Transmission upper limitofband-pass double-layer FSS and method oftransmission performance improvement

Schoolof Aeronautic Science and Engineering,Beihang University,Beijing 100191,China

Transmission upper limitofband-pass double-layer FSS and method oftransmission performance improvement

Minjie Huang and Zhijun Meng*

Schoolof Aeronautic Science and Engineering,Beihang University,Beijing 100191,China

The transmission upperlimitofa double-layerfrequency selective surface(FSS)with two in fi nitely thin metalarrays is presented based on the study ofthe generalequivalent transmission line modelof a double-layer FSS.Results of theoreticalanalyses, numerical simulations and experiments show that this transmission upper limitis independentofthe array and the element,which indicates that it is impossible to achieve a transmission upper limit higher than this one under a given incident and dielectricsupporting condition by the design of the periodic array.Both the applicable condition and the possible application of the transmission upperlimitare discussed.The results show thatthe transmission upper limitnotonly has a good reachability,butalso provides a key to effectively improve the transmission performance of a double-layer FSS or more complex frequency selective structures.

frequency selective surface(FSS),equivalent circuit, transmission line theory,dielectric interfaces.

1.Introduction

The frequency selective surface(FSS)has been widely used as fi lters from microwave band to opticalband[1-6]. Inband transmission is one of the most important design indices of those fi lters,such as FSS antennas and radomes [5,6].Generally,for the inband performance,the requirementis the highertransmission the better.However,forthe outband performance,lower transmission and sharp switch are preferred.Considering these requirements,the doublelayer FSS has attracted wide attention forits nearly perfect transmission performance[2,6].

Although the double-layer FSS has a quasi-perfectfrequency selective characteristic,itis usually hard to achieve a high inband transmission.Many researchers are devoted to improving the inband transmission of FSSs.

Dielectric layers are viewed as one of the most important factors of the electromagnetic performance of FSSs besides the shape of the array.A lot of investigation focused on improving the transmission performance by the dielectric-supportform design[7-18].

Luebbers and Munk investigated the effectofa covering layer on the resonance frequency,bandwidth,and theirangle stability of an array of narrow rectangular slots.They noted thatwith proper design,the angle stability of the resonance frequency and bandwidth could be improved by the dielectric layers;however,some other effects might be encountered,such as a reduction in the frequency of resonance and Wood’s anomaly(surface wave)null[8]. In an earlier study[9],Munk and Fulton demonstrated a double-layer FSS design that has an almost perfect frequency response achieved by appropriately arranging the dielectric layers.Callaghan,Parker,and Langley investigated the change in the resonance frequency,bandwidth, and the shape of transmission/re fl ection curves with the increase in the dielectric thickness.Both the cases of arrays bonded on one side and embedded centrally in the dielectrics were considered,and some importantconclusions were presented[10].Munk and Wu summarized the effects of the dielectrics on the electromagnetic performance of single-layerand multilayer FSSs,based on previous works [2,11].Nowadays,the design work of FSS has become increasingly complicated,and novel FSS structures are being continuously developed.However,dielectrics are still playing importantroles in those novelFSS structures,such as active FSSs,tunable FSSs,dielectric periodic structures [12-20].For the irreplaceable ability,the in fl uence of dielectrics on the performance of FSSs has become a hotspot in the FSS area for recentyears.

Although there is an overallunderstanding aboutthe effects of dielectrics on FSSs,some problems are still waiting to be solved,such as how to improve the transmission of a band-pass FSS by the metalarray designing,and whether there is an impassable transmission upper limit lowerthan 1 foran FSS with given dielectric-loading form. Barlevy and Rahmat-Samiiinvestigated the re fl ection analytic constraints of a dielectric substrate supported aper-ture array.They noted that there was a theoretical limit on the amount of energy that can be lost and this theoretical limit can be predicted by an approach presented in [21].They also pointed outthattheir analysis could be extended to any number of substrates and superstrates,but could not be extended to multiple periodic surfaces.The author of this paper discussed the maximum transmission thata single-layer metal array with an arbitrary dielectricloading form can reach in[22,23]and provided a speci fi c transmission upperbound for single-layer FSSs.However, allof the above have notprovided a perfect answer to the design problem of a complex frequency selective structure. Considering these,this paper discusses the effects of dielectrics on double-layer FSSs and tries to provide an analytic constraintfor the effects on an arbitrary frequency selective structure.The technique of improving the performance of a double-layer or more complex frequency selective structure is also introduced.

2.Transmission upper limit of double-layer band-pass FSS

The method of analyzing FSSs by using the equivalent transmission line theory was fi rstly introduced by Marcuvitz and Wait[24,25].In the transmission line theory,the equivalentmodelofa periodic metalarray is a combination of capacitance,inductance and resistance[1,2,6,24-28].It is hard to deal with the progressional solution of an FSS directly but easy to obtain some useful information from this quasi-static approximation model.The following discussion is builton this approximation model.

2.1 Equivalent transmission line model of doublelayer FSS

Fig.1(a)describes a double-layer FSS structure comprised of M dielectric superstrates,N interlayers and L substrates.The aperture arrays at z=zMand z=zM+Nare assumed to be in fi nitely thin and of a perfect conducting area.We de fi ne the permittivity and the permeability ofthe i th layerasε(i)andμ(i)respectively,and the impedance in the equivalenttransmission line modelas Zi.According to the equivalenttransmission line theory,we can assume that the equivalent modelof the aperture array at z=zMis a parallelconnection of inductive impedance L1and capacitive impedance C1,and the equivalent model of the array at z=zM+Nis a parallelconnection of L2and C2.Thus the equivalenttransmission modelof the double-layer FSS shown in Fig.1(a)can be given as Fig.1(b).

Fig.1 Description ofa double-layer FSS

2.2 Transmission coefficient of double-layer FSS

Considering the continuity of the electric voltage and current on the nodes at z=zMand z=zM+Nin the equivalent transmission line model shown in Fig.1(b)respectively,we can obtain the equation as follows:

where the superscript“(n)”indicates the n th layer,which usually be ignored in vacuum(n=0 or M+N+L+1). T(n)is the transmission coef fi cient while R(n)is the refl ection coef fi cientin the n th layer.Ynandξ(n)are different denotations of the admittance of medium in the equivalenttransmission line model and the FSS structure, respectively.Obviously,Yn=ξ(n)=1/Zn.β(n)means the normalcomponentof the wave vector.In addition,

The continuity of the electric voltage and currenton other nodes can also be written as the similarform as(1)and(2):

By substituting(3)into(1)and(2),and noticing

By solving(5)we obtain the transmission coef fi cient

and XYYmndenotes the element of the m th row and the n th column of matrix XYY.

2.3 Transmission upper limit of double-layer FSS

Equation(10)means thatthe transmission coef fi cient T is a function of G1and G2.

Assuming the inductive impedances L1,L2and the capacitance impedances C1,C2are unconstrained positive values,then the equivalentmodelshown in Fig.1(b)must have a maximum transmission coef fi cient Ttopwhich isnotmore than 1.If the metal array in the FSS structure is not perfect electric conductive(PEC),the pure resistance impedance component will not be zero in its equivalent model[2,29].However,itcan be proved theoretically that the transmission coef fi cient will never exceed Ttopeven though the pure resistance component is not zero.Based on this fact,Ttopis obtained by solving(10).are de fi ned as the natural frequencies of the two aperture arrays,and fincis de fi ned as the incident frequency,we can obtain

which means thatboth the value ranges of G1and G2are real number fi eld R,theoretically.Thus it is sure that T has an extremum independentof G1and G2.We de fi ne:

The extremum of T can be written as

The solution of Ttopis involved with the minimum problem

The minimum problem(26)has an analytical solution which can be obtained by translating(26)to a simple cubic equation.The expression of the analytical solution is too complex to print here but is easy to calculate,so it is ignored in this paper.

As this derivation is not relevant to the incidence condition,dielectrics,and the shape of array,the transmission upper limitgiven by(25)is suitable for any planardoublelayer FSS ofan in fi nite extent.The expression of Ttopalso shows that Ttopis independentof the array and the shape of the unit cell,which means it is impossible to obtain a higher transmission than Ttopby designing the array and the unit cell for a given FSS structure excited by certain plane waves.

3.Verification and practicaleffects

3.1 Testing by simulated and experimentalresults

Numerical simulation is conducted to test the reliability and the practicability of the transmission upperlimitgiven by(25).The mode analysis method(MAM),the mutualadmittance approach(MAA)and the fi nite elements method (FEM)are employed to ensure thatthe conclusions are independentof the solving technique.

A lot of literature mentioned a high transmission A-sandwich FSS structure[2,6,30].Reference[30]presented the performance of an A-sandwich FSS radome which appeared thatthere is a high transmission coef fi cientfrom the design and experimentalresults.Fig.2 shows the doublelayer FSS presented in[30],while Fig.3 shows a comparison between the transmission given by[30]and the corresponding transmission upper limit calculated by(25).Obviously,both the experimentalresults and the results simulated using MAA are constrained by(25).

Fig.2 Double-layer FSS presented in[30]

Fig.3 Comparison between the transmission coefficient and the transmission upper limit

3.2 Rationality of the transmission upper limit

The transmission upper limitgiven by(25)faces two challenges.One is the transmission of the corresponding pure dielectric structure;the othercomes from the perfecttransmission of the multilayered dielectric structure.The transmission coef fi cientof the multilayered dielectric structure can be easily deduced by the equivalenttransmission line theory:

We can de fi ne the perfect transmission coef fi cient as the transmission coef fi cient in an ideal state when there is no re fl ection on each dielectric interface.Thus,

Fig.4 shows a comparison of the transmission upper limit of the FSS structure given by[30],the transmission of the dielectric structure and the perfecttransmission determined by(28).Itis obvious that

Equation(29)has been proved to satisfy any condition by further works,which means that the transmission of the double-layer FSS is still constrained by(25)even if the shape of the aperture is in fi nite or the aperture area ratio (Saperture/Sunitcell)approximates 1,while FSS degenerates to a multilayered dielectric structure.It also shows thatthe presence of the conductive array signi fi cantly improvesthe impedance-matching performance ofthe dielectric structure.

Fig.4 Comparison of the transmission upper limitofthe FSS structure given by[30],the transmission ofthe dielectric structure and the perfect transmission determined by(28)

3.3 Reachability ofthe transmission upper limit

Two techniques have been tried to check the reachability of the transmission upper limit.One is checking the reachability of the values of G1and G2while function|F|in the minimum problem(26)reaches its minimum value,and the other is designing a double-layer FSS structure whose transmission is close to the transmission upper limitgiven by(25).

Fig.5 shows the values of G1and G2when the transmission reaches its upper limit for TE incidentwave at angle θ=15?.The dielectric loading form is the same as Fig.2, so G1is equal to G2for the symmetrical loading condition.Fig.5 indicates that G1and G2need to be adjusted to -5e-4-4e-4 if we want to obtain a transmission as high as(25)in frequency band of9-14 GHz.According to(19),to improve the transmission performance,f01and f02can be designed to adjust G1and G2if the equivalentinductance L1and L2are small,or L1and L2are designed if they are big enough when keeping f01and f02constants to regulate the transmission accurately.

Fig.5 Values of G1 and G2 when the transmission reaches its upper limit for TE incident wave at angleθ=15°

Fig.6 shows the simulated transmission of the doublelayer FSS with two rectangle-loop arrays.The dielectric loading form is the same as Fig.2.The transmission coef fi cientcalculated by MAM can well approximate to the upper limit,while the result simulated by FEM is a little lower than the upper limit.Generally,elaborate design can effectively improve the transmission performance of a double-layer FSS.

Fig.6 Transmission characteristic of high transmission doublelayer FSS design

4.Applicable condition and possible application

4.1 Applicable condition

Based on the derivation in Section 2,we can conclude the scope of application of the transmission upper limitgiven in(25)as follows:

(i)The double-layer FSS is of an in fi nite extent.

(ii)The array is in fi nitely thin and perfectconductive. (iii)The FSS structure is excited by plane wave incidence.

(iv)The properties ofthe dielectrics do notchange in the tangentdirection(x-and y-directions).

If the metalarray has a non-zero thickness,one can divide the interfaceand right interfaceA similar process also can be conducted on the interface z=zM+N.Then the equivalent impedance of the metal array still has the form of Zr+j Zi.Thus if one adds the phasic term induced byinto the derivation process,the transmission upper limit(25)will be applicable to the thick metalscreen FSS.The forms of(25)and(26) signify thatif the equivalentadmittances of the dielectrics can be obtained,then the transmission upper limit should exist.In other words,constraint(iv)can be relaxed,such that the properties of the dielectrics have the same period as the array.

4.2 Possible application

The transmission upper limit presented in this paper not only has a good reachability,but also provides a key to effectively improve the transmission performance of a double-layer FSS.

We know thatthe solving process willnotonly produce the transmission upper limit,but also give the equivalent impedance G1and G2while the transmission reaches its maximum value.Generally,the pure resistance,inductance and capacitance components of the equivalent impedance ofthe metalarrays can be estimated by severalways which can be found in[25-28].According to[27-29],the pure resistance component depends on the conductivity of the metalarray.The pure resistance componentwill in fi nitely approach 0 if the metal array has a nearly perfect electric conductivity.The inductance component mainly depends on the geometry of the metal array,but is independent of the dielectric layers.However,the capacitance component is changing with both the geometry of the array and the dielectric loading form,but can be estimated by the free-standing metalarrays and the dielectric loading form. Thus,(25)providesnotonly a transmission upperlimit,butalso a method of designing high performance double-layer FSSs.

Fig.7 shows the change in the equivalent inductive impedance and the capacitance impedance with the length of the dipole elementof a single-layer FSS,which is calculated by fi tting the frequency response curve.The calculated results shown in Fig.7 have the same trend with the wave guide theory and the equivalenttransmission line theory.The authors’future work will focus on improving the transmission performance of the double-layer FSS by designing the equivalentimpedance of the metalarrays.

Fig.7 Equivalent inductive impedance and capacitance impedance of a dipole array

5.Conclusions

Dielectric is one ofthe mostimportantfactors to determine the frequency behavior of FSS.This study shows that for a double-layer FSS structure in a given dielectric loading form,there is a transmission upper limit for each planewave incidence.It is observed that it is impossible to obtain a highertransmission than the upperlimitby designing the array and elementof FSS.The transmission upperlimit for the double-layer FSS is presented as(25).This transmission upperlimitis found to be independentofthe solution technique,which is proved by both theoreticalanalyses and experiments.

The transmission upperlimitpresented in this paperdescribes the possible degree of the effectof dielectrics on the frequency response of FSS.The expression of this upper limit is much simple and easy to compute,which makes it possible to make a rapid evaluation of different dielectric loading plans in the design of FSS.This result also provides a noveltechnique of designing high transmission performance double-layer FSSs.

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Biographies

Minjie Huang was born in 1982.He received his B.S.degree in aircraft design in 2003 and Ph.D. degree in 2008,from Beihang University,Beijing, China.Currently he is an instructor and Master tutor in the Departmentof Aircraft Design,Beihang University.His research interests include aircraft design,low observable technology and FSSs.

E-mail:huangmingjiebuaa@163.com

Zhijun Meng was born in 1982.He received his B.S.degree in 2004 and Doctor degree in 2009 from the School of Aeronatic Science and Engineering, Beihang University.After the postdoctoralresearch, he became a lecturer of Beihang University in July, 2010,and currently holds a position as an associate professor,in the Department of Aircraft Design,Beihang University.His research interests include aircraftdesign,modeling/control/simulation ofunmanned aerialvehicle(UAV)and rotorcraft.

E-mail:mengzhijun@buaa.edu.cn

10.1109/JSEE.2015.00027

Manuscriptreceived June 09,2013.

*Corresponding author.