Absorption efficiency and physical bounds on antennas

The all spectrum absorption e(cid:30)ciency appears in the physical bounds on antennas expressed in the polarizability dyadics. Here, it is shown that this generalized absorption e(cid:30)ciency is close to 1 / 2 for small idealized dipole antennas and for antennas with a dominant resonance in their absorption. Also, the usefulness of this parameter is analyzed for estimation of antenna performance. The results are illustrated with numerical data for several antennas.


Introduction
A new set of physical bounds on antennas was introduced in [47,16].These bounds relate the performance of the antenna to the electro-and magneto-static polarizability dyadics of a circumscribing geometry.This generalizes the classical bounds by Chu [3] for spherical geometries to geometries of arbitrary shape.The new bounds are valid for lossless and linearly polarized [4,6,7,16] and elliptically polarized [5] antennas.Moreover, the approach can be used to estimate the performances of many small antennas if the polarizabilities of the antennas are used instead of the circumscribing geometries [4,7,16].
The only parameter in the bound that depends on the dynamic properties of the antenna is the generalized (or all spectrum) absorption eciency, η.This is the generalization of the frequency dependent absorption eciency analyzed in [1] given by integration of the absorbed and total power, independently, over all wavelengths.
In [4,6,7,16], it is demonstrated that η is close to 1/2 for many small antennas that are connected to a frequency independent resistive load and matched at their rst resonance.This is motivated by the minimum scattering property that small matched antennas often possess, i.e., they scatter as much power as they absorb at the resonance frequency giving an absorption eciency of 1/2 at the resonance frequency [1,12].Here, it is shown that small idealized dipole antennas with a dominant rst single resonance have an all spectrum absorption eciency η 1/2.
The region around the resonance is minimum scattering but the contributions from regions away from the resonance scatter slightly more power than is absorbed giving a generalized (all spectrum) absorption eciency close to but less than 1/2.
Minimum scattering is a property that many non electrically small resonant antennas also possess.Numerical simulation results of common antennas, both electrically small and not small, verify the theoretical results.

Absorption eciency
The physical bounds analyzed in [4,6,7,16] are derived for single port, linearly polarized, reciprocal, and lossless antennas with the reection coecient Γ (k) and the directivity D(k; k, ê), where k denotes the free-space wavenumber, k the direction, and ê the electric polarization.The forward scattering sum rule [6] gives the antenna identity ∞ 0 where γ e and γ m are the electro-and magneto-static polarizability dyadics, respectively.The integral (2.1) is bounded in various ways to produce bounds for dierent applications, e.g., resonant and constant partial-realized gain in [4,6,7] and ultrawide band cases in [16].The resonant case is applicable for antennas with a dominant rst single resonance [7].It is given by where k 0 is the resonance wavenumber and Q denotes the Q-factor at the resonance, i.e., it has the half-power fractional bandwidth B ≈ 2/Q.
The polarizability dyadics in the right-hand sides of (2.1) and (2.2) are easily determined for the antenna or, as an upper bound, for an arbitrary circumscribing geometry 1 by the solution of the corresponding electro-and magneto-static equa- tions [4,6,7,16].This leaves the generalized absorption eciency, η, as the only quantity in the right-hand sides of (2.1) and (2.2) that depends on the dynamic properties of the antenna.It is an all spectrum measure of the absorption and scattering properties of the object, that is dened by where σ ext = σ a + σ s , σ a , and σ s denote the extinction, absorption, and scattering cross sections, respectively and λ = 2π/k is the wavelength.It is clear that 0 ≤ η < 1 for all objects as σ a ≥ 0 and σ s ≥ 0. In [4,7], it is observed that η ≈ 1/2 for many small antennas that are matched at a dominant rst resonance k 0 .This is partly explained by the fact that the absorption eciency σ a (k 0 )/σ ext (k 0 ) = 1/2 for minimum scattering antennas, i.e., small single mode antennas absorb and scatter the same amount of power at the resonance frequency [1,12].The weighting factor, 3) emphasizes the dynamics of the antenna for low wavenumbers.Thus, the lower the resonance frequency, the closer η is to 1/2 as the resonance region will dominate the integrals.As a consequence, the theory derived here is useful if the analyzed resonance has the lowest frequency.The contributions to η in (2.3) away from the resonance are small due to the fact that scattering dominates the behavior of the antenna in the regions where the mismatch is high.
Here, the case with an idealized lossless antenna that radiates an electric dipole mode is considered to explicitly determine η and illustrate how σ ext (k) and σ a (k) depend on the wavenumber around the resonance.A spherical dipole mode at the radius a has the impedance [3] Z TM = 1/(jωC) + jωL/(1 + jωL/η 0 ), where L = µ 0 a, C = 0 a, ω = kc 0 , the time convention e jωt is used, and 0 , µ 0 , c 0 , and η 0 denote the free space permittivity, permeability, speed of light, and impedance, respectively.
Evaluation of η in (2.3) requires a model of the extinction cross section, σ ext (k), that is consistent with (2.4).Consider a single port antenna with incoming signal u and outgoing signal v.The electromagnetic eld is expanded in incoming and outgoing spherical modes with coecients a and b, respectively.This gives the scattering matrix [11] Γ R T S where Γ is the reection coecient, R is an 1 × ∞ matrix with elements R n , T is an ∞ × 1 matrix with elements T n , and S is an ∞ × ∞ matrix with elements S mn .
For simplicity, order the modes such that the idealized dipole antenna radiates the rst mode, i.e., R n = T n = 0 for n > 1.
where k n denote the zeros of S 11 in Re{jk} > 0.
The extinction cross section is often expressed in the transition matrix.It is related to the S-matrix in (2.6), S, via T mn = (S mn − 1)/2.Consider an idealized dipole antenna that is resonant for k 0 a 1.The scattering from higher order modes is negligible for k 0 a 1 so the extinction cross section is well approximated with the dipole mode in this region.The extinction cross section from the dipole mode is hence approximated by where T 11 denotes the diagonal dipole element of the transition matrix [14].Consider the simplest possible case with a single zero k 1 .The value of k 1 is determined by (2.8) and (2.9) inserted into the low-frequency expansions [13] (2.10) The cross sections σ ext , σ a , and σ s are depicted in Fig. 2 for the same cases as in Fig. 1.
It is observed that the areas under the curves are concentrated to the resonances and that σ a (k) ≈ σ s (k) around the resonances for k 0 a = 0.25.
For minimum scattering, Re {T 11 (k)} = −1/2 in (2.9), we obtain the envelope σ ext (k)/a 2 ≈ 3π/k 2 a 2 = 3λ 2 /4πa 2 , also plotted in Fig. 2. The more dominant a resonance, the closer the obtained value of the extinction cross section is to this envelope at the resonance frequency.
The generalized (all-spectrum) absorption eciency (2.3) for the idealized dipole is nally determined by dk . (2.11) In Fig. 3, the generalized absorption eciency η is evaluated for the idealized dipole model (2.4) and (2.10) as well as various other parameter values on C, L, and R 1 .It is observed that η 1/2 for k 0 a 1.The deviation from 1/2 is due to the region with small σ s but negligible σ a for λ/a < 8 as seen in Fig. 2.
The particular impedance (2.4) is not crucial for this result.It is sucient that the contributions to the integrals in (2.3) are dominated by the region around the resonance k 0 and that σ a and σ ext have similar bandwidths and shapes.It is common to assume antennas with a single resonance structure [8] to relate the bandwidth with the antenna Q.Similarly, assume that the transition matrix element T 11 (k) has a single resonance at k 0 and is minimum scattering, i.e., S 11 (k 0 ) = 0 implying T 11 (k 0 ) = −1/2.The resonance model has complex valued poles at k ≈ ±k 0 and the shape of the classical Lorentz or resonance circuit [8,14] around the resonance, i.e., around the resonance wavenumber, see the k 0 a = 1/4 case in Fig. 2. For antennas with negligible σ a away from the resonance, e.g., the dipole model (2.4), the σ ext ≈ σ s contribution to the integral (2.3) away from the resonance gives η 1/2.

Numerical Examples
The above theoretical results have been analyzed for a number of geometries by numerical simulations; the numerical results show very good agreement with the theory.For each of the examples the approach was the same; rst we started with the design and simulation of a radiating structure using the Method of Moments (MoM) simulator in Eeld 2 .Then the radiation resistance at the rst resonance was used as a load at the feeding point in a forward scattering simulation, performed using the same software.The results of the antenna and scattering simulations have been used to numerically compute the theoretical parameters using Matlab.Note that not all the available digits are presented in the text as the numerical accuracy does not justify them.However the formulas are computed without truncation.

Folded Spherical Helix
We rst describe the results for the folded spherical helix [2] depicted in Fig. 4. It comprises a closed loop of perfectly electric conducting wire of radius R w = 2 mm that is folded on the surface of a sphere of radius 58 mm thus obtaining a structure with the radius of the smallest circumscribing sphere a = 62 mm.The structure has two arms of equal length (approximately l a = 646 mm) symmetric with respect to the z axis.
The rst step in the analysis is to simulate this structure with Eeld.The antenna parameters are determined with an ideal voltage source connected at point F (see Fig. 4).The resulting input impedance is plotted in Fig. in [4,6,7,16].At this frequency the antenna radiates a z dipole type pattern.
First we compute the D/(Qk 3 0 a 3 ) value using the simulation data from Eeld and the method proposed in [8,18] to approximate the Q factor.The computed values D = 1.5, Q = 43 and k 0 a = 0.38 result in the quotient D/(Qk 3 0 a 3 ) = 0.63.
The second step is to evaluate the right hand side in (2.2).Here, the polarizability dyadics reduce to the high contrast polarizability dyadic of the perfect electric conductor which is computed using a MoM algorithm (see e.g., [4]) that solves the electrostatic problem associated with the wire geometry.With ê = ẑ and only high contrast electric material present ê The generalized absorption eciency is computed from the Eeld simulation data with (2.3) and the denitions of absorption and extinction cross sections from [7,17].
After an integration of 5999 absorption and extinction cross sections samples taken equidistantly between 1 MHz and 3 GHz (see Fig. 6) we obtain η ≈ 0.51 and write: Moreover, the integrated extinction cross section is related to the polarizability of the structure, as stated in [17] i.e., The physical bounds in (2.2) create a link between the dynamic properties of the radiating structure and its static properties described by the electric and magnetic polarizability dyadics.Because many common antennas have a generalized absorption eciency of approximately 1/2 obtaining the bounds for an antenna reduces to a static problem of computing the polarizability dyadics for the geometry, which is easily solved using a Method of Moments algorithm.
It is very important to distinguish the geometry of the radiating structure from its smallest circumscribing sphere.The antenna can be optimized in the limit given by its own polarizability, e.g., the smallest circumscribing sphere of the helix in Fig. 4 has the radius a = 62 mm, which gives ẑ • γ ∞ • ẑ = 4πa 3 ≈ 3 • 10 −3 m 3 thus the maximum attainable value for the D/Q quotient is: D/(Qk 3 0 a 3 ) ≤ 1, whereas the wire structure simulated here has a maximum attainable value D/(Qk 3 0 a 3 ) ≤ 0.67.The presence of the (k 0 a) 3 term allows radiating structures to be directly compared even though they do not have the same size.It can be stated that the wire structure of the helix in Fig. 4 can only reach 67% of the best attainable performance of an antenna circumscribed by a sphere with equal radius.Hence, it is neccesary to use a structure with high polarizability to improve the performance, e.g., the polarizability of the spherical helix increases with the number of arms and with the wire radius.

Folded Dipole Array
The second considered structure is a folded dipole array [10].The dimensions in Fig. 8 are the following: l = 492.9mm, d = 470 mm, h = 40 mm and s = 65.8 mm.
The structure is assumed to be fed at point F with an ideal voltage source.All wires have a radius of R w = 4 mm thus simulating one possible realization of a simple and common array design using the same type of conductor for all the elements.The smallest circumscribing sphere has the radius a = 347 mm.
First we simulate the structure in transmission in order to obtain the impedance behavior in the frequency range of interest, see Fig. 9.The rst resonance with practical relevance is close to 233 MHz and, at this frequency the antenna has an input impedance of 59 Ω.We shall illustrate the bounds using the characteristics of the structure at this frequency.The far eld radiation pattern at this resonance consists of two linearly polarized ( x-direction) pencil beams in the broad sides, with a maximum directivity D = 2.6.The quality factor of this resonance is Q = 4.2 and with k 0 a = 1.7 we obtain the quotient D/(Qk 3 0 a 3 ) = 0.13.The high contrast polarizability for ê = x polarization has the value γ ∞,11 = 67.4• 10 −3 m 3 .We note here that the structure is also highly polarizable on the ŷ-direction but the ŷ-polarization does not contribute to the radiation because of the choice of feeding.
We now turn to the analysis of the cross sections and use 5999 absorption and extinction cross section samples from equidistantly spaced frequencies between 1 MHz and 3 GHz.Gathering the results, we rewrite (2.2) in numbers as: 0.13 ≤ 0.13 thus making this array a structure that is close to the optimal D/Q performance of the wire structure.The perfect matching is explained by the deviation of the structure from the assumed models for the Q-factor (Q 1) and the rst single and dominant resonance.Compared with a smallest circumscribing sphere, the array only reaches 13% of its D/Q performance.For evaluating the reliability of the generalized absorption eciency we integrate the extinction cross section over the wavelength and obtain the value 66.8•10 −3 m 3 which is less than 1% from the previous γ ∞,11 = 67.4•10−3 m 3 ; so the frequency interval is well chosen as to not signicantly deviate the resulted η from its correct value.

Meanderline
We now analyze a planar meander line antenna depicted in Fig. 11.The notations follow [7]: l 1 is the long and l 2 is the short dimension of the antenna, w is the metal width and h is the base meander height.We only analyze situations where the E eld is polarized along l 1 .
First we simulate the antenna in transmission and use for this purpose an ideal voltage source distributed along line F .The impedance behavior is shown in Fig. 12.
At low frequencies the antenna is a resonating structure with the rst resonance approximately at frequency 322 MHz; at high frequencies the input impedance shows very little variations both in the real and imaginary parts ultra wide band behavior.
The radiated pattern shows an almost dipole mode radiation at 322 MHz polarized along y with a directivity D = 1.6 and a radiation resistance of approximately 45 Ω.Small variations in the directivity aect the omnidirectionality with the maximum directivity obtained on a direction slightly deviated from z (θ ≈ 10 • , φ = 0 • ).The other parameters that characterize radiation at 322 MHz are: Q = 4.4 and k 0 a = 1.14 and the quotient D/(Qk 3 0 a 3 ) = 0.247.
Next, we use a MoM code to compute the static high contrast electric polarizability dyadic of the structure and obtain: γ ∞,22 = 14.667 • 10 −3 m 3 .This value serves as a reference for the integrated extinction cross section and thus as a reliability gure for the generalized absorption eciency.
The last step in the analysis consists of a forward scattering simulation that provides data for the cross sections of the structure.We use the same value for the feed load as the radiation resistance (approximately 45 Ω).The structure is excited with a plane wave impinging the structure from the direction of the maximum directivity (here θ = 10 • , φ = 0 • ).In total 4000 frequency samples were simulated between 1 MHz and 5 GHz, equally spaced at 1 MHz up to 3 GHz and at 2 MHz above.Fig. 13 shows that this antenna does not have a single dominant resonance behavior; this fact is reected in the following gures.The generalized absorption eciency is 0.52 and the integrated extinction cross section has a value of 14.204 • 10 −3 m 3 which deviates less than 3.3% from γ ∞,22 = 14.667 • 10 −3 m 3 .We rewrite (2.2) in numbers as: 0.247 ≤ 0.248.This structure performs close to the optimal performance of the planar antenna.Compared to the smallest circumscribing sphere, this meanderline antenna achieves approximately 25% of the performance of the optimal sphere structure.
The bounds (2.2) do not depend on the choice of direction of radiation and polarization i.e., k and ê.The previous results considered the situation where k was the direction of the maximum radiation and η was computed based on a scattering simulation data with a plane wave impinging from the same direction (see [7]).The polarization of the wave was the polarization most favored by the structure.
There aren't so many choices for these directions because the structure radiates almost omnidirectionally.The dierence from the previous approach is that scattering simulation uses a plane wave impinging from the directions given by (θ, φ).
Two things can be checked in Table 1: rst D Qk 3 0 a 3 ≤ ηγ 2πa 3 , in agreement with (2.2), and second, the integrated extinction cross section (IECS) is less than or equal to γ where γ = ê • γ ∞ • ê is the electric polarizability corresponding to the chosen incidence.The gures are in good agreement, showing, as before, that the antenna performs close to the optimal performance of the planar antenna.The dierences in the integrated extinction cross section come from a limited number of samples, only 3000, 1 MHz apart, from 1 MHz to 3 GHz.

Meanderline
This meanderline structure is a modied version of the previous design that improves radiation resistance.To achieve this, the antenna is featured with a loop in the feeding area, see Fig.In the forward scattering simulation we use a load along the feeding line with the same value as the radiation resistance and excite the antenna with a plane wave impinging from the direction θ = 20 • and φ = 0 • .Cross sections are computed from 3000 frequency samples of the far eld, uniformly distributed between 1 MHz and 3 GHz (see Fig. 16).Integrating these variables gives a value η = The integrated extinction cross section has a value of 15.867 • 10 −3 m 3 which is 3% away from γ ∞,22 = 16.357• 10 −3 m 3 .
We rewrite the bounds (2.2) with numbers as: 0.258 ≤ 0.267.This structure performs close to the optimal planar structure (but not as close as the previous antenna).Compared to the smallest circumscribing sphere this structure is comparable with the previous one achieving, at most, approximately 27% of the performance of the optimal sphere structure.
The structure is considered in other directions of radiation/scattering and the results are shown in Table 2.The conditions are the same as in the previous section.

Other Structures
A number of other structures have been analyzed and the results are gathered in Table 3.The rst eight rows in the table correspond to planar geometries circumscribed by dierent l 1 × l 2 rectangles with l 1 /l 2 respectively equal to: 500, 100, 25, 9, 3.6, 2, 1 and 0.5.The polarization is always directed along l 1 , the long dimension of the antenna in most of the cases.The two meander type antennas on rows ve and six are described in 3.3 and 3.4, respectively.The ninth and tenth rows correspond to the structures described respectively in 3.1 and 3.2.
The eleventh and twelfth rows correspond to four element arrays fed in phase to obtain two broadside pencil shaped lobes.For the rst one the elements are represented by simple dipoles of length l = 500 mm spaced at d = 500 mm and wire radius R w = 1 mm and for the second the elements are folded dipoles with length l = 502 mm and height h = 6 mm spaced at d = 470 mm and wire radius R w = 2 mm.Both structures are fed through transmission lines made from the same wire as the radiating elements.
The last line corresponds to a two element array, each element being a Yagi antenna with a reector (l r = 510 mm), driven element (l f = 500 mm) and director (l d = 420 mm) spaced at l s = 200 mm.The distance between the elements in the array is d = 470 mm and the wire radius is R w = 5 mm.Feeding is realized with a transmission line made from the same wire as the elements.
It must be noted that the table contains some examples that fail to obey the resonance model and thus cannot be characterized properly by their static properties.
In this category fall loop structures, e.g., folded dipole with aspect ratio of 9 and the loops with aspect ratio of 1 and 0.5.

Conclusions
We demonstrate that η 1/2 for small, k 0 a 1, idealized dipole antennas and for minimum scattering antennas with a dominant rst single resonance.As observed in [4,7] this is also valid for several antennas that are of the order k 0 a ≈ 1.Here, it is important to realize that the identity (2.1) is not restricted to electrically small antennas and that η in general cannot be replaced by 1/2.Many antennas, e.g.,

Figure 3 :FFigure 4 :Figure 5 :
Figure 3: The generalized absorption eciency for the idealized dipole antenna (2.4) and (2.10) and various values of C, L, and R 1 as function of the normalized resonance wavenumber k 0 a.
This is a true relation showing that the antenna performs close to the D/Q bound of the wire structure.

Figure 7 :
Figure 7: Extinction, absorption and scattering cross sections of the spherical helix depicted in Fig. 4 as function of wavelength, λ = 2π/k = c 0 /f .

Figure 8 :
Figure 8: Two element array of folded dipoles with l = 492.9mm and d = 470 mm.

Figure 9 :
Fig.6with Fig.10we expect to have dierences between the two generalized ab-

Figure 13 :
Figure 13: Extinction, absorption and scattering cross sections of the antenna depicted in Fig. 11.

Figure 15 :
Figure 15: Input impedance of the antenna depicted in Fig. 14.

Figure 16 :
Figure 16: Extinction, absorption and scattering cross sections of the antenna depicted in Fig. 14.
Conservation of energy shows that the amplitudes of the reection coecient and the scattering coecient, S 11 , are identical in this case, i.e.,

Table 1 :
Numerical results for the antenna depicted in Fig.11.

Table 2 :
Numerical results for the antenna depicted in Fig.14.
The quality factor for these structures is very small due very close rst and second resonances.There are two structures that show a low Q but still obey the bounds, namely the folded dipole arrays.In these cases the structures are not thoroughly designed yielding lower performances at their rst resonances.

Table 3 :
Numerical results of the antennas in 3.5.