Maximum Gain of Lossy Antennas Without and With Q-Bounds

We have recently formulated a maximum bound of superdirectivity of self-resonant antennas for a given Q (inverse of frequency bandwidth). This article complements the above work treating the influence of the losses. The problem is faced by assuming small losses in terms of surface resistance over the metalized surface of the minimum sphere circumscribing the antenna. The final closed-form formula shows that the maximum gain is obtained by a summation that resembles the well-known Harrington’s sum for maximum directivity, except that the expansion coefficients are weighted by the radiation efficiency of each spherical harmonic. The formulation is then generalized to the case of self-resonant antennas, providing a tighter bound for any losses. For small antennas, we provide a simple interpretation of the field corresponding to the maximum gain in terms of dipolar and quadrupolar source contributions, weighted by the appropriate efficiency, offering a physical insight into the phenomenon. The formulation is then extended to also account for a Q-bound, deriving a final series expression as a function of the loss resistance and the antenna electrical size. This expression seamlessly merges to the previously derived Q-bounded maximum directivity as losses tend to zero and converges to Q-unbounded maximum gain for very large values of Q.

Abstract-We have recently formulated a maximum bound of superdirectivity of self-resonant antennas for a given Q (inverse of frequency bandwidth).This article complements the above work treating the influence of the losses.The problem is faced by assuming small losses in terms of surface resistance over the metalized surface of the minimum sphere circumscribing the antenna.The final closed-form formula shows that the maximum gain is obtained by a summation that resembles the well-known Harrington's sum for maximum directivity, except that the expansion coefficients are weighted by the radiation efficiency of each spherical harmonic.The formulation is then generalized to the case of self-resonant antennas, providing a tighter bound for any losses.For small antennas, we provide a simple interpretation of the field corresponding to the maximum gain in terms of dipolar and quadrupolar source contributions, weighted by the appropriate efficiency, offering a physical insight into the phenomenon.The formulation is then extended to also account for a Q-bound, deriving a final series expression as a function of the loss resistance and the antenna electrical size.This expression seamlessly merges to the previously derived Q-bounded maximum directivity as losses tend to zero and converges to Q-unbounded maximum gain for very large values of Q.Index Terms-Antenna bandwidth, antenna efficiency, quality factor, superdirectivity, super-gain.

I. INTRODUCTION
T HE relationship between super-gain and superdirectiv- ity has been a highly debated issue in antenna theory, prompting extensive research and numerous publications.Many papers have been published since the 1950s [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22] aiming to provide guidance to antenna designers on achieving the maximum directivity and gain compatible with a given space constraint.Assuming sources fitting inside a minimum sphere of radius r min , the maximum directivity D max can be found as a function of r min as suggested by Harrington [5].His method is based on the expansion of the radiated field in a finite number of spherical wave (SW) modes excited on the minimum sphere and on the maximization of the directivity with respect to the coefficients of the expansion.This procedure leads to D max = N max n=1 (2n + 1) = (N max ) 2 + 2N max , where N max ≥ 1 is the maximum polar index of the SWs that contribute to the far field for the given minimum sphere.However, there are antennas that exhibit directivities larger than the value suggested by Harrington.These antennas are occasionally referred to as superdirective antennas.However, achieving such a high directivity often comes at the expense of a narrow bandwidth and a low efficiency.The Harrington process does not explicitly impose any constraints on bandwidth and losses, and the bound of directivity is obtained just by imposing a truncation of the series at the number of degrees of freedom (DoFs) of the field.It is not possible, indeed, to establish a rigorous limit of superdirectivity without introducing a limit on bandwidth or on losses.Recently, a limit of superdirectivity constrained by a given quality factor Q (inverse of relative bandwidth) has been provided by Passalacqua et al. [1].
However, work [1] does not address the extent to which small losses can affect the performance of superdirective antennas.Indeed, it is widely acknowledged that superdirectivity does not always translate into "super-gain," especially in cases involving high Q-factors.The primary objective of this article is to examine this specific aspect by assessing the impact of losses on the maximum super-gain.
In [22], a fundamental bound on antenna gain is found by solving a problem of convex optimization of the current density.The optimal current distribution is found expressing the antenna gain and the product bandwidth-gain as a quadratic form of the corresponding matrix operators.This procedure leads to the definition of a convex optimization problem, which is finally solved via an eigenvalue problem.However, the results in [22] do not explicitly impose frequency bandwidth maximization in the limit of maximum gain.
We articulate the presentation of this article as follows.Section II briefly summarizes the results obtained in [1].Section III presents a derivation of the maximum antenna gain without bandwidth limitations.A closed-form formula is derived and compared with the numerical approach presented in [22].Section IV focuses on small antennas, providing an interpretation in terms of dipolar and quadrupolar contributions and simple approximations valid for maximum gain smaller than 7. Section V formulates the gain maximization problem by introducing a limit on the total Q of the antenna and derives new exact analytical expressions.Section VI draws some conclusions.

II. MAXIMUM DIRECTIVITY WITH Q-BOUNDS
Recently, a limit of superdirectivity constrained by a given quality factor Q (inverse of relative bandwidth) has been provided by Passalacqua et al. [1] as where ξ 0 is the value that minimizes the series ] and Q n are the quality factors of each individual harmonic according to Fante's definition for self-resonant antennas [2].These correspond to , where Q ′ n and Q ′′ n are defined in Appendix C and will be introduced later.The analytical form (1) is obtained by applying the Lagrange multipliers method to a convex problem of directivity maximization bounded to a given constant quality factor Q. The value of ξ that minimizes the series has been approximated in [1], leading to simple closed-form expressions for certain values of the antenna size, as given in the following: ( The two terms kr 1 = 0.4(Q −1 + 2Q −1/3 ) and kr quad = 1.42(Q + Q −1/5 ) correspond to the values at which the maximum directivity is 3 (Chu limit) and 7.5, respectively.

III. MAXIMUM GAIN WITHOUT Q-BOUNDS
We present here a formulation for the maximum gain without Q-bound.This formulation resorts to spherical wave expansion (SWE), using the notation of Hansen [9], summarized in Appendix A, for the representation of the electric field.According to this notation, the polar index n refers to the order of the Hankel functions, the index m refers to the azimuthal angular wavenumber, and the index s = 1 and 2 denotes the TE and TM polarization with respect to the radial direction r , respectively.Whenever convenient, we compact the notation, using the single index i, which renumbers the term of indexes s, m, and n by the rule i = 2(n(n + 1) + m − 1) + s.

A. Equivalent Currents and SW Radiation Resistances
The Love formulation of the equivalence theorem is first applied to the smallest spherical surface which includes all the sources [Fig.1(a)].The equivalent electric and magnetic currents radiate zero field inside the surface.The external field is the same as that provided by magnetic currents radiating on a perfectly conducting sphere of radius r min , as predicted by the Schelkunoff formulation [see Fig. 1(b)].In the latter case, the induced electric currents J over the conducting sphere are the same as the electric currents J in Fig. 1(a).
In Appendix A, the conventional SWE of the radiated field is presented.It is, however, convenient writing the SWE of the field in a form that is related to the radiation resistance of the individual harmonics.Due to the orthonormality of the SWs, one has , where J i = I i f i is the electric current of the ith SW harmonics over the minimum sphere and the orthonormal functions f i are defined in Appendix A. |I i | represents the mean square over the minimum sphere of the electric currents of the ith SW harmonic.The radiated power can be expressed in terms of radiation resistance R (s)  rad,n , associated with the ith harmonic, as P r = (1/2) i R (s)  rad,n |I i | 2 .This radiation resistance does not depend on the azimuthal index m, but only on the polar index n and on the polarization index s, and it can be explicitly obtained through the spherical Hankel's function of the second type h (2)  n (kr min ), i.e., , for s = 2 (TM). ( Note that these functions have a weak dependence on kr min for n > kr min and tend to ζ for n ≫ kr min .We also observe that even if P r = (1/2) i R (s) rad,n |I i | 2 involves only the electric current coefficients I i , it represents the total power associated with both the equivalent electric and magnetic current radiating together in free space [Fig.1(a)] or, equivalently, the power radiated by the magnetic current alone on top of the perfect electric surface [Fig.1(b)], on which it induces the electric current.This formulation brings to (3) as a definition of the radiation resistance and makes the radiated power congruent with a zero field (and zero total energy) inside the minimum sphere.Considering instead only the contribution of the electric currents radiating alone in free space leads to a different expression of radiation resistance as well as a nonzero value of field inside the surface.This aspect will be investigated further in Section III-F.
It is seen that for small values of kr min , R (2)  rad,1 and R (2)   rad,2 (TM) go to zero as (kr min ) 2 and (kr min ) 4 , respectively, while R (1)  rad,1 and R (1)  rad,2 (TE) go to zero as (kr min ) 4 and (kr min ) 6 , respectively.The plot of radiation resistance for n = 1, 2, 3, 4 as a function of the electrical size is shown in Fig. 2.

B. Gain Representation as a Function of the Current Coefficients
To estimate the gain, we assume that the magnetic currents of Fig. 1(c) radiate in the presence of a lossy conductor with square surface resistivity R .This resistance, also known as sheet resistance, has a value that depends on √ f , (for copper, it is approximately R = 2.82 × 10 −7 √ f ).We assume that Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.the electric currents induced on the conductor by the forced magnetic currents will not change significantly with respect to the currents on a PEC [Fig.1(b)].The total power P r + P (i.e., radiated plus dissipated power) is given by The gain is obtained in Appendix A as where K i are the normalized far-field functions defined in Appendix A. In deriving (5), we have assumed that the modal currents on the surface are not perturbed by the presence of the small losses.Due to the symmetry of the problem, the gain can be maximized in an arbitrary observation direction and with an arbitrary polarization.It is therefore not restrictive to assume (θ, φ) = (0, 0) as well as a θ -polarization.Therefore, K i in ( 5) can be substituted by its projection along θ evaluated at (θ, φ) = (0, 0); this projection depends only on the polar index n and will be hereinafter denoted by K n .This value is a complex number whose magnitude is Applying the Schwarz identity to the numerator of (5) yields where the equality is valid when the phase of K i is equal to the phase of I * i .

C. Maximum Gain for Antennas With External Tuning
Equation ( 6) is well suited to make a maximization of the gain in the absence of Q-bounds, but with an external tuning circuit for reaching the resonance.This is different from the case presented in [1], relevant to self-resonant antennas.In order to find the maximum value G 0 (|I q |) for any possible set of amplitudes I q , we impose ∂G 0 /∂|I q | = 0 for any q.This approach follows the method applied by Harrington [3] to maximize the directivity, but without imposing a priori a truncation on the harmonic series, which was necessary in Harrington's process to get a finite result.In Appendix B, it is demonstrated that the solution for the maximum gain where η (TE,TM) n are the radiation efficiencies of any individual TE and TM SW harmonic and R (s) rad,n is defined in (3).It is noted that a similar formula, with a different efficiency definition, was derived in [19].This maximum gain is obtained with the current coefficients over the minimum sphere given in Appendix B. These correspond to SWE electric field coefficients (with Hansen's normalization) where and the superscripts TE and TM stand for s = 1 and s = 2, respectively.In deriving (8), we have used the relationship rad,n .The constant I 0 is arbitrary since it disappears in calculating G max .The radiation efficiency η = P r /(P r + P ) in condition of maximum gain is indeed given by Fig. 3 shows the values of maximum gain obtained by implementing (7) for various values of the surface resistance R (continuous lines).It also shows the directivity (dashed lines) obtained by dividing the maximum gain in (7) by the efficiency in (10).
The maximum gain and the corresponding directivity show inflection points at the maxima of the radiation efficiency.The lower frequency inflection point occurs at G max = 1.5.For smaller electrical sizes, the directivity saturates to 1.5.It is worth noting that this value corresponds to the directivity of an electric dipole and not to the directivity of a Huygens' source.This behavior was already underlined in [22], and its meaning stems from the higher radiation resistance exhibited by the first TM dipolar mode (electric dipole) for very small antennas.In contrast, the radiation resistance of the first TE mode is much lower and therefore more affected by the losses.It is also worth noting that the second inflection points of the maximum gain correspond to the fact that in the summation in (7) one more mode becomes significantly excited.

D. Value of Q on the Maximum Gain Curve
The Q-factor can be defined in two different ways depending on whether the antenna is self-resonant or it is matched by providing external reactive energy from a lossless circuit.In the first case, one has W E = W H and Q = 2ωW E /P r = 2ωW H /P r , where ω is the angular frequency, P r is the radiated power, and W E and W H are the electric and magnetic energies stored in the external region, respectively.For nonresonant antennas, one has Q = 2ωW E /P r for capacitive antennas and Q = 2ωW H /P r for inductive antennas.This definition assumes that an external energy has been added to the system to get the resonance.In both cases, the Q-factor can be interpreted as the reciprocal of the fractional bandwidth BW = 1/Q when it is larger than 10 [16].
The calculation of the stored energies W E,H of a general SWE is an old topic [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30].Essentially, the most used approaches are the ones provided by Chu [25], Collin and Rothschild [16], and Fante [2], the latter generalized to the case of arbitrary field internal to the minimum sphere in [28].In [16], the quality factor of each individually tuned SW is defined, for a unit radiated power, as where W E,n and W H,n are the electric and magnetic energy associated with the nth mode.Fante [2] introduced additional subdominant terms needed for the calculation of the Q-factor of a generic nonresonant antenna with unit power as The exact expressions of Q ′ n and Q ′′ n are reported in Appendix C. It is noted that they are independent of the azimuthal index m.A general expression of Q of a nonresonant antenna is given by where C ′ n and C ′′ n are the coefficients of the TM and TE modes, respectively.In the specific case we are dealing with, the antenna is not resonant due to the unbalancing provided by the different efficiency of TE and TM modes.This implies that till a certain dimension of the antenna the reactive electric Fig. 4.
Quality factor Q of SWE with coefficients associated with the maximum gain for various values of the loss resistance (values in Ohm).The two dashed-dotted lines represent energy dominates, and therefore, we should always apply the first of (13); therefore, using (8) leads to The value of Q on the maximum Gain curve as a function of the antenna size is presented in Fig. 4 for various values of the loss resistances.It is seen that the bandwidth (the inverse of Q tot ) becomes extremely small for small resistance, even if it corresponds to a very high maximum gain limit.Furthermore, all the curves tend asymptotically for small electrical size to the value of Q that satisfies the Chu limit, namely It is worth noting that for resonant antennas that maximize the directivity, the convergence as kr min → 0 is like This behavior is due to the lower efficiency of the TE modes in the quasi-static limit.

E. Maximum Gain for Self-Resonant Antennas
The solution in ( 7) is not self-resonant; namely, there should be an additional external reactive tuning circuit to reach the maximum gain.For self-resonant antennas (which is also the case treated in [1]), the maximum gain is obtained by using a method similar to the one given in Appendix B of [1], based on the dual (Lagrange) problem.We do not repeat here the process, showing only the result, that is, where ξ 0 is the value that minimizes the series G max (ξ ) in the interval Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.The gain in ( 16) is obtained with field coefficients where δ i is defined in ( 9) and the upper (lower) sign applies to TE (TM).The coefficients for the current expansion are obtained dividing by the square root of the individual TE and TM n-indexed radiation resistance.Comparisons between ( 16) and ( 7) are given in Fig. 5.The resonant antenna maximum gain is tighter than the one for an externally tuned antenna, especially for small antennas, and it drops rapidly to zero for a gain approximately equal to 3.This aspect has also been underlined in [22].

F. Comparison With Gustafsson-Capek Results
It is important to compare our result with the one obtained by Gustafsson and Capek [22].The formulation presented there is based on a method moments (MoM) applied to the of an arbitrary body and by a convex optimization procedure.They found the maximum super-gain by imposing maximization of the power intensity with constant radiated power for any coefficient of the MoM basis functions, assuming small losses on the metallic surface.
This procedure is quite general and can be applied to arbitrary shapes.Gustafsson and Capek [22, Fig. 1] apply the procedure to a spherical shape, using spherical modes as basis functions.In the externally tuned case, the convex optimization is carried out without conditions on the reactance of the MoM matrix.In the self-resonant case, they also impose a vanishing reactive average power through the imaginary part of the MoM matrix.We have reimplemented the convex optimization procedure of Gustafson-Capek for both cases, and we have found very close results (but not equal) to the results provided by ( 7) and ( 16).Fig. 6 shows the discrepancy between our results and the ones in [22] for the externally tuned case (a similar situation is found for the self-resonant case).Although this has initially puzzled us, finally, we have found the reason for it.We found indeed that the results from our procedure become identical to those obtained in [22] if one uses Fig. 6.Comparison between the maximum gain for the externally tuned case of our formulation (continuous lines) and the formulation in [22] (dashed lines), also obtained by using ( 19) in (7).
the radiation resistance produced by electric currents only.More specifically, we can reproduce exactly the results in [22], changing the SW radiation resistances in (7) with the following expression: (kr min j n (kr min ))

2
, for s = 1 (TE) These radiation resistances are the ones corresponding to harmonics of electric currents only radiating in free space.It should be noted that using the electric currents only provides a field (and energy) different from zero inside the minimum sphere, which leads to a slightly tighter value of the maximum gain.
G. Distribution of the Coefficients of the Currents Fig. 7 shows the histogram of the n-indexed current coefficients' amplitude corresponding to the maximum gain for the externally tuned case in (8).The coefficients are normalized in such a way to have unitary radiated power.Fig. 7(a) and (b) correspond to different values of surface resistance (R = 1 and R = 0.1 , respectively).Both cases of externally tuned coefficients and self-resonant coefficients are reported.The calculation is carried out for kr min = 2.It can be seen that the maximum coefficient amplitude is given at the index at which the ohmic loss resistance approaches the resistance of the harmonic, that is, R ≈ R (1,2)  rad,2 , namely, when the efficiency of the harmonic is 50%, i.e., η (TE,TM) 2 ≈ (1/2).It is apparent that for smaller values of the loss resistance, the optimal current coefficients are concentrated on superreactive harmonics, and it means SWs with polar index larger than kr min = 2.This makes it challenging to achieve their excitation on the minimum sphere.Consequently, the bound described in ( 7) is difficult to achieve.It is also apparent from Fig. 7 that the TE optimal coefficients for the resonant case are higher, aligning with their reduced efficiency for small antennas.Furthermore, lower losses correspond to larger optimal coefficient values, as noted by the disparity in the vertical scales between Fig. 7(a) and (b).

IV. EXTERNALLY TUNED SMALL ANTENNAS
For small antennas, the interpretation of the previous formulas becomes interesting.We limit the analysis to the externally tuned case, but similar considerations can be carried out for the self-resonant antennas.We first observe that the nth TE and TM spherical harmonics with azimuthal index m = ±1 have different coefficients, and this implies a different impact on both bandwidth and gain.This is in contrast with what happens for the maximum directivity, namely, in the absence of losses [1].In the latter case and for n = 1, the maximum directivity field can be interpreted as the one produced outside the minimum sphere by an elementary Huygens' dipole (HD) located at the origin.By duality, the energy density of the HD is balanced outside the minimum sphere.In the same lossless case, the combination of the SW harmonics for n = 2 provides the field of a Huygens' quadrupole (HQ) combined with the one of a dual vertical quadrupole (DVQ) [1].In the presence of losses, the maximum gain is obtained by unbalanced coefficients, which renders the antenna nonresonant.This means that outside the minimum sphere, electric and magnetic energies are not balanced, thus requiring the use of ( 13).In particular, the coefficients associated with the electric and magnetic dipoles and the quadrupoles field are weighted by their efficiency, as prescribed by the general form (8). In the far zone, this leads to the electric field proportional to where In particular, h (TM) 1 are the electric far-field pattern of an x-directed electric and y-directed magnetic dipole, respectively, and h (TM) 2 and h (TE) 2 are electric and magnetic quadrupoles, respectively.The latter are obtained by in phase combination of x-directed and z-directed electric quadrupoles, and y-directed and z-directed magnetic quadrupoles, respectively.The situation is illustrated in Fig. 8.
The maximum gain for this approximation is given by which is associated with a total quality factor Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.In ( 16)-( 18), the explicit expressions of the SW efficiency posing x = kr min are x 4 where the first equality is derived directly by (3); the approximation at the last members are valid for any value of x from 0 to infinity and for R < 10 −1 with maximum relative error e max limited by the following values: e max < 2 × 10 −4 , e max < 8 × 10 −3 , e max < 8 × 10 −3 and e max < 2 × 10 −3 for ( 27)- (30), respectively.The exact values of the quality factors are given by It is interesting to compare the maximum gain obtained considering the two source contributions in (25) and the full series in (7).The directivity is also reported in the same plot.It is worth noting that the insertion in (25) of the approximation in the last expressions in ( 27)-( 30) does not practically provide numerical differences.For completeness, Fig. 10 shows similar results as those of Fig. 9 for self-resonant antennas.16) (continuous lines).Dotted lines: first two terms of the series in (16).Dasheddotted lines: directivities obtained by G max /η with the full series.

V. MAXIMUM GAIN WITH Q-BOUNDS
In [1], the exact formula for Q-bound and maximum directivity has been derived through a convex optimization for the radiation density bounded to have a unit power and a given Q.Following the same approach, we can obtain the Q-bounded super-gain applying Lagrange's multiplier method shown in Appendix C of [1].In obtaining the bounded maximum gain, we assume that the condition W E > W H is respected, so as to impose as a bound the first expression in (13).Since the procedure is like the one in [1], we omit the demonstration, and we present only the final expression, that is, where ν 0 is the values that minimize the series In (32), Q is the desired total quality factor of the antenna.It is worth noting that the above expression changes when W E < W H ; in such a case, in accordance with (13), it is sufficient to interchange In the implementation, one should therefore switch the formula at the resonance (W E = W H ). The field coefficients for the Q-bounded maximum gain are given by The curves obtained with (32) are compared with a numerical convex optimization obtained by expanding electric and magnetic currents in terms of small-domain basis functions (dotted lines); excellent agreement has been found.We should highlight that the intricate interplay between bandwidth and losses implies that, for a fixed antenna size, a larger required bandwidth may necessitate a slight increase in losses to minimize the disparity between maximum gain and maximum directivity.Eventually, the Q-bounded parametric curves of maximum gain gradually tend to the Q-bounded maximum directivity in (1) for vanishing losses [dashed-dotted lines in Fig. 11(a)-(c)], with level of admissible losses depending on the required Q.More precisely, the discrepancy between maximum directivity and maximum gain vanishes for smaller losses when Q is larger.This aspect is quantified in Fig. 12, where the percentage error ε = (D max − G max )/D max = 1 − η between the Q-bounded gain in (32) and the Q-bounded directivity in (1) has been calculated as a function of the antenna size for different values of R .Fig. 12(a)-(c) correspond to Q = 10, 100, and 1000, respectively.A more exhaustive analysis revealed that ε less than 7% (which corresponds to an overall radiation efficiency larger than 93%) is found for Q < 9/R 0.8 + 3.This means that in this range, one can calculate the maximum gain with the simple formula in (2) for maximum  directivity.The situation is illustrated with the help of a picture in Fig. 13.Finally, we should also emphasize the fact that the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
direct use of (32) may provide inconsistencies in the region Q < 9/R 0.8 + 3 since (32) is not valid when W E < W H , where one has to interchange Q ′ n and Q ′′ n to get the right gain maximization.

VI. CONCLUSION
This article has delineated various fundamental characteristics of the maximum gain exhibited by arbitrary antennas with specified electrical sizes.Analytical expressions for the maximum antenna gain, contingent upon losses and subject to Q-bounds, have been presented for both resonant and nonresonant (externally tuned) cases.The exact solution is given in terms of a series that should be minimized with respect to a single parameter.For smaller antennas, a concise closed form has been furnished, delivering accuracy up to a maximum gain of 7. In addition, an intriguing interpretation in terms of magnetic and electric dipoles has been provided.The findings showcased in this study have also unveiled the range of losses and Q values wherein superdirectivity corresponds to super-gain.

APPENDIX A SW EXPANSION
The equivalent electric and magnetic currents of the Love formulation [Fig. 1 s,m,n × r , respectively, where ζ is the freespace impedance, C s,m,n are the expansion coefficients of the electric field, and F (3)  s,m,n are the SW functions evaluated over the minimum sphere, as defined in Hansen's book [9], except for a different time dependency [which is here exp( jωt)].In this notation, the superscript "3" corresponds to the spherical Hankel of the second kind r -dependent function.The polar index n refers to the order of the Hankel function and the index m refers to the azimuthal angular wavenumber.The subscript s = 1, 2 denotes TE and TM polarization with respect to the radial direction r , respectively.
The proper normalization of the SW functions of Hansen and their orthogonality imply that the radiated power is given by the summation of the squared amplitude of the coefficients, P r = (1/2) s,m,n |C s,m,n | 2 .The expansion of the currents can be rewritten through the (θ, φ)-dependent functions f i = (1/r min )T s,m,n (θ, φ)(1/r min )r × F (3)  s,m,n × r /R (3)  s,n , where R (3)  s,n (kr ) are defined through the spherical Hankel functions [9].To compact the notation, we have used the index i that renumbers the term of indexes s, m, and n in a single index i = 2(n(n + 1) + m − 1) + s.
The far-field radiation density for unit solid angle can be written as U = lim r →∞ (1/2ζ )|r E| 2 = (1/8π )| i C i K i | 2 , where K i = 4πT i lim r →∞ kr R (3)  3−s,n .The same quantity can be expressed in terms of the electric current coefficients rad,n where R (s) rad,n is the radiation resistance of the individual harmonics given in (3).Using the above expressions, the antenna gain in direction (θ, φ) is given by (5).

APPENDIX B MAXIMUM GAIN COEFFICIENTS
The maximization of the function G 0 (|I q |) in ( 6) with respect to the coefficients |I q | is obtained by imposing the vanishment of the derivative with respect to |I q |, namely, The above is equivalent to which is respected if and only if K q R (s) rad,q = I q R (s) rad,q + R s (36) for any q.When substituted in ( 6), (36) leads to (7) and, together with the phase condition ̸ I i + ̸ K i = 0, yields the current coefficients rad,n R (1)  rad,n + R The relationship between current coefficients and electric fields coefficients |C i | = |I i | R (s) rad,n leads to (8).

APPENDIX C Q-FACTORS FOR SWS
The analytical expression of Fante's dominant and subdominant Q-factors of n-indexed SWs is given by x 2 (2n + 3)( j n (x) j n+1 (x) + y n (x)y n+1 (x)) (38) where x = kr min and h n , j n , and y n are the spherical Hankel of second kind, Bessel, and Neumann functions of order n, respectively.For n = 1 and 2, they assume the exact expressions in (31).

Fig. 1 .
Fig. 1.Application of the equivalence principle to the minimum sphere surface.(a) Equivalent surface electric and magnetic currents radiating in free space with zero field inside (Love formulation).(b) Magnetic currents radiating over a perfect electric conducting sphere.(c) Magnetic currents radiating over a metallic sphere with small losses.

Fig. 2 .
Fig. 2. Radiation resistance of individual harmonics for some values of n as a function of the normalized radius of the minimum sphere.

Fig. 3 .
Fig. 3. Continuous lines: maximum gain of nonresonant (externally tuned) antennas calculated for different values of R as a function of kr min .Dashed lines: corresponding directivity obtained by G max /η.

Fig. 5 .
Fig. 5. Maximum antenna gain calculated for different values of R as a function of kr min .Dashed lines: externally tuned antennas.Continuous lines: self-resonant antennas.

Fig. 7 .
Fig. 7. Histograms of current coefficients for TE (inductive, left-hand side) and TM (capacitive, right-hand side) harmonics; both externally tuned case (blue bars) and self-resonant case (red bars) are reported.(a) R = 1 .(b) R = 10 −1 .The maximum gain is in this case approximately equal for self-resonant and tuned cases.The amplitudes are normalized to have a unit radiated power (P r = 1 W).The maximum coefficients are obtained for n where the harmonics efficiency is about 50%.Note the difference of vertical scale for (a) and (b).

Fig. 8 .
Fig. 8. Elementary sources associated with the far-field pattern of dipolar (a) h (TM) 1 and (c) h (TE) 1 and quadrupolar (b) h (TM) 2 and (d) h (TE) 2 , contributions.Electric dipoles (TM) are denoted in blue with a single arrow and magnetic dipoles (TE) in red with a double arrow.The vertical doublet is aligned along x for electrical dipoles and along y for magnetic dipoles.

Fig. 9 .
Fig.9.Maximum gain calculated for different values of R as a function of kr min using the full series in (7) (continuous line) and the two terms approximation in (25) (dotted line).Dashed-dotted lines are the corresponding directivities obtained by G max /η by using the full series.

Fig. 9
shows this comparison for values R ∈ [10 −4 , 1 ].It is seen that the two expressions coincide below G max = 6 and agree till G max = 7.

Fig. 10 .
Fig.10.Maximum gain of self-resonant antennas calculated for different values of R (in Ohm) as a function of kr min using the full series in (16) (continuous lines).Dotted lines: first two terms of the series in(16).Dasheddotted lines: directivities obtained by G max /η with the full series.

Fig. 11 .
Fig. 11.Q-bounded maximum gain G max as a function of the antenna size calculated for (a) Q = 10, (b) Q = 100, and (c) Q = 1000 and different values of the ohmic losses resistance R ; the curves tend smoothly to the Q-bounded maximum directivity (black line) when the losses tend to zero.R = 0 corresponds to the maximum Q-bounded superdirectivity D max as in (1).

Fig. 12 .
Fig. 12. Percentage difference between maximum Q-bounded directivity and maximum Q-bounded gain for different values of the loss resistance.The green curves correspond to an efficiency larger than 93% in the overall range from the Chu limit.(a) Q = 10.(b) Q = 100.(c) Q = 1000.The horizontal scale starts from the Chu limit.

Fig. 13 .
Fig.13.Region of validity of the formulas for Q-bounded maximum gain with losses (32) and without losses (2).For Q < 9/R 0.8 + 3, (2) can be used with a maximum error of less than 7% for all antenna sizes.