IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 62, NO. 5, MAY 2014 2361

Improved Bandwidth Formulas for Fabry-PérotCavity Antennas Formed by Using a Thin

Partially-Reflective SurfaceAli Hosseini, Student Member, IEEE, Filippo Capolino, Senior Member, IEEE, Franco De Flaviis, Fellow, IEEE,Paolo Burghignoli, Senior Member, IEEE, Giampiero Lovat, Member, IEEE, and David R. Jackson, Fellow, IEEE

Abstract—The power bandwidth of Fabry-Pérot-cavity an-tennas comprised of a thin partially-reflective-surface (PRS)above a perfectly conducting ground plane, based on its trans-verse-equivalent-network model and the simple susceptancemodel of a thin PRS, is studied. Considering the frequency varia-tion of the PRS susceptance model, a new formula is proposed toestimate the power density bandwidth and thus (approximately)the gain bandwidth of such cavities. The application and accuracyof the proposed formula are investigated using both numerical(i.e., based on full-wave simulations) and analytical (i.e., basedon a transmission-line model of the antenna) methods. Finally,the accuracy of the proposed formula is investigated for cavitiesformed using a finite versus infinite PRS.

Index Terms—Fabry-Pérot cavity (FPC) antenna, leaky-waveantenna (LWA), thin metallic frequency-selective-surface (FSS),thin partially-reflective-surface (PRS).

I. INTRODUCTION

S INCE the seminal work of G. Von Trentini [1], the useof partially-reflecting surfaces (PRSs) as an effective way

to enhance the directivity of simple sources placed above aground plane has been extensively investigated. Realizationsof PRSs in the form of single or multiple dielectric layers[2]–[4], or frequency-selective surfaces (FSSs) comprised ofmetal patches or slots cut in a metal plate [5]–[9] have beenproposed and studied. The operating principle of the resultingantenna can be explained in different ways. With reference toits receiving mode, the strong enhancement of directivity at aprescribed angle (e.g., broadside, which is normal to the PRSplane) can be interpreted as due to the resonant response ofthe Fabry-Pérot-like cavity formed by the PRS and the ground

Manuscript received December 11, 2012; revised November 11, 2013; ac-cepted February 05, 2014. Date of publication February 20, 2014; date of cur-rent version May 01, 2014.A. Hosseini, F. Capolino, and F. De Flaviis are with the Henry Samueli

School of Engineering, University of California, Irvine, CA 92697 USA(e-mails: sahossei@uci.edu; f.capolino@uci; edu and franco@uci.edu).P. Burghignoli is with the Department of Information Engineering, Elec-

tronics and Telecommunications, “La Sapienza” University of Rome, 00184Rome, Italy (e-mail: burghignoli@ die.uniroma1.it).G. Lovat is with the Department of Astronautic, Electrical and Energetic En-

gineering, “La Sapienza” University of Rome, 00184 Rome, Italy (e-mail: gi-ampiero.lovat@uniroma1.it).D. R. Jackson is with the Department of Electrical and Computer Engi-

neering, University of Houston, Houston, TX 77204-4005 USA.Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TAP.2014.2307337

plane when excited by an impinging plane wave. Therefore,the structure can be classified as a Fabry-Pérot Cavity (FPC)antenna. Alternatively, with reference to the transmitting modeof the antenna, directivity enhancement can be related to theexcitation of weakly-attenuated leaky modes, as first notedand then extensively studied by Jackson, Oliner, and theirco-workers [10]–[14]. Such leaky modes dominate the aperturefield of the antenna and give rise to strongly peaked patterns inthe far field; the structure can thus be classified as a leaky-waveantenna (LWA). When excited by a localized source, such asa horizontal electric or magnetic dipole, the relevant leakymodes propagate radially as cylindrical waves in the transverseplane (i.e., the plane of the PRS). A critical aspect of FPCantennas is their operational bandwidth (BW), which may bedefined in terms of either input impedance or radiated powerdensity (or alternatively, gain). In particular, with reference toan FPC antenna designed to radiate at broadside, the relative 3dB power density bandwidth, PBW, is defined as the frequencyrange over which the broadside power density of the antennaremains within 3 dB of its maximum value, divided by thefrequency of maximum broadside radiated power density. Asimilar definition holds for the 3 dB gain bandwidth GBW,which uses the broadside gain instead of radiated power den-sity. The two bandwidths are approximately equal providedthe total power radiated by the source is constant over thebandwidth of the structure, which is usually the case as seen inthe results provided later. Whereas the impedance bandwidthis mainly determined by the feeding structure of the antenna(not considered in this work), the power or gain bandwidths areessentially determined by the cavity behavior and hence by thePRS features and the material filling the cavity.For highly-directive FPCs (i.e., cavities formed by a highly

reflective FSSs), the frequency dependence of the FSS has anegligible impact on the power bandwidth of the antenna asdiscussed in [6], [7], [12]. It has to be noted that in this work,only FPCs formed by a ‘thin’ PRS (i.e., with a thickness muchless than the wavelength at the operating frequency of the an-tenna that can be modeled as a susceptance) are considered.In [15], using rather simple analytical approximations, and ne-glecting the frequency-dependence of a thin FSS susceptance, anew power bandwidth formula was discussed with gives an im-proved accuracy for low/moderate-gain FPCs relative to the for-mula derived in [6], [7], [12] [presented here as formula (20)].By considering simple LC models, the frequency dependenceof the shunt susceptance representing a thin FSS was studied in

0018-926X © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

2362 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 62, NO. 5, MAY 2014

Fig. 1. (a) Side-view of a FPC antenna formed by a thin PRS, being excited bya magnetic current (slot) on its ground plane. (b) TL model of the FPC antennain (a).

[16]. In [17], using the transmission line (TL) model of a FPCantenna covered by a thin FSS, an analytical expression for thebroadside radiated power density of such FPCs was derived.Considering the frequency-dependent model of a thin PRS asdiscussed in [16], the power expression was used to derive anapproximate power bandwidth formula.In this work, the goal is to further improve on the accuracy

of the proposed bandwidth formula in [17], by resorting to anew look into the radiation behavior of low- and moderate-gainFPC antennas and deriving some new and exact mathematicalexpressions for some of the terms used in the derivation. Thiswill lead to a new formula with improved accuracy, given hereas formula (19). Furthermore, considering FPCs made of com-plex thin PRSs, it will be shown the proposed formula providesa valuable tool for antenna engineers to design and study suchcavities and even optimize them for enhanced power BW (asdiscussed in [18], [19]). Note that this method can be extendedto FPCs covered by thick FSS structures as discussed in [20],[21]. The accuracy of the proposed formula will be studied usingboth analytical and numerical simulations and finally, the PBWbehavior of FPCs formed by using finite and infinite PRSs willbe investigated.

II. ANTENNA POWER BANDWIDTH

The formula proposed in [17] was derived assuming the peakvalue of the power density and its frequency can be approxi-mated by their respective values at the resonance frequency ofthe cavity. However, as will be shown here, this assumption isnot valid for moderate and low-gain FPC antennas. Moreover,based on the resonance condition of FPC antennas, the sinu-soidal functions required to express the TL model of the cavity,will be expressed here through an exact formula related to thecavity design parameters (i.e., the reflectivity of the PRS and thepermittivity and permeability of the material filling the cavity).As previously stated, similar to [17], this work studies FPC an-tennas formed by a thin PRS, and the assumption here is thatthe cavity is excited by an ideal magnetic current (i.e., slot) onthe ground plane of the cavity, as shown in Fig. 1(a), whereand are the relative permittivity and permeability of the ma-terial filling the cavity. The general TL model of this antennais shown in Fig. 1(b), where the thin PRS is modeled as a pureimaginary admittance (i.e., a susceptance , as previously dis-

cussed in [6], [7], [12]). With reference to Fig. 1(b), a thin PRScan be defined as a PRS which satisfies the condition

(1)

where is the free-space characteristic admit-tance. The bar over the PRS susceptance indicates that the valuehas been normalized by multiplication with . Following thediscussion in [17], the main goal of this work is to propose aneffective design tool (i.e., a formula) for such cavities to esti-mate their PBW. The authors would like to refer the readers to[17] for preliminary analytical studies carried-out on this topic.Referring to [17], the broadside radiation power density of a

FPC antenna covered by a thin PRS, can be written as

(2)

with and . More-over, , is the free-space wavenumber( is the free-space speed of light), is thewavenumber of the material filling the cavity and is theresonance height of the cavity. At the resonance frequency ofthe cavity, , and thus, the broadside powerdensity radiated from the cavity at the resonance frequency ofthe antenna can be expressed as

(3)

where and is the speed of lightinside the cavity medium. The 3 dB power bandwidth is definedby the radian frequencies at which .By neglecting the frequency variation of the term , from (2)the half-power bandwidth condition can be simplified as

(4)

where

(5)

, and can be found from (2) as willbe discussed shortly. Assuming now thatis a small fraction of , the functions and can beapproximated through a Taylor expansion in ,around (i.e., around ); taking also into account that

, it is found that

HOSSEINI et al.: IMPROVED BANDWIDTH FORMULAS FOR FABRY-PÉROT CAVITY ANTENNAS 2363

(6)

where can be expressed as a function of thecavity parameters (as will be shown shortly), ( at

), and evaluated at . It can be seenthat using approximation (6), (4) can be expressed as a quadraticequation as a function of , the solving of which results in thePBW of the cavity. The bandwidth quadratic equation can beexpressed as a function of the FSS model parameters and asfollows:

(7)

where

(8)Solving (7) for , the fractional power bandwidth is then ob-tained as , which is expressed as

(9)

By expressing , and as a function of cavityparameters, (9) can be simplified as will be shown in the nextsubsection.

A. Sinusoidal Functions as a Function of Cavity Parameters

Using the resonance condition, where and , the

terms and can be expressed exactly as a function of thecavity parameters at the cavity resonance frequency. The cosinefunction assumes only negative values at the fundamental FPCresonance, where the cavity is approximately one-half of adielectric wavelength in thickness. Hence, .Solving the above resonance equation then results in

, which yields

(10)

It can be seen that at has positive and negative valuesfor inductive and capacitive FSS layers, respectively. Therefore,(10) can be modified as

(11)

After calculating the exact value of the sine function at ,the function can be expressed as

(12)

B. Approximate Maximum Power Density and its Frequency

Using numerical calculations, it can be seen that the max-imum value of the power density ( ) is not exactly the sameas , as will be shown here. Moreover, the radian frequency

at which the broadside power density attains its maximumvalue, , is different than . This difference becomesmore significant for low/moderate-gain FPC antennas (as in thepresent case of interest). Therefore, in order to have a accuratePBW formula, these values must be estimated accurately. Here,we use the resonance condition of the antenna, in order to ex-press and as functions of the cavity parameters aswill be shown below.In order to estimate the maximum value of the power density

and the corresponding frequency, the reciprocal of the powerdensity expression is approximated using a quadratic expression

, resulting in

(13)

Using (13) and considering , the unknowncoefficients ( for ,1,2) can be found, at , as

(14)

where

(15)

which yields

(16)

By substituting (16) and (14) into (13), the approximatevalues of and can be expressed as

(17)

Table I shows a comparison of the normalized maximumpower density and the corresponding frequency of maximumpower density between the approximate values, from (17), and

2364 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 62, NO. 5, MAY 2014

TABLE ICOMPARISON BETWEEN THE EXACT AND APPROXIMATE VALUES OF AND

FOR CAVITIES FILLED WITH AIR AND DESIGNED FOR GHz

the exact values obtained numerically from (2), showing a verygood agreement.It can be seen that for larger absolute values of ,

. Also, it can be seen that the broadside radiationpower density of the antenna at converges to its value atthe resonance frequency ( ) as the magnitude ofthe PRS susceptance increases.Considering (16) and (17), can be expressed as

(18)

and thus using (18), (9) can be simplified as

(19)

Formula (19) is the main formula that is proposed in thiswork, which estimates the PBW of FPC antennas formed by athin PRS, as a function of the cavity parameters.A simpler but more approximate formula can be obtained for

high-gain FPC antennas because they require .This corresponds to thus . Therefore, fornonresonant FSSs as discussed in [17], (8) can be simplified to

. As a result, the high-gain approximationof (9), considering and ,can be expressed as , which simplifiesto

(20)

The result (20) is the formula proposed in [6], [7], [12] to es-timate the relative power bandwidth of high-gain FPCs formedby thin FSSs. In the next two sections, application of the newproposed formula (19) and its ability to estimate the PBW (thusthe GBW) of FPCs formed by conventional (i.e., thin metallicFSS made of periodic holes or patches) and more complex thinPRSs (i.e., satisfying (1)), is studied.

III. ANALYTICAL AND NUMERICAL COMPARISONS

In this section, the accuracy of the proposed formula (19)will be investigated by comparing its predicted values to values

obtained from both analytical (i.e., based on TL model of theantenna) and numerical methods (i.e., using full-wave simula-tions) as will be explained shortly. In this section, the versatilityand accuracy of this new formula in estimating the PBW andGBW of FPCs formed by an arbitrary thin PRS will be demon-strated. Only FPCs formed by infinite PRSs will be discussed.The impact of using a finite PRS on the PBW/GBW of suchcavities will be discussed in the next section. For each antenna,four values of PBW, estimated using four different methods, arecompared to each other. The first value is obtained from a nu-merical calculation based on power formula (2). This gives anumerically exact result based on the model shown in Fig. 1(b)(which assumes that the PRS can be modeled as a pure losslessshunt susceptance). These two PBW values are also comparedto PBW/GBW results obtained from two other analytical/nu-merical methods, discussed next, which are in conduction withAnsys HFSS™.In the first method (denoted as M1), the equivalent normal-

ized susceptance of any arbitrary thin PRS is numerically ob-tained as a function of frequency by surrounding the PRS/FSSunit-cell by periodic boundaries, being illuminated by a planewave impinging from broadside (i.e., from the -direction, as-suming that the FSS is located in the -plane), and calculatingthe relevant scattering parameters. In the next step, based on theformulas proposed in [22, Appendix A] and considering the fre-quency-dependent susceptance of the PRS, both the broadsidegain and the radiated power density of an FPC antenna fed byan ideal feed (i.e., a magnetic current on the ground plane ofthe cavity) can be found as a function of the cavity material (and ), , and the frequency, and this is used to determinethe PBW and GBW of the designed cavities. This method thususes a combination of both analytical and numerical analysis(based on the TL model of the antenna and full-wave simula-tions, respectively).In the second method (denoted as M2), the TL model of the

antenna is used to derive a novel simple formula leading toa single-step method which is based on full-wave simulationsonly. This is explained next. Based on TL formulas that holdfor the broadside direction, the real part of the upward admit-tance looking into the cavity just above the ground plane of thecavity, as shown in Fig. 1(b), can be found as

(21)

Looking at the general power density formula (2), (21) canbe used to find the broadside power density of the antenna as

(22)

Using (22), the PBW of the antenna can be found using full-wave simulations as follows. Based on [17, formula (2)] andthe susceptance value of the PRS at the desired resonance fre-quency, the resonance frequency/height of the cavity is first de-termined. In the next step, a unit-cell of the PRS is surroundedby the periodic boundaries and terminated on the top by anair-impedance-boundary modeling the radiation area on top of

HOSSEINI et al.: IMPROVED BANDWIDTH FORMULAS FOR FABRY-PÉROT CAVITY ANTENNAS 2365

TABLE IIFPCS FORMED BY THIN METALLIC FSS, RESONATING AT GHz

the antenna. Below the FSS, the box is extended by the reso-nance height of the cavity being terminated by a waveport andfilled with a material forming the cavity. Finally, the upwardadmittance of the waveport looking into the cavity is simulated,which using (22) results in the broadside power density of theantenna versus frequency which can be used to calculate thePBW.In the next parts, PBW and GBW of FPC antennas formed

by three different thin PRS layers are investigated. The dif-ferent PRSs include: (1) a thin PRS made of a conventional thinmetallic FSSs, (2) a thin PRS made of a stack of thin dielectricslabs, and (3) a thin PRSs made of a stack of thin metallic FSSsseparated by thin dielectric slabs.

A. Thin PRS Made of a Conventional Thin Metallic FSS

The PBW and GBW of FPCs formed by thin metallic FSSsare investigated here. Two forms of FSS layers are selected:capacitive (i.e., periodic patches) and inductive (i.e., periodicholes in a thin metallic sheet). Moreover, the cavities are filledwith a homogenous material with a relative permittivity ofor 2. In order to use (19), the first frequency-derivative of thenormalized susceptance of the FSS is required, which for an FSSmodeled as a capacitor or an inductor can be expressed as [16]

(23)

Table II shows the simulated values of GBW and PBW forFPCs formed by conventional metallic FSSs and resonating at 5GHz, obtained from methods 1 and 2, compared to their valuesestimated using a numerical solution involving the powerformula (2) and the proposed BW formula (19). Rectangularpatches and slots are used to form the FSSs. The dimensionsof the FSS features (in millimeter) are given in Table II where‘cell’ denotes the unit-cell dimension of the FSS. Moreover,‘ ’ and ‘ ’ respectively correspond to the length and widthsof the rectangular slots or patches, for inductive and capacitiveFSSs, respectively.

Fig. 2. Layer-view of a PRS made of thin multilayer dielectric slabs.

A good agreement can be observed among the numerical andsimulated values and the values estimated from the proposedformula (19). Also, these values are compared to those obtainedfrom formula (20), showing the significant improvement of (19)over (20). It is also seen that the GBW is reasonably close to thePBW in all cases. The GBW is thus an improvement over thesimpler formula reported in [23].Plots of the reflection and transmission coefficients of the FSS

(omitted) show interesting effects such as perfect transmissionand reflection at nearby frequencies, which can be explainedusing an accurate CAD model [24].

B. Thin PRS Made From a Stack of Thin Multilayer DielectricSlabs

In this subsection, the PBW and GBW of FPCs formed bya PRS made of stack of thin multilayer dielectric slabs will beestimated using the proposed formula (19), and the results willbe compared to results obtained from methods 1 and 2. Fig. 2shows the layered-view of an arbitrary PRS made of stack ofthin multilayer dielectric slabs where is thecharacteristics impedance of the th layer of the PRS, and andare the relative permittivity and permeability of the layer, re-

spectively. As previously discussed, each layer is much smallerthan the wavelength at the resonance frequency of the cavity,i.e., where and isthe speed of light inside the th layer.Using (1), the upward admittance looking into the PRS, as

shown in Fig. 2, can be then simplified as

(24)

In these calculations the dielectric layers are assumed to belossless. It can be seen that for a stack of thin dielectric slabs,since , to , the normalized susceptance ( )modeling the PRS is positive (i.e., capacitive) and a nonlinearfunction of the frequency. Note that different positions of thelayers in the stack results in the same susceptance and hencereflectivity of the PRS.In order to use (19), the first derivative of the susceptance

modeling the PRS should is derived as

(25)

2366 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 62, NO. 5, MAY 2014

TABLE IIIFPCS FORMED BY A 3-LAYER PRS AND FILLED WITH AIR, GHz

Fig. 3. Side-view of a thin PRS made of a stack of thin metallic FSSs sepa-rated by distances much smaller than the wavelength. (a) Unit cell of PRS1,with layers of capacitive metallic dipoles on the top and the bottom of the slab.(b) Unit cell of PRS2, which has inductive slots instead of dipoles.

Table III shows some numerical examples of PBW/GBW ofair-filled FPCs formed by a stack of thin dielectric slabs, de-signed to resonate at 5 GHz. Two different thicknesses are usedfor the slabs, 1 mm and 1.5 mm, as shown in Table III. A goodagreement can be observed between the numerical and simu-lated PBW values and the one estimated from (19). Moreover,as previously discussed, (20) cannot predict an accurate PBWand (thus GBW) of such low/moderate gain FPCs.

C. Thin PRS Made From Stack of Multilayer Metallic FSSs

The PBW and GBW values of FPC antennas formed by athin PRS made of a stack of two thin metallic FSSs separated bydistances much smaller than the operating wavelength of the an-tenna is now investigated. The thickness of the separating slab,1 mm, is much smaller than that the wavelength at the reso-nance frequency of the cavity (i.e., 42.4 mm inside the dielec-tric slab with a relative permittivity of 2); therefore, it can beconsidered as a thin PRS. Two designs of PRS layers are con-sidered here, namely a PRS made of a thin dielectric slab cov-ered by either periodic dipoles or slots on both sides, as shownin Fig. 3(a) and (b), respectively (henceforth named PRS1 andPRS2).In order to use (19) we need to extract the derivative (with re-

spect to radian frequency) of the normalized susceptance at theresonance frequency of the antenna (i.e., 5 GHz). the derivative

TABLE IVFPCS FORMED BY A THIN METALLIC FSS, RESONATING AT GHz

TABLE V

is calculated numerically using a standard central-difference ap-proximation.Table IV shows a summary of the results obtainedfrom formulas (2) and (19), along with numerical values re-sulting from methods 1 and 2. The dimension of the designedPRS layers are shown in Fig. 3.It can be seen that the results obtained from the proposed for-

mula (19) show a good agreement with the numerical resultsobtained from methods 1 and 2. Hence, the proposed PBW for-mula can be used numerically, by estimating the susceptancevalue of any arbitrary thin PRS and its first frequency derivativeat resonance frequency of the cavity. Hence, formula (19) can beused together with a simple numerical calculation of a frequencyderivative to estimate the PBW of the antenna, which allows forthe method to be applied in a variety of circumstances.

IV. FINITE VERSUS INFINITE PRS

In this section, the PBW will be studied for FPCs formed bya finite versus infinite PRS. Note in the TL model of the an-tenna, the PRS and the cavity are extended indefinitely hori-zontally, which cannot be simulated using a full-wave simula-tion tool. Therefore, the PRS must be truncated transversely inorder to simulate the antenna using full-wave simulations or tofabricate the antenna. In order to minimize the feeding impacton the full-wave BW results, infinitesimal dipoles (being ex-cited using lumped ports) are used here. From Table II, an FPCformed by using an inductive FSS (i.e., made of periodic slotswith in a square unit-cell square that is18 18 and a cavity filled with air) is used as an examplehere. Table V shows the GBW of the FPC antenna (based onfull-wave simulations) using a finite-size FSS where corre-sponds to the number of unit-cells used for the PRS/FSS in eachsimulation. It can be seen that by increasing the number of ele-ments, the GBW values approach the values obtained based onan infinite PRS.

V. CONCLUSION

The main result of this paper, namely formula (19), has beenintroduced to accurately estimate the 3 dB radiated power den-sity bandwidth and thus also (approximately) the 3 dB gainbandwidth of Fabry-Pérot cavity antennas formed by any arbi-trary thin PRS. The formula is based on the susceptance modelof a thin PRS and considers its frequency variation, which playsan important role in the PBW/GBW behavior of low or mod-erate-gain cavities. This paper provides a valuable tool for the

HOSSEINI et al.: IMPROVED BANDWIDTH FORMULAS FOR FABRY-PÉROT CAVITY ANTENNAS 2367

study of antennas based on generalized PRSs, either havingnon-negligible thickness, or active loading, or both. This willbe the subject of future investigations.

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[23] D. R. Jackson, P. Burghignoli, G. Lovat, F. Capolino, J. Chen, D.R. Wilton, and A. A. Oliner, “The fundamental physics of directivebeaming at microwave and optical frequencies and the role of leakywaves,” Proc. IEEE, vol. 99, no. 10, pp. 1780–1805, Oct. 2011.

[24] M. García-Vigueras, F. Mesa, F. Medina, R. Rodríguez-Berral,and J. L. Gómez-Tornero, “Simplified circuit model for arrays ofmetallic dipoles sandwiched between dielectric slabs under arbitraryincidence,” IEEE Trans. Antennas Propag., vol. 60, no. 10, pp.4637–4649, Oct. 2012.

Ali Hosseini, photograph and biography not available at the time of publication.

Filippo Capolino, photograph and biography not available at the time ofpublication.

Franco De Flaviis, photograph and biography not available at the time ofpublication.

Paolo Burghignoli, photograph and biography not available at the time ofpublication.

Giampiero Lovat, photograph and biography not available at the time ofpublication.

David R. Jackson, photograph and biography not available at the time ofpublication.