1

Maximum Gain, Effective Area, and DirectivityMats Gustafsson, Senior Member, IEEE and Miloslav Capek, Senior Member, IEEE

Abstract—Fundamental bounds on antenna gain are found viaconvex optimization of the current density in a prescribed region.Various constraints are considered, including self-resonance andonly partial control of the current distribution. Derived formulasare valid for arbitrarily shaped radiators of a given conductivity.All the optimization tasks are reduced to eigenvalue problems,which are solved efficiently. The second part of the paper dealswith superdirectivity and its associated minimal costs in efficiencyand Q-factor. The paper is accompanied with a series of examplespractically demonstrating the relevance of the theoretical frame-work and entirely spanning wide range of material parametersand electrical sizes used in antenna technology. Presented resultsare analyzed from a perspective of effectively radiating modes.In contrast to a common approach utilizing spherical modes,the radiating modes of a given body are directly evaluated andanalyzed here. All crucial mathematical steps are reviewed inthe appendices, including a series of important subroutines to beconsidered making it possible to reduce the computational burdenassociated with the evaluation of electrically large structures andstructures of high conductivity.

Index Terms—Antenna theory, current distribution, eigenval-ues and eigenfunctions, optimization methods, directivity, an-tenna gain, radiation efficiency.

I. INTRODUCTION

A question of how narrow a radiation pattern can be or,in terms of standard antenna terminology [1], what are thebounds on directivity and gain, has been in the spotlight ofantenna theorists’ and physicists’ for many years.

Early works studied needle-like radiation patterns [2]. Aseries of works starting in the 1940s revealed the fact that thedirectivity is unbounded [3] but also predicted the enormouscost in other antenna parameters, namely in Q-factor [4],related sensitivity of feeding network [5], and radiation ef-ficiency in case that the antenna is made of lossy material [6].Consequently, as pointed out by Hansen [7], the superdirectiveaperture design requires additional constraint, replacing fixedspacing in array theory [8], [9].

In order to tighten the bounds on directivity, Harring-ton [10], [4] proposed a simple formula which predicts thedirectivity from the number of used spherical harmonics as afunction of aperture size. The number of modes radiating welland the pioneering works on bounds [11] became popular inantenna design and hold in many realistic cases, therefore, this

Manuscript received December 19, 2018; revised December 19, 2018.This work was supported by the Swedish Foundation for Strategic Re-search (SSF) grant no. AM13-0011. The work of M. Capek was sup-ported by the Ministry of Education, Youth and Sports through the projectCZ.02.2.69/0.0/0.0/16 027/0008465.

M. Gustafsson is with the Department of Electrical and Infor-mation Technology, Lund University, 221 00 Lund, Sweden (e-mail:mats.gustafsson@eit.lth.se).

M. Capek is with the Department of Electromagnetic Field, Faculty ofElectrical Engineering, Czech Technical University in Prague, 166 27 Prague,Czech Republic (e-mail: miloslav.capek@fel.cvut.cz).

approach demarcated the avenue of further research. Improvedformula has been proposed in [12], suggesting that, in general,the maximum directivity in the electrically small region isequal to three. The maximum directivity is studied in [13]considering a given current norm. For antenna arrays, direc-tivity bounds are shown in [14]. Trade-off between maximumdirectivity and Q-factor for arbitrarily shaped antennas ispresented in [15]. Upper bounds for scattering of metamaterial-inspired structures are found in [16]. Recently, a compositionof Huygens multipoles has been proposed [17] to increasethe directivity. Notice, however, that no losses other than theradiation were assumed which re-opens the question of theactual cost of super-directivity.

Another way to limit the directional properties is a pre-scribed, non-zero, material resistivity of the antenna body [18],[19]. A quantity to deal with is the antenna gain, which isalways bounded if at least infinitesimal losses are assumed.It may seem reasonable at this point to argue that the lossescan be overcame with a concept of superconducting antennas,however, as shown in [6], the increase in gain with decreaseof resistivity embodies slow (logarithmic) convergence. Con-sequently, even tiny losses, which are always present at RF,restrict the gain to a finite number.

Tightly connected is the question of maximum achievableabsorption cross-section. The capability to effectively radiateenergy in a certain direction can reciprocally be understoodas a potential to absorb energy from that direction [20], [21].This can be interpreted as an ability of a receiver to distort thenear-field so that the incoming energy is effectively absorbedin the receiver’s body or concentrated at the receiving port. Ithas been realized that such an area can be huge as comparedto the physical size of the particle or the physical antennaaperture [22], [23]. Fundamental bounds on absorption cross-sections are proposed in [24], [25].

The importance to establish fundamental bounds on gainand absorption cross-section are underlined by recent devel-opment in design of superdirective (supergain) antennas andarrays [26], [27], [28], [29], [30], [31], partly fueled by theadvent of novel materials and technologies [32], [33].

The procedure developed in this paper relies on convexoptimization [34] of current distributions [15]. In order tofind the optimal current distribution in a prescribed region,the antenna quantities are expressed as quadratic forms ofcorresponding matrix operators [35], [36]. This makes itpossible to solve the optimization problems rigorously viaeigenvalue problems [37], [38]. The procedure is generalas arbitrarily shaped regions can be investigated. Additionalconstraints are enforced, e.g., self-resonance and restrictedcontrollability of the current [15], [39]. Much work in thisarea has already been done in determining bounds on Q-factor [37], radiation efficiency [39], superdirectivity [15],

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gain [11], and capacity [40]. The recent trend, followed by thispaper, is to understand the mutual trade-offs between variousparameters [41], [42], [38].

The original approach from [11] and [35] maximizingthe Rayleigh quotient for antenna gain via a generalizedeigenvalue problem is recast here into an eigenvalue prob-lem of reduced rank. Such a formulation is compatiblewith fast numerical methods [43], therefore, the results canbe presented in a wide frequency range, ka ∈

[10−3, 103

],

where ka is used throughout the paper to denote the di-mensionless frequency with k being the wavenumber and abeing the radius of a sphere circumscribing all the sources.The surface resistivity used spans the interval from extremelylow values, Rs = 10−8 Ω/, reachable in RF superconduct-ing cavities [44], through values valid for copper at RF(Rs ≈ 0.01 Ω/, f = 1 GHz), to poor conductors of surfaceresistivity Rs = 1 Ω/.

Optimal currents presented in this paper maximize theantenna gain. Therefore, taking reciprocity into account, theydelimit the maximum effective area of any receiver designed inthat region as well. For this reason, the proportionality betweengain and effective area is utilized, making it possible to judgethe real performance of designed and manufactured antennas,arrays, scatterers, and other radiating systems.

The behavior of the optimal solution evolves markedlywith electrical size. Huygens source formed by electric andmagnetic dipoles is strictly preferred in electrically small (sub-wavelength) region and a large effect of self-resonance, if en-forced, is observed. End-fire radiation and negligible effect ofself-resonance constraint is observed in an intermediate region.Finally, broadside radiation dominates in the electrically largeregion with the effective area being proportional to the cross-section area.

The paper is organized as follows. Antenna gain and ef-fective area are introduced in Section II and expressed asquadratic forms in the currents. The optimal currents are thenfound for maximum gain in Sections II-A and II-B, includingcases with additional constraints like self-resonance. Examplescovering various aspects of antenna design are presented inSection II-C. Superdirective currents are found in Section IIIand presented as a trade-off between required directivity andminimum ohmic losses or Q-factor. All presented examplesreveal the enormous cost of superdirectivity. The maximumgain is reinterpreted in Section IV in terms of number ofsufficiently radiating modes of a structure and the results arelinked back to Harrington’s formula. The paper is concludedin Section V. All required mathematical tools are reviewedand key derivations are presented in the Appendices.

II. GAIN AND EFFECTIVE AREA

Antenna gain describes how an antenna converts inputpower into radiation in a specified direction r, [45]. The gainin a direction r is determined as 4π times the quotient betweenthe radiation intensity P (r) and the dissipated power Pr +PΩ,

G(r) = 4πP (r)

Pr + PΩ, (1)

where Pr and PΩ denote the radiated power and power dissi-pated in ohmic and dielectric losses, respectively. The effectivearea, Aeff , is an alternative quantity used to describe directiveproperties for receiving antennas, which is for reciprocalantennas simply related to the gain as [20]

Aeff =Gλ2

4π, (2)

where λ = 2π/k denotes the wavelength. It is seen that max-imization of gain is equivalent to maximization of effectivearea [35].

The optimized parameters are expressed in the currentdensity J(r) which is expanded in a set of basis functionsψn (r) as [35]

J(r) ≈N∑n=1

Inψn(r), (3)

where the expansion coefficients, In, are collected in thecolumn matrix I. This substitution yields algebraic expressionsfor radiation intensity, radiated power, and ohmic losses asfollows [36]

P (r) ≈ 1

2|FI|2 =

1

2IHFHFI, (4)

Pr ≈1

2IHRrI, (5)

PΩ ≈1

2IHRΩI. (6)

The matrices used in (4)–(6) are reviewed in Appendix A.Substitution of (4)–(6) into (1) yields

G(r) ≈ 4π|FI|2

IH(Rr + RΩ)I= 4π

IHUI

IH(Rr + RΩ)I, (7)

where we also introduced the matrix U = FHF to simplifythe notation and highlight the expression of the gain G(r) asa Rayleigh quotient.

A. Maximum Gain: Tuned Case

The maximum gain for antennas confined to a region r ∈ Ωis formulated as the optimization problem

maximize IHUI

subject to IH(RΩ + Rr)I = 1,(8)

where for simplicity the dissipated power is normalized tounity. This problem is equivalent to the Rayleigh quotient

Gub ≈ 4πmaxI

IHFHFI

IH(Rr + RΩ)I, (9)

to which a solution is found via the generalized eigenvalueproblem [35]

FHFI = γ(Rr + RΩ)I. (10)

In order to reduce the computational burden, the for-mula (10) is further transformed to

(Rr + RΩ)−1

FHFI = γI (11)

3

and multiplied from left by the matrix F. By introduc-ing I = FI we readily get

F(Rr + RΩ)−1FHI = γI. (12)

Taking into account that the far-field matrix F can be ex-pressed using two orthogonal polarizations, see Appendix A,the original N × N eigenvalue problem (10) is reduced intothe 2× 2 eigenvalue problem (12) which can be written as

Gub ≈ 4πmax eig(F(Rr + RΩ)−1FH) (13)

with the optimal current determined as

I = γ−1(Rr + RΩ)−1FHI. (14)

The corresponding case with the partial gain containsone polarization direction and hence the eigenvalueγ = F(Rr + RΩ)−1FH and current

I ∼ (Rr + RΩ)−1FH. (15)

Here, the FH part can be interpreted as phase conjugationof an incident plane wave from the r-direction, and hence thecurrent corresponding to the maximum gain is found by phaseconjugation of the incident wave modified by (Rr + RΩ)−1.

B. Maximum Gain: Self-Resonant Case

The solution to (13) is in general not self-resonant. Selfresonance is enforced to (13) by adding the constraint ofzero reactance, IHXI = 0, see Appendix A, producing theoptimization problem

maximize IHUI

subject to IHXI = 0

IH(RΩ + Rr)I = 1.

(16)

This optimization problem is a quadratically constrainedquadratic program (QCQP), see Appendix B, that is trans-formed to a dual problem by multiplication of IHXI with ascalar parameter ν and adding the constraints together, i.e.,

maximize IHUI

subject to IH(νX + RΩ + Rr)I = 1,(17)

which is solved as a generalized eigenvalue problem analo-gously to Section II-A. The solution to this problem is greateror equal to (16) and taking its minimum value produces thedual problem [34]

Gub,r ≈ 4πminν

max eig(U, νX + RΩ + Rr)

= 4πminν

max eig(F(νX + RΩ + Rr)−1FH)

(18)

which is convex and easy to solve, e.g., with the bisectionalgorithm [46]. The derivative of the eigenvalue γ with respectto ν is [47]

∂γ

∂ν= −γ2 IHXI

IHUI

≤ 0 for ν ≤ νopt, inductive= 0 for ν = νopt, resonant≥ 0 for ν ≥ νopt, capacitive

(19)

for cases with non-degenerate eigenvalues. Degenerate eigen-values are often related to geometrical symmetries and solved

10−3 10−2 10−1 100 101 102 10310−1

100

101

102

103

104

105

106

aa

ka

G

10−8 Ω/10−6 Ω/10−4 Ω/10−2 Ω/1 Ω/HarringtonGO

Fig. 1. Maximum gain for a spherical shell of radius a with surface resis-tivity Rs = 10−n Ω/, n = 0, 2, 4, 6, 8, both for externally tuned (13),Gub, (solid lines) and for self-resonant (18), Gub,r, (dashed lines) currents.

by decomposition of the current I into orthogonal sub-spaces [37]. The range ν ∈ [νmin, νmax] in (18) is determinedfrom the condition νX+RΩ+Rr 0 which can be computedfrom the smallest and largest eigenvalues, eig(X,RΩ + Rr),i.e.,

−1

max eig(X,RΩ + Rr)≤ ν ≤ −1

min eig(X,RΩ + Rr), (20)

see Appendix E for details.The minimal eigenvalue min eig(X,RΩ + Rr) is related

to the Q-factor of the maximal capacitance in the geometrywhich is very large for all considered cases giving an upperlimit very close to zero and νmax → 0 as the mesh is refined.The maximal eigenvalue is related to the maximal inductiveQ-factor which is a fixed value depending on shape of theobject and gives the lower bound νmin, cf. with the inductorQ-factor in [38].

C. Examples of Maximum Gain and Effective Area

The following section presents maximum gain and effectivearea for examples of various complexity:

1) spherical shell both for externally and self-resonantcurrents, Section II-C1,

2) comparison of end-fire and broadside radiation from arectangular region, Section II-C2,

3) maximization of effective area if different parts of acylinder are considered, Section II-C3,

4) limited controllability of currents for a parabolic dishwith spherical prime feed, Section II-C4.

1) Externally tuned and self-resonant currents (sphericalshell): Expansion of the current density on a spherical shellin vector spherical harmonics [48] produces diagonal reac-tance X, radiation resistance Rr, and loss RΩ matrices withclosed form expressions of the elements. The direction ofradiation can without loss of generality be chosen to r = zfor which the elements F are zero for azimuthal Fourierindices |m| 6= 1. It is hence sufficient to consider |m| = 1for the radiation, see Appendix F.

4

−5

−20

−5

10

25

ka = 1

−5

−20

−5

10

25

ka = 10

10−8 Ω/10−6 Ω/10−4 Ω/10−2 Ω/1 Ω/

Fig. 2. Radiation patterns for a spherical shell of radius a with surface resis-tivity Rs = 10−n Ω/, n = 0, 2, 4, 6, 8 corresponding to the externallytuned case in Fig. 1. The radiation patterns are shown in terms of gain Gfor a ϑ-cut and ϕ = 0. The two electrical sizes, ka = 1 and ka = 10, aredepicted.

The maximum gain for a spherical shell with surface re-sistivity Rs = 10−n Ω/ for n = 0, 2, 4, 6, 8 is determinedusing (13), (18) and depicted in Fig. 1. The results are com-pared with the estimates GH = (ka)2 + 2ka by Harrington [4]and from the geometrical cross section GGO = 4πAcross/λ

2.It is observed that the additional constraint on self-resonance,i.e., IHXI = 0, in (16) has a large effect for small structures(ka < 1) but negligible effect for electrically large struc-tures. The tuned and self-resonant cases have D = 3/2 andD = 3, respectively, in the limit of electrically small structures(ka → 0), see Appendix F. Onset of spherical modes forsmall ka gives a step-wise increasing directivity and gain, seefigures in Appendix F. Dependence on Rs diminishes and thegain approaches GGO as ka increases. The radiation patternsand the influence of the surface resistivity on the maximumgain G is shown in Fig. 2 for the externally tuned case andelectrical sizes ka = 1 and ka = 10. The electrically largelimit is more clearly seen by plotting the effective area (2) inFig. 3, where it is observed that the effective area approachesthe cross-section area as ka→∞.

2) Broadside and end-fire radiation (rectangular plate):The symmetry of the sphere is ideal for analytic solution ofthe optimization problem but cannot be used to investigateimportant cases such as broadside and end-fire radiation [21].Let us, therefore, consider a planar rectangular plate withside lengths ` and `/2 placed at z = 0 having surfaceresistivity Rs = 10−4Z0 per square. The maximum effectivearea is depicted in Fig. 4 for radiation in the cardinal di-rections. Three regions can be identified: electrically small(ka 1) with large difference between the externally tunedand self-resonant cases, intermediate region with dominantend-fire radiation, and electrically large ka 1 with dominantbroadside radiation.

Negligible directional differences are observed for the elec-trically small (ka 1) externally tuned case which can be ex-plained by radiation patterns originating from electric dipoles.The effective area for the self-resonant case deceases as (ka)2

10−3 10−2 10−1 100 101 102 10310−1

100

101

102

103

104

105

106

aa

ka

Aeff/πa

2

10−8 Ω/10−6 Ω/10−4 Ω/10−2 Ω/1 Ω/HarringtonGO

Fig. 3. Maximum effective area for a spherical shell of radius a withsurface resistivity Rs = 10−n Ω/, n = 0, 2, 4, 6, 8, both for externallytuned (13) (solid lines) and for self-resonant (18) (dashed lines) currents.

10−2 10−1 100 101 102 10310−1

100

101

102

103

ka ≈ 3.51`/λ

Aeff/(`2/2)

end-fire short sideend-fire long sidebroadsideHarringtonGO, GO/2

Fig. 4. Maximum effective area in the cardinal directions for a rectangularplate with size ` × `/2 and surface resistivity Rs = 10−4Z0 per square.Bounds for externally tuned (13) (solid lines) and self-resonant (16) (dashedlines) currents are depicted.

and consists of a combination of electric and magnetic dipoles.Huygens sources are obtained for the end-fire cases where thegain is higher for radiation along the longest side than for theshorter side due to its lower amount of stored electric energy.Gain in the broadside direction is lower due to its up-downsymmetric radiation pattern.

The difference between the externally tuned and self-resonant cases decreases as ka increases and become negligi-ble around ka ≈ 1. Here, it is also seen that the effective areafor the self-resonant case has a maximum around the samesize. The end-fire directions have higher effective area (andgain) than the broadside direction up to ka ≈ 50. Approachingthe electrically large region (ka→∞), the broadside radiationconverges to one half of the physical cross section area sincethe electric currents produce symmetric radiation patterns inup-down direction, and the end-fire directions are observed todecay approximately as (ka)−1/2.

3) Contribution to the maximum effective area (cylinder):The maximum effective area is studied in this example fora single disc Ωt, two separated discs Ωt ∪ Ωb, a mantelsurface Ωm and a cylinder Ωt ∪Ωb ∪Ωm.

5

10−1 100 101 102

100

101

102

103

104

ka ≈ 8.89r/λ

Aeff/πr2

Ωt

Ωt ∪Ωb

Ωm

Ωt ∪Ωb ∪Ωm

GO

Fig. 5. Maximum effective area for a disc, two discs, a mantel surface, and acylindrical structure with surface resistivity Rs = 10−2 Ω/ in the axial (z)-direction. Bounds for externally tuned (13) (solid lines) and self-resonant (16)(dashed lines) currents are depicted.

The performance of a single disc Ωt with radius r depictedin Fig. 5 confirms the broadside limit Aeff → Across/2 inthe electrically large region as observed for the rectanglein Fig. 4. The stepwise decrease for smaller sizes can beinterpreted as the onset of spherical modes in agreement withthe sphere in Fig. 1. Addition of a second disc separated by thedistance 2r from the first disc breaks the up-down symmetryof the radiation pattern. The effective area is depicted in Fig. 5with the curve labeled Ωt ∪Ωb. A rapid oscillatory behavioris observed for electrically large structures. These oscillationsare due to the up-down symmetry for disc distances of integermultiples of the wavelength, i.e., the radiation in the ±z-directions are identical, where z denotes the axis of rotation.For other distances the radiation from the discs can contributeconstructively in the z-direction and destructively in the −z-direction. This produces an effective area approaching Across

on average in the electrically large (ka→∞) region.End-fire radiation is considered from the mantel surface Ωm

(the hollow cylindrical structure without top and bottomdiscs) in Fig. 5. The effective area decreases approximatelylinearly in the log-log scale giving the approximate scal-ing Aeff ∼ (ka)−1/2 as also seen in Fig. 4. Here, it is alsoobserved that the effect of resistivity is larger for the end-firecase as compared to the broadside cases.

Adding the bottom and top discs to the cylinder mantelsurface forms a cylindrical shell as shown in Fig. 5. Theeffective area approaches Across similar to the discs case butwith most of the oscillations removed.

4) Controllable currents (parabolic reflector): A parabolicreflector is used to illustrate the effective area for controllablesubstructures, see Fig. 6. The parabolic reflector is rotation-ally symmetric and has radius a, focal distance a/2, anddepth a/2. A sphere with radius r = a/20 is placed in thefocal point. Maximum effective area is depicted for threecases: controllable currents on the parabolic reflector andsphere, controllable currents on the reflector, and controllablecurrents on the sphere. The induced currents are determinedfrom the method of moments (MoM) impedance matrix [15].

1 10 1000.5

0.7

1

2

3

4

567

ka

Aeff/πa

2

sphere and reflectorreflectorspherecontrollable sphere

Fig. 6. Maximum effective area from (13) for a parabolic reflectorcombined with a sphere placed in the focal point with surface resistivityRs = 10−2 Ω/ in the axial (z)-direction. The parabolic reflector hasradius a, focal distance a/2, and depth a/2 and the sphere has radiusr = a/20.

Controlling both the reflector and sphere gives the largesteffective area and approaches the cross section area for electri-cally large structures as seen in Fig. 6. The oscillations startingaround ka ≈ 55 originates in the internal resonances of thesphere, where it is noted that kr ≈ 2.74 in agreement with theTE dipole resonance [49]. This is also confirmed by negligibleimpact on the overall behavior of the effective area from usingsmaller and larger spheres except for shifting of the resonancesup and down. However, the scenario with both reflector andthe prime feeder controllable is unrealistic.

Removing the sphere and optimizing the currents on thereflector lowers the effective area with approximately a factorof two for large ka. This might at first seem surprising asthe cross-section area of the reflector is 400 times largerthan for the sphere having radius r = a/20. Moreover, theeffective area of the sphere is close to its cross-section area,i.e., Aeff ≈ πr2 ≈ Across/400 as seen in Fig. 3. The effectivearea of the reflector is better explained by its similarity to thedisc in Fig. 5 and rectangle in Fig. 4, where the asymptoticlimit Across/2 is explained by the up-down symmetry of theradiation pattern. The limit Aeff ≈ Across for the reflectortogether with the sphere is hence explained by elimination ofthe backward radiation.

Replacing the controllable currents on the reflector withinduced currents from the sphere produces an effective areajust below Across for high ka. The reduction for small kais similar to the short circuit of the currents above a groundplane. Internal resonances for the sphere are more emphasizedas all radiation originates from the sphere in this case.

III. SUPERDIRECTIVITY

Directional properties of the radiation pattern are quantifiedby the directivity

D(r) = 4πP (r)

Pr≈ 4π

IHUI

IHRrI. (21)

Here, it is seen that the directivity (21) only differs from thegain (1) by its normalization with the radiated power instead

6

of the total dissipated power. This difference is the radiationefficiency η = Pr/(Pr+PΩ), which is related to the dissipationfactor

δ =PΩ

Pr≈ IHRΩI

IHRrI. (22)

Directivity higher than a nominal directivity is often referredto as superdirectivity and associated with low efficiency andnarrow bandwidth [7]. The trade-off between the Q-factor anddirectivity was shown in [15] and further investigated in [36],[42], [50]. Superdirectivity is also associated with decreasedradiation efficiency or equivalently an increased dissipationfactor (22).

A. Trade-off Between Dissipation Factor and Directivity

The trade-off between losses and directivity for a self-resonant antenna can be analyzed by separating the radiatedpower Pr and losses PΩ in (16) giving the optimizationproblem

maximize IHUI

subject to IHXI = 0

IHRrI = 1

IHRΩI = δ.

(23)

The constraint IHXI = 0 is dropped for the correspondingnon-self resonant case (7). The Pareto front is formed byadding the constraints weighted by scalar parameters, i.e.,

maximize IHUI

subject to IH(νX + αRΩ + Rr)I = 1,(24)

where the right-hand side is re-normalized to unity withoutrestriction of generality. This problem is identical to themaximum gain problem (17) if the Pareto parameter α ≥ 0 isincluded in the surface resistivity Rs and is hence solved asthe eigenvalue problem (18). Here, α = 0 solely weights theradiated power regardless of ohmic losses and increasing αstarts to emphasize ohmic losses. The maximal directivity(α = 0) is in general unbounded [2], [3] but has low gain.The other extreme point α → ∞ neglects the radiated powerand maximizes D/δ, i.e., the quotient between the directivityand dissipation factor.

The minimum dissipation factor for the rectangular platefrom Fig. 4 as a function of the directivity in the cardinaldirections and its corresponding case with maximum gainas a function of surface resistivity Rs are shown in Fig. 7and Fig. 8, respectively. Although the physical interpretationof these two problems is different, they are both solvedusing the same eigenvalue problem and have identical currentdensities, i.e., the optimal currents were found using (12)which is identical to (24) without the X-term. Consider,e.g., the blue curve depicting end-fire radiation along theshort side. The normalized dissipation factor is monotonicallyincreasing with D from approximately 10 for D ≈ 2 to107 for D ≈ 25 showing that an increased directivity comeswith a high cost in losses. The corresponding blue curve inFig. 8 decreases monotonically with the surface resistivity Rs

from G ≈ 22 for Rs = 10−8Z0 to G ≈ 0.1 for Rs = Z0. Thecurrent density is depicted for Rs ∈ 10−6, 10−4, 10−2Z0

5 10 15 20101

102

103

104

105

106

107

D

(Z0/R

s)δ

end-fire short sideend-fire long sidebroadside

Fig. 7. Minimum externally tuned dissipation factor for a rectangular plate ofside aspect ratio 2 : 1 and electrical size ka = 1 as a function of directivity Din the cardinal directions. The corresponding case with maximum gain isdepicted in Fig. 8.

10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 1010

5

10

15

20

Rs/Z0

G

end-fire short sideend-fire long sidebroadside

Fig. 8. Maximum externally tuned gain for a rectangular plate of side aspectratio 2 : 1 and electrical size ka = 1 as a function of surface resistivity Rs.The current density is depicted for Rs ∈ 10−6, 10−4, 10−2, 1Z0 andRs = 10−5Z0 for radiation in end-fire short side and broadside directions,respectively, see also Fig. 7.

in Fig. 8 and Rs ∈ 10−5, 10−3Z0 in Fig. 7, where it isseen that the oscillations in the current density increase forhigh D and low Rs. Moreover, the markers on each curve inFig. 7 and Fig. 8 correspond to points with identical currentdensities. Here, it is seen that the uniform spacing in Fig. 8is not preserved in Fig. 7, e.g., the green curve depictingbroadside radiation has two almost overlapping points aroundD ≈ 8 and (Z0/Rs)δ ≈ 105. These two points also haveclose to orthogonal current densities as seen by the insetsand correspond to cases where the eigenvalue problem (24)has degenerate eigenvalues. For these cases we use linearcombinations between the eigenvectors to span the Paretocurve [37].

The minimum dissipation factor [39], [51], [38] is lowerthan the dissipation factor obtained from the α→∞ case forelectrically large structures. These limit cases are connectedby reformulating the problem (23) by either minimizing theohmic losses or maximizing the radiated power. Minimizationof ohmic losses subject to fixed radiation intensity and radiated

7

power isminimize IHRΩI

subject to IHXI = 0

IHUI = 2P

IHRrI = 2Pr,

(25)

which is relaxed to

minimize IHRΩI

subject to IH(νX + αU + Rr)I = 1,(26)

where again the right-hand side is re-normalized to unity.

B. Trade-off Between Q-factor and Directivity

Superdirectivity is also associated with narrow bandwidthand high Q-factor [15], [42]. Adding constraints on the storedenergy to the optimization problem (23) results in the opti-mization problem

maximize IHUI

subject to IHXI = 0

IH(Xe + Xm)I = 2Q

IHRΩI = δ

IHRrI = 1,

(27)

where Xe + Xm = k∂X/∂k are matrices used to determinethe stored energy [52], [36]. Forming linear combinationsbetween the constraints is used to determine the Pareto frontand analyzing the trade-off between directivity, Q-factor, anddissipation factor.

Although (27) can be used to analyze the trade-off, it isillustrative to focus on the constraints on the dissipation factorand Q-factor separately. Dropping the constraint on the ohmiclosses reduces (27) to the problem of lower bounds on theQ-factor for a given directivity [15] which is relaxed to

maximize IHUI

subject to IH(νX + α(Xe + Xm) + Rr)I = 1,(28)

and solved analogously to (18) for fixed α. Here, α = 0 solelyweights the radiated power regardless of ohmic losses andincreasing α starts to emphasize ohmic losses. The maximaldirectivity (α = 0) is in general unbounded [2], [3] but hasa high Q-factor. Here, reformulations similar to (25) can beused to reach the lower bound on the Q-factor.

The trade-offs between directivity and dissipation factorand Q-factor are compared in Fig. 9 for a spherical shellsand a rectangular plate of size ka ∈ 0.5, 2. The boundsare normalized with the lower bounds on the dissipationand Q-factors for the structure. The stored energy matricesare transformed to be positive semidefinite for the Q-factorcalculation [15].

IV. RADIATION MODES AND DEGREES OF FREEDOM

The maximum gain was expressed by Harrington in spher-ical mode expansion as [4]

GH = L2 + 2L =NDoF

2, (29)

2 4 6 8 10100

101

102

103

104

D

Q/Q

lb,δ/δ lb

plate, ka = 0.5plate, ka = 2sphere, ka = 0.5sphere, ka = 2

aa

Fig. 9. Lower bounds on dissipation (solid lines) and Q-factors (dashedlines) for prescribed directivity D normalized with respect the lower bounds.The results were calculated for a spherical shell of radius a and a rectangularplate of side aspect ratio 2 : 1. The electrical size used is ka ∈ 0.5, 2 andthe currents are self-resonant.

where L is the order of the spherical modes and NDoF

degrees of freedom, i.e., total number of modes [12]. Themaximum gain is related to the size of an antenna aperture kaby a cut off limit for modes L = ka [49], but should becorrected for ka < 1 as L ≥ 1, see also [12]. This sphericalmode expansion is most suitable for spherical geometries butoverestimates the number of modes for other shapes.

In order to take a specific shape of an antenna into account,the modes maximizing the radiated power Pr over the lostpower PΩ, i.e., those minimizing dissipation factor δ, are foundfrom an eigenvalue problem [4], [39], [38] as

RrIn = %nRsΨIn, (30)

where RΩ = RsΨ was substituted on the right-hand side, andonly modes with δn = %−1

n < 1 are considered here as well-radiating. It can be seen in (30) that the eigenvectors In donot change with the surface resistivity and only the eigenvalueshave to be rescaled with Rs. Formula (30) can be simplifiedusing

eig(Rr,Ψ) = eig(SΥ−1Υ−HSH) = svd(SΥ−1)2, (31)

where we also used the factorization Rr = SHS based onthe spherical mode matrix S, [53], see Appendix A, anda Cholesky factorization Ψ = ΥHΥ to reduce the compu-tational burden. The radiation modes in (30) produce anexpansion in modes with orthogonal far fields and increasingdissipation factors. They also appear in the analysis of theeigenvalue problems for the radiation operator [54] and forMIMO capacity problems. Notice, that for a spherical shellthey form a set of properly scaled spherical harmonics.

Radiation modes (30) are evaluated for a rectangu-lar plate of side aspect ratio 2 : 1 and electricalsizes ka ∈ 0.1, 0.32, 1, 3.2, 10 and the normalized eigen-values %nRs are depicted in Fig. 10. The low-order modesare emphasized in the inset, where it is seen that the modesappear in groups with similar amplitudes for small ka. This isconfirmed via spherical mode expansion, see Appendix F, for

8

50 100 150 200 250 300 350 40010−24

10−18

10−12

10−6

100

n

Rs%n

0.1 0.32 1 3.2 10

5 10 15 20

100

10−2

10−4

10−6

10−8

Fig. 10. Radiation modes for a rectangular plate of side aspect ratio 2 : 1and electrical sizes ka ∈ 0.1, 0.32, 1, 3.2, 10.

which the rectangular plate supports only half of the sphericalmodes, e.g., x- and y- electrical and z-directed magnetic dipolemodes. This characteristic is most emphasized for electricallysmall structures and the increasing cost of higher order modesvanishes with increasing electrical size, e.g., the first ten modesfor ka = 3.2 differ only in a factor of ten compared to 105

for ka = 0.32.

V. CONCLUSION

Maximum gain and effective area for arbitrarily shapedantenna regions are formulated as quadratically constrainedquadratic programs (QCQP) which are effectively solved aslow-rank eigenvalue problems. The approach is general andincludes constraints on self-resonance and parasitic objects,such as reflectors and ground planes. Radiation modes are usedto interpret the results and simplify the numerical solution ofthe optimization problems.

The results are illustrated for a variety of shapes, electricalsizes ranging from subwavelength objects to objects hundredsof wavelengths long, and resistivities covering a wide rangefrom superconductivity to lossy resistive sheets. Plotting themaximal gain versus electrical size reveals three regions.Dipole and Huygens sources dominate in the electrically smallregion, where the gain depends strongly on the resistivityand whether self-resonance is enforced or not. The effectof self-resonance diminishes as the electrical size approachesa wavelength. End-fire radiation dominates over broadsideradiation for objects of wavelength sizes. This changes inthe limit of electrically large objects where the effective areais proportional to geometrical cross-section and broadsideradiation dominates over endfire radiation.

Superdirectivity is analyzed from the perspective of de-termining the trade-off between directivity and efficiency.Here, it is shown that the problem of maximum gain for agiven resistivity is solved by the same eigenvalue problem asminimum dissipation factor for a given directivity. Moreover,numerical results suggest that the increase in the dissipationand Q-factor are similar for superdirectivity.

The results presented in this paper are of general interestas they can be utilized to evaluate the actual performance

of designed and manufactured antennas and scatterers withrespect to the fundamental bounds. Together with the previ-ously published bounds on Q-factor and radiation efficiency,this work completes the rigorous study of electrically smallantenna limits and extends the fundamental bounds towardsthe electrical large antennas. Understanding of fundamentalbounds and knowledge in optimal currents reopen a call foroptimal antenna designs.

APPENDIX AMATRIX REPRESENTATION OF USED OPERATORS

The matrices used in the optimization problems are con-structed by expansion of the current density J(r) accord-ing (3) for r ∈ Ω.

The far-field matrix for direction r reads [36]

F =

(Fe

Fh

), (32)

where e = h × r and h = r × e denote two orthogonalpolarizations with elements

Fe,n =−jk√Z0

4π

∫Ω

e ·ψn(r1)ejkr1·r dS1, (33)

and similarly for Fh.The radiation resistance matrix Rr and reactance ma-

trix X form the MoM electric field integral equation (EFIE)impedance matrix Z = Rr + jX of a structure modeled asperfect electric conductor (PEC) [35].

The ohmic loss matrix RΩ = RsΨ for a region with ahomogeneous surface resistivity, i.e., Rs, is given by the Grammatrix [55], defined as

Ψmn =

∫Ω

ψm(r) ·ψn(r) dS. (34)

The expansion matrix between basis functions used andspherical waves reads [53]

Sυn = k√Z0

∫Ω

u(1)υ (kr) ·ψn(r) dS , (35)

where u(1)υ denotes the regular spherical vector waves with

index υ [48]. The matrix S is a low-rank factorization of theradiation resistance matrix Rr = SHS.

APPENDIX BQCQP

Maximum gain for self-resonant currents is determined froma QCQP [34], [56] of the form (16)

maximize IHUI

subject to IHRI = 1

IHXI = 0,

(36)

where U = UH 0, R = RT 0, and X = XT beingindefinite. This formulation can be relaxed to a dual problem

minimizeν

maximizeI

IHUI,

subject to IH(νX + R)I = 1,(37)

9

−12 −10 −8 −6 −4 −2 010−3

10−2

10−1

ν1ν2 ν3ν4

upper bound on Gresonan

t

IHXI=

0

νopt

capacitiveIHXI < 0

inductiveIHXI > 0

range for G

νmin νmax

ν

G

Fig. 11. Solution of the QCQP (36) using the dual formulation (38). Therange [νmin, νmax] ≈ [−13.7, 0.02] for the dual parameter ν is determinedin Appendix E by (49) and the optimal parameter value νopt ≈ −8.74is determined from the sequence νn, n = 1, 2, . . . using the bisectionalgorithm [46].

analogously to the analysis in Section II-B with the solution

minimizeν

max eig(U, νX + R). (38)

The range ν ∈ [νmin, νmax] is restricted such that

νX + R 0, (39)

and an efficient procedure to find νmin and νmax is outlinedin Appendix E.

The minimization problem (38) is solved iteratively usinga line-search algorithm, e.g., the bisection algorithm [46],where also the derivative (19) is used, see Fig. 11 showingthe optimization setup. Note that the Newton algorithm [34]can be used if the Hessian is evaluated as, e.g., in the casewith partial gain [36].

The explicit form of the derivative (19) also shows that thederivative is zero for the optimal value νopt if the eigenvaluedepends continuously on ν as the derivative changes signaround νopt. Hence, the solution to (38), Iopt, at the extremepoint νopt is self resonant IH

optXIopt = 0 and satisfies thesecond constraint in the QCQP (36). This implies that theduality gap is zero and that the QCQP (36) is solved byits dual (38). Moreover, non-degenerate eigenvalues dependcontinuously on parameters [57] so the problem is solved forthis case. For a treatment of modal degeneracies and otherimplementation issues, see Appendix D.

APPENDIX CALTERNATIVE SOLUTIONS TO QCQP

The Lagrangian dual [34] is convex and offers an alternativeapproach to solve the QCQP (36). It is given by the semidef-inite program (SDP)

minimize µ,

subject to −U + νX + µR 0,

µ ≥ 0, ν ∈ R,(40)

which can be solved efficiently [34]. The semidefinite con-straint in the Lagrangian dual (40) can be written

IH(νX + µR)I = µIH(ν1X + R)I ≥ IHUI (41)

for all currents I and νl = ν/µ. Here, it is seen thatνlX + R 0 and

µ ≥ IHUI

IH(νlX + R)I≥ min

νlmax eig(U, νlX + R) (42)

and hence the solution of the Lagrangian dual in (40) is similarto the solution (38) of the relaxation (37). The only differenceis in the range for νl that is a subset of (49) due to the IHUIterm in the right-hand side of (41). However, self-resonantsolutions of (24) satisfies (41). Semidefinite relaxation isanother standard relaxation technique [34] for the QCQP (36).

APPENDIX DNUMERICAL EVALUATION OF QCQP

In this paper, the implementation is as follows. We usethe eigenvalue problem (37) together with the factoriza-tion U = FHF due to its simplicity and computational ef-ficiency. The computational complexity is dominated by thesolution of the linear system (νX + R)−1FH which requiresof the order N3 operations for direct solvers, where N is thenumber of basis functions, cf. (3). Here, we also note that theadditional computational cost of using multiple directions Fis negligible. For electrically large structures we use iterativealgorithms to solve the linear system [58]. The rectanglein Fig. 4 was, e.g., solved iteratively using a matrix-freeFFT-based formulation [43] using N ≈ 4.2 · 106 unknownsfor ka ≈ 103. The fast multipole method (FMM) and similartechniques can also be used [43] to reduce the computationalburden.

Whenever possible, symmetries are used to simplify thesolution by separating the eigenvalue problem into orthog-onal subspaces which are solved separately and combinedanalytically [38]. This reformulates the optimization problemsinto block diagonal form, where each block corresponds to asubspace. The problem is further simplified for cases wheresome of the subspaces do not contribute to the radiationintensity in the considered direction as, e.g., for radiation inthe normal direction k = z for the rectangle in Fig. 4, wherecurrents with odd inversion symmetry J(r) = −J(−r) donot contribute and similarly for the cylinder in Fig. 5 whereazimuthal Fourier indices |m| 6= 1 do not contribute. For thesecases, the currents in the non-contributing subspace can onlybe used to tune the currents into self resonance. Hence, theyare quiescent in the externally tuned case (8) and determinedby the eigenvectors associated with the largest eigenvalues ofthe eigenvalue problems in (20), where the matrices X,Rr,and RΩ are restricted to the non-contributing subspace.

Expansion in radiation modes (30) is also useful in the so-lution of the maximum gain optimization problem (13), whichcontains the solution of the linear system (Rr + RsΨ)−1FH

and often is solved for many values of Rs as in Section III.Using Rr = SHS and RΩ = RsΥ

HΥ together with the sin-gular value decomposition (SVD) UΣVH = SΥ−1 reducesthe inversion of the linear system to inversion of a diagonal

10

κ

ν

0

0

κmax

νmin

Indefinite PSD

κmin

νmax

Fig. 12. Determination of the range of ν : νX+R 0; eigenvalues κ aresolutions to XI = κRI.

matrix, i.e.,

(Rr + RΩ)−1 = (SHS +RsΥHΥ)−1

= Υ−1(Υ−HSHSΥ−1 +Rs1)−1Υ−H

= Υ−1(V−HΣHΣVH +Rs1)−1Υ−H

= Υ−1V(ΣHΣ +Rs1)−1VHΥ−H, (43)

where 1 denotes the identity matrix. The computational cost ofsweeping the maximum gain versus Rs is hence traded to com-putation of the SVD of SΥ−1 that only requires N2

s N +N2

operations, where Ns denotes the number of spherical modes.Note, that we also use that matrix Ψ is a sparse matrixwith approximately 3N non-zero elements that reduces thecomputational cost to compute Υ.

APPENDIX EDETERMINATION OF νmin AND νmax

The task here is to find a range of ν, delimited by νmin

and νmax, such that (39) holds, R = RT 0, and X = XT

is indefinite. Equivalently, we can state that

IH(νX + R)I ≥ 0 ∀ I (44)

which can be reformulated to the Rayleigh quotient

νIHXI

IHRI≥ −1 ∀ I. (45)

First, consider the case with ν > 0, for which the Rayleighquotient satisfies

IHXI

IHRI≥ min

I

IHXI

IHRI= min eig(X,R) = κmin ≥

−1

ν(46)

that implies the upper limit of the interval

ν ≤ νmax =−1

κmin, (47)

where κmin is the smallest eigenvalue, see Fig. 12.Second, consider the case ν < 0. Analogously to (46) we

get

IHXI

IHRI≤ max

I

IHXI

IHRI= max eig(X,R) = κmax =

−1

ν(48)

and the range is given by (47) and (48) as

−1

κmax= νmin ≤ ν ≤ νmax =

−1

κmin. (49)

10−3 10−2 10−1 100 101 102 103100

101

102

103

104

105

106

aa

ka

D

10−8 Ω/10−6 Ω/10−4 Ω/10−2 Ω/1 Ω/HarringtonGO

Fig. 13. Resulting directivity from the maximum gain for a spherical shellwith surface resistivity Rs ∈ 10−nΩ/ for n = 0, 2, 4, 6, 8 in Fig. 1.Dotted lines depicts directivitiesD ∈ 3/2, 3, 11/2, 8, 23/2 associated withthe lowest order modes.

10−3 10−2 10−1 100 101 102 10310−3

10−2

10−1

100

101

102

103

104

aa

ka

δ

10−8 Ω/10−6 Ω/10−4 Ω/10−2 Ω/1 Ω/

Fig. 14. Resulting dissipation factor from the maximum gain for a sphericalshell with surface resistivity Rs ∈ 10−nΩ/ for n = 0, 2, 4, 6, 8 inFig. 1.

APPENDIX FMAXIMUM GAIN FOR A SPHERE

The optimization problems (8) and (16) are solved ana-lytically for spherical structures using spherical waves [48].The matrices in the eigenvalue problems (13) and (18)are diagonalized for spherical modes offering closed-formsolutions of the eigenvalue problems. The radiation resis-tance and ohmic loss matrices have elements (kaR

(1)lτ (ka))2

and Rs, respectively, giving normalized dissipation fac-tors δlτ/Rs = (kaR

(1)lτ (ka))−2, where l is the order of

the spherical mode, τ the TE or TM type, and R(p)lτ ra-

dial functions [59]. Radiation dominates for modes with(kaR

(1)lτ (ka))2 > Rs. This resembles (29) with the obser-

vation that the radial functions are negligible for ka l.The directivity associated with the maximum gain and

effective area in Fig. 1 is depicted in Fig. 13. The directiv-ity increases stepwise as additional modes are included. Inthe electrically small limit, the self-resonant case combineselectric and magnetic dipoles to form a Huygens sourcewith directivity D = 3. The radiation efficiency is, however,

11

10−1 100100

101

102

103

104

ka

(Z0/R

s)δ

spherical shellrectangular platemeandered dipole

aa

Fig. 15. Minimum dissipation factors for a spherical shell (blue lines), arectangular plate (red lines), and a meanderline (green lines) are depicted bothfor externally tuned (solid lines) and self-resonant (dashed lines) currents.

low as seen in the much lower gain in Fig. 1. Inclusionof quadrupole modes increases the directivity to D = 8 asseen by (29) for L = 2. The externally tuned case startsat D = 3/2, where the radiation is caused by a sole elec-tric dipole. It increases to D = 11/2 when the magneticdipole and electric quadrupole starts to contribute. This step-wise increase is explained by the lower losses for the TMmodes, see Fig. 14, causing the modes to appear in or-der as TM1, TM2,TE1, TM3,TE2 giving directivityD = L2 + L− 1/2 as compared with (29).

APPENDIX GSELF-RESONANT AND TUNED CASES

The large difference between the gain (the effective area)for the externally tuned and self-resonant cases for electricallysmall structures originates in much higher dissipation factorfor the loop current forming the TE dipole mode than for thecharge separation producing the TM dipole mode [60], [39],[51], [38], [61]. The difference reduces as the electrical sizeincreases and is negligible for the electrically large structures.The minimum dissipation factor for the two cases can beused as an estimate of the size when the self-resonancecondition becomes irrelevant, which is demonstrated in Fig. 15for examples of a spherical shell, a rectangular plate, and ameanderline.

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Mats Gustafsson received the M.Sc. degree in En-gineering Physics 1994, the Ph.D. degree in Electro-magnetic Theory 2000, was appointed Docent 2005,and Professor of Electromagnetic Theory 2011, allfrom Lund University, Sweden.

He co-founded the company Phase holographicimaging AB in 2004. His research interests are inscattering and antenna theory and inverse scatteringand imaging. He has written over 90 peer reviewedjournal papers and over 100 conference papers. Prof.Gustafsson received the IEEE Schelkunoff Transac-

tions Prize Paper Award 2010 and Best Paper Awards at EuCAP 2007 and2013. He served as an IEEE AP-S Distinguished Lecturer for 2013-15.

Miloslav Capek (M’14, SM’17) received his M.Sc.degree in Electrical Engineering and Ph.D. degreefrom the Czech Technical University, Czech Repub-lic, in 2009 and 2014, respectively. In 2017 he wasappointed Associate Professor at the Department ofElectromagnetic Field at the CTU in Prague.

He leads the development of the AToM (AntennaToolbox for Matlab) package. His research interestsare in the area of electromagnetic theory, electricallysmall antennas, numerical techniques, fractal geom-etry and optimization. He authored or co-authored

over 75 journal and conference papers.Dr. Capek is member of Radioengineering Society, regional delegate of

EurAAP, and Associate Editor of Radioengineering.