Hutton, Jeremy M

Kurtz, Paul Michael

Morrow, Amanda R

English

Ghent University Academic Bibliography

A Perfectly Matched Layer Based Technique for theScattering from 1-D Periodic Microstrip StructuresDries Vande Ginste #1 and Hendrik Rogier #2# Electromagnetics Group, Department of Information Technology, Ghent UniversitySint-Pietersnieuwstraat 41, B-9000 Gent, Belgium1 dries.vandeginste@intec.UGent.be2 hendrik.rogier@intec.UGent.beAbstract—An efficient technique is presented to computethe scattering from one-dimensional (1-D) periodic microstripstructures, illuminated by a plane wave under perpendicularincidence. The technique relies on a Mixed Potential IntegralEquation (MPIE), discretized by the Method of Moments (MoM),solving for the unknown current density flowing within a unitcell of the periodic structure. The pertinent 1-D periodic Green’sfunctions are obtained by invoking the Perfectly Matched Layer(PML)-paradigm. The proposed formalism is illustrated andvalidated by evaluating the scattering from a 1-D periodicmicrostrip patch array.I. INTRODUCTIONThe study of one-dimensional (1-D) and two-dimensional (2-D) periodic metallizations residing inlayered background media, e.g. microstrip substrates, hasbeen a research topic for many years [1]–[3]. The rangeof applications comprise antenna arrays, electromagneticbandgap structures, frequency selective surfaces, leaky-waveantennas, high impedance metamaterial surfaces, etc.The periodic character of these structures can be exploitedby invoking the Floquet-Bloch theorem, allowing to restrictthe analysis of the complete structure (with infinite extent) toone representative unit cell. Consequently, simulation methodsoften rely on a boundary integral equation combined with theMethod of Moments (MoM) [4], solving for the unknowncurrent density flowing within a unit cell of the periodicstructure [2]. This is also the case in this contribution. Atthe crux of the formalism are the pertinent 1-D periodicGreen’s functions. Unfortunately, these Green’s functions arenot (immediately) available in closed form. Classically, thecalculation of layered medium Green’s functions necessitatesthe time-consuming numerical evaluation of Sommerfeld-typeintegrals. In [2], the 1-D periodic Green’s functions in thespatial domain are obtained by means of an efficient sum of in-verse Fourier transforms. Here, a different approach is adopted,based on the Perfectly Matched Layer (PML)-paradigm [5].This paradigm allows to convert the open microstrip structureinto a closed waveguide configuration, leading to closed-formseries representations of the pertinent 1-D periodic Green’sfunctions.The outline of this paper is as follows. In Section II, thenew formalism is detailed. This new technique is illustratedand validated in Section III. Conclusions are summarized inSection IV.In the sequel, all sources and fields are assumed to betime harmonic with angular frequency ω and time dependen-cies ejωt are suppressed. Also, transverse to z restrictions ofvectors v are denoted v̌ ≡ vxx̂ + vyŷ = −ẑ × [ẑ × v]; herex̂, ŷ, and ẑ are unit Cartesian vectors.II. DESCRIPTION OF THE TECHNIQUEA. 1-D periodic microstrip geometry......EsEPWd εr, µrPEC x̂ŷẑMbbbbbunit cell SmucFig. 1. 1-D periodic microstrip structure, illuminated by a plane wave EPWunder perpendicular incidence, inducing a scattered field Es.Consider the 1-D periodic microstrip structure of Fig. 1.This structure consists of a dielectric layer with thickness d,relative permittivity εr, and relative permeability µr, placedon a perfect electrically conducting (PEC) ground plane. A1-D periodic PEC metallization M resides at the substrate-airinterface z = d. This metallization M is periodic along the x-direction and the period is b. In the example of Fig. 1, one unitcell consists of three square patches. A plane wave EPW(r ≡xx̂ + yŷ + zẑ) = E0ejk0z , with wavenumber k0 = ω/c andwhere c is the speed of light, impinges perpendicularly uponthe substrate. This plane wave causes an incident field Ei(r),which in turn induces unknown current densities J̌(ρ ≡ xx̂+yŷ + dẑ) flowing on M. In response, a scattered field Es(r)is generated. The total electric field Et(r) comprises both theincident and the scattered fields, viz. Et(r) = Ei(r) + Es(r).B. Construction of the MPIE-MoMUpon knowledge of the pertinent 1-D periodic layeredmedium Green’s function GperA (ρ|ρ′) for the magnetic vectorpotential and GperV (ρ|ρ′) for the electric scalar potential,a Mixed Potential Integral Equation (MPIE) is constructedby demanding that the total electric field tangential to themetallization M vanishes:−Ěi(ρ) = Ěs,r(ρ)≡ −jω∫∫MmGperA (ρ|ρ′)J̌(ρ′) dρ′+1jω∇̌∫∫MmGperV (ρ|ρ′)(∇̌′ · J̌(ρ′))dρ′,∀ρ ∈ Mm, (1)with ∇̌ = ∂∂x x̂ + ∂∂y ŷ and where Mm is that part ofthe metallization that lies within one unit cell Smuc ={ρ ≡ xx̂ + yŷ + dẑ : mb ≤ x < (m + 1)b,−∞ < y < ∞},m ∈ Z. The integral equation (1) can be solved by theMoM. Thereto, the metallization Mm is approximated bya (potentially nonuniform) rectilinear mesh with N interioredges. Next, J̌(ρ) is expanded into a set of vector rooftop basisfunctions [6] w̌j(ρ) = wx,j(ρ)x̂ + wy,j(ρ)ŷ, j = 1, . . . , N ,with support Mmj ⊂ Mm comprising two patches that arejoined by the mesh’s jth interior edge:J̌(ρ) =N∑j=1Ijw̌j(ρ). (2)Inserting (2) into (1) and applying a Galerkin testing procedureyields an N × N linear systemV = Z · I. (3)In (3), the N -vector I contains the unknown expansion coeffi-cients Ij , j = 1, . . . , N , and the elements of the N -vector Vand the N × N matrix Z are given byVi =∫∫MmiĚi(ρ) · w̌i(ρ) dρ, (4)Zij = jω∫∫Mmi∫∫MmjGperA (ρ|ρ′) (w̌i(ρ) · w̌j(ρ′)) dρ′ dρ+1jω∫∫Mmi∫∫MmjGperV (ρ|ρ′)[(∇̌ · w̌i(ρ))(∇̌′ · w̌j(ρ′))] dρ′ dρ. (5)The linear system (3) is solved using direct or iterative solutionschemes.C. PML-based 1-D periodic Green’s functionsAs stated before, the MPIE-MoM formalism requires theknowledge of GperA (ρ|ρ′) and GperV (ρ|ρ′). These 1-D periodicGreen’s functions can be determined through application ofthe PML-paradigm, described in [5], [7]. To this end, thesemi-infinite layer of air z > d is closed by a PEC-backedPML, which corresponds to placing a PEC-plate at a complexdistance z = d + D above the microstrip configurationairdDεr, µrPECPECPMLx̂ ŷẑFig. 2. Frontal view of a PEC-backed PML-closed microstrip substrate.(Fig. 2). The complex value of D must be chosen such thatthe PML provides damping along the direction perpendicularto the layer interfaces, i.e. the direction z. This damping isachieved when the real part (D) > 0 and the imaginarypart (D) < 0. Due to the reflectionless absorption ofwaves impinging upon the PML, the PEC-backed PML-closedmicrostrip structure closely mimics the behavior of the originalopen structure. However, whereas the open structure’s modalspectrum contains a continuous set of radiation modes, ne-cessitating the numerical evaluation of Sommerfeld-integrals,the closed structure only exhibits discrete sets of eigenmodes.Consequently, the Green’s functions can be written as closed-form series expansions:GperA (ρ|ρ′) = −j2∞∑n=1+∞∑m=−∞H(2)0 (βTE,n∆m)MTE(βTE,n), (6)GperV (ρ|ρ′) = −jω22∞∑n=1+∞∑m=−∞H(2)0 (βTE,n∆m)β2TE,nMTE(βTE,n)+j2∞∑n=1+∞∑m=−∞H(2)0 (βTM,n∆m)β2TM,nMTM(βTM,n). (7)Here, ∆m =√(x − x′ − mb)2 + (y − y′)2, H(2)0 (·) is thezeroth-order Hankel function of the second kind, andMTE(β) =dµ11sin2 γ1d− cot γ1dµ1γ1+Dµ01sin2 γ0D− cot γ0Dµ0γ0, (8)MTM(β) =ε1 cot γ1dγ31+ε1dγ21 sin2 γ1d+ε0 cot γ0Dγ30+ε0Dγ20 sin2 γ0D, (9)with γ0 =√k20 − β2 and γ1 =√k20εrµr − β2. βTE,n andβTM,n represent the propagation constants of the TransverseElectric (TE) and the Transverse Magnetic (TM) polarizedeigenmodes of the PEC-backed PML-closed waveguide. Allthese propagation constants have a negative imaginary part,and hence, only a limited set of modes needs to be retained inthe summations. Via the Poisson summation, (6) and (7) areconverted intoGperA (ρ|ρ′)= −jb∞∑n=1+∞∑m=−∞e−j2πmb (x−x′)e−j√β2TE,n−( 2πmb )2|y−y′|MTE(βTE,n)√β2TE,n −(2πmb)2 ,(10)GperV (ρ|ρ′)= −jω2b∞∑n=1+∞∑m=−∞e−j2πmb (x−x′)e−j√β2TE,n−( 2πmb )2|y−y′|β2TE,nMTE(βTE,n)√β2TE,n −(2πmb)2+jb∞∑n=1+∞∑m=−∞e−j2πmb (x−x′)e−j√β2TM,n−( 2πmb )2|y−y′|β2TM,nMTM(βTM,n)√β2TM,n −(2πmb)2 .(11)Both kinds of series are useful. On the one hand, series (6)and (7) are required to calculate interactions Zij (5) whenever|y − y′| is small. On the other hand, series (10) and (11)converge very rapidly for large enough |y − y′| and thisproperty is exploited in the implementation of the new 1-Dperiodic MPIE-MoM. For m = 0, the series (6) and (7)reduce to the PML-based series for non-periodic microstripsubstrates [8] and it is known that these series become im-practical for ∆0 = |ρ − ρ′| → 0. This issue can be easilyalleviated by replacing the term m = 0 in the summation withthe classical Sommerfeld-integral based non-periodic Green’sfunction. This yields alternative series, valid for small |ρ−ρ′|,leading to very accurate calculations of the self-patch interac-tions Zii, i = 1, . . . , N .III. NUMERICAL EXAMPLE AND VALIDATIONThe above described PML-based MPIE-MoM is now illus-trated and validated by considering a representative example.The results are obtained using a Linux-based 64-bit AMDOpteron 270 computer with 8 GB of RAM running at 2 GHz.A BiCGstab iterative solver is used to solve linear system (3).Again consider the configuration of Fig. 1, where themicrostrip substrate has a thickness d = 3.17 mm, a relativepermittivity εr = 11.7, and a relative permeability µr = 1.The metallization is the patch array shown in Fig. 1. Eachof the three PEC patches in the unit cell have a sizeof 10 mm × 10 mm. The period is b = 40 mm and the spac-ing between the patches along the y-direction is 10 mm. Thestructure is illuminated by a plane wave EPW(r) = ejk0zŷat an angular frequency of ω = 2π10 GHz. The currentdensity J̌(ρ), induced by the plane wave, is shown in Fig. 3.Obviously, as the plane wave’s electric field is y-oriented,the y-component of the current density (Fig. 3(b)) is muchlarger than its x-component (Fig. 3(a)). Our approach is nowvalidated by considering the far-field Es,ff(r) scattered fromthe antenna array, expressed asEs,ff(r) ≈(Fθ(θ, φ)θ̂ + Fφ(θ, φ)φ̂) e−jk0RR. (12)  −65−60−55−50−45−40−3520log|J̌(ρ)·x̂|[dB](a) x-oriented current density  −65−60−55−50−45−40−3520log|J̌(ρ)·ŷ|[dB](b) y-oriented current densityFig. 3. Current density within one unit cell of the patch array.In (12) the usual spherical coordinate systemis used: x = R sin θ cos φ, y = R sin θ sin φ,z = R cos θ, θ̂ = cos θ cos φ x̂ + cos θ sin φ ŷ − sin θ ẑ,and φ̂ = − sin φ x̂ + cos φ ŷ. The far-field pattern’scomponents Fθ(θ, φ) and Fφ(θ, φ) are calculated basedon the current density on the metallization using thestationary phase method [9]. After solving the periodicproblem, to calculate the scattering pattern, here, we onlyconsider seven unit cells, with seven identical current densitiesas shown in Fig. 3, spaced b = 40 mm apart from eachother. The far-field, scattered from these seven unit cells, isshown in Fig. 4. The results are compared with the solutionof a reference program, the SVD-PML-MLFMA [10]. Thisis a tool developed for the simulation of finite, non-periodicstructures. With the SVD-PML-MLFMA a finite array of7 × 3 patches is simulated. Although only seven periodsare used, good agreement is already observed, as suchvalidating the approach described in this contribution. InFig. 4(a), apart from the specular reflection at θspec = 0◦,two grating lobes at angles θgr,1 = arcsin 2πbk0 = 48.5◦and θgr,2 = arcsin −2πbk0 = −48.5◦ are observed. In thexz-plane (Fig. 4(a)), the θ-component of the scatteredfar-field is very small. In the yz-plane (Fig. 4(b)), there is nosignificant φ-component.Notwithstanding the fact that the SVD-PML-MLFMA isa MoM-based simulation tool that is hugely accelerated bymeans of a multilevel fast multipole technique, the 1-D peri-odic MPIE-MoM approach is still faster. Both solvers are setto reach 6 digits of accuracy (leading to the results presented inFig. 4). The total solution time using the 1-D periodic MPIE-MoM was 42.60 sec. The solution time of the SVD-PML-−80 −60 −40 −20 0 20 40 60 80−120−100−80−60−40−200  θ [◦]20log 10|Fφ(θ,φ)|[dB]1-D Periodic MPIE-MoMSVD-PML-MLFMA [10](a) Far-field component Fφ(θ, 0◦) in the xz-plane−80 −60 −40 −20 0 20 40 60 80−120−100−80−60−40−200  θ [◦]20log 10|Fθ(θ,φ)|[dB]1-D Periodic MPIE-MoMSVD-PML-MLFMA [10](b) Far-field component Fθ(θ, 90◦) in the yz-planeFig. 4. Scattered far-field for the configuration of Fig. 1. The 1-D periodicMPIE-MoM results (full line) are calculated by applying the stationary phasemethod to seven identical current densities as shown in Fig. 3, spaced b =40 mm from each other. The reference results (circles), calculated using theSVD-PML-MLFMA [10], correspond to the scattered far-field of a finite arrayof 7 × 3 patches.MLFMA was 2387.90 sec. Obviously, the speed-up factorof 56 is obtained because the matrix Z (3) of the 1-D periodicMPIE-MoM is 7× 7 = 49 times smaller than the SVD-PML-MLFMA’s matrix.From the unit cell solution for the current density, otherrapid analyzes can be made. For example, when increasing thenumber of periods from 7 to 100, it is seen that the scatteringpattern more and more resembles discrete Dirac-like lobes atthe angles θspec, θgr,1, and θgr,2 (Fig. 5). This new result isobtained without having to recompute the current density, thisin contrast to classical methods for finite patch arrays, such asthe SVD-PML-MLFMA.IV. CONCLUSIONTo compute the scattering from 1-D periodic microstrippatch arrays, the PML-paradigm is invoked, leading to closed-form 1-D periodic Green’s functions. These Green’s functions−80 −60 −40 −20 0 20 40 60 80−120−100−80−60−40−200  θ [◦]20log 10|Fφ(θ,φ)|[dB]1-D Periodic MPIE-MoM: 100 unit cellsFig. 5. Scattered far-field component Fφ(θ, 0◦) in the xz-plane for theconfiguration of Fig. 1, calculated by applying the stationary phase method to100 identical current densities as shown in Fig. 3, spaced b = 40 mm fromeach other.form the basis of an MPIE-MoM formalism, allowing todetermine the currents flowing on 1-D periodic metallizationsresiding on a microstrip substrate. Application of a stationaryphase method yields the scattered far-field induced by thesecurrents. The method is implemented and tested by comparingit with the scattered far-field from a finite array of patches,showing good agreement and high efficiency.REFERENCES[1] J. McVay, N. Engheta, and A. Hoorfar, “High impedance metamaterialsurfaces using Hilbert-curve inclusions,” IEEE Microwave WirelessComp. Lett., vol. 14, no. 3, pp. 130–132, March 2004.[2] P. Baccarelli, C. Di Nallo, S. Paulotto, and D. R. Jackson, “A full-wave numerical approach for modal analysis of 1-D periodic microstripstructures,” IEEE Trans. Microwave Theory Tech., vol. 54, no. 4, pp.1350–1362, Apr. 2006.[3] P. Baccarelli, S. Paulotto, and C. Di Nallo, “Full-wave analysis of boundand leaky modes propagating along 2D periodic printed structures witharbitrary metallisation in the unit cell,” IET Microw. Antennas Propag.,Special Issue on Metamaterials EBG, vol. 1, no. 1, pp. 217–225, 2007.[4] R. F. Harrington, Field computation by moment methods. Piscataway,NJ: IEEE Press, 1993.[5] H. Rogier, “New series expansions for the 3D Green’s function of multi-layered media with 1D periodicity based on perfectly matched layers,”IEEE Trans. Microwave Theory Tech., vol. 55, no. 8, pp. 1730–1738,Aug. 2007.[6] J. Sercu, N. Faché, F. Libbrecht, and P. Lagasse, “Mixed potentialintegral equation technique for hybrid microstrip-slotline multilayeredcircuits using a mixed rectangular-triangular mesh,” IEEE Trans. Mi-crowave Theory Tech., vol. 43, no. 5, pp. 1162–1172, May 1995.[7] F. Olyslager, “Discretization of continuous spectra based on perfectlymatched layers,” SIAM J. Appl. Math., vol. 64, no. 4, pp. 1408–1433,May 2004.[8] D. Vande Ginste, L. Knockaert, and D. De Zutter, “Error control inthe perfectly matched layer based multilevel fast multipole algorithm,”Journal of Computational Physics, vol. 228, no. 13, pp. 4811–4822, July2009.[9] L. B. Felsen and N. Marcuvitz, Radiation and scattering of waves.Piscataway: IEEE Press, 1994.[10] D. Vande Ginste, E. Michielssen, F. Olyslager, and D. De Zutter, “Ahigh-performance upgrade of the perfectly matched layer multilevelfast multipole algorithm for large planar microwave structures,” IEEETransactions on Antennas and Propagation, vol. 57, no. 6, pp. 1728–1739, June 2009.

The morphological development of the 3 m. sg. pronominal suffix on plural nouns in classical Hebrew, part 2

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The morphological development of the 3 m. sg. pronominal suffix on plural nouns in classical Hebrew, part 2

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