IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 61, NO. 3, MARCH 2013 1109
Optimal Antenna Currents for Q, Superdirectivity,and Radiation Patterns Using Convex Optimization
Mats Gustafsson, Member, IEEE, and Sven Nordebo, Senior Member, IEEE
Abstract—The high Q-factor (low bandwidth) and low efficiencymake the design of small antennas challenging. Here, convex op-timization is used to determine current distributions that provideupper bounds on the antenna performance. Optimization formula-tions for maximal gain Q-factor quotient, minimal Q-factor for su-perdirectivity, and minimum Q for given far-fields are presented.The effects of antennas embedded in structures are also discussed.The results are illustrated for planar geometries.
Index Terms—Antenna Q, antenna theory, convex optimization,physical limitations, small antennas, superdirectivity.
I. INTRODUCTION
T HERE are many advanced small antenna designs, such asfolded helices, folded meander lines, and concepts based
on metamaterials, fractals, and genetic algorithms. The highQ-factor (low bandwidth) and low efficiency make the designof small antennas challenging as the Q-factor, efficiency, andradiation resistance must be controlled simultaneously [1], [2].It is well known that the antenna performance deteriorates withdecreasing physical size (measured in wavelengths) of the an-tenna. The fundamental trade-off between performance and sizeis expressed by physical bounds. Physical bounds are useful be-cause they provide bounds on the performance based solely onthe shape and size of the design volume.Chu [3] used the stored and radiated energies outside a sphere
that circumscribes the antenna to determine physical bounds onthe Q-factor, , see also [1] for an overview. The stored energyin the interior of the sphere was added in [4]. Physical bounds onthe directivity Q-factor quotient were introduced for arbi-trary sized and shaped antennas in [5], [6] under the assumptionof . Related bounds on the Q-factor are investigated forsmall antennas in [7], [8] and for finite sizes in [9]. In [10], op-timal currents and physical bounds on are formulated asan optimization problem using the expressions for the stored en-ergies presented by Vandenbosch [11], see also [12]–[14]. Thebounds in [5]–[8], [10] are similar for the case of small electricdipole antennas composed of non-magnetic materials.
Manuscript received April 02, 2012; revised June 30, 2012; accepted Oc-tober 22, 2012. Date of publication November 15, 2012; date of current versionFebruary 27, 2013. This work was supported by the Swedish Research Council(VR).M. Gustafsson is with the Department of Electrical and Informa-
tion Technology, Lund University, SE-221 00 Lund, Sweden (e-mail:[email protected]).S. Nordebo is with the School of Computer Science, Physics and Mathe-
matics, Linnaeus University, Växjö, Sweden (e-mail: [email protected]).Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TAP.2012.2227656
Here, it is shown that convex optimization [15], [16] can beused as a tool to formulate and solve several fundamental ques-tions for small antennas. The approach offers many opportuni-ties to derive new physical bounds on antennas. Here, we e.g.,present results for minimum of superdirective antennas andminimum for antennas with a prescribed far field. We alsoillustrate how antennas embedded in metallic structures can beincluded in the bounds. This generalizes the results in [3]–[10]to many new and important antenna problems. It also general-izes the optimization formulation for in [10] to include themaximum of the stored electric and magnetic energies.The presented results are for arbitrary shaped structures but
restricted to antennas composed of non-magnetic materials.The convex optimization problems are also only valid for func-tionals that are positive semidefinite. Here, we limit the size toapproximately half a wavelength to obtain positive semidefiniteenergy expressions, see also [10], [14]. It is also important torealize that the quality factor loses its practical meaning whenis small. This restricts the interpretation of the presented
results to that coincides with the size restrictions on theantennas considered in this paper.Optimization is used in many areas including antenna design
[17], synthesis of array patterns [18], and inverse problems [19],[20]. The formulation as a convex optimization problem is ad-vantageous as: it has a well-developed theory [15], there are ef-ficient solvers [21], [22], and the solution gives error estimates.Moreover, a local minimum is also the global minimum, so thereis no risk of getting trapped in local minima. This is very dif-ferent from general global optimization problems [17] and onecan often state that a problem is solved if it is formulated as aconvex optimization problem [15].This paper is organized as follows. The considered antenna
parameters are introduced in Section II. The used method ofmoments formulation is presented in Section III. Section IVcontains the main results of the paper. It is divided into fivesubsections containing various convex optimization formula-tions giving antenna bounds. The results are also illustrated withbounds for planar rectangular geometries. The paper is con-cluded in Section V. Polarizability and spherical modes are dis-cussed in Appendices A and B, respectively.
II. ANTENNA PARAMETERS
We consider antennas in a volume composed of non-mag-netic materials with free space in the region exterior to , seeFig. 1. The radiated fields and stored energies are expressedin the antenna current in . The radiation intensity in the-direction is , where is the
1110 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 61, NO. 3, MARCH 2013
Fig. 1. Structure in a region with the antenna confined to . Theantenna current for induces the current density forwith a radiation pattern evaluated in the direction and polarization .
tangential components of the radiation vector [23]
(1)
denotes the free space impedance, thewavenumber, the speed of light in free space, , andthe time convention is used. The corresponding electricfield is , where denotes theposition vector, , and .We also use the unit vector , with , to evaluate
the partial radiation intensity for the polarization . The field islinearly polarized for and circularly polarized for
, where the superscript, , denotes the complexconjugate. This gives the partial radiation intensity
, where
(2)
The partial directivity, , and partial gain, , aredefined as [24]
(3)
respectively, where is the total radiated power and isthe dissipated power in the antenna structure. The quality factor(or antenna Q), , is
(4)
where and denote the stored electric and magnetic en-ergies, respectively.Follow the approach in [10] and use the results by Van-
denbosch [11], to express the stored electric energy as, where
(5)
and , and isthe permeability of free space. The stored magnetic energy is
, where
(6)
The corresponding expression for the total radiated power iswith
(7)
The normalized quantities, , in (6) and in (7) areintroduced to simplify the optimization approach used in thispaper. They have dimensions given by volume, , times thedimension of , i.e., . The corresponding dimensionof the radiation vector (1) is volume times the dimension of .
III. MOM FORMULATION
We use local basis functions, , analogous with ordinaryMethod of Moments (MoM) solutions of the electric field inte-gral equation [25] to approximate the radiation vector (1), storedenergies (5), (6), and radiated power (7). Expand the currentdensity in local basis functions
(8)
and introduce the matrix with elements to simplifythe notation. The basis functions are assumed to be real valued,divergence conforming, and having vanishing normal compo-nents at the boundary [25]. In this paper, we use piecewise linearbasis functions on quadrilateral elements. The discretization isnon-equidistant to capture edge singularities of the charge den-sity.The radiation vector projected on , cf., (2), defines the
matrix from
(9)
where the superscript, , denotes the Hermitian transpose andthe dependence of the matrix on and is suppressed. Thenormalized stored electric energy is approximated as
(10)
where the matrix has the elements
(11)
The normalized stored magnetic energy,, and the normalized radiated power,
GUSTAFSSON AND NORDEBO: OPTIMAL ANTENNA CURRENTS FOR Q, SUPERDIRECTIVITY, AND RADIATION PATTERNS 1111
are defined analogously. The matrices , andare real-valued and symmetric. It is observed that andcan be indefinite for electrically large structures [10],
[14]. In the numerical examples in this paper, we restrict theelectrical size to be approximately less than half a wavelength.The eigenvalues are also computed to verify that andare positive semidefinite. Here, it is observed that there can bea few negative eigenvalues. These negative eigenvalues arehowever due to the used finite numerical precision and their rel-ative amplitude is compared to the positive eigenvalues.We transform the matrices to become positive semidefinite bysetting these eigenvalues to zero.
IV. CONVEX OPTIMIZATION FORMULATIONS
We use convex optimization [15] to determine fundamentalbounds on the antenna performance and their corresponding op-timal current densities. We assume that , and arepositive semidefinite for the electrical sizes considered in thispaper. First, bounds on for small antennas as and
are analyzed. It is followed by bounds on forantennas with prescribed far fields, and for superdirective an-tennas, all with . The final case is bounds on forembedded antennas where , see Fig. 1.
A. Bounds on for Small Antennas
Explicit bounds on the directivity -factor quotient, ,(and equivalently ) are presented for small antennas in[10]. The low-frequency expansion of the current density is
as , whereand . The expansion simplifies the energy ex-pressions (5) and (6) for small antennas [8], [10], [12]. The
bound separates into electric dipoles, magnetic dipoles,and mixed modes antennas [10]. In [10], it is also shown thatit is sufficient to consider surface currents for small antennas.The gain Q-factor quotient for small electric dipole antennas isbounded as [10]
(12)
subject to the constraint of zero total charge .Use that the quotient is invariant for scaling to rewritethe bound (12) as the optimization problem
(13)
where is included for convenience. Use basis functions sim-ilar to (8) to approximate the charge density as ,let denote the matrix with elements , and ,
and the corresponding matrix representations for the integraloperators in (13), see also Appendix A. This gives the convexoptimization problem
(14)
This is a convex optimization problem in the form of a linearlyconstrained quadratic program [15] that e.g., can be solved using
[22]. It is also illustrative to use Lagrange multipliers [15],[26] to rewrite (14) as the linear system
(15)
where and are the Lagrange multipliers. The linearsystem (15) is identical with the MoM solution for the polariz-ability, using Galerkin’s method, see Appendix A.This illustrates that the convex optimization can be numericallyidentical to the solution of the integral equation in [10].The bound for the magnetic dipole case is reduced to an
integral equation involving an arbitrary function in [10]. Here,we use a convex optimization problem to derive a simple linearsystem for the bound. The gain Q-factor quotient for amagnetic dipole antenna is bounded as
(16)
where . We scale to reformulate (16) as the opti-mization problem
(17)
Use the basis functions (8) to get the convex optimizationproblem
(18)
where is an matrix for the stored magnetic energy,an matrix representation of the divergence operator
on the basis (8) and the corresponding row matrix for thefirst constraint in (17). Use Lagrange multipliers, and , totransform (18) to the linear system
(19)
1112 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 61, NO. 3, MARCH 2013
Fig. 2. Upper bounds on the partial gain Q-factor quotient for cur-rents confined to planar rectangles with sides and for small antennas using(15), (19), and (20). The radiation patters of the and modes are de-picted. The bound is normalized with , where is the radius of the smallestcircumscribing sphere, i.e., .
In [10], it is also shown that the constraint on is relaxedfor combined electric and magnetic dipole antennas:
(20)
We consider a planar rectangle to illustrate the physical boundson for small antennas, see Fig. 2. The rectangle has sidelengths and with the radius of thesmallest circumscribing sphere. The bound on the electric dipoleis identical to the results in [5], [6], [10] and many small dipoleantennas perform close to the bound [6]. The magnetic dipolecase is more constrained. In particular, the bound shows that it isdifficult to utilize the magnetic dipole for elongated structures.The combined mode case (20) offers a substantially improvedperformance [10].The bounds in Fig. 2 are computed using piecewise linear
basis functions on rectangular elements. We use a non-equidis-tant mesh for the electric dipole case, where the mesh is con-structed to have approximately equal charge on each elementfor improved convergence, see also [27]. The magnetic dipolecase is computed on an equidistant mesh. We also use the con-straint to reduce the size of the linear system.
B. Maximal
Maximization of for finite sized antennas is formulatedas a convex optimization problem. Combine (3) and (4) to ex-press the gain Q-factor quotient as
(21)
In [10], the quotient is maximized under the assumptionusing a Lagrangian formulation. To solve the gen-
eral case (21), we rewrite the quotient as a convex op-timization problem. We follow [10] and note that is in-variant for multiplicative scaling with arbitrary com-plex valued . It is hence sufficient to consider real-valuedquantities , see (9). Moreover, maximization of
can be replaced by maximization of .This gives the convex optimization problem
(22)
This is a quadratically constrained linear program (QCLP)giving the upper bound on (21) as , where
and a solution of (22). Note, that the currentmatrix, , is rescaled such that is dimensionless.There are many alternative convex formulations to maximize
(21), e.g., the Lagrange dual or using that the maximum of twoconvex functions is convex [15] to minimize the stored energy,i.e.,
(23)
giving , whereand a solution of (23).We consider currents confined to planar rectangles to illus-
trate the results. The bound on and its correspondingand for lossless antennas are depicted in Fig. 3 for rectan-gles with side lengths and . Thepartial gain is evaluated for the polarization and the di-rections and . The two optimization formulations(22) and (23) give similar results when solved using [22].The bound on is normalized with the electrical sizeto simplify comparison with the results in [6], [10], wheredenotes the radius of the smallest circumscribing sphere. Thelow-frequency limit for is given by the polarizability asshown in [10]. It is also observed that is almost in-dependent of the electrical size for or .The case with radiation in the direction offers an
increased . In particular, the case increasesfrom 0.29 to 0.63. Here, we note that the bound
in [5], [6], [27] is sharper for this case. It is also important to re-alize that the bound in [5], [6], [27] are for the bandwidth of theantenna and it is not guaranteed that the optimal current distri-butions considered here can be generated from single-port an-tennas with a half-power fractional bandwidth , seealso [28]. In Fig. 3(a), it is further seen that the improvementfor the direction diminishes as decreases. The resultingcurrent distribution on a coarse mesh is depicted for
, and . The current is composed of one -di-rected component and one loop type component, see also Fig. 4.It is similar for the thinner structures but the loop current getsweaker. The corresponding currents for the cases are sim-ilar to the -directed component. It is also known that the cur-rent distribution is not unique [10]. There are antenna designsthat perform close to the bound for the cases [6]. The
cases are more involved and there are, to our knowledge,presently no design that reach these bounds. It is also knownthat the loop current is associated with low radiation resistance.The corresponding partial directivities, , and Q-fac-
tors for lossless structures are depicted in Figs. 3(b), (c), respec-tively. Here, it is observed that the directivity differs between
GUSTAFSSON AND NORDEBO: OPTIMAL ANTENNA CURRENTS FOR Q, SUPERDIRECTIVITY, AND RADIATION PATTERNS 1113
Fig. 3. Upper bounds on the partial gain Q-factor quotientfor currents confined to planar rectangles with sides and
for using (22). The bound is nor-malized with , where is the radius of the smallest circumscribing sphere,i.e., . (a) maximal . The resulting current distributionis shown for , and . (b) resulting for losslessantennas. The radiation patterns with their mode expansions (48) are depictedfor and , where it is seen that and dominates,cf., (50). (c) resulting for lossless antennas.
the and cases. There is also a decrease in the Qfor the case except for the larger structures, ,where is very low.We expand the far-field in spherical modes to analyze the ra-
diated field, see Appendix B. It is noted that the radiation patternfor the case is dominated by mode number , i.e.,
or an -directed electric dipole (50), see the histograms inFig. 3(b) for the magnitude distribution at . Thereare also small contributions from higher order modes as ana-lyzed in Section IV.C. The improved performance for thecase is due to the additional excitation of a -directed magneticdipole, . This is consistent with the explicit solu-tion for small mixed mode antennas in (20), see also [10].
Fig. 4. The regular spherical vector waves for , i.e., mag-netic and electric dipoles in the - and -directions, respectively, evaluated ona planar rectangle with side lengths and , see also Fig. 2for illustrations of their radiation patterns.
C. Minimal for Given Radiation Pattern
Consider the case with a desired radiated field repre-sented with the radiation vector (2). We search for the currentdensity, , with the radiation vector that approximates, i.e., , and has minimal stored energy. The deviation
of from can be quantified by the projection of on orby some norm . We start by maximizing projectedon , i.e., the real valued part of
(24)
where denotes the unit sphere and
(25)
is a representation of the desired radiated field in the currentdensity on the structure.This gives the convex optimization problem
(26)
It is common to expand the radiated far field in vector sphericalharmonics, , or modes, see Appendix B, i.e.,
(27)
where is sufficiently large [29] and the expansion coeffi-cients are
(28)
and similarly for the expansion coefficients of . Here, themulti-index for
, and is introduced to simplify the notation.
1114 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 61, NO. 3, MARCH 2013
The index, , is also ordered such that, see Appendix B. The current in (26) is
(29)
where denotes the regular spherical vector waves, seeFig. 4 and Appendix B.We can also minimize the stored energy for a radiated field of
the form , i.e.,
(30)
where quantifies the deviation of the desired radiation patternand the least-square norm is used for simplicity. It is conve-nient to expand the radiated field in spherical vector waves andrewrite the deviation as
(31)
where
(32)
This gives the optimization problem
(33)
It is noted that arbitrary weight functions and norms can be usedin (33).A planar rectangle with is considered to illustrate
the results for given far-fields (26) and (33). The factors aredepicted in Fig. 5 for projections (26) and norm bounds (33) forthe cases of the -directed electric dipole and -directedmagnetic dipole patterns, see Fig. 4.We observe that the is lower for the electric dipole mode
than the magnetic dipole mode. The Q is also lowest for theprojection cases (26). Moreover, tend to increase as de-creases, i.e., the lowest is for radiation patterns that are close
Fig. 5. Q-factors for a lossless planar rectangle (side lengths and ) withelectric and magnetic dipole radiation patterns using (26) and(33). The radiation patterns with their mode expansions (48) are depicted for
, where it is seen that and dominate for the andcases, respectively.
to but not exact dipoles. The radiation patterns are depicted for. It is hard to distinguish between the patterns for
the cases, but the partial directivity for the projection caseand is lower than for for the casein the region around . The radiation patterns differmore for the case, where again the projection formulation(26) offers the lowest .
D. Maximal for
The Chu bound [3] shows that the radiation is dominated bydipole modes for small antennas . Consequently, thedirectivity is low, i.e., for single mode antennas andin general bounded as for mixed electric and magneticdipole modes. Higher directivity requires higher order modesthat imply a higher , e.g., the of quadrupolemodes is propor-tional to for . It is known to be difficult to designand utilize high (or super-) directivity for small antennas [2],[30]–[33]. It is hence interesting to investigate the boundfor antennas with directivities for some .These bounds give an estimate of the increased Q-factor for su-perdirective antennas.The partial directivity (3) is included in the optimization
problem (23) with the constraint giving
(34)
where the factor is due to the normalization of and.The bounds are illustrated in Fig. 6 for planar rectangles with
and for the polarization . The constraintsand are considered for
and , respectively. These values for are chosenas they require excitation of higher order modes. The additionof the constraints reduces for small structures when theconstraint is active. The resulting partial directivities are seen inFig. 6(b) together with the mode distribution for .The superdirectivity for the cases are due to
GUSTAFSSON AND NORDEBO: OPTIMAL ANTENNA CURRENTS FOR Q, SUPERDIRECTIVITY, AND RADIATION PATTERNS 1115
Fig. 6. Upper bounds on the partial gain Q-factor quotient for antennas withfor a planar rectangle with side lengths
and . (a) . (b), (c) resulting and for lossless antennas. Theradiation patterns with their mode expansions (48) are depicted for ,where it is seen that higher order modes appear in addition to forthe superdirective cases.
the excitation of electric quadrupole terms. This also explainsthe increased Q-factors.Fig. 7 illustrates the corresponding results for usingand radiation in the directions with polariza-
tions for a lossless structure. Here, the cost ofsuperdirectivity is clearly seen. The Q-factor is highest for the
case where the symmetry causes the current to radiatein both the and directions. The mode expansion indicatesthat many higher order modes are excited. The caseshave pencil beams and much lower factors. The end-fire case
has the lowest with for . The re-sulting current distributions are oscillatory as for superdirectivearrays [2].
E. Embedded Antennas
It is useful to analyze the case when the antenna is embeddedin a structure. In this case the currents on the entire structure, ,
Fig. 7. Lower bounds on the Q-factor for lossless superdirective antennashaving constrained to a planar rectangle with sidelengths and using (34). The radiation patterns with their modeexpansions (48) are depicted for , where it is seen that the radiationpatterns are composed of many higher order modes.
contribute to the radiation but we can only control the currentsin the volume , see Fig. 1. Here, we consider the casewhere the structure is perfectly electric conducting(PEC), see Fig. 8. The induced currents on the surface of aredetermined from the electric field integral equation (EFIE) thathas the matrix elements [25]
(35)
where the similarities with (5), (6), (7), and (11) are noted. Theintegration in (35) is over the PEC surface of the structure. Thedriving sources of the EFIE are confined to the region andthey are unknown. Moreover the EFIE is not necessarily validin . Decompose the current density as , whereis the current density in . The EFIE gives two
rows corresponding to test functions in and . Here, the firstrow is unknown but the second row gives the constraint
(36)
that can be added as a constraint to the convex optimizationproblems in this paper, e.g., the bound in (22).It is convenient to use (36) to express the induced current den-
sities in , i.e., and eliminatein the optimization problem. Decompose the matrices and
according to
(37)
that gives
(38)
and
(39)
1116 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 61, NO. 3, MARCH 2013
Fig. 8. Illustration of antennas embedded in planar metallic rectangles. Arbi-trary currents in and induced currents on the metallic structure. (a) antenna region in the center. (b) antenna region in the upper corner.
Fig. 9. Bound on for planar strip dipoles with length and width. Arbitrary currents in the central region, with length , of
the strips for and PEC in .
The optimization problem on becomes
(40)
We illustrate the bound on , normalized with , forembedded antennas using the structures in Fig. 8. The stripdipole in Fig. 8(a) has length and width . Letthe center, be the region, , where the currents areoptimized. In Fig. 9, we observe that the performance decreaseswith decreasing except for , where the centerfed strip dipole is self-resonant. This shows that the inducedcurrents are optimal for short dipoles. This is consistent withthe analysis of strip dipoles in [10]. Note that only havingcurrents in corresponds to a shorter dipole, so it is clearthat the induced currents on the surrounding PEC structure, ,improves the performance compared to only having currents in.The second example is for antennas embedded in a planar
PEC rectangle with length and width . We con-sider the cases with arbitrary feed currents in rectangular re-gions in the upper corner of the structure with size times
for , see 8(b). Note that feed region is notnecessarily PEC and only modeled by its current density. The
quotient is optimized for and using (40), seeFig. 10. We observe that the performance deteriorates for smallregions and small antennas. There is however a region around
where the performance is close to the case of
Fig. 10. Bounds on and resulting for antennas embedded in planarmetallic rectangles. Arbitrary currents in and induced currentsin with the antenna region in the upper corner with ,see Fig. 8(b). The results without induced currents are also shown. (a) boundson using (40). (b) resulting for a lossless structure.
utilizing the entire structure. It is also noted that the resonanceof the first characteristic mode [34] is at . The cor-responding bounds for the cases without the surrounded PECstructure, i.e., for rectangles with size times , are alsodepicted in Fig. 10. Here, it is seen that the induced currentson the surrounding PEC structure improves the performancecompared to the cases without the surroundingPEC. Consider e.g., and where and
with and without the PEC structure, respectively. Thecorresponding values for and areand .
V. CONCLUSION
We show that several performance limits for antennas can beformulated as convex optimization problems. Standard software[15], [22] is used to solve the convex optimization problems.The results for are consistent with the bounds in [6], [10].The new bounds offer physical insight into the design of smallantennas, see also [10]. They also offer the possibility of sys-tematic studies of how and directivity are related for smallsuperdirective antennas. Moreover, properties of antennas em-bedded in structures, such as mobile phones and other terminals,are discussed.It is important to realize that the convex optimization problem
determines an optimal current distribution. This current is ingeneral not unique although the minimum of the convex opti-mization problems is unique. It is also not known if there areantennas performing close to the bounds except for the case ofelectric dipole type antennas [6]. Moreover, the optimal perfor-mance can be useful in global optimization of antennas [17].
GUSTAFSSON AND NORDEBO: OPTIMAL ANTENNA CURRENTS FOR Q, SUPERDIRECTIVITY, AND RADIATION PATTERNS 1117
APPENDIX APOLARIZABILITY
The physical bounds on in [5], [6] are expressed inthe polarizability of the antenna structure. In [10], it is alsoshown that the bound on small antennas (12) can be expressedin the polarizability. Here, we further show that the solution ofthe convex optimization problem (14) using (15) is identical tocomputing the high-contrast polarizability [10] using Galerkin’smethod [26].The high-contrast polarizability for the polarization can be
determined from the first moment of the induced normalizedcharge density as
(41)
Here, we keep the notation with complex conjugates on tosimplify the comparison with (15), although it is sufficient toconsider real valued unit vectors to determine the electrostaticpolarizability. The charge density is the solution of the integralequation
(42)
where the constant is determined from the constraint of zerototal charge . It turns out that it is convenientto set for comparison with (15). Expand the chargedensity in basis functions , whereand are matrices, to rewrite (41) as
(43)
Solving the integral equation (42) with the Galerkin’s method[26] gives the linear system of equations
(44)
where the matrix is
(45)
Finally, the constraint of zero total charge is
(46)
Written as a linear system, (43), (44), and (46) becomes
(47)
We note that this system is identical to (15).
APPENDIX BMODE EXPANSION
The radiated electromagnetic field is expanded inspherical vector waves [29] (or modes) outside a circum-scribing sphere. The corresponding far field is expanded inspherical vector harmonics as
(48)
giving for the expansion coefficients in (32),where is the spatial coordinate, , and thewavenumber. The multi index for
, and is introduced to simplifythe notation. The index, , is also ordered such that
, see [35].There are a few alternative definitions of the spherical vector
waves in the literature [29], [36], [37]. Here, we follow [36] anduse and as basis functions in the azimuthal co-ordinate. This choice is motivated by the interpretation of thefields related to the first 6 modes as the fields from differentHertzian dipoles. For , we use spherical vector har-monics
(49)
and where denotes the spher-ical harmonics [37]. The modes labeled by (odd ) iden-tify TE modes (or magnetic -poles) and the terms labeled by
(even ) correspond to TM modes (or electric -poles).Moreover, the dipoles corresponding to are directed inthe -direction, in the -direction, and in the-direction having the explicit representation
(50)
where . The regular, , spherical vectorwaves are given by and
, where denotes the sphericalBessel function.
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Mats Gustafsson (M’05) received the M.Sc. degreein engineering physics and the Ph.D. degree in elec-tromagnetic theory from Lund University, Sweden,in 1994 and 2000, respectively.In 2005, he was appointed was appointed Docent
and, in 2011, Professor of electromagnetic theoryat Lund University, Sweden. He co-founded thecompany Phase Holographic Imaging AB in 2004.His research interests are in scattering and antennatheory and inverse scattering and imaging withapplications in microwave tomography and digital
holography. He has written over 60 peer reviewed journal papers and over 75conference papers.Prof. Gustafsson received the best antenna poster prize at EuCAP 2007 and
the IEEE Schelkunoff Transactions Prize Paper Award 2010. He serves as anAP-S Distinguished Lecturer for 2013–2015.
Sven Nordebo (SM’05) received the M.S. degreein electrical engineering from the Royal Instituteof Technology, Stockholm, Sweden, in 1989 andthe Ph.D. degree in signal processing from LuleåUniversity of Technology, Luleå, Sweden, in 1995.Since 2002, he is a Professor of signal processing
at the School of Computer Science, Physics andMathematics, Linnæus University. His researchinterests are in statistical signal processing, electro-magnetic wave propagation, inverse problems andimaging.
