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Planar FabryPerot directive antenna: asimplified analysis by equivalent circuitapproachG. Di Massaa, S. Costanzoa & H.O. Morenoba Dipartimento di Ingegneria Informatica, Modellistica, Elettronicae Sistemistica, Universit Della Calabria, Rende (Cs), 87036, Italy.b Facultad de Informatica y Electronica, Escuela SuperiorPolitecnica De Chimborazo, Riobamba, Ecuador.Published online: 08 Jan 2015.

To cite this article: G. Di Massa, S. Costanzo & H.O. Moreno (2015): Planar FabryPerot directiveantenna: a simplified analysis by equivalent circuit approach, Journal of Electromagnetic Wavesand Applications, DOI: 10.1080/09205071.2014.997838

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Journal of Electromagnetic Waves and Applications, 2015http://dx.doi.org/10.1080/09205071.2014.997838

Planar FabryPerot directive antenna: a simplified analysis byequivalent circuit approach

G. Di Massaa, S. Costanzoa and H.O. Morenob

aDipartimento di Ingegneria Informatica, Modellistica, Elettronica e Sistemistica, Universit DellaCalabria, Rende (Cs) 87036, Italy; bFacultad de Informatica y Electronica, Escuela Superior

Politecnica De Chimborazo, Riobamba, Ecuador(Received 3 June 2014; accepted 8 December 2014)

A new approach is proposed for the analysis and design of a planar FabryPerot antenna.The complete modal analysis of the field into the cavity leads to a simplified equivalentcircuit, able to provide a reliable description of the coupling with the feeding waveguide,as well as to compute the equivalent currents on the radiating apertures, thus obtaining theradiated far-field. The proposed approach is preliminary validated on a metallic FabryPerot antenna structure. Then, a modified configuration, based on a cavity partially-filledwith a dielectric substrate, is assumed to obtain a FabryPerot antenna with improvedbandwidth features. Experimental validations on array prototypes are successfully dis-cussed.

Keywords: FabryPerot cavity; antennas; millimeter waves

1. IntroductionMillimeter-wave frequencies are very attractive resources for telecommunications, as beinguseful in many applications which include the realization of high data rate links in picocellular networks, local multipoint data services, automotive radars, inter-satellite commu-nications, and so on.

Spherical or hemispherical FabryPerot [1] antennas [2,3] give a very interesting solu-tion, as being able to provide a high agility in the design-synthesis process. In the existingconfigurations, one or both reflecting mirrors are composed by metal strip gratings, andthe basic idea is that the gaussian field distribution on the mirrors could provide a far-fieldin the absence or with very low sidelobes. This advantage is obtained at the expense of acomplicated mechanical structure.

In [4], an optimized partially reflecting surface, placed in front of a waveguide apertureon a ground plane, is used to obtain a high gain antenna with a useful frequency bandwidthequal to 100 MHz.

In [5], an empty planar metallic FabryPerot antenna is studied using a 3-D MoM(method of moments) simulation code with a thin wire approximation. The ground plane istaken into account by adopting the image theory, and the resonance condition is excited by apatch antenna placed into the cavity in the proximity of the ground plane. The experimentalstudy shows a good agreement with the simulation results, but a very little bandwidth isobserved.

Corresponding author. Email: [email protected]

2015 Taylor & Francis

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2 G. Di Massa et al.

In [6], the directivity and bandwidth of FabryPerot resonator antennas made of twodifferent ground plates with a partially-reflecting surface, are theoretically studied. A sim-plified model considering the plane waves propagation in the parallel plate waveguide isadopted, and the directivity of FabryPerot resonator is shown to obtain its maximum whenthe primary source is located at a specific position.

In [7], a FabryPerot cavity is used to design a 60 GHz single-feed directive antennagiven by a ground plane covered by a frequency selective surface, with the adoption of asimple synthesis process based on a transmission line model.

In this paper, a FabryPerot antenna composed by an open resonator with plane mirrors isconsidered. An equivalent circuit based on a modal analysis of the open resonator is adoptedfor the characterization of the proposed structure, in order to optimize the coupling betweenthe feeding rectangular waveguide and the planar open cavity. Preliminary results on themetallic FabryPerot antenna are discussed as predicted by the adopted equivalent circuit.Then, a modified FabryPerot configuration based on the insertion of a dielectric substrate,is proposed to significantly improve the operation bandwidth of the original metallic FabryPerot structure. Experimental results on a Ku-band partially-filled 8 8 elements array aresuccessfully reported to demonstrate a radiation bandwidth improvement of about 48%.

The paper is organized as follows. First, in Section 2, the theoretical background ofthe proposed method for an open resonator composed by metallic rectangular mirrorsis presented. In Section 3, the feeding-waveguide-to-cavity and the cavity-to-free spacecouplings are analyzed for the case of metallic FabryPerot antenna. In Section 4, theproblem of antenna bandwidth improvement is addressed by introducing the partially-filledantenna configuration and the relative experimental results assessing the proposed approachare presented. Finally, conclusions and future works are outlined in Section 5.

2. Open resonator composed by rectangular mirrors2.1. Outline of open resonator theoryLet us consider an open resonator composed by two parallel metallic rectangular mirrors,as illustrated in Figure 1.

In order to give a modal analysis for the assumed structure, let us consider the waveequation for a rectangular field component:

2u + k2u = 0 (1)with the boundary condition u = 0 on the mirrors.

2l

2a

2b

z

x

y

Figure 1. Rectangular open resonator.

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Journal of Electromagnetic Waves and Applications 3

The solution of Equation (1), assuming a e jt time dependence, can be written as:u = v(x, y, z)e jkz (1)qv(x, y,z)e jkz (2)

where q is the longitudinal mode number.Assuming that function v has a very slow variationwith respect to variable z, so that its second derivative could be neglected with respect tothe term

k vz , it is easy to obtain the well-known parabolic approximation to the waveequation:

2v

x2+

2v

y2 2 jk v

z= 0 (3)

with the boundary conditions:{v(x, y,l) = 0, for |x | > a or |y| > b;v(x, y,l) = e j (2klq)v(x, y, l), for |x | < a and |y| < b. (4)

Equation (3) admits a solution of the form:v = va(x, z)vb(y, z) (5)

where the functions va and vb satisfy the equations:

2vax2

2 jk vaz

= 0 (6)2vby2

2 jk vbz

= 0 (7)

with the proper boundary conditions.The eigenfrequency of the resulting mode is then given by:

kl = (q

2+ p

), p = pa + pb (8)

where

pa = m2

4 (Ma + + j )2, m = 1, 2, . . . (9)

pb = n2

4 (Mb + + j )2, n = 1, 2, . . . (10)

Ma =

2ka2l

, Mb =

2kb2l

(11)

= R

(12

)

= 0.824 (12)

R being the Riemann Zeta function.The parameter p = p + j p into Equation (8) yields the complex mode frequency,

with the following meaning:

2p is the additional phase shift, during the time = 2l/c, due to the fact that thefrequency is not exactly equal to the cut-off frequency;

4p is the relative decrease in energy of the mode during the same time interval.

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The value of quality factor Q due to the only radiation loss is given by:

Q = q4p"

(13)

with

p" m2

2M2a+ n

2

2M2b(14)

Let us introduce the dimensionless coordinates:

=

k2l

x, =

k2l

y, = z2l

(15)

In the above coordinate system, the variables , , at the end of the mirror x = a, y = b,assume the values:

e = Ma2 e = Mb2

(16)

where the parameter Ma,b is related to the Fresnel number N [8]:

M = 8 N (17)In the (, , ) coordinates, we have:

u (, , ) = 2e jpva () vb () cos (q ) , (q odd) (18)u (, , ) = 2e jpva () vb () sin (q ) , (q even) (19)

where

va () = cos m(Ma + + j ) , (m = 1, 3 . . .)

va () = sin m(Ma + + j ) , (m = 2, 4 . . .)

vb () = cos n(Mb + + j ) , (n = 1, 3 . . .)

vb () = sin n(Mb + + j ) , (n = 2, 4 . . .) (20)

2.2. Electromagnetic field into the cavityThe electromagnetic field into the cavity can be expressed in terms of (quasi)-transverseelectromagnetic modes as follows:

E =mn

Vmnemn (21)

H =mn

Imnhmn (22)

The relation between function u = u(x, y, z) and the electromagnetic field can be estab-lished with the aid of Hertz vectore by the following relationships:

e = e + k2e (23)h = j k

Zw e (24)

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where k = and Zw =

.

Let us set:

ex = u; ey = ez = 0 (25)

into Equations (23) and (24). Neglecting the losses and considering the (1 1 1) mode, weobtain the modal solution for the y component of the magnetic field as:

Hy = j kZw

le jp

cos

(

Ma + )

cos

(

Mb + )

sin( z

2l

)(26)

and the x component of the electric field as:

Ex =[

k2 k2l

2

(Ma + )2]

2e jp

cos

(

Ma + )

cos

(

Mb + )

cos( z

2l

)(27)

Finally, from Equations (26) and (27) we obtain the equivalent surface impedance at the zabscissa, given as:

ZT = j Zw 2l

[k 1

2l2

(Ma + )2]

cot

2lz (28)

3. Metallic FabryPerot antennaThe antenna is a parallelepipedal flat structure with a square flat metallic base that is coupledby a slot to a rectangular waveguide, and a radiating face composed by a metallic sheet werethe radiating slots are cut (Figure 2).

Figure 2. Metallic FabryPerot antenna.

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Figure 3. Equivalent circuit of metallic cavity.

3.1. Rectangular waveguide-to-cavity couplingFollowing what developed in [911], the coupling between the field inside the cavity,Equations (23) and (24), and the field into a rectangular metallic waveguide, assumedas cavity feed, is matched on the coupling aperture, thus allowing an equivalent circuitrepresentation where only one cavity resonant mode is taken into account, while all T En0modes of the exciting waveguide are considered. Under the above hypotheses, the equivalentcircuit modeling both the cavity behavior and the cavity-to-waveguide coupling is derived(Figure 3).

In Figure 3, parameters L0 and C give the inductance and the capacity of the equivalentcavity, respectively; the term R is the resistance due to the losses on the metallic sheet,parameter L gives the inductance representing the length increase due to non perfectconducting sheets, 01 is the cavity-waveguide coupling factor, Le is the inductance dueto higher non propagating modes in the waveguide, and Z0 represents the characteristicimpedance of the feeding waveguide. In the Appendix 1 a more detailed description isreported.

When the second metallic sheet in Figure 2 is replaced with a partially reflectingsurface,[12] an equivalent impedance is inserted into the circuit of Figure 4. The valueof ZS is obtained following the approach given in [13]

1ZS

= 1j X L 1

j XC +1

ZW(29)

where with reference to Figure 2,

X LZW

= dy

ln csc[

2dy W

dy

](30)

XCZW

= {

4dx

ln csc[

2Ldx

]}1(31)

and ZW = 120 is the free space impedance.The reported circuit leads to optimize the transition and to compute the field, inside the

cavity, which impinges on the radiating sheet, thus allowing the radiated field computationfrom the equivalent currents on the radiating apertures.

3.2. Cavity-to-free space couplingIn order to compute the radiated field from the slots cut into one wall of the resonator, theelectromagnetic field in the resonator (26) is used to obtain the incident field on the slotsand subsequently derive from it the equivalent magnetic current.

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Journal of Electromagnetic Waves and Applications 7L

C

Le

Lo

R

ZS

Zg

1: 01

Figure 4. Equivalent circuit of radiating cavity.

In the Bethes original theory,[14] the incident field is considered in the absence of theaperture. The magnetic dipole moment is related to the incident field as follows:

M = mHt (32)where Ht is the tangential magnetic field at the center of the aperture, and the magneticpolarizability, for small rectangular aperture, is given by [15]:

m = 0.132lg(

1 + 0.66WL)W 3 (33)

W and L being the aperture dimensions, with L W .In the presence of a slot grid, the array factor can be expressed as:

F (, ) =M

g=1

Nh=1

Ig,he jk (r rgh) (34)

where Ig,h is the amplitude for each point-source (g, h), which is equal to the magneticdipole moment M evaluated at the point:

(x, y) =(

g M + 12

)dx ,

(h N + 1

2

)dy (35)

while

r = x sin cos + y sin sin + z cos (36)rgh =

(g M + 1

2

)dx x +

(h N + 1

2

)dy y (37)

M and N being the number of elements along x and y directions.

3.3. Numerical resultsFollowing the reasoning of previous paragraphs, the antenna configuration in Figure 2 isconsidered. An array of 8 8 elements is assumed, with dimensions 2l = 10 mm, dx =12.5 mm, dy = 12.5 mm, W = 6.35 mm, and L = 2 mm. A WR62 feeding waveguide, inthe Ku band, with dimensions Wa = 15.8 mm, W b1 = 7.9 mm is considered.

The analysis of the coupling between a rectangular cavity feeding waveguide and theplanar open cavity is performed by taking into account the results of previous paragraphs. In

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0 1 2 3 4 5 6 720

15

10

5

0

b [mm]

S11

[dB]

Proposed approachHFSS simulation

Figure 5. Return loss versus waveguide height.

14.5 14.6 14.7 14.8 14.9 15 15.1 15.2

35

30

25

20

15

10

5

Frequency (GHz)

S11

(dB)

HFSS simulationProposed approach

Figure 6. Return loss versus frequency.

particular, for the considered cavity of Figure 2, the result reported in Figure 5 is obtainedfor a frequency of 14.85 GHz. The resonant frequency established from (8) is used tocompute the circuit elements under Table A1. Results obtained with the equivalent circuitare compared with a full wave simulation. It is exploited to maximize the coupling betweenthe waveguide and the cavity, thus terminating the feeding section into an aperture of sizeWa = 15.8 mm, W b = 1.3 mm.

In Figure 6, the return loss of the cavity as function of the frequency, computed by theproposed equivalent circuit, is reported.

Afirst preliminary result of the radiation diagram using the simplified analysis, comparedwith full wave simulation, of the proposed antenna is reported in Figure 7 where, due tothe tapering of the exciting field of the slots, a substantial reduction of the sidelobes can beobserved.

4. FabryPerot antenna with improved bandwidthIn order to improve the operation bandwidth of the FabryPerot antenna configuration inFigure 2, a modified structure partially-filled with a dielectric substrate is considered, asillustrated in Figure 8. The gain and half-power fractional bandwidth are primarily deter-mined by the cover reflection coefficient.[1618] The method for bandwidth enlargement

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100 80 60 40 20 0 20 40 60 80 10035

30

25

20

15

10

5

0

Angle (deg)

Nor

mal

ized

pat

tern

(dB)

Proposed approachHFSS simulation

Figure 7. Radiated field of metallic FabryPerot antenna.

Figure 8. Partially-filled FabryPerot antenna.

focuses on the design of the cover. By adjusting the phase of the reflection coefficient, theresonance condition of the cavity can be satisfied in a wider range of frequencies. Such anoptimized reflection response can be realized by introducing a dielectric layer.

In the following paragraphs, the expression of the electromagnetic field into the partially-filled cavity are derived, and experimental results on a Ku-band prototype are reported toshow a significant bandwidth improvement with respect to the case of metallic FabryPerotstructure of Figure 2.

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Figure 9. Photograph of partially-filled FabryPerot antenna.

4.1. Electromagnetic field into the partially-filled cavityWhen assuming a partially-filled cavity, the quasi-transverse electromagnetic field in thedielectric, for h1 l z h1 + h2 l, is given as:

H2y = j A kZw

le jp

cos

(

Ma + )

cos

(

Mb + )

sin(

z

2l2

)(38)

14.4 14.5 14.6 14.7 14.8 14.9 15 15.10

2

4

6

8

10

12

14

16

18

20

Frequency [GHz]

[dB]

Empty Cavity (Simulation)Partially Filled Cavity (Simulation)Partially Filled Cavity (Measurement)

Frequency [GHz]

Figure 10. Boresight gain versus frequency: comparison between metallic and partially-filled FabryPerot antenna.

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Figure 11. Radiation patterns of partially filled FabryPerot antenna (comparison betweenmeasurements and proposed analysis method): (a) f = 14.6 GHz, (b) f = 14.85 GHz.

E2x = B[k2

k

2l2

(Ma + )2]

2e jp

cos

(

Ma + )

cos

(

Mb + )

cos

( z

2l2

)= ZT 2 H2y (39)

where l2 = h1 + h2 l.For z = h1 l, the field into the empty part (Equations (26) and (27)), and the field in

the dielectric-filled part must be equal, thus applying the continuity of tangential fields atthe air-dielectric interface, it is easy to derive the expressions for the terms A and B:

A = 1

sin[

2l (h1 l)

]sin

[

2l2 (h1 l)] (40)

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B = ZT 2ZT

A (41)

where ZS2 and ZS are derived from (28) and (39) for z = h1 l.

4.2. Experimental resultsIn order to assess the analysis method outlined in the previous paragraphs, a partially-filledFabryPerot antenna composed by an array of 8 8 elements, with inter-element spacingsdx = dy = 12.5 mm and slot dimensions W = 6.35 mm L = 2 mm, is realized andexperimentally tested. Referring to Figure 8, a plane distance l = h1 + h2 = 9.662 mmis considered, which is given by the sum of an empty space h1 = 8.9 mm and a dielectric(r = 2.33) with thickness h2 = 0.762 mm. The optimal coupling is obtained for a feedingwaveguide with dimensions Wa = 18.8 mm, Wb = 2 mm, at a frequency f = 14.85 GHz.The partially-filled FabryPerot antenna is realized and tested into the Microwave Labora-tory at University of Calabria (Figure 9).

The enhanced bandwidth behavior of the partially-filled FabryPerot antenna can beeasily observed from the boresight gain results illustrated in Figure 10, where an improve-ment of about 48% is obtained with the insertion of the dielectric substrate, and a goodagreement between simulations and measurements is demonstrated in the case of partially-filled antenna configuration.

The wideband feature of the proposed antenna structure is further proved by the verysimilar behavior of the radiation patterns (obtained from near-field to far-field transforma-tion) at different frequencies, as illustrated in Figure 11, where the successful agreementbetween results obtained from measurements and those derived from the proposed analysismethod can be also observed.

5. Conclusions and future workAn open planar cavity antenna has been presented in this work, by providing an equivalentsimplified circuit and obtaining both input characteristics and radiation diagrams of theantenna. In particular, a modified configuration based on a cavity partially-filled with adielectric substrate is proposed to significantly improve the antenna radiation bandwidth.

For future developments, the following comments are in order:

the development of a synthesis procedure using the positions and the dimensions ofthe radiating slots will be considered. In particular, a change in the slots dimensionwill modify the amplitude of the magnetic current, thus leading to control the shapeof the radiation pattern;

several types of radiating elements and feeding structures will be analyzed in futuredevelopments.

References

[1] Fabry C, Perot A. Theorie et Applications dune Nouvlle Method de SpectroscopieInterfrentielle [Theory and applications for a new method of spectroscopy interferential]. Ann.Chim. Phys. 1899;16:115.

[2] Di Massa G, Boccia L,Amendola G. Gaussian beam antennas based on open resonator structures.In: 28th ESA Antenna Workshop on Space Antenna Systems and Technologies; Olanda; 2005.

[3] Sauleau R, Coquet P, Matsui T, Daniel J.Anew concept of focusing antennas using plane-parallelFabryPerot cavities with nonuniform mirrors. IEEE Trans. Ant. Propag. 2003;51:31713175.

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nloa

ded

by [U

niv St

udi D

ella C

alabr

ia] at

09:33

12 Ja

nuary

2015


[4] Feresidis AP, Vardaxoglou JC. High gain planar antenna using optimised partially reflectivesurface. IEE Proc. Microw. Ant. Propag. 2001;148:345350.

[5] Gurin N, Enoch S, Tayeb G, Sabouroux P, Vincent P, Legacy H.Ametallic FabryPerot DirectiveAntenna. IEEE Trans. Ant. Propag. 2006;54:220224.

[6] Liu Z-G. Effect of primary source location on FabryPerot resonator antenna. In: Asia PacificMicrowave Conference, APMC; Singapore; 2009. p. 18091812.

[7] Hosseini SA, Capolino F, De Flaviis F. Design of a single-feed 60 GHz planar metallic FabryPerotcavity antenna with 20 dB gain. In: IEEE International Workshop on Antenna Technology,iWAT; Santa Monica, CA; 2009.

[8] Fox AG, Li T. Resonant modes in a maser interferometer. BeIl Sys. Tech. J. 1961;40:453488.[9] Bucci O, Di Massa G. Open resonator powered by rectangular waveguide. IEE Proc.-H.

1992;139:323329.[10] Di Massa G, Costanzo S, Moreno OH. Open resonator system for reflectarray elements

characterization. Int. J. Ant. Propag; 2012:7 p. Article ID: 912809. doi:10.1155/2012/912809.[11] Di Massa G, Costanzo S, Moreno OH. Accurate circuit model of open resonator system for

dielectric material characterization. J. Electromagn. Waves Appl. 2012;26:783794.[12] Von Trentini G. Partially reflecting sheet arrays. IRE Trans. Ant. Propag. 1956;4:666671.[13] Whitbourn LB, Compton RC. Equivalent-circuit formulas for metal grid reflectors at dielectri

boundary. Appl. Opt. 1985;24:217220.[14] Bethe HA. Theory of diffraction from small holes. Phys. Rev. 1944;66:163182.[15] McDonald NA. Simple approximations for the longitudinal magnetic polarizabilities of some

small apertures. IEEE Trans. Microw. Tech. 1988;MTT36:11411144.[16] Feresidis AP, Vardaxoglou JC. High gain planar antenna using optimised partially reflective

surfaces. IEE Proc-Microiv. Ant. Propag. 2001;148:345350.[17] Costa F, Genovesi S, Monorchio A. On the bandwidth of high-impedance frequency selective

surfaces. IEEE Ant. Wireless Propag. Lett. 2009;8:13411344.[18] Wang N, Zhang C, Zeng Q, Wang N, Xu J. New dielectric 1-D EBG structure for the design of

wideband resonator antennas. Prog. Electromagn. Res. 2013;141:233248.[19] Di Massa G. Microwave open resonator techniques part I: theory. In: Costanzo S, editor.

Microwave materials characterization; Rijeka (HRV); InTech Web; 2012.

Appendix 1. Equivalent circuitThe details of the following derivation are given in [19], which is the reference for the meaning ofthe symbols.

Let us consider our system under the assumption of negligible intercoupling between cavitymodes, i.e.:

nm = 12

Mhn hmd S = 0 i f n = m (A1)

The coupling factor between the waveguide and cavity modes,

nm =

Ae

gm hn nd S =

A

hn hgmd S. (A2)

Considering, with reference to Figure 2, the waveguide y component magnetic field for oddmodes,

Hy = j KzK 2t

m

Wasin

[m

Wa

(y Wa

2

)](A3)

and the magnetic field of the cavity (26) considered constant on the coupling aperture,

Hy = j kZW

le jp (A4)

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Table A1. Expressions for the circuit elements of Figure 3.

L0 0l (H) C l/(k0l)2 (F)L 0 (H) R 2/ ()

we can express the coupling factor (A2) as:

0m = KzkKt ZW1l

e jp 2Wa Wbm

(A5)

with p = j p at resonant frequency of cavity.By enforcing the continuity of magnetic field tangential component over the coupling aperture,

we get for the equivalent terminal impedance relative to the fundamental mode [19]:

Z = 01 1 + 1 = 0

201 F0+ 0

k =1

(0k01

)2k (A6)

Hence, for the sum at the right hand side of (A6), we have:

0

k=3,5,

(0k01

)2k = j0

k=0

1(2k + 3)2

1(2k+32Wa

)2 1 j0

(2Wa

) k=0

1(2k + 3)3 = jWa

1

[(1 1

8

)(3) 1

]= jLe

(A7)wherein () denotes the Rieman zeta function. From Equation (A7), we have the value of Le:

Le = 16.5 1030 a [henry] (A8)The explicit expression for the elements of equivalent circuit reported under Figure 3 are collectedunder Table A1.

In Table A1 k0 = 000 with 0 resonant frequency of cavity.

Dow

nloa

ded

by [U

niv St

udi D

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alabr

ia] at

09:33

12 Ja

nuary

2015

Abstract1. Introduction2. Open resonator composed by rectangular mirrors2.1. Outline of open resonator theory2.2. Electromagnetic field into the cavity

3. Metallic FabryPerot antenna3.1. Rectangular waveguide-to-cavity coupling3.2. Cavity-to-free space coupling3.3. Numerical results

4. FabryPerot antenna with improved bandwidth4.1. Electromagnetic field into the partially-filled cavity4.2. Experimental results

5. Conclusions and future workReferencesAppendix 1. Equivalent circuit

articulofabrypeort.pdf

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