the basis unit vectors ˆ ξ , ˆ η , ˆz can be obtained as ˆ ξ = h ξ ∂ r ∂ξ , ˆ η = h η ∂ r ∂η , ˆz = h z ∂ r ∂z , where h ξ = h η = h (cid:112) sinh ξ + sin η and h z = 1 are the scale factors. Thus, the basis unit vectorsin elliptic cylindrical coordinates are expressed as ˆ ξ ˆ η ˆz = hh ξ sinh ξ cos η cosh ξ sin η − cosh ξ sin η sinh ξ cos η
00 0 h ξ h ˆxˆyˆz (20)Furthermore, the surface differential dS can be derived byapplying the formulas of an orthogonal coordinate system.Thus, dS = h ξ h η dξdη = h (sinh ξ + sin η ) dξdη. (21)Other differential operators as the Laplacian of a scalar field ∇ φ , the gradient of a scalar field ∇ φ , the divergence of avector field ∇ · Φ , or the curl of a vector field ∇ × Φ can bederived in the same manner [58].A PPENDIX BM ODAL F UNCTIONS IN THE E LLIPTICAL H OLE
As previously discussed, the elliptical hole can be seen asa short-circuited elliptical waveguide. The i -th modal functionis expressed in terms of elliptical coordinates as Φ i = φ ξi ˆ ξ + φ ηi ˆ η . For TM modes ( Y i = k / ( η k zi ) ): φ ξi ( ξ, η ) = 1 h ξ (cid:26) Ce (cid:48) m ( ξ, q e mn ) ce m ( η, q e mn ) (even) Se (cid:48) m ( ξ, q o mn ) se m ( η, q o mn ) (odd) (cid:27) (22) φ ηi ( ξ, η ) = 1 h ξ (cid:26) Ce m ( ξ, q e mn ) ce (cid:48) m ( η, q e mn ) (even) Se m ( ξ, q o mn ) se (cid:48) m ( η, q o mn ) (odd) (cid:27) (23)For TE modes ( Y i = k zi / ( k η ) ): φ ξi ( ξ, η ) = 1 h ξ (cid:26) Ce m ( ξ, q (cid:48) e mn ) ce (cid:48) m ( η, q (cid:48) e mn ) (even) Se m ( ξ, q (cid:48) o mn ) se (cid:48) m ( η, q (cid:48) o mn ) (odd) (cid:27) (24) φ ηi ( ξ, η ) = − h ξ (cid:26) Ce (cid:48) m ( ξ, q (cid:48) e mn ) ce m ( η, q (cid:48) e mn ) (even) Se (cid:48) m ( ξ, q (cid:48) o mn ) se m ( η, q (cid:48) o mn ) (odd) (cid:27) (25)where ce m ( η, q e mn ) and se m ( η, q o mn ) are the even and oddangular Mathieu functions of order m , Ce m ( η, q (cid:48) e mn ) and Se m ( η, q (cid:48) o mn ) are the even and odd radial Mathieu functionsof the first kind of order m [52]. The prime symbol ( (cid:48) ) denotes Fig. 16 : Elliptic cylindrical coordinate system describing an ellipticwaveguide. the derivative with respect to ξ or η in the Mathieu functions.As a difference with the circular waveguide [59], the equationsof the tangential fields (22)–(25) show that there are twopropagating solutions (even and odd) for each mode. Thus, anadditional index is needed to differentiate them. We will usethe subscript “ e ” to indicate that the mn -th mode is even and“ o ” to indicate that the mn -th mode is odd, being the completenotation TM/TE e / o mn . Note that the even and odd solutionsdegenerate into a unique solution when the eccentricity of theellipse is zero and the elliptic waveguide transforms into acircular waveguide.The q -parameters ( q e mn , q o mn , q (cid:48) e mn and q (cid:48) o mn ) are cal-culated by imposing the boundary condition on the metallicwall of the waveguide. They are directly related with the cutoffwavelength λ c in the elliptic waveguide as [48]: λ c = πae √ q . (26)In the case of TM modes, the parameters q e mn , q o mn are the n -th roots of Mathieu Radial (cosine and sine) functions: (cid:26) Ce m ( ξ , q e mn ) Se m ( ξ , q o mn ) (cid:27) = 0 . (27)In the case of TE modes, the parameters q (cid:48) e mn , q (cid:48) o mn are the n -th roots of the first derivative of the Mathieu Radial functions: (cid:26) Ce (cid:48) m ( ξ , q (cid:48) e mn ) Se (cid:48) m ( ξ , q (cid:48) o mn ) (cid:27) = 0 . (28)The radial value ξ delimits the metallic surface of thewaveguide. This parameter is related with the eccentricity ofthe elliptical hole according to the expression cosh ξ = 1 /e [48]. Extracting the q -roots from the boundary conditions(27)–(28) is not easy, since the Mathieu radial functions Ce m and Se m are expressed as an infinite sum of hyperbolicfunctions [60], [61] and dependent on the eccentricity of thehole. The normalized cut-off frequencies of the first fifteenmodes inside the elliptical hole, obtained for different valuesof the eccentricity, are represented in Fig. 17. For the sakeof clarity, the even and odd versions of the same mode havebeen plotted with the same color. Additionally, the even modesare displayed with solid lines and the odd modes with dashedlines. EEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. X, NO. X, 2020 11
Fig. 17 : Normalized cut-off wavelength λ c /a for the first fifteenmodes as a function of the eccentricity e of the elliptical hole. (a) (b) (c)(d) (e) Fig. 18 : Normalized absolute value of the electric field distributionof the first five modes inside an elliptical waveguide with eccentricity e = 0 . : (a) TE e11 , (b) TE o11 , (c) TM e01 , (d) TE e21 , (e) TE o21 . The fundamental mode inside the elliptic hole is the TE e11 .However, the order of the higher-order modes depends on theeccentricity of the hole. Thus, the first higher-order mode isTE o11 for e < . , and TE e21 for e > . . As previouslydiscussed, both even and odd versions of the same modedegenerate into a unique mode as the eccentricity of the ellipseis near zero.Afterwards, the modal functions are computed using (22)–(25) according to the q -parameters derived from Fig. 17. Thenormalized absolute value of the electric field distribution ofthe first five modes in the elliptic hole (TE e11 , TE o11 , TM e01 ,TE e21 , TE o21 ) is shown in Figs. 18 and 19 for two differenteccentricities: e = 0 . and e = 0 . . For the case of e = 0 . (Fig. 18), the elliptic waveguide behaves similarly to a circularwaveguide. Thus, the odd modes (Fig. 18(b) and Fig. 18(e))are essentially rotated versions of the even modes (Fig. 18(a)and Fig. 18(d)). However, as the eccentricity becomes higher,the even and odd versions of the same mode are not the rotated (a) (b) (c)(d) (e) Fig. 19 : Normalized absolute value of the electric field distributionof the first five modes inside an elliptical waveguide with eccentricity e = 0 . : (a) TE e11 , (b) TE o11 , (c) TM e01 , (d) TE e21 , (e) TE o21 . version of each other (see Figs. 19(d) and 19(e)).R EFERENCES[1] N. I. Zheludev and Y. S. Kivshar. “From metamaterials to metadevices,”
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