Multimodal propagation of the electromagnetic wave on a structuredperfect electric conductor (PEC) surface

Sayak Bhattacharya a,b,1, Kushal Shah a,b,n

a Department of Electrical Engineering, Indian Institute of Technology (IIT) Delhi, Hauz Khas, New Delhi-110016, Indiab Bharti School of Telecommunication Technology and Management, Indian Institute of Technology (IIT) Delhi, Hauz Khas, New Delhi-110016, India

a r t i c l e i n f o

Article history:Received 24 February 2014Received in revised form23 April 2014Accepted 28 April 2014Available online 9 May 2014

Keywords:FDTDWaveguideMetamaterialComputational electromagneticsHole array

a b s t r a c t

Spoof surface plasmon (SSP) on structured perfect electric conductor (PEC)–air interface is usuallymodeled by an effective plasma frequency in the long wavelength limit. In this limit, the effectivepermittivity model suggests a purely exponential decay of the field profile on both sides of the interface,drawing an analogy with a surface plasmon polariton (SPP)-like single mode (TM) propagation.However, most of the practical applications and experiments involving SSP fall in the regime wherethe feature-size of the structure is comparable to the wavelength. In this wavelength regime of practicalinterest, we show that in contrast to surface plasmon polariton (SPP), the propagation of the surfacewave is hybrid involving both TM and TE modes. By combining angular spectrum representation andvector nature of the electromagnetic fields, we show that the field profile of SSP on patterned PEC can bedescribed accurately by superposition of multiple evanescent orders caused by the diffraction from sub-wavelength features at the interface. This multimodal nature of the surface wave readily explains theanisotropic propagation observed earlier. Also, inclusion of at least one mode from higher order Brillouinzone is necessary for describing the near-field above the interface with reasonable accuracy.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

Surface plasmon polariton (SPP) refers to the propagation ofelectromagnetic energy along a metal–dielectric interface with anexponential decay of the field strength along the perpendicular tothe interface [1–4]. The first scientific observation related tosurface plasmon dates back to 1902, when Wood made theremarkable discovery of an anomaly [5]. Wood's anomaly man-ifests itself as the presence of rapid variations in the spectralintensity distribution of the diffracted wave from a reflection typegrating within a certain narrow band of frequencies (even thoughthe grating is illuminated by a source with a slowly varyingspectral intensity distribution). Wood also noted that this phe-nomenon occurs only with p-polarized light i.e. when the mag-netic field is parallel to the grating grooves. Fano connected thisanomaly with the excitation of a “leaky wave supportable by thegratings” [6]. Finally in 1957, the concept of surface plasmonwas first theoretically explained by Ritchie [7]. 11 years later, healong with others also explained the connection between SPP and

the grating anomaly [8]. Propagation of such a surface wave isattributed to the interaction of electromagnetic energy with theelectron gas of the metal, described by Drude's model [3]. In thefrequency range where this surface plasmon wave exists, thefrequency dependence of the real part of the permittivity of ametal is exactly same as that of a cold unmagnetized plasma [9]and is given by ϵ=ϵ0 ¼ 1�ω2

p=ω2, where ωp is the plasma fre-

quency. For metals, this plasma frequency usually falls in the ultra-violet region. Since SPP can exist only when ω≲ωp, the interactionbetween electrons of the metal and electromagnetic energy togenerate surface plasmon becomes prominent for infra-red andhigher frequencies. For ω5ωp, most metals tend to behave as aperfect conductor with their permittivity approaching towards avery high negative value (effectively �1, at microwave frequen-cies) and hence, at these frequencies SPP cannot exist.

The ability of surface plasmon to confine electromagneticenergy at a length-scale below diffraction limit has attracted theattention of many researchers to the problem of finding a way tomake propagation of surface plasmon possible at lower frequen-cies (such as microwave frequencies) also. An attempt to achievethis goal was made by using a 3D network of thin wires [10,11].Possibility of mimicking surface plasmon on structured perfectelectric conductor (PEC) surfaces was predicted through anapproximate derivation of the corresponding dispersion relationin the large wavelength limit [12,13]. Finally, this “designer surfaceplasmon” (also known as the spoof surface plasmon or SSP) was

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Optics Communications

http://dx.doi.org/10.1016/j.optcom.2014.04.0680030-4018/& 2014 Elsevier B.V. All rights reserved.

n Corresponding author at: Department of Electrical Engineering, Indian Institute ofTechnology (IIT) Delhi, Hauz Khas, New Delhi-110016, India. Tel.: þ91 11 26581102;fax: þ91 11 26581606.

E-mail addresses: sayak.electromagnetics@ee.iitd.ac.in (S. Bhattacharya),kkshah@ee.iitd.ac.in (K. Shah).

1 Tel.: þ91 11 26586582; fax: þ91 11 26581606.

Optics Communications 328 (2014) 102–108

modeled using finite element method (FEM) and verified experi-mentally [14]. In [14], the plot of the electric field at the center of asquare hole with distance z from the structured surface was fittedwith decaying exponential to draw a parallel between this surfacemode and surface plasmon. Further, finite difference time domain(FDTD) method [15–17] was used to numerically investigate theproperties of SSP [18,19]. In [19], the experimentally measuredelectric field intensity was plotted as a function of distance abovethe structured surface and fitted to a decaying exponential andthus, strengthening the notion of the existence of a single SPP-likemode with the electric field decaying exponentially both insidethe holes as well as above the interface. Also, numerical results of[18] has shown that the propagation of SSP is anisotropic and aconsequence of this anisotropy was demonstrated in form of self-collimation [19].

In this paper, with the help of FDTD simulation we show thatthe SSP wave is a combination of both TM and TE modes unlike theSPP wave, which is purely TM. Further, in case of SPP, the fieldprofile is not differentiable with respect to the normal at theinterface. But for SSP on a structured PEC, as can be seen from theFEM simulation of [14], the field profile is differentiable every-where on the air hole with respect to the normal. This difference inthe field profile between SSP on a structured PEC and SPP can beunderstood with the help of boundary conditions of the fieldcomponents, while taking into account the fact that there is nodiscontinuity in the permittivity of the medium inside and outsidethe air hole. In the experiment and simulation demonstrated in[14], the hole-size is greater than 0.2λ at 12.3 GHz. Thus, the long-wavelength assumption made in the theory of [12,13] does notapply to this structure. Also, Fig. 4c of [14] shows that the fieldprofile above the interface starts to agree with a single exponentialonly after a distance of almost 0.25λ from the interface at thespecified frequency (12.3 GHz). Thus, the gap is a significantfraction of the wavelength. In fact, we have used the samegeometry but with different dimensions (refer to Section 2 fordetails) and found this gap to be much larger (0.7347λ for Ez at15.1 GHz).

In order to characterize the decay profile of the field correctly,we take a note of the fact that many relaxation processes in thenature involve multiple independent exponential decays andthe resultant relaxation behavior can be well approximated byWilliam–Watts function [20,21]:

ϕðxÞ ¼ϕ0 exp½�ðαxÞβ� ð1Þwhere, 0oβr1 is the stretching parameter. Although this func-tion makes the slope at x¼0 smooth, it has maxima at x¼0 andcannot be used to approximate the SSP field profile since the peakof the field profile in [14] as well as in our FDTD simulation isshifted off the interface. However, superposition of multipleexponentials can shift the maxima off the interface if some ofthe coefficients of the exponentials posses opposite sign. Forexample, ϕðxÞ ¼ fe�0:1x�0:2e�0:9xg has its maxima at x¼0.77. Todescribe the field profile we use the angular spectrum representa-tion [22,23] which is based on scalar diffraction theory. Sincescalar diffraction theory does not account for the boundaryconditions (which is an important issue in connection to the fieldprofile), we incorporate the boundary conditions with scalartheory to conclude that there must be multiple decay constantsassociated with the diffracted field from a sub-wavelength feature.The field profile below the interface is shown to arise fromsuperposition of exponentials with decay constants correspondingto TE10, TE=TM11, TE=TM12=21, TE=TM22… modes of the waveguide.From the angular spectrum representation we calculate the decayconstants above the interface and identify those exponentials outof a large distribution which are sufficient to describe the fieldprofile with reasonable accuracy. In this context, we note that at

least one mode from higher order Brillouin zone is necessary todescribe the field profile above the interface accurately.

Section 2 contains the details of the geometry and the compu-tation method used. In Section 3, we show that the significantdisagreement between the actual field profile and the exponentialfit stems from the fact that the SSP wave propagating on struc-tured PEC surface involves multiple modes. This multimodalpropagation is not only important from the dispersion point ofview but also provides an explanation of the anisotropic propaga-tion observed in [18,19]. Finally, Section 4 contains the conclusion.

2. Geometry and computation details

The geometry under consideration is shown in Fig. 1 andconsists of a PEC block (410 mm�410 mm�40 mm) with a 2-Darray of air-filled square holes drilled into it. Dimension of thecomputation box is 410 mm�410 mm�60 mm, implying thatthe height of the air above the hole-array is 20 mm. Numericalcomputation has been carried out using the commercially avail-able Remcom XFdtds software [24] that implements the 3D FDTDtechnique on GPU (Graphics Processing Unit) architecture to makethe computations much faster.

The inner side, b, of each square hole is 8.75 mm (same as thatused in [19]). The lattice constant of the structure along both x andy directions is a¼0 mm and d¼

ffiffiffi2

pa is the diagonal dimension of

each unit cell. The cut-off frequencies for different modesðTEmn or TMmnÞ of each air-filled square waveguide can be calcu-lated from the following relation [25]:

f c ¼c2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimb

� �2þ n

b

� �2r

ð2Þ

where c is the velocity of light in free space and m, n are non-negative integers. The cut-off frequency of the dominant modeðTE10Þ is calculated as 17.14 GHz. Surface waves can exist on theair–hole interface only below this cut-off frequency since above it,a large fraction of the electromagnetic energy will flow throughthe waveguides. However, if we choose a frequency far below thiscut-off, the power in the radiation modes increases making it

Fig. 1. The 41�41 array of air-holes patterned on a PEC block. At the top rightcorner, the inset shows a magnified view of the cell with a pair of thin strips fed bya current source at 15.1 GHz.

S. Bhattacharya, K. Shah / Optics Communications 328 (2014) 102–108 103

difficult to observe the surface wave. For this reason, the simula-tion was carried out over a frequency range of 14.5–15.6 GHz andthe procedure illustrated in this paper was repeated for the entirefrequency range. However, in the subsequent sections we discussthe results corresponding to 15.1 GHz only since near this fre-quency, the interesting phenomenon of self-collimation of SSP [19]takes place.

The length of each waveguide has been chosen to be 40 mm(approximately 2λ at the simulation frequency of 15.1 GHz). Forour frequency range of operation, the modes inside the waveguidehave a very short decay-length, thereby making it possible to treata waveguide of length 2λ as an infinite one. Thus, the structuredinterface is represented by the z¼40 mm plane.

To discretize the geometry, the spatial grid dimension ð▵xÞalong each direction has been chosen to be 0.5 mm (approximatelyλ=40 at 15.1 GHz) along with a time step▵t ¼ 6:809� 10�13 s. So,in our simulation, c▵t=▵x¼ 0:041o1=

ffiffiffi3

p, ensuring the Courant–

Freidrichs–Lewy (CFL) stability [15] of the FDTD solution. Also, allthe results reported in this paper are for a time when the systemhas reached steady-state. The corresponding convergence calcu-lated by the XFdtd software was �42 dB, clearly signifying thehigh degree of accuracy of our results.

To get an idea about the effect of grid dispersion, we considerthe simple case of wave propagation in xy-plane with kz ¼ 0. Wehave followed the method outlined in [15] with same▵x and▵tthat we have used for our simulation and calculated the numericalphase velocity ðvpÞ to be 0.99587c (i.e. an error of 0.413% in thephysical phase velocity). Thus, a resolution of 40 cells per wave-length is sufficient to provide a reasonable accuracy. Each of thefaces of the computation box is bounded by 7 layers of PerfectlyMatched Layer (PML). A very small gap ð � 1 mmÞ between a pairof thin, diagonal PEC strips, placed at z¼40 mm plane, is fed by acurrent source at 15.1 GHz to excite the surface wave on thestructure (inset of Fig. 1).

Unless otherwise stated, all lengths in this paper are in mm andall decay constants are in mm�1.

3. Hybrid mode and multi-modal propagation

3.1. Existence of hybrid mode

Fig. 2 plots the relative amplitudes of Ez and Hz compared to jEjand jHj respectively. This shows that both Ez and Hz are present inthe surface wave and have a significant amplitude. It was verifiedin our simulation that for all points on the diagonal of each of theholes along which the propagation takes place, Hz ¼ 0. However, ifthe point is offset from the diagonal along which propagation

takes place (and not on the PEC), both Ez and Hz exist. Thus the SSPwave is hybrid in nature, characterized by the existence of both TMand TE modes. This is a marked difference from the SPP wavewhich is purely TM. Also, the peak of Hz is located below theinterface, a behavior which is complementary to that of Ez.

3.2. Angular spectrum representation and boundary conditions

Angular spectrum representation of a scalar field [22,23] is anapproach based on Helmholtz equation and spatial Fourier trans-form. Given the scalar field distribution at the reference plane,Zð ¼ z�40Þ ¼ 0, this technique predicts the field distribution forZ40 planes according to

Uðf x; f y; zÞ ¼ Aðf x; f yÞeiβZþBðf x; f yÞe� iβZ ð3Þ

where, 2πf x ¼ kx and 2πf y ¼ ky and β2 ¼ ½k2�4π2ðf 2x þ f 2yÞ�, k beingthe wavenumber in the dielectric filling the half-space Z40 andthe speed of light in free-space is taken to be unity. For a sub-wavelength feature at Z¼0 plane, all the non-zero fx and fy resideoutside the f 2x þ f 2y ¼ 1=λ2 circle and Eq. (3) gives rise to anevanescent solution [22]:

Uðf x; f y; ZÞ ¼Uðf x; f y;0Þe�αZ ð4Þ

where, α2 ¼ ½4π2ðf 2x þ f 2yÞ�k2�. Thus, an accurate specification ofthe field in the Z¼0 plane will produce the accurate results forZ40. However, angular spectrum approach, being based onHelmholtz equation, does not take the boundary conditions intoaccount. But, even when the vector nature of the field is con-sidered, the resulting field must satisfy Helmholtz equation [26].So, the solution has to be exponentially decaying even when it isobtained by a vector approach. At this point, for a sub-wavelengthhole at Z¼0 plane with decaying field profiles on both sides, weask the following question: Can there be a single exponentialdecay associated with the field profile? In other words, canðf 2x þ f 2y Þ be a constant?

For an arbitrary uðx; y;0Þ, scalar theory does not forbid such apossibility. So, let us assume, (if possible) α1 and α2 are the decayconstants to be used to describe the tangential components of theelectric field for Z40 and Zo0 respectively. But this will intro-duce a discontinuity in the slopes of the fields at Z¼0, which willbe a violation of boundary conditions because, there is nodiscontinuity in the permittivity of the medium within and out-side the hole. So, when we consider the vector nature of the field,the angular spectrum representation will still be valid but therewill be multiple evanescent orders to maintain the continuity ofthe slope of the fields across the interface. This also says that at aparticular frequency, ðk2x þk2yÞ is not a constant. Thus, the scattering

Fig. 2. Relative magnitudes of Ez and Hz at a point offset from the center of the hole by 2.5 mm and �2.5 mm along x and y-directions respectively: (a) Ez (red/dash-dottedline) and jEj (black/solid line). (b) Hz (red/dash-dotted line) and jHj (black/solid line). This shows that both Ez and Hz exists in the propagating mode giving rise to a hybridmode propagation. Operating frequency is 15.1 GHz. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of thispaper.)

S. Bhattacharya, K. Shah / Optics Communications 328 (2014) 102–108104

characteristics of a sub-wavelength hole must be anisotropic in theazimuth plane to conform with Maxwell's equations. The argu-ment presented here is also applicable to any individual sub-wavelength hole of the periodic geometry under consideration

(Fig. 1). Thus, in order to be a valid solution of Maxwell's equations,the wave propagation on the structured PEC surface must bemulti-modal and anisotropic.

3.3. Evanescent orders above the interface

For the geometry under investigation, we assume thatZ ¼ ðz�40Þ. For a single exponential decay when we take aninverse Fourier transform of Eq. (4), it reduces to a simple form:

uðx; y; zÞ ¼ uðx; y;0Þe�αZ ð5ÞThe above form clearly says that the same decay constant appliesto both Uðf xf y; ZÞ and uðx; y; ZÞ. But this is not very obvious whenmultiple exponential decays are involved since in case of multipleexponentials the inverse transform takes the following form:

uðx; y; zÞ ¼ uðx; y;0Þnϝ�1fe�αZg ð6Þwhere, n denotes convolution and ϝ�1 denotes inverse Fouriertransform. So, instead of using the exact angular spectrum wemake certain approximations to mathematically quantify the fieldprofile above the interface. The approximation is based on the factthat some spectral orders are dominant as compared to the others.Although we elaborate only on the procedure for u¼ Ez, the sameprocedure is applicable to any field component. First, we extractthe distribution of Ez in Z¼0 plane from the FDTD result and findits Fourier transform. The corresponding log-scaled and normal-ized amplitude spectrum has been plotted in Fig. 3. This plotshows the presence of multiple localized clusters of spectralcomponents with high amplitudes outside the k0 circle. Thesespectral components correspond to the dominant evanescentorders of the multi-modal surface wave. We select n of suchdominant spectral components. Let, ðf xi; f yiÞ denote the location of

Fig. 3. Magnitude plot (log-scaled and normalized) of angular spectrum of Ez of theself-collimated beam propagation at 15.1 GHz. The black circle shows the circlewith a radius k0 ¼ :3163 rad/mm. The dominant spectral components (red andyellow regions in the plot), located outside the k0 circle, are responsible formultiple evanescent orders of the field profile. (For interpretation of the referencesto color in this figure caption, the reader is referred to the web version of thispaper.)

Fig. 4. Fitting of (a), (c) real and (b), (d) imaginary parts of Ez above the interface corresponding to the same point on a hole used in Fig. 2a with Eq. (9). FDTD results areshown with black/solid line and fitted results are shown with red/dash-dotted line. In (a) and (b) the decay constants used are 0:162;0:173;0:153;0:144. In (c) and (d) thedecay constants are 0:162;0:173;0:153;0:144;0:906. Note the improved accuracy due to the inclusion of 0.906, which corresponds to a mode from higher order Brillouinzone. The values of these decay constants are obtained from the angular spectrum using an approach as described in Section 3.3. (For interpretation of the references to colorin this figure caption, the reader is referred to the web version of this paper.)

S. Bhattacharya, K. Shah / Optics Communications 328 (2014) 102–108 105

the ith spectral component. With these notations Eq. (4) can beapproximated as

Uðf x; f y; ZÞ � Uðf x; f y;0Þ ∑n

i ¼ 1δðf x� f xi; f y� f yiÞe�αZ ð7Þ

Inverse transform of the above expression gives

uðx; y; ZÞ � uðx; y;0Þn ∑n

i ¼ 1ei2πðf xixþ f yiyÞe�αiZ ð8Þ

where, αi ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4π2ðf 2xiþ f 2yiÞ�k20

q. The above equation can be rewrit-

ten as

uðx; y; ZÞ � ∑n

i ¼ 1Biðx; yÞe�αiZ ð9Þ

where, Biðx; yÞ ¼ fuðx; y;0Þnei2πðf xixþ f yiyÞg. Thus, the field profileabove the interface can be approximated as a superposition ofexponentials whose decay constants are obtained from the angularspectrum. Different signs of the coefficients Bi easily account forthe peak-shift off the interface. The position of the peak ðz¼ zpÞcan be obtained by equating the partial derivative of uwith respectto z to 0 at z¼ zp:

∑1

n ¼ 0Biðx; yÞαie

�αiðzp �40Þ ¼ 0 ð10Þ

Since Bi are functions of (x,y), the peak position of the Ez willvary from point to point in the xy-plane. This result explains theshifting of peak away from the interface observed in our simula-tion as well as in the FEM simulation of [14]. From the angularspectrum of Ez we identify the dominant decay constants as0:162;0:173;0:153;0:144;0:906. With these values of αi we havefitted the field profile (obtained by FDTD simulation) of real andimaginary part of Ez corresponding to a point on a hole which issame as that used for plotting Fig. 2 (shown in Fig. 4c and d). Thefitting was done using the curve fitting tool of MATLABs [27].Fig. 4a and b shows the same fitting with the decay constant 0.906excluded. This shows that the mode with the decay constant 0.906is necessary to describe the field profile accurately. In the angularspectrum the particular combination of ð∣kx∣; ∣ky∣Þ which givesrise to this particular decay constant is either ð0:9436;0:1765Þ or

Table 1SSE and RMSE corresponding to the fits of RefEzg and ImfEzg above the interfacedue to successive inclusion of modes as described in Section 3.3. The highlightedentry shows the improvement in SSE and RMSE due to inclusion of the mode fromthe higher order Brillouin zone.

Decay constant (s) (in mm�1) SSE forRefEzg

RMSE forRefEzg

SSE forImfEzg

SRMSE forImfEzg

0.162 164.90 0.090800 247.5 0.1112000:162;0:173 15.520 0.027860 34.58 0.0415900:162;0:173;0:153 13.440 0.025930 8.194 0.0202400:162;0:173;0:153;0:144 1.8790 0.009694 4.027 0.0141900:162;0:173;0:153;0:144;0:906 0.0572 0.001700 0.101 0.002247

Table 2Decay constants of different evanescent waveguidemodes at 15.1 GHz used for fitting the field profilesbelow the interface.

Mode Decay constant (αmn in mm�1)

TE10 0.1700TE=TM11 0.397252TE=TM12=21 0.737864TE=TM22 0.965076TE=TM23=32 1.2523TE=TM33 1.4901

Fig. 5. Fitting of (a) real, (b) imaginary parts of Ex and (c) real, (d) imaginary parts of Ez below the interface (corresponding to the same point on a hole used for Fig. 2) withEq. (11). FDTD results are shownwith black/solid line and fitted results are shownwith red/dash-dotted line. The decay constants have been taken from Table 2. For obtaininga good fit, inclusion of the decay constant corresponding to the TE10 mode is very important (see Table 3). (For interpretation of the references to color in this figure caption,the reader is referred to the web version of this paper.)

S. Bhattacharya, K. Shah / Optics Communications 328 (2014) 102–108106

ð0:9436;0:1765Þ. Since the periodicity of the structure a¼10 mm,the first Brillouin zone corresponds to ∣kx∣oπ=10 and ∣ky∣oπ=10.Clearly, the mode with the decay constant 0.906 does not belongto first Brillouin zone. Table 1 illustrates the improvement of thesum of squares due to error (SSE) and root mean squared error(RMSE) of the fit due to successive inclusion of each of the modes.

3.4. Evanescent orders below the interface

Field distribution of different field components below theinterface ðzo40Þ can be found out by solving the Helmholtzequation subjected to appropriate boundary conditions at thePEC boundaries. These solutions simply correspond to differentmodes of the rectangular waveguide. Since there must be multipleexponentials for matching of the fields at z¼40, we express thefields inside the hole by modal expansion [25] of differentevanescent TMz and TEz modes:

ϕðx; y; zÞ ¼ ∑1

m ¼ 1∑1

n ¼ 0Amnh

mπxb

� �h

nπyb

� �eαmnðz�40Þ ð11Þ

where, na0 for TM modes and hð�Þ is to be replaced by propercombinations sin ðÞ or cos ðÞ corresponding to TE and TM modes.Table 2 lists the decay constants of different evanescent waveguidemodes at 15.1 GHz. For fitting the field profile to the sum ofexponentials, we need to incorporate the TE10 mode also since TMmodes alone do not give a good fit, particularly as we approach thecorners. This shows that the SSP wave is composed of both TE andTM mode unlike the SPP wave which is purely TM.

Using these decay constants, the field profiles of real andimaginary parts of Ex and Ez (obtained by FDTD calculation) havebeen fitted with functions of the form:

ψ ðzÞ ¼ ∑m

i ¼ 1Cie

αiðz�40Þ ð12Þ

The coordinate in the xy-plane is same as that used for Fig. 2. Thecomparison of the curve-fitted result with FDTD data have beenshown in Fig. 5. The slower decay of Ex inside the hole as comparedto Ez is due to the existence of TE modes. Ez can arise only from TMmode but Ex can arise from both TE and TM modes. Since, TE10mode has the lowest decay constant, it makes the decay of Exslower than Ez. This result corroborates our observation of thehybrid mode. The SSE and RMSE of the fitting have been illustratedin Table 3. Note the significant increase in the SSE and RMSE inTable 3 when the decay constant 0.17 corresponding to TE10 modeis excluded.

In this context, note that above the interface there can be somecontamination of radiation modes from the current-strip and from

the reflection at the computation boundaries (due to reflectionfrom PML layers). However, the agreement of the multiple expo-nential fitting (corresponding to the modes strictly outside k0circle) with the FDTD result (which contains both decaying modesfrom outside and oscillatory modes from inside the k0 circle) inFig. 4c,d and Fig. 5 shows that the profile of the surface wave wasnot affected much by the radiation modes.

4. Conclusion

In summary, we have incorporated the vector nature of theelectromagnetic field in the angular spectrum representation toprove that the electromagnetic wave propagating on structuredPEC surface is multimodal. Each of these modes is evanescent innature with a characteristic decay length of its own and originatesdue to diffraction from the sub-wavelength features at the PECinterface. The decay length associated with each mode can bedetermined by angular spectrum approach. Such a multi-modalnature readily explains why a patterned PEC surface exhibitsanisotropic behavior. The decay lengths below the interfacecorrespond to different waveguide modes. One important featurein this case is the presence of TE mode, which makes the surfacewave on structured PEC hybrid. Finally, the transmission ofmicrowave below cut-off was demonstrated in [28], whichinvolved periodic square holes drilled on a PEC and the hole-dimension was comparable to the wavelength. Our results describ-ing the surface-wave behavior in this wavelength regime can alsolead to a better understanding of the microwave transmissionprocess below cut-off.

Acknowledgments

We would like to thank Uday Khankhoje, Anuj Dhawan andKedar Khare for interesting discussions.

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Table 3SSE and RMSE corresponding to the fits of the field components below the interface due to successive inclusion of modes as described in Section 3.4. The highlighted entriesshow that the exclusion of TE10 mode leads to a significant increase in the values of SSE and RMSE in the fitting of RefExg and ImfExg.

Decay constant (s) (in mm�1) RefExg ImfExg

SSE RMSE SSE RMSE

TE=TM11 : 0:397252 761.70 0.138000 443.60 0.105300TE10 : 0:17 86.140 0.046400 7.3360 0.0135400:17;0:397252 1.7560 0.006626 3.0214 0.0087000:17;0:397252;0:737864 1.5790 0.006282 2.9060 0.0085000:17;0:397252;0:737864;0:965076 0.7202 0.004200 0.6224 0.0039450:17;0:397252;0:737864;0:965076;1:2523 0.4191 0.003237 0.4711 0.0034320:17;0:397252;0:737864;0:965076;1:2523;1:4901 0.2761 0.002627 0.3554 0.0029810:397252;0:737864;0:965076;1:2523;1:4901 70.210 0.041900 42.780 0.032700

RefEzg ImfEzg

0.397252 9.4770 0.017770 10.3092 0.0185400:397252;0:737864 0.2456 0.002861 0.27460 0.0030260:397252;0:737864;0:965076 0.0991 0.001817 0.14870 0.002226

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