zigzag metallic conductors as frequency selective surfaces

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Published in IET Microwaves, Antennas & PropagationReceived on 11th October 2012Accepted on 7th April 2013doi: 10.1049/iet-map.2013.0107

22The Institution of Engineering and Technology 2013

ISSN 1751-8725

Zigzag metallic conductors as frequency selectivesurfacesKemal Delihacioglu1, Cumali Sabah2, Muharrem Karaaslan3, Emin Ünal3

1Electrical & Electronics Engineering Department, Kilis 7 Aralık University, Kilis, Turkey2Middle East Technical University-Northern Cyprus Campus, Department of Electrical and Electronics Engineering,

Kalkanli, Guzelyurt, KKTC, Mersin 10, Turkey3Electrical & Electronics Engineering Department, University of Mustafa Kemal, Iskenderun, Hatay, Turkey

E-mail: [email protected]

Abstract: Reflection and transmission properties of a new type of frequency selective surface (FSS) made of zigzag metallicelements placed periodically on a dielectric substrate are investigated theoretically for arbitrary polarisation. The electric fieldintegral equation technique with the subdomain overlapping piecewise sinusoidal basis functions are employed to computethe unknown current coefficients induced on the metallic elements. The zigzag FSS arrays act as a low-pass filter in S-(1–5 GHz), X-(8–12 GHz) and Ku-(12–18 GHz) bands. Owing to multiple resonances the zigzag FSS arrays can be used inmultiband frequency applications. Between the resonances there exist transmission windows that allow full energy transfer.By removing the first and third arms, the power reflection coefficient of three-arm zigzag FSS is compared with afreestanding strip FSS, and it is compatible with the measured and calculated results found in the literature.

1 Introduction

Theoretical and experimental investigations on arrays ofdifferent frequency selective surface (FSS) elements havebeen carried out over four decades. FSS usually has atwo-dimensional (2D) planar periodic structure eitherconstructed by metallic elements or slots on metallicscreens. The frequency response is characterised by theshape of the FSS structure in the unit cell. FSScomprised of periodic structures act as a band-stop filterclose to the resonance frequency which is determined bythe polarisation of the incident wave, the shape and theunit cell dimensions of conducting elements. In antennaapplications a zigzag FSS array can be designed toreflect frequencies in X-, Ku-, K- (18–26 GHz) and Ka-(26–40 GHz) bands depending on the value of armlength and unit cell dimensions. It can be used totransmit electromagnetic waves in S- and X-bands. Thesubstrate is used to support conducting elements andmodify the spectral response of FSS. The mainapplication area of periodic structures is to filtermicrowave and optical signals. In microwave regionperiodically arranged elements, either aperture orconducting, are used as a band pass or bandstop filterdepending on the type of elements [1, 2]. FSS isexploited to make more efficient use of reflector antenna[3] and radomes [4]. In Sub-millimeter wave region [5]FSS-based split slot ring is used as a linear to circularpolarisation converter. FSS is used for photonic bandgapstructures which prohibit electromagnetic wavepropagation within a certain frequency range [6].

The behaviour of the new electromagnetic bandgapmaterial formed by combining different FSSs, and theone of the new electromagnetic band gap resonator isbriefly described in [7]. In [8], a quasi-square openmetallic ring is studied using wave concept iterativeprocedure and compared with simulation andexperimental results. Tunable frequency selective surfacewith a shorted ring slot based on the spectral domainmethod is realised in [9]. A triple-band FSS-based onthree concentric circular metallic rings on chiral slab isstudied in [10]. The concept of non-uniform FSS backedas a reflection plane for amplitude controlled reflect arrayis proposed in [11]. Ortiz et al. [12] proposed a newtype of surface made of split ring resonators (SRRs) and,at the same time, complementary SRRs, placed in such away that the surface is self-complementary. A full waveanalysis for the scattering problem of infinite periodicarrays on dielectric substrates excited by circularlypolarised incident wave [13], and electromagneticscattering by infinite double 2D periodic array ofresistive upper and lower elements with dielectricsubstrate are presented by Wakabayashi et al. [14]. Thescattering characteristics of, periodic arrays of two-turnsquare shaped conductors on a dielectric slab [15],L-shaped FSS on chiral slab [16] and L-shaped andone-turn helix shaped conductors backed by dielectricslab for transverse electric (TE) and transverse magnetic(TM) excitations [17] are analysed using the modalexpansion method. In this study, a novel structure FSScomprised of perfectly conducting array of zigzagmetallic elements is considered and discussed in detail.

IET Microw. Antennas Propag., 2013, Vol. 7, Iss. 9, pp. 722–728doi: 10.1049/iet-map.2013.0107

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2 Theoretical study
Consider the perfectly conducting zigzag array as shown inFig. 1. It is assumed that the dielectric substrate issandwiched between two free space regions (air) andperfectly conducting zigzag elements are placed periodicallyon a face of dielectric substrate, as shown in Fig. 1b. Atime harmonic field with an ejwt time dependence isassumed and suppressed throughout the study. Theconducting zigzag array is illuminated by a plane wave ofarbitrary polarisation (either TE or TM) from the free spaceregion propagating along the z-axis. The total transverseincident electric and magnetic fields in the region z≤ 0 inthe absence of scatterers can be written as

Ei =∑2r=1

[e−jgz + Grejgz]Are

−jk·rkr00 (1)

H i =∑2r=1

[e−jgz − Grejgz]ArYre

−jk·raz × kr00 (2)

where r = 1 corresponds to the TM wave incidence and r = 2to the TE wave incidence, Ar is the amplitude of the incidentwave (A1 = 1, A2 = 0 for TM incidence and A1 = 0, A2 = 1 forTE incidence), Γr is reflection coefficient because of dielectricsubstrate in the absence of scatterer and k100 = cosφiax +sinφiay, k200 = az × k100, ρ = xax + yay

g pq =(k2 − |k pq|2)1/2 k2 . |k pq|2

−j(|k pq|2 − k2)1/2 k2 , |k pq|2{

p, q = −1, . . . , −2, −1, 0, 1, 2, . . . , 1

k pq = ko sin ui cosfi +2pp

d1

( )ax

+ ko sin ui sin fi +2pq

d1

( )ay

ko is the free space propagation constant, θi and φi are the

incident angles, Y1pq =kY

g pq

, Y2pq =g pqY

k, Y =

��1/m

√, ɛ, μ

are the permittivity and permeability of medium and k isthe propagation constant of dielectric substrate.The incoming electromagnetic wave induces the current on

the zigzag conductor surfaces and this current results inscattered fields. Since the array is periodic and is assumed

Fig. 1 Layout of three-arm zigzag metyallic conductor FSS

a Unit cellb Front view


to be infinite in the xy-plane, the fields inside the dielectricsubstrate and in the air on either side can be expanded intoFloquet modes [18]. Thus, the fields can be represented byan infinite summation of plane waves in the three regions.The tangential components of the scattered fields in the airand inside the dielectric substrate can be written as follows:In region 1 (z≤ 0)

E1 =∑rpq

R−rpqe

jg1pqze−jk pq·rkrpq (3)

H1 = −∑rpq

YrpqR−rpqe

jg1pqze−jk pq·r(az × krpq) (4)

In region 2 (0≤ z≤ d )

E2 =∑rpq

(Trpqe−jg2pqz + Rrpqe

jg2pqz)e−jk pq·rkrpq (5)

H2 =∑rpq

(Trpqe−jg2pqz − Rrpqe

jg2pqz)Yrpqe−jk pq·r(az × krpq)

(6)

In region 3 (z≥ d )

E3 =∑rpq

T+rpqe

−jg3pqze−jk pq·rkrpq (7)

H3 =∑rpq

YrpqT+rpqe

−jg3pqze−jk pq·r(az × krpq) (8)

where R−rpq, Trpq, Rrpq and T+

rpq are the unknown scatteredfield amplitudes. Each Floquet mode in the scattered fieldshould satisfy the boundary conditions given below

1. The tangential component of the electric field iscontinuous at z = 0.2. The tangential component of electric and magnetic fieldsare continuous at z = d.3. The tangential component of magnetic field isdiscontinuous at z = 0 and is equal to the surface currentdistribution J(x, y).

Using these boundary conditions, an electric field integralequation (EFIE) can be obtained by imposing the totalelectric field as zero over the conducting area of unit cell.

723& The Institution of Engineering and Technology 2013

Fig. 2 Schematic layout of PWS basis functions on three-armzigzag FSS

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Therefore at z = 0 we have
Ei(x, y, 0)+ E1(x, y, 0) = 0 (9)

This is an integral equation for the unknown currentdistribution and can be solved by the Galerkin type momentmethod (MM) [19] if J(x, y) is approximated as follows

J (x, y) =∑Nn=1

cnf n(x, y) (10)

where cn is the unknown current coefficients to be calculated,N is the finite value for computability and fn(x, y) is the basisfunctions. Substituting (10) into (9) and then integrating itover a unit cell, after multiplying both sides by a complexconjugate of the basis function (f ∗m (x, y)) yields thefollowing system of equation

∑2r=1

(1+ Gr)Arkr00 · g∗m(k00)

=∑Nn=1

cnd1d2

∑rpq

krpq · g∗mpqkrpq · gnpqY eqrpq

(11)

where

gnpq=∫∫

fn(x,y)ejkpq·rdxdy, k1pq=kpq/|kpq|, k2pq=az×k1pq

m = 1, 2,…, N and the asterisk denotes the complexconjugate,

Y eqrpq=Y air

rpq+Y dielrpq

1−Grpq

1+Grpq

( ), Rrpq=

Y dielrpq Y

airrpq

Y dielrpq +Y air

rpqe−j2gpqd

Equation (11) represents an infinite linear system equation forthe unknown current coefficients. If this infinite system oflinear equations is truncated, the solution to the truncatedsystem approximates the exact solution. The truncatedlinear system can be written in matrix form as

[Vm]N×1= [Zmn]N×N [cn]N×1 (12)

where the excitation vector is

[Vm]N×1= (1+Gr)Arkr00 ·gm00The elements of the impedance matrix are

[Zmn]N×N =1

d1d2

∑pq

k1pq ·g∗mpqk1pq ·gnpqY eq1pq

+k2pq ·g∗mpqk2pq ·gnpqY eq2pq

The unknown current coefficients are [cn]N × 1 = [c1 c2 …cN]

T, with the superscript T denotes transpose.Once we select the basis functions, the unknown current

coefficients can be computed by taking the matrix inversionof (12). The basis functions are expanded in the direction ofcurrent flow as an overlapping piecewise sinusoidal (PWS)as shown in Fig. 2. The basis function fn, in an arbitrary


direction as is defined in parametric form as

f n(s) = as

sinb(s− sn−1)

sinb(sn − sn−1)Sn−1 ≤ s , sn

sinb(sn+1 − s)

sinb(sn+1 − sn)sn ≤ s ≤ sn+1

⎧⎪⎪⎨⎪⎪⎩ (13)

where β is the free space phase constant and as shows a unitvector, either in x- or y-direction. The variation of thecurrent across the width is ignored because of its smallvalue compared with the arm length and the wavelength.The current flows only along the lengths of structures,which is axially directed and only x- or y-dependent. Thereference current direction is assumed to flow top down.The zigzag FSS is divided into three arms denoted by s1, s2

and s3 and s3 with lengths h1, h2 and h3, respectively, asshown in Fig. 1a. Each arm is divided into segments suchas {s1(0), s1(1),… , s1(N1)}, {s2(0), s2(1),… , s2(N2)} etc.as shown in Fig. 2. The total number of segments (N ) is thesum of segments on each arm (N =N1 +N2 +N3). At thecorner points, the last segment of the former arm iscoincident with the first segment of the latter arm (e.g.s1(N1) and s2(0)). To satisfy the continuity, the current isformed by two halves at the corner points (red lines inFig. 2), one is in x-directed, and the other is in y-directed.Since there are no current coefficients at the end points ofthe zigzag element, the number of basis functions and thedimension of the impedance matrix is equal to N− 1.Having computed the current coefficients the reflection

coefficient (R) in region 1 and transmission coefficient (T )in region 3 can be calculated from the following relations

R =∑2r=1

GrAr −1

d1d2Yeqr00

∑Nr=1

cngn00 · kr00

{ }kr00 (14)

T =∑2r=1

tr00(1+ Gr)Ar −tr00

d1d2Yeqr00

∑Nn=1

Cngn00 · kr00

{ }kr00

(15)


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where
tr00 =ej(gair00 − gdieal00 )d + Rr00e

j(gair00 + gdieal00 )d

1+ Rr00

3 Numerical results and discussions

The reflection and transmission characteristics of perfectlyconducting zigzag FSS elements are analysed theoreticallyfor arbitrary polarisations at normal incident case (θi = φi =0°). The zigzag elements are periodically arranged on aplanar dielectric substrate as depicted in Fig. 1b. Thepermittivity (ɛr), thickness (d ) of substrate and the width ofthe elements are taken as 2.1, 5 mm and h1/10 mm,respectively. The number of Floquet modes is determinedby inclusion of more Floquet modes until there is littlechange in the results, as illustrated in Fig. 3 for the TMreflection coefficient because of three-arm zigzag FSSbacked by dielectric substrate. The convergence is achievedfor the values of p = q = 20. Therefore the total number ofFloquet modes used is (2p + 1)(2q + 1), which is equal to

Fig. 3 Convergence of Floquet modes for three-arm zigzag FSSbacked by dielectric substrate

d1 = d2 = 5.5 mm, h1 = h3 = 2.5 mm, h2 = 5 mm, d = 5 mm, N = 40

Fig. 4 Comparison of the method with measure and calculatedresults in literature


1641. The distant scattered fields only consist of zero-orderpropagating Floquet modes and all other modes are madeevanescent.In order to verify the correctness of our method, the power

reflection coefficient of a three-arm zigzag FSS (by removingthe arms s1 and s3) was compared with a freestanding stripFSS at normal incident plane wave and found excellentagreement with measured [20] and calculated [21] results asshown in Fig. 4. Secondly, we used the power conservationprinciple as a criterion to test the accuracy of the obtainedresults and thus prove the validity of our model.The reflection and transmission coefficients of three-arm

zigzag FSS backed by dielectric substrate are comparedwith strip FSS as depicted in Fig. 5 for TM polarisation atnormal incidence. As is apparent from Fig. 5, there is asmall difference between the two results. This is anexpected result because an x-directed incident field inducesmore current on the arm s2. When the arms s1 and s3 areremoved the two curves would be coincident. As shown inFig. 6, the three-arm zigzag FSS elements resonate at

Fig. 6 Three-arm zigzag FSS reflection (RTE, RTM) andtransmission (TTE, TTM) coefficient magnitudes for TE and TMpolarisations

d1 = d2 = 5.5 mm, d = 5 mm, h1 = h3 = 2.5 mm, h2 = 5 mm, N = 40

Fig. 5 TM polarisation reflection (RTM) and transmission (TTM)coefficients for strip and three-arm zigzag FSS

h = 5 mm, d = 5 mm, d1 = d2 = 5.5 mm, h1 = h3 = 2.5 mm, h2 = 5 mm, N = 10(strip), N = 40 (three-arm zigzag)


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different frequencies for TM and TE polarisations. A singleresonance is appears at 22.45 and 32 GHz corresponding toTM and TE polarisations. The bandwidth (−3 dB points) ofTM case is wider than that of TE case.In order to understand the physics behind the proposed
three-arm zigzag FSS and for the possible practicalapplications, the structure is simulated by a commercialfull-wave EM simulation software based on finiteintegration technique. The geometrical parameters of thestructure are the same as in the first example (respectivelyFigs. 5 and 6). The electric field distribution on theconducting part of the structure at the resonance frequenciesfor TE and TM polarisations is shown in Fig. 7. As seenfrom the figure, the electric field strongly concentrates onthe horizontal strip for both polarisations. In addition, thereis also electric field concentration in the vertical arms of thestructure depending on the polarisation state of the incidentwave. This concentration is more pronounced for TEpolarisation case because of the strong coupling of the TEelectric field to the vertical arms. This means that thecontribution of the vertical arms to the resonance propertiesof the structure for TE polarisation is more prominent. Ifone compares the results between the strip and three-armzigzag FSS (Fig. 5), the additional contribution of thevertical arms (three-arm zigzag FSS with respect to strip)clearly explains the small difference in the reflection andtransmission properties of the structure and downshift in theresonances. Furthermore, it can be seen that the horizontal

Fig. 7 Electric field distribution on the conduction part of thethree-arm zigzag FSS at the resonance frequency of

a 22.45 GHz for TM polarisationb 32 GHz for TE polarisation

Fig. 8 TM polarisation reflection and transmission coefficients forvariation of substrate thickness

d1 = 5.5 mm, d2 = 10.5 mm, h1 = h2 = h3 = 5 mm, N = 60


arm works as a main mode with the contribution of thevertical arms as coupling modes for both TE and TMpolarisations. Note that the main mode of the horizontalarm is moderately smeared in TE case (relative to TM case)since the direction of the electric field is parallel to thevertical arms. Finally, the general feature of theelectromagnetic response of the proposed FSS can beexplained that horizontal arm with strong contribution ofthe vertical arms (relatively low and/or high) exhibits unityreflection property at some frequencies for the electric fieldvector with different directions.In Figs. 8 and 9, we examine the effect of substrate

thickness on the reflection and transmission coefficients forTM and TE polarisations. In Fig. 8, there are two resonancepeaks in K-band (18–26 GHz) for freestanding case.Between these two resonances, there exists a reflection gapor transmission window which means all the incidentenergy passes through the FSS. It is also seen from thesame figure that the entire incident wave passes up to 20GHz for the freestanding case. When the FSS is backed bythe dielectric substrate (d = 5 mm), three resonance peaksare observed in Ku − and K-bands. Between the resonances,

Fig. 9 TE polarisation reflection (RTE) and transmission (TTE)coefficients for variation of substrate thickness

d1 = 5.5 mm, d2 = 10.5 mm, h1 = h2 = h3 = 5 mm, N = 60

Fig. 10 TM polarisation reflection and transmission coefficientsfor variation of d1d = 5 mm, d2 = 10.5 mm, h1 = h2 = h3 = 5 mm, N = 60


Fig. 11 TE polarisation reflection and transmission coefficientsfor variation of d1d = 5 mm, d2 = 10.5 mm, h1 = h2 = h3 = 5 mm, N = 60

Fig. 12 TM polarisation reflection and transmission coefficientsfor five-arm zigzag FSS

h1 = h5 = 2.5 mm, h2 = h3 = h4 = 5 mm, d1 = d2 = 10.5 mm, N = 80, d = 5 mm

Fig. 13 TE polarisation reflection and transmission coefficientsfor five-arm zigzag FSS

h1 = h5 = 2.5 mm, h2 = h3 = h4 = 5 mm, d1 = d2 = 10.5 mm, N = 80, d = 5 mm

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there are transmission windows which allow fulltransmission. As is seen from the same figure, waves passthrough the FSS up to 15 GHz. In Fig. 9, there is a singleresonance at 25.5 GHz for the freestanding case. For d =5 mm, two resonances appeared with a reflection gapbetween them. The three-arm zigzag FSS passes the wavesup to 20 GHz for the freestanding case, and up to 15 GHzfor d = 5 mm.The effect of unit cell dimension d1 on the reflection and

transmission coefficients is illustrated in Figs. 10 and 11 forTM and TE polarisations. In Fig. 10, the incident TM waveis totally transmitted up to 15 GHz for both rectangular (d1 =5.5 mm, d2 = 10.5 mm) and square (d1 = d2 = 10.5 mm)lattice. Three resonance peaks appeared in Ku − and K-bandsfor both cases. The first resonance bandwidth is wider thanthe others. Between the resonance frequencies, there is areflection gap, which indicates complete incident energytransfer. In Fig. 11, the incident TE wave is also totallytransmitted up to 16 GHz for both rectangular and squarelattice. For rectangular lattice two resonances, in K-band,are seen whereas three resonances appear in Ku- and


K-band frequency regions for square lattice. There is awideband resonance around 20 GHz for the rectangularlattice. As shown in Figs. 10 and 11, the resonancefrequency is affected by the variation of unit celldimensions (periodicity of zigzag elements). When theperiodicity is switched from square to rectangular theresonance frequencies shift to lower one for TM incidence.However, a reverse situation is valid in the case of TEincidence.Finally, the reflection and transmission coefficients are

plotted for the five-arm zigzag FSS as shown in Figs. 12and 13. As in Fig. 12, four narrow band resonances areseen in K- and Ka- (26–40 GHz) bands for the case of TMpolarisation. The five-arm zigzag FSS passes all theincident energy up to 19 GHz. As shown in Fig. 13, thereare three resonance frequencies in Ku- and K-bandfrequency regions for TE polarisation. The incident energypasses through the FSS up to 13 GHz. Transmissionwindows exist between the resonances for both types ofpolarisations.

4 Conclusions

Reflection and transmission properties of a new type of FSSmade of zigzag metallic elements placed periodically on adielectric substrate is examined theoretically for arbitrarypolarisation. The overlapping PWS basis functions are usedto determine the unknown current coefficients induced onmetallic structures. Multiband resonances are observed forzigzag FSS. The periodic zigzag conductor elements passthe incident energy in S- and X-band frequency regionswhile reflecting in Ku- and K-bands. Between theresonances there exist transmission windows which allowfull transmission. Multiple resonances are seen for theproposed structure consequently it can be used in multibandantenna applications, subreflector systems and multibandbandstop or bandpass filter applications.

5 References

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zigzag metallic conductors as frequency selective surfaces

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