970 IEEE TRANSACTIONS ON ANTENNAS AND …cem01.ucsd.edu/papers/IEEE_TRANS_AP_54_3_970_2006.pdf · Transmission of Transient Plane Waves Through Perfect Electrically Conducting Plates

970 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 54, NO. 3, MARCH 2006

Transmission of Transient Plane Waves ThroughPerfect Electrically Conducting Plates Perforated

by Periodic Arrays of Subwavelength HolesVitaliy Lomakin and Eric Michielssen, Fellow, IEEE

Abstract—Phenomena associated with the transmission of tran-sient plane waves through plates perforated by subwavelengthholes are studied. Specifically, the time domain transmitted field(TDTF) or transient response of a thin plate (sandwiched inbetween two dielectric slabs) that is perforated periodically bysubwavelength holes and illuminated by a transient plane wave(TPW) is calculated by inverse Fourier transforming the productof the plate’s frequency domain transmission coefficient (FDTC)and the TPWs Fourier transform. The incident TPW comprisessmoothed and damped exponentials. The FDTC has pole andbranch point (BP) singularities associated with resonant andRayleigh–Wood anomalies. Special choice of branch cuts associ-ated with the BPs allows choosing the poles on a single Riemannsheet. The TDTF is represented in terms of incident TPW andFDTC pole residues, and branch cut (BC) integrals. The latter areevaluated asymptotically for arbitrary small pole-branch point(BP) separations. The obtained expressions are simplified andinterpreted for narrow- and wide-band incident fields, and keycharacteristics of the transient fields scattered from the abovestructure are identified. For narrowband fields, only the incidentTPW pole contributions are significant and the resulting TDTF isan attenuated and delayed replica of the incident TPW. When thefield frequency support is near a FDTC pole, significant positivedelays are obtained. When the field frequency support is near aBP, small positive and noticeable negative delays are obtained forlarge and small pole-BP separation, respectively. For wide-bandfields all contributions in the TDTF are significant. The incidentTPW pole contributions are significant only for early time. TheFDTC pole contributions are the counterparts of frequency do-main leaky waves and only are significant for much later time.The BC contributions are the counterparts of frequency domainlateral wave and decay slowly, thereby resulting in a very longTDTF temporal tail. Their magnitudes are substantially larger forsmaller pole-BP separations than for large ones. The investigatedphenomena occur not only on hole-perforated plates but also onother types of periodic gratings. They can find use in constructingnovel probes, antennas, and microwave and optical filters.

I. INTRODUCTION

PHYSICAL phenomena involving the transmission of elec-tromagnetic fields through metal plates perforated by sub-

wavelength size holes are important in numerous applications

Manuscript received August 6, 2004; revised August 3, 2005. This researchwas supported by DARPA VET Program under AFOSR contract F49620-01-1-0228.

V. Lomakin is with the Department of Electrical and Computer Engineering,University of California, San Diego, La Jolla, CA 92093 USA (e-mail:[email protected]).

E. Michielssen is with the Department of Electrical Engineering and Com-puter Science, University of Michigan, Ann Arbor, MI 48109 USA (e-mail:[email protected]).

Digital Object Identifier 10.1109/TAP.2006.869896

in physics and engineering. For example, small holes are usedto excite and couple energy between various waveguide andcavity systems [1]. Wire meshes and plates perforated by sub-wavelength holes often serve as cost-effective ground planesand electromagnetic shields [2], [3]. Finally, probes comprisingsubwavelength apertures are used in near-field imaging and mi-croscopy [4], [5].

Electromagnetic scattering from perforated metal plates hasbeen studied both theoretically and experimentally for decades[6]–[12]. As a result, it is known that plates perforated byjudiciously sized, shaped, and spaced wavelength-size holesmay allow a large fraction of incident power to be resonantlytransmitted [6]–[12]. Such resonant transmission typically isassociated with resonances and propagating modes supportedin the holes. Until very recently, however, plates perforatedby subwavelength-size holes were thought to only allow forweakly transmitted fields. This likely is due to the fact thatclassical Bethe theory [13] predicts that electromagnetic powertransmitted through a free standing infinitesimally thin, flat,and perfect electrically conductive (PEC) plate perforated bysubwavelength noninteracting holes is very weak and scales as

times the power density of the incident field; here isthe hole size and the wavelength of the illumination. Due tocutoff phenomena, the power transmitted through thick platesis even weaker and decays exponentially with the plate thick-ness [6]–[8], [10]–[12]. When the holes interact resonantly,however, the transmitted power may be substantially enhanced.This phenomenon was exemplified by a recent experiment byEbbesen demonstrating that a metal plate perforated by a peri-odic array of subwavelength holes may transmit a large fractionof incident optical power provided that its frequency and angleof incidence are carefully tuned [14]. Ebbesen’s experimentserved as a catalyst to many other studies into enhancedtransmission phenomena beyond the realm of experimental op-tics [15]–[24]. By now, enhanced transmission phenomena areknown to occur also in the terahertz and microwave regimes andnumerous studies have build solid experimental and theoreticalfoundations for their existence [25]–[33]. Optical, terahertz,and microwave enhanced transmission phenomena have beenassociated with strong coupling between the incident field andsource-free fields supported by the plates; the latter arise due tothe existence of a periodic perforation and surface waves (SWs)supported by dense arrays of small holes in plates [28], [29],by dielectric slabs residing on top of plates [30], [31], or bycorrugations in plate surfaces [25], [26]. In the optical regime,the SWs also can be so-called surface plasmon polaritons,

0018-926X/$20.00 © 2006 IEEE

LOMAKIN AND MICHIELSSEN: TRANSMISSION OF TPW THROUGH PEC 971

viz. slow waves supported by metal surfaces with a negativepermittivity—these waves were at the root of the phenomenaobserved by Ebbesen [14]–[21]. Transmission enhancementoccurs when the wavenumber of one of the Floquet modes gen-erated upon illumination of the periodic grating approximatelymatches that of a SW.

Phenomena of enhanced transmission through periodic arraysof subwavelength holes illuminated by time-harmonic planewaves often are manifestations of Wood anomalies [34]–[36].Wood anomalies reveal themselves as rapid variations in theangular or frequency dependence of the scattering coefficientsof periodic structures. Hessel and Oliner demonstrated the ex-istence of two types of Wood anomalies [37], viz. resonant andRayleigh ones. Resonant Wood anomalies occur when a scat-tering coefficient has poles in the complex angular or frequencyplane due to resonant excitation of a structure’s SWs. Whenthese poles are close to the real angular or frequency axis, thenthe scattering coefficients exhibit rapid variations for certainranges of the angle of arrival or frequency of the illuminatingplane wave. Resonant Wood anomalies have been identifiedas the origin of enhanced transmission through arrayed holesin metal plates [22], [28]. Rayleigh–Wood anomalies occurwhen a propagating Floquet mode appears/disappears as theangle of incidence or frequency of the illuminating planewave changes, as such naturally leads to a redistribution ofscattered power between Floquet modes and rapid variations ofscattering coefficients. Angles or frequencies of resonant andRayleigh–Wood anomalies may be distinct or reside near oneanother. In the latter case, only the combined effect of bothanomalies is observed.

Phenomena of enhanced transmission through, and the some-times surprising properties of fields scattered from, periodicarrays of subwavelength holes illuminated by a spectrum oftime-harmonic plane waves often too can be traced to Woodanomalies. Consider a transient plane wave (TPW) describedby a frequency spectrum of plane waves of set direction ofpropagation. Understanding the phenomena associated withthe scattering of TPWs from subwavelength hole perforatedplates is important for many applications. For example, tran-sient excitations can be used to obtain wide-band responsesof perforated plates by means of a single measurement aspracticed in terahertz time domain spectroscopy [8], [27]. Inaddition, perforated plates can be used as narrow band filtersand multiplexers that divide a wide-band signal into narrowband sub-signals [6]–[8], [27]. Finally, femtosecond transienttransmission responses of thin metallic plates perforated bysubwavelength holes were used recently for the excitation andobservation of short-pulse surface plasmon polaritons [21]. Inpast, it has been observed that the scattered TPW scatteredfrom periodic gratings may be delayed [21], [38]–[42] and/ordistorted in shape [8], [21] compared to the incident TPW whenthe latter’s frequency spectral support is near Wood anomalyfrequencies. However, despite extensive studies on scatteringof TPWs from periodic gratings, including plates perforatedby subwavelength holes, several uninvestigated phenomenaand open-ended questions remain. For instance, to the authors’knowledge, effects of Rayleigh–Wood anomalies on TPWstransmitted through metal plates perforated by subwavelength

holes have yet to be characterized. In addition, the combinedeffect of both resonant and Rayleigh–Wood anomalies, in-cluding the influence of the separation of their characteristicfrequencies, on a wide-band TPW transmitted through a peri-odic grating remains unexplored.

This paper studies the transient field scattered by periodicsubwavelength hole perforated PEC plates illuminated by aTPW. The specific focus is on thin plates with circular holessandwiched in between two identical dielectric slabs. Thisparticular structure was selected as it supports phenomenarepresentative of those occurring on many other subwavelengthhole structures and a wide class of periodic gratings, whileallowing for a relatively straightforward analysis.1 Four facetsof the scattering problem are presented in turn. First, expres-sions for the plate’s frequency domain transmission coefficient(FDTC), viz. the ratio of time-harmonic transmitted and inci-dent field amplitudes, which are valid throughout the complexfrequency plane, are proffered. Second, the dispersion graphsand the FDTC’s intricate structure along the real frequency axisare elucidated by means of several numerical examples. Whilethe archival literature abounds with studies of periodic structureFDTC’s, our discussion is tailored to support the ensuing char-acterization of transient fields scattered from such structures.Third, the singularity expansion method [43], [44] and asymp-totic analysis [45] are invoked to derive uniform asymptoticexpressions for the time domain transmitted field (TDTF), viz.the inverse Fourier transform of the FDTC multiplied by theFourier transform of a particular type of incident TPW com-prising a modulated sum of damped exponentials. This choiceof TPW was prompted by the fact that it leads to closed-formasymptotic expressions of transmitted fields that reveal a widearray of scattering phenomena occurring on subwavelengthhole-perforated plates. The derived expressions are uniformlyvalid for narrow- and wide-band fields and account for thepossible proximity of resonant and Rayleigh–Wood anomalycharacteristic frequencies. Fourth, the obtained expressions forthe transmitted field are simplified and interpreted for narrow-and wide-band incident TPWs. The effects of the separationbetween the resonant and Rayleigh–Wood anomaly character-istic frequencies on the TDTF are demonstrated. The paper isorganized as follows. Section II presents the problem configu-ration. Sections IIIthrough VI present material pertaining to thefour above-delineated problem facets. Section VII summarizesour findings.

II. PROBLEM CONFIGURATION

The structure in Fig. 1 comprises an infinitesimally thin PECplate loaded with a doubly periodic Cartesian array of circularholes that is sandwiched in between two identical dielectricslabs. The plate resides in the plane and the slabs havethickness ; the top face of the upper slab thus coincides with

1This structure previously was studied by several authors [6]–[10]. However,for subwavelength holes, the transmission was presumed weak and no enhancedtransmission phenomena were reported (except of our recent works [30], [31]).It is interesting to note that Chen [10] provided approximate expressions for thefield transmitted through thick perforated plates that can predict strong resonantpeaks in the field frequency dependence even for subwavelength holes; unfortu-nately, these phenomena were not identified probably because nothing “special”was expected to occur in this regime.


Fig. 1. Problem configuration.

the plane. The holes are of radius and their center spacingalong and is and , respectively. The slab relative per-mittivity is .

This structure is excited by a polarized TPW with mag-netic field

(1)

Here, with the angle of incidence andis the free-space speed of light. The incident field’s temporal

signature is zero for and has vanishingderivatives at by virtue of the Heaviside step functionand multiplicative factor . For , it comprises two oscil-lating and decaying exponentials with modulation frequencies

and decay constantsand ; and determine the pulse’sduration and rise time, respectively. The temporal offset

ensures that the total field on linecommences at . The incident field’s spectral signature

contains two poles of order at and that residein the upper half complex -plane. The integer leads torapid decay of the spectrum for largeand real . As a result, the frequency content of the TPW is con-centrated between and , where

is a real constant leading to a prescribed smalland .

The transient field scattered by the structure comprises aninfinite number of Floquet (diffraction) modes with normal-

ized transverse wavenumbersand , . In whatfollows, normalized transverse Floquet wavenumbers aredenoted . The magnetic field of

the zeroth-order transmitted TPW is defined as, where is re-

ferred to as the time domain transmitted field (TDTF) andis the structure’s ze-

roth-order frequency domain transmission coefficient (FDTC);here and are the Fourier transforms of and

, respectively.Several parametric relationships that constrain the nature of

the field scattered from the perforated plate are introduced next.First, as this study focuses on scattering from small holes, it isassumed that where . Second, it isassumed that the slab thickness as thisguarantees that only a single SW is supported by the slabsfor all ; the focus is on excitations as they canefficiently excite this SW. Furthermore, it is assumed that thehole periodicity along the -direction satisfiesas this condition guarantees that enhanced transmission of theabove specified incident field always is associated with reso-nances of Floquet modes with transverse propagation vectorsalong , that is, with . No restrictions are imposedon . Finally, it is noted that is complex-valued to allowfor (i) compact derivations of scattered transient field expres-sions and (ii) simple representations of signal envelopes as fieldmagnitudes; real-valued transient fields can be obtained by con-sidering ’s real or imaginary parts.

III. MODEL OF THE FREQUENCY DOMAIN

TRANSMISSION COEFFICIENT

This section explores the mathematical properties of theFDTC on the real -axis and throughout a specificcomplex -plane Riemann sheet (to be defined below). TheFDTC of the above described structure and excitation can beexpressed as

(2)

The first and second terms on the right hand side of (2) rep-resents the FDTC’s resonant and nonresonant components, re-spectively. Each is detailed next.

The resonant FDTC component comprises of a sum of terms,each of which is characterized by a complex pole and acorresponding residue . The causality principle dic-

tates that ’s poles reside in the complex upper half-plane. Symmetry of the Floquet mode structure for positive andnegative requires that and

; here the asterisk denotes complex conjuga-tion (Fig. 2). The nature of these poles is elucidated next.

When , the structure in Fig. 1 supports the lowestorder antisymmetric mode in a slab of height thatpropagates along the structure with normalized transversewavenumber , where . When ,


Fig. 2. Singularities and integrations paths in the complex ! plane. (a) TheBCs are chosen vertically according to (6). (b) The BCs are chosen along thereal ! axis to render Imfk g < 0.

the perforated plate supports source-free fields with com-ponents that comprise Floquet (diffraction) modes

; hereand are Floquet mode indices, are modal

amplitudes, and are complex characteristic frequen-cies. Since and , the characteristicfrequencies can be expressed as ,where are real-valued frequencies determined by

and is a small complex-valued displacement. Since, the above condition for cannot be satisfied

for when . From here on forward, therefore,our focus is on resonant frequenciesgiven by

(3)

and corresponding characteristic frequencies .Note that from the above discussion it follows that no reso-nance occurs for . Interaction of an incident plane wavecharacterized by a normalized wavenumber and (real) fre-quency with the source-free fieldsresults in a strongly enhanced transmitted field. The character-istic frequencies appear as complex poles to the transmis-sion coefficient. These poles are typically associated with res-onant Wood anomalies [37], that is, changes fast for real

near . (While the above line of thinkingbreaks down for large , that is, when the holes themselvesmight become resonant, the corresponding will fall out-side of ’s spectral support).

The second term in (2) represents the nonresonantcomponent of the FDTC. Even though its magnitude typicallyis smaller than that of the resonant component, cannotbe neglected since it has real branch point (BP) singularities.

The BPs are obtained by equating the -directed Floquet modewavenumbers

(4)

to zero; just as with the resonant component of the FDTC, themodes can be safely ignored as their corresponding

BP frequencies will fall outside ’s spectral support. Theresulting BPs are

(5)

The frequencies are real-valued with and(Fig. 2). The BPs determine the condition

when an additional propagating Floquet mode appears/disap-pears as the frequency of the incident field is scanned. Thisphenomenon is accompanied by re-distribution of the trans-mitted energy between Floquet modes and, as a result, by rapidvariations of the FDTC near . The BPs give rise toRayleigh–Wood anomalies; that is, changes fast for real

near BPs [36], [37], [39]. To fully define the FDTCin the complex -plane, branch cuts (BCs) associated withthe z-directed wavenumbers and corresponding upper andlower Riemann sheets must be specified. Following [45], isrepresented as , where

. Then the BC associated withand the corresponding upper Riemann sheet is chosen

according to

(6)

The BCs corresponding to the BPs are shown inFig. 2(a) as vertical wiggly lines. The definition in (6) ensuresthat , for realand that , for realor . The upper Riemann sheet of is thatwhere all are chosen according to (6). It is found that

when ; the andsigns correspond to proper decay and improper growth ofthe field away from the plate, respectively. It is importantto mention that these proper and improper poles adequatelyrepresent the FDTC in (2) while (mostly2) residing on a singleRiemann sheet. The BC definition in (6) also leads to a compactderivation of expressions in Section IV for the TDTF. It is notedthat, if (2) is interpreted only as a representation of forreal as opposed to one valid through an entire Riemann sheet,

2Except near Rayleigh–Wood anomaly frequencies—see next section.


Fig. 3. Dispersion graphs for the structure in Fig. 1 with L = 2:6 mm, L = 0:4 mm, s = 0:19 mm. Circle, diamonds and triangles designate first, secondand third passbands, respectively. (a), (b) Ref! g, ! and Imf! g, Res fT̂g versus � for h = 0:3 mm and " = 3. (c), (d) Ref! g, ! and

Imf! g, Res fT̂g versus � for h = 0:6 mm and " = 1:1.

then other branch cut choices become possible. For example,the upper Riemann sheet of the can be chosen to yield

for all [46]; doing so would lead to decayof the corresponding Floquet modes as they propagate awayfrom the structure and result in BCs defined via ,

; these BCs are depicted in Fig. 2(b) by horizontalwiggly lines. Unfortunately, this BC specification does not suitthe purpose of our study, as the poles that govern ’sresonant behavior on the real -axis then all reside on differentRiemann sheets. In addition, when plotting dispersion diagrams,this choice would require switching between different Riemannsheets to pick up physical poles thus making the computation ofdispersion diagrams inconvenient.

IV. ANALYSIS OF THE FREQUENCY DOMAIN

TRANSMISSION COEFFICIENT

This section explores the physical content of ’s mathe-matical description through a number of concrete examples. Alldata was generated using the modal approach of [9], in whichfields outside the plate and in its apertures are expanded in termsof Floquet and circular waveguide modes, respectively. Modalexpansion coefficients are solved for by enforcing field (deriva-tive) continuity across the aperture interfaces. The poles arefound from the by allowing to be complex.

A. Poles and Residues

Fig. 3 depicts typical dispersion graphs of ’s lowest-order poles, their corresponding residue values, and trajecto-ries for the lowest-order BPs, all versus positive . Problemparameters are , , ,

and . Fig. 3(a) tracks andin the first three passbands. Pole trajectories near(broadside illumination) and delineate stopbands.The pole governs ’s behavior throughout the entirefirst passband. Other poles, in contrast, may cross over fromone passband to another when frequencies approach stopbandsand; nomenclature consistency requires that different ’sswitch trajectory. The trajectory never crosses any BCtrajectory whereas trajectories of others poles do. For instance,in the second passband ’s trajectory crosses that of theBC associated with near ; similarly, in thethird passband the and trajectories cross those ofBCs associated with and near and

, respectively. When this happens, pole trajecto-ries are discontinuous. Indeed, after hitting a BC from oneside, a pole will migrate into that BCs lower Riemann sheet.Virtually simultaneously, however, a pole will migrate fromthe other side of the BC in the lower Riemann sheet into theupper Riemann sheet. In other words, when a pole is close


to a BP not one pole but two, namely the one disappearinginto the lower Riemann sheet and the one migrating into theupper Riemann sheet, significantly contribute to forreal in (2) (and either the resonant contribution toshould be completed with poles from a lower Riemann sheet,or the becomes locally resonant). The discontinuity inpole trajectory is especially visible in Fig. 3(b), which shows

and the residue magnitude for posi-tive . It should be noted, however, that poles continuouslymigrate between Riemann sheets. The discontinuities in thepole trajectories in Fig. 3 appear because only the (relevant)poles on the upper Riemann sheet are tracked and poles onlower Riemann sheets are disregarded. In the first passband

for any . The magnitude of

the residue in the second passband changes from

to when

crosses the BC associated with , i.e., whenchanges from to . Forall other , . Note also that in thestopbands the imaginary part and residue of one of the polesvanishes. For example, and approachzero when .

The separation between the pole and BP definedas depends on the structure parameters. As willbe shown below, this separation significantly affects the TDTF.Two cases should be distinguished. Case 1: The separation

is small. This situation occurs, for example,when is small and is close to unity, i.e., when the slabs areelectrically thin, as this leads to in (3). Fig. 3(c) and (d)depicts dispersion graphs when the plate parameters are chosenas in Fig. 3(a) and (b) but the slab thickness andpermittivity (i.e. ). It is seen that the trajectoriesof track those of much more closely than inFig. 3(a) and (b); for yet smaller values of and these tra-jectories are found to be visually almost indistinguishable ondispersion graphs like those in Fig. 3. Case 2: The separation

, is small. This situations may occur whenthe trajectories of and cross as was discussed in con-nection with Fig. 3(a) and (b). Clearly, in Case 1, the separation

is small for all whereas Case 2 occurs only forspecific and and in a special narrow range of . Moreover,unlike Case 2, Case 1 may occur for the lowest order poleand BP , which may be of interest in applications. Forthis reason, in the remainder of this paper, only the influence ofCase 1 separations on the FDTC and TDTF is studied in detailand only minor attention is paid to Case 2.

B. FDTC Magnitude and Phase for Real

It follows from (2) that exhibits resonant peaks andrapid variations for near or . To fully grasp

’s behavior, consider a range of frequencies near a poleand assume that this pole is not close to any BP; the latter

condition is satisfied when the slabs in Fig. 1 are sufficiently

Fig. 4. Singularities of T̂ (s) and f̂(s) and the integration path in the complexs plane.

electrically thick and do not lead to the pole trajectory inter-section in Fig. 3. Then, only the contribution of the poleaffects the FDTC resonantly and the FDTC can be expressed as

(7)

where .

From (7), has a simple zeroresiding near the pole ; it can be

shown that this zero is real valued [30]. As a consequence,when the incident field frequency is near , then hasan asymmetric profile and the phase of denoted byvaries rapidly.

The situation is different when the pole is close to a BP;this situation occurs e.g. when the slabs are electrically verythin. In this case, both the resonant and nonresonant contribu-tions affect the transmission coefficient simultaneously. Assumethat the dielectric slabs are electrically thin such that the pole

on the top Riemann sheet is close to the BP . To takeinto account the combined effect of the poles and BPs, a changeof variables

(8)

is introduced and the FDTC is approximated as

(9)

Here, is a pole that resides in the thirdquadrant of the complex plane (Fig. 4),

, and

. Introducing the variable assures thatthe first (resonant) component in (9) takes into account both thepole and BP and allows approximating the second(nonresonant) term by a constant. It also indicates an intimaterelation between the existence of the resonant and RayleighWood anomaly types in this case. From (9), the FDTC has poles


Fig. 5. FDTC T̂ (!) for � = 0:2, L = 2:6 mm, L = 0:4 mm, s = 0:19 mm, " = 1:1, and h = 0:6 mm and h = 0:1 mm. (a) Magnitude jT̂ (!)jand (b) phase �(!). The resonant regime correspond to the poles ! = 5:9968� 10 + 1:9� 10 j and ! = 6:0345� 10 + 7:6� 10 j for thick(h = 0:6 mm) and thin (h = 0:1 mm) slabs, respectively. The BP is ! = 6:04152� 10 .

and zeros in the complex plane. The effect of the poles, zeros,and BPs results in maxima and minima in the FDTC. However,for real , these maxima and minima can be displaced furtherfrom those predicted by the poles or disappear. The phaseexperiences rapid variations due to a combined effect of thepole and the closely residing BP.

To verify the above discussions, the FDTC was calcu-lated for , , , ,

and two values of and .The values of and were chosen to be small such that isfairly close to . For instance, the lowest positive BP

and the poleand forand , respectively. They correspond to separa-tions and

for and , respectively;the former and latter values of in all numericalexamples that follow are referred to as large and small (pole-BP)separations corresponding to thick and thin slabs, respectively.

Fig. 4(a) and (b) shows the magnitude and phase of forthe real frequency range covering and . Thisfrequency range was chosen to demonstrate enhanced transmis-sion peaks and rapid variations in the FDTC associated with thepoles and BPs. In Fig. 5(a), a transmission peak in isobserved at and for

and , respectively; it derives from thecorresponding pole . The minimum of is found tobe a real zero between and for(large separation ), whereas it is obtained at

for (small separation ). InFig. 5(b), rapid variations/discontinuities of are observedwhen passes through ’s pole, real zero, or BP. For realnear the slope of is negative. For real near

, the slope of is negative for (large) whereas it is positive (for ) for

(small ). The change of the FDTFmagnitude and phase behavior also has been observed for otherpoles and BPs and other structure parameters. Finally, it is notedthat the approximate results obtained via (7), (9) and the resultsin Fig. 5 cannot be distinguished visually if plotted together;they therefore are not shown.

V. MODEL OF THE TIME DOMAIN TRANSMITTED FIELD

This section uses the singularity expansion method [43], [44]and asymptotic analysis [45] to represent the TDTF andderive closed-form asymptotic expressions that permit physicalinterpretations. It follows from (1) that can be calculatedfrom and via

(10)

The integration path in (10) runs slightly below the real -axisto pass below the BPs [Fig. 2(a)].

For , the integration path in (10) can be closed in thelower half complex -plane. As this half plane is devoid of polesand BCs, it follows that for , in agreement withthe causality principle. For the integration path in (10) canbe closed in the upper half complex -plane. It follows from theCauchy theorem that the integral in (10) can be expressed as

(11)

The functions and in (11) representcontributions to resulting from the poles inand in , respectively. These functions thus can be ex-pressed as

(12)

Note that the FDTC and exponential termscannot be separated from in the expression for

since the order of the poles can be largerthan one.


The function in (11) represents contributions of in-tegrals along BCs associated with the BPs and is expressedas

(13)

where is an integration contour tracking the th BC de-fined in (6) [Fig. 2(a)].

For large , the BC integrals in (13) can be eval-uated asymptotically as explained next. The arguments belowpertain to the case where pole (potentially) resides nearthe BC (Section 4.ii, Case 1), but are easily extended toaccount for the case where poles cross branch cut trajectories(Section 4.ii, Case 2). Direct evaluation of is hin-dered by the fact that has unbounded fast variations when

. These variations however can be avoided by intro-ducing the variable in (8), which maps the BPinto the origin . With this change of variable theintegral becomes

(14)Because of the change of variable (8), now is an ana-

lytic function near . A map of the integrand of (14) and thenew integration path in the complex -plane are shown in Fig. 4.The integration path is a straight line that passes through theorigin at an angle of radians w.r.t. the real -axis. The partsof the integration path that are below and above the real -axiscorrespond to the left and right sides of the -plane BCs. The ex-ponential factor in the integrand in (14) introduces a saddlepoint at and the integrand possesses polesand arising from and , respec-tively. Because the pole resides on the upper Riemann sheetassociated with the BP , it is mapped into a single polein the third quadrant of the -plane. In contrast, the polesare mapped into symmetrical poles in the first and thirdquadrants. All the poles can reside arbitrarily close to each otherand to the saddle point . When , the inte-gral in (14) can be evaluated asymptotically using the steepestdescent path (SDP) technique [45]. Due to the BC definitionin (6), the original integration path is the SDP and no addi-tional path deformation is called for. Since the pole can re-side arbitrary close to the saddle point , its effect mustbe accounted for explicitly. To this end, is expressedas . Then, the numerator anddenominator are multiplied by leading to

(15)

with ; this seemingly artificial in-troduction of an additional pole at (Fig. 4) renders thedenominator symmetric around the -origin, which simplifiesthe evaluation of the integral, as demonstrated next. Since the

numerator in (15) is an analytic function around ,it can be expanded as , where

. Substituting thefirst anti-symmetric term in this expansion along with (15) intothe expression (14) for , it follows that

(16)Next, recalling the expression for in (1) and the map-

ping in (8), the ratio and hence the integrandin (16) can be expressed as a sum in which each term comprisesonly a single pair of symmetric poles, or no poles at all. Suchintegrands allow a uniform asymptotic SDP evaluation of the in-tegral [45] that accurately accounts for the (possible) proximityof the poles to the saddle point. The resulting uniform asymp-totic expression for (16) is

(17)

Here is a transitionfunction; erfc is the complementary error function with com-plex argument [47]. This transition function asymptotically be-haves as when and as

when . The latter property ren-ders all three terms in (17) bounded as the poles ,which happens for small and when the poles andare close to the BP . For large and when the are farremoved from the origin, then it is possible to simplify (17) bysetting . Finally, using transformation (8), expression(17) can be recast in terms of , and resulting in

(18)


To summarize, is given by (11) with andspecified by (12) and by (13) or (18)

provided that . Note that the latter condition doesnot severely restrict usage of (18) as long as . Thisis because the pulse duration is and therefore (18) onlyfails inside a temporal window comprising a very small fractionof the incident pulse duration.

Finally, it is noted that when in the dispersion curve inFig. 3 is such that a pole crosses a BC , (Sec-tion 4.ii, Case 2), two poles from both Riemann sheets can re-side close to the BP. In this case, the integral evaluation is to beslightly modified to account for all possible poles. In still othersituations two BPs can reside close to each other. For instance,the BPs and are close when the incident anglebut (Fig. 3). When this occurs, (18) is to be modified for

, although it remains valid for sufficiently latetime . Finally, it is noted that, (18) remainsvalid for the important case of broadside incidence sincethe BPs ( and ) coincide resulting in a single BC.

VI. ANALYSIS OF THE TIME DOMAIN TRANSMITTED FIELD

This section provides physical interpretations of the math-ematical expressions for provided by (11) along with(12), (13), and (18) by considering a variety of limiting cases.For simplicity, it is assumed that the incident signal’s spectralsupport covers, or is close to, only a single pole

and/or BP ; for convenience, the subscript andsuperscript are omitted in the remainder of the section.Section VI-A explores when is narrowband and theBP resides away from the incident signal’s spectral support

. Understanding the field behavior in this case isimportant in applications where perforated plates are used asnarrow band filters [38], [39]. Section VI-B studies when

is wideband and/or the interval containsthe BP . Understanding the field behavior in this case isimportant in applications where transient excitations are usedto obtain wide-band responses of perforated plates, e.g., in timedomain spectroscopy [8], [11], [27].

A. Narrowband Fields

1) Expressions for : Assume that the following condi-tions are satisfied:

(19)

The first condition implies that is narrowband com-pared to the resonant term in (2). The second conditionensures that resides outside the interval .Under conditions (19), can be considered slowly varyingcompared to within and the contribu-tions and to can be neglected,that is . Next, is expressed

aswhere the prime denotes a derivative w.r.t. . Approximating

by assuming that is approxi-mately constant near and using (12) yields

(20)

It follows that, to first order, is a scaled and delayedreplica of ; the delay of the carrier and envelope parts of theTDTF can be referred to as phase and group delay, respectively.In reality, slightly differs in shape from ascannot be assumed perfectly constant near .

2) Discussion of the Time Delay: By expressing the FDTCas [see (8) and (15)], canbe expressed as

(21)The first component in the parenthesis of (21) is strong fornear, and explicitly contains, both the pole and BP. This indi-

cates that both poles and BPs may affect the time delay, simulta-neously, in the regimes of resonant and Rayleigh Wood anoma-lies. The second component is strong only for near the BP. Toclarify the importance of the time delay in the transmitted field,it is instructive also to relate the absolute delay to the incidentpulse width or rise time. To this end, the relative delay is definedas . The expression for the absolute and relative time delaysare simplified and interpreted in several limiting cases furthernext.

Time delay for (the regime of resonant Woodanomalies): Absolute time delay : Consider the situationwhere the slabs are thick enough such that the pole-BP separa-tion is large. When the field frequency support is nearthe resonant Wood anomaly angle, i.e., when ,then the second component in (21) can be neglected and

. From here, is positive and itreaches the maximum of for .

Relative time delay : Clearly, for a fixed , the rela-tive delay scales with . It should be noted, however, that isbounded by conditions (19). From the first condition in (19), itfollows that with , i.e., the maximalbandwidth of the field is bounded by the bandwidth of the reso-nant regime. Therefore, in the regime of resonant Wood anoma-lies, i.e., when , it is found that ; that is,the maximal relative delay is positive and determined entirelyby .

Time delay for (the regime of Rayleigh–Woodanomalies): Absolute time delay : When the field fre-quency support is near the Rayleigh–Wood anomaly angle,i.e., when , then the first component in (21) can beneglected and , where

,for . From here, due to the dependence ,

is unboundedly large as . When the slabs are thicksuch that pole-BP separation is fairly large,such that is dictated by the ratio and it canbe positive or negative depending on this ratio. When the slabsare thin such that is small, then is dominant and

is dictated by the distance between the pole and BPvia . Since for the current case is in the third quadrant ofthe complex plane, it follows that a positive or negative timedelay is obtained for or , respectively. Sinceis a smooth function of , it is expected that the time delay in


Fig. 6. Delay and FDTF when f̂(!) is narrowband and ! 2 (! ; ! ) includes Ref! g = Ref! g. The structure parameters are as in Fig. 5. (a) �t

versus ! for h = 0:6 mm and h = 0:1 mm. (b) Normalized by their maximal values jT (t)j and jf(t)j for h = 0:1 mm and � = 2:5 � 10 , � = 2� �0:66Imf! g and ! = 6:0345 � 10 = Ref! g.

the case of small pole-BP separation is much larger than in thecase of large one.

Relative time delay : From the second condition in (19),it follows that with , i.e., the maximalbandwidth is bounded by the closeness between the carrier fre-quency and the BP. Therefore, in the regime of Rayleigh–Woodanomalies, i.e., when , it is found that

. From here, unlike the absolute time delay, therelative time delay as .

Note that the fact that the time delay can be negative meansthat the group velocity of the transmitted TPW can exceed thefree space wave velocity. Such superluminal group velocity,however, does not violate the causality principle because, whileapproximately replicating , in reality is still reshapedand attenuated [38]–[40]. Moreover, the causality is impliedexplicitly in the derivations of Section V. It should also be notedthat the phenomena of superluminal group velocity in scatteredfields were associated with evanescent fields supported byvarious structures [8].

3) Numerical Simulations: In the following numerical sim-ulations the structure parameters were chosen as in Fig. 5 and thederivative order in (1) was . The incident signal’s spectralsupport was chosen nearor . For , the TDTF magni-tude and time delay are expected to be large as, from the Sec-tion V, VI and Fig. 5, and has alarge slope. For , the time delay in the TDTF is ex-pected to be large as has an unboundedly large slope.It should be noted, however, that the magnitude of the TDTF for

is weaker than that for .Fig. 6(a) shows when is near , i.e., when the

carrier frequency approximately matches the resonant condi-tion/Rayleigh Wood anomaly angle. The delay is positive forboth thin and thick slabs and itsmaximum value is obtained for . Fig. 6(b) depicts

and normalized by their maximal magnitudes forthin slabs , ,and . The results obtainedvia (11) with (12) and (18) agree well with those obtained via(20); some shape distortion is observed because was chosenfairy large to render the relative delay large. Note that thedirect integration (10) provides an essentially identical result

for , which, therefore, is not shown. In agreement withthe discussion in Section VI-A.2, the delayis positive and constitutes a significant fraction of the pulsewidth: .

Fig. 7(a) shows when is near , i.e., when the carrierfrequency approximately matches the Rayleigh–Wood anomalyangle. As expected, is unbounded as . It is foundthat is positive for thick slabs (large separation

) whereas is negative in the range forthin slabs (small ). In addition, in thelatter case, is considerably larger (by more than an orderof magnitude). Fig. 7(b) depicts corresponding to inFig. 7(a) as a function of for . Asexpected, as for both values of . However,for thick slabs (large ), is positiveand increases monotonically with whereas for thin slabs

(small ), has a maximum atand its magnitude significantly exceeds that for thick

slabs . Fig. 7(c) shows normalized magnitudes ofand for thin slabs (small ),

, , .A noticeable negative delay is observed; itcorresponds to . The relative delay for thick slabs

(larger ) is very small and the normalizedmagnitudes of and cannot be distinguished visuallyif plotted together. Note that the behavior of the TDTF and thetime delay agree with the discussions in Section VI-A.1 andVI-A-2.

Finally, it should be noted that delays of narrowband TPWswhen propagating through various periodic gratings includingsubwavelength hole structure have been studied theoreticallyand experimentally in past [38]–[42]. Here some of these re-sults have been reinterpreted and clarified within the frameworkof the material presented in the previous sections and some newobservations have been made. For instance, the delay of nar-rowband TPWs with frequency support covering both types ofWood anomalies transmitted through hole-perforated plates wasstudied and a significant increase was observed in the time delayassociated with Rayleigh–Wood anomalies as the separation be-tween the resonant and Rayleigh–Wood anomalies character-istic frequencies shrinks. In addition, differences in the behav-iors of absolute and relative time delays were clarified.


Fig. 7. Delay and normalized by its maximal magnitude FDTF when f̂(!) is narrowband and ! 2 (! ; ! ) is near ! = ! . The structure parametersare as in Fig. 5. (a) �t versus ! for h = 0:6 mm and h = 0:1 mm. (b) ��t versus ! for h = 0:6 mm and h = 0:1 mm when � = 0:33j! � ! j.(c) Normalized jT (t)j and jf(t)j for � = 3� 10 , � = 2� = 0:33j! � ! j and ! = 6:04171� 10 for h = 0:6 mm and (d) h = 0:1 mm.

B. Wide-Band Fields

When conditions (19) are not satisfied then all three contri-butions in (11) can be significant. Below these contributions areexplored and their effects on are elucidated.

1) Incident Signal Pole Contribution —“Direct”Response: This contribution is given by the first line in (12).The time dependence of is expressed as a weightedsum of two oscillating and damped exponentialsand . Therefore, the function contributessignificantly for , where is a constant; thatis, and are of the same duration. Therefore,

represents the direct response of the structureto the incident TPW When the field is wideband such that

, the function decays much faster than. The magnitude of is significant only

when the incident TPW frequency support overlaps with thesupport of the resonant (enhanced) transmission regime, i.e.,when the signal carrier frequency is near the pole.

2) FDTC Pole Contributions —Reso-nances: FDTC pole contribution is given by the secondline in (12). The time dependence of is

, where is the carrier frequencyand determines the decay rate. The contribution of

is significant for . Whenthe field is wideband such that , the function

affects the field much longer than .Physically, it arises due to a resonance supported by the perfo-

rated plate. It can be considered as the time domain counterpartof a frequency domain leaky wave [23], [24], [48]. The existence

of these contribution and corresponding resonances are linkedto the enhanced transmission peaks in the FDTC.

From here, it is clear that if one uses transient measurementsto obtain wide-band information about the FDTC [8], he needsto record the TDTF for long enough time in order to resolvethe frequency domain variations associated with poles/reso-nances/resonant Wood anomalies. Finally, incidentally, if thestructure in Fig. 1 is truncated, then its dimension along the

-direction should measure at least forto be noticeable.

3) The BC Contribution —Diffraction Field: BCcontribution is given by (13) and evaluated asymptoticallyin (18) for . For time the BCcontribution contains the error function that describes themutual influence between the poles and BP . Theuniform expression in (18) can be simplified for the late time

(provided that ) by approximatingthe transition function in (18) as

(22)

It follows that the magnitude of depends on thestructure parameters via and the pole-BP separationas well as on the incident TPW via . The time depen-dence , however, is only determined by the period

and does not depend on other structure parameters. Thecarrier frequency of is given by the BP and itstime decay is algebraic. Due to this slow decay, eventually(that is after and become negligible


Fig. 8. Transmitted field for h = 0:6 mm (large j! � ! j), ! = 5:9717 � 10 , � = 1:3 � 10 and � = 2�. These parameters are chosen such that� � Imf! g and ! 2 (! ; ! ) overlaps with Ref! g but not ! . Other structure parameters are as in Fig. 5 and h = 0:6 mm. (a) jT (t)j and scaledjf(t)j. (b) Magnitude of different contributions in T (t).

Fig. 9. Transmitted field for h = 0:6 mm, ! = 6:0415 � 10 , � = 6:2 � 10 and � = 2�. These parameters are chosen such that j! � ! j is large,� � Imf! g and both Ref! g and ! are within ! 2 (! ; ! ). Other structure parameters are as in Fig. 5. (a) jT (t)j and scaled jf(t)j. (b) Magnitudeof different contributions in T (t).

for and the contribution ofbecomes dominant and leads to a long time tail in

.Physically the BC contribution arises due to power inter-

change between different Floquet (diffraction) modes; it canbe therefore referred to as diffraction field. The BC contri-bution can be considered as the time domain counterpart offrequency domain lateral waves supported by periodic gratings[31], [45]. The existence of the BC contribution is linked toRayleigh–Wood anomalies.

It follows that the ability to resolve the fine features inthe FDTC for the frequencies near the BPs/Rayleigh–Woodanomalies from wide-band time domain measurement is limitedby the recording time of the TDTF. For instance, when theTDTF is not recorded for long enough time the fine structureof the FDTF in the regime of Rayleigh–Wood anomalies canbe lost [8].

4) Numerical Study: In the following numerical simulationsthe TDTF was calculated via (11) with (12) and (18);integration (10) provided identical results, which therefore arenot shown. The structure parameters were chosen as in Fig. 5and derivative order in (1) was . The field frequencysupport covered and/or .These parameters were chosen to demonstrate combined ef-

fects of poles/resonances and BC contribution/diffraction fieldon wide-band transmitted fields.

Fig. 8(a) shows the magnitude of and for thickslabs , and , and

. These parameters were chosen such that the pole-BPseparation is large, the incident field’s bandwidth islarge such that , and the incident field’s spectralsupport contains but not such that .This choice allows demonstrating effect of the pole/resonanceon the TDTF. It is observed that the shape of is distortedfrom that of ; it has two peaks due to a combination ofthe functions and . Fig. 8(b) shows themagnitudes of the contributions of , and

composing the TDTF . It is seen that the magni-tude of is negligibly small since is large. Forthe early time the magnitudes of andare of the same order. However, the duration of islonger than that of since the former has narrowerband .

Fig. 9(a) and (b) shows and , and the contribu-tions composing for , ,

and . These parameters were chosensuch that the pole-BP separation is large, the inci-dent field’s bandwidth is large such that and both


Fig. 10. Transmitted field for h = 0:1 mm, ! = 6:0415 � 10 , � = 6:2 � 10 and � = 2�. These parameters are chosen such that j! � ! j is small,� � Imf! g and both Ref! g and ! are within ! 2 (! ; ! ). Other structure parameters are as in Fig. 5. (a) jT (t)j and scaled jf(t)j. (b) Magnitudeof different contributions in T (t).

Fig. 11. Transmitted field for h = 0:1 mm, ! = ! = 6:04152 � 10 , � = 3 � 10 and � = 2�. These parameters are chosen such that j! � ! j issmall, j! � ! j � � but j! � ! j � �. Other structure parameters are as in Fig. 5 and h = 0:1 mm. (a) jT (t)j and scaled f(t). (b) Magnitude of differentcontributions in T (t).

and are within the incident field’s spectral support.This choice allows demonstrating effect of the pole/resonanceas well as BC contribution/diffraction field on the TDTF. InFig. 9(a), the function has a long time tail and severalmaxima resulting from the combination of various contributionsin . Fig. 9(b) shows that all contributions are of the sameorder for early time. The contribution of lasts muchlonger than since . For late time, how-ever, is dominated by , as seen in the insert.

Fig. 10 repeats the experiment leading to Fig. 9 but forthin slabs (i.e. for small separation ).In Fig. 10(a), the function has a long time tail and isdistorted from . As shown in Fig. 10(b), due the smallseparation , the magnitudes of the contributions of

and increase significantly compared tothose in Fig. 9(b) while the contributions of isdeemphasized. The FDTC pole contribution lastsmuch longer than that in Fig. 9 since is smaller. TheBC contribution again is dominant for late time.

Finally, Fig. 11 shows and for thin slabs, , and .

These parameters were chosen such that the incident field’s fre-quency support covers the BP but the band-width is smaller than that of the FDTC resonant bandwidth, i.e.,

. It is seen that due to latter condition, theFDTC pole contribution is negligible. The magni-tudes of and are of the same order for earlytime, but is dominant for late time.

VII. SUMMARY AND DISCUSSION

Phenomena associated with transmission of TPWs throughinfinitesimally thin PEC plates sandwiched between dielectricslabs and perforated by doubly-periodic arrays of sub-wave-length holes were studied. The TDTF characterizing transmittedTPWs was represented as the inverse Fourier transform of theproduct of the Fourier transform of the incident TPW and theFDTC. The incident TPW studied comprised of a sum of twosmoothed and damped exponentials; its Fourier transform con-tains two higher order poles. The FDTC was represented interms of resonant and nonresonant part associated with com-plex pole and BP singularities. By a proper choice of BCs thepole singularities (mostly) reside on a single Riemann sheet inthe complex frequency plane.

By closing the Fourier integration contour around the singu-larities for the positive time, the TDTF was represented in termsof incident TPW and FDTC pole residues, and BC integrals. The


BC integrals were evaluated asymptotically via the SDP tech-nique. The derived expressions account uniformly for the pos-sible proximity of the poles and BPs.

The obtained expressions were simplified and interpreted fornarrow- and wide-band incident TPWs. When the incident TPWis narrowband and its frequency support does not contain a BP,then the contribution to the TDTF of the incident TPW polesdominates those of the FDTC poles and BC integrals. The TDTFis an attenuated and delayed replica of the incident TPW. Sig-nificant positive delays were obtained when the incident TPWfrequency content was near a pole. Strong dependence of thedelay on the pole-BP separation was found when the incidentTPW frequency content was near a BP. Very small positive de-lays were observed for large pole-BP separations whereas sig-nificant negative delays were obtained for small pole-BP sepa-rations.

When the incident TPW is wideband and/or its frequencysupport overlaps with a BP, all pole and BP contributions in theTDTF can be significant. Each contribution has a distinct carrierfrequency and decay rate and can be identified separately. Theincident TPW pole contribution represents the direct responseof the periodic structure to the incident field. It is significantonly for early time. The FDTC pole contributions represent res-onances existing on the structure and are the time domain coun-terparts of frequency domain leaky waves. For wide-band fieldsthey are significant for much later time than the incident TPWpole contributions. The BC contributions are the time domaincounterparts of frequency domain lateral waves. They have analgebraic decay rate and lead to a very long time tail in theTDTF. The magnitude of the BC contribution was found to bestronger for smaller pole-BP separations.

The present study allows for a better understanding of phe-nomena associated with scattering of TPWs from not only sub-wavelength hole-perforated metal plates but also other typesof periodic gratings. Indeed, the expressions derived for theTDTF remain valid without major modifications for most peri-odic gratings. Many observed phenomena, including those re-lating to delays of transmitted narrowband TPWs associatedwith Rayleigh–Wood anomalies, are expected to occur on otherperiodic configurations as well. Wide-band TPWs transmittedthrough, or scattered from, many periodic gratings will containresonant and BC contributions as well.

It is noted that small losses in the dielectric slabs and metalplates are not expected to lead to qualitative changes in the be-havior of FDTC and TDTF. It may lead however to a quantita-tive decrease in magnitude and change in the shape of the fields.When the plate has a finite thickness, additional interesting ef-fects, including the appearance of additional resonances sup-ported by the structure due interactions between two faces ofthe plate; these phenomena for FDTF’s are investigated in de-tail in [30]. It is also noted that asymptotic expressions for scat-tered fields due to other shapes of incident TPW, e.g. Gaussianshapes, can be derived. The procedure used in this paper can beinvoked to evaluate and study the scattered field of higher ordertime domain Floquet modes [46]. When the structure in Fig. 1is illuminated by a beam, leaky and lateral waves can be excitedand their combined influence may lead to extraordinary proper-ties of scattered fields.

In past, transient excitations have been used to studywide-band responses of perforated plates in terahertz timedomain spectroscopy [8], [27]. Recently, short pulse (fem-tosecond) fields transmitted through metallic plates perforatedby subwavelength holes were used for imaging transient surfaceplasmon polariton propagation [21]. Applications of the studiedstructure and phenomena in target identification and near-fieldimaging are being studied. The investigated phenomena canfind use in the construction of novel probes, antennas, periodicgrating filters, frequency selective surfaces, multiplexers, aswell as in time-domain spectroscopy.

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[46] F. Capolino and L. B. Felsen, “Time-domain green’s function for an in-finite sequentially excited periodic planar array of dipoles,” IEEE Trans.Antennas Propag., vol. 51, no. 1, pp. 160–170, Jan. 2003.

[47] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Prod-ucts, 5 ed. New York: Academic Press, 1994.

[48] R. E. Collin and F. J. Zucker, Antenna Theory, Part Two. New York:McGraw-Hill, 1969.

Vitaliy Lomakin received the M.Sc. degree in elec-trical engineering from Kharkov State University,Ukraine, in 1996 and the Ph.D. degree in electricalengineering from Tel Aviv University, Israel, in2003.

From 2002 to 2005 he was a Postdoctoral As-sociate and Assistant Professor in the Departmentof Electrical and Computer Engineering, Postdoc-toral Associate at the Center for ComputationalElectromagnetics, University of Illinois at Ur-bana-Champaign. Currently he holds the position of

Assistant Professor in the Department of Electrical and Computer Engineering,University of California, San Diego. His research interests include computa-tional electromagnetics, the analysis of metal-dielectric structures, the analysisof periodic structures, as well as high-frequency and time-domain techniquesfor propagation and scattering.

Eric Michielssen (M’95–SM’99–F’02) receivedthe M.S. degree in electrical engineering (summacum laude) from the Katholieke Universiteit Leuven(KUL), Belgium, in 1987 and the Ph.D. degree inelectrical engineering from the University of Illinoisat Urbana-Champaign (UIUC), in 1992.

He was a Research and Teaching Assistant inthe Microwaves and Lasers Laboratory at KUL andthe Electromagnetic Communication Laboratory atUIUC from 1987 to 1988 and 1988 to 1992, respec-tively. He joined the Faculty of the Department of

Electrical and Computer Engineering at the University of Illinois as a VisitingAssistant Professor in 1992, became Assistant Professor of electrical andcomputer engineering in 1993, and became an Associate Professor in 1998.Since 1995 he has been Associate Director of the Center for ComputationalElectromagnetics at UIUC. In 2005, he joined the University of Michigan asProfessor of Electrical Engineering and Computer Science. He is the BeckmanFellow in the Center for Advanced Studies. He authored or co-authored over130 journal papers and book chapters and over 190 in conference proceed-ings. His research interests include all aspects of theoretical and appliedcomputational electromagnetics. His principal research focus has been onthe development of fast frequency and time domain integral-equation-basedtechniques for analyzing electromagnetic phenomena, and the development ofrobust optimizers for the synthesis of electromagnetic/optical devices. He wasan Associate Editor for Radio Science..

Prof. Michielssen is a Member of International Union of Radio Scientists(URSI) Commission B. He received a Belgian American Educational Founda-tion Fellowship in 1988 and a Schlumberger Fellowship in 1990. He receiveda 1994 URSI Young Scientist Fellowship, a 1995 National Science FoundationCAREER Award, and the 1998 Applied Computational Electromagnetics So-ciety (ACES) Valued Service Award. In addition, he was named 1999 URSI-United States National Committee Henry G. Booker Fellow and was selectedas the recipient of the 1999 URSI Koga Gold Medal. He received UIUC’s 2001Xerox Award for Faculty Research, was appointed 2002 Beckman Fellow in theUIUC Center for Advanced Studies, named 2003 Scholar in the Tel Aviv Uni-versity Sackler Center for Advanced Studies, and selected as the UIUC 2003University Scholar. He was Technical Chairman of the 1997 ACES Sympo-sium (Review of Progress in Applied Computational Electromagnetics, March1997, Monterrey, CA), and from 1998 to 2001 and 2002 to 2003, served onthe ACES Board of Directors and as ACES Vice-President from 1998 to 2001.From 1998 to 2005, he served as Associate Editor for the IEEE TRANSACTIONS

ON ANTENNAS AND PROPAGATION.

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970 IEEE TRANSACTIONS ON ANTENNAS AND …cem01.ucsd.edu/papers/IEEE_TRANS_AP_54_3_970_2006.pdf · Transmission of Transient Plane Waves Through Perfect Electrically Conducting Plates