1 Maximum Gain, Effective Area, and Directivity Mats Gustafsson, Senior Member, IEEE and Miloslav Capek, Senior Member, IEEE

Abstract—Fundamental bounds on antenna gain are found via approach demarcated the avenue of further research. Improved convex optimization of the current density in a prescribed region. formula has been proposed in [12], suggesting that, in general, Various constraints are considered, including self-resonance and the maximum directivity in the electrically small region is only partial control of the current distribution. Derived formulas are valid for arbitrarily shaped radiators of a given conductivity. equal to three. The maximum directivity is studied in [13] All the optimization tasks are reduced to eigenvalue problems, considering a given current norm. For antenna arrays, direc- which are solved efficiently. The second part of the paper deals tivity bounds are shown in [14]. Trade-off between maximum with superdirectivity and its associated minimal costs in efficiency directivity and Q-factor for arbitrarily shaped antennas is and Q-factor. The paper is accompanied with a series of examples presented in [15]. Upper bounds for scattering of metamaterial- practically demonstrating the relevance of the theoretical frame- work and entirely spanning wide range of material parameters inspired structures are found in [16]. Recently, a composition and electrical sizes used in antenna technology. Presented results of Huygens multipoles has been proposed [17] to increase are analyzed from a perspective of effectively radiating modes. the directivity. Notice, however, that no losses other than the In contrast to a common approach utilizing spherical modes, radiation were assumed which re-opens the question of the the radiating modes of a given body are directly evaluated and actual cost of super-directivity. analyzed here. All crucial mathematical steps are reviewed in the appendices, including a series of important subroutines to be Another way to limit the directional properties is a pre- considered making it possible to reduce the computational burden scribed, non-zero, material resistivity of the antenna body [18], associated with the evaluation of electrically large structures and [19]. A quantity to deal with is the antenna gain, which is structures of high conductivity. always bounded if at least infinitesimal losses are assumed. Index Terms—Antenna theory, current distribution, eigenval- It may seem reasonable at this point to argue that the losses ues and eigenfunctions, optimization methods, directivity, an- can be overcame with a concept of superconducting antennas, tenna gain, radiation efficiency. however, as shown in [6], the increase in gain with decrease of resistivity embodies slow (logarithmic) convergence. Con- sequently, even tiny losses, which are always present at RF, I.INTRODUCTION restrict the gain to a finite number. A question of how narrow a radiation pattern can be or, Tightly connected is the question of maximum achievable in terms of standard antenna terminology [1], what are the absorption cross-section. The capability to effectively radiate bounds on directivity and gain, has been in the spotlight of energy in a certain direction can reciprocally be understood antenna theorists’ and physicists’ for many years. as a potential to absorb energy from that direction [20], [21]. Early works studied needle-like radiation patterns [2]. A This can be interpreted as an ability of a receiver to distort the series of works starting in the 1940s revealed the fact that the near-field so that the incoming energy is effectively absorbed directivity is unbounded [3] but also predicted the enormous in the receiver’s body or concentrated at the receiving port. It cost in other antenna parameters, namely in Q-factor [4], has been realized that such an area can be huge as compared related sensitivity of feeding network [5], and radiation ef- to the physical size of the particle or the physical antenna ficiency in case that the antenna is made of lossy material [6]. aperture [22], [23]. Fundamental bounds on absorption cross- Consequently, as pointed out by Hansen [7], the superdirective sections are proposed in [24], [25]. aperture design requires additional constraint, replacing fixed The importance to establish fundamental bounds on gain spacing in array theory [8], [9]. and absorption cross-section are underlined by recent devel- In order to tighten the bounds on directivity, Harring- opment in design of superdirective (supergain) antennas and ton [10], [4] proposed a simple formula which predicts the arrays [26], [27], [28], [29], [30], [31], partly fueled by the directivity from the number of used spherical harmonics as a advent of novel materials and technologies [32], [33].

arXiv:1812.07058v1 [physics.class-ph] 11 Dec 2018 function of aperture size. The number of modes radiating well The procedure developed in this paper relies on convex and the pioneering works on bounds [11] became popular in optimization [34] of current distributions [15]. In order to antenna design and hold in many realistic cases, therefore, this find the optimal current distribution in a prescribed region, the antenna quantities are expressed as quadratic forms of Manuscript received December 19, 2018; revised December 19, 2018. This work was supported by the Swedish Foundation for Strategic Re- corresponding matrix operators [35], [36]. This makes it search (SSF) grant no. AM13-0011. The work of M. Capek was sup- possible to solve the optimization problems rigorously via ported by the Ministry of Education, Youth and Sports through the project eigenvalue problems [37], [38]. The procedure is general CZ.02.2.69/0.0/0.0/16 027/0008465. M. Gustafsson is with the Department of Electrical and Infor- as arbitrarily shaped regions can be investigated. Additional mation Technology, Lund University, 221 00 Lund, Sweden (e-mail: constraints are enforced, e.g., self-resonance and restricted [email protected]). controllability of the current [15], [39]. Much work in this M. Capek is with the Department of Electromagnetic Field, Faculty of Electrical Engineering, Czech Technical University in Prague, 166 27 Prague, area has already been done in determining bounds on Q- Czech Republic (e-mail: [email protected]). factor [37], radiation efficiency [39], superdirectivity [15], 2

gain [11], and capacity [40]. The recent trend, followed by this where Pr and PΩ denote the radiated power and power dissi- paper, is to understand the mutual trade-offs between various pated in ohmic and dielectric losses, respectively. The effective parameters [41], [42], [38]. area, Aeff , is an alternative quantity used to describe directive The original approach from [11] and [35] maximizing properties for receiving antennas, which is for reciprocal the Rayleigh quotient for antenna gain via a generalized antennas simply related to the gain as [20] eigenvalue problem is recast here into an eigenvalue prob- Gλ2 lem of reduced rank. Such a formulation is compatible Aeff = , (2) with fast numerical methods [43], therefore, the results can 4π  3 3 be presented in a wide frequency range, ka 10− , 10 , where λ = 2π/k denotes the wavelength. It is seen that max- where ka is used throughout the paper to denote∈ the di- imization of gain is equivalent to maximization of effective mensionless frequency with k being the wavenumber and a area [35]. being the radius of a sphere circumscribing all the sources. The optimized parameters are expressed in the current The surface resistivity used spans the interval from extremely density J(r) which is expanded in a set of basis functions 8 low values, Rs = 10− Ω/, reachable in RF superconduct- ψn (r) as [35] ing cavities [44], through values valid for copper at RF { } N (Rs 0.01 Ω/, f = 1 GHz), to poor conductors of surface X ≈ J(r) Inψn(r), (3) resistivity R = 1 Ω/ . ≈ s  n=1 Optimal currents presented in this paper maximize the antenna gain. Therefore, taking reciprocity into account, they where the expansion coefficients, In, are collected in the delimit the maximum effective area of any receiver designed in column matrix I. This substitution yields algebraic expressions that region as well. For this reason, the proportionality between for radiation intensity, radiated power, and ohmic losses as gain and effective area is utilized, making it possible to judge follows [36] the real performance of designed and manufactured antennas, 1 1 P (rˆ) FI 2 = IHFHFI, (4) arrays, scatterers, and other radiating systems. ≈ 2 | | 2 The behavior of the optimal solution evolves markedly 1 P IHR I, (5) with electrical size. Huygens source formed by electric and r ≈ 2 r magnetic dipoles is strictly preferred in electrically small (sub- 1 P IHR I. (6) wavelength) region and a large effect of self-resonance, if en- Ω ≈ 2 Ω forced, is observed. End-fire radiation and negligible effect of The matrices used in (4)–(6) are reviewed in AppendixA. self-resonance constraint is observed in an intermediate region. Substitution of (4)–(6) into (1) yields Finally, broadside radiation dominates in the electrically large region with the effective area being proportional to the cross- FI 2 IHUI r (7) G(ˆ) 4π H | | = 4π H , section area. ≈ I (Rr + RΩ)I I (Rr + RΩ)I The paper is organized as follows. Antenna gain and ef- where we also introduced the matrix U = FHF to simplify fective area are introduced in SectionII and expressed as the notation and highlight the expression of the gain G(rˆ) as quadratic forms in the currents. The optimal currents are then a Rayleigh quotient. found for maximum gain in Sections II-A and II-B, including cases with additional constraints like self-resonance. Examples covering various aspects of antenna design are presented in A. Maximum Gain: Tuned Case Section II-C. Superdirective currents are found in Section III The maximum gain for antennas confined to a region r Ω and presented as a trade-off between required directivity and is formulated as the optimization problem ∈ minimum ohmic losses or Q-factor. All presented examples reveal the enormous cost of superdirectivity. The maximum maximize IHUI gain is reinterpreted in SectionIV in terms of number of (8) subject to IH(R + R )I = 1, sufficiently radiating modes of a structure and the results are Ω r linked back to Harrington’s formula. The paper is concluded where for simplicity the dissipated power is normalized to in SectionV. All required mathematical tools are reviewed unity. This problem is equivalent to the Rayleigh quotient and key derivations are presented in the Appendices. IHFHFI (9) Gub 4π max H , ≈ I I (Rr + RΩ)I II.GAINAND EFFECTIVE AREA to which a solution is found via the generalized eigenvalue Antenna gain describes how an antenna converts input problem [35] H power into radiation in a specified direction rˆ,[45]. The gain F FI = γ(Rr + RΩ)I. (10) in a direction rˆ is determined as 4π times the quotient between the radiation intensity P (rˆ) and the dissipated power Pr +PΩ, In order to reduce the computational burden, the for- mula (10) is further transformed to P (rˆ) G(rˆ) = 4π , (1) 1 H − (11) Pr + PΩ (Rr + RΩ) F FI = γI 3 and multiplied from left by the matrix F. By introduc- 106 8 ˜ 10− Ω/ ing I = FI we readily get 5 6 10 10− Ω/ 4 1 H˜ ˜ 10− Ω/ F(Rr + RΩ)− F I = γI. (12) 2 104 10− Ω/ 1 Ω/ Taking into account that the far-field matrix F can be ex- 3 Harrington pressed using two orthogonal polarizations, see AppendixA, 10 GO G the original N N eigenvalue problem (10) is reduced into 2 × 10 the 2 2 eigenvalue problem (12) which can be written as a × 101 1 H Gub 4π max eig(F(Rr + RΩ)− F ) (13) ≈ 100 with the optimal current determined as 10 1 1 1 H − 3 2 1 0 1 2 3 I = γ− (Rr + RΩ)− F ˜I. (14) 10− 10− 10− 10 10 10 10 ka The corresponding case with the partial gain contains one polarization direction and hence the eigenvalue Fig. 1. Maximum gain for a spherical shell of radius a with surface resis- 1 H tivity R = 10 n Ω/ , n = {0, 2, 4, 6, 8}, both for externally tuned (13), γ = F(Rr + RΩ)− F and current s −  Gub, (solid lines) and for self-resonant (18), Gub,r, (dashed lines) currents. 1 H I (R + R )− F . (15) ∼ r Ω H Here, the F part can be interpreted as phase conjugation by decomposition of the current I into orthogonal sub- of an incident plane wave from the rˆ-direction, and hence the spaces [37]. The range ν [νmin, νmax] in (18) is determined current corresponding to the maximum gain is found by phase from the condition νX+R∈ +R 0 which can be computed 1 Ω r conjugation of the incident wave modified by (Rr + RΩ)− .  from the smallest and largest eigenvalues, eig(X, RΩ + Rr), i.e., B. Maximum Gain: Self-Resonant Case 1 1 − ν − , (20) The solution to (13) is in general not self-resonant. Self max eig(X, RΩ + Rr) ≤ ≤ min eig(X, RΩ + Rr) resonance is enforced to (13) by adding the constraint of zero reactance, IHXI = 0, see AppendixA, producing the see AppendixE for details. optimization problem The minimal eigenvalue min eig(X, RΩ + Rr) is related to the Q-factor of the maximal capacitance in the geometry maximize IHUI which is very large for all considered cases giving an upper H subject to I XI = 0 (16) limit very close to zero and νmax 0 as the mesh is refined. The maximal eigenvalue is related→ to the maximal inductive IH(R + R )I = 1. Ω r Q-factor which is a fixed value depending on shape of the This optimization problem is a quadratically constrained object and gives the lower bound νmin, cf. with the inductor quadratic program (QCQP), see AppendixB, that is trans- Q-factor in [38]. formed to a dual problem by multiplication of IHXI with a scalar parameter ν and adding the constraints together, i.e., C. Examples of Maximum Gain and Effective Area maximize IHUI The following section presents maximum gain and effective H (17) subject to I (νX + RΩ + Rr)I = 1, area for examples of various complexity: which is solved as a generalized eigenvalue problem analo- 1) spherical shell both for externally and self-resonant gously to Section II-A. The solution to this problem is greater currents, Section II-C1, or equal to (16) and taking its minimum value produces the 2) comparison of end-fire and broadside radiation from a dual problem [34] rectangular region, Section II-C2, 3) maximization of effective area if different parts of a Gub,r 4π min max eig(U, νX + RΩ + Rr) ≈ ν cylinder are considered, Section II-C3, 1 H (18) = 4π min max eig(F(νX + RΩ + Rr)− F ) 4) limited controllability of currents for a parabolic dish ν with spherical prime feed, Section II-C4. e.g. which is convex and easy to solve, , with the bisection 1) Externally tuned and self-resonant currents (spherical algorithm [46]. The derivative of the eigenvalue with respect γ shell): Expansion of the current density on a spherical shell to is [47] ν in vector spherical harmonics [48] produces diagonal reac-  0 for ν ν , inductive tance X, radiation resistance Rr, and loss RΩ matrices with H  opt ∂γ 2 I XI ≤ ≤ closed form expressions of the elements. The direction of = γ = 0 for ν = νopt, resonant (19) ∂ν − IHUI radiation can without loss of generality be chosen to rˆ = zˆ  0 for ν ν , capacitive ≥ ≥ opt for which the elements F are zero for azimuthal Fourier for cases with non-degenerate eigenvalues. Degenerate eigen- indices m = 1. It is hence sufficient to consider m = 1 values are often related to geometrical symmetries and solved for the radiation,| | 6 see AppendixF. | | 4

6 25 8 25 10 10− Ω/ 8 6 10− Ω/ 10− Ω/ 5 6 4 10 10− Ω/ 10− Ω/ a 4 2 10− Ω/ 10− Ω/ 2 10  10 4 10− Ω/ 1 Ω/ 10   1 Ω/ 2 3 Harrington 10 GO

5 5 /πa

− − eff 102 A

20 20 101 − − 100 5 5 − − 1 10− 3 2 1 0 1 2 3 10− 10− 10− 10 10 10 10 ka ka = 1 ka = 10 Fig. 3. Maximum effective area for a spherical shell of radius a with n Fig. 2. Radiation patterns for a spherical shell of radius a with surface resis- surface resistivity Rs = 10− Ω/, n = {0, 2, 4, 6, 8}, both for externally n tivity Rs = 10− Ω/, n = {0, 2, 4, 6, 8} corresponding to the externally tuned (13) (solid lines) and for self-resonant (18) (dashed lines) currents. tuned case in Fig.1. The radiation patterns are shown in terms of gain G for a ϑ-cut and ϕ = 0. The two electrical sizes, ka = 1 and ka = 10, are depicted. 3 end-fire short side 10 end-fire long side broadside Harrington The maximum gain for a spherical shell with surface re- 102 GO, GO/2 2)

n / sistivity Rs = 10− Ω/ for n = 0, 2, 4, 6, 8 is determined 2 `

{ } ( using (13), (18) and depicted in Fig.1. The results are com- / 101 pared with the estimates G = (ka)2 + 2ka by Harrington [4] eff

H A 2 and from the geometrical cross section GGO = 4πAcross/λ . It is observed that the additional constraint on self-resonance, 100 i.e., IHXI = 0, in (16) has a large effect for small structures

(ka < 1) but negligible effect for electrically large struc- 1 10− 2 1 0 1 2 3 tures. The tuned and self-resonant cases have D = 3/2 and 10− 10− 10 10 10 10 D = 3, respectively, in the limit of electrically small structures ka 3.51`/λ ≈ (ka 0), see AppendixF. Onset of spherical modes for → small ka gives a step-wise increasing directivity and gain, see Fig. 4. Maximum effective area in the cardinal directions for a rectangular 4 figures in AppendixF. Dependence on R diminishes and the plate with size ` × `/2 and surface resistivity Rs = 10− Z0 per square. s Bounds for externally tuned (13) (solid lines) and self-resonant (16) (dashed gain approaches GGO as ka increases. The radiation patterns lines) currents are depicted. and the influence of the surface resistivity on the maximum gain G is shown in Fig.2 for the externally tuned case and electrical sizes ka = 1 and ka = 10. The electrically large and consists of a combination of electric and magnetic dipoles. limit is more clearly seen by plotting the effective area (2) in Huygens sources are obtained for the end-fire cases where the Fig.3, where it is observed that the effective area approaches gain is higher for radiation along the longest side than for the the cross-section area as ka . shorter side due to its lower amount of stored electric energy. 2) Broadside and end-fire→ radiation ∞ (rectangular plate): Gain in the broadside direction is lower due to its up-down The symmetry of the sphere is ideal for analytic solution of symmetric radiation pattern. the optimization problem but cannot be used to investigate The difference between the externally tuned and self- important cases such as broadside and end-fire radiation [21]. resonant cases decreases as ka increases and become negligi- Let us, therefore, consider a planar rectangular plate with ble around ka 1. Here, it is also seen that the effective area side lengths ` and `/2 placed at z = 0 having surface for the self-resonant≈ case has a maximum around the same 4 resistivity Rs = 10− Z0 per square. The maximum effective size. The end-fire directions have higher effective area (and area is depicted in Fig.4 for radiation in the cardinal di- gain) than the broadside direction up to ka 50. Approaching rections. Three regions can be identified: electrically small the electrically large region (ka ), the≈ broadside radiation (ka 1) with large difference between the externally tuned converges to one half of the physical→ ∞ cross section area since and self-resonant cases, intermediate region with dominant the electric currents produce symmetric radiation patterns in end-fire radiation, and electrically large ka 1 with dominant up-down direction, and the end-fire directions are observed to  1/2 broadside radiation. decay approximately as (ka)− . Negligible directional differences are observed for the elec- 3) Contribution to the maximum effective area (cylinder): trically small (ka 1) externally tuned case which can be ex- The maximum effective area is studied in this example for  plained by radiation patterns originating from electric dipoles. a single disc Ωt, two separated discs Ωt Ωb, a mantel The effective area for the self-resonant case deceases as (ka)2 surface Ω and a cylinder Ω Ω Ω . ∪ m t ∪ b ∪ m 5

4 10 7 sphere and reflector 6 Ωt reflector Ωt Ωb 5 sphere ∪ controllable sphere 103 Ωm 4 Ωt Ωb Ωm GO∪ ∪ 3 2 2 102 /πa 2 /πr eff eff A A 101 1

100 0.7 0.5 1 10 100 10 1 100 101 102 − ka ka 8.89r/λ ≈ Fig. 6. Maximum effective area from (13) for a parabolic reflector Fig. 5. Maximum effective area for a disc, two discs, a mantel surface, and a combined with a sphere placed in the focal point with surface resistivity 2 2 cylindrical structure with surface resistivity Rs = 10− Ω/ in the axial (zˆ)- Rs = 10− Ω/ in the axial (zˆ)-direction. The parabolic reflector has direction. Bounds for externally tuned (13) (solid lines) and self-resonant (16) radius a, focal distance a/2, and depth a/2 and the sphere has radius (dashed lines) currents are depicted. r = a/20.

The performance of a single disc Ωt with radius r depicted Controlling both the reflector and sphere gives the largest in Fig.5 confirms the broadside limit Aeff Across/2 in effective area and approaches the cross section area for electri- the electrically large region as observed for→ the rectangle cally large structures as seen in Fig.6. The oscillations starting in Fig.4. The stepwise decrease for smaller sizes can be around ka 55 originates in the internal resonances of the interpreted as the onset of spherical modes in agreement with sphere, where≈ it is noted that kr 2.74 in agreement with the the sphere in Fig.1. Addition of a second disc separated by the TE dipole resonance [49]. This is≈ also confirmed by negligible distance 2r from the first disc breaks the up-down symmetry impact on the overall behavior of the effective area from using of the radiation pattern. The effective area is depicted in Fig.5 smaller and larger spheres except for shifting of the resonances with the curve labeled Ωt Ωb. A rapid oscillatory behavior up and down. However, the scenario with both reflector and is observed for electrically∪ large structures. These oscillations the prime feeder controllable is unrealistic. are due to the up-down symmetry for disc distances of integer Removing the sphere and optimizing the currents on the multiples of the wavelength, i.e., the radiation in the zˆ- reflector lowers the effective area with approximately a factor directions are identical, where zˆ denotes the axis of rotation.± of two for large ka. This might at first seem surprising as For other distances the radiation from the discs can contribute the cross-section area of the reflector is 400 times larger constructively in the zˆ-direction and destructively in the zˆ- than for the sphere having radius r = a/20. Moreover, the − direction. This produces an effective area approaching Across effective area of the sphere is close to its cross-section area, 2 on average in the electrically large (ka ) region. i.e., Aeff πr Across/400 as seen in Fig.3. The effective → ∞ ≈ ≈ End-fire radiation is considered from the mantel surface Ωm area of the reflector is better explained by its similarity to the (the hollow cylindrical structure without top and bottom disc in Fig.5 and rectangle in Fig.4, where the asymptotic discs) in Fig.5. The effective area decreases approximately limit Across/2 is explained by the up-down symmetry of the linearly in the log-log scale giving the approximate scal- radiation pattern. The limit Aeff Across for the reflector 1/2 ≈ ing Aeff (ka)− as also seen in Fig.4. Here, it is also together with the sphere is hence explained by elimination of observed∼ that the effect of resistivity is larger for the end-fire the backward radiation. case as compared to the broadside cases. Replacing the controllable currents on the reflector with Adding the bottom and top discs to the cylinder mantel induced currents from the sphere produces an effective area surface forms a cylindrical shell as shown in Fig.5. The just below Across for high ka. The reduction for small ka effective area approaches Across similar to the discs case but is similar to the short circuit of the currents above a ground with most of the oscillations removed. plane. Internal resonances for the sphere are more emphasized 4) Controllable currents (parabolic reflector): A parabolic as all radiation originates from the sphere in this case. reflector is used to illustrate the effective area for controllable substructures, see Fig.6. The parabolic reflector is rotation- III.SUPERDIRECTIVITY ally symmetric and has radius a, focal distance a/2, and Directional properties of the radiation pattern are quantified depth a/2. A sphere with radius r = a/20 is placed in the by the directivity focal point. Maximum effective area is depicted for three P (rˆ) IHUI cases: controllable currents on the parabolic reflector and r (21) D(ˆ) = 4π 4π H . sphere, controllable currents on the reflector, and controllable Pr ≈ I RrI currents on the sphere. The induced currents are determined Here, it is seen that the directivity (21) only differs from the from the method of moments (MoM) impedance matrix [15]. gain (1) by its normalization with the radiated power instead 6 of the total dissipated power. This difference is the radiation 107 efficiency η = Pr/(Pr+PΩ), which is related to the dissipation 6 factor 10 H PΩ I RΩI δ = . (22) 5

H δ 10

Pr ≈ I RrI ) s

Directivity higher than a nominal directivity is often referred /R 4

0 10 to as superdirectivity and associated with low efficiency and Z ( narrow bandwidth [7]. The trade-off between the Q-factor and 103 directivity was shown in [15] and further investigated in [36], 2 end-fire short side [42], [50]. Superdirectivity is also associated with decreased 10 end-fire long side radiation efficiency or equivalently an increased dissipation broadside 101 factor (22). 5 10 15 20 D A. Trade-off Between Dissipation Factor and Directivity The trade-off between losses and directivity for a self- Fig. 7. Minimum externally tuned dissipation factor for a rectangular plate of side aspect ratio 2 : 1 and electrical size ka = 1 as a function of directivity D resonant antenna can be analyzed by separating the radiated in the cardinal directions. The corresponding case with maximum gain is power Pr and losses PΩ in (16) giving the optimization depicted in Fig.8. problem maximize IHUI 20 end-fire short side subject to IHXI = 0 end-fire long side broadside H (23) I RrI = 1 15 H I RΩI = δ. H The constraint I XI = 0 is dropped for the corresponding G 10 non-self resonant case (7). The Pareto front is formed by adding the constraints weighted by scalar parameters, i.e., 5 maximize IHUI H (24) subject to I (νX + αRΩ + Rr)I = 1, 0 8 7 6 5 4 3 2 1 0 1 where the right-hand side is re-normalized to unity without 10− 10− 10− 10− 10− 10− 10− 10− 10 10 restriction of generality. This problem is identical to the Rs/Z0 maximum gain problem (17) if the Pareto parameter α 0 is ≥ included in the surface resistivity Rs and is hence solved as Fig. 8. Maximum externally tuned gain for a rectangular plate of side aspect ratio 2 : 1 and electrical size ka = 1 as a function of surface resistivity Rs. the eigenvalue problem (18). Here, α = 0 solely weights the 6 4 2 The current density is depicted for Rs ∈ {10− , 10− , 10− , 1}Z0 and 5 radiated power regardless of ohmic losses and increasing α Rs = 10− Z0 for radiation in end-fire short side and broadside directions, starts to emphasize ohmic losses. The maximal directivity respectively, see also Fig.7. (α = 0) is in general unbounded [2], [3] but has low gain. The other extreme point α neglects the radiated power → ∞ 5 3 and maximizes D/δ, i.e., the quotient between the directivity in Fig.8 and Rs 10− , 10− Z0 in Fig.7, where it is ∈ { } and dissipation factor. seen that the oscillations in the current density increase for The minimum dissipation factor for the rectangular plate high D and low Rs. Moreover, the markers on each curve in from Fig.4 as a function of the directivity in the cardinal Fig.7 and Fig.8 correspond to points with identical current directions and its corresponding case with maximum gain densities. Here, it is seen that the uniform spacing in Fig.8 as a function of surface resistivity Rs are shown in Fig.7 is not preserved in Fig.7, e.g., the green curve depicting and Fig.8, respectively. Although the physical interpretation broadside radiation has two almost overlapping points around of these two problems is different, they are both solved D 8 and (Z /R )δ 105. These two points also have ≈ 0 s ≈ using the same eigenvalue problem and have identical current close to orthogonal current densities as seen by the insets densities, i.e., the optimal currents were found using (12) and correspond to cases where the eigenvalue problem (24) which is identical to (24) without the X-term. Consider, has degenerate eigenvalues. For these cases we use linear e.g., the blue curve depicting end-fire radiation along the combinations between the eigenvectors to span the Pareto short side. The normalized dissipation factor is monotonically curve [37]. increasing with D from approximately 10 for D 2 to The minimum dissipation factor [39], [51], [38] is lower 107 for D 25 showing that an increased directivity≈ comes than the dissipation factor obtained from the α case for with a high≈ cost in losses. The corresponding blue curve in electrically large structures. These limit cases→ are ∞ connected Fig.8 decreases monotonically with the surface resistivity Rs by reformulating the problem (23) by either minimizing the 8 from G 22 for Rs = 10− Z0 to G 0.1 for Rs = Z0. The ohmic losses or maximizing the radiated power. Minimization ≈ ≈ 6 4 2 current density is depicted for R 10− , 10− , 10− Z of ohmic losses subject to fixed radiation intensity and radiated s ∈ { } 0 7 power is 104 H plate, ka = 0.5 minimize I RΩI plate, ka = 2 subject to IHXI = 0 sphere, ka = 0.5 (25) 103 sphere, ka = 2 IHUI = 2P lb H

I RrI = 2Pr, δ/δ

, a 102 which is relaxed to lb

H Q/Q minimize I RΩI (26) 1 H 10 subject to I (νX + αU + Rr)I = 1, where again the right-hand side is re-normalized to unity. 100 2 4 6 8 10 B. Trade-off Between Q-factor and Directivity D Superdirectivity is also associated with narrow bandwidth and high Q-factor [15], [42]. Adding constraints on the stored Fig. 9. Lower bounds on dissipation (solid lines) and Q-factors (dashed lines) for prescribed directivity D normalized with respect the lower bounds. energy to the optimization problem (23) results in the opti- The results were calculated for a spherical shell of radius a and a rectangular mization problem plate of side aspect ratio 2 : 1. The electrical size used is ka ∈ {0.5, 2} and the currents are self-resonant. maximize IHUI subject to IHXI = 0 where L is the order of the spherical modes and N H (27) DoF I (Xe + Xm)I = 2Q degrees of freedom, i.e., total number of modes [12]. The H I RΩI = δ maximum gain is related to the size of an antenna aperture ka H I RrI = 1, by a cut off limit for modes L = ka [49], but should be corrected for ka < 1 as L 1, see also [12]. This spherical where Xe + Xm = k∂X/∂k are matrices used to determine mode expansion is most suitable≥ for spherical geometries but the stored energy [52], [36]. Forming linear combinations overestimates the number of modes for other shapes. between the constraints is used to determine the Pareto front In order to take a specific shape of an antenna into account, and analyzing the trade-off between directivity, Q-factor, and the modes maximizing the radiated power Pr over the lost dissipation factor. power PΩ, i.e., those minimizing dissipation factor δ, are found Although (27) can be used to analyze the trade-off, it is from an eigenvalue problem [4], [39], [38] as illustrative to focus on the constraints on the dissipation factor and Q-factor separately. Dropping the constraint on the ohmic RrIn = %nRsΨIn, (30) losses reduces (27) to the problem of lower bounds on the where RΩ = RsΨ was substituted on the right-hand side, and Q-factor for a given directivity [15] which is relaxed to 1 only modes with δn = %n− < 1 are considered here as well- maximize IHUI radiating. It can be seen in (30) that the eigenvectors I do (28) n H not change with the surface resistivity and only the eigenvalues subject to I (νX + α(Xe + Xm) + Rr)I = 1, have to be rescaled with Rs. Formula (30) can be simplified and solved analogously to (18) for fixed α. Here, α = 0 solely using weights the radiated power regardless of ohmic losses and 1 H H 1 2 increasing α starts to emphasize ohmic losses. The maximal eig(Rr, Ψ) = eig(SΥ− Υ− S ) = svd(SΥ− ) , (31) directivity (α = 0) is in general unbounded [2], [3] but has where we also used the factorization H based on a high Q-factor. Here, reformulations similar to (25) can be Rr = S S the spherical mode matrix ,[53], see AppendixA, and used to reach the lower bound on the Q-factor. S a Cholesky factorization H to reduce the compu- The trade-offs between directivity and dissipation factor Ψ = Υ Υ tational burden. The radiation modes in (30) produce an and Q-factor are compared in Fig.9 for a spherical shells expansion in modes with orthogonal far fields and increasing and a rectangular plate of size ka 0.5, 2 . The bounds dissipation factors. They also appear in the analysis of the are normalized with the lower bounds∈ { on} the dissipation eigenvalue problems for the radiation operator [54] and for and Q-factors for the structure. The stored energy matrices MIMO capacity problems. Notice, that for a spherical shell are transformed to be positive semidefinite for the Q-factor they form a set of properly scaled spherical harmonics. calculation [15]. Radiation modes (30) are evaluated for a rectangu- lar plate of side aspect ratio 2 : 1 and electrical IV. RADIATION MODESAND DEGREESOF FREEDOM sizes ka 0.1, 0.32, 1, 3.2, 10 and the normalized eigen- ∈ { } The maximum gain was expressed by Harrington in spher- values %nRs are depicted in Fig. 10. The low-order modes ical mode expansion as [4] are emphasized in the inset, where it is seen that the modes N appear in groups with similar amplitudes for small ka. This is G = L2 + 2L = DoF , (29) H 2 confirmed via spherical mode expansion, see AppendixF, for 8

0 10 100 of designed and manufactured antennas and scatterers with respect to the fundamental bounds. Together with the previ- 2 10− ously published bounds on Q-factor and radiation efficiency, 10 6 − 4 10− this work completes the rigorous study of electrically small antenna limits and extends the fundamental bounds towards 10 6

n −

% 12 the electrical large antennas. Understanding of fundamental s 10−

R 8 10− bounds and knowledge in optimal currents reopen a call for

5 10 15 20 optimal antenna designs. 18 10− APPENDIX A 0.1 0.32 1 3.2 10 MATRIX REPRESENTATION OF USED OPERATORS 10 24 − 50 100 150 200 250 300 350 400 The matrices used in the optimization problems are con- n structed by expansion of the current density J(r) accord- ing (3) for r Ω. Fig. 10. Radiation modes for a rectangular plate of side aspect ratio 2 : 1 The far-field∈ matrix for direction rˆ reads [36] and electrical sizes ka ∈ {0.1, 0.32, 1, 3.2, 10}. F  F = eˆ , (32) Fhˆ which the rectangular plate supports only half of the spherical where eˆ = hˆ rˆ and hˆ = rˆ eˆ denote two orthogonal modes, e.g., x- and y- electrical and z-directed magnetic dipole polarizations with× elements × modes. This characteristic is most emphasized for electrically √ Z small structures and the increasing cost of higher order modes jk Z0 jkr1 rˆ Fe = − eˆ ψ (r )e · dS , (33) vanishes with increasing electrical size, e.g., the first ten modes ˆ,n 4π · n 1 1 for ka = 3.2 differ only in a factor of ten compared to 105 Ω for ka = 0.32. and similarly for Fhˆ . The radiation resistance matrix Rr and reactance ma- V. CONCLUSION trix X form the MoM electric field integral equation (EFIE) impedance matrix Z = R + jX of a structure modeled as Maximum gain and effective area for arbitrarily shaped r perfect electric conductor (PEC) [35]. antenna regions are formulated as quadratically constrained The ohmic loss matrix R = R Ψ for a region with a quadratic programs (QCQP) which are effectively solved as Ω s homogeneous surface resistivity, i.e., R , is given by the Gram low-rank eigenvalue problems. The approach is general and s matrix [55], defined as includes constraints on self-resonance and parasitic objects, Z such as reflectors and ground planes. Radiation modes are used Ψmn = ψ (r) ψ (r) dS. (34) to interpret the results and simplify the numerical solution of m · n the optimization problems. Ω The results are illustrated for a variety of shapes, electrical The expansion matrix between basis functions used and sizes ranging from subwavelength objects to objects hundreds spherical waves reads [53] of wavelengths long, and resistivities covering a wide range p Z S = k Z u(1)(kr) ψ (r) dS , (35) from superconductivity to lossy resistive sheets. Plotting the υn 0 υ · n maximal gain versus electrical size reveals three regions. Ω Dipole and Huygens sources dominate in the electrically small (1) where uυ denotes the regular spherical vector waves with region, where the gain depends strongly on the resistivity index υ [48]. The matrix S is a low-rank factorization of the and whether self-resonance is enforced or not. The effect H radiation resistance matrix Rr = S S. of self-resonance diminishes as the electrical size approaches a wavelength. End-fire radiation dominates over broadside APPENDIX B radiation for objects of wavelength sizes. This changes in QCQP the limit of electrically large objects where the effective area Maximum gain for self-resonant currents is determined from is proportional to geometrical cross-section and broadside a QCQP [34], [56] of the form (16) radiation dominates over endfire radiation. Superdirectivity is analyzed from the perspective of de- maximize IHUI termining the trade-off between directivity and efficiency. subject to IHRI = 1 (36) Here, it is shown that the problem of maximum gain for a IHXI = 0, given resistivity is solved by the same eigenvalue problem as minimum dissipation factor for a given directivity. Moreover, where U = UH 0, R = RT 0, and X = XT being numerical results suggest that the increase in the dissipation indefinite. This formulation can be relaxed to a dual problem and Q-factor are similar for superdirectivity. minimize maximize IHUI, The results presented in this paper are of general interest ν I (37) as they can be utilized to evaluate the actual performance subject to IH(νX + R)I = 1, 9

for all currents I and νl = ν/µ. Here, it is seen that 1 10− νlX + R 0 and inductive = 0 capacitive  H H

I XI > 0 XI I XI < 0 H

H I UI resonant I (42) µ H min max eig(U, νlX + R) ≥ I (νlX + R)I ≥ νl G 10 2 − ν2 ν4 ν3 ν1 and hence the solution of the Lagrangian dual in (40) is similar upper bound on G to the solution (38) of the relaxation (37). The only difference H is in the range for νl that is a subset of (49) due to the I UI range for G term in the right-hand side of (41). However, self-resonant νmin νopt νmax solutions of (24) satisfies (41). Semidefinite relaxation is 10 3 − 12 10 8 6 4 2 0 another standard relaxation technique [34] for the QCQP (36). − − − − − − ν

Fig. 11. Solution of the QCQP (36) using the dual formulation (38). The range [νmin, νmax] ≈ [−13.7, 0.02] for the dual parameter ν is determined APPENDIX D in AppendixE by (49) and the optimal parameter value νopt ≈ −8.74 NUMERICAL EVALUATION OF QCQP is determined from the sequence νn, n = {1, 2,...} using the bisection algorithm [46]. In this paper, the implementation is as follows. We use the eigenvalue problem (37) together with the factoriza- analogously to the analysis in Section II-B with the solution tion U = FHF due to its simplicity and computational ef- minimize max eig(U, νX + R). (38) ficiency. The computational complexity is dominated by the ν 1 H solution of the linear system (νX + R)− F which requires The range ν [νmin, νmax] is restricted such that of the order N 3 operations for direct solvers, where N is the ∈ νX + R 0, (39) number of basis functions, cf. (3). Here, we also note that the  additional computational cost of using multiple directions F and an efficient procedure to find νmin and νmax is outlined is negligible. For electrically large structures we use iterative in AppendixE. algorithms to solve the linear system [58]. The rectangle The minimization problem (38) is solved iteratively using in Fig.4 was, e.g., solved iteratively using a matrix-free a line-search algorithm, e.g., the bisection algorithm [46], FFT-based formulation [43] using N 4.2 106 unknowns where also the derivative (19) is used, see Fig. 11 showing for ka 103. The fast multipole method≈ (FMM)· and similar the optimization setup. Note that the Newton algorithm [34] techniques≈ can also be used [43] to reduce the computational can be used if the Hessian is evaluated as, e.g., in the case burden. with partial gain [36]. Whenever possible, symmetries are used to simplify the The explicit form of the derivative (19) also shows that the solution by separating the eigenvalue problem into orthog- derivative is zero for the optimal value if the eigenvalue νopt onal subspaces which are solved separately and combined depends continuously on as the derivative changes sign ν analytically [38]. This reformulates the optimization problems around . Hence, the solution to (38), , at the extreme νopt Iopt into block diagonal form, where each block corresponds to a point is self resonant H and satisfies the νopt IoptXIopt = 0 subspace. The problem is further simplified for cases where second constraint in the QCQP (36). This implies that the some of the subspaces do not contribute to the radiation duality gap is zero and that the QCQP (36) is solved by intensity in the considered direction as, e.g., for radiation in its dual (38). Moreover, non-degenerate eigenvalues depend the normal direction kˆ = zˆ for the rectangle in Fig.4, where continuously on parameters [57] so the problem is solved for currents with odd inversion symmetry J(r) = J( r) do this case. For a treatment of modal degeneracies and other not contribute and similarly for the cylinder in Fig.− 5 −where implementation issues, see AppendixD. azimuthal Fourier indices m = 1 do not contribute. For these | | 6 APPENDIX C cases, the currents in the non-contributing subspace can only ALTERNATIVE SOLUTIONSTO QCQP be used to tune the currents into self resonance. Hence, they are quiescent in the externally tuned case (8) and determined The Lagrangian dual [34] is convex and offers an alternative by the eigenvectors associated with the largest eigenvalues of approach to solve the QCQP (36). It is given by the semidef- the eigenvalue problems in (20), where the matrices X, R , inite program (SDP) r and RΩ are restricted to the non-contributing subspace. minimize µ, Expansion in radiation modes (30) is also useful in the so- subject to U + νX + µR 0, (40) lution of the maximum gain optimization problem (13), which −  1 H µ 0, ν R, contains the solution of the linear system (Rr + RsΨ)− F ≥ ∈ and often is solved for many values of Rs as in Section III. which can be solved efficiently [34]. The semidefinite con- H H Using Rr = S S and RΩ = RsΥ Υ together with the sin- straint in the Lagrangian dual (40) can be written H 1 gular value decomposition (SVD) UΣV = SΥ− reduces IH(νX + µR)I = µIH(ν X + R)I IHUI (41) the inversion of the linear system to inversion of a diagonal 1 ≥ 10

κmin κmax 106 κ 8 10− Ω/ 0 6 5 10− Ω/ 10 4 10− Ω/ a Indefinite PSD 2 ν 10− Ω/ 104 νmin νmax 1 Ω/ 0 Harrington GO 3 Fig. 12. Determination of the range of ν : νX + R  0; eigenvalues κ are D 10 solutions to XI = κRI. 102 matrix, i.e., 101

1 H H 1 0 (Rr + RΩ)− = (S S + RsΥ Υ)− 10 3 2 1 0 1 2 3 10− 10− 10− 10 10 10 10 1 H H 1 1 H = Υ− (Υ− S SΥ− + Rs1)− Υ− ka 1 H H H 1 H = Υ− (V− Σ ΣV + Rs1)− Υ− Fig. 13. Resulting directivity from the maximum gain for a spherical shell 1 H 1 H H n = Υ− V(Σ Σ + Rs1)− V Υ− , (43) with surface resistivity Rs ∈ {10− } Ω/ for n = {0, 2, 4, 6, 8} in Fig.1. Dotted lines depicts directivities D ∈ {3/2, 3, 11/2, 8, 23/2} associated with where 1 denotes the identity matrix. The computational cost of the lowest order modes. sweeping the maximum gain versus Rs is hence traded to com- 4 1 2 2 10 putation of the SVD of SΥ− that only requires Ns N + N 8 10− Ω/ operations, where denotes the number of spherical modes. 3 6 Ns 10 10− Ω/ a 4 Note, that we also use that matrix Ψ is a sparse matrix 10− Ω/ 2 2 with approximately 3N non-zero elements that reduces the 10 10− Ω/ 1 Ω/ computational cost to compute . Υ 101 δ 100 APPENDIX E 1 DETERMINATION OF νmin AND νmax 10−

2 The task here is to find a range of ν, delimited by νmin 10− T T and νmax, such that (39) holds, R = R 0, and X = X 3  10− 3 2 1 0 1 2 3 is indefinite. Equivalently, we can state that 10− 10− 10− 10 10 10 10 ka IH(νX + R)I 0 I (44) ≥ ∀ Fig. 14. Resulting dissipation factor from the maximum gain for a spherical n which can be reformulated to the Rayleigh quotient shell with surface resistivity Rs ∈ {10− } Ω/ for n = {0, 2, 4, 6, 8} in Fig.1. IHXI ν 1 I. (45) IHRI ≥ − ∀ APPENDIX F First, consider the case with ν > 0, for which the Rayleigh MAXIMUM GAINFORA SPHERE quotient satisfies The optimization problems (8) and (16) are solved ana- lytically for spherical structures using spherical waves [48]. IHXI IHXI 1 min = min eig(X, R) = κmin − (46) The matrices in the eigenvalue problems (13) and (18) IHRI I IHRI ν ≥ ≥ are diagonalized for spherical modes offering closed-form that implies the upper limit of the interval solutions of the eigenvalue problems. The radiation resis- (1) tance and ohmic loss matrices have elements (ka R (ka))2 1 lτ ν νmax = − , (47) and Rs, respectively, giving normalized dissipation fac- ≤ κmin (1) 2 tors δlτ /Rs = (ka Rlτ (ka))− , where l is the order of (p) where κmin is the smallest eigenvalue, see Fig. 12. the spherical mode, τ the TE or TM type, and Rlτ ra- Second, consider the case ν < 0. Analogously to (46) we dial functions [59]. Radiation dominates for modes with (1) 2 get (ka Rlτ (ka)) > Rs. This resembles (29) with the obser- vation that the radial functions are negligible for ka l. IHXI IHXI 1 The directivity associated with the maximum gain and max = max eig(X, R) = κmax = − (48) IHRI ≤ I IHRI ν effective area in Fig.1 is depicted in Fig. 13. The directiv- ity increases stepwise as additional modes are included. In and the range is given by (47) and (48) as the electrically small limit, the self-resonant case combines 1 1 electric and magnetic dipoles to form a Huygens source − = νmin ν νmax = − . (49) with directivity . The radiation efficiency is, however, κmax ≤ ≤ κmin D = 3 11

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Schab, “Trade-off between antenna Mats Gustafsson received the M.Sc. degree in En- efficiency and Q-factor,” Lund University, Department of Electrical and gineering Physics 1994, the Ph.D. degree in Electro- Information Technology, P.O. Box 118, S-221 00 Lund, Sweden, Tech. magnetic Theory 2000, was appointed Docent 2005, Rep. LUTEDX/(TEAT-7260)/1–11/(2017), 2017. and Professor of Electromagnetic Theory 2011, all [39] L. Jelinek and M. Capek, “Optimal currents on arbitrarily shaped from Lund University, Sweden. surfaces,” IEEE Trans. Antennas Propag., vol. 65, no. 1, pp. 329–341, He co-founded the company Phase holographic 2017. imaging AB in 2004. His research interests are in [40] C. Ehrenborg and M. Gustafsson, “Fundamental bounds on MIMO scattering and antenna theory and inverse scattering antennas,” IEEE Antennas Wireless Propag. Lett., vol. 17, no. 1, pp. and imaging. He has written over 90 peer reviewed 21–24, January 2018. journal papers and over 100 conference papers. 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Operations Research and Financial Engineering. New York, NY: Springer-Verlag, 2006. [47] P. Lancaster, “On eigenvalues of matrices dependent on a parameter,” Numerische Mathematik, vol. 6, no. 1, pp. 377–387, 1964. [48] G. Kristensson, Scattering of Electromagnetic Waves by Obstacles. Edison, NJ: SciTech Publishing, an imprint of the IET, 2016. Miloslav Capek (M’14, SM’17) received his M.Sc. [49] R. F. Harrington, Time Harmonic Electromagnetic Fields. New York, degree in Electrical Engineering and Ph.D. degree NY: McGraw-Hill, 1961. from the Czech Technical University, Czech Repub- [50] M. Gustafsson, D. Tayli, and M. Cismasu, Physical bounds of antennas. lic, in 2009 and 2014, respectively. In 2017 he was Springer-Verlag, 2015, pp. 1–32. appointed Associate Professor at the Department of Electromagnetic Field at the CTU in Prague. [51] H. L. Thal, “Radiation efficiency limits for elementary antenna shapes,” He leads the development of the AToM (Antenna IEEE Trans. Antennas Propag., vol. 66, no. 5, pp. 2179 – 2187, 2018. Toolbox for Matlab) package. His research interests [52] G. A. E. Vandenbosch, “Reactive energies, impedance, and Q factor of are in the area of electromagnetic theory, electrically radiating structures,” IEEE Trans. Antennas Propag., vol. 58, no. 4, pp. small antennas, numerical techniques, fractal geom- 1112–1127, 2010. etry and optimization. He authored or co-authored [53] D. Tayli, M. Capek, L. Akrou, V. Losenicky, L. Jelinek, and M. Gustafs- over 75 journal and conference papers. son, “Accurate and efficient evaluation of characteristic modes,” IEEE Dr. Capek is member of Radioengineering Society, regional delegate of Trans. Antennas Propag., vol. 66, no. 12, pp. 7066–7075, 2018. EurAAP, and Associate Editor of Radioengineering. [54] K. R. Schab, “Modal analysis of radiation and energy storage mecha- nisms on conducting scatterers,” Ph.D. dissertation, University of Illinois at Urbana-Champaign, 2016. [55] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis. Cambridge University Press, 1991. [56] J. Park and S. Boyd, “General heuristics for nonconvex quadratically constrained quadratic programming,” arXiv preprint arXiv:1703.07870, 2017. [57] T. Kato, Perturbation Theory for Linear Operators. Berlin: Springer- Verlag, 1980. [58] Y. Saad, Iterative Methods for Sparse Linear Systems. Boston: PWS Publishing Company, 1996.