A circular cylindrical metallic resonator half filled with DPS material and half with DNG metamaterial is analyzed, in the frequency domain. The two materials are linear, lossless, homogeneous, and anti-isorefractive to each other. The electric field is assumed to be parallel to the cylinder axis. It is shown that the resonator performs independently of diameter size. Numerical results are presented and discussed for a resonator excited by a line source parallel to the axis

Daniele V.G.

Graglia R.D.

Lombardi G.

Uslenghi P.L.E.

English

DANIELE, Vito

GRAGLIA, Roberto

LOMBARDI, Guido

Uslenghi P. L. E.

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

04 August 2020POLITECNICO DI TORINORepository ISTITUZIONALESize-independent cylindrical resonator half-filled with DNG metamaterial / Daniele V.G.; Graglia R.D.; Lombardi G.;Uslenghi P.L.E.. - 1(2013), pp. 295-297. ((Intervento presentato al convegno 2013 International Symposium onElectromagnetic Theory (EMTS 2013) tenutosi a Hiroshima, Japan nel May 20-24, 2013.OriginalSize-independent cylindrical resonator half-filled with DNG metamaterialPublisher:PublishedDOI:Terms of use:openAccessPublisher copyright(Article begins on next page)This article is made available under terms and conditions as specified in the  corresponding bibliographic description inthe repositoryAvailability:This version is available at: 11583/2518675 since:IEEE - INST ELECTRICAL ELECTRONICS ENGINEERS INCSize-independent Cylindrical ResonatorHalf-filled with DNG MetamaterialV.G. Daniele #1 , R. D. Graglia #2, G. Lombardi #3, P. L. E. Uslenghi 4# Dipartimento di Elettronica e Telecomunicazioni, Politecnico di TorinoCorso duca degli Abruzzi 24, Torino, Italy1 vito.daniele@polito.it2 roberto.graglia@polito.it3 guido.lombardi@polito.it∗ Department of ECE, University of Illinois at Chicago851 S. Morgan M/C 154, Chicago, IL, USA4 uslenghi@uic.eduAbstract—A circular cylindrical metallic resonator half filledwith DPS material and half with DNG metamaterial is analyzed,in the frequency domain. The two materials are linear, lossless,homogeneous, and anti-isorefractive to each other. The electricfield is assumed to be parallel to the cylinder axis. It is shown thatthe resonator performs independently of diameter size. Numericalresults are presented and discussed for a resonator excited by aline source parallel to the axis.I. INTRODUCTIONA metallic cylindrical resonator half-filled with anti-isorefractive lossless DNG metamaterial is analyzed in thefrequency domain, with time-dependence factor exp(+jωt)omitted throughout. The geometrical configuration of theresonator is independent of the direction along the axis z ofthe cylinder, that is, the cross section of the structure in aplane z = constant is independent of z. Half of the resonatoris filled with a double-positive (DPS) medium characterizedby a real positive electric permittivity ε and a real positivemagnetic permeability µ, while the other half is filled with adouble-negative (DNG) metamaterial characterized by a realnegative permittivity −ε and a real negative permeability −µ.Consequently, the DNG medium has a real negative refractiveindex that is the opposite of the refractive index of the DPSmedium, while the two media have the same real positiveintrinsic impedance. The interface separating the two mediais a plane containing the cylinder axis, so that the resonatorconsists of two equal half cylinders, each filled with one ofthe two media. The electric field is assumed to be parallel tothe cylinder axis, so that the boundary conditions at the endsz = constant of the resonator are satisfied and the resonatorlength plays no role in the analysis.The concept of utilizing DNG metamaterial inclusions torender a resonator size-independent was introduced by En-gheta [1]-[2] for a one-dimensional structure, to show thatit is possible to build resonators that functions independentlyof dimensions at those frequencies for which the metamaterialbehaves as postulated. This concept was later extended to fullythree dimensional cavity resonators by Couture et al. et al. [3]-[4] and by Uslenghi [5]-[7]. Recently, Daniele et al. [8]-[9]studied in detail a cylindrical resonator sectorally filled withmetamaterial; in particular, their analysis showed that phasecompensation leading to size independence is possible onlyif the cylinder is half-filled with metamaterial. This is thecase examined in the present work, in which the excitationis provided by an electric line source located anywhere insidethe DPS half cylinder.II. GEOMETRY OF THE PROBLEMWith reference to rectangular coordinates (x, y, z), themetallic cylindrical resonator has the z axis as symmetryaxis, the inner radius of its circumference in any planez = constant is a, and the length of the resonator in thez direction does not come into play because the resonator isassumed to be excited by an electric line source parallel to itsaxis, leading to an electric field that is everywhere parallel to z,so that the boundary conditions at the two circular bases of thecylinder are always satisfied. The half-cylinder occupying thevolume x < 0 is filled with DPS material characterized by awavenumber k = ω√εµ and an intrinsic impedance Z =√µε .The half-cylinder occupying the volume x > 0 is filled with aDNG material whose wavenumber −k is the opposite of thewavenumber in the DPS region, and whose intrinsic impedancehas the same value Z of the DPS region. The planar interfacex = 0 separates the DPS and DNG regions.With reference to circular cylindrical coordinates (ρ, ϕ, z),the DPS material fills the cross-sectional area (0 ≤ ρ ≤ a,pi/2 ≤ ϕ ≤ 3pi/2), whereas the DNG material fills the cross-sectional area (0 ≤ ρ ≤ a, −pi/2 ≤ ϕ ≤ pi/2). The electricline source J0 located inside the DPS region at ρ0 (ρ = ρ0,ϕ = ϕ0) is assumed to be:J0 = Iδ(ρ− ρo)zˆ = I 1ρoδ(ρ− ρo)δ(ϕ− ϕo)zˆ (1)where I is the intensity (in A), zˆ is a unit vector parallel tothe z axis, and δ is the delta function.A cross section of the resonator in a plane perpendicular toits axis is shown in Fig. 1.Fig. 1. Cross section of the resonatorIII. DESCRIPTION OF THE ANALYSISThe dispersion relation for any homogeneous penetrablewedge is obtained by expressing the solution to the boundary-value problem in terms of circular cylindrical coordinates andimposing the continuity of the tangential electric and magneticfields across the two wedge faces. As shown by Osipov [10],such a relation always results in the product of trigonometricfunctions being equal to zero. In the particular case of a wedgeof semi-aperture angle α, whose faces separate two regions ofspace that are anti-isorefractive to each other, the dispersionrelation takes the simple form (see [9]):sin((pi − 2α)ν) = 0 (2)where ν is a constant arising in the solution of the waveequation by separation of variables. In the present work,α = pi/2, and therefore equation (2) is satisfied for any valueof ν. It follows that the boundary condition on the PEC wallof the resonatorJν(ka) = 0 (3)where Jν is the Bessel function, yields the allowed values ofν for any given frequency, that is, the cylinder resonates at allfrequencies for which the DNG material behaves as postulated,and therefore the resonator can be miniaturized.For a resonator partially filled by a DNG sectoral wedgewhose semiaperture angle α is different from pi/2, the solutionfor line source excitation has been obtained in [9] by employ-ing a Green resolvent technique [11]. However, this techniquefails when α = pi/2 because the characteristic Green functioncannot be defined in such a case. Hence, a different approachis introduced in this work.We introduce an image line source J1 having the samemagnitude but opposite sign of the line source (1), and locatedat the image line ρ1 (ρ = ρ0, ϕ = pi−ϕ0) with respect to theinterface x = 0 separating the DPS and DNG regions:J1 = −Iδ(ρ− ρ1)zˆ (4)Let us designate with E0 = E0zˆ the electric field producedby the line source J0 when the resonator is completely filledwith DPS material, and with E1 = E1zˆ the electric fieldproduced by the image line source J1 when the resonator iscompletely filled with DNG metamaterial.It can be proven that the electric field in the half-filledresonator considered in this work is everywhere given byE = Ez zˆ = E0p0 +E1p1zˆ (5)where p0 and p1 are window functions vanishing in the DNGhalf-cylinder and in the DPS half-cylinder, respectively.Slightly modifying the procedure proposed in [12], the fieldsEn (n = 0 or 1) are given byEn(ρ, ϕ) = jkZIgn(ρ,ρo) (6)where g0 and g1 are:go(ρ, ρo) =12pi∞∑m=0εmgˆom(ρ, ρo) cos[m(ϕ− ϕo)] (7)g1(ρ, ρ1) =12pi∞∑m=0εmgˆ1m(ρ, ρ1) cos[m(ϕ− ϕ1)] (8)where ε0 = 1, εn = 2 for n > 0, andgˆom(ρ, ρo) =piJm(koρ <)Dm(koρ >)2Jm(koa)(9)gˆ1m(ρ, ρ1) =piJm(−koρ <)Dm(−koρ >)2Jm(−koa) (10)withDm(kρ) = Jm(kρ>)Ym(kρ>)− Jm(kρ<)Ym(kρ<) (11)and ρ< (ρ>) is the smaller (larger) between ρ and ρ0. In (9)-(11) Jm and Ym are respectively the Bessel functions of thefirst kind and second kind.The magnetic field components are obtained from the elec-tric field via Maxwell’s equations. It can be verified that theabove solution satisfies all the boundary conditions.Numerical validations and results will be presented at theconference and they will be published in [13].ACKNOWLEDGMENTThis work was sponsored in part by the Italian Ministryof Education, University and Research (MIUR) under PRINgrant 20097JM7YR, and in part by the College of Engineeringin the University of Illinois at Chicago.REFERENCES[1] N. Engheta, “Compact cavity resonators using metamaterials with nega-tive permittivity and permeability,” Proc. Intl. Conf. Electromagnetics inAdvanced Applications (ICEAA’01), Torino, Italy, Sept. 2001, pp. 739–742.[2] N. Engheta, “An idea for thin subwavelength cavity resonators using meta-materials with negative permittivity and permeability,” IEEE AntennasWireless Propag. Lett., vol. 1, pp. 10–13, 2002.[3] S. Couture, A. Parsa and C. Caloz, “Design of a size-independent zero-permittivity rectangular resonator and possible antenna applications,” 14thIntl. Symposium on Antenna Technology and Applied Electromagnetics(ANTEM), Ottawa, Canada, July 2010, pp. 1–4.[4] S. Couture, A. Parsa and C. Caloz, “Size-independent zeroth-order electricplasmonic cavity resonator,” Microwave Opt. Tech. Lett., vol. 53, pp. 927–932, April 2011.[5] P.L.E. Uslenghi, “A miniaturized parallelepipedal cavity resonator,” IEEEAntennas and Wireless Propag. Lett., vol. 9, pp. 807–808, 2010.[6] P.L.E. Uslenghi, “Size-Independent Prism Resonator Partially Filled WithDNG Metamaterial,” IEEE Antennas Wireless Propag. Lett., vol. 10,pp.746–747, 2011.[7] P.L.E. Uslenghi, “A fully three-dimensional size-independent paral-lelepipedal resonator partially filled with DNG metamaterial,” Inter-national Conference on Electromagnetics in Advanced Applications(ICEAA), 2-7 Sept. 2012, Cape Town, South Africa, pp. 11-14, doi:10.1109/ICEAA.2012.6328575[8] V.G. Daniele, R. D. Graglia, G. Lombardi and P. L. E. Uslenghi,“Cylindrical resonators partially filled with a DNG metamaterial sector,”International Conference on Electromagnetics in Advanced Applications(ICEAA), 2-7 Sept. 2012, Cape Town, South Africa, pp. 1295-1297, doi:10.1109/ICEAA.2012.6328828[9] V.G. Daniele, R. D. Graglia, G. Lombardi and P. L. E. Uslenghi,“Cylindrical Resonator Sectorally Filled With DNG Metamaterial andExcited by a Line Source,” IEEE Antennas Wireless Propag. Lett.,, vol.11,pp.1060-1063, 2012, doi: 10.1109/LAWP.2012.2215300[10] A. Osipov, “Analysis of electromagnetic diffraction by wedges withthe method of edge functions,” Proc. 27th URSI General Assembly,Maastricht, The Netherlands, August 17-24, 2002.[11] L.B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves. NewYork: IEEE Press, 1994.[12] V. Daniele, M. Gilli, and R. D. Graglia, “Alternative representation ofdyadic Green’s functions for circular cylindrical cavities with applicationsto the EMC characterization of space station modules,” Radio Sci., vol.35, pp. 621-637, 2000.[13] V.G. Daniele, R. D. Graglia, G. Lombardi and P. L. E. Uslenghi, “Size-independent Cylindrical Resonator Half-filled with DNG Metamaterialand Excited by a Line Source,” IEEE Antennas Wireless Propag. Lett.,to be submitted, 2012

Size-independent cylindrical resonator half-filled with DNG metamaterial

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Size-independent cylindrical resonator half-filled with DNG metamaterial

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