JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 28, NO. 10, MAY 15,

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1 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 28, NO. 10, MAY 15, A Unified FDTD Lattice Truncation Method for Dispersive Media Based on Periodic Boundary Conditions Dongying Li and Costas D. Sarris, Senior Member, IEEE Abstract A unified treatment for the truncation of finite-difference time-domain lattices, applicable to dispersive and conductive media alike, is proposed. The method is based on periodic boundary conditions, hence necessitating that the medium under study be periodic along the direction of truncation. When this condition (which is satisfied in many practical cases) is met, a much simpler but equally effective alternative to the PML is provided by the combination of periodic boundaries with an array-scanning method. The proposed formulation does not need any additional auxiliary variables when applied to dispersive media, unlike the PML. Applications include a Bragg filter and a negative-refractive-index super lens. Index Terms Finite-difference time-domain (FDTD), periodic boundary conditions. I. INTRODUCTION T HE development of absorbing boundary conditions for the finite-difference time-domain method (FDTD) has been a topic of continuing interest for many years [1]. Trading accuracy for complexity, the state-of-the-art in this area has evolved from first and higher order absorbing boundary conditions stemming from the factorization of the wave equation [2] [4] to the PML absorber [5]. Presently, perfectly matched layer (PML) has established itself as the method of choice for FDTD lattice termination. It is also worth recognizing that dispersive media PMLs employ multiple auxiliary variables in addition to field vectors, which produce nontrivial memory and execution time overhead. Moreover, the performance of the PML in problems involving spatially dispersive media and backward-wave metamaterials has recently come under scrutiny [6], [7]. These questions are particularly important for the application of FDTD to the modeling of optical structures and motivate research into alternative techniques that may strike a better balance between accuracy, complexity, and versatility. In the applied mathematics literature, the simple 1-D periodic boundary condition is a widely used absorbing boundary condition. In computational electromagnetics and optics, this option Manuscript received September 29, 2009; revised December 29, 2009; accepted February 06, First published February 22, 2010; current version published April 28, This work was supported by the Natural Sciences and Engineering Research Council of Canada and by the Nortel Networks, through a Strategic Grant. The authors are with the Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON M5S 3G4, Canada ( dongyingli@waves.utoronto.ca; cds@waves.utoronto.ca). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /JLT has not been explored as it is incompatible with multidimensional wave propagation and the presence of complex source and boundary conditions. However, recent work has enabled the integration of nonperiodic sources and boundaries with periodic boundary conditions in FDTD [8], [9], allowing for the reconsideration of those as an alternative technique for FDTD lattice truncation. This paper presents a feasibility study on the use of periodic boundary conditions for FDTD lattice truncation. An obvious prerequisite for the applicability of this approach is that the working volume be periodic in the direction of termination. This condition is met in many practical cases of temporally and spatially periodic media either used as substrates or as standalone devices. The following important points are made clear through benchmark examples and applications. First, this approach offers a unified framework for the termination of FDTD lattices, regardless of the dispersion properties of the media enclosed. Hence, the issues of reduced PML absorptivity for periodic structures [6] and instability for negative-index media [7] are naturally resolved. Second, its accuracy performance is typically similar to that of the PML, except when the source is very close to the absorbing boundary. Then, it actually becomes better, as PML reflectivity increases dramatically due to the strong presence of near-grazing incident waves. II. METHODOLOGY A. Sine-Cosine Array-Scanning FDTD A brief overview of how nonperiodic sources can be embedded in an FDTD domain terminated in periodic boundary conditions is given in the following [8], [9]. To that end, consider the 2-D periodic structure, shown in Fig. 1(a), with a lattice vector, excited by a nonperiodic source. Instead of following the conventional method of simulating a finite number of unit cells until the field solution converges, an alternative path is followed. In particular, the computational domain of Fig. 1(b), enclosing just one unit cell around the source, is used. The periodic boundaries are terminated by means of periodic boundary conditions: where is a Floquet wave vector. The simulated s are uniformly sampled within the irreducible Brillouin zone of the periodic structure. As for the translation of (1) into the time domain, the sine-cosine method of [10] is used. (1) /$ IEEE

2 1448 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 28, NO. 10, MAY 15, 2010 this equivalent unbounded structure is driven by the original localized source. Hence, when an open structure is periodic or infinite in a certain direction, the same formulation can act as a lattice termination scheme. Note that the application of (2) is not limited by the presence of dispersion or conductivity, nor do the actual complexity and computational work involved grow in these cases. This feature renders the sine-cosine array-scanning FDTD a promising alternative to PML for many practical cases. On the other hand, there is no straightforward extension of this methodology that may be applicable to nonlinear media. B. Error Estimation Absorbing boundary conditions suffer from errors related to the angle of incidence and the velocity of impinging waves. The PML's main source of error is the wave reflected from the perfect electric conductor that typically is used to terminate it, in addition to reflections at the interface with the working volume. In the case of periodic boundary conditions augmented by the array-scanning method, the error is related to the number of the wave vectors used in the sum of (2), which approximates the continuous array-scanning integral. This sampling error can be expressed in terms of alias terms in the space domain due to the finite sampling rate in the spectral domain as [12] (3) Fig. 1. (a) x 0 y plane cross section of a 3-D periodic structure with a lattice vector d = d ^x+d ^y+d ^z excited by a source; (b) computational domain used in the sine-cosine FDTD; (c) effective computational domain corresponding to the setup in (b) with a specific Floquet wave vector k. Note that the interaction of the source with the periodic boundaries leads to the simulation of a problem that is different from the original one, as shown in Fig. 1(c). Namely, each of the aforementioned simulations of the reduced domain of Fig. 1(b) corresponds to the problem of a phased array of sources with progressive phase difference in the directions, respectively, exciting the periodic structure. The solution to the original problem can be recovered by an additional step, implementing the array-scanning method of [11]. In particular, let be either the electric or the magnetic field vector; be this vector, in the time domain, at a point, within the domain of Fig. 1(b), with the Floquet wave vector ; and, where, the position vector of an arbitrary point within the periodic structure. Then, the field at can be expressed as follows: When this method is applied, the unit cell is effectively extended to infinity in the directions of periodicity, while (2) In (3), the vector represents a reference solution to the problem at hand. If an analytical solution exists, it can be directly used in (3). Alternatively, can be obtained from an FDTD simulation with a dense mesh and a large number of cells, to prevent waves reflected from the terminating boundaries from affecting the working volume. Both alternatives are useful primarily for benchmarking the method in canonical problems. Moreover, (3) implies that as long as the actual fields tend to zero periods away in the -direction and periods away in the -direction, respectively, the error related to the proposed lattice termination should also go to zero. This observation can actually guide the choice of the number of points in practical cases. A detailed study of the error of the sine-cosine array-scanning FDTD itself can be found in [9]. C. Discussion Before we proceed to numerical examples of the periodic boundary condition-based termination of FDTD lattices for spatially and temporally dispersive media, it is necessary to clarify the tradeoffs involved in its application. Notably, this method allows for a significant reduction of the computational domain. On the other hand, it requires multiple simulations of a reduced domain, in order to complete the sampling of the wave vectors needed for the array-scanning integral (2) to converge. It is also important to observe that these simulations are totally independent and perfectly parallelizable as such. Hence, this technique can be interpreted as a spectral decomposition method, whereby a given source is decomposed into wave vectors that are individually modeled (in parallel if possible) in a small computational domain. Let us compare this approach to its conventional alternative, namely, the termination of a finite periodic

3 LI AND SARRIS: UNIFIED FDTD LATTICE TRUNCATION METHOD 1449 Fig. 2. Current source in a 2-D conducting half-space. structure in a PML, for single and multiple processor environments, respectively. For a single processor, periodic boundary conditions are preferable when the size of the corresponding finite problem is large, either exceeding the available memory resources or becoming extremely time consuming. For multiple processors, the spectral decomposition approach compares favorably to a domain decomposition of the finite periodic problem, as it totally eliminates the communication overhead between subdomains. While state-of-the-art domain decomposition techniques may lead to almost linear speedups, the sine-cosine-based lattice termination leads to a perfectly linear speedup due to the independence of each -simulation. III. NUMERICAL RESULTS: VALIDATION A. 2-D Conducting Half-Space A 2-D benchmark problem from [13] is used here to assess the performance of the proposed method. The geometry consists of a conducting half-space with and S/m, over a free-space region (see Fig. 2). The excitation is a modulated Gaussian current source, spectrally supported from 0.5 to 10 GHz. The 24 mm wide computational domain is discretized by 40 Yee cells in the -direction. As the geometry is infinite in that direction, we can also consider it as infinitely periodic with a period of 24 mm. To that end, periodic boundary conditions are applied and 16 to 32 -wavenumbers are uniformly sampled within the Brillouin zone. For error comparison, an identical domain terminated in a 10-cell uniaxial PML in the -direction is simulated. The PML conductivity profile is constructed using polynomial grading and, where is the thickness of the PML,, and is the characteristic wave impedance of the region terminated in the PML. Notably, the presence of conductivity necessitates the augmentation of the conventional PML formulation with additional auxiliary variables, as detailed in [13]. As for the -direction, 2000 Yee cells are used to eliminate reflections from the terminal boundaries, in order to ensure that our error study will include the -boundary alone. Finally, the time step is set to ps and 4096 time Fig. 3. Frequency-domain relative error of the structure in Fig. 2(a) using the proposed method with 16 and 32 k samples, compared with the relative error of a 10-cell uniaxial PML. steps are run. Each wave vector simulation takes 533 s, while the PML simulation takes 712 s. The error of both termination methods is evaluated by comparing to a reference solution in a domain, and computing the norm where is the -component of the electric field obtained by the reference simulation, in two cases. First, the current source is placed at point A at the center of the interface between the two half-spaces, and second, at point B in the upper-half-space one Yee cell away from the boundary, and 12 cm from the conducting medium interface. Fig. 3 shows the corresponding errors in the frequency domain. It is clear that the change of the accuracy level of the sine-cosine array-scanning FDTD caused by the proximity of the source to the boundaries is much smaller than that of the PML. In the case of 32 sample points with source A, the accuracy level of the method is comparable to that of the PML. With a source close to the boundary, the performance of the proposed method clearly surpassed that of the PML. The effect of the size of the computational domain on efficiency and accuracy of the proposed method is further examined by applying the source at point A and gradually decreasing the domain size in the -direction. Fig. 4(a) shows the maximum time-domain error within the computational domain, using (4) with respect to the number of Yee cells in the -direction. It is clear that the proposed method is relatively insensitive to the change of the size of the working volume. Furthermore, Fig. 4(b) shows the relationship between the CPU time with respect to the change of the working volume, using the PML termination and the proposed method, with both a single and the complete simulation if the program is executed serially. This toy problem, considered for benchmarking purposes, can be efficiently solved by the PML. Hence, while single -simulations remain faster than the PML-based ones, the total time spent on all s exceeds the PML simulation time for the same (4)

4 1450 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 28, NO. 10, MAY 15, 2010 x0 y plane and x0 z plane pattern of the geometry of Fig. 5 at 20 GHz Fig. 6. using the proposed method, compared with a finite structure simulation. Fig. 4. (a) Maximum error and (b) computational time of the structure in Fig. 2(a), using the proposed method with 32 k samples and excitation at point A, compared with results using a 10-cell uniaxial PML. Fig. 5. Geometry of a Hertzian dipole source embedded in a dispersive substrate. level of accuracy. In this case, the proposed method would be more appropriate for a parallel rather than a serial computational environment. B. Dipole Antenna Within a Dispersive Substrate The second example is a Hertzian dipole embedded in a dispersive substrate [14], as shown in Fig. 5. The substrate permittivity follows the Drude model:, where GHz. The height of the substrate is h mm, with a horizontally oriented dipole source placed at below the air substrate interface. The size of the FDTD computational domain in the - and the -directions is 3.6 cm 3.6 cm, and is discretized in Yee cells. The dipole is represented by a GHz modulated Gaussian current source excitation. The time step is set to 1.38 ps and time steps are run. This geometry is periodic (as infinite) in the - and -directions, hence the proposed method is applicable. To that end, 16 and 16 samples are considered. A 10-cell uniaxial PML is used in the -direction. The reference solution is extracted by a structure that is 12 cm 12 cm long in the - and -directions, respectively, and terminated in 10-cell uniaxial PMLs in all directions, using the same polynomial grading profile as in the previous example. The simulations are executed in parallel, each one taking 7740 s, while the reference solution simulation is run on a single console and takes s. Several representative sets of results are shown. The radiation patterns of the dipole on the plane and the plane are shown in Fig. 6. The agreement between the sine-cosine array-scanning FDTD results, and the corresponding reference simulation is very good. Moreover, to compare the error between the proposed method and the PML termination, an alternative working volume is set up with uniaxial PML termination along the directions of periodicity, and with an identical computational domain size of 3.6 cm 3.6 cm in the - and the -directions. The -component of the electric field is sampled at point A, which is 1 cm above the air substrate interface. The time-domain normalized error is computed using (4) and is plotted in Fig. 7. The result clearly demonstrates that close proximity of the source to the periodic boundaries does not compromise the accuracy of the proposed method. On the other hand, the error of the PML-based solution is substantially higher than the proposed method. In particular, it is noted that the accuracy of the PML-based simulation is not substantially improved even when the number of PML cells increases significantly. The tradeoff between the simulation time and the corresponding maximum error achieved using both the proposed method and the PML termination is further illustrated in Fig. 8. This is done by gradually increasing the size of the working volume, using both the proposed method and the PML termination, and recording the simulation time associated with the particular size as well as the maximum time-domain error at point A. Fig. 8 shows the relationship between the CPU time and the maximum normalized error achieved, both with respect to a single and to the complete simulation, if the program is

5 LI AND SARRIS: UNIFIED FDTD LATTICE TRUNCATION METHOD 1451 Fig. 9. Computational domain of a structure with 1-D periodic permittivities excited by an infinite line source, terminated in periodic boundaries or PMLs in the y-direction. Fig. 7. Normalized time-domain error of E sampled at point A in the geometry of Fig. 5 with PML terminations of different numbers of cells and the proposed method, using a computational domain of cm. Fig. 10. Numerical error with respect to time of the array-scanning method with different sampling densities, compared with 10-cell PMLs, in the geometry of Fig. 9. Fig. 8. Required CPU time of the proposed method versus the maximum normalized error of E sampled at point A in the geometry of Fig. 5, compared with the 10-cell PML termination. executed serially. The result is compared to the performance of a domain terminated in 10-cell PMLs. Again, the insensitivity of the proposed method to the working volume change is observed. In terms of execution time, the proposed method is preferable if multiple processors are available, as its serial execution remains slower than the PML. IV. NUMERICAL RESULTS: APPLICATIONS A. 1-D Bragg Filter The first application, which has been studied in [6], is shown in Fig. 9(b). The objective is to simulate a 1-D Bragg filter with a periodic dielectric permittivity of the form in the -direction, where cm, within a computational domain of 10 cm 10 cm. The presence of an inhomogeneous dielectric permittivity raises a question as to which particular should the PML be matched to. Studies of various PML-based alternatives were carried out in [6], indicating a substantially increased level of reflection errors in all cases. On the other hand, the application of periodic boundary conditions along the -direction seems a natural way to terminate the FDTD lattice in this case, as the presence of a finite source can be modeled. This is indeed possible via the proposed method, whereby the computational domain is terminated by periodic boundary conditions in the -direction and in perfect magnetic conductors in the -direction. For comparison, an alternative setup with 10-cell uniaxial PMLs terminating the -direction, with fourth-order polynomial grading of the conductivity profile, is also simulated. A uniform line source, of a 5 25 GHz modulated Gaussian waveform in time, is applied at cm. The time step is set to ps and 2048 steps are run. For the proposed method, s are sampled uniformly within the Brillouin zone in both the - and the -directions. Each of these simulations takes 32 s. The alternative setup with PML termination takes 41 s to execute. The reference fields, used for error estimation, are obtained using a large computational domain of Yee cells, so that no reflections from the boundary can reach the positions of interest during the complete simulation time. The results of these simulations are shown in Fig. 10, which corroborates the significantly increased reflections from the PML reported in [6]. On the other hand, the sine-cosine-based array scanning FDTD delivers again a relative error of about 0.1%, with 16 and 32 samples. The effect of the sampling density of on the accuracy of the proposed method is also shown in the figure. The results indicate that the numerical error tends to decrease as the sampling density increases, as expected.

6 1452 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 28, NO. 10, MAY 15, 2010 Fig D dispersive metamaterial lens with NRI n = 01 at 16 GHz. B. Negative Refractive Index Lens The proposed method is applied to simulate the geometry of a doubly dispersive negative-refractive-index (NRI) lens [7], in the time domain. Fig. 11 shows the 2-D computational domain. A dispersive slab with 2 cm thickness is placed in free space, with both magnetic and electric plasma response, following the Drude model: with GHz. The setting yields an NRI at GHz, with a small loss introduced by the term. The fact that yet another modification of the conventional PML formulation is necessary for backward wave media, to avoid numerical instability, has been discussed in [7]. On the contrary, the sine-cosine array-scanning FDTD is readily applicable in this case as well. In FDTD, the computational domain is discretized with Yee cells. The dispersive slab is modeled using the z-transform method of [15]. The time step is set to 0.83 ps. For the proposed method, the computational domain is terminated in 10-cell uniaxial PMLs in the -direction, and in periodic boundary conditions in the -direction, which also includes the NRI region occupied by the slab. The array-scanning integral is approximated with 16 s, which are simulated in parallel. For comparison, an identical domain is terminated in 10-cell dispersive uniaxial PMLs in the -directions. The form of the complex permittivity in the PML region is where is used as a parameter to control the numerical instability observed in [7]. Finally, a time-harmonic current source is placed 1 cm from the first interface between free space and the slab, while the parameter. Fig. 12 shows the time evolution of the transverse electric field at the first and the second interface of the slab using the proposed method, which indeed remains absolutely stable. On the other hand, Fig. 13 shows at the second interface, as computed with the dispersive PML in the -direction for different values of. Evidently, increasing the value of delays the onset of the numerical instability observed in [7] for, yet it cannot eliminate it totally. With a stable simulation technique at hand, some interesting aspects of this super-lens geometry can be further explored. For (5) Fig. 12. Electric field E at the first and second interfaces of the dispersive slab of Fig. 11 and at x = 2:95 cm, using the proposed method for FDTD lattice termination in the 6y-directions. Fig. 13. Electric field E at the second interfaces of the dispersive slab of Fig. 11 and at x = 2:95 cm, using conventional dispersive PMLs for FDTD lattice termination in the 6y-directions, with different. example, Fig. 12 indicates the growth in amplitude of at the second interface, compared to the first. This resonant effect is due to multiple reflections between the two interfaces; its transient evolution can be clearly observed in the time domain. Fig. 14 depicts the magnitude of, recorded throughout the computational domain at various time steps. Evidently, in the beginning, starts as a space wave attenuating away from the source (steps ). However, as multiple reflections build up, the wave attenuation is still featured in the free-space regions, but starts being inverted within the NRI slab, until the steady state of Fig. 12 is eventually reached. More specifically, this growth is indicated in Fig. 15, which demonstrates the temporal evolution of the exponentially growing pattern of the field inside the NRI slab. In this case,, while the theoretical result has been obtained from [16]. As in every resonant effect, the timing of the exponential field growth depends on the losses. This dependence is illustrated in Fig. 16, which shows the temporal evolution of the electric field at the second interface of the lens for different imaginary parts of the refractive index of the slab. Three cases are considered, tuning :, and. Obviously, the higher the losses, the faster the steady state is reached. Moreover, the timing predictions from FDTD are in agreement with the Laplace transform-based calculation of [17],

7 LI AND SARRIS: UNIFIED FDTD LATTICE TRUNCATION METHOD 1453 Fig. 14. Electric field E in the computational domain of Fig. 11, at several time steps. Fig. 15. Electric field E in the computational domain of Fig. 11 along the x axis at y = 2:95 cm, at different time steps (given in terms of the excitation period). which indicated that the time required for an NRI lens to reach steady state is in the order of. V. CONCLUSION The use of periodic boundaries, effected by the sine-cosine FDTD method, was studied as a potential alternative to absorbing boundary conditions and PMLs for FDTD lattice termination. It was found that as long as periodic boundary conditions are applicable, they can deliver at least comparable and potentially better absorptivity than the PML, overcoming existing constraints of conventional absorbers. Moreover, the proposed formulation remains the same regardless of the dispersion or loss of the working volume. This feature is in contrast with PML, which needs additional variables to properly account for electric or magnetic dispersion. To summarize, there are two aspects of the periodic boundary condition-based FDTD lattice termination that the present paper has attempted to illuminate, in order to clarify when this approach may be preferred over existing alternatives. First, in terms of efficiency, this method would be useful in any case periodic methods would be preferable, i.e., when a finite periodic Fig. 16. Electric field E in the computational domain of Fig. 11, at the second interface and at y =2:95 cm, with refractive index of the NRI slab being n = (a) :1j, (b) :01j, and (c) :001j. structure that needs a large number of unit cells to converge to the behavior of its infinite periodic counterpart is simulated. Such cases arise mostly in photonic bandgap structures (see our Bragg reflector example). Note that substantial literature has been dedicated to the question of what kind of PML would be applicable in this case when the dielectric permittivity is inhomogeneous and hence the choice of the PML parameters ambiguous. This method provides a simple answer to this question that works. Moreover, if multiple processors are available to a user, the spectral decomposition would be clearly preferable over classical domain decomposition, as it suffers from no communication overhead and needs no special parallel programming for message passing between sub domains (all wave vector simulations are completely independent). On the other hand, regular PML terminations, if applicable, mostly outperform periodic boundary conditions in a single processor.

8 1454 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 28, NO. 10, MAY 15, 2010 Second, it was demonstrated that a periodic FDTD code, augmented with the array-scanning method, can be recycled as a lattice termination technique for cases where ordinary PMLs would either fail (negative index media) or need substantial modifications (conducting/dispersive media). The very same formulation applies to all linear media, regardless of dispersion, offering FDTD users a convenient, if not always faster, alternative to the PML absorber. REFERENCES [1] A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. Boston, MA: Artech House, 2000, ch. 7, pp , Perfectly matched layers absorbing boundary conditions. [2] A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. Boston, MA: Artech House, 2000, ch. 7.7, pp , Efficient implementation of UPML in FDTD. [3] G. Mur, Absorbing boundary conditions for the finite-difference approximation of the time domain electromagnetic field equations, IEEE Trans. Electromagn. Compat., vol. EMC-23, no. 4, pp , Nov [4] Z. P. Liao, H. L. Wong, B. P. Yang, and Y. F. Yuan, A transmitting boundary for transient wave analysis, Scientia Sinica (Series A), vol. XXVII, pp , [5] J. P. Berenger, Perfectly matched layer for the FDTD solution of wave-structure interaction problems, IEEE Trans. Antennas Propag., vol. 51, no. 1, pp , Jan [6] A. F. Oskooi, L. Zhang, Y. Avniel, and S. G. Johnson, The failure of perfectly matched layers, and towards their redemption by adiabatic absorbers, Opt. Exp., vol. 16, no. 15, pp , Jul [7] S. A. Cummer, Perfectly matched layer behavior in negative refractive index materials, IEEE Antennas Wireless Propag. Lett., vol. 3, no. 1, pp , Jan [8] R. Qiang, J. Chen, F. Capolino, D. R. Jackson, and D. R. Wilton, ASM-FDTD: A technique for calculating the field of a finite source in the presence of an infinite periodic artificial material, IEEE Microw. Wireless Compon. Lett., vol. 17, no. 4, pp , Apr [9] D. Li and C. D. Sarris, Efficient finite-difference time-domain modeling of driven periodic structures and related microwave circuit applications, IEEE Trans. Microw. Theory Tech., vol. 56, no. 8, pp , Aug [10] P. Harms, R. Mittra, and W. Ko, Implementation of the periodic boundary condition in the finite-difference time-domain algorithm for FSS structures, IEEE Trans. Antennas Propag., vol. 42, no. 9, pp , Sep [11] B. Munk and G. A. Burrell, Plane-wave expansion for arrays of arbitrarily oriented piecewise linear elements and its application in determining the impedance of a single linear antenna in a lossy half-space, IEEE Trans. Antennas Propag., vol. 27, no. 9, pp , May [12] A. V. Oppenheim, Signals and Sysmtems, 2nd ed. Englewood Cliffs, NJ: Prentice Hall, [13] A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. Boston, MA: Artech House, [14] G. Lovat, P. Burghinmoli, F. Capolino, and D. R. Jackson, Combinations of low/high permittivity and/or permeability substrates for highly directive planar metamaterial antennas, IET Microw. Antennas Propag., vol. 1, pp , [15] D. M. Sullivan, Frequency-dependent FDTD methods using Z-transforms, IEEE Trans. Antennas Propag., vol. 40, no. 10, pp , Oct [16] W. C. Chew, Waves and Fields in Inhomogeneous Media. Piscataway, NJ: IEEE Press, 1990, pp [17] D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, Limitations on sub-diffraction imaging with a negative refractive index slab, Appl. Phys. Lett., vol. 82, pp , Dongying Li received the M.A.Sc. degree in electrical engineering from Mc- Master University, Hamilton, ON, Canada in 2006, and the B.Sc. degree in electrical engineering from Shanghai Jiao Tong University, Shanghai, China, in He is currently working toward the Ph.D. degree at the Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON, Canada. During , he was a Research Assistant at the Computational Electromagnetics Laboratory, McMaster University, where he was engaged in research on the sensitivity analysis and engineering optimization of microwave structures. In 2006, he joined the Electromagnetics Group, University of Toronto, where he is currently engaged in research on computational electromagnetics, with emphasis in periodic structure modeling for metamaterial applications. Costas D. Sarris (M 02 SM 08) received the M.Sc. and Ph.D. degrees in electrical engineering and the M.Sc. degree in applied mathematics from the University of Michigan, Ann Arbor, in 1998 and 2002, respectively. He is currently an Associate Professor and the Eugene V. Polistuk Chair in Electromagnetic Design at the Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON, Canada. His research interests are in the area of numerical electromagnetics, with emphasis in high-order, multiscale/multi-physics computational methods. He is involved with basic research in novel numerical techniques, as well as applications of time-domain analysis to wireless channel modeling, wave-propagation in complex media and meta-materials, electromagnetic compatibility/interference (EMI/EMC) problems and modeling under uncertainty. Prof. Sarris was the recipient of the Early Researcher Award from the Ontario Government in 2007 and the Gordon R. Slemon (teaching of design) award from the ECE Department of the University of Toronto. His students have received paper awards at the 2009 IEEE MTT-S International Microwave Symposium, the 2008 Applied Computational Electromagnetics Society conference and honorable mentions at the 2008, 2009 IEEE International Symposia on Antennas and Propagation. He serves as an Associate Editor for the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, the vice Chair of the IEEE Toronto section in Electromagnetics and Radiation (a joint MTT/AP/EMC section) and the Technical Program Committee co-chair for the 2010 IEEE International Symposium on Antennas and Propagation. He is the Chair of the Sub-Committee on Time-Domain Methods of the Technical Program Committee of the IEEE MTT-S International Microwave Symposium and the Guest Editor of the IEEE Microwave Magazine Special Issue on Time-Domain Methods for Microwave CAD (April 2010).