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Int. J. Electron. Commun. (AEÜ) ( ) –www.elsevier.de/aeue

Definition of unifying decoupling efficiency of different array antennas—Case study of dense focal plane array feed for parabolic reflector

Malcolm Ng Mou Kehna,∗, Marianna V. Ivashinab, Per-Simon Kildalc, Rob Maaskantb

aDepartment of Electrical and Computer Engineering, University of Manitoba, Winnipeg, Canada MBR3TSV6bNetherlands Institute for Radio Astronomy (ASTRON), The Netherlands Organization for Scientific Research (NWO), The NetherlandscDepartment of Signals and Systems, Antenna Group, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden

Received 10 November 2008; accepted 17 February 2009

Abstract

This paper emphasizes the importance of recognizing the major role played by mutual coupling effects in array antennas,especially for dense arrays with element spacing within half a wavelength. For array operation in transmit mode, backwardwaves emanate both from element ports that are excited by forward waves, and from non-excited ports. This overall andexcitation-dependent mismatch is herein characterized in terms of a decoupling efficiency. It is shown how this decouplingefficiency reduces to the well-known impedance mismatch factor for a conventional phase-steered array (all elements ex-cited), and to the embedded element efficiency (only one element excited) used to characterize multiple input and multipleoutput (MIMO) arrays. Generally, this decoupling efficiency can be used to characterize the combined effect of mismatch andcoupling for any excitation of an array, for instance, the Airy pattern excitation of focal plane array (FPA) feeds for reflectors,as showcased here. The abovementioned forms of decoupling efficiencies are evaluated and discussed for an example arrayof idealized rectangular apertures with special emphasis on the Airy excitation.� 2009 Elsevier GmbH. All rights reserved.

Keywords: Antenna array; Element efficiency; Aperture efficiency; Radiation efficiency; Decoupling efficiency

1. Introduction

Array antennas may be employed in two common formsof applications. They can either be conventional direct-radiating phased arrays in which beams are formed directlyby the array aperture itself, or multi-beam cluster feedsfor reflector antennas [1,2] in which there is one beam perelement in the cluster. In recent times, there has also beenan increased interest in an alternative form: the so-called“dense” wide-band focal plane array (FPA) feeds [3–7] in

∗Corresponding author.E-mail address: malcolm.ng@ieee.org (M. Ng Mou Kehn).

1434-8411/$ - see front matter � 2009 Elsevier GmbH. All rights reserved.doi:10.1016/j.aeue.2009.02.011

which several possibly overlapping subarrays are used togenerate multiple sector beams, each one efficiently illumi-nating the reflector. A fourth kind of array is the so-calledsignal processing or multiple input and multiple output(MIMO) array for use in multipath environments with fad-ing, in which multiple channels are formed between MIMOports of a transmitting and a receiving array, respectively,with the signals of each channel being processed indepen-dently and incoherently to maximize the capacity. Both thecluster feeds and the MIMO arrays are characterized mainlyby an embedded element efficiency, for which only onearray element is excited while all the other surrounding el-ements are still present but passively match-terminated (i.e.unexcited) [8]. This embedded element is also known to be

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Excitation amplitude distribution ofcontinuous Airy pattern

Airy pattern truncated to two sidelobesand discretized

by 11x11 array of rectangular apertures

Fig. 1. Airy pattern defined by (13) and truncated to two sidelobes, (a) continuous and (b) discretized by an 11× 11 array of rectangularapertures, as an example.

an auxiliary scenario for characterizing all kinds of arrayantennas and for understanding the so-called element-gainparadox [9]. In this paper, we will consider three classes ofexcitations of the array elements: (i) the “all-excited” classi-cal phase-steered array in which all elements are excited toform a pencil beam, (ii) the “singly-excited” embedded el-ement, and (iii) the “FPA-excited” subarray in which someelements in the array are excited to form a sector beam.The present paper aims at highlighting the impact of mu-

tual coupling on the total received or transmitted power forthe abovementioned classes of array excitations. In partic-ular, we will consider FPA-fed reflectors and demonstratethat mutual coupling between elements poses a fundamentallimitation to achieving high total aperture efficiency of thereflector. In fact, to generate various simultaneous beams,the corresponding excited subarrays may partially overlap,implying that the same array elements may contribute to sev-eral beams with different weights. Therefore, it is not possi-ble to impedance match all elements simultaneously for allbeam excitations. Consequently, there will be a reductionof the total aperture efficiency due to losses related to mis-match and mutual coupling that are herein characterized bya general decoupling efficiency. This represents a unifyingefficiency concept that applies to all types of array excita-tions. Therefore, it reduces to a mismatch efficiency factorfor the all-excited case, and to the embedded element ef-ficiency for the single-excited element case. The latter hasbeen defined in a seminal paper [9]. In a recent work [10],this element efficiency is also claimed to be a unifying con-cept for phased array antennas. The present decoupling effi-ciency is therefore even more fundamental than the elementefficiency in [9,10] as it applies to arrays of any excitation.Traditionally, the power losses due to absorption in the

materials of both the reflector and feed constitute the ra-diation efficiency, whereas the impedance mismatch factoraccounts for the losses due to impedance mismatch of thefeed, which we hereafter represent by the so-called mis-

match efficiency. The mismatch efficiency can be evaluatedfor a conventional large phased array via the so-called “activeimpedance” (now preferably called scan impedance) whenall elements are excited with uniform amplitude and linearphase progression to attain a steered main beam. However,such active/scan impedances cannot be defined for FPAs,because there are both strong mutual coupling and largevariations in excitation amplitudes between elements. Con-sequently, the “active reflection coefficient” may becomelarger than unity on weakly excited elements. This leads toa negative real part of the active impedance that cannot bematched by standard techniques. Therefore, the decouplingefficiency introduced here to characterize FPAs replaces theconventional mismatch efficiency applicable only to phasedarrays. The decoupling efficiency, like the mismatch effi-ciency, will be present even if the arrays are made of ideallossless materials.The excitation of a subarray for a specific beam of an

FPA-fed reflector can be determined through conjugate-fieldmatching of the FPA aperture fields to the focal plane fields,generated when a plane wave from the desired beam di-rection is incident on the reflector. For paraxial plane waveincidence (paraxial beam) and paraboloids with small sub-tended angles, the focal region field approaches the idealAiry diffraction pattern. This Airy pattern, defined later in(13), will in practice need to be truncated and, consequently,leads to a reduced aperture efficiency, see Fig. 3 in [11]. Inaddition, the Airy pattern can only be synthesized in a step-wise manner by the array elements located at discrete arraypositions. Fig. 1(b) shows how the Airy pattern can be re-produced in a discrete manner by mapping the excitationcoefficients of an 11×11 FPA of rectangular apertures ontothe truncated Airy pattern function of Fig. 1(a).In this paper, results are presented for a hypothetical array

of idealized rectangular apertures, each excited with uniformfield distribution and linear polarization defined over theentire unit cell. Such apertures, synonymous with ideal hard

Please cite this article as: Ng Mou Kehn M, et al. Definition of unifying decoupling efficiency of different array antennas—Case studyof dense focal plane array feed for parabolic reflector Int J Electron Commun (AEU) (2009), doi: 10.1016/j.aeue.2009.02.011

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rectangular waveguides [12] with two magnetic side walls,represent a simple geometry that can be miniaturized to fitinto unit cells that are fractions of a wavelength in size andstill be an efficient radiator in an array environment. Thesimple and clean geometry makes it possible to accuratelymodel the array for any excitation using a moment-methodapproach, thereby being a good theoretical reference casefor illustrating fundamental properties of dense arrays.

2. Definitions of efficiencies

The performance of an FPA-fed reflector antenna can becharacterized by the total aperture efficiency �tot, which isdefined here as a product of the IEEE standard apertureefficiency �ap and the total radiation efficiency �totrad,

�tot = �ap�totrad. (1a)

The total radiation efficiency is

�totrad = �rad�decoup (1b)

and accounts for different types of losses in the antenna. Thefirst factor �rad accounts for the dissipations due to finiteconductivity and loss tangents of practical conductive or di-electric materials and is called radiation efficiency accordingto the IEEE standard. The second factor �decoup is introducedin the present paper as a decoupling efficiency and accountsfor the power lost in backward waves at the element portsof the array (due to mismatch and mutual coupling). As aresult, the total aperture efficiency �tot of the entire reflectorsystem will be a product of all these subefficiencies, i.e.

�tot = �ap�rad�decoup. (1c)

It is pointed out that all the subefficiencies of �tot, in particu-lar, �decoup, depend on the excitations of the array elements.

Fig. 2. General block scheme of FPA in which M elements are active and (N − M) elements are passively match-terminated.

2.1. General form of decoupling efficiency

The decoupling efficiency �decoup is generally defined asthe ratio of the array’s accepted power Pacc to the incidentpower Pinc for a given excitation. For an array with N ele-ment ports in total, of which a subarray of M (� N ) elementports are used to generate the beam, this gives

�decoup = PaccPinc

= 1 − PcPinc

= 1 −∑N

n=1|bn|2∑Mm=1|am |2

, (2a)

bn =M∑

m=1

Snmam , (2b)

where Pc is the total power lost in the terminations on allelement ports, Pinc is the sum of the incident powers on Mexcited element ports of the antenna, |am |2 is the forwardpower incident on the excited element port m, |bn|2 is thebackward power leaving the element port n, and Snm is theusual scattering matrix element between the element portsmand n when all the other element ports are match-terminated.Hence, N − M element ports are unexcited. The wave am-plitudes an and bn are illustrated in Fig. 2 for a generic ar-ray. The total power lost in the terminations on the elementports includes both the reflected and dissipated power onexcited and non-excited element ports, respectively. Underdissipationless conditions in which ideal lossless materialsare involved, �rad = 1, and hence, the decoupling efficiencybecomes �decoup = Prad/Pinc, where Prad = Pacc is the totalradiated power.It is illustrative to demonstrate how �decoup reduces to

well-known simpler formulas for special array excitationcases, such as the uniformly all-excited array (broadsidearray), and the singly-excited embedded element.

2.2. Uniformly all-excited array

This case corresponds to a large conventional phase-steered array (M = N and large) in which all elements are

Please cite this article as: Ng Mou Kehn M, et al. Definition of unifying decoupling efficiency of different array antennas—Case studyof dense focal plane array feed for parabolic reflector Int J Electron Commun (AEU) (2009), doi: 10.1016/j.aeue.2009.02.011

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excited with uniform amplitude and linear phase progressionto attain a steered main beam. For this case the nth forwardwave amplitude an = e− j�n and the nth backward waveamplitude bn are related to the nth element scan reflectioncoefficient

�n = bnan

=∑N

m=1 Snmaman

=∑N

m=1 Snme− j�m

e− j�n

=N∑

m=1

Snmej(�n−�m ). (3)

If N → ∞, so that we can neglect the truncation effectsdue to the finiteness of the array, the backward wave ampli-tudes become identical for all elements and the decouplingefficiency �decoup reduces to the mismatch efficiency,

�decoup = limN→∞

{1 −

∑Nn=1 |bn|2∑Nn=1 |an|2

}

= limN→∞

{1 − N |bn|2

N |an|2}

= 1 − |�n|2

= 1 − |�|2 = �mismatch, (4)

where � is the well-known active reflection coefficient ofphased arrays.

2.3. Singly excited embedded element

In this case, only one element is excited and the others arematch-terminated. The corresponding embedded element ef-ficiency characterizes each single element in a MIMO arraythat is located in a statistical field environment with Rayleighfading, see [8,13]. For such arrays the received voltages onall the element ports are supposedly uncorrelated and willtherefore be processed independently of each other by theMIMO detection algorithm, which combines them in an op-timum way to increase the capacity of the communicationchannel. Then, each element port will be characterized interms of its singly-excited embedded element efficiency. Thisis (for the case of lossless antennas) the same as the decou-pling efficiency introduced in the present paper evaluatedfor a singly-excited element m when the remaining array el-ements are match-terminated, i.e. we only retain a single mterm in the summation of (2), yielding

�decoup = 1 −∑N

n=1 |bn|2|am |2 = 1 −

∑Nn=1 |Snmam |2

|am |2

= 1 −N∑

n=1

|Snm |2. (5)

Note that the analysis in [11] is done in terms of mutualimpedances rather than S-parameters, but the results are thesame. This definition is also in full agreement with the fun-damental results in Eq. (13) of [9], which shows that such anembedded element efficiency may be used even for correctcharacterization of classical all-excited phased arrays, and

in particular when this is performed on element level. Theembedded element efficiency is then defined from Eq. (15)in [9] to be the gain per element of the uniformly excitedarray (i.e. the maximum available gain per element) dividedby the directivity of an embedded element. This gives

�decoup,m = Gmaxe /Dmax

e , (6)

where the maximum available gain per element is

Gmaxe = Gmax

array/N = 4�

�2A

N= 4�

�2(dx dy) (7)

in which Gmaxarray is the maximum available gain of the whole

uniformly excited array, A is the aperture area of the wholearray with a total of N elements, and dx and dy the elementspacings in the x and y directions, respectively. For denseplanar arrays (with the element spacing < 0.5�), the em-bedded element power pattern has approximately a cos(�)shape, see [9], so the directivity of the embedded elementpattern is

Dmaxe = 4�P(� = 0◦)

Prad

= 4� cos(�)|�=0◦∫ 2�0

∫ �/20 cos(�) sin(�)d�d�

= 4. (8)

Hence, the embedded element efficiency becomes

�decoup|elem = �dx dy/�2 (9)

which is an approximate formula for the case that elementspacings are smaller than 0.5�. This efficiency may be verylow for small spacings, and is already �decoup = �/4 forconventional arrays with 0.5� element spacings.We recall that, generally, the embedded element efficiency

has contributions both from reflections at the singly-excitedelement port, and from lost powers coupled into the portsof neighboring elements, see e.g. (5). The latter is normallythe larger in dense arrays.

2.4. FPA excited subarray

The FPA employs a more complex excitation scheme thanthe above two special cases, so the decoupling efficiencymust be computed from the general formulas in (2a) and(2b) using the appropriate excitations of the elements in thesubarray for each beam. Therefore, the decoupling efficiencywill be different for each beam.Basically, the decoupling efficiency for the lossless case,

i.e. �rad = 1, can be measured by taking the ratio betweenthe total radiated power and the maximum available powerat the beam port of an ideal power divider with isolated (i.e.uncoupled) output ports providing the desired excitationsat the element ports within the subarray of the consideredbeam. This corresponds to the method used to measure thetotal radiation efficiency at an antenna port in a reverbera-tion chamber [13]. However, on account of (1b), note that

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such a measurement will also include losses due to ohmicattenuation. Fig. 2 shows a general block scheme in whichM elements are active and (N − M) elements are passivelymatch-terminated. The active elements are excited using anideal power splitter including the vector W containing thecomplex weights to provide the required beam. The wavesaact incident on the active ports are defined as

aact = [a1, . . . , aM ]T = [W1, . . . ,WM ]T = W , (10)

where T is the transposition operator and {W1,W2, . . . ,WM }are the complex weights of the M ports. Because the unex-cited elements are match-terminated, we have

apas = [aM+1, aM+2, . . . , aN ]T = 0. (11)

The reflected waves b= [bact; bpas]T are related to the inci-dent waves a = [aact; apas]T via(bact

bpas

)= SA

(aact

0

)= SA

(W0

), (12)

where SA is the antenna scattering matrix. With these, thecoupling efficiency of (2a) may be easily computed. Thisapproach can also be used if the forward and backward waveamplitudes have been found by an infinite array approach,but we then have to truncate the infinite array.

3. Construction of the FPA excitation

In the present paper the excitation of the FPA is mappedonto the so-called Airy diffraction pattern given by

2J1(k� sin �0)/k� sin �0, (13)

where J1 is the Bessel function of the first kind and order1, k is the wavenumber, �0 is the half-subtended angle ofthe paraboloid, and � is the radial coordinate on the FPAaperture. This represents the calculated focal plane fieldsarising from an on-axis plane wave that is impingent ona paraboloid with large focal ratio F/D and no apertureblockage. The Airy diffraction pattern comprises a main lobeand an infinite number of sidelobes. In practice, the Airypattern must be truncated. In the following examinations, wewill only retain the main lobe and first two sidelobes (see alsoFig. 1a). The truncated Airy pattern is chosen to have a fixedsize in terms of wavelengths in order to provide the samesector beam width at all frequencies. Since the decouplingefficiency depends strongly on the element spacing, we havechosen to plot the various parameters as a function of theelement spacing in wavelengths rather than of the frequency.

4. Results and discussion

The results presented below are computed for an arrayconsisting of ideally hard rectangular waveguide elements,having two opposite perfect electric conducting (PEC) walls

as well as two opposite perfect magnetic conducting (PMC)walls orthogonal to the first two. The basic mode of prop-agation has uniform field distribution over the waveguideaperture, so it represents a reference case in which gratinglobes (if present) appear in the nulls of the element patterns.The results have been obtained by a moment-method ap-proach built upon that in [12], in which mutual couplingsbetween all considered elements for a multitude of waveg-uide modes are accounted for, i.e. higher order modes areincluded. The PMC walls are in the computations achievedby using dielectric walls with so high permittivity that thehard PMC condition appears when the walls are 1

80 of theelement width.The powers coupled into various surrounding (u, v) el-

ements centered at (x = udx , y = vdy) due to a singlyexcited element (0, 0) are shown in Fig. 3 as a func-tion of element spacing (i.e. frequency). The coupled lostpower for the (0, 0) element corresponds to the returnloss of the excited element. Besides, the coupling lossesinto the eight closest elements in one quadrant are shown,and constitute the first two tiers [elements at (u, v) =(1, 0), (0, 1), (1, 1), (2, 0), (0, 2), (2, 1), (1, 2), (2, 2)]. Theprogressive fall in mutually coupled power loss with in-creasing element separation is apparent. In addition, forany fixed element spacing, the mutual coupling between in-creasingly displaced elements along any particular directiondecays rapidly.Fig. 4 shows a summarized plot for the decoupling effi-

ciency of the waveguide array for three excitation scenar-ios: singly-excited embedded element, uniform all-excitedbroadside array, and an FPA excited according to the Airypattern (truncated to two or four sidelobes) for feeding a30◦, 50◦, and 60◦ reflector. Consider first, the embedded ele-ment scenario. The graph shows the variation of the elementefficiency with electrical element spacing produced by twoapproaches. One of them pertains to the rigorously com-puted embedded hard-waveguide element, while the otheris obtained from the theoretical approximate formula of(9). For element separations below 0.5�, the agreement be-tween the two traces is very good. It is concluded that theembedded element efficiency rises strongly with increasingcell size (frequency) because the coupling effects becomeweaker when the element spacing increases, thus resultingin reduced coupled losses. The slight reduction in efficiencybeyond a wavelength of element separation is due to the on-set of grating lobes as well as higher order modes that startto propagate backwards within the large waveguides, thusdrawing power back into the generators.The efficiency of the broadside excited array corresponds

to the mismatch efficiency as discussed previously, andit is seen to be very high because the chosen magneticside-walled waveguide elements are inherently matched forthis excitation condition [10]. In fact, the uniform aperturedistribution of each element combines ideally to a planewave, thus providing an ideal match to a plane wave in freespace over the whole array aperture. Next, we examine the

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Fig. 3. Variation with element spacing of the ratio of total lost power in the (u, v)th element (as indicated by the legend) to the incidentfundamental modal power into the singly-excited (0, 0)th element.

Fig. 4. Variation of decoupling efficiency of the magnetic side-walled waveguide array with element spacing, for four excitation scenarios:singly-excited (embedded element), uniform all-excited (broadside array), and FPA Airy-excitation for 30◦, 60◦, and 50◦ subtended halfangles of the parabolic reflector, truncated to two or four Airy sidelobes (ASL).

decoupling efficiencies of FPAs that are excited accord-ing to the Airy pattern. One observes from Fig. 4 that thedecoupling efficiencies of the FPA-excited arrays are mod-erately low at around −1.6dB when the array density ishigh (small element spacing). Nonetheless, it is still signif-icantly higher than the corresponding embedded elementefficiencies, which are as low as −9dB. Furthermore, it canbe seen that the curves for the decoupling efficiencies of allthree FPA excitations are bound within those pertaining tothe all-excited and singly-excited array scenarios. Typically,the all-excited and singly-excited arrays constitute two ex-treme cases, whereas the FPA is something in between. The

merits and demerits of the three types of array excitationare summarized in Table 1 .Fig. 5 shows how the various subefficiencies are affected

by the electrical element spacing. Two cases of reflector an-tennas are analyzed, with focal ratios F/D = 0.93 and 0.36corresponding to 30◦ and 60◦ subtended half angles, respec-tively. The usual reflector subefficiencies of Fig. 5 have beenevaluated using the formulas in [14]. For the 30◦ case ofFig. 5(a), the investigated array densities range from a dens-est subarray of 47-by-47 elements (dx = dy = 0.14�) to aleast-dense one of 7-by-7 subarray (dx=dy=1.08�). As seenin Fig. 5(a), the aperture efficiency is high at about −0.5dB

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Table 1. Summary of merits and demerits of the three forms of array excitations (valid for identical elements).

Array excitation type Merits Demerits

All-excited Can be matched for �decoup = 0dB Can only generate pencil beamsfor given beam directions and frequencies Blind scan angles for certain

directions and frequenciesSingly excited Fundamental building-block �decoup>0dB

for characterizing arrays for small element spacingswith an arbitrary excitation

FPA-excited �decoup is higher w.r.t. singly excited case and �decoup is lower w.r.t all-excited array(Airy excitation) varies between −0.1 and −0.7dB for element for the broadside beam direction

spacings 0.3�.0.6� and reflector angles 30.70◦ Matching is generally not possibleAllows for complex beam shaping for the array for all beam directions

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1-2.5

-2

-1.5

-1

-0.5

0

Sub

effic

ienc

y [d

B]

decoup effspillover effillumin effpolariz effphase effaperture efftotal eff

47x4745x45

43x4341x41 7x7

17x17

19x19

21x21

23x23

25x25

27x27

29x2931x31

33x33

35x3537x37

39x39

15x1513x13

11x11

9x9

30deg paraboloid, 2 Airy sidelobes included

Element spacing [λ]

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

Element spacing [wavelength]

Sub

effic

ienc

y [d

B]

rad (coup) effspillover effillumin effpolariz effphase effaperture efftotal eff

5x5

7x7

9x9

11x1113x13

15x15

17x17

19x19

21x21

23x23

25x2527x27

60 deg paraboloid, 2 Airy sidelobes included

Fig. 5. Variation of various subefficiency contributions and the overall aperture efficiency of (a) 30◦ and (b) 60◦ parabolic reflector whichis fed by an FPA of magnetic side-walled rectangular waveguide elements.

Please cite this article as: Ng Mou Kehn M, et al. Definition of unifying decoupling efficiency of different array antennas—Case studyof dense focal plane array feed for parabolic reflector Int J Electron Commun (AEU) (2009), doi: 10.1016/j.aeue.2009.02.011

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0 20 40 60 80-30

-20

-10

0

10

Theta [deg]

Dire

ctiv

e ga

in [d

Bi]

Continuous

E-planeH-planeco-polar 45cross-polar 45

0 20 40 60 80-30

-20

-10

0

10

Theta [deg]D

irect

ive

gain

[dB

i] 21x21 subarray

E-planeH-planeco-polar 45cross-polar 45

0 20 40 60 80-30

-20

-10

0

10

Theta [deg]

Theta [deg] Theta [deg] Theta [deg]

Dire

ctiv

e ga

in [d

Bi] 15x15 subarray

E-planeH-planeco-polar 45cross-polar 45

0 20 40 60 80-30

-20

-10

0

10

Dire

ctiv

e ga

in [d

Bi]

11x11 subarray

E-planeH-planeco-polar 45cross-polar 45

0 20 40 60 80-30

-20

-10

0

10

Dire

ctiv

e ga

in [d

Bi]

9x9 subarray

E-planeH-planeco-polar 45cross-polar 45

0 20 40 60 80-30

-20

-10

0

10

Dire

ctiv

e ga

in [d

Bi] 7x7 subarray

E-planeH-planeco-polar 45cross-polar 45

Fig. 6. Co- and cross-polar radiation patterns in E-plane, H -plane, and 45◦ plane of an FPA excited to illuminate a paraboloid with30◦ subtended angle. The excitation function is the truncated Airy pattern with main lobe and two first sidelobes which is discretizedfor different subarray sizes. The continuous theoretical case is included as a reference, (a) continuous, (b) 21 × 21 subarray, (c) 15 × 15subarray, (d) 11 × 11 subarray, (e) 9 × 9 subarray, (f) 7 × 7 subarray.

for small element separations, attributed to the high valuesof the spillover efficiency as well as the other subefficien-cies of �ap, the latter resulting from fine sampling resolu-tions of the Airy pattern. Consequently, the fairly low overallefficiency at small element spacing (i.e. low frequency) ismainly caused by the low decoupling efficiency. On the otherhand, when the element separation is large (high frequency),the decoupling efficiency is high, whereas the aperture ef-ficiency is low due to the coarsely sampled Airy patternsand sharp decline in the spillover efficiency as shown inFig. 5(a). Therefore, due to the opposing effects which theelement separation has on the decoupling and aperture subef-ficiencies, an optimal overall efficiency exists, i.e. a trade-off condition. This is indeed observed in Fig. 5(a), whichdistinctively shows an optimal total aperture efficiency of−0.7dB (or 85%) corresponding to an intermediate FPAdensity of 15 × 15 and about 0.5� element spacing. Thereis a useful frequency range of 5:1 over which the overallefficiency is better than −2dB. Resonance phenomena ob-

served at around 0.2� spacing are explained by resonancesin the mutual coupling occurring around that element spac-ing. This gives rise to surges in coupled power losses anddistortions in the feed radiation pattern.The graph of subefficiencies versus element spacing for

the 60◦ paraboloid is shown in Fig. 5(b). The overall effi-ciency does not degrade with increasing element spacing asseverely as for the shallower reflector in Fig. 5(a). This isbecause the spillover efficiency does not drop so drasticallywhen the feed patterns are distorted from the idealized ones.In fact, due to the large subtended angle of the reflector, aconsiderable amount of the radiated feed power is capturedregardless of how distorted the feed pattern is. An optimalcondition is again found to be at around half-wavelength el-ement separation, similar to the case in Fig. 5(a). Note, how-ever, that the highest overall efficiency is lower for this deepreflector than for the former shallow reflector. The same ex-planation for the resonances occurring now at around 0.25�reapplies. The co- and cross-polar directive gain patterns of

Please cite this article as: Ng Mou Kehn M, et al. Definition of unifying decoupling efficiency of different array antennas—Case studyof dense focal plane array feed for parabolic reflector Int J Electron Commun (AEU) (2009), doi: 10.1016/j.aeue.2009.02.011

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the FPA for feeding the 30◦ paraboloid in E , H , and �=45◦planes are shown in Figs. 6(a)–(f), for various array densi-ties. For convenience, the respective step-wise Airy excita-tions corresponding to the actual subarray sizes are showntogether as inset plots. Fig. 6(a) shows the gain patterns thatwere computed by integrating the aperture-field distributionfor the ideal linearly polarized “continuous”, and truncatedAiry pattern (main lobe and two sidelobes). As can be seen,array densities of 15×15 and higher have almost completelysynthesized the ideal far-field patterns, including the cross-polar sidelobe in the 45◦ plane, which is more than 20dBbelow the main lobe. On the contrary, the two least-densearrays (7×7 and 9×9) have generated feed patterns that aregrossly dissimilar from the ideal case, and large spilloverlosses are observed.

5. Conclusion

The decoupling efficiency of FPAs has been defined toaccount for mutual coupling and impedance mismatch. Thisdecoupling efficiency, as defined in (2), is formulated interms of forward and backward wave amplitudes on the el-ement ports of the array. The decoupling efficiency is usedto quantify the power loss due to coupling and mismatch ef-fects of all different types of arrays, such as phased arrayswhere each element is used/excited coherently (i.e. depen-dently), or MIMO arrays where each element is used/excitedcompletely independently, or even FPAs where the elementsare located in overlapping subarrays that are used/excitedindependently, so that each element may take on many dif-ferent excitations simultaneously. This work has conducteda comparative investigation into the decoupling efficienciesof all these array scenarios. The decoupling efficiency hasbeen evaluated for a hypothetical array consisting of rect-angular waveguides with two magnetic sidewalls. Such el-ements, each capable of providing a uniform aperture fieldfor a single linear polarization, are “clean” elemental sam-plers of the focal plane Airy field patterns. The results showthat the decoupling efficiency can be as low as −9dB ifonly one element is excited (embedded element) and the el-ement spacing is smaller than 0.2�. This value is in agree-ment with the fundamental results reported by Hannan [7].It has been shown that the decoupling efficiency of an FPAthat is excited to create a sector beam can be much higher,and the example FPA feeds analyzed here give numericalvalues better than −0.3dB.

Still, this value is so significant that the decoupling effi-ciency must be accounted for when designing and predict-ing the performance of FPAs, particularly in deep reflectors.Although not described in this paper, it has been shown thatthe overall aperture efficiency of the reflector system can beslightly improved by optimizing the excitation of the FPA[15].Main conclusions of the present study are that FPA-fed

shallow reflectors may have a maximum total aperture ef-

ficiency of −0.7dB appearing for optimum 0.5� elementspacing, and there is a useful frequency range of about 5:1over which the overall efficiency is better than −2dB.

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[8] Kildal P-S, Rosengren K. Correlation and capacity ofMIMO systems and mutual coupling, radiation efficiencyand diversity gain of their antennas: simulations andmeasurements in reverberation chamber. IEEE Commun Mag2004;42:102–12.

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feeding a reflector antenna. IEEE Trans Antennas Propagat2009;57:373–82.

Malcolm Ng Mou Kehn (S’02) wasborn in Singapore on September 26,1976. He received the B.Eng. (honors)degree from the National Universityof Singapore, Singapore, in 2001 andthe Licentiate and Ph.D. degrees inelectrical engineering from ChalmersUniversity of Technology, Gothenburg,Sweden, in 2004 and 2005, respec-tively. During 2006–2008, he was aPostdoctoral Fellow at the University

of Manitoba, Winnipeg, MB, Canada, and is currently with theDepartment of Electrical and Computer Engineering at ConcordiaUniversity, in Montreal, Canada. Dr. Ng Mou Kehn received theURSI (Union Radio-Scientifique Internationale) Young ScientistAward in 2007.

Marianna V. Ivashina was born inUkraine in 1975. She received the com-bined B.Sc./M.Sc. degree and the Ph.D.degree in electrical engineering fromthe Sevastopol National Technical Uni-versity (SNTU), Sevastopol, Ukraine, in1995 and 2000, respectively. From 1999to 2001, she was a Professor Assistantat SNTU. From 2001 to 2004, she wasa Postdoctoral Researcher and from2004 until now an Antenna System

Scientist at ASTRON in Dwingeloo, the Netherlands. She has car-ried out research on an innovative focal plane array technologyfor a new generation radio telescope, known as the square kilome-ter array (SKA). In 2002, she was also a Visiting Scientist at theEuropean Space Agency (ESA), ESTEC, The Netherlands, whereshe studied multiple-beam array feeds for satellite telecommuni-cation system large deployable antenna (LDA), which is beingrealized by Alenia. Her research interests are focal plane arrays,array feeds for reflector antennas, wideband receiving arrays, andradio astronomy. Dr. Ivashina received the URSI Young ScientistsAward for GA URSI, Toronto, ON, Canada (1999), APS/IEEETravel Grant, Davos, Switzerland (2000), and the Second Best Pa-per Award (best team contribution) at the ESA Antenna Workshop(2008).

Per-SimonKildal (M’82–SM’84–F’95)was born in Molde, Norway, on July4, 1951. He received the M.S.E.E.,Ph.D., and Doctor Technicae degreesfrom the Norwegian Institute of Tech-nology (NTH), Trondheim, Norway,in 1976, 1982, and 1990, respectively.From 1979 to 1989, he was with theELAB and SINTEF, research institutesin Trondheim, Norway. Since 1989,he has been a Professor at Chalmers

University of Technology, Gothenburg, Sweden, where he has ed-ucated 17 persons to a Ph.D. in antennas. He authored or coau-thored more than 237 papers in IEEE or Institute of Electrical En-gineers (IEE) journals and conferences, concerning antenna the-ory, analysis, design and measurement. He gives short courses andorganizes special sessions at conferences, and he has given in-vited lectures in plenary sessions at several conferences. His text-book Foundations of Antennas—A Unified Approach, Studentlit-teratur, Sweden, 2000, has been well received. He holds severalgranted patents and patents pending. He has done the electricaldesign and analysis of two very large antennas: the 40m × 120mcylindrical parabolic reflector antenna of the European IncoherentScatter Scientific Association (EISCAT), located in North Nor-way, and the Gregorian dual-reflector feed of the 300-m diame-ter radio telescope in Arecibo, on a contract for Cornell Univer-sity. He is currently involved in the design of wideband so-called11 feeds for the VLBI2010 and square kilometer array (SKA)projects. Based on his former hat feed design, he has founded thecompany Comhat (www.comhat.se), now acquired by ArkivatorAB. He has pioneered the reverberation chamber to an accuratetool for testing small antennas and wireless devices, being com-mercialized in the company Bluetest AB (www.bluetest.se). Dr.Kildal has done several services to the IEEE Antennas and Prop-agation Society: elected member of the administration committee1995–1997, Distinguished Lecturer 1991–1994, Associate Editorof the Transactions 1995–1998, and Associate Editor of a specialissue in the Transactions 2005. ELAB awarded his work in 1984.He has received two best paper awards in the IEEE Transactionon Antennas and Propagation.

Rob Maaskant received the M.Sc.degree (cum laude) in electrical engi-neering from the Eindhoven Universityof Technology, Eindhoven, The Nether-lands, in 2003. He is currently work-ing towards the Ph.D. degree in thefield of numerically efficient integral-equation techniques for large finitearray antennas at the same university.Concurrently, he has been an AntennaResearch Engineer at the Netherlands

Institute for Radio Astronomy (ASTRON) in Dwingeloo, theNetherlands, where his research is carried out in the frameworkof the square kilometer Array (SKA) radio telescope project.His research interest includes the characterization and design ofantenna array receiving systems.

Please cite this article as: Ng Mou Kehn M, et al. Definition of unifying decoupling efficiency of different array antennas—Case studyof dense focal plane array feed for parabolic reflector Int J Electron Commun (AEU) (2009), doi: 10.1016/j.aeue.2009.02.011