We present an analytical model to study the reflection properties of a multilayered wire media loaded with 2-D arrays of thin imperfect conductors. Based on charge conservation, generalized additional boundary conditions (ABCs) for the interface of two uniaxial wire mediums loaded with thin arbitrary imperfect conductors at the junction are derived. It is observed that by proper selection of the structural parameters, the mushroom structure acts as a wideband absorber for an obliquely incident TM-polarized plane wave. The presented model along with the new ABCs are validated using the full-wave numerical simulations

Glisson, Allen W.

Hanson, George W.

Kaipa, Chandra S.R.

Medina Mena, Francisco

Mesa Ledesma, Francisco Luis

Padooru, Yashwanth R.

Yakovlev, Alexander B.

English

Yashwanth R. Padooru

Alexander B. Yakovlev

Chandra S. R. Kaipa

George W. Hanson

Francisco Medina

Francisco Mesa

Allen W. Glisson

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Generalized additional boundary conditions and analytical model for multilayered mushroom-type wideband absorbers

idUS. Depósito de Investigación Universidad de Sevilla

Generalized Additional Boundary Conditions andAnalytical Model for Multilayered Mushroom-TypeWideband AbsorbersYashwanth R. Padooru∗, Alexander B. Yakovlev∗, Chandra S. R. Kaipa∗, George W. Hanson†,Francisco Medina‡, Francisco Mesa‡and Allen W. Glisson∗∗Dept. of Electrical Engineering, The University of Mississippi, University, MS 38677-1848, USA†Dept. of Electrical Engineering and Computer Science, University of Wisconsin-Milwaukee, Milwaukee, WI 53211, USA‡Dept. of Electronics and Electromagnetism, University of Seville, Seville, 41012, SpainAbstract—We present an analytical model to study the re-flection properties of a multilayered wire media loaded with2-D arrays of thin imperfect conductors. Based on chargeconservation, generalized additional boundary conditions (ABCs)for the interface of two uniaxial wire mediums loaded with thinarbitrary imperfect conductors at the junction are derived. It isobserved that by proper selection of the structural parameters,the mushroom structure acts as a wideband absorber for anobliquely incident TM-polarized plane wave. The presentedmodel along with the new ABCs are validated using the full-wave numerical simulations.I. INTRODUCTIONIn recent years, characterization of metamaterial structures,which constitute wire media has attracted special attentiondue to their ability in enabling anomalous phenomena suchas negative refraction and sub-wavelength imaging, amongothers. It has been shown in [1] that wire media exhibitsstrong spatial dispersion at microwave frequencies, and thatthe constitutive relations between the macroscopic fields andthe electric dipole moment are non-local. In [2]–[5] the role ofspatial dispersion has been discussed and it was demonstratedthat nonlocal homogenized models with additional boundaryconditions (ABCs) become essential in solving electromag-netic problems associated with wire media.In this work we study the reflection properties of a mul-tilayered mushroom structure with thin (resistive) patchesusing an analytical model, which is an extension of theanalysis presented in [5] for a single-layered mushroom.Based on charge conservation two-sided ABCs are derivedat the interface of two uniaxial wire mediums with thinimperfect conductors at the junction. To illustrate the validityof the ABCs, we characterize the reflection properties of themultilayer structure, demonstrating that such a configurationwith proper choice of the geometrical parameters acts as anabsorber. Interestingly, it is noticed that the presence of viasresults in the enhancement of the absorption bandwidth andan improvement in the absorptivity performance for increasingangles of the obliquely incident TM-polarized plane wave.II. ANALYTICAL MODELThe geometry of a multilayered mushroom structure isshown in Fig. 1 (a is the period of the patches and vias). It isassumed that each dielectric layer, perforated with thin losslessmetallic vias of radius rl ≪ a, is homogeneous and isotropicof thickness hl, characterized by relative permittivity εr,l andpermeability of free space, and loaded with 2-D periodic thinconductive patches of conductivity σ2D,l at the interface dl, l =1, 2, . . . ,m. Consider a time-harmonic plane wave incident onFig. 1. Schematic of a multilayer mushroom structure formed by periodicallyloading grounded wire medium with thin metallic square patches.the multilayer mushroom structure as shown in Fig. 1. Eachwire-medium slab is characterized by the nonlocal dielectricfunction [2] εeff,l = ε0εr,l [εxx,l(ω, kx)x̂x̂+ ŷŷ + ẑẑ], whereεxx,l(ω, kx) = 1 − k2p,l/(k2h,l − k2x), kh,l = k0√εr,l is thewavenumber in the host material, k0 is the wavenumber in freespace, kp,l is the plasma wavenumber which depends on theperiod and radius of the vias: k2p,l = (2π/a2)/[ln(a/2πrl) +0.5275], and kx is the x-component of the wave vector k =(kx, 0, kz). Let Jw,l be the currents induced on the metallicwires. It is known that for a TM plane-wave incidence, thewire medium excites both transverse electromagnetic (TEM)and TMx modes, and thus, following [2], the total magneticfields in the air region (x > dm) are given by:η0Hy = eγ0(x−dm) +Re−γ0(x−dm) (1)and in the wire-medium slab (dl−1 < x < dl) are written as,η0H(l)y = A+TM,leγTM,l(x−dl−1) +A−TM,le−γTM,l(x−dl−1)978-1-4673-0462-7/12/$31.00 ©2012 IEEEAuthorized licensed use limited to: Universidad de Sevilla. Downloaded on June 02,2020 at 14:03:49 UTC from IEEE Xplore.  Restrictions apply. brought to you by COREView metadata, citation and similar papers at core.ac.ukprovided by idUS. Depósito de Investigación Universidad de Sevilla+B+TEM,leγTEM,l(x−dl−1) +B−TEM,le−γTEM,l(x−dl−1) (2)where dl = h1 + h2 + . . . + hl, l = 1, 2, . . . ,m, η0 =√µ0/ϵ0 is the free-space impedance, R is the reflection coef-ficient, A±TM,l, B±TEM,l are the amplitude coefficients of the TMand TEM fields, γ0 =√k2z − k20 , εTMxx,l = 1−k2p,l/(k2z +k2p,l),γTEM,l = jk0√ϵr,l, γTM,l =√k2p,l + k2z − k2h,l, and kz =k0 sin θi.To calculate the unknown coefficients, R, A±TM,l, B±TEM,l,we impose the following boundary conditions at x =0, d1, d2, . . . , dm: 1. At the thin metal patch interfaces (x =d±l , l = 1, 2, . . . ,m), the macroscopic two-sided impedanceboundary conditions establish that the tangential electric (E(l)z )and magnetic fields (H(l)y ), can be related via sheetimpedance [4]. 2. At the ground plane interface (x = 0+),we have two boundary conditions [2]: i) tangential macro-scopic total electric field vanishes (E(1)z |x=0+ = 0) and ii)derivative of current is zero (dJw,1(x)/dx|x=0+ = 0), [3].3. At the top patch interface x = d−m, following [5], wehave Jw,m(d−m) + (σ2d,m/jωε0εr,m)(dJw,m(x)/dx)|d−m = 0.However, these conditions are not sufficient to calculate theunknown coefficients. Hence, new ABCs for the microscopiccurrent are derived at the interface of two uniaxial wiremediums with thin metallic patches at the junction resultingin:σ2d,l2jωε0[1εr,ldJw,l(x)dx∣∣∣∣d−l+1εr,l+1dJw,l+1(x)dx∣∣∣∣d+l]+[Jw,l(d−l )− Jw,l+1(d+l )]= 0 (3)1εr,ldJw,l(x)dx∣∣∣∣d−l− 1εr,l+1dJw,l+1(x)dx∣∣∣∣d+l= 0 . (4)Using these boundary conditions (3) and (4), and the onesdiscussed above, we can easily obtain the reflection coefficientR.III. NUMERICAL RESULTS AND DISCUSSIONSIn order to validate the new ABCs, here we study the re-flection properties of the two-layer mushroom structure usingthe analytical model discussed in Section II. The absorberconsists of two resistive patch arrays separated by dielectricslabs perforated with metallic vias, with a ground plane atthe bottom. The parameters of the dielectric slabs used inthe design together with the dimensions and sheet resistivityvalues (Rs1 and Rs2) of the square patches are given in thecaption of Fig. 2. Here Rs is the sheet resistance, whichdepends on the bulk conductivity (σ3D, S/m) of the material,i.e., Rs = 1/σ2D = 1/(σ3Dt), with t being the materialthickness. The analytical results obtained using the new ABCs(3) and (4) agree well with the full-wave HFSS [6] resultsshown in Fig. 2(a). In Fig. 2(b) we present the behavior ofthe reflection coefficient for a similar structure without vias.It is observed that for large angles of incidence there is anenhancement in the absorption bandwidth and an increase inthe absorption with respect to the structure without vias.(a)(b)Fig. 2. Comparison of analytical (solid lines) and full-wave HFSS results(crosses, circles, and plus signs) of the reflection coefficient for the two-layerstructure excited by a TM-polarized plane wave at oblique angles of incidenceθi: (a) with vias and (b) without vias. Structural parameters used in this work(with a general case shown in Fig. 1): h1 = h2 = 3.2mm, εr,1 = 2.2,εr,2 = 1.33, r1 = r2 = 0.05mm, a1 = a2 = 5mm, g1 = g2 = 0.1mm,Rs1 = 196Ω, Rs2 = 1078Ω.IV. CONCLUSIONA simple analytical model has been presented to analyze thereflection properties of multilayered mushroom HIS structureswith thin resistive patches. Additional boundary conditions forthe double-sided junctions of wire media with thin resistivepatches at the interface have been obtained. The new boundaryconditions have been applied and verified for the two-layermushroom-type structure using full-wave simulations.REFERENCES[1] P. A. Belov, R. Marques, S. I. Maslovski, I. S. Nefedov, M. G. Silveir-inha, C. R. Simovski, and S. A. Tretyakov, “Strong spatial dispersion inwire media in the very large wavelength limit,” Phys. Rev. B, vol. 67,2003, 113103(1-4).[2] M. G. Silveirinha, C. A. Fernandes, and J. R. Costa, “Electromagneticcharacterization of textured surfaces formed by metallic pins,” IEEETrans. Antennas Propag., vol. 56, no. 2, pp. 405-415, Feb. 2008.[3] M. G. Silveirinha, C. A. Fernandes, and J. R. Costa, “Additionalboundary condition for a wire medium connected to a metallic surface,”New J. Phys., vol. 10, 2008, 053011(1-17).[4] A. B. Yakovlev, M. G. Silveirinha, O. Luukkonen, C. R. Simovski,I. S. Nefedov, and S. A. Tretyakov, “Characterization of surface-waveand leaky-wave propagation on wire-medium slabs and mushroomstructures based on local and nonlocal homogenization models,” IEEETrans. Microw. Theory Tech., vol. 57, no. 11, pp. 2700-2714, Nov. 2009.[5] A. B. Yakovlev, Y. R. Padooru, G. W. Hanson, A. Mafi, and S. Karbasi,“A generalized additional boundary condition for mushroom-type andbed-of-nails-type wire media,” IEEE Trans. Microw. Theory Tech., vol.59, no. 3, pp. 527-532, Mar. 2011.[6] High Frequency Structure Simulator (HFSS) ver. 12.0, Ansoft Corpora-tion, PA, 2010. [Online]. Available:http://ansoft.comAuthorized licensed use limited to: Universidad de Sevilla. Downloaded on June 02,2020 at 14:03:49 UTC from IEEE Xplore.  Restrictions apply. 

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Generalized additional boundary conditions and analytical model for multilayered mushroom-type wideband absorbers

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