Periodic FDTD Characterization of Guiding and Radiation Properties of Negative Refractive Index Transmission Line Metamaterials

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1 Periodic FDTD Characterization of Guiding and Radiation Properties of Negative Refractive Index Transmission Line Metamaterials Costas D. Sarris The Edward S. Rogers Sr. Department of Electrical and Computer Engineering University of Toronto, Toronto, ON, M5S 3G4, Canada Abstract The application of the Finite Difference Time Domain technique, equipped with periodic boundary conditions, can lead to the fast full-wave characterization of negative refractive index (NRI) structures. Employing the so-called sine-cosine method, the guiding and radiating properties of the NRI transmission-line medium are determined. Surface wave effects are captured, while the operation of the medium as a leaky-wave antenna (within its fast-wave region) is also modeled. The efficiency of this approach stems from the fact that all these results are obtained by simulating up to two unit cells of the periodic NRI medium under study. 1 Introduction Recent periodic implementations of artificial dielectrics exhibiting simultaneously negative dielectric permittivity ǫ and magnetic permeabilty µ have enabled the exploration of a domain of unconventional applications first predicted by Veselago [1], as well as Pendry s concept of a perfect lens [2]. The theoretical characterization of such periodic implementations as the negative refractive index transmission line (NRI-TL) medium of [3] can be efficiently carried out by means of time- or frequency-domain techniques, equipped with periodic boundary conditions. The latter enable the reduction of the computational domain to a single unit cell of the periodic medium under consideration and hence, of the associated memory and execution time requirements. Furthermore, the time-domain modeling of metamaterials, via the Finite-Difference Time-Domain (FDTD) method, is motivated by the richness of the transient behavior that it captures. As an example, the NRI-TL implementation of Pendry s perfect lens concept, designed according to the guidelines of [4], is numerically demonstrated in Fig. 1. The structure consists of cascaded slabs of positive/negative/positive refractive index, respectively. The causal evolution of a focal spot at GHz is depicted in the four successive vertical field snapshots. A ray diagram indicating the negative refraction taking place at the interface between the positive and negative index media is also superimposed. On the other hand, this paper presents recent work on the application of a periodic FDTD technique, based on the so-called sine-cosine method of [5], to the full-wave analysis of wave guidance and radiation in the NRI-TL medium of [3]. The unit cell of this medium is shown in Fig. 2, where the loading of a conventional microstrip transmission-line structure with series capacitors and shunt inductors is indicated, along with the position of the periodic boundaries. Moreover, the incorporation of the lumped elements in an FDTD grid can be achieved by means of the extended FDTD technique [6, 7]. First, periodic FDTD simulations are employed for the extraction of the Brillouin diagram of the NRI-TL medium [8]. Then, upon identifying this medium, within its fast-wave frequency band, as a two-dimensional leaky-wave antenna, a study of its radiation properties is pursued and backward to forward end-fire radiation patterns are extracted [9]. Research supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) through a Discovery Grant.

Figure 1: Image evolution in a NRI-TL-based planar lens structure, simulated by FDTD. L (z) (y) (x) Figure 2: Unit cell for the simulation of the planar loaded transmission line metamaterial of [3].

coordinate system, state that phasor field components one period away in either direction differ only by a constant attenuation and phase shift term exp( jk x d x ), exp( jk y d y ) respectively (k

2 Figure 1: Image evolution in a NRI-TL-based planar lens structure, simulated by FDTD. L (z) (y) (x) Figure 2: Unit cell for the simulation of the planar loaded transmission line metamaterial of [3]. 2 Brillouin diagram of the NRI-TL medium via a periodic FDTD analysis Floquet s boundary conditions for two-dimensional periodic media of period d x and d y along the x and y axes of a rectangular coordinate system, state that phasor field components one period away in either direction differ only by a constant attenuation and phase shift term exp( jk x d x ), exp( jk y d y ) respectively (k x, k y ) being complex propagation constants). Several ways have been proposed for the translation of this frequency-domain relationship to the time-domain. In this work, the propagation constants are limited to be real, while the sine/cosine method of [5] is implemented. According to it, two separate grids are considered, corresponding to field components with sin(ωt) and cos(ωt) time dependence, E s and E c. For example, if p = ˆxd x + ŷd y is the lattice vector and k p = ˆxk x + ŷk y the lattice wavevector, Floquet s conditions on electric field phasors assume the form: E(r + p) = Ẽ(r)exp( jk p p). (1) Then, if a frequency-to-time domain transform is considered for E c and E s, the following set of timedomain conditions is derived: E s (r + p, t) = E s (r, t)cos(k p p) E c (r, t)sin(k p p) E c (r + p, t) = E s (r, t)sin(k p p) + E c (r, t)cos(k p p)

3 The propagation constants k x, k y in these conditions are chosen to represent a point on the -X, X-M, M- axis of the Brillouin diagram. For each such point, the time-domain waveforms of fields are sampled within the unit cell. Subsequently, their Fourier transform reveals the mode resonances that represent the modal frequencies ω(k x, k y ). The three-dimensional structure, simulating the NRI medium of [3], is depicted in Fig. 2. The open boundary in the z direction is terminated in Mur s first order absorbing boundary conditions. Periodicity in x and y directions is represented by the periodic boundary conditions. An example of how lumped element equations are coupled to FDTD updates is given for the case of a capacitor C, along a parallel to the x axis edge of Yee s cell, with a voltage-current characteristic dv x (t)/dt = I x (t)/c. Assuming the E x is the x component of the electric field sampled on a node at the center of the capacitor, its update equation can be written as: Ex n+1 = Ex n t y z ( n+1/2 + ˆx H) (2) C x + ǫ y z Similarly, electric field components along z oriented inductors are updated as: E n+1 z = E n z + t ǫ ( ) n+1/2 z t 2 ẑ H ǫl x y n Ez k (3) Finally, as mentioned in [5], a point source can excite all the modes for a set of (k x, k y ), implicitly introduced by the boundary conditions. Line sources can be used as well, properly oriented in order to excite TE z or TM z modes. Based on the these concepts, the dispersion analysis of the NRI medium of [3] is performed. The unit cell is discretized in N x N y N z cells, with N x = N y = N z = 16. The periodicity in both x and y is d x = d y = 5.65 mm and hence, x = d x /N x, y = d y /N y, while z is also chosen to be the same. Three cells model the substrate of ǫ r = 2.2 and the characteristic impedance of the microstrip lines interconnecting the lumped elements is Ohms. The time step is set to t = 0.61 ps (0.9 of the Courant limit). As an excitation, a Gabor function s(t) = exp( (t t 0 ) 2 /T 2 s ) {sin(ω 0 t), cos(ω 0 t)}, with T s = 2 6/π(f max f min ) and ω 0 = π(f min +f max ) is used in the sine, cosine grid respectively, with f min =.1 GHz and f max = 35 GHz. Fast convergence was obtained, with the necessary number of time steps being limited to 7,000. The cases under consideration wwas also treated in [], that offers a basis for comparison. The Brillouin diagram for the case under consideration is shown in Fig. 3 and corresponds to lumped element values = pf and L = nh. Modeling of the same structure was pursued in []. The FDTD-derived Brillouin diagram is in excellent agreement with the one in [], which was derived via Ansoft s HFSS. It is noted that in addition to the forward and backward TM waves supported in the structure, TM surface waves and TE waves were deduced as well. This is the fundamental difference between carrying out this dispersion analysis via a full-wave technique, as opposed to employing transmission-line theory. 3 FDTD Modeling of the NRI-TL medium as a two-dimensional leaky-wave antenna Recently, the sine-cosine method of [5] was extended to model leaky-wave structures [11], providing an efficient way of calculating attenuation/propagation constants in such structures and characterizing their behavior as leaky-wave antennas (LWAs). This development was utilized in [9] to analyze the radiation properties of the NRI medium of [3], within its leaky-wave zone, when β < k 0. Determining the propagation and attenuation constants within this structure, one can employ array theory to deduce its radiation pattern. k=1

4 40 35 Μ Χ Light Line Frequency (GHz) TM Surface Wave TM Wave TE Wave 5 TM Backward Wave 0 Χ Μ Figure 3: Brillouin diagram for the case study of the NRI grid of [3]. Backward, forward and surface waves are indicated. In Figs. 4, the radiation patterns corresponding to backward waves, developing within the leaky-wave zone of the NRI medium of [3], around the stop-band forming at β = 0 are shown. The dimensions of the NRI metamaterial unit cell are d x = d y = 5 mm, the height of the substrate is h = mm, the dielectric permittivity of the substrate is ǫ r = 2.94 and the characteristic impedance of the hosting transmission-line segments is Z o = Ohms. The values of the lumped elements are assumed to bel = 5.6nH and C = 1pF. These values of the structure parameters result in a dispersion diagram with a stop-band (band gap) in the case where the phase shift per unit cell is 0 (βd ξ = 0, ξ = x, y). According to the patterns of Fig. 4, the antenna starts to leak radiation at 35 0 (2.688 GHz), while the gain increases toward broadside, reaching its maximum value at 11 0 (2.78 GHz). However the gain starts decreasing just before broadside, while radiation ceases within the stop-band at broadside. In Fig. 5 the values of the attenuation constant α, along the x axis of the LWA, together with the corresponding scanning angles as a function of frequency are also presented. These results are in agreement with the experimental observations of [12]. 4 Conclusion The FDTD technique enhanced with periodic boundary conditions can provide an effective means of achieving the complete characterization of NRI metamaterial structures fast and accurately. This characterization includes surface-wave modes developing at air-dielectric interfaces of practical implementations of NRI media and radiation characteristics (gain and pattern) of such media, operating in their leaky-wave zone. Future extensions of the periodic FDTD technique will enable the extraction of the response of truncated structures as well as the determination of the Green s function of NRI/PRI media from periodic simulations.

5 Frequency vs. α/k o 40 Scanning Angle vs. α/k o Frequency (GHz) Scanning Angle ( o ) α / k o Figure 4: Backward radiation patterns of the NRI-TL medium of [3], operating as a leakywave antenna. Figure 5: Calculated values of α/k 0 for the backward radiation region. 5 Acknowledgement The research reported in this paper is the outcome of a collaboration of the author with Titos Kokkinos and Prof. George Eleftheriades. References [1] V. G. Veselago, The electrodynamics of substances with simultaneously negative values of ǫ and µ, Soviet Phys. USPEKHI, vol., pp , [2] J. B. Pendry, Negative refraction makes a perfect lens, Phys. Rev. Lett, vol. 85, pp , Oct. 00. [3] G.V. Eleftheriades, A.K. Iyer and P.C. Kremer, Planar negative refractive index media using periodically loaded transmission lines, IEEE Trans. Microwave Theory Tech., vol. 50, pp , Dec, 02. [4] A. Grbic, G.V. Eleftheriades, Negative Refraction, growing evanescent waves and sub-diffraction imaging in loaded transmission line metamaterials, IEEE Trans. Microwave Theory Tech., vol. 51, pp , Dec. 03. [5] P. Harms, R. Mittra, W. Ko, Implementation of the periodic boundary condition in the finite-difference time-domain algorithm for FSS structures, IEEE Trans. Antennas Prop., vol. 42, 1994, pp [6] W. Sui, D.A. Christensen, C.H. Durney, Extending the two-dimensional FDTD method to hybrid electromagnetic systems with active and passive lumped elements, IEEE Trans. Microwave Theory Tech., vol. 40, 1992, pp [7] A. Taflove, S. Hagness, Computational Electrodynamics : The Finite Difference Time Domain Method, ch. 15: High Speed Electronic Circuits with active and passive lumped elements, Artech House, [8] T. Kokkinos, R. Islam, C. D. Sarris, and G. V. Eleftheriades, Rigorous analysis of negative refractive index metamaterials using FDTD with embedded lumped elements, in Proc. IEEE MTT-S Int. Microwave Symp. Dig., Ft. Worth, TX, Jun. 612, 04, pp [9] T. Kokkinos, C. D. Sarris, and G. V. Eleftheriades, Finite-Difference Time-Domain Analysis of Metamaterial-Based Leaky-Wave Antennas, in IEEE AP-S Int. Symp. Dig., July 05. [] A. Grbic, G.V. Eleftheriades, Dispersion analysis of a microstrip based negative refractive index periodic structure, IEEE Microwave Wireless Comp. Lett., vol. 13, no. 4, pp , April 03. [11] T. Kokkinos, C.D. Sarris, and G.V. Eleftheriades, Efficient Finite-Difference Time-Domain (FDTD) Modeling of Periodic Leaky-Wave Structures, in Proc. IEEE International Microwave Symposium Digest, Jun. 05. [12] A.K. Iyer, and G.V. Eleftheriades, Leaky-wave radiation from planar negative-refractive-index transmission-line metamaterials, in Proc. IEEE Antennas and Propagation Symposium, vol. 2, Jun. 04, pp

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES 1

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES 1 Periodic FDTD Analysis of Leaky-Wave Structures and Applications to the Analysis of Negative-Refractive-Index Leaky-Wave Antennas Titos Kokkinos,