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1 Annular Apertures in Metallic Screens as Extraordinary Transmission and Frequency Selective Surface Structures Rodríguez-Ulibarri, Pablo; Navarro-Cia, Miguel; Rodriguez-Berral, Raul; Mesa, F; Medina, F; Beruete, Miguel DOI: /TMTT License: None: All rigts reserved Document Version Peer reviewed version Citation for publised version (Harvard): Rodríguez-Ulibarri, P, Navarro-Cia, M, Rodriguez-Berral, R, Mesa, F, Medina, F & Beruete, M 2017, 'Annular Apertures in Metallic Screens as Extraordinary Transmission and Frequency Selective Surface Structures', IEEE Transactions on Microwave Teory and Tecniques, vol. PP, no. 99, pp ttps://doi.org/ /tmtt Link to publication on Researc at Birmingam portal Publiser Rigts Statement: (c) 2017 IEEE. Personal use of tis material is permitted. Permission from IEEE must be obtained for all oter users, including reprinting/ republising tis material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrigted components of tis work in oter works. General rigts Unless a licence is specified above, all rigts (including copyrigt and moral rigts) in tis document are retained by te autors and/or te copyrigt olders. Te express permission of te copyrigt older must be obtained for any use of tis material oter tan for purposes permitted by law. Users may freely distribute te URL tat is used to identify tis publication. Users may download and/or print one copy of te publication from te University of Birmingam researc portal for te purpose of private study or non-commercial researc. User may use extracts from te document in line wit te concept of fair dealing under te Copyrigt, Designs and Patents Act 1988 (?) Users may not furter distribute te material nor use it for te purposes of commercial gain. Were a licence is displayed above, please note te terms and conditions of te licence govern your use of tis document. Wen citing, please reference te publised version. Take down policy Wile te University of Birmingam exercises care and attention in making items available tere are rare occasions wen an item as been uploaded in error or as been deemed to be commercially or oterwise sensitive. If you believe tat tis is te case for tis document, please contact UBIRA@lists.bam.ac.uk providing details and we will remove access to te work immediately and investigate. Download date: 27. Mar. 2019

2 JOURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, MONTH YEAR 1 Annular Apertures in Metallic Screens as Extraordinary Transmission and Frequency Selective Surface Structures Pablo Rodríguez-Ulibarri, Miguel Navarro-Cía, Senior Member, IEEE, Raúl Rodríguez-Berral, Francisco Mesa, Fellow, IEEE, Francisco Medina, Fellow, IEEE, and Miguel Beruete Abstract A two dimensional periodic array of annular apertures (or ring slots) is studied using an accurate circuit model. Te model accounts for distributed and dynamic effects associated wit te excitation of ig-order modes operating above or below cut-off but not far from teir cut-off frequencies. Tis study allows to ascertain te substantial differences of te underlying pysics wen tis structure operates as a classical frequency selective surface or in te extraordinary-transmission regime. A discussion of two different designs working at eac regime is provided by means of te equivalent circuit approac, full wave simulation results, and experimental caracterization. Te agreement between te equivalent circuit calculation applied ere and te simulation and experimental results is very good in all te considered cases. Tis validates te equivalent circuit approac as an efficient minimal-order model and a low computational-cost design tool for frequency selective surfaces and extraordinarytransmission based devices. Additional scenarios suc as oblique incidence and parametric studies of te structural geometry are also considered. Index Terms frequency selective surfaces, extraordinary transmission, equivalent circuit, annular aperture, ring slot, millimeter waves. I. INTRODUCTION EXTRAORDINARY Transmission (ET) as been a widely studied penomenon in te last two decades. In 1998, T. W. Ebbesen et al. publised a seminal paper describing ig-transmission peaks in perforated plates in te optical regime [1]. Before [1], Betzig et al. ad observed similar P. Rodríguez-Ulibarri is wit te Antennas Group-TERALAB, Department of Electrical and Electronic Engineering, Universidad Pública de Navarra, Pamplona 31006, Spain ( pablo.rodriguez@unavarra.es) M. Navarro-Cía is wit te Scool of Pysics and Astronomy, University of Birmingam, Birmingam B15 2TT, U.K. ( m.navarrocia@bam.ac.uk) R. Rodríguez-Berral and F. Mesa are wit te Microwaves Group, Department of Applied Pysics 1, ETS de Ingeniería Informática, Universidad de Sevilla, Sevilla 41012, Spain ( rrberral@us.es, mesa@us.es) F. Medina is wit te Microwaves Group, Department of Electronics and Electromagnetism, Faculty of Pysics, Universidad de Sevilla, Sevilla 41012, Spain ( medina@us.es) M. Beruete is wit te Antennas Group-TERALAB and Institute of Smart Cities, Department of Electrical and Electronic Engineering, Universidad Pública de Navarra, Pamplona 31006, Spain ( miguel.beruete@unavarra.es) P. Rodríguez-Ulibarri acknowledges funding from Universidad Pública de Navarra via a predoctoral scolarsip. M. Navarro-Cía was supported by University of Birmingam [Birmingam Fellowsip]. Tis work as been partially supported by te Spanis Ministerio de Economía y Competitividad wit European Union FEDER funds (project TEC P and TEC C2-2-R) and by te Spanis Junta de Andalucía (project P12-TIC- 1435) peaks in metallic plates periodically perforated wit subwavelengt oles and ad found a relation between tese peaks and te excitation of surface plasmons [2], altoug tey did not go as deeply as [1] into te underlying pysics. Tese ET peaks were considered extraordinary because tey appear well below te cut-off frequency of te apertures and still below te first Rayleig-Wood (RW) anomaly [3], [4] (a deep null in te transmission spectrum related to te periodicity of te structure due to te onset of te first diffraction order). Ebbesen et al. explained ET as te coupling of te incident ligt to surface plasmons polaritons (SPP); i.e., a type of surface waves sustained by a metal-dielectric interface at optical frequencies [5]. Tis discovery gave a strong impulse to te scientific discipline termed plasmonics, leading ultimately to te so-called plasmon resurrection in te early years of te twenty-first century [6]. Wen a new pysical penomenon is reported, some time is needed to fully understand te underlying pysics. In ET te first source of controversy was te exact role played by SPPs in te penomenon. Remarkably, ET was also observed all along te electromagnetic spectrum including millimeter waves [7], were metals are modeled as finite conductivity materials tat do not support SPPs. A more general teory to explain ET was ten developed based on te excitation of complex surface waves in te periodic structure, wic in te case of optical frequencies coincide wit leaky SPPs [8] [10]. Following a similar reasoning, equivalent circuits were crucial to give a deep insigt into te underlying pysics and ave now become a fast analytical tool for te design and analysis of ET structures. Te first equivalent circuit for ET was discussed in [11]. Tere, te periodic problem was reduced to a single unit cell defined by a diapragm located inside a virtual waveguide, and te ET peaks were explained in terms of te reactive elements accounting for te ig-order modes scattered by te diapragm. A more elaborated equivalent circuit approac (ECA) was later proposed in [12]. It was noticed tat tis ECA can also be applied to more classical periodic structures like frequency selective surfaces (FSS), discussed below. In fact, FSSs ave been istorically studied by equivalent circuit tecniques well before te appearance of ET penomenon. Usually, circuit-model solutions were oriented toward reducing te electromagnetic problem to a simplified network of lumped elements (see, e.g., te papers of Langley et al. [13] [15]). Oter works relied on multimodal equivalent networks in order

3 JOURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, MONTH YEAR 2 to caracterize planar strip gratings [16] [18]. Recently, fully analytical ECAs ave been derived by some of te autors of tis work for one-dimensional (1D) and two-dimensional (2D) periodic metallic screens printed on dielectric slabs [19], [20], stacked fisnet structures [21] or T-saped corrugated surfaces [22]. A great number of papers dealing wit equivalent circuits applied to FSS structures can be found elsewere; see, for instance, te review presented in [23] and references terein. Anoter source of controversy concerning ET as been ow to delimit exactly wat is and wat is not ET. Due to teir similarity, ET resonances ave sometimes been identified as standard resonances of FSSs. Tese selective surfaces are 2Dperiodic arrays of metallic patces (or apertures in metallic screens) tat present a stopband (passband) associated wit te resonance of te patc (aperture) [24]. It sould be noted tat even wit a single patc (aperture), ig reflection dips (transmission peaks) can be obtained due to self-resonances of te element. Te addition of more elements in a periodic array affects mainly te amplitude of te transmittance and te bandwidt of te stopband (passband). In contrast, genuine ET peaks appear well below te cut-off frequency of te aperture and are associated wit te excess of energy accumulated by te first diffraction order mode of te periodic structure as it approaces te onset from evanescent to propagating. Tis means tat ET is a resonance related to te periodic nature of te structure rater tan to te single element geometry itself. Terefore, a necessary condition for ET is tat te aperture of te metallic screen as to be in cutoff. Altoug ET penomena can also be observed in nonperiodic structures (suc as diapragms on circular waveguides [12], [25]), periodic structures are te most typical geometry analyzed in te literature, wit teir lattice size being of utmost importance. Regarding t Te structure under study in te present work, as a similar geometry as te coaxial ole arrays ave previously been lately studied in te optical regime in te framework of ET [26] [36]. In some of tese works, te igtransmission peaks observed in te spectrum were labeled as ET, altoug a more careful scrutiny demonstrates tat tey are associated wit a resonance of te coaxial aperture; i.e., tey appear at te cut-off frequency of te TE 11 mode of te coaxial line. From our previous discussion it is apparent tat tis peak sould not be called ET. Following tis formalism, Orbons and Roberts correctly set te boundaries between FSS and ET regimes [30]. In addition, Lomakin et al. studied wave guidance on sub-wavelengt coaxial cross-section metallic oles witin te microwave and teraertz regimes [37]. In tis work, we use an ECA [20] to model and analyze coaxial ole arrays annular apertures (also called ring slots) drilled on metallic screens in order to explore wen te observed peaks correspond to ET excitation. As it was done in [20] for a different structure, te equivalent circuit topology is derived from very first principles giving rise to a fully analytical metod wit low computational cost, wic additionally provides a compreensive understanding of te underlying pysics of te considered penomena. Te coaxial geometry studied ere along wit te perspective provided by te ECA allow for an easy identification of te ET or FSS regime in terms of its structural parameters. In most practical situations concerning FSS and ET structures, a dielectric substrate is required as mecanical support of te patterned metallic screen. Terefore, knowing te effect of a dielectric slab on te spectral response is very relevant for design purposes. Someow naively, one may just take te beavior of a free standing FSS/ET structure and only expect a downsift of te resonance frequency due to te presence of te dielectric environment. However, a layered dielectric medium gives rise to additional features tat need be taken into account [38], [39]. In fact, te response of a dielectric-backed FSS/ET structure may be relatively complex, especially in te frequency range selected ere (millimeter waves) were te available commercial substrates can be considered electrically tick. Te transmission/reflection results may sow complex details due to multiple resonances occurring inside te dielectric slab. All tese features are accurately accounted for by te equivalent circuit developed in tis work. Te paper is organized as follows. Sec. II presents te analyzed geometry as well as te equivalent circuit employed. Sec. III deals wit analytical and numerical comparison of cases selected to igligt te differences between FSS and ET regimes. Besides, several parametric studies of te structural parameters are presented. In Sec. IV experimental results at millimeter waves are presented under bot normal and oblique incidence. Finally, a summary of te main ideas discussed trougout te paper is given. II. GEOMETRY AND EQUIVALENT CIRCUIT APPROACH BASICS Te structure under consideration in tis work is a periodic array of coaxial annular apertures in a metallic screen. Bot free standing and dielectric slab backed structures are studied. Since te latter structures are more interesting for practical applications, a greater empasis is put on tem. Te structure backed wit dielectric slab along wit its top and side views are sown in Fig. 1. Te geometrical parameters of te unit cell are te internal radius a, external radius b, periodicity along x and y directions, d x and d y, respectively, and te dielectric substrate eigt s. Rectangular unit cells (d x d y ) are also used in order to distinguis between vertical and orizontal polarization and discern te different mecanisms involved in te transmission spectrum. Given te periodic nature of te problem under consideration, Floquet analysis is employed (a time convention of te type e jωt is assumed trougout tis work). Wit tis formalism, depending on te polarization and incidence angle of te impinging wave, te infinitely periodic problem is reduced to a generalized waveguide problem in wic te boundary conditions can be eiter periodic, electric, or magnetic walls [11], [19]. Te assumed field in te aperture can be expanded in terms of Floquet modes leading to a multimodal network equivalent circuit model werein te armonics are represented by transmission lines connected in parallel wit an associated coupling coefficient [20]. Tese coupling coefficients can be interpreted as turn ratio values

4 JOURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, MONTH YEAR 3 as long as te slot widt is sufficiently small (b/a 1). As a furter simplifying assumption, te angular component ˆφ can be neglected and te radial component can be taken constant, independent on te radial distance (for larger b/a values, te 1/ρ dependence of te radial component sould be taken into account). As our main purpose is to study te different FSS and ET operation regimes, we will apply tis simplification in te following analysis. Terefore, taking te aperture field profile as [41]: E a = A cos (ϕ ϕ 0 ) ˆρ, (2) Fig. 1. (a) 2-D periodic structure wit a rectangular array of coaxial annular apertures on a metallic screen backed wit a dielectric slab. (b) Top view of a portion of te array structure. (c) Side view of a single unit cell. of standard transformers, N. Te field expansion in terms of Floquet modes is equivalent to evaluating te corresponding Fourier transform of te field profile at te aperture. Te value of N can be calculated as te following dot product of te Fourier transform of te assumed field at te aperture, Ẽ a, and te corresponding unit transverse wavevector (ˆk t, for TM armonics or its cross-product wit te unit vector ẑ for TE armonics): N = { Ẽa (ˆk t, ẑ) TE armonics Ẽ a ˆk t, TM armonics. To acieve congruent and accurate results, te assumed field profile of te aperture as to be consistent wit te aperture itself, te excitation, and te boundaries of te unit cell problem. Based on te previous experience of some of te autors of tis work, for normal incidence and geometrically simple apertures, it is sufficient to set te aperture field profile as te first propagating mode of te equivalent ollow waveguide. Two modes would be required for more complex situations, appearing in our case wen oblique incidence is considered [40]. First we consider a normally-incident plane wave wit linear vertical (i.e., parallel to y) polarization. In tis For normallyincident plane wave scenario (wit electric field paralell to y axis), te original periodic problem can be reduced to te scattering of an aperture discontinuity inside a virtual waveguide wit perfect electric top and bottom walls and perfect magnetic sidewalls. Given te polarization of te impinging wave, te first waveguide mode tat can be excited in te coaxial annular aperture is te TE 11. (note tat te symmetry of te incident wave precludes te excitation of te fundamental TEM mode It sould be noticed tat te symmetry of te incident wave precludes te excitation of any mode wit even parity. Ten, since te fundamental TEM mode cannot be excited, te TE 11 is te mode wit odd parity of lowest order tat can be considered. Tus, te first guess for te aperture field profile sould resemble te field distribution of te TE 11 coaxial mode. Wit tese prescriptions, Dubrovka et al. proposed a field aperture wit only radial component and an angular dependence wit a sine/cosine function [41]. It sould be taken into account tat tis approximation works (1) were A is a constant, ϕ te azimut angle and ϕ 0 te reference azimut angle. Te problem of oblique incidence over a coaxial an array of annular apertures FSS as already been studied in [42]. In tat work, te TEM mode field distribution is incorporated to te field profile of te aperture as a constant witout neiter radial nor angular dependence as: E TEM a = B ˆρ (3) were B is a constant. It sould be noticed tat te TEM mode can be uniquely excited under TM polarization for symmetry reasons. Since te ECA used ere is largely based on te approaces presented in Refs. [20], [40] [42] te explicit equations and circuit topologies are presented in te Appendix A. It sould also pointed out tat ere te contribution of te iger order modes is incorporated in a distributed manner oppositely to te discretization tecniques carried out in Ref. [41]. III. ANALYTICAL AND NUMERICAL RESULTS In tis section, numerical and analytical results for te structure depicted in Fig. 1 are compared for different design configurations. First, a selected range of examples are studied to validate our ECA results in different scenarios. Next, two studies of a free-standing coaxial array of annular apertures are conducted in order to delimit and observe bot FSS and ET operation regimes. In te first study te mean radius of te aperture is varied wile keeping te rest of parameters constant and in te second one te lattice period is swept wile aperture remains uncanged. In addition, two different dielectric backed designs are studied wit geometrical parameters tuned to ave eiter classical FSS or ET performance. It will be found tat te ECA is able to accurately predict te response of structures working at different pysical regimes. A. Equivalent Circuit: General Results Some particular free-standing structures wit square unit cell (d x = d y ) and several internal radii a, and dielectric backed substrates wit different eigts, s, and relative permittivity ε r are analyzed by means of equivalent circuits and full wave simulations (see Fig. 2), assuming a normally incident plane wave. Te frequency is normalized to te diffraction limit, i.e. f diff = c 0 /d wit d being te larger lattice period. Te numerical results are computed using te full wave commercial software CST MICROWAVE STUDIO (CST MWS ) [43]. For te simulation settings, te unit

5 JOURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, MONTH YEAR 4 Fig. 2. Transmission coefficient results obtained by ECA (blue line) and CST (red line). Free standing structure: (a) d x = d y = 1.5 mm; b = 0.65 mm; s = 0 mm and several internal radii from a = 0.3 mm to a = 0.6 mm. Dielectric backed structure: d x = 1.5 mm; d y = 3 mm; b = 0.65 mm; a = 0.5 mm; [ ] ε r = 5, s = 0.4 mm; [ ] ε r = 3, s = 0.2 mm; [ ] ε r = 2.1 j0.018, s = 1.6 mm. (b) Vertical polarization (ϕ = 90 ). (c) Horizontal polarization (ϕ = 90 ). cell option is selected in CST boundary conditions for te frequency domain (FD) solver. Te metallic screen is made of perfect electric conductor (PEC) wit infinitesimal tickness. A tetraedral mes wit adaptive refinement is selected wit a maximum of eigt steps and a stop criterion of 0.02 as tresold for consecutive absolute values of S-parameters. Te stop criterion for te frequency sweep is again selected wit respect to te absolute value of te S-parameters difference between consecutive calculations and set to Figure 2(a) sows te results for te free standing structure. It can be seen tat te accuracy of te circuit model is better for narrow slots (b/a 1), in concordance wit te assumed approximation tat te field dependence wit te radial distance is neglected. In any case, te agreement is quite good until b/a 2. Results for dielectric-backed structures are sown in Figs. 2(b) and (c). In tis case, te unit cell is rectangular leading to a polarization-dependent structure. Panels (b) and (c) sow te response under vertical and orizontal polarization respectively. It can be observed tat te agreement is good in bot cases, wit a better agreement for te orizontal polarization. Different dielectric slabs ave been tested wit similar level of agreement between ECA and CST results (note tat dielectric loss can be easily modeled by setting te permittivity as a complex number). In all te analyzed cases, te computation resources used by te ECA are negligible in comparison wit full wave simulations. For instance, for te cases selected in tis section, a 2000-point frequency sweep is run witin 1 or 2 seconds wile te simulation may last several minutes. Certainly, tis difference becomes even more crucial wen parametric sweeps and optimization of te structure is required. Fig. 3. (a) Resonant wavelengt λ res versus mean radius (r m) normalized to te lattice period d for free-standing coaxial array of annular apertures wit b/a = 1.3 and d = 1.5 mm. (b) Transmission coefficient results obtained by ECA (blue line) and CST (red line). [ ] b = 0.25 mm (r m = 0.13d mm). [ ] b = 0.3 mm (r m = 0.177d mm). [ ] b = 0.4 mm (r m = 0.236d mm). B. Discerning FSS and ET regimes In tis section our goal is to explore te limits between FSS and ET operation regimes. To tis end, a square unit cell and free-standing coaxial array of annular apertures is analyzed in wic te b/a ratio and te period, d, are kept constant (b/a = 1.3 and d = 1.5 mm respectively) and te mean radius, r m = (a + b)/2, is varied. Te resonance wavelengt, λ res, is defined as te wavelengt were te maximum transmission is obtained. Tis resonance is represented versus r m in Fig. 3(a). It sould be noticed tat λ res and r m are normalized to te lattice period, d. It can be observed ow for small radii (0.1d < r m < 0.15d), λ res is practically equal to 1 regardless te aperture size. ET penomenon can be claimed witin tis operation range. In contrast, for large r m values (r m > 0.2d), λ res is linearly proportional to te aperture size, wic is te classical beavior of FSS regime. Between tese two regions tere is a range (0.15d < r m < 0.2d), werein λ res is neiter equal to 1 nor proportional to r m and tat can be considered as te boundary between FSS and ET regimes. Figure 3(b) sows te transmission results obtained by te ECA and CST for tree selected cases: ET (r m = 0.2 mm), FSS (r m = 0.35 mm) and an intermediate case (r m = mm). Note tat ET transmission peaks are inerently narrowband wile FSS peaks are relatively wide. Next, a rectangular (d x d y ) free-standing coaxial array of annular apertures is studied in order to evaluate te influence of te lattice period, d y, in te transmission properties and te operation regime (FSS or ET). Te rest of parameters are set to d x = 1.5 mm, b = 0.65 mm, and a = 0.5 mm. Wit tese dimensions te cut-off frequency of te aperture is f ap = c 0 /[(a + b)π] = GHz. Te results for vertical and orizontal polarization are sown in Figs. 4(a) and (b), respectively. Te figures also include wite dased lines marking f ap as well as te limits of te FSS/ET regimes. Fig. 4(a) sows tat, for vertical polarization, te transmission peak sifts to lower frequencies as d y increases. It is also wort noting te

6 JOURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, MONTH YEAR 5 TABLE I GEOMETRICAL PARAMETERS OF THE STUDIED FSS/ET STRUCTURES ALONG WITH THE APERTURE CUT-OFF FREQUENCY (f AP). Design a b d x d y s ε (1) r f ap (GHz) FSS ET FSS (prot.) ET (prot.) dimensions are given in mm. Fig. 4. ECA results of te transmission coefficient for FSS design versus frequency and d y; d x = 1.5 mm, b = 0.65 mm, and a = 0.5 mm. (a) Vertical polarization. (b) Horizontal polarization. Vertical wite-dased line accounts for te cut-off frequency aperture. Te wite-dased curves delimit ig transmission regions and ence, te ET and FSS regimes. appearance of ig transmission bands below and above f ap in te region 3 mm < d y < 3.5 mm. Tis corresponds to te aforementioned transition region were te ig transmittance band is due to a combination of bot FSS and ET regimes. Wen d y 3.5 mm te transmission above f ap vanises, and for d y > 3.5 mm te ig transmittance peak is completely due to te ET resonance. For te orizontal case in Fig. 4(b) it can be observed tat ET does not take place, fact tat will be discussed later in tis section. Tus, only te FSS regime region is igligted in Fig. 4(b). Tese results clearly set te boundaries between ET and FSS regimes. Wile te FSS regime is related to te aperture in te metallic screen and its exact location depends on its size and geometry, te ET peak always appears rigt before te first diffraction order, independently of te aperture geometry. In fact, for te FSS case, te ig-transmission frequency is independent of te period and te transmission bands are broad. On te oter and, for ET, te frequency of te transmission peak is governed by te period and its bandwidt is always narrow. In te next study we analyze structures printed on a dielectric slab, designed specifically to ave eiter FSS or ET performance. Tese designs will be experimentally analyzed subsequently in Sec. IV. Te geometrical parameters of te teoretical designed and final fabricated prototypes are outlined in Table I. Due to te dielectric slab loading, te cut-off frequency of te aperture is modified. Here, it is approximated as te cut-off frequency of te TE 11 mode of te equivalent coaxial waveguide divided by te square root of te effective permittivity, ε eff [44]: c 0 f ap = (a + b)π (4) ε eff wit c 0 being te speed of ligt in te vacuum. As it is well known, te magnitude of ε eff depends on te surrounding media, wit 1 < ε eff < [ε (1) r + 1]/2, were ε (1) r = 2.4 is te relative permittivity of te dielectric slab. Te exact value of ε eff is not easy to estimate since it depends on te electrical dimensions of te dielectric slab and te metallic geometry. For te sake of simplicity, te upper-bound value of tis parameter (ε eff = 1.7) is assumed for evaluation of te f ap values sown in Table I. It sould be noted tat te given f ap ten takes its lowest possible value. Figure 5 sows a comparison between numerical and analytical results for te FSS and ET designs outlined in Table I. Normal incidence and bot vertical and orizontal polarizations [ϕ = 90 - Fig. 5(a)] and orizontal polarization [ϕ = 0 - Fig. 5(b)] of te incident wave is considered. Magnitude and pase results are sown for te FSS [Figs. 5(a) and (c)] and ET [Figs. 5(b) and (d)] designs, respectively. In order t To elp to discern te nature of te transmission peaks, te frequency is normalized to te diffraction limit frequency, f > f diff = c 0 /d y, for eac case, i.e. 100 GHz and 60 GHz for te FSS and ET cases, respectively. In addition, te aperture cut-off frequency for eac design as been labeled in Figs. 5(a) and (b), as fap FSS and fap ET and marked wit vertical dased-lines. As it can be seen, te agreement between CST simulation and ECA results is very good for bot designs and polarizations, even above te diffraction limit (f > f diff ). For tis reason, only te ECA is used for analyzing te impact of te dielectric eigt, s, on te transmission features in bot FSS and ET structures (see Fig. 6). Te transmission coefficient for bot FSS and ET structures and bot vertical and orizontal polarization are presented. Te substrate tickness, s,was varied from 0 to 2 mm. Te particular cases sown in Fig. 5 are depicted in Fig. 6 by means of orizontal wite-dased lines. For te vertical polarization case sown in Fig. 5(a), te first ig transmission band of te FSS design appears near 65 GHz, near and sligtly above fap FSS 63.8 GHz, and is located well below fdiff FSS 100 GHz. Tis transmission band precedes a null of transmission and a second transmission band tat appear at 79 GHz and 84 GHz, respectively. Te null is due to te presence of te dielectric slab and can easily be explained wit te ECA in terms of te first iger order mode of te equivalent virtual waveguide loaded wit te dielectric medium wit ε (1) r. In te present case, tis role is played by te TM 02 mode, wose cut-off frequency inside te dielectric (1) is f c,tm02 = c 0 /(d y ε r ) = 64.6 GHz. Te input admittance seen by TM 02 diverges to infinity [pole in (12)] at a frequency tat depends on te substrate caracteristics and tat is located between f c,tm02 and f diff. Indeed, an admittance divergence is considered as a sort-circuit in te ECA leading to a null of transmission tat can be related to te RW anomaly inside te dielectric region [38]. As it can be seen in Fig. 6(a), wen s = 0, a single transmission band is obtained but, as s

7 JOURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, MONTH YEAR 6 Fig. 6. ECA results of te transmission coefficient for FSS and ET designs versus frequency and s. (a) FSS - Vertical polarization (ϕ = 90 ). (b) FSS - Horizontal polarization (ϕ = 0 ). (c) ET - Vertical polarization (ϕ = 90 ). (d) ET - Horizontal polarization (ϕ = 0 ). Horizontal wite dased lines denote te particular cases sown in Fig. 5. Fig. 5. Transmission coefficient for FSS and ET designs. ECA (blue line) and CST (red line) results are sown for vertical polarization, i.e. ϕ = 90 (squares), and orizontal polarization, i.e. ϕ = 0 (circles). (a) FSS - Magnitude. (b) ET - Magnitude. (c) FSS - Pase. (d) ET - Pase. increases, te null is sifted to lower frequencies, wit its lower bound given by f c,tm02. It can be observed ow te null appearance splits te transmission peak into two different transmission bands as s increases. For te ET case and vertical polarization sown in Fig. 5(a), two transmission peaks arise at approximately 51 and 59.9 GHz, well below fap ET 91.6 GHz. Te lower peak can be explained again by te te presence of te dielectric slab and te role of te TM 02 mode of te equivalent virtual waveguide. Te input admittance TM 02 mode diverges to infinity leading to a null in te transmission spectrum at 52.8 GHz. Below tis frequency, te total inductance due to te aperture can be compensated by te capacitance of te TM 02 mode. For te second transmission peak, te situation is similar. In tis case, it is due to te divergence of te TM 02 admittance as it approaces to propagation regime in te air [12]. Again, te TM 02 mode contribution adds te required capacitance at a certain frequency near and below te first RW anomaly (60 GHz) to compensate te inductance introduced by te aperture. Te particular origin of tese two peaks can be better discerned wen te substrate eigt, s, is varied [see Fig. 6(c)]. Two narrow-band transmission peaks arise between 45 and 60 GHz. Te peak related to te dielectric appears for s > 0 and sifts to lower frequencies as s increases. Tis sift finds its limit at approximately 45 GHz. On te oter and, as expected, te transmission peak related to te air region remains always rigt before te diffraction limit regardless of s. Furtermore, peaks wit weak transmission arise witin te diffraction regime between 90 an 120 GHz for te particular case sown in Fig. 5(a). Tese two low transmission peaks rely on te same principles as te ordinary transmission peaks of te FSS structure. In fact, tey are broader tan tose obtained at 51 and 59.9 GHz, denoting tat tey are linked to te aperture resonance. Here, te transmission peak splits into two because of te divergence to infinity of a iger order mode (te TM 04 mode in tis case). Tis feature is corroborated by te results sown in Fig. 6(c) wen te slab eigt, s, is swept. Indeed, te partial transmission arising at lower frequencies follows te trend of te null due to te dielectric. In any case, as tese transmission peaks occur witin te diffraction regime, part of te power is transferred to te diffraction modes and, ence, total transmission in te fundamental mode is not possible. From tese results we can attribute to ET penomena te two ig and narrow transmission peaks arising at 51 and 59.9 GHz. Since tere is an admittance tat diverges to infinity in bot cases, ig-transmission peaks arise regardless te selected aperture geometry and size. Indeed, even for very small apertures in wic very ig capacitance is required to compensate small inductance values, te resonance condition is attainable due to te divergence to infinity [12]. It sould be pointed out tat in te ET case te inductance is low and terefore te quality factor, Q, is ig. Hence, te transmission bands are inerently narrowband. On te oter and, for te FSS case, te inductance is usually large and te total capacitance required for resonance does not need to rely on singularities to attain te convenient value. Ten, te transmission peaks obtained are broadband. Moreover, tey can be regarded as ordinary because tey are due to te

8 JOURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, MONTH YEAR 7 resonance of te aperture. In te case of orizontal polarization in Fig. 5(b), te results are quite different. Probably, te most remarkable difference comes from te absence of transmission peaks for te ET case witin te non-diffractive regime. Tis can be explained again wit te ECA. Now, due to te different polarization state, te first relevant ig order mode is te TE 02. Below cut-off tis mode is inductive, so its contribution is just adding an extra inductance and, terefore, te cancellation of te inductance due to te aperture does not seem possible. However, as explained in [45], wit a proper selection of ε (1) r and s, one could get an anomalous ET resonance [46], [47]. Te condition for tis penomenon to occur is aving F = s [ε (1) r 1] 1/2 /d y > In te case presented in Fig. 5(b) tis is not satisfied, since F = Te only transmission feature tat appears in te spectrum is an abrupt slope cange at te first RW anomaly (60 GHz). Focusing now on te s study for tis particular case [see Fig. 6(d)], it can be clearly seen tis feature and ow anomalous ET only occurs for sufficiently tick slabs, i.e., F 0.25 ( s = 0.25d y /( ε r 1) 1.05 mm). Te F parameter is linked to grounded slab configurations as detailed in [45]. As mentioned above, in tis case te first ig order mode tat becomes propagating inside te dielectric slab is te TE 02. Unlike te TM fundamental mode tat propagates even wit s = 0 [notice te null excursion from s = 0 and f = 60 GHz to s = 2 mm and f 40 GHz in Fig. 6(c)], te fundamental TE mode requires a minimum s so tat te transcendental modal equation as a valid solution [44]. In addition, a weak transmission peak is observed at approximately 105 GHz in Fig. 5(b). As in te vertical case, tis transmission feature is due to te aperture resonance influenced by te TE 04 mode dispersion. In fact, it can be observed in Fig. 6(c) tat for tin substrates ( s < 0.5 mm) a relatively broad and weak transmission band appears, reinforcing te argument about its connection wit te aperture resonance. At te second RW anomaly frequency (120 GHz), a cange in te slope of te transmission is again observed for all te s values. Te rest of transmission features arising above 120 GHz and for slab eigts s > 1 mm rely on te subsequent ig order modes excitation, and ten a furter detailed discussion will not be addressed. For te FSS design [Fig. 5(b)], a ig-transmission band appears in te spectrum similarly to te vertical polarization case but at a iger frequency (around 75 GHz). Tis transmission peak can again be directly linked to te coaxial annular aperture resonance, and its frequency sift wit respect to te vertical polarization case can be attributed to te different contribution of te first iger order mode of te equivalent virtual waveguide. Tis contribution results in a different value of te effective permittivity; namely, ε eff is iger for te vertical case tan for te orizontal one. As ε eff accounts for te static-capacitance variation due to te evanescent fields of cut-off TM modes, and tese modes play a major role for te vertical polarization case tan for te orizontal one, it leads to a iger value of ε eff for te former polarization case. It is wort noting tat TE modes at cut-off are inductive Fig. 7. Transmission coefficient results for te FSS design for TM polarized impinging wave and θ = 45. ECA wit one mode in te aperture (blackdased-dotted line), ECA wit two modes (blue-solid line) and CST (reddased line) are sown. (a) ϕ = 90. (b) ϕ = 0. and are not affected by te surrounding medium [see (19)]. By inspecting Fig. 6(b) it is observed tat, indeed, te differences in te performance are greater for tin substrates. As before, te first ig order mode tat becomes propagating inside te dielectric slab is te TE 02 of te equivalent virtual waveguide. It can be observed tat for tin substrates (up to s = 0.65 mm) a single band operation is retained wile for iger s values te band is split in two. Tis is again directly connected to te F parameter presented before. In fact, s = 0.25d y /( ε r 1) 0.63 mm. Tis feature can be tracked by observing te first null evolution in Fig. 6(b). It starts at s 0.65 mm, f = 100 GHz and reaces a frequency of f 75 GHz wen s = 2 mm. In contrast, for te vertical polarization, a null excursion was observed from s = 0 and f = 100 GHz to s = 2 mm and f 70 GHz [seefig. 6(a)]. Te results obtained in tis study yield a quite complete understanding for accurately discerning between FSS and ET regimes. In addition, useful guidelines are given for practical design purposes were dielectric slabs are present. For instance, for te case of classical FSS design, if a single peak wit nearly total transmission is required, tin substrates sould be selected. For te studied case, witin te range 0.5 mm < s < 0.75 mm, a wider transmission band but wit lower amplitude can be obtained. For larger s, double band solutions for bot polarizations are possible. Similarly, for te ET case, te dependence of te ig-transmission peak frequency location wit s as been demonstrated. In addition, it sould be mentioned tat, despite te obvious differences between te geometrical parameters and te transmission spectra of te two studied designs, FSS and ET, te ECA results agree quite well wit te full-wave simulation ones. C. Oblique Incidence Te case of oblique incidence is briefly treated in tis section. Te structure under study is now te FSS design analyzed in Sec. III-B (te structural parameters can be found in Table I). Figure 7 sows a comparison between simulation and ECA results wen one and two modes in te aperture are

9 JOURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, MONTH YEAR 8 Fig. 8. Experimental set-up. Insets: Unit cell microscope picture of fabricated FSS and ET prototypes. considered under TM polarization. Wen only one mode is included, a single function compatible wit te TE 11 mode of te coaxial waveguide is used for approximating te field at te aperture. For te case of two modes, anoter function compatible wit te TEM coaxial waveguide mode is added. A TM polarized wave wit θ = 45 is considered for bot principal incidence planes, ϕ = 90 and ϕ = 0. For te case wen te incidence plane is parallel to te long period (d y ), it can be seen in Fig. 7(a) tat te inclusion of te second mode sligtly improves te accuracy of te equivalent circuit (specially in te prediction of te null at 55 GHz). For te ortogonal polarization sown in Fig. 7(b), te accuracy of te model at low frequencies is qualitatively improved wen te contribution of te TEM mode is taken into account. In addition, te transmission peak observed after te null at approximately 95 GHz is muc better predicted by te ECA considering two modes. IV. EXPERIMENTAL RESULTS Te numerical and analytical results obtained in previous sections are corroborated ere by experiments. To tis end, two prototypes featuring te FSS and ET geometries considered in Sec. III-B were fabricated by milling macining. Microscope potograps of te unit cells of bot fabricated prototypes can be found in te insets of Fig. 8. Te selected commercial substrate was te ULTRALAM 2000 from Rogers Corp. Te dielectric properties given by te manufacturer are a dielectric constant ε r = and a loss tangent tan δ = (measured at 10 GHz). Te tickness of te copper cladding is 35 µm. Te measurements were made wit an AB-Millimetre vector network analyzer (VNA) equipped wit a quasi-optical (QO) benc [48]. Te experimental setup consists of two corrugated orn antennas [transmitter (TX) and receiver (RX)], four ellipsoidal mirrors to get a focused Gaussian beam illumination at te sample position, and an automatic rotatory platform were te sample is placed to perform oblique incidence measurements; a picture of te experimental setup is given in Fig. 8. Te experimental caracterization was carried out troug a frequency sweep between 45 and 110 GHz in steps of 25 MHz. Two independent sets of transmitters and receivers were employed to cover te V band (45-75 GHz) and te W band ( GHz). Figure 9 sows a comparison between te obtained experimental and full wave simulation results of te structure under normal incidence. Te structural parameters of te fabricated prototypes differ sligtly from tose selected in te design section due to fabrication tolerances. Te prototype parameters are sown in te Table I, were te values are obtained from direct observation wit a microscope. Tese are te parameters used in te CST full-wave simulation sown in Fig. 9. In addition, te dielectric permittivity is taken as ε r = 2.5 and losses are included wit a loss factor tan δ = Tese values come from fitting wit te experimental results and tey are congruent wit te values reported in te literature. In fact, dielectrics can sow an increment of ε r and tan δ at frequencies ranging between millimeter and teraertz waves [49]. In order t To acieve a reliable full-wave simulation of te actual finite structure, a Gaussian beam source and te time domain solver were cosen to model te experimental illumination. Te Gaussian-beam field profile was set to ave te focus exactly at te sample surface. Te beam-waist diameter (2w 0 ) was 23 mm in agreement wit te experimental value, so tat it covers approximately alf of te sample (te sample diameter is about 52 mm). Te cosen frequency for te Gaussian beam definition was 75 GHz, rougly te center frequency of te range of interest. Te finite simulation results correspond to te field recorded at a distance of 45 mm away from te center of te prototype sample. As it can be observed in Fig. 9, tere is an overall good agreement between experimental and simulation results. For te FSS structure [Fig. 9(a)], an excellent concordance is acieved for bot incidence planes despite a small frequency downsift observed in te simulation. Tis disagreement can be attributed to possible errors in te assumption of ε r and/or s or to te deficiency in modelling te frequencydependent beamwaist of te Gaussian beam. In any case, te frequency sift is about 2.5% at 80 GHz. For te vertical polarization case (red lines), it can be observed tat te second transmission band is seriously degraded in te experiment (red-solid line). A transmission dip appears splitting te transmission peak into two. Tis penomenon is also observed in te finite structure simulation (red-dased line). Tis feature is due to te non-uniformity of te wave tat impinges onto te sample. Actually, due to te intrinsic propagation caracteristics of te Gaussian beam, oblique wavevectors will be present at te illumination plane even under normal incidence case. For tis reason, te obtained response in tis case sows te unfolding of te RW anomaly into two dips, as it would appen (and will be sown later) for oblique incidence under TM polarization and ϕ = 90. In order t To reinforce tis argument, a field inspection of te full-wave simulation results for te FSS case at vertical polarization is presented in Fig. 10, were it is sown te electric field distribution of te y and z components at different frequencies, f = 63, 78 and 84 GHz. Te yz-plane (x = 0) is cosen wit te impinging wave propagating positively along te z axis. It can be seen tat at 63 GHz te impinging wave is transmitted troug te array wit little reflections. Te field is mostly accounted for by te y component [see Figs. 10(a) and (b)]. Conversely, a very ig reflection appears at 78 and 84 GHz. In addition, waves running along te surface are observed at bot frequencies (wic are related to te RW anomaly unfolding), wit a wave propagating along te

10 JOURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, MONTH YEAR 9 Fig. 10. Electric field distribution for te FSS design ϕ = 0 incidence plane. (a) E y - f = 63 GHz. (b) E z - f = 63 GHz. (c) E y - f = 78 GHz. (d) E z - f = 78 GHz. (e) E y - f = 84 GHz. (f) E z - f = 84 GHz. Fig. 9. Finite simulation and experimental results of fabricated samples for ϕ = 0 and ϕ = 90 incidence planes. (a) FSS prototype. (b) ET prototype. surface eventually reacing te prototype edges and radiating at endfire. Anoter feature observed in bot simulation and experimental results is a smooter RW anomaly compared to te one predicted by te ECA results sown in te previous section; a fact tat can be directly related to te finite size of te fabricated samples [50] and te focused Gaussian beam illumination. For te orizontal polarization case [blue lines in Fig. 9(a)], te agreement is again good despite a sligt frequency sift. Regarding te ET design results sown in Fig. 9(b), it can be seen tat te differences for te ET transmission peaks between simulations and measurements are greater tan in te previous case. Te first transmission peak as lower amplitude tan te one predicted by ECA results (see Fig. 5), and te second one disappears bot in te simulations and measurements. Te poor transmission at te first ET resonance as well as te vanising of te second one can be attributed to an insufficient illumination of te coaxial annular apertures, as discussed in previous papers [51], [52]. In addition, tese two resonances ave a very ig Q and tus losses affect dramatically te transmission amplitude, as lossy simulation confirm (not sown ere). It sould be remarked ere tat te coaxial aperture dimensions ave been cosen to work deeply in te ET regime, so te apertures are very small compared to te wavelengt resulting in an enanced quality factor of te ET resonance [25]. Te large differences above f diff between te simulation and te measurements of te finite sample are minimized wen e detection plane is sligtly tilted can be attributed to te grating lobes contribution. At suc a sort distance, te energy associated to te grating lobes can be important for wide range of frequency and angular regions. In contrast, in te experimental measurements, te receiver is placed at a muc larger distance, preventing te detection of tose grating lobes falling outside te narrow acceptance angle of te receiver. Tis implies tat small alignment errors may lead to big discrepancies in te recorded transmission coefficient. Next, we study te oblique-incidence response for bot prototypes. Te rotatory platform is used for selecting te angle of incidence (θ in ) and, by switcing te TX and RX antennas from vertical to orizontal position, TE and TM polarization can be selected. Finally, te rotation angle of te sample determines te plane of incidence; i.e., te alignment wit ϕ = 0 or ϕ = 90 planes. Te angular step is set to 5 and θ in is swept from 0 to 45. To fully caracterize bot prototypes, eigt independent measurements were made in total (2 incidence planes 2 polarizations (TE/TM) 2 prototypes). For te sake of brevity, only results concerning TE incidence at ϕ=0 and TM incidence at ϕ=90 are sown for bot FSS and ET designs in Figs. 11 and 12, respectively. In addition, CST simulation results of te infinite structure are sown. Te computational effort required to caracterize te oblique incidence is uge and, for a qualitative comparison, simulations of te infinite sample ave been found sufficiently accurate. For completeness, ECA results are also included, and due to its low computational cost an angular step of 1 as been taken. Regarding te FSS design in Fig. 11, an overall good agreement is obtained for te TE incidence at ϕ = 0. Te sift of te first null wit increasing θ in is clearly observed in te experiment [Fig. 11(a)]. As θ in approaces 45, te sample older starts to block te impinging wave, resulting in a reduction of te transmitted power. For te ET case in Fig. 11, te differences are iger. As predicted in te finite-structure simulation presented in Fig. 9(b), te amplitude of te first ET peak is igly reduced and te second one (just before f diff ) vanises. Tus, we ave found some transmission features in te CST unit-cell simulation and ECA results [Figs. 11(d) and (f)] tat are not present in te experimental results [Fig. 11(b)]. Despite tis, te low transmission peak before te null located near 92 GHz is retained in te experiment. For te case of TM polarization and ϕ = 90 sown in Fig. 12, tere is also a qualitative good agreement between te experiment, simulations, and ECA results. However, in tis case, an additional error source arises from te experimental setup since now te electric field emanating from te antenna is orizontally polarized. Altoug absorbent material is placed on te metallic benc, a ground effect on te measurements is ardly avoidable. In any case, for te FSS design [Figs. 12(a), (c) and (e)], te sift of te transmission band is retained in te experiment. Specifically, it is clearly observed te angle dependence of te dip tat appears rigt after te transmission peaks. Te discontinuities observed for te CST

11 JOURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, MONTH YEAR 10 Fig. 11. Transmission coefficient versus frequency and θ in for TE polarized impinging wave and ϕ = 0 incidence plane for FSS and ET prototypes. (a) Experiment-FSS. (b) Experiment-ET. (c) CST-FSS. (d) CST-ET. (e) Equivalent Circuit-FSS. (f) Equivalent Circuit-ET. ave been studied by means of an equivalent circuit approac (ECA), full wave simulation, and experiments. In addition, it as been sown tat te ECA accurately predicts te transmission features of tis kind of structures regardless of te operation regime (FSS or ET). By some modifications of te ECA, oblique incidence as been tested as well. A relevant result of te discussion on te FSS/ET operation regime is tat ET penomenon can only be claimed wen te ig-transmission peak is located well below te first resonance frequency of te aperture; oterwise, a typical FSS beavior is found. Our results and discussions ave been sustained by te ECA tus endowing a meaningful understanding of te underlying pysics. In addition, te results of tis approac ave found to be very accurate regardless of te FSS/ET nature of te structure. Terefore, te ECA can be used as an efficient and time-saving analytical tool wen compared wit full-wave numerical solutions. Regarding te experimental caracterization, te obtained results corroborate te analytical and numerical findings. Despite te differences observed, wic are mainly related to te finite nature of te structure and te non-uniformity of te illumination wavefront, te experimental results ratify te utility of te ECA as a design tool for FSS/ET structures as key elements in te development of devices suc as spatial filters, dicroic filters, and polarizers. APPENDIX A EQUIVALENT CIRCUIT APPROACH In tis appendix te explicit equations required to implement te ECA for normal and oblique incidence cases are provided. Fig. 12. Transmission coefficient versus frequency and θ in for TM polarized impinging wave and ϕ = 90 incidence plane for FSS and ET prototypes. (a) Experiment-FSS. (b) Experiment-ET. (c) CST-FSS. (d) CST-ET. (e) Equivalent Circuit-FSS. (f) Equivalent Circuit-ET. results [Fig. 12(c)] come from te angular step considered, and te subsequent interpolation. Tis artifact does not appear for te equivalent circuit model due to te smaller angular step tat is taken. Regarding te ET design [Figs. 12(b), (d), and (f)], again very low transmission levels are obtained. Neverteless, te experimental results still keep a good agreement wit te simulations and EC results for te most important features. In fact, te diamond-sape regions delimited by te nulls are present in bot simulations and experiments. V. CONCLUSIONS A 2-D periodic structure based on a metallic screen wit coaxial annular apertures and backed wit a dielectric substrate as been analyzed. Te analysis includes free-standing structures and parametric studies of te lattice period and dielectric slab eigt. A special empasis as been put on accurately setting te terminology wen referring to structures aving eiter a frequency selective surface (FSS) or an extraordinary transmission (ET) operation regime. In tis way, two different designs working at different operation regimes (FSS and ET) A. Normal Incidence Given te field aperture of Eq. 2 te following expressions for N are found [41]: N TE = 2π k xn sin(ϕ 0 ) + k ym cos(ϕ 0 ) k ρ, N TM = 2π k xn cos(ϕ 0 ) + k ym sin(ϕ 0 ) k ρ, J 0(k ρ, b) + J 0 (k ρ, a) k 2 ρ, J 0(k ρ, b) J 0 (k ρ, a) + k ρ, [bj 1 (k ρ, b) aj 1 (k ρ, a)] k 2 ρ, (6) wit (5) k xn = k 0 sin θ cos ϕ + 2πn d x (7) k ym = k 0 sin θ sin ϕ + 2πm (8) d y k ρ, = k t, = kxn 2 + kym 2 (9) were k 0 is te vacuum wavenumber, corresponds to a nm armonic, J ν (x) is te Bessel function of order ν and

12 JOURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, MONTH YEAR 11 Fig. 13. (a) Equivalent network for a 00 impinging armonic. (b) Equivalent network for 00 impinging armonic in wic te equivalent admittance is categorized into lo and o modes. argument x and θ and ϕ correspond to te elevation and azimut angle respectively. Te inclusion of θ and ϕ allow us to present te equations in a general manner. For te particular case of normal incidence treated ere, we simply set θ = 0 and ϕ = 0 or ϕ = 90. Wen θ 0, te only modification tat is required is to substitute te perfect electric or magnetic (depending on te polarization) walls of te virtual waveguide by periodic boundary conditions instead. Now, te corresponding N turn ratios for TE and TM armonics can be directly introduced into te equations governing te circuit in Fig. 13. Te admittance in parallel labeled as Y eq (in) accounts for te infinite summation of te modal admittances of all te armonics (bot TM and TE) excluding te impinging mode (typically te TE 00 /TM 00 mode). In tis case, te parallel admittance, Y eq (in), can be written as te summation of te admittance for eac mode looking at te left, Y (L) (R) in,, and rigt, Y in,, sides of te circuit as follows: were Y (in) eq Y (R) in, = Y (1) = N 2 [Y (L) in, + Y (R) in, ] (10) Y (L) in, = Y (0) (11) Y (0) + jy (1) tan(β (1) s) Y (1) + jy (0) tan(β (1) s). (12) Note tat for a free standing structure Y (R) in, = Y (0). Te modal admittances Y (i) are given by were Y (i) = 1 η { β /k (i) (i) k (i) /β TE armonics TM armonics (13) k (i) = ε (i) r k 0 (14) β (i) η (i) = = ε (i) η 0 ε (i) r r k 2 0 k2 ρ, (15) (16) wit η 0 = µ 0 /ε 0 being te vacuum wave impedance and ε (i) r te relative permittivity of te medium (i). As explained in [20], a convenient distinction between low order (lo) and ig order (o) modes sould be done. Te former group, lo modes, are tose in propagation (or evanescent but close to cut-off) at te frequency range of interest. Terefore, teir explicit expression keeping teir distributed nature (frequency dependent admittance) is taken. On te oter and, te contribution of ig order TM/TE modes can be seen as regular capacitors/inductors, wic are frequency independent. Tis fact allows us to save computational resources wen performing a frequency sweep given tat te circuit elements associated wit ig order modes are computed only once. It sould be noted tat te presence of a layered dielectric environment is explicitly taken into account for bot low and ig order modes [see Fig. 13(b)]. Te final expression for te equivalent input admittance yields wit Y eq (in) = N 2 [Y (L) in, + Y (R) in, ] + jωc 1 o + (17) M jωl o C o = 1 = L o N TM 2 M+1 ε 0 + ε 0ε (1) r ε (0) r + ε (1) r tan(k ρ,o s ) k ρ, k ρ, ε (1) r + ε (0) r tan(k ρ,o s ) M+1 (18) 2µ 0 N TE 2. (19) k ρ, As it can be seen from (17)-(19), te mode categorization is determined by te factor M wic sets an upper limit for low order modes (in most practical cases M is a small integer number). In addition, te infinite series sown in (18) and (19) can be truncated by setting an upper bound N wic fixes te maximum number of evanescent modes considered. Once all te circuital parameters are known, one may apply classical network teory to obtain te scattering parameters of te equivalent circuit [44]. B. Oblique Incidence Provided te field profile in Eq. 3 tat caracterizes te TEM mode field distribution, te transformer turn ratio for tis case can be are evaluated as follows [42]: N TEM = j 2π kρ, 2 [ a b J 0 (k ρ, ρ)dρ k ρ, bj 0 (k ρ, b) + k ρ, aj 0 (k ρ, a)]. (20) Te first term in (20) is evaluated by using Struve functions as it can be found in [53]. It sould be pointed out tat due to symmetry conditions, te TEM mode can only be excited at oblique incidence under TM polarization. Even under suc condition, te degree of excitation of te TEM mode can be very low depending on te geometrical parameters of te perforated screen. PRU:Ya se dice en el texto principal.

13 JOURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, MONTH YEAR 12 Fig. 14. Equivalent circuit model taking into account two independent spatial field profiles for caracterizing oblique incidence under TM polarization. Te inclusion of an additional mode in te assumed aperture field profile leads to a different circuit topology tan te one presented in te previous section. Te equivalent circuit presented in Sec. A-A uses a single term for caracterizing te field profile at te aperture. An ECA tat takes an aperture field profile wit two terms can be derived in order to enlarge te upper frequency bound of te model in metallic patc arrays and extend te approac to unit cell containing more tan one scatterer [40]. Tis approac can also be applied to model te TM oblique incidence of a single coaxial annular aperture. One term caracterizes te field profile of te TE 11 mode and te oter one te TEM mode. Te corresponding equivalent circuit topology is sown in Fig. 14, were te Y 11 and Y 22 admittances account, respectively, for te first and second mode considered, and Y 12 for te coupling between bot modes. Tese admittances can be evaluated as Y ij = N i, N j, N i,00 N j,00 [Y (L) in, + Y (R) in, ] (21) were (i, j) = 1, 2 and te same considerations outlined in Sec. A-A apply ere (including te LO/HO mode distinction). Again, te scattering parameters can be evaluated by applying classical network teory. REFERENCES [1] T. W. Ebbesen, H. J. Lezec, H. Gaemi, T. Tio, and P. Wolff, Extraordinary optical transmission troug sub-wavelengt ole arrays, Nature, vol. 391, no. 6668, pp , Feb [2] E. Betzig, A. Lewis, A. Harootunian, M. Isaacson, and E. Kratscmer, Near field scanning optical microscopy (NSOM): development and biopysical applications, Biopys. J., vol. 49, no. 1, pp , Jan [3] R. W. Wood, On a remarkable case of uneven distribution of ligt in a diffraction grating spectrum, Pilos. 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14 JOURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, MONTH YEAR 13 [33] P. Banzer, J. K. (née Müller), S. Quabis, U. Pescel, and G. Leucs, Extraordinary transmission troug a single coaxial aperture in a tin metal film, Opt. Express, vol. 18, no. 10, pp , May [34] G. Si, Y. Zao, H. Liu, S. Teo, M. Zang, T. Jun Huang, A. J. Danner, and J. Teng, Annular aperture array based color filter, Appl. Pys. Lett., vol. 99, no. 3, Jul [35] Z. Wei, J. Fu, Y. Cao, C. Wu, and H. Li, Te impact of local resonance on te enanced transmission and dispersion of surface resonances, Potonics Nanostruct. Fundam. Appl., vol. 8, no. 2, pp , May [36] Z. Wei, Y. Cao, Y. Fan, X. Yu, and H. Li, Broadband transparency acieved wit te stacked metallic multi-layers perforated wit coaxial annular apertures, Opt. Express, vol. 19, no. 22, pp , Oct [37] V. Lomakin, S. Li, and E. 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Dover Publications, Pablo Rodríguez-Ulibarri was born in Pamplona, Spain, in He received te M.Sci. degree in Telecommunication Engineering, and M.Res. degree in Communications from Universidad Pública de Navarra, Spain, in 2010, and 2013, respectively. From June 2011 to June 2013, e worked as a Researc Assistant in te Electrical and Electronic Engineering Department, Universidad Pública de Navarra. From August 2013 to June 2014 e worked as a Researcer in te Fraunofer Institute of Hig Frequency and Radar in Wactberg, Germany. Since July 2014 e is working towards te P.D. in te Antennas Group - TERALAB, Universidad Pública de Navarra. He was a Visiting researcer at Universidad de Sevilla from September 2016 to December His current researc interests are focused on metamaterials, antennas, frequency selective surfaces and advanced devices ranging from microwave to teraertz frequencies. Miguel Navarro-Cía (S 08-M 10-SM 15) was born in Pamplona, Spain, in He received te M.Sci. and P.D. degrees in Telecommunication Engineering, and M.Res. degree in Introduction to Researc in Communications from Universidad Pública de Navarra, Spain, in 2006, 2010 and 2007, respectively. From September 2006 to January 2010, and from February 2010 until Marc 2011, e worked as a Predoctoral Researcer (FPI fellowsip recipient) and a Researc & Teacing Assistant in te Electrical and Electronic Engineering Department, Universidad Pública de Navarra, respectively. He was a Researc Associate at Imperial College London and University College London in 2011 and 2012, respectively, and a Junior Researc Fellow at Imperial College London from December 2012 until November Currently e is a Birmingam Fellow in te Scool of Pysics and Astronomy, University of Birmingam. He is also affiliated as a Visiting Researcer wit Imperial College London and University College London. He worked as Visiting Researcer at University of Pennsylvania for 3 monts in 2010, at Imperial College London in 2008, 2009 and 2010 for 4, 6 and 3 monts, respectively, and at Valencia Nanopotonics Tecnology Center for 2 monts in His current researc interests are focused on plasmonics, near-field time-domain spectroscopy/microscopy, metamaterials, antennas and frequency selective surfaces at millimeter-wave, teraertz and infrared. Dr. Navarro-Cía is a Senior Member of te Optical Society of America (OSA), and a Member of Te Institute of Pysics (IOP). He was awarded te Best Doctoral Tesis in Basic Principles and Tecnologies of Information and Communications, and Applications corresponding to te XXXI Edition of Awards Telecommunication Engineers 2010, and twice te CST University Publication Awards for te best international journal publication using CST Microwave Studio TM (in 2012 and 2016) and was recipient of te 2011 Junior Researc Raj Mittra Travel Grant. PLACE PHOTO HERE Raúl Rodríguez-Berral was born in Casarice, Seville, Spain, in He received te M.Sc. (Licenciado) and P.D. degrees in pysics from te University of Seville, Seville, Spain, in 2001 and 2008, respectively. In January 2002, e joined te Department of Applied Pysics 1, University of Seville, were e is currently an Associate Professor. His researc interests include te study of te spectrum and te excitation of periodic and non periodic planar structures and ig-frequency circuit modeling.

15 JOURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, MONTH YEAR 14 PLACE PHOTO HERE Francisco Mesa (M 93-SM 11-F 14) was born in Cádiz, Spain, in April He received te Licenciado and Doctor degrees in pysics from te Universidad de Sevilla, Seville, Spain, in 1989 and 1991, respectively. He is currently a Professor wit te Departamento de Física Aplicada 1, Universidad de Sevilla, Seville, Spain. His researc interests include electromagnetic propagation/radiation in planar structures. PLACE PHOTO HERE Francisco Medina (M 90-SM 01-F 10) was born in Puerto Real, Cádiz, Spain, in November He received te Licenciado and Doctor degrees in pysics from te University of Seville, Seville, Spain, in 1983 and 1987 respectively. He is currently a Professor of Electromagnetism wit te Department of Electronics and Electromagnetism, University of Seville, and Head of te Microwaves Group. He as co-autored more tan 130 journal papers and book capters on tose topics and more tan 260 conference contributions. His researc interests include analytical and numerical metods for planar structures, anisotropic materials, and artificial media modeling. Dr.Medina acts as Reviewer for more tan 40 IEEE, IET, AIP, and IOP journals and as been a member of te TPCs of a number of local and international conferences. Miguel Beruete was born in Pamplona, Spain, in He received te M.Sci. and P.D. degrees in telecommunication engineering, from te Public University of Navarre (UPNA), Navarre, Spain, in 2002 and 2006, respectively. From September 2002 to January 2007, e was working as Predoctoral Researcer (FPI fellowsip recipient) in te Electrical and Electronic Engineering Department, Public University of Navarre. From January to Marc 2005 e worked as visiting researcer at te University of Seville, as a part of is doctoral researc. From February 2007 to September 2009 e was at te electronics department of te tecnological center CEMITEC in Noain (Navarre), developing, designing and measuring ig frequency communication devices. In September 2009 e joined te TERALAB at UPNA, as a postdoc Ramón y Cajal fellow researcer under te supervision of Prof. Mario Sorolla. In Marc 2014, e joined te Antennas Group-TERALAB of UPNA. He worked in tis group as a Distinguised Researcer from January to Marc 2017, and since Marc 2017 e is Assitant Lecturer were e supervises several P.D. and M.Sci. Teses and leads te TERALAB laboratory. Dr. Beruete as autored more tan 120 JCR articles, 4 book capters, 3 patents, near 250 conference communications and acts as a reviewer for more tan 40 international journals. His researc interests are directed towards teraertz sensing and communication tecnology, including metamaterials, plasmonics, extraordinary transmission structures, leaky-wave antennas, nanoantennas, and in general quasioptical devices. Dr. Beruete was awarded te P.D. Prize from te Public University of Navarre ( ) for te best Doctoral Tesis in te year , tree CST University Publication Awards for te best international journal publication using CST in te years 2005, 2012 and 2016, te XII Talgo Award of Tecnological Innovation in 2011 and several awards in international conferences.