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Keywords:

  • inverse source problem;
  • antenna substrate;
  • antenna performance

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Inverse Source Theory Based on Constrained Optimization
  5. 3. Numerical Results and Case Studies
  6. Appendix A:: Determination of Vector Fields equation imagel,m(j) for a Core-Shell System
  7. Appendix B:: Calculation of DFℰ(J) and DF((equation imagel,m(j), J) − al,m(j))
  8. Acknowledgments
  9. References

[1] We present a comparative study on radiation enhancement due to metamaterial antenna substrates. The study is performed via electromagnetic inverse source theory and constrained optimization. The aim of the study is to understand the effect of reference antennas on enhancement level estimates. The geometry of the problem is that of a piecewise-constant, radially symmetric, three-region system (spherical core-shell system). Particular attention is given to the case when the core and the shell are made up of lossless materials having oppositely signed constitutive parameters.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Inverse Source Theory Based on Constrained Optimization
  5. 3. Numerical Results and Case Studies
  6. Appendix A:: Determination of Vector Fields equation imagel,m(j) for a Core-Shell System
  7. Appendix B:: Calculation of DFℰ(J) and DF((equation imagel,m(j), J) − al,m(j))
  8. Acknowledgments
  9. References

[2] An electromagnetic wave with wavelength λ much larger than a certain object does not really “see” that object. Consequently a wave propagating in a homogeneous substrate with small inclusions that are, let us say, ∼λ/10 will interact effectively not with the individual inclusions but with the system {substrate,inclusions} as a whole. The wave, thus, sees a new effective medium having electromagnetic properties different from those of its components. This suggests that effective media with properly chosen inclusions may exhibit electromagnetic responses not found in nature. Metamaterials are such media. They are engineered materials whose effective constitutive parameters can, in principle, have any value, even negative.

[3] The onset of the “metamaterials era” is frequently attributed to the landmark theoretical article by Veselago [1968], though the study of complex media has a long history (see, for instance, the review by Ziolkowski and Engheta [2006]). Yet, until just a few years ago, the study of these materials was largely ignored by scientists and engineers alike. Then, at the turn of the century, Smith et al. [2000] announced the fabrication of a metamaterial with negative index of refraction in the microwave regime. This discovery instigated several studies that resulted in the identification of several novel and exciting applications of metamaterials. For example, metamaterials have been suggested for the realization of perfect lenses [Pendry, 2000], subwavelength cavities [Engheta, 2002], highly resonant electrically small antennas [Ziolkowski and Kipple, 2003], subwavelength waveguides [Alù and Engheta, 2004], cloaking devices [Alù and Engheta, 2005a], and ultrathin laser cavities [Ziolkowski, 2006a].

[4] In this paper we present a comparative study of radiation enhancement due to metamaterial substrates. Antenna radiation enhancement due to metamaterial substrates is a very active field of research [see Engheta et al., 2006, and references therein] and several leading groups have addressed several important aspects of the problem. Here, we wish to address the issue of “fairness” in antenna performance comparisons which has been the subject of some controversy [Kildal, 2006; Ziolkowski, 2006b]. By “fairness” we mean the suitability of the reference antennas with respect to which the comparisons are to be carried out.

[5] This study complements our investigation of radiation enhancement with metamaterials for the core-shell system [Khodja and Marengo, 2008] which is a generalization of the homogeneous substrate case [Marengo and Khodja, 2007, 2008]. However, in the work of Khodja and Marengo [2008] we considered only the effects that adding a shell would have on the performance of an existing antenna. The novelty of the present treatment resides in the fact that it explores the effects that different definitions of the reference antenna would have on enhancement estimates (in particular three classes of reference antennas shall be considered.) As in the work of Khodja and Marengo [2008], the underlying formulation is a non-antenna-specific, full-vector, inverse-theoretic formulation in substrate media that is a generalization of the scalar inverse source theory by Devaney et al. [2007].

[6] The configuration investigated in the following is that of a piecewise-constant, radially symmetric, lossy core-shell system immersed in free space (three-region system). The core, of radius a, has relative permittivity ɛa and relative permeability μa. This core is the smallest spherical volume V that circumscribes the largest physical dimension of the original antenna. This antenna is treated, under a suppressed time dependence eiωt, as an arbitrary primary current density J(r). The core is surrounded by a spherical shell, of inner radius a and outer radius b. The constitutive parameters of the shell are relative permittivity ɛb and relative permeability μb. The core and the shell are assumed to be generally lossy; their relative constitutive parameters are, in general, complex.

[7] We formulate an inverse source problem in substrate media, whose objective is to deduce an unknown impressed, or primary, current density J(r) that is contained, along with the substrate, in the spherical volume V, and that generates a prescribed exterior field for r > b. The solution of the inverse problem is nonunique due to the possible presence of nonradiating source within V [Devaney and Wolf, 1973; Marengo and Khodja, 2006]. The uniqueness of the solution is guaranteed if, for instance, one requires that the square of the L2 norm of the source (as defined by equation (2)) be minimum. This quantity is usually termed “the source energy.” It is widely used in inverse theory literature [Porter and Devaney, 1982a, 1982b; Langenberg, 1987; Kaiser, 2005; Devaney et al., 2008]. A minimized source energy would indicate that the resources of the antenna generating the given field pattern are optimally used within the prescribed volume of the source.

[8] Comparison of the required minimum source energies for different substrate configurations enables quantification of the enhancement due to such structures. This substrate enhancement characterization is non-device-specific. In particular, one would, in this framework, be able to compare the optimal source for a given substrate with the optimal source for another (including the free-space case).

2. Inverse Source Theory Based on Constrained Optimization

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Inverse Source Theory Based on Constrained Optimization
  5. 3. Numerical Results and Case Studies
  6. Appendix A:: Determination of Vector Fields equation imagel,m(j) for a Core-Shell System
  7. Appendix B:: Calculation of DFℰ(J) and DF((equation imagel,m(j), J) − al,m(j))
  8. Acknowledgments
  9. References

[9] For two arbitrary vector functions f(r) and g(r) we define the inner product (f, g) ≡ Vdrf*(rg(r) where the asterisk * denotes the complex conjugate. By borrowing from well-known results in classical radiation theory [Harrington, 1958; Devaney and Wolf, 1974], it follows that the electric (TM) and magnetic (TE) multipole moments of the radiated electromagnetic field, al,m(1) and al,m(2), respectively, can be expressed as (see Appendix A)

  • equation image

i.e., they are the projections of the source current distribution J onto a set of (substrate-dependent) vector fields equation imagel,m(j) (r), where j, l, m are, for the time being, general indices that are to be specified when equation imagel,m(j) are determined. These source-free fields equation imagel,m(j) need to be determined for the particular antenna substrate under consideration. They are closely related to the familiar source-free multipole fields that appear in the free-space case [Harrington, 1958; Devaney and Wolf, 1974]. The derivation of the exact expressions for the source-independent fields equation imagel,m(j)(r) for our particular background is presented in Appendix A. Next we tackle the constrained optimization problem.

[10] The problem of determining the minimum-energy source of support V generating a given exterior field (for ∣r∣ > b) can be mathematically cast into a constrained optimization problem: minimize the source energy functional ℰ(J) defined as

  • equation image

subject to the constraints

  • equation image

where I is the index set. The problem can be concisely stated as

  • equation image

where

  • equation image

The objective functional ℰ is coercive and bounded from below; its lower bound is zero and occurs at inf L2(V; equation image3) = 0. This lower bound is also the global minimum of the unconstrained problem.

[11] Problem (4), (5) is a nonlinear convex programming problem. The convexity of the problem stems from the convexity of the objective functional ℰ and the convexity of the closed constraint set S. The convexity of ℰ is written as [Angell and Kirsch, 2004]

  • equation image

The inequality is in fact strict unless J1 = J2 implying that ℰ is a strictly convex functional. The convexity of ℰ and S in addition to the continuity of ℰ at some point, guarantee its continuity on the whole space L2(V; equation image3) [Jahn, 2007].

[12] Since ℰ is a continuous and convex functional on a Hilbert space (i.e., L2 (V; equation image3)) then it is also weakly sequentially lower semi-continuous [Angell and Kirsch, 2004]. This property of ℰ, along with the fact that it is coercive and the fact that S is a closed and convex subset of a Hilbert space, guarantee the existence of at least one minimizer JME [Kurdilla and Zabarankin, 2005]. The uniqueness and global minimality of JME are, then, insured by the strict convexity of ℰ [Angell and Kirsch, 2004].

[13] To solve the problem and find the global minimizer JME we note that (see Appendix B): (1) the objective functional ℰ as well as the constraints [(equation imagel,m(j),) −al,m(j)] are Fréchet differentiable, (2) their Fréchet derivatives are continuous, and (3) DF [(equation imagel,m(j), JME) −al,m(j)] is surjective (where DF stands for the Fréchet derivative). Therefore there exist Lagrange multipliers cl,m(j)∈C such that [Kurdilla and Zabarankin, 2005]

  • equation image

where the generalized Lagrangian ℒ(J,cl,m(j)) is given by

  • equation image

[14] Writing equation (7) explicitly in terms of the gradients of the objective functional and the constraints (see Appendix B) yields the solution

  • equation image

where we have introduced the positive-definite singular values

  • equation image

Solution (9) is the sought unique and global minimizer of problem (4), (5).

3. Numerical Results and Case Studies

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Inverse Source Theory Based on Constrained Optimization
  5. 3. Numerical Results and Case Studies
  6. Appendix A:: Determination of Vector Fields equation imagel,m(j) for a Core-Shell System
  7. Appendix B:: Calculation of DFℰ(J) and DF((equation imagel,m(j), J) − al,m(j))
  8. Acknowledgments
  9. References

[15] In this section a MATHEMATICA code is used to study the impact of reference antenna definitions on performance level estimates. The code has been validated against some well-known cases such as the free-space case [Marengo et al., 2004] and the homogeneous spherical substrate case [Marengo and Khodja, 2008]. We shall limit ourselves to a few illustrative examples. We focus on the case when the core does not contain any material, i.e., ɛa = 1 = μa, and the shell is a lossless double-negative medium (DNG), i.e., a medium for which ɛ < 0 and μ < 0 In particular ɛb = −4 and μb = −1. Two types of antennas are investigated: a quarter-wavelength antenna (i.e., 2a/λ =1/4) and a full-wavelength antenna (i.e., 2a/λ =1). The driving frequency of the antennas is set to f = 10 GHz. This corresponds to a = 3.75 mm for the quarter-wavelength antenna and a = 1.5 cm for the full-wavelength antenna. (Note that because the frequency f is fixed these two values of antenna size would clarify the effect, if any, of electric antenna size on radiation performance as measured with respect to a particular reference antenna.) These particular choices of the numerical values of f and a are, of course, arbitrary. However, they lie well within the range of values used in the scientific and engineering literatures [Ziolkowski and Kipple, 2003, 2005; Ziolkowski and Erentok, 2006; Alù and Engheta, 2005b, 2006].

[16] For a given substrate antenna operating at a given electromagnetic mode, we define the normalized singular values

  • equation image

i.e., they are the singular values that correspond to the configuration of interest normalized to those of the free-space case. By the free-space case we mean the situation in which a reference antenna is radiating in free-space. Three “reasonable” definitions of the reference antenna are considered: (1) it is an antenna which resides within a sphere of radius a, (2) it is an antenna which resides within a sphere of radius b, or (3) it is an antenna which resides within a sphere of radius bmax, where bmax is the outer radius of the shell that maximizes the singular value [σl(1)]2. These three classes of reference antennas will subsequently be referred to as RA1, RA2, and RA3, respectively. The idea behind the first definition is a simple answer to the question of how the addition of a metamaterial shell would affect the performance of an existing antenna radiating in free space. This definition proved to be adequate when the antenna substrate was merely a homogeneous sphere of given radius [Marengo and Khodja, 2007, 2008]. The second definition is motivated by the realization that adding a shell to an existing antenna creates, in fact, a new antenna with new dimensions. As for the third definition the idea behind it is to compare optimal radiation in either configuration, i.e., to compare different antennas when they operate at their best [Ziolkowski and Kipple, 2003; Kildal, 2006; Ziolkowski, 2006b; Ziolkowski and Kipple, 2005; Ziolkowski and Erentok, 2006].

[17] It follows from equations (9), (2), and (11) that, generally, the larger the normalized singular values [ρl(j)]2 the smaller the minimum source energy ℰME ≡ (JME, JME) required for the launching of a given radiation pattern with a source of a given size. Therefore the larger the normalized singular values [ρl(j)]2 the greater the associated improvement, due to the associated substrates, of radiation of the lth multipole order field with given resources. Resonant peaks in the plots of [ρl(j)]2 will indicate improved radiation for such operational modes or conditions, with the given resources. It is to be noted that the presence of these resonant peaks may also be viewed as an indication that the metamaterial core and shell act like an impedance matching network for the antenna resulting in an enhanced performance for the system [Ziolkowski and Kipple, 2003]. Because of the noted similarity between the behavior of the electric singular values and the magnetic singular values we concentrate our attention in what follows on the study of the electric singular values [Devaney et al., 2007; Marengo and Khodja, 2007, 2008; Khodja and Marengo, 2008].

[18] In Figures 1–3 we plot the normalized singular values [ρl(1)]2 for a quarter-wavelength antenna with radius a = 3.75 mm versus the radii ratio db/a. In Figure 1 the reference antenna is assumed to be an antenna radiating in free space and circumscribed by a sphere of radius a; in Figure 2 the reference antenna is assumed to be an antenna radiating in free space and circumscribed by a sphere of radius b; in Figure 3 the reference antenna is assumed to be an antenna radiating in free-space and circumscribed by a sphere of radius bmax, where bmax is the outer radius of the shell that maximizes the singular values [σl(1)]2. The maxima occur at bmax/a = 1.40,1.29,1.17,1.13,1.1, for l = 1,2,3,4,5, respectively. In Figures 4–6we show plots similar to those of the quarter-wavelength antenna case. The outer radii ratios that maximize the singular values [σl(1)]2are in this case bmax/a = 1.35,1.16,1.16,1.18,1.13, for l = 1,2,3,4,5.

image

Figure 1. Logarithmic plot of the normalized singular values [ρl(1)]2 for a quarter-wavelength antenna with radius a = 3.75 mm versus the radii ratio db/a. The shell is a lossless DNG material with ɛb = −4 and μb = −1. The reference antenna is assumed to be RA1 antenna.

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image

Figure 2. Logarithmic plot of the normalized singular values [ρl(1)]2 for a quarter-wavelength antenna with radius a = 3.75 mm versus the radii ratio db/a. The shell is a lossless DNG material with ɛb = −4 and μb = −1. The reference antenna is assumed to be an RA2 antenna.

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image

Figure 3. Logarithmic plot of the normalized singular values [ρl(1)]2 for a quarter-wavelength antenna with radius a = 3.75 mm versus the radii ratio db/a. The shell is a lossless DNG material with ɛb = −4 and μb = −1. The reference antenna is assumed to be an RA3 antenna.

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image

Figure 4. Logarithmic plot of the normalized singular values [ρl(1)]2 for a full-wavelength antenna with radius a = 1.5 cm versus the radii ratio db/a. The shell is a lossless DNG material with ɛb = −4 and μb = −1. The reference antenna is assumed to be an RA1 antenna.

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image

Figure 5. Logarithmic plot of the normalized singular values [ρl(1)]2 for a full-wavelength antenna with radius a = 1.5 cm versus the radii ratio db/a. The shell is a lossless DNG material with ɛb = −4 and μb = −1. The reference antenna is assumed to be an RA2 antenna.

Download figure to PowerPoint

image

Figure 6. Logarithmic plot of the normalized singular values [ρl(1)]2 for a full-wavelength antenna with radius a = 1.5 cm versus the radii ratio db/a. The shell is a lossless DNG material with ɛb = −4 and μb = −1. The reference antenna is assumed to be an RA3 antenna.

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[19] Reference antennas that belong to RA2, i.e., reference antennas for which the radius of the circumscribing volume is equal to b, tend to shift the positions of the resonant peaks. For instance, the first electric dipole peak appears for the singular value [σl(1)]2 at dmaxbmax/a = 1.40, as mentioned earlier, but when an RA2 antenna is adopted the position of the peak shifts slightly: it appears at d = 1.32 for the normalized singular value [ρl(1)]2. However, no relative shifts in the positions of the peaks are produced by the adoption of RA1 or RA3 antennas. This is due to the fact that with these antennas radiation improvement is nothing but the singular values [σl(1)]2 scaled by a constant.

[20] Adopting an RA1 reference antenna, i.e., a reference antenna for which the circumscribing sphere has radius a, yields relatively higher estimates of the performance levels with respect to the performance levels obtained by means of the remaining two reference antenna classes (see Figures 1–6). This is quite understandable because the size of an RA1 reference antenna is, by definition, smaller that the sizes of the corresponding RA2 and RA3 antennas.

[21] The simulations show also that the discrepancies between the estimates yielded by the three reference antenna classes are not significant in the immediate vicinity of the resonant peak which corresponds to bmax: the ratios between any two estimates, in particular those obtained by means of RA1 and RA3 antennas on one hand and RA2 antennas on the other hand, are less than one order of magnitude in favor of the estimates obtained by means of RA1 and RA3. (Note that RA2 and RA3 antennas yield identical singular values at d = dmax.) It is only when one attempts to compare estimates yielded by the three reference antenna classes far away from this resonant peak that the discrepancies become significant (the ratios between the estimates obtained by means of the RA2 and RA3 antennas, for instance, are about one to two orders of magnitude larger, in favor of RA3 estimates, for radii ratios d > 1.8). Note also that the discrepancies are more pronounced for higher multipole modes (see Figures 1–6). Thus for RA3 reference antennas, i.e., for those which have a circumscribing volume of radius bmax, to yield a fair comparison they have to be used locally. In other words, it is not suitable to use a value of bmax which corresponds to a maximum enhancement that appears for a specific range of shell thickness values as an absolute reference for enhancement level characterization. The discrepancies would be even larger if bmax corresponded to an infimum of the set of shell thickness values at which resonant peaks appear in the singular values spectrum.

[22] These observations are true for all multipolar modes and for small (quarter-wavelength) and large (full-wavelength) antennas alike. (In the simulations we also investigated 5λ-antennas but the results are not shown here.) They lead us to the conclusion that RA2 antennas, i.e., the antennas radiating in free space and having a circumscribing volume of radius b, are more convenient as reference antennas, especially when one is interested in performance characterization over a wide range of shell thicknesses, though the adoption of an RA2 reference antenna may result in a slight shift of the resonant peaks position.

Appendix A:: Determination of Vector Fields equation imagel,m(j) for a Core-Shell System

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Inverse Source Theory Based on Constrained Optimization
  5. 3. Numerical Results and Case Studies
  6. Appendix A:: Determination of Vector Fields equation imagel,m(j) for a Core-Shell System
  7. Appendix B:: Calculation of DFℰ(J) and DF((equation imagel,m(j), J) − al,m(j))
  8. Acknowledgments
  9. References

[23] The aim of this appendix is twofold: (1) determine the exact expressions for the source-independent fields equation imagel,m(j) (r), and (2) show that the multipole moments al,m(j) are indeed given by equation (1).

[24] In the exterior region corresponding to ∣r∣ = r > b the electric field E(r) can be represented by the multipole expansion [Jackson, 1999; Devaney and Wolf, 1974]

  • equation image

where equation imager/r, al,m(j), j = 1,2, are the complex-valued multipole moments of the field, hl(+) denotes the spherical Hankel function of the first kind and order l (as defined by Arfken and Weber [2005]), and Yl,m is the vector spherical harmonic of degree l and order m (as defined by Devaney and Wolf [1974]).

[25] One way of arriving at the desired results is to invoke Lorentz's reciprocity theorem and the concept of reaction [see, e.g., Balanis, 1989]. We conveniently consider, without loss of generality, the following two classes of canonical sources:

  • equation image

and

  • equation image

where in both expressions R > b represents the radius of the helper source centered around the origin. Ultimately, our calculation of the multipole moments al,m(j) will be independent of R. It can easily be shown that if R > b is the radius of a sphere centered about the origin, then

  • equation image

where Vl (k0R) is defined such that

  • equation image

The field [El,m(j)]inc that would be produced in free space by the source [Jl,m(j)]0 (defined in equations (A2) and (A3)) is given by

  • equation image

where equation image0(r, r′) is the free-space electric dyadic Green function [Tai, 1971].

[26] The fields [El,m(j)]0, j = 1,2, produced by [Jl,m(j)]0, j = 1,2, respectively, are obtained by imposing the continuity of the tangential components of the electric and magnetic fields on the inner and outer boundaries of the spherical shell. This yields two systems (one for j = 1 and one for j = 2) of four equations each linear in a set of four unknown coefficients. Upon solving the two linear systems of equations, one finds the expressions of [El,m(1)]0 and [El,m(2)]0 (they are omitted for the sake of space).

[27] By applying the reciprocity theorem to the preceding results one finds that the multipole moments al,m(j) are indeed independent of R and given by equation (1) with the source-free wave functions equation imagel,m(j) given by

  • equation image

where η0 is the free space impedence, ji is the spherical Bessel function of the first kind and order l (as defined by Arfken and Weber [2005]), equation image is the inner sphere substrate wave number, and where we have defined

  • equation image

where [cf. Aden and Kerker, 1951, equations (26) and (27); Alù and Engheta, 2005b, 2006, equations (8) and (9)]

  • equation image

and

  • equation image

In (A9) and (A10) the quantities Vl have already been defined in equation (A5) and the quantities Ul are defined such that

  • equation image

Note that in equations (A8), (A9), and (A10), equation image and equation image are the propagation constants in free space and in the shell, respectively. One also notes that with equation imagel,m(j) being given by equation (A7) one can express the singular values introduced in equation (10) as

  • equation image

where

  • equation image

Appendix B:: Calculation of DFℰ(J) and DF((equation imagel,m(j), J) − al,m(j))

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Inverse Source Theory Based on Constrained Optimization
  5. 3. Numerical Results and Case Studies
  6. Appendix A:: Determination of Vector Fields equation imagel,m(j) for a Core-Shell System
  7. Appendix B:: Calculation of DFℰ(J) and DF((equation imagel,m(j), J) − al,m(j))
  8. Acknowledgments
  9. References

[28] The Fréchet derivative and the Gateaux derivative of a functional f : L2(V; C3) [RIGHTWARDS ARROW] C at J0 are equal if the Gateaux derivative is continuous from L2(V; C3) into the space of bounded linear operators ℒ(L2(V; C3), C). This, in fact, will turn out to be the case for all the functionals considered here. In the sequel we shall use the Gateaux derivative because it is easier to compute.

[29] The Gateaux derivative of f at J0 in the direction v is defined as [Kurdilla and Zabarankin, 2005]

  • equation image

We should note, however, that when f : L2(V; C3) [RIGHTWARDS ARROW] R, as is the case for ℰ, there emerge difficulties related to the definition of linearity of the differentiation operator [Angell and Kirsch, 2004]. In that case, L2(V; C3) is viewed as a space over R instead of C [Angell and Kirsch, 2004], and the definition of the Gateaux derivative takes on the modified form

  • equation image

[30] Given these definitions, it is easy to show that

  • equation image

whereby we identify the Gateaux derivative of ℰ at J0, i.e., DGℰ(J0) with 2J0. Similarly, for the constraints we obtain

  • equation image

whereby we identify the Gateaux derivative of [al,m(j) − (equation imagel,m(j), J0)] with (−equation imagel,m(j)).

[31] Note that (B3) and (B4) clearly show that the Gateaux derivatives are continuous from L2(V; C3) into ℒ(L2(V; C3), C) which justifies our approach. Thus we can safely replace the Gateaux derivative with the Fréchet derivative in (B3) and (B4).

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Inverse Source Theory Based on Constrained Optimization
  5. 3. Numerical Results and Case Studies
  6. Appendix A:: Determination of Vector Fields equation imagel,m(j) for a Core-Shell System
  7. Appendix B:: Calculation of DFℰ(J) and DF((equation imagel,m(j), J) − al,m(j))
  8. Acknowledgments
  9. References
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