This report presents the progress on activities carried out on the topic of Metamaterial lenses used in antenna arrays

Andrea, Massa

Douglas, H. Werner

Ephrem Teshale, Bekele

Federico, Viani

Lorenzo, Poli

Luca, Manica

English

University of Trento : UNITN-eprints

Metamaterial Lenses For Antenna Arrays Circular To Linear Array Transformation

This report presents the progress on activities carried out on the topic of Metamaterials

Unitn-eprints Research

Metamaterial Lenses For Antenna Arrays Conforma To Circular Array Transformation

       UNIVERSITY OF TRENTO  DIPARTIMENTO DI INGEGNERIA E SCIENZA DELL’INFORMAZIONE   38123 Povo – Trento (Italy), Via Sommarive 14 http://www.disi.unitn.it            METAMATERIAL LENSES FOR ANTENNA ARRAYS CIRCULAR TO LINEAR ARRAY TRANSFORMATION  E. T. Bekele, F. Viani, L. Manica, Douglas H. Werner and A. Massa  March 2013  Technical Report # DISI-13-007                                                                  .  Contents1 Introduction 32 Mathematical Formulation 42.1 +qTo Library . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Grid Orthogonality Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Numerical Results 83.1 Circular to Linear Array Transformation . . . . . . . . . . . . . . . . . . . . . . . . . 821 IntroductionThis report presents the progress on activities carried out on the topic of Metamaterial lenses used in antennaarrays.32 Mathematical FormulationEach one step transformation involves two domains. In this report, the first domain is called “virtual domain” or“virtual space” while the other one is referred to as “physical domain” or “physical space”. In addition, the terms“virtual” and “physical” are used to describe entities in virtual and physical spaces respectively. The rectangularcoordinate system in virtual space is labeled as (x′, y′, z′) whereas in the physical space the labels (x, y, z) areused.If the transformation from (x′, y′, z′) to (x, y, z) is defined as:(x, y, z) = Γ (x′, y′, z′) (1)x = x (x′, y′, z′) (2)y = y (x′, y′, z′) (3)z = z (x′, y′, z′) (4)the Jacobian matrix of the transformation Λ will be:Λ =∂x∂x′∂x∂y′∂x∂z′∂y∂x′∂y∂y′∂y∂z′∂z∂x′∂z∂y′∂z∂z′ . (5)For the inverse transformation i.e. (x, y, z) to (x′, y′, z′),(x′, y′, z′) = Γ′ (x, y, z) (6)x′ = x′ (x, y, z) (7)y′ = y′ (x, y, z) (8)z′ = z′ (x, y, z) (9)the corresponding Jacobian matrix will beΛ′ =∂x′∂x∂x′∂y∂x′∂z∂y′∂x∂y′∂y∂y′∂z∂z′∂x∂z′∂y∂z′∂z . (10)and the following relations can be established.Λ′ = Λ−1 (11)4det(Λ′)=1det (Λ)(12)If !′ and µ′ represent permittivity and permeability tensors in virtual medium respectively,!′ =!′xx !′xy !′xz!′yx !′yy !′yz!′zx !′zy !′zz (13)µ′ =µ′xx µ′xy µ′xzµ′yx µ′yy µ′yzµ′zx µ′zy µ′zz (14)corresponding permittivity and permeability tensors in physical space can be computed as follows:! =Λ !′ ΛTdet (Λ)(15)µ =Λµ′ ΛTdet (Λ). (16)If there is a source with current I ′ and current density J ′ in virtual space its corresponding image in the physicalspace can be computed as [2]J =ΛJ ′det (Λ). (17)I = I ′. (18)For a cascade of transformations: Γˆ {(x′′, y′′, z′′)→ (x′, y′, z′)} followed by Γ˜ {(x′, y′, z′)→ (x, y, z)}, the overalltransformation Γ {(x′′, y′′, z′′)→ (x, y, z)} can be formulated as follows. In the following discussion, and in theremaining of this report, when dealing with cascade of transformations, the space defined by the coordinates(x′, y′, z′) will be termed as the Intermediate space and objects defined in this space will be called intermediateobjects. Let Λˆ and Λ˜ represent the Jacobian matrices of the transformations Γˆ and Γ˜ respectively defined as:Λˆ =∂x′∂x′′∂x′∂y′′∂x′∂z′′∂y′∂x′′∂y′∂y′′∂y′∂z′′∂z′∂x′′∂z′∂y′′∂z′∂z′′ (19)Λ˜ =∂x∂x′∂x∂y′∂x∂z′∂y∂x′∂y∂y′∂y∂z′∂z∂x′∂z∂y′∂z∂z′ . (20)Further more, let{!′′, µ′′},{!′, µ′}and {!, µ} represent sets of permittivity and permeability tensors in(x′′, y′′, z′′), (x′, y′, z′) and (x, y, z) spaces respectively, while the corresponding currents are represented as J ′′,5J ′ and J . Considering the transformation: Γ˜ {(x′, y′, z′)→ (x, y, z)}, the following relations can be established:! =Λ˜ !′ Λ˜Tdet(Λ˜) (21)µ =Λ˜µ′ Λ˜Tdet(Λ˜) (22)and for the other transformation, Γˆ {(x′′, y′′, z′′)→ (x′, y′, z′)},!′ =Λˆ !′′ ΛˆTdet(Λˆ) (23)µ′ =Λˆµ′′ ΛˆTdet(Λˆ) . (24)Substituting (23) and (24) in (21) and (22) respectively and rearranging terms gives the relationship betweenmaterial properties for the overall transformation! =(Λ˜Λˆ)!′′(Λ˜Λˆ)Tdet(Λ˜Λˆ) (25)µ =(Λ˜Λˆ)µ′′(Λ˜Λˆ)Tdet(Λ˜Λˆ) . (26)Following similar analysis, the current sources for the complete transformation can be related as:J =(Λ˜Λˆ)J ′′det(Λ˜Λˆ) . (27)I = I ′ = I ′′. (28)2.1 +qTo LibraryThe +qTo software library is a numerical implementation of 2D Transformation. It takes boundary contour ofvirtual space as input and generates grid of transformation to a rectangular region. It first selects points onthe input contour in virtual space corresponding to uniformly distributed points on the contour of rectangle inphysical space. The internal grid is generated by taking these points as boundary conditions and solving the2D Laplacian equation. The detail of the solution is presented in [4].Equations (15) and (16) are used to compute material permittivity and permeability tensors for generic trans-formation. Under the following assumptions, the expression for these quantities can be simplified[5].• TE or TM mode of propagation.6• Grid lines are orthogonal.• Size of mesh elements is equal (square mesh).• Isotropic medium in virtual space.Under such assumptions, permittivity and permeability in physical space will be simplified as:• For TE mode of propagation:– Constant permeability: µr (x, y) = µr.– Isotropic permittivity computed as ratio of the area of the cells of the transformation grid.• For TM mode of propagation:– Constant permittivity: !r (x, y) = !r.– Isotropic permeability computed as ratio of the area of the cells of the transformation grid.2.2 Grid Orthogonality AssessmentSince the orthogonality of the transformation grid is the basis for isotropic approximation, it is quantified asfollows. Figure 1 shows a sample grid intersection in the complex plane.θz1z2Figure 1. Description of grid orthogonality measure: A sample unit cell of a grid in the complex plane.Refering to Figure 1, and using Euler’s notation, z1 = |z1| ej[arg(z1)], z2 = |z2| ej[arg(z2)], the internal angle θcan be computed asθ = arg (z1)− arg (z2) = arg(z1z2).The offset from orthogonality χ can then be evaluated asχ = θ − 90 (29)73 Numerical Results3.1 Circular to Linear Array TransformationGoal: To demonstrate Transformation of an array using the transformation library, andTo verify results using full wave simulation.Test Case DescriptionThe transformation is described pictorially in Figure 1. Figure 1(a) shows the setup of the virtual array underinvestigation. The region included with the solid curve (Figure 1(a)) is transformed to a rectangular region inthe physical space (Figure 1(b)). As a result, the radiating elements that are located on circular contour in thevirtual space will be projected to a linear geometry in the physical space. The shaded region in the physicalspace (Figure 1(b)) shows the location of metamaterial lens.t′w′x′y′a′(a)twyx(b)Figure 1. Description of (a) Virtual and (b) Physical spaces.8In the virtual space (Figure 1(a)), a traditional circular array is used and the medium is empty space. Thesources in virtual space are uniformly distributed along the circumference of the circle while the physical arrayhas linear distribution.Simulation Parameters• Array of Isotropic radiators.• Number of Array Elements: N = 9.• Frequency of operation: ν = 600MHz• Wavelength in free space: λ = 0.5mVirtual Object Parameters:• Circular Array of point sources in 2D.• Radius of the circle: a′ = 2λ = 1m.• Width of transformation region: w′ = 16λ = 8m.• Height of transformation region: t′ = 8λ = 4m.• Element Spacing: d′ = 0.5λ = 0.25m. (Measured along the circumference.)• Uniform Amplitude Excitation: I ′n = 1.• Pattern steered to φ′s = 300 with phase excitation defined as[6]:ϕ′n =−2piλa′ cos (φ′s − φ′n) . (30)where φ′n is the angular position of the nth radiating element.Transformation Parameters:• Number of grid lines along x′ − axis: xgrid = 201.Full wave simulation Parameters:• Two dimensional problem.• TE waves.There are some differences between the test case presented here and the one presented in the reference paper.The reference paper uses magnetic currents having length of λ40 as sources, however, in the test case presentedin this report ideal point sources in 2D space are used. Another difference is, in the reference paper, a groundplane is used on one side of the geometry; in contrast scattering boundary conditions are used in all sides of thegeometry of the problem presented in this report.9RESULTSTransformation Grid 0 1 2 3 4 5 6 7 8-8 -7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6  7  8y/λx/λ(a) 0 1 2 3 4 5 6 7-8 -7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6  7  8y/λx/λ(b)Figure 2. Transformation Grids: (a) Virtual Space and (b) Physical space.Observations:• Number of horizontal grid lines obtained( including the boundaries): ygrid = 93.• The metamaterial lens is discretized into 92× 200 cells.• Each cell has dimensions: 0.08λ× 0.08λ.• As a result the dimension of the transformation region in physical space is: w = 200 × 0.08λ = 16λ,t = 92× 0.08λ = 7.36λ.10RESULTSGrid orthogonality assessment.(a) 0 2 4 6 8 10 12 14 16 18-14 -12 -10 -8 -6 -4 -2  0  2  4  6  8  10  12  14% of Grid Intersectionsχ[Degrees](b)Figure 3. Transformation grid orthogonality test.Observations:• Some degree of offset from orthogonality is observed.• The offset is severe around the transition region from circular to linear. (Figure 3(a)).11RESULTSExact values of permittivity.Because of the two dimensionality of the problem, some of the parameters related to the z−axis are zero valued(i.e !′vz = !vz = !′zv = !zv = µ′vz = µvz = µ′zv = µzv = 0 where v ∈ {x, y}). The non-zero valued parameters arereported next.(a) (b)(c) (d)(e)Figure 4. Components of the permittivity tensor in the physical medium.12Observations:• As expected off-diagonal entries have near zero values at almost all parts of the region: Figures 4(c), 4(d).• Diagonal entries are near unity: Figures 4(a), 4(b).• Since the virtual medium is free space (!′r = µ′r = 1), the permeability tensor in the physical medium, µis the same as that of permittivity, i.e µ = !.• For TE waves, the components of ! and µ that contribute are !zz, µxx, µxy, µyy and µyx.• The quantities µxy = !xy and µyx = !yx are near zero whereas µxx = !xx and µyy = !yy are near unity.• As a result, pure Isotropic approximation can be made for TE waves.13RESULTSIsotropic approximationFigure 5. Isotropic permittivity (!r) of the physical medium computed as ratio of area of transformation grid.Observations:• For TE waves, this Isotropic gradient of permittivity (!r) is used.• Values are bounded: 0.2680 ≤ !r ≤ 5.8178.14RESULTSSimulation ResultsThe following results report the output of full-wave simulation performed assuming isotropic approximation.The simulation setup considers 2D, TE mode hence, the perpendicular component of the electric field (Ez) isreported. The beam steering direction (φs = 300) is shown by a broken line.(a) (b)(c) (d)(e) (f)15(g) (h)(i) (j)Figure 6. Normal component of the electric field for (a) Virtual Uniform circular array, (b) Ordinary uniformhalf wavelength array , (c) Transformed linear array, (d) Transformed linear array in the absence of themetamaterial lens.Observations:• The transformed linear array along with the lens is is able to steer the beam to wards the expecteddirection φs = 300 (Figure 6(c), Figure 6(d)).• The amplitude the field in the transformed array is considerably smaller than that of the virtual and halfwavelength arrays (Figure 6(e), Figure 6(f)).16RESULTSComparison of power patternsThe patterns are obtained from COMSOL far-field computing tool.-40-30-20-10 0 0  30  60  90  120  150  180Normalized Power Pattern [dB]φ [degrees]Circular to Linear Array TransformationVirtualPhysical: With MetamaterialPhysical: Without MetamaterialHalf Wavelength Linear Array(a)(b)Figure 7. Comparison between normalized power patterns (a) simulation results and (b) from Kwon.2012[1].17RESULTSComparison of power patterns-40-30-20-10 0 0  15  30  45  60  75  90  105  120  135  150  165  180Normalized Power Pattern [dB]φ [degrees]Circular to Linear Array TransformationVirtual Mediaφs =  00φs =  450φs =  900φs =  1350φs =  1800-40-30-20-10 0 0  15  30  45  60  75  90  105  120  135  150  165  180Normalized Power Pattern [dB]φ [degrees]Circular to Linear Array TransformationHalf Wavelength Arrayφs =  00φs =  450φs =  900φs =  1350φs =  1800(a) (b)-40-30-20-10 0 0  15  30  45  60  75  90  105  120  135  150  165  180Normalized Power Pattern [dB]φ [degrees]Circular to Linear Array TransformationPhysical Mediaφs =  00φs =  450φs =  900φs =  1350φs =  1800-40-35-30-25-20-15-10-5 0 0  15  30  45  60  75  90  105  120  135  150  165  180Normalized Power Pattern [dB]φ [degrees]Circular to Linear Array TransformationPhysical Media Without Metamaterialφs =  00φs =  450φs =  900φs =  1350φs =  1800(c) (d)Figure 8. Plots of normalized power patterns at different steering angles: (a) Virtual circular array, (b)Transformed physical array, (c) Half-wavelength linear array, (d) Transformed physical array withoutmetamaterial.Observations:• The shape of the main beam of the linear array with metamaterial lens approximates the semicircularvirtual array(Figure 8(c)).• When the metamaterial medium is removed, the pattern deteriorates (Figure 8(d)).18Comparison of power patterns: 3dB Beam WidthThe 3dB beam width of the the patterns of Figure 8 are reported here. 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 0  15  30  45  60  75  90  105  120  135  150  165  1803dB Beam Width [degrees]φs [degrees]Circular to Linear Array TransformationVirtualPhysical: With MetamaterialHalf WavelengthPhysical: Without MetamaterialFigure 9. Comparison between normalized power patterns: Plot of beam width at different steering angles.19AcknowledgmentThis work has been developed within the EMERALD Project funded by Autonomous Province of Trento - Callsfor proposal “Team 2011”.References[1] D.-H. Kwon, “Quasi-conformal transformation optics lenses for conformal arrays,” IEEE Antennas WirelessPropag. Lett., vol. 11, pp. 1125-1128, Sept. 2012.[2] S. A. Cummer, N. Kundtz, and B.-I. Popa, “Electromagnetic surface and line sources under coordinatetransformations,” Physical Review Letters A, 80 033820, pp. 033820 1-7, Sept. 2009.[3] Y. Luo, J. Zhang, L. Ran, H. Chen, and J. A. Kong, “New concept conformal antennas utilizing metama-terial and transformation optics,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 509-512, Jul. 2008.[4] D. C. Ives and R. M. Zacharias, “Conformal mapping and orthogonal grid generation,” InAIAA/SAE/ASME/ASEE 23rd Joint Propulsion Conference, San Diego, California, Paper number 87-2057, June 1989.[5] W. Tang, C. Argyropoulos, E. Kallos, W. Song and Y. Hao, “Discrete coordinate transformation for de-signing all-dielectric flat antennas,” IEEE Trans. Antennas Propag., vol. 58, no. 12, pp. 3795-3804, Dec.2010.[6] C. A. Balanis, Antenna Theory: Analysis and Design (3rd Ed.). Wiley, 2005.[7] P. Rocca, M. Benedetti, M. Donelli, D. Franceschini, and A. Massa, "Evolutionary optimization as appliedto inverse problems," Inverse Problems - 25th Year Special Issue of Inverse Problems, Invited TopicalReview, vol. 25, pp. 1-41, Dec. 2009.[8] P. Rocca, G. Oliveri, and A. Massa, "Differential Evolution as applied to electromagnetics," IEEE Antennasand Propagation Magazine, vol.53, no. 1, pp. 38-49, Feb. 2011.[9] G. Oliveri and A. Massa, "GA-Enhanced ADS-based approach for array thinning," IET Microwaves, An-tennas & Propagation , vol. 5, no. 3, pp. 305-315, 2011.[10] G. Oliveri, F. Caramanica, and A. Massa, "Hybrid ADS-based techniques for radio astronomy array design,"IEEE Transactions on Antennas and Propagation - Special Issue on "Antennas for Next Generation RadioTelescopes," vol. 59, no. 6, pp. 1817-1827, Jun. 2011.[11] L. Poli, P. Rocca, G. Oliveri, and A. Massa, "Harmonic beamforming in time-modulated linear arrays,"IEEE Transactions on Antennas and Propagation, vol. 59, no. 7, pp. 2538-2545, Jul. 2011.20[12] L. Poli, P. Rocca, L. Manica, and A. Massa, "Handling sideband radiations in time-modulated arraysthrough particle swarm optimization," IEEE Transactions on Antennas and Propagation, vol. 58, no. 4,pp. 1408-1411, Apr. 2010.[13] L. Poli, P. Rocca, L. Manica, and A. Massa, "Time modulated planar arrays - Analysis and optimizationof the sideband radiations," IET Microwaves, Antennas & Propagation, vol. 4, no. 9, pp. 1165-1171, 2010.[14] L. Lizzi, F. Viani, R. Azaro, and A. Massa, "Optimization of a spline-shaped UWB antenna by PSO,"IEEE Antennas and Wireless Propagation Letters, vol. 6, pp. 182-185, 2007.[15] L. Lizzi, F. Viani, R. Azaro, and A. Massa, "A PSO-driven spline-based shaping approach for ultra-wideband (UWB) antenna synthesis," IEEE Transactions on Antennas and Propagation, vol. 56, no. 8, pp.2613-2621, Aug. 2008.[16] P. Rocca, L. Manica, and A. Massa, "An improved excitation matching method based on an ant colonyoptimization for suboptimal-free clustering in sum-difference compromise synthesis," IEEE Transactionson Antennas and Propagation, vol. 57, no. 8, pp. 2297-2306, Aug. 2009.[17] P. Rocca, L. Manica, and A. Massa, "Ant colony based hybrid approach for optimal compromise sum-difference patterns synthesis," Microwave and Optical Technology Letters, vol. 52, no. 1, pp. 128-132, Jan.2010.[18] P. Rocca, L. Manica, F. Stringari, and A. Massa, "Ant colony optimization for tree-searching based synthesisof monopulse array antenna," Electronics Letters, vol. 44, no. 13, pp. 783-785, Jun. 19, 2008.[19] G. Oliveri and L. Poli, "Optimal sub-arraying of compromise planar arrays through an innovative ACO-weighted procedure," Progress in Electromagnetic Research, vol. 109, pp. 279-299, 2010.[20] G. Oliveri, "Improving the reliability of frequency domain simulators in the presence of homogeneousmetamaterials - A preliminary numerical assessment," Progress In Electromagnetics Research, vol. 122,pp. 497-518, 2012.[21] I. Martinez, A. H. Panaretos, D. Werner, G. Oliveri, and A. Massa "Ultra-thin reconfigurable electro-magnetic metasurface absorbers," 2013 European Conference on Antennas and Propagation (EUCAP),(Gothenburg, Sweden), 8-12 April 2013.[22] G. Oliveri, P. Rocca, D. H. Werner, E. Bekele, M. Salucci, and A. Massa, "Design and synthesis of innovativemetamaterial-enhanced arrays" 2013 IEEE Antennas and Propagation Society International Symposium,(Orlando, USA), July 7-13, 2013.[23] E. Martini, G. M. Sardi, P. Rocca, G. Oliveri, A. Massa, and S. Maci "Optimization of metamaterial WAIMfor planar arrays", 2013 USNC-URSI National Radio Science Meeting, (Orlando, USA), July 7-13, 2013.21

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Metamaterial Lenses For Antenna Arrays Circular To Linear Array Transformation

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