1928 IEEE TRANSACTIONS ON MICROWAVE THEORY … · Efﬁcient Finite-Difference Time-Domain Modeling...

1928 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 56, NO. 8, AUGUST 2008

Efficient Finite-Difference Time-Domain Modelingof Driven Periodic Structures and Related

Microwave Circuit ApplicationsDongying Li and Costas D. Sarris, Senior Member, IEEE

Abstract—The sine–cosine method for the finite-differencetime-domain-based dispersion analysis of periodic structures isextended to incorporate the presence of nonperiodic widebandsources. A new formulation of this method is presented to clearlydemonstrate that it can be employed for the characterization ofperiodic structures over a broad bandwidth. Moreover, its cou-pling with the array-scanning technique enables the incorporationof nonperiodic sources, thus enabling the fast characterization ofdriven periodic structures in the time domain via a small numberof low-cost simulations. The convergence, accuracy, and efficiencyof the proposed method is demonstrated with its applicationto the analysis of a negative-refractive-index transmission-line“perfect lens” and the successful comparison of simulated withexperimental results. Finally, a modified version of this methodis proposed for the accelerated simulation of microwave circuitgeometries printed on periodic substrates.

Index Terms—Finite-difference time-domain (FDTD) methods,microstrip circuits, periodic structures.

I. INTRODUCTION

T HE STUDY of periodic structures is motivated by themany applications they can support, either as frequency-

selective surfaces [1], photonic- and electromagnetic-bandgapcrystals [2], [3], or artificial dielectrics [4]. The interest in thelatter area has been recently enhanced by the extensive researchactivity aimed at synthesizing media with unusual macroscopicproperties (metamaterials [5]). Along with this activity, numer-ical tools that can capture unconventional wave effects observedin metamaterial geometries, such as negative refraction, and il-luminate the underlying physics have been proposed. To thisend, time-domain techniques, such as the finite-difference time-domain (FDTD) method [6], are particularly useful because theyeffectively model the rich transients involved with the evolutionof these effects. The potential of the FDTD method to signif-icantly contribute to understanding the nature of wave propa-gation in synthesized media has been demonstrated in severalpapers, including [7]–[10].

Manuscript received March 11, 2008; revised May 15, 2008. First publishedJuly 15, 2008; last published August 8, 2008 (projected). This work was sup-ported by the Natural Sciences and Engineering Research Council of Canada(NSERC) under a Strategic Grant.

The authors are with the Edward S. Rogers Sr. Department of Electrical andComputer Engineering, University of Toronto, Toronto, ON, Canada M5S 3G4(e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMTT.2008.927386

Furthermore, the dispersion analysis of periodic structurescan be carried out by simulating a single unit cell of those,terminated with periodic boundary conditions. Originally castin the frequency domain, periodic boundary conditions can betranslated into the framework of the FDTD method [6]. In [11],the well-known sine–cosine method [12] was employed to an-alyze a recently proposed 2-D negative-refractive-index trans-mission-line structure [13]. In [14], this method was extendedto account for leaky-wave radiation from the same structure,indicating an efficient FDTD-based methodology for the con-current computation of attenuation and phase constants of fastwaves in periodic geometries. Finally, in an effort to investigatethe possibility of transferring the concepts of negative-refrac-tive-index transmission lines from the microwave to the opticalregime (along the lines of [15]), a conformal periodic FDTDanalysis of plasmonic nanoparticle arrays in a triangular meshwas presented in [16].

Building on this earlier work and the research reported in[17], the problem of modeling driven periodic structures withinthe same simulation framework that employs the sine–cosinemethod to implement periodic boundary conditions is consid-ered in this paper. Since the presence of a nonperiodic sourceis not compatible with the use of periodic boundary condi-tions, this problem would be typically handled by simulatinga finite version of the periodic structure, up to the number ofcells necessary to achieve the convergence of the solution.Evidently, the efficiency of this approach largely depends onthe nature of the problem at hand and may be quite costlyin terms of execution time and computer memory. A similarquestion, arising in the context of the method of moments, wasaddressed in [18] by invoking the array-scanning method of[19], to model the interaction of a printed microstrip line withan electromagnetic bandgap substrate. Recently, [17] suggestedthat the same methodology can enable the modeling of drivenperiodic structures by means of the sine–cosine method. Inde-pendently, Yang et al. [20] and Qiang et al. [21], [22] combineda spectral FDTD method with array scanning, thus presentingan alternative methodology for the same class of problems.

In particular, this paper makes the following contributions.First, it is rigorously shown that the sine–cosine method of [12]can be applied for the broadband characterization of periodicstructures, although it had been originally suggested that its ap-plicability was limited to monochromatic simulations [6]. Onthe contrary, a new formulation of the method offers new in-sights to its broadband character and the sources necessary toexcite Floquet modes in a sine–cosine-based FDTD mesh. This

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LI AND SARRIS: EFFICIENT FDTD MODELING OF DRIVEN PERIODIC STRUCTURES AND RELATED MICROWAVE CIRCUIT APPLICATIONS 1929

Fig. 1. Geometry of the problem under consideration: a nonperiodic sourceexciting a 2-D infinite periodic structure of spatial period d along the x-axis.

paves the way for the coupling of the sine–cosine with the array-scanning technique, which results in an efficient modeling toolfor the interaction of broadband nonperiodic sources with peri-odic geometries based on a small number of low-cost simula-tions. The effectiveness of this tool is tested in the challengingproblem of the negative-refractive-index transmission-line “per-fect lens” [23], which is used as a vehicle for the demonstra-tion of its accuracy and convergence properties. Finally, theapplication of the proposed technique to the modeling of mi-crowave circuits printed on periodic substrates is discussed. Itis shown that the presence of aperiodic metallic boundaries (inaddition to the source that excites the microstrip) cannot be ac-counted for by the array-scanning method alone. Instead, a com-posite boundary is proposed, where both periodic and absorbingboundary conditions are applied. Thus, a highly efficient simu-lation approach for this class of problems is offered.

II. ARRAY-SCANNING SINE–COSINE METHOD:FORMULATION AND WIDEBAND VALIDITY

A. Problem Statement

The problem under consideration is shown in Fig. 1, wherethe interaction of a broadband nonperiodic source with an infi-nite 2-D periodic geometry is shown. Instead of approximatingthe infinite periodic structure by a truncated version of it, theproposed solution is based on the computational domain ofFig. 2(a), where periodic boundary conditions are applied tothe electric field phasors (denoted by ) at the boundaries alongthe two directions of periodicity

(1)

where is the lattice vector of the periodic struc-ture and is a Floquet wave vector. How-ever, the computational domain of Fig. 2(a) leads to the so-lution of the problem shown in Fig. 2(b), where the response

Fig. 2. Computational domain for the solution of the problem of Fig. 1 and itsequivalent problem. (a) Computational domain. (b) Problem corresponding tothe computational domain shown above.

of the structure to an array, consisting of phase-shifted peri-odic replicas of the original source is determined. In Fig. 2(b),

.The derivations and numerical results of this section are

aimed at showing that the problem of Fig. 2(a) can be solvedby means of the sine–cosine method [12], and to clarify thesource conditions needed. Moreover, the sine–cosine method iscoupled with the array-scanning technique to isolate the effectof the original source from the combined effect of the phasedarray of sources shown in Fig. 2(b).

B. New Derivation for the Sine–Cosine Method

Consider a field expansion in terms of Floquet modes in a pe-riodic structure of lattice vector , inverse Fourier transformedfrom the frequency to the time domain

(2)


where is an either discrete or continuous spectrum of fre-quencies corresponding to the Floquet wave vector and

(3)

Note that these two waves have identical frequency spectra (asthey share a common temporal dependence). Moreover,

(4)

Similarly,

(5)

Therefore, the Floquet waves are shown tosatisfy the “sine–cosine” boundary conditions of [12]. Note thatit is straightforward to implement (4) and (5) in the discrete-timeframework of the FDTD since all terms on the two sides of theequation are evaluated at the same time step. This formulationoffers new insights into the sine–cosine method. Clearly, thesetwo waves are neither monochromatic, nor at phase quadraturein time. In fact, our sine/cosine waves are distinguished basedon their spatial rather than temporal dependence. Therefore,they can be excited by identical broadband sources (instead ofsine/cosine modulated ones), provided that the frequency spec-trum of such sources includes . Withbeing excited (in their respective meshes), their spectral anal-ysis yields all frequencies at once. This is demonstratedthrough the numerical results of Section II-C.

C. Wideband Validity of the Sine–Cosine Method:Numerical Results

Consider the unit cell of the 2-D negative-refractive indextransmission-line structure that was originally presented in [13],shown in Fig. 3(a). The corresponding positive-refractive indextransmission-line unit cell is also appended in Fig. 3(b). Thisunit cell resides on a substrate of thickness 1.52 mm and relativepermittivity . The spatial periods and (indicated inFig. 3) are both equal to 8.4 mm. The width of the microstriplines is 0.75 mm. In the FDTD mesh, the negative-refractiveindex unit cell is discretized by 22 22 16 Yee cells. Threeof the 16 cells in the -direction model the substrate. The openboundary in the vertical direction is simulated by a uniaxial per-fectly matched layer (PML) absorber [6]. This absorber consistsof ten cells with a fourth-order polynomial conductivity grading.The maximum conductivity value is with

being the Yee cell size in the direction of mesh truncation(hence, in this case, the open boundary being parallel to theplane, ). The same absorber has been used to simulate

Fig. 3. Unit cell of the 2-D negative and positive-refractive index transmissionlines. (a) Negative-refractive index transmission-line unit cell. (b) Positive-re-fractive index transmission-line unit cell.

open boundaries in all numerical simulations included in thispaper.

Moreover, the series capacitor and shunt inductor, shown inFig. 3(a), are chosen to be pF and nH. Thetwo sine–cosine grids are excited by a 0.5–3-GHz Gabor pulse

applied to the components in cellsand inside the substrate. The Gabor pulse parame-ters are ps and . The time step is set to0.723 ps and 60 000 time steps are performed for three cases of

and , while . Hence,all three points are along the portion of the Brillouindiagram of the structure that is occupied by three TM waves,as shown in previous studies as well [11]: a backward, forward,and surface wave. In Fig. 4(a)–(c), the part of theBrillouin diagram for the negative-refractive index transmis-sion-line unit cell, independently determined by Ansoft’s HighFrequency Structure Simulator (HFSS), is shown along withthe magnitude of the Fourier transform (normalized to its max-imum) of a vertical electric field component determined bythe sine–cosine FDTD method and sampled within the substratefrom 0 to 5 GHz. For each case of , the FDTD-calculatedfield presents multiple resonances, which correspond to thefrequencies , given by the intersections of the diagramwith the constant lines. Hence, the FDTD and HFSScalculated resonant frequencies are in excellent agreement.Moreover, it is clearly shown that a single run of the sine–co-sine FDTD, with the same excitation for each grid, is sufficientto determine all resonant frequencies at once. Note that theboundary conditions (4) and (5) enforce the Floquet wave


Fig. 4. (left) Magnitude of the Fourier transform (normalized to its maximum)of a vertical electric field component E within the substrate of the nega-tive-refractive index transmission-line unit cell of Fig. 3(a), determined by thesine–cosine FDTD for three cases of k d . (right) Dispersion diagram (��X)for the unit cell of Fig. 3(a) determined by HFSS. (a) k d = 0:0833� rad.(b) k d = 0:167� rad. (c) k d = 0:333� rad.

vector, while they are independent of frequency, thus settingup an eigenvalue problem in the time domain, where onlythe modes with that given wave vector are excited. This isanalogous to the way the FDTD can be used to characterize

cavity resonances or waveguide dispersion [24] over a broadbandwidth.

D. Coupling the Array-Scanning and Sine–Cosine Methods

The combination of periodic boundary conditions with abroadband source leads to the solution of the problem shownin Fig. 2(b), where the response of the structure to an array,consisting of phase-shifted periodic repetitions of the originalsource, is determined. It is the purpose of the array-scanningtechnique to isolate the effect of the original source, as de-scribed below.

Let be the electric field determined by thesine–cosine method, at a point within the unit cell, for a Flo-quet wave vector within the Brillouin zone ofthe structure (hence, and

). The electric field at this point that is only due to theoriginal source can be found by integrating over [19]

(6)

Since (6) is a continuous integral, while only discreteand discrete points are sampled, (6) is approximated ata time (the th time step of the FDTD method) by thesum

(7)

A modified form of (7) can be employed to determine theelectric field at points outside the simulated unit cell by invokingthe periodic boundary conditions (1). In particular,

(8)

with for integer .The fundamental limit that guides the choice of the number

of points , which need to be sampled inside the Bril-louin zone, is the sampling theorem. If the fields in the drivenperiodic structure under study are spatially limited within thearea , then the samplingrates samples/(rad/m) andsamples/(rad/m) should obey the inequalities

(9)

which leads to

(10)


Practically, safe bounds for and can be deduced fromthe physics of the problem at hand. For example, in the presenceof a driven microstrip line printed over a periodic structure, thespatial extent of the fields can be estimated by the well-knownWheeler formula for the microstrip width correction due to fieldfringing. On the other hand, if there is significant field couplingin the direction of periodicity, a larger number of maybe needed. A convergence study is then necessary. It is notedthough that all sine–cosine FDTD simulations for different ’sare independent from each other, and therefore, lend themselvesto perfect parallelization with no inter-processor communica-tion overhead. In consequence, in a parallel environment, thereis no added cost as the number of sampled Floquet wave vectorsincreases. In the following examples, driven periodic structuresof the artificial dielectric type (with ) and of theelectromagnetic-bandgap type (with ) are examined.Since these are driven along one of the two directions of period-icity, they are treated as 1-D structures, where 10–20 pointsare always sufficient for convergence of the calculated fields.

III. ANALYSIS OF A NEGATIVE-REFRACTIVE-INDEX

TRANSMISSION-LINE-BASED MICROWAVE “PERFECT LENS”

In this section, the proposed method is applied to a microwaveimplementation of Pendry’s concept of a “perfect lens” [25] thathas been recently proposed and experimentally demonstrated[23]. The structure utilizes the unit cells of 2-D positive andnegative-refractive-index transmission lines, shown in Fig. 3.The parameters of both cells and of their FDTD discretizationare the same as in Section II.

To evaluate the convergence properties of the methodology ofSection II, a domain consisting of 18 positive-refractive indextransmission-line cells in the -direction is used. The geometryis treated as a 1-D periodic structure in the -direction and sim-ulated by the sine–cosine method, using one unit cell along thedirection of periodicity, terminated at periodic boundary condi-tions. The vertical electric field nodes within the substrateof the first cell are excited by a sinusoidal hard source at 1 GHz.The vertical electric field at the center of each subsequentcell is then determined in the middle of the substrate. The stan-dard approach to this problem, namely, the use of a finite numberof cells along the -direction until a convergent solution is at-tained, has also been implemented. In both simulations, the timestep is again set to 0.723 ps and the total number of time stepsis 16 384.

The results of the two approaches are presented in Figs. 5 and6, respectively, which include diagrams of the computationaldomains used. It is noted that the sine–cosine method basedarray-scanning converges with points, or a sam-pling rate of rad/m in the wavenumber domain. On theother hand, 17 cells are needed for the field in the finite struc-ture in the -direction to converge within 1% of the fields ofthe infinite periodic one. The convergence properties of the twomethods are summarized in Fig. 7, which depicts the relativeerror norm

(11)

Fig. 5. Vertical electric field E in the middle of the substrate and along they-axis in a positive-refractive index transmission line that is infinite in the x-di-rection. The sine–cosine-based array-scanning method with a variable numberof k points,N , is used. A sinusoidal hard source of amplitude 1 V/m is appliedat the E nodes within the substrate of the first unit cell.

Fig. 6. Vertical electric field E in the middle of the substrate and along they-axis in a positive-refractive index transmission line for 5-31 unit cells in thex-direction. A sinusoidal hard source of amplitude 1 V/m is applied at the Enodes within the substrate of the first unit cell.

where is the -component of the electric field in themiddle of the substrate and along the -axis, calculated with thearray-scanning sine–cosine FDTD and finite structure simula-tions (plotted in Figs. 5 and 6, respectively) and is the samefield calculated with a 32-cell finite structure FDTD simulation.The error norm is plotted with respect to the number ofpoints used for the array-scanning-based field calculation andwith respect to the number of cells in the transverse directionused for the finite structure field calculation.

The electric field amplitude decays away from the source,as expected. As discussed in [23], this amplitude decay, andthe resulting loss of the evanescent spectral components of thesource, can be compensated for by introducing a layer occu-pied by the negative-refractive-index transmission-line cells ofFig. 3(a). To achieve the matching of this layer to the posi-tive-refractive index transmission-line half-spaces (to the leftand right of it) at 1 GHz, the loading elements of Fig. 3(a) are


Fig. 7. Error norm E of (11) with respect to the number of k points used forthe array-scanning-based field calculation and with respect to the number ofcells in the transverse direction used for the finite structure field calculation.

Fig. 8. Vertical electric field E in the middle of the substrate and along they-axis in a planar microwave lens geometry, calculated via the sine–cosine-based array-scanning method (N = 16) and a finite structure simulation, using17 cells in the x-direction.

pF and nH, as in the case studies ofSection II. The characteristic impedance of both lines then be-comes 50 . The domain is excited by a 1-GHz sinusoidal hardsource that is placed 2 1/2 unit cells away from the first inter-face. Note that the negative-refractive index region occupies fivecells, twice as many as the distance of the source and the imageplane from the positive-to-negative index interfaces.

The expected electric field amplitude growth withinthe negative-refractive-index slab is verified by the sine–co-sine-based array scanning, as shown in Fig. 8, which includes a

Fig. 9. Vertical electric field E in the middle of the substrate and along thex-axis in a planar microwave lens geometry, calculated via the sine–cosine-based array-scanning method (N = 16) and a finite structure simulation, using17 cells in the x-direction, at the source and the image (focal) plane. All fieldshave been normalized to their maximum amplitude.

diagram of the computational domain. For these results, 16points have been calculated. For comparison, the results of afinite structure simulation, employing 17 cells in the transverse( -direction) direction, are appended, being in good agreementwith the sine–cosine-based array-scanning results. It is notedthat the field amplitude growth effect is due to resonant cou-pling between the two interfaces, and therefore, builds up ratherslowly during the time-domain simulation. The steady stateis reached in 60 000 time steps, in a total simulation time of2454 s with the sine–cosine-based array-scanning method, asopposed to 20513 s with the finite structure simulation. Hence,the value of reducing the computational domain of the problemis even higher in this case.

The field pattern of Fig. 8 indicates that the matching of thepositive- and negative-refractive-index regions is imperfect,mainly because of the fringing capacitance at the microstripgaps where the lumped capacitors are placed, which contributesto the total gap capacitance. As a result, the field growthstarts outside the negative-refractive-index slab because of theinteraction between incident and reflected waves, somethingalso evident in the experimental results of [23]. The mismatch,along with the fact that the structure has a finite spatial period,leads to an imperfect restoration of the source at the imageplane (2 1/2 unit cells from the second interface), which is stillbetter than the conventional diffraction-limited case. Indeed,Fig. 9 shows the electric field amplitude at the source andimage plane (along the transverse direction), determined via theaforementioned sine–cosine-based array-scanning and finitestructure simulations. The diffraction-limited source image (foran all positive-index space) is also appended. It is noted thatthe half-power beamwidth of the source image extends overfour cells, whereas the diffraction-limited image extends oversix cells. These patterns are in excellent agreement with theexperimental results of [23] for the same structure and offer thefirst full-wave validation of those.


Fig. 10. Geometry of a microstrip line printed on a three-layer electromagnetic-bandgap substrate, introduced in [18].

The vertical field values in the transverse direction, beyondthe simulated unit cell, shown in Fig. 9 have been calculated bymeans of (8). Note that there are significant field values in upto approximately six unit cells in the -direction. As a result,applying (10) with yields as a limit for thenumber of points needed for the reconstruction of the fieldprofile in the space domain, which is consistent with the resultsof our convergence study. This is an a posteriori verification ofthe bounds of (10).

IV. MICROSTRIP LINE PRINTED ON AN

ELECTROMAGNETIC-BANDGAP SUBSTRATE

A. Problem Statement

The second example is the electromagnetic-bandgap sub-strate microstrip line that was studied in [18]. The three-layerperiodic substrate is shown in Fig. 10. All three layers are

mm high and their dielectric constantsare 9.8, 3.2, and 9.8, respectively. The center layer includes pe-riodic rectangular air blocks of 6.5 mm 6.5 mm 0.635 mm.The spacing between the neighboring blocks is mmin both directions. The width of the microstrip line, which isaligned with the air blocks underneath it, is 3 mm.

This structure belongs to a highly practical class of prob-lems, where nonperiodic planar waveguides are printed on pe-riodic substrates. For such applications, a crucial difference be-tween the FDTD and integral-equation method of [18] exists[26]. In the FDTD, surface current densities on metallic guidesare not modeled as sources, while field components tangen-tial to metallic surfaces are set to zero. Hence, if the compu-tational domain is terminated in periodic boundary conditions,these metallic surfaces are periodically reproduced; their pres-ence cannot be eliminated by array scanning. The aforemen-tioned situation is demonstrated in Fig. 11, where representsthe phase progression for a Floquet mode across one unit cell(in the -direction).

To numerically confirm that this problem does accompanythe application of the sine–cosine method, the following simula-

Fig. 11. Problem of simulating a microstrip over a periodic substrate in theFDTD with periodic boundary conditions: array-scanning eliminates the effectof the periodic sources, but not that of the strip boundary conditions (contraryto the integral-equation technique [18]).

tions are performed. First, a single unit cell of the periodic sub-strate is terminated with sine–cosine-based periodic boundaryconditions in the -direction, as shown in Fig. 11(b), and excitedwith a 2–10-GHz modulated Gaussian hard source. The -com-ponent of the electric field is sampled at the plane of the air–sub-strate interface, which is the plane where the microstrip line liesas well. This is compared to the component in a structureconsisting of seven unit cells of the periodic substrate in the

-direction, including the microstrip [similar to Fig. 11(d)]. TheEucledian norms of the two

are shown to be identical in Fig. 12. The plot of this normalso clearly shows that the middle strip (and the associatedboundary condition ) is periodically reproduced by theperiodic boundary conditions, after the application of arrayscanning (which, in [18], was sufficient to eliminate the pres-ence of these strips). Hence, this important difference betweenintegral-equation methods and the FDTD needs to be taken intoaccount when the modeling of printed waveguides on periodicsubstrates is pursued through an approach that is presented inSection IV-B.

B. Proposed Methodology

While previous research on periodic FDTD formulationshas focused on the application of either periodic or absorbingboundary conditions at each boundary of a given computationaldomain (a feature inherited by commercial packages as well),the problem at hand is best served by terminating the substrateat periodic boundary conditions and the space above it, in-cluding the nodes of the metallic guide, in absorbing boundaryconditions (or PMLs). This approach corresponds to the con-figuration shown in Fig. 13. Thus, the periodic imaging ofthe metallic boundaries is prevented, while the array scanning


Fig. 12. Eucledian norm of the x-component of the electric field on the air–sub-strate interface of: a microstrip line over a unit cell of the periodic substrate ofFig. 10 terminated in periodic boundary conditions; a finite structure consistingof unit cells of the same periodic substrate, with microstrip lines printed on eachone of these cells. The position of the microstrip lines in this finite structure isalso shown.

Fig. 13. Combination of periodic and absorbing boundary conditions witharray scanning ensures that the original structure can be simulated through thereduced computational domain.

method can still be employed to isolate the effect of the originalsource excitation.

When a PML absorber is used for the implementation of theabsorbing boundary condition over the periodic boundary, spe-cial care needs to be taken for the update of the electric fieldnodes that are tangential to the interface between the absorberand periodic substrate. Our approach is explained in Fig. 14,where auxiliary magnetic field nodes are introduced within theperiodic substrate along the interface with the absorber to en-sure that the regular Yee updates for the tangential electric fields(which employ these nodes) can be carried out. These auxiliarynodes are easily updated by periodic boundary conditions, usingmagnetic field values within the unit cell. Hence, their presencedoes not add any significant computational cost. Note that at thisregion of the substrate, which is beyond the boundaries of the

Fig. 14. Update scheme for the tangential electric field components at the in-terface between the PML absorber and periodic substrate.

simulated single unit cell, no other nodes (except for these aux-iliary ones) need to be updated.

Subsequently, the sine–cosine-based array-scanning analysisof the structure of [18] is revisited. The structure is considered asa 1-D periodic geometry in the -direction. With one unit cellin that direction, terminated in sine–cosine periodic boundaryconditions within the substrate and the aforementioned PML,the scattering parameters of five unit cells in the -directionare computed. One unit cell, without air blocks, is added be-fore and after these five cells, to provide space for the exci-tation and probe points, giving rise to a domain of seven unitcells in total, terminated at a PML as well. The Yee cell sizeis 0.5 mm 0.5 mm 0.318 mm, and hence, a single unit cellof the structure contains 28 28 30 cells. A 2–10-GHz mod-ulated Gaussian hard source excitation is applied 14 Yee cellsfrom the first set of air blocks in the propagation direction andthe vertical electric field is probed one cell beneath the mi-crostrip at the two boundaries of the perforated substrate. Thetime step is ps and 16384 time steps are run.

The scattering parameters are then computed withpoints that take a total simulation time of 863 s. For comparison,a finite structure with seven cells in the -direction is also mod-eled in 4107 s. The results of the two methods are in good agree-ment, as shown in Fig. 15, and are corroborated by the theoret-ical and experimental results of [18]. This agreement can also beobserved in the time domain. To that end, Fig. 16 presents thetime-domain waveform of the transmitted vertical elec-tric field at the output of the simulated five unit cell geometry(one cell beneath the microstrip), as determined by the sine–co-sine-based array-scanning method and the corresponding finitestructure simulation.

Finally, to confirm that the spurious periodic reproduction ofthe microstrip line (observed in Fig. 12) is avoided, the Eucle-dian norm of the -component of the electric field is sampledone cell below the air–substrate interface and plotted in Fig. 17.Note that, in this case, the plane of the air–substrate interface isnot terminated in periodic boundary conditions; therefore, thevalues of the field beyond the limits of one unit cell in the trans-verse direction at the interface are not determined. Hence, isnow sampled just one cell below this plane within the periodic


Fig. 15. Scattering parameters of the electromagnetic-bandgap substrate mi-crostrip line of Fig. 10, calculated by the sine–cosine-based array-scanning tech-nique (withN = 16k points) and a finite structure simulation, with seven cellsin the x-direction. (a) S . (b) S .

Fig. 16. Time-domain waveform of the transmitted vertical (E ) electric fieldat the output of the simulated five unit cell structure of the electromagnetic-bandgap substrate microstrip line of Fig. 10 (one cell beneath the microstrip), asdetermined by the sine–cosine-based array-scanning method (with N = 16 kpoints) and a finite structure simulation, with seven cells in the x-direction.

Fig. 17. Eucledian norm of the x-component of the electric field one Yee cellbelow the air–substrate interface of: a microstrip line over a unit cell of the pe-riodic substrate of Fig. 10 terminated in periodic boundary conditions withinthe substrate and an absorber from the air–substrate interface on. A finite struc-ture consisting of seven unit cells of the same periodic substrate, with microstriplines printed on the center cell. The position of the microstrip line is also shown.

substrate. Clearly, the field pattern is now free of the spikes thatappeared in Fig. 12, correctly decaying to zero away from themicrostrip.

V. CONCLUSION

The sine–cosine method, which enables the FDTD modelingof periodic structures by simulating a single unit cell, was com-bined with the array scanning technique. Thus, a fast, yet accu-rate, approach for the time-domain modeling of driven periodicstructures was formulated and its potential to achieve signifi-cant savings in both computation time and memory was demon-strated. In addition, the application of this method to the par-ticularly interesting class of geometries, consisting of metallicstrips printed on periodic substrates, has been discussed. It hasbeen shown that FDTD allows for the stable localized applica-tion of periodic boundary conditions and their combination withabsorbing boundary conditions to form composite periodic/ab-sorbing boundaries. This result is extremely important for theefficient treatment of periodic structure-based multilayer circuitand antenna problems since it shows that it is possible to restrictthe application of Floquet conditions in periodic layers only.

There is one key difference between this approach and its ex-isting alternative, which is modeling a sufficiently large finitenumber of unit cells. In the former, convergence depends on thenumber of simulated wave vectors within the Brillouin zone.However, these wave-vector simulations are independent fromeach other and can be readily performed in parallel at the sametime. In addition, their memory cost remains constant withoutscaling with the number of wave vectors. In the latter, on theother hand, convergence depends on the number of unit cellsemployed. Accordingly, the memory and execution time of theassociated problem grows as more cells are added. While struc-tures with weak coupling between unit cells may converge fast,others (artificial dielectrics, such as the first example of thispaper) may need several unit cells before finally converging.For those cases, the method presented here can be particularlyuseful.


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Dongying Li received the B.Sc. degree in electricalengineering from Shanghai Jiao Tong University,Shanghai, China, in 2004, the M.A.Sc. degree inelectrical engineering from McMaster University,Hamilton, ON, Canada, in 2006, and is currentlyworking toward the Ph.D. degree in the area ofcomputational electromagnetics (with an emphasisin periodic structure modeling for metamaterialapplications) at the University of Toronto, Toronto,ON, Canada.

From 2004 to 2006, he was a Research Assistantwith the Computational Electromagnetics Laboratory, McMaster University,where he was involved with the sensitivity analysis and engineering optimiza-tion of microwave structures. In 2006, he joined the Electromagnetics Group,University of Toronto.

Costas D. Sarris (M’02–SM’08) received theDiploma in electrical and computer engineering(with distinction) from the National Technical Uni-versity of Athens (NTUA), Athens, Greece, in 1997,and the M.Sc. degree in electrical engineering, M.Sc.degree in applied mathematics, and Ph.D. degreefrom The University of Michigan at Ann Arbor, in1998, 1998, and 2002, respectively.

In November 2002, he joined the Edward S.Rogers Sr. Department of Electrical and ComputerEngineering (ECE), University of Toronto, Toronto,

ON, Canada, where he is an Associate Professor. He has been involved withbasic research in novel numerical techniques, as well as applications of time-do-main analysis to wireless channel modeling, wave-propagation in complexmedia and metamaterials, and electromagnetic compatibility/electromagneticinterference (EMI/EMC) problems. He authored Adaptive Mesh Refinement forTime-Domain Numerical Electromagnetics (Morgan-Claypool, 2007).

Prof. Sarris is an associate editor for the IEEE MICROWAVE AND WIRELESS

COMPONENTS LETTERS. He was the recipient of an Early Researcher Award pre-sented by the Ontario Government in 2007, a Canada Foundation for InnovationNew Opportunities Fund Award in 2004, and a Student Paper Award presented atthe 2001 IEEE Microwave Theory and Techniques Society (IEEE MTT-S) Inter-national Microwave Symposium (IMS). He was also the recipient of a TeachingAward (for excellence in undergraduate teaching) in 2006 and the Gordon R.Slemon Award (for teaching of design) in 2007 presented by the Electrical andComputer Engineering (ECE) Department, University of Toronto.

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