Maximum Gain, Effective Area, and Directivity Mats Gustafsson, Senior Member, IEEE and Miloslav Capek, Senior Member, IEEE
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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 denote2 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- n (r) as [35] ing cavities [44], through values valid for copper at RF f g N (Rs 0:01 Ω=, f = 1 GHz), to poor conductors of surface X ≈ J(r) In n(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 (r^) FI 2 = IHFHFI; (4) arrays, scatterers, and other radiating systems. ≈ 2 j j 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(^) 4π H j j = 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(r^) 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 2 minimum ohmic losses or Q-factor.