Research on the Negative Permittivity Effect of the Thin Wires Array in Left-Handed Material by Transmission Line Theory

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1 96 Progress In Electromagnetics Researc Symposium 25, Hangzou, Cina, August Researc on te Negative Permittivity Effect of te Tin Wires Array in Left-Handed Material by Transmission Line Teory Qun Wu, Fan-Yi Meng, and Ming-Feng Wu Harbin Institute of Tecnology, Cina Jian Wu Cina Researc Institute of Radioave Propagation, Cina Le-Wei Li National University of Singapore, Singapore Abstract In tis paper, te negative permittivity effect of te tin ire array in Left-anded material is clearly described by te transmission line teory. Firstly, te transmission line system equivalent to te tin ire array is presented, according to te periodic modulation action of te tin ire array on te incident electromagnetic ave and te equivalence relation of free space and transmission line. Secondly, te effective permittivity model of te asymmetric ire array is derived based on te transmission line teory, and te validity of te model is proven by numerical simulations. Introduction left-anded material is also called negative refractive index material, ic represents a dielectric medium tat exibits negative refractive index penomenon and as simultaneously negative permittivity and negative permeability [-4]. In 996, Pendry et al derived te effective permittivity model of te symmetric tin ire array and stated tat te negative permittivity effect can be exibited by te array [5]. In 999, te negative permeability effect of te periodic SRR s (Split Rings Resonators) array as also presented by Pendry et al. [6]. Tese results are considered classic orks, because it resolved key problems for fabrication of te left-anded material. In 2 Smit firstly fabricated left-anded material by reasonably arranging te tin ires and SRR s [7].Wort noting is tat te Pendry s effect permittivity model as derived troug making use of te plasmon teory and leading te concept of te effective density and effective mass of electrons, so it is difficult to understand te model from te angle of macropysics. In addition, te Pendry s model is only limited to te symmetric tin ire array. In vie of tis consideration, te effective permittivity model of te asymmetric array is derived in tis paper based on te transmission line teory and is verified by numerical simulations. Te description of teoretical derivation exibits clear pysical concepts and provides a ne approac to understand te negative permittivity effect of te tin ire array from te macropysics. Effective Permittivity Model of Tin Wire Array from Plasmon Teory In 96, Rotman stated tat te tin ire array son in Fig., can be used to simulate te plasma, because teir effective permittivity is expressed in te same form [8] ε eff ω2 p ω 2 () Here, ω p is te plasma frequency, ω is te frequency of te incident electromagnetic ave. From tis point, in 996, Pendry stated tat tere is negative permittivity effect in te tin ire array ile ω is less tan ω p, and derived te relation beteen te plasma frequency ω p and te array parameters troug leading te effective density n eff and effective mass of electrons m eff in form [5] ω 2 p n effe 2 ε m eff 2πc2 a 2 ln(a/r) (2)

2 Progress In Electromagnetics Researc Symposium 25, Hangzou, Cina, August Fig. Tin ire array in vacuum Figure : Tin ire array in vacuum n eff nπr 2 /a 2 (3) m eff µ e 2 πr 2 n ln(a/r) (4) 2π Here, a is te ire spacing, c is te velocity of ligt in vacuum, r is te ire radius. Tis is just te Pendry s effective permittivity model, it plays an important role in te researc on left-anded material. Modeling of te Effective Permittivity from Transmission Line Teory As son in Fig. 2, te tin ire array can be described by an inductor array. Here L is te ire self- Figure 2: Te inductor array equivalent to te tin ire array inductor per unit lengt in Z direction, and x and yare te ire spacing in X and Y direction, respectively. In Pendry s model [3] it is assumed tat x y a. Hoever, ere e propose x y to obtain more general results. In tis case, te average inductance of te array in XY plane is described by te inductance area density L s L ln( x/r)ln( y/r) x y µ x y (5) π[ln( x/r) + ln( y/r)] Note tat tis parameter is very important, because it relates te macro effect of te tin ire array to its size parameters. On te oter and, for a plane electromagnetic ave in te vacuum, its electric field E and magnetic fieldh can be expressed by E α z E e jβy (6) H α x H e jβy (7) ere, α z and α x are te unit vectors, β is te propagation constant in te vacuum, E and H are te amplitude of te E and H, respectively. Obviously, it as to caracters: firstly, te electromagnetic field distribution in te plane perpendicular to te direction of propagation is uniform; secondly, as son in Fig. 3(a), te internal field distribution is not canged if a parallel-plate aveguide infinite in te XY plane is placed perpendicularly to te electric field direction. In tis case, te plane electromagnetic ave can be represented by te internal electromagnetic ave, and te internal field can be described by te voltage U beteen te to plates and te

3 Progress In Electromagnetics Researc Symposium 25, Hangzou, Cina, August (a) (b) Figure 3: Te inductor array equivalent to te tin ire array surface current density j on te internal faces of te parallel-plate aveguide, respectively, as son in Fig. 3(b). In Fig. 3(b), te field in te region ABCD is proposed as te representative because of te uniformity of te plane ave. Te I is te total current in te region ABCD, it is given by I j αz H αy H e jβy (8) ere, is te lengt of te line segment AD. And te voltage U is given by U E αz E e jβ y (9) ere, is te distance beteen te to plates. So tere is a transmission line system, it is equivalent to te free space. Based on te transmission line teory and equations (8), (9), te caracteristic impedance is given by r r E µ L () Z H ε C I ere, C and L are te distributed capacitance and inductance per unit lengt of te equivalent transmission line, respectively; ε and µ are vacuum permittivity and permeability, respectively. In addition, te propagation constant β can be expressed by p β ω ε µ ω L C () U From equations () and (), te equivalence beteen te transmission line and te free space is derived µ ε and C (2) According to te above results te equivalent circuit of te tin ire array is sketced, as son in Fig. 4, and te circuit parameters are related to te parameters of te array son in equation (5) and te vacuum permeability /permittivity son in (2). In tis case, te sunt admittance per unit lengt is expressed as L Z iω ε G iω + ω 2 (ε 2 ) iωls ω Ls Figure 4: Equivalent circuit of te tin ire array (3)

4 Progress In Electromagnetics Researc Symposium 25, Hangzou, Cina, August Furtermore, te equivalent sunt capacitance per unit lengt is given by L eq ω 2 (ε (4) ω 2 ) L s So according to te left-anded transmission line teory of Caloz et al [9] and equation (2), te effective permittivity of te tin ire array is expressed by ε eff ω 2 L eq ω 2 ω2 p L s ε ω 2 (5) ere, ω p is te resonant frequency of te dispersion relation Wen ω 2 p /(L s ε ) /( x yl ε ) (6) x y a, ω 2 p /(a 2 L ε ) (7) Obviously, tis is just te plasmon model [3]. In oter ords, Pendry s model is a specific case of te derived formula (6). From above derivation, it can be seen tat te effective permittivity model of te tin ire array derived based on te transmission line teory can be applied not only to a symmetric ire array, but also to an asymmetric array. So te expression of te formula is more general tan plasmon model. Numerical Simulation and Discussion Figure 5: Wire array vertically standing beteen opposing metal plates To demonstrate te effective permittivity effect of te asymmetric ire array, e simulate, using CST MW Studio simulation tool, te array, ic vertically stands beteen opposing metal plates, as son in Fig. 5. In simulations, te perfect electric boundary conditions in Z direction is used to simulate te opposing metal plates, and te perfect magnetic boundary conditions in X direction is used to simulate te infinite ire array in X direction. In tis case, e can set to ports in Y direction and calculate te S-parameters of te to ports to simulate te transmission of te ave troug te infinite ire array in XZ plane. So te plasma frequency can be determined by te magnitude and pase of te S2, because on te one and te ave can t propagate in te ire ile te ave frequency is under te plasma frequency, and on te oter and tere ill be discontinuity in te pase of te S2 ile te ave frequency equals te plasma frequency. Fig. 6 sos te S 2 tendency for different array parameters and te location of te corresponding resonant frequency. In addition, Tab. sos te comparison beteen te simulated resonant frequencies and calculated one by equation (6) for different x during y 3 mm. It can be seen tat te maximum error less tan 5%, easily itin te error introduced by our approximations and numerical procedure. Table : Te comparison of simulated resonant frequencies vs calculation by equation (7) for different x during y 3 mm x(mm) Simulation Results (GHz) Calculation results (GHz) Relative Error 2.96% 2.76% 2.56% 4.46%

5 2 Progress In Electromagnetics Researc Symposium 25, Hangzou, Cina, August Figure 6: Te calculation results of S2 for different array parameters Conclusion In tis paper, an effective permittivity model for an asymmetric tin ire array is proposed by using te equivalent circuit metod. Simulation results are in a very good agreement it te numerical simulation results obtained by te commercial softare package CST. Te present analysis sos clearly and explicitly te principle of te negative permittivity effect of te tin ire array from te macropysics. Acknoledgement Tis ork as partly supported by a grant from te National Natural Science Foundation of Cina (No ), and Fund for te National Key Laboratory of Electromagnetic Environment (No ). REFERENCES. Veselago, V. G., Te Electrodynamics of Substances it Simultaneously Negative Values of ε and µ, Soviet Pysics Uspeki, Vol., No. 4, 59-5, Selby, R. A., D. R. Smit, et al, Microave Transmission troug a To-Dimensional, Isotropic, Left- Handed Metamaterial, Appl. Pys. Lett., Vol. 78, No. 4, , Pendry, J. B., Negative Refraction Makes a Perfect Lens, Pys. Rev. Lett., Vol. 85, No. 8, , Xu, Wei, Le-Wei Li and Qun Wu, Design of Left-Handed Materials it Broad Bandidt and Lo Loss Using Double Resonant Frequency Structure, Proceedings of te 24 IEEE Antennas and Propagation Symposium, California, USA, Pendry, J. B., A. J. Holden, W. J. Steart and I. Youngs, Pys. Rev. Lett., Vol. 76, , Pendry, J. B., A. J. Holden, D. J. Robbins and W. J. Steart, Magnetism From Conductors and Enanced Nonlinear Penomena, IEEE Trans. on Microave Teory and Tecniques, Vol. 47, No., , Smit, D. R., Willie J. Padilla, D. C. Vier, S. C. Nemat-Nasser and S. Scultz, Composite medium it simultaneously negative permeability and permittivity, Pys. Rev. Lett., Vol. 84, , Rotmant, W., Plasma Simulation by Artificial Dielectrics and Parallel-Plate Media, IRE Trans. Antennas Propagat., Vol. AP-, 82 95, Caloz, C and T Ito, Application of te Transmission Line Teory of Left-Handed (LH) Materials to te Realization of a Microstrip LH line, IEEE Antennas and Propagation Society International Symposium, 42-45, 22.

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