Inverse Source Problem in an Oblate Spheroidal Geometry - CiteSeerX

Inverse Source Problem in an Oblate Spheroidal Geometry Johan C.-E. Sten1 and Edwin A. Marengo2 1

2

Technical Research Centre of Finland VTT Information Technology P. O. Box 1000 FI-02044 VTT, Finland

Department of Electrical and Computer Engineering Northeastern University Boston, MA 02115, USA

Abstract The canonical inverse source problem of reconstructing an unknown source whose region of support is describable as a spheroidal (oblate or prolate) volume from knowledge of the far-field radiation pattern it generates is formulated and solved within the framework of the inhomogeneous scalar Helmholtz equation via a linear inversion framework in Hilbert spaces. Particular attention is paid to the analysis and computer illustration of flat, aperture-like sources whose support is approximated by an oblate spheroidal volume.

Key words: Inverse source problem, minimum energy source, nonradiating source, spheroidal wave.

1

1

Introduction

The linear inversion scheme established in [1]-[3] for the solution of the inverse source problem, consisting of the reconstruction of the timeharmonic source distribution having spatial dependence Q(r) that is localized within a given spatial support region V0 and that generates a prescribed (exterior) field ψ(r) for r ∈ / V0 is applied in this paper to the reconstruction of scalar sources confined within a spheroidal support region. In the past, treatments of this problem have focused on the case of a spherical source volume [2, 3, 4, 5] for which the problem has been naturally formulated in terms of spherical (multipole) wave expansions [6]. The present research, which is a continuation of a previous contribution [7] authored by one of us (J.C.-E.S.), is a generalization of those formulations in that it allows the shape of the source to be taken into account via the adoption of the more “conformally suited” spheroidal coordinate system. This approach is motivated mainly by the fact that the spheroid comprises a wide variety of technically interesting shapes, ranging from a thin needle (e.g., a linear source or array), to a sphere, to a flat, circular disk (e.g., a radiating aperture). The present treatment along with the earlier work in [7] then expands analytic and computational schemes for the formulation and solution of the inverse source problem into a broader class of source configurations than those formally tractable with existing approaches. Unlike in [7] where the focus was the full vector case and prolate spheroidal configurations, in the present work the focus is the scalar case and oblate spheroidal geometries. In the adoption of the scalar model significant simplifications arise which facilitate the route for more formal (closed form) developments than those in [7] which, on the other hand, do not severely limit the conceptual framework and certain applications. The present effort is motivated mainly by potential applications to antenna characterization and synthesis, including calculations of antenna quality factor and communication capacity for certain antenna systems which due to their shape can be ideally characterized using the spheroidal coordinate system. Thus, questions of Cram´er-Rao bounds for the estimation error in inverse source problem reconstructions involving noise [10] and of the diffraction of electromagnetic information from volumes [11, 12, 13, 14, 15] can also be more generally tackled using the general approach of the current work which holds not only in the familiar spherical source region case which has been the focus in many of these efforts but also in other situations where the use of more general spheroidal source regions is preferable. In fact, in [16, 17] where the spheroidal coordinate system has been used, tighter bounds in antenna quality factor have been found relative to those corresponding to the same device in the spherical coordinate description [18, 19, 20, 21]. In particular, the problem at hand, involving sources and fields

2

with suppressed harmonic time-dependence, is governed by the inhomogeneous scalar Helmholtz equation in three-dimensional free space, (∇2 + k 2 )ψ(r) = −Q(r),

(1)

where ∇2 denotes the Laplacian operator and where the wavenumber of the field k = ω/c0 , where ω is the angular oscillation frequency and c0 is the free space speed of wave propagation. The solution of (1) that obeys the Sommerfeld radiation condition [22, p. 479] is given by the well-known Green function integral Z ψ(r) = Q(r0 )G(r, r0 )dV 0 (2) V0 0

where dV denotes volume differential element and G is the outgoing wave Green function of the Helmholtz operator in (1). It has been known for a long time [23, 24] that there are certain source distributions, called nonradiating sources, that do not radiate. Despite their zero radiation nature, such nonradiating sources store reactive field energy within the source region, and thereby influence the source radiation impedance, quality factor and energy dissipation [1, 25]. As a corollary to the impossibility to “detect” nonradiating sources, the inverse source problem is inherently nonunique [4, 8, 9]. It is well known [1]-[7], nevertheless, that a unique solution to the inverse problem can be found by imposing the additional requirement of minimizing the source L2 norm, i.e., the quantity  1/2 Z  |Q(r)|2 dV  .

(3)

V0

This unique source of minimum L2 norm corresponds to the normal solution to the inverse problem [1, 26] and is, in general, termed the minimum-energy source [1, 2, 9], where “energy” refers here to functional energy and is not to be confounded with real, physical energy. Other solution constraints, such as functional regularity or smoothness constraints, can be imposed which also lead to uniqueness [27], as well as constraints on the reactive power supported by the source. In this work we formulate the inverse problem within the scalar and spheroidal coordinate frameworks as the reconstruction of the minimum-energy part of the source along the lines of a linear inversion formalism in Hilbert spaces, leaving the more involved question of extension of this work to the reactive power constraint case applicable to the respective fully vectorial framework for the future.

3

2

The Forward Problem

To solve any inverse problem it is necessary to clearly state first the associated forward problem relating the unknown quantity (in this work, the source Q) to the available data (in this work, the external field ψ(r) for r ∈ / V0 where V0 is a rather arbitrary source region which in the following will be taken to be a spheroidal volume centered about the origin). We formulate next the pertinent forward or radiation problem.

2.1

Geometry and Coordinates

It is assumed that the source Q(r) is non-zero only if r ∈ V where V is the interior volume bounded by the spheroid defined in terms of the Cartesian coordinates (x, y, z) by x2 + y 2 z2 + =1 b2 c2

(4)

where c and b denote, respectively, the polar and equatorial radius of the spheroid. The source shape is called an oblate if c < b and a prolate if c > b, and the case b = c corresponds to a spherical region of support which is the case considered in most past treatments of the inverse problem [2, 3, 4, 5]. In the following general framework many of the results will hold for both oblates and prolates, but later as the discussion advances we shall focus mainly on oblates. Thus the numerical simulation part, Sect. 4, will be devoted exclusively to oblates. A natural choice of coordinate system for the corresponding analysis is the spheroidal system (f ; ξ, η, ϕ) defined in relation to the more familiar Cartesian system by the transformations [28, 29] p x + jy = f ejϕ (ξ 2 ± 1)(1 − η 2 ), z = f ξη, (5) √ where f denotes the semifocal distance | c2 − b2 | and where here and henceforth the upper sign relates to oblates while the lower sign applies to prolates. The “angular coordinates” η ∈ [−1, 1] and ϕ ∈ [0, 2π], while the “radial coordinate” ξ ∈ [1, ∞) in the prolate system, and ξ ∈ [0, ∞) in the oblate system. Surfaces of constant ξ are spheroids. For example, the spheroid ∂V defined by (4) has ξ = ξ0 ≡ c/f .

2.2

Spheroidal Wave Expansion Solution

In spheroidal coordinates the Helmholtz operator (∇2 + k 2 ) in (1) is separable; in particular, the free space Green function G can be expanded in a set of spheroidal wave modes as [28, 29]

4

∞ n jk X X 2 G(r, r ) = − γ 4π n=0 m=−n mn

(

0

(4)

(1)?

Ψmn (kf, r)Ψmn (kf, r0 ), (1) (4)? Ψmn (kf, r)Ψmn (kf, r0 ),

where: 1) ? denotes complex conjugation; 2) the term s (2n + 1)(n − |m|)! γmn = (n + |m|)!

ξ > ξ0 ξ < ξ0 (6)

(7)

is a normalization factor; (4) 3) the functions Ψmn are the spheroidal wavefunctions of outgoing fields which are defined by [29] (4) −jmϕ Ψ(4) mn (kf, r) = Rmn (kf ; ξ)Smn (kf ; η)e

(8)

where the functions Smn represent angular spheroidal functions that are derived from the associated Legendre functions Pnm [30] via [28, 29] Smn (kf ; η) =

∞ X

|m|

d|m|n (kf )P|m|+p (η) p

(9)

p=0 |m|n

are defined using methods established, e.g., where the coefficients dp in [28], and normalized as in [30, Eq. (21.7.12)] and where the functions (4) Rmn are radial spheroidal functions of the fourth kind, defined in terms (2) of the spherical Hankel function hn [30] as (4) Rmn (kf ; ξ)

·

∞ X

j p+|m|−n

p=0

µ ¶|m|/2 (n − |m|)! ξ 2 ± 1 = (n + |m|)! ξ2

(2|m| + p)! |m|n (2) dp (kf ) h|m|+p (kf ξ); p!

(10)

and (1) 4) the functions Ψmn are the (source-free) spheroidal wavefunctions (4) of the first kind, given by (8) and (10) with the substitutions Ψmn → (1) (2) (4) (1) Ψmn , Rmn → Rmn and hn → jn where jn denotes the spherical Bessel function (of the first kind) of order n. It follows from the radiation integral (2) and the Green function expansion (6) in terms of spheroidal wavefunctions that the field ψ radiated by a source Q confined within the spheroidal volume V defined in connection with (4) can be expressed everywhere outside V as ψ(r) =

∞ X n X

amn Ψ(4) mn (kf, r)

n=0 m=−n

5

(if r ∈ / V)

(11)

where the spheroidal mode expansion coefficients amn or “generalized multipole moments” are given by Z jk 2 0 0 amn = − γmn Q(r0 )Ψ(1)? (12) mn (kf, r )dV . 4π V

The latter expression is a complete statement of the solution of the forward or radiation problem for this class of sources, and will, in fact, be our starting point in formulating in the following the associated inverse problem as the inversion of a linear mapping between continuous source and discrete data Hilbert spaces. To formalize this, it is useful to briefly examine also the mapping from the source to the far field.

2.3

Source-to-Far-Field Radiation Pattern Mapping (2)

Just as the spherical Hankel functions hn (kr), also the radial spher(4) oidal functions Rmn (kf ; ξ) satisfy the Sommerfeld radiation condition, in particular, Z ∂ (4) (4) 2 lim | Rmn + jkRmn | dS = 0 (13) ξ→∞ ∂ξ Sr

where Sr is a sphere of radius r centered at the origin. In fact, at large (2) distances from the spheroid, that is, when kf ξ ∼ kr → ∞, hn and (4) Rmn approach the same asymptotic form ) (2) j n+1 −jkr hn (kr) → e (14) (4) kr Rmn (kf ; ξ) By employing (14) in (11) when kf ξ ∼ kr → ∞, while letting η → cos θ, where θ is the spherical polar coordinate, one arrives at the far zone asymptotic behaviour ψ(r) ∼ where F(θ, ϕ) = j

∞ X n X

e−jkr F(θ, ϕ) kr

(15)

amn j n Smn (kf ; cos θ)e−jmϕ

(16)

n=0 m=−n

is the far-field radiation pattern. Owing to the orthogonality property of the angular spheroidal functions Smn [28], it also follows from (16) that the set of coefficients {amn } corresponding to a given far-field radiation pattern F(θ, ϕ) can be obtained from amn

(−j)n+1 =− 2πΛmn

Z2πZπ F(θ, ϕ)Smn (kf ; cos θ)ejmϕ sin θdθdϕ 0

0

6

(17)

where Λmn is the constant [28] Λmn =

∞ X

2(p + 2m)! (d|m|n )2 . p (2p + 2m + 1)p! p=0

(18)

It follows that not only the expansion coefficients amn uniquely define via (11) the external fields ψ(r) everywhere outside the spheroidal source volume V , along with their associated far-field radiation pattern F(θ, ϕ) via (16), but also the far fields uniquely define via (17) the same expansion coefficients. The latter observation assumes noiseless conditions. In contrast, in the presence of noise or other perturbations, including numerical round-off errors, the informational equivalence of far-field data and near-field data does not hold due to the ill-posedness of the problem of inverse diffraction from far-field data. In particular, in practice the evanescent portion of the plane wave spectra which is theoretically contained (due to analyticity) in the far-field radiation pattern (the propagating spectrum) cannot be reliably deduced from it so that the super-resolution near-field information about the source is lost. In the following we formulate the inverse source problem in essentially finite-dimensional data spaces, restricting attention to a finite number of multipoles which can be reliably extracted from the data, be it in the near or far zone, corresponding to the particular noise level and application at hand.

3

Linear Inversion in Hilbert Spaces

In this section the minimum-energy solution to the inverse source problem whose L2 norm or functional energy is minimal among all solutions to the inverse problem is derived within the framework of a linear inversion formalism in Hilbert spaces. Borrowing from (11), (16) and (17), one can formulate the inverse source problem as being that of deducing an unknown source Q from knowledge of the set of all generalized multipole moments, which we denote by the vector a ¯ ≡ {amn }. We will consider this problem in an infinite-dimensional data space first, but following on the practical considerations outlined at the end of the preceding section we will at a future point in the development approximate the formulation as an inversion from a finite number of effective multipole moments only.

3.1

Hilbert Spaces

It is convenient to introduce next the Hilbert space X of L2 source functions Q located within the spheroidal volume V to which we assign the inner product

7

Z hQ|Q0 iX =

Q? (r0 )Q0 (r0 )dV 0 ,

(19)

V 0

valid for any Q ∈ X and Q ∈ X. Similarly, we introduce the Hilbert space Y of square summable vectors a (representing an infinite sequence whose squared amplitudes, when summed, form a convergent series [31]) for which the inner product ha|a0 iY =

∞ X n X

a?mn a0mn ,

(20)

n=0 m=−n

where a ∈ Y and a0 ≡ {a0mn } ∈ Y , is defined. The forward problem (12) can now be stated in terms of the linear mapping as LQ = a where the mapping L : X → Y is defined such that Z jk 2 0 0 (LQ)mn = − γmn Q(r0 )Ψ(1)? mn (kf, r )dV . 4π

(21)

(22)

V †

The adjoint mapping L : Y → X of L, being defined through hLQ|a0 iY = hQ|L† a0 iX

(23)

is found from (19), (20) and (22) to be defined by ∞ n jk X X 2 (L a)(r) = γ amn M (r)Ψ(1) mn (kf, r) 4π n=0 m=−n mn †

(24)

where we have introduced the indicator or masking function M (r) whose value is 1 if r ∈ V and is zero otherwise.

3.2

The Inverse Problem

The inverse problem associated to (21), i.e., that of deducing Q for a given a, does not generally admit a unique solution due to the mathematical existence within the source support V of nonradiating sources QNR whose generated external fields (and the respective generalized multipole moments) vanish identically, i.e., (LQNR )mn = 0 for all n ∈ {0, 1, · · · ∞} and m ∈ {−n, −n + 1, · · · , 0, · · · , n − 1, n} [24]. Consequently, if Q is a solution to the inverse problem, then Q + QNR is also a solution. On the other hand, the inverse problem becomes unique if one imposes the additional constraint of minimizing the source energy as

8

described in connection with (3). The minimum-energy solution in question is given by [1, 2, 3, 26] ˜ QME = L† a

(25)

which accounts for a generalized filtered backpropagation (via the adjoint mapping L† ) of the field data a where a ˜ = (LL† )−1 a

(26)

is the “filtered data vector” (obtained by filtering the data vector a via the operation (LL† )−1 ). In order to determine the filtered data vector a ˜ one has to form the equation LL† a ˜ = a, in particular, from (22) and (24) one deduces that amn =

∞ X ∞ X

αmnµν a ˜µν

(27)

ν=0 µ=−∞

where the coefficients αmnµν are defined by µ αmnµν = Iµν

kγmn γmν 4π

¶2 D

E 0 (1) 0 Ψ(1) mn (kf, r )|Ψµν (kf, r )

(28)

where Iµν is an indicator function that is equal to one if |µ| ≤ ν and is zero otherwise. Now, from orthogonality of the wavefunctions in the ϕ-coordinate it follows that αmnµν = 0 unless µ = m. Therefore, for a given index m one finds from (27) that      am0 αm0m0 . . . αm0mν a ˜m0  ..     ..  .. .. .. (29)  . =  .  . . . amn

|

αmnm0

. . . αmnmν {z LL† [m]

}

a ˜mν

where n → ∞ and ν → ∞. To compute the filtered data vector by matrix inversion it is necessary to truncate the equation so that n ≤ no ∼ kb for an oblate or n ≤ no ∼ kc for a prolate, where the value no measures the electrical radius of the smallest sphere enclosing the spheroid. The truncation in the condition n ≤ no ∼ max(kb, kc) corresponds to standard numerical filtering or regularization which stabilizes the source inversion against data fluctuations (noise), as is well known [14, 26, 32]. The elements αmnmν of the “filtering matrix” LL† can be written as

9

αmnmν

1 2 2 = Imν k 2 f 3 γmn γmν 8π

Zξ0 Z1 (1) (1) Rmn Rmν Smn Smν (ξ 2 + η 2 )dη dξ 0 −1

(30) in the oblate case and as

αmnmν

1 2 2 = Imν k 2 f 3 γmn γmν 8π

Zξ0 Z1 (1) (1) Rmn Rmν Smn Smν (ξ 2 − η 2 )dη dξ 1 −1

(31) in the prolate case. For brevity, the arguments ξ and η of the functions Rmn (kf ; ξ) and Smn (kf ; η) have been omitted. Also, we remind the reader that ξ0 = c/f which corresponds to the constant coordinate surface for the spheroid in (4). A computer code for spheroidal functions developed by Li et al. [33] is used for computing the matrix elements (30) in the examples given in Section 4. Because Smn (kf ; η) is per se orthogonal to Smν (kf ; η) over the interval η ∈ [−1, 1], but not in combination with the weight η 2 , modes of different n and ν generally have a nonzero coupling. There is an interesting exception that is worthwhile pointing out. In particular, for the largest value of n in the data, i.e., n = no , then m runs from −no to no and for m = no or −no the quantity Imν = 0 unless ν = no so that ano no = αno no no no a ˜no no and a−no no = α−no no −no no a ˜−no no and thus for this special case the data and filtered coefficients are conveniently related by a simple factor (which acts as a corresponding singular value). Thus, the whole filtering operation established in Eqs. (26)-(29) and (30),(31) implies the general structure   a 00

. . .

   a  0no   a11     . .    a.     1no K   · · · 00· · · . .    0 .  a   0  = 0 −1 1    . . 0    a .  0 −1 no   0   . .   . a   no −1 no −1   ano −1 no  a−no +1 no −1   a−no +1 no 

···0··· K1 0 0 0 0 0 0

0 0 K−1 0 0 0 0 0

ano no a−no no

0 0 0 ··· 0 0 0 0

0 0 0 0 Kno −1 0 0 0

0 0 0 0 0 K−no +1 0 0

0 0 0 0 0 0 Kno 0

a ˜ 00 . . . a ˜ 0no a ˜ 11 . . . a ˜ 1no . . . a ˜ −1 1 . . . a ˜ −1 no . . .



                 0   0   0    0   0   0    0   K−no      a˜   no −1 no −1   a˜no −1 no  a˜−no +1 no −1   a˜−no +1 no  (32)

where Km , m = −no , −no + 1, · · · , 0, · · · no + 1, no is an (no − m + 1) × (no − m + 1) matrix that needs to be inverted in order to complete the

10



a ˜ no no a ˜ −no no

data filtering step Eq. (26). For |m| = no the respective matrix Km is 1 × 1, i.e., a scalar, as we have discussed above, while for |m| = no − 1 the corresponding matrix is 2 × 2 and so on. By inverting the matrix ˜ and substitute in (32) one can put the data vector a in filtered form a this result into (25) which completes the inversion algorithm. In the particular case that the spheroid is turned into a sphere of radius r0 , there is a singular correspondence between the filtered and unfiltered data. By taking the limit f → 0, and consequently ξ → ∞ of both (31) and (30) we get

αmnmν

1 2 2 = Imν k 2 γmn γmν 8π

Zr0

Zπ 2

Pnm (cos θ)Pνm (cos θ) sin θdθ

jn (kr)jν (kr)r dr 0

0

(33) for the sphere. Finally, as a result of the associated Legendre functions of different n, ν being orthogonal [30], i h 1 2 (34) jn2 (kr0 ) − jn−1 (kr0 )jn+1 (kr0 ) Imn k 2 r03 γmn 8π when ν = n, but otherwise αmnmν = 0. This result for the sphere has been derived earlier [2] using a slightly different notation. αmnmn =

4

Application to a Thin Spheroidal Source

Our next goal is to illustrate the workings of the theory in Sect. 3 from both the analysis (radiation) and source synthesis points of view for a number of canonical source distributions that can be handled partly analytically and partly numerically. In particular, using the portion of the theory in Sect. 3 corresponding to essentially finitedimensional data spaces, we shall recover the minimum-energy part of the sought-after source which is consistent with the given finite number of multipole moments which we assume to be reliably measurable. The minimum-energy source obtained this way will not in general coincide with the original source (the inverse problem is nonunique). In particular, since the original source and its respective minimum energy component produce the same measurable multipole moments (the field data), the generally non-trivial source obtained by subtracting the minimum-energy source from the original can be regarded as “essentially nonradiating” since its measurable multipole moments are zero. We wish to note that this source is not a true nonradiating source in the same sense as those studied in, e.g., [1, 4, 5, 24]; in particular, this “essentially nonradiating” source is nonradiating only to a finite number of multipole modes while the truly nonradiating sources are nonradiating to all of them.

11

4.1

Thin Disk Approximation

Specifically, let us apply the previous formulation to the special case of radiators having the structure of a thin circular disk as is relevant, e.g., in applications using radiating apertures, planar broadside antenna arrays or even quasi-planar three-dimensional antennas [34]. Thus we shall emphasize oblate sources. A similar presentation based on prolate spheroidal wavefunctions (see also [7]) applies to elongated radiators such as certain traveling-wave antennas and endfire antenna arrays. Referring to the source geometry depicted in Fig. 1, we will model the radiating disk as a radiator confined within a thin oblate spheroid for which h ¿ a where h is the thickness of the radiator and a is its radius. To match the shape of an oblate spheroid the thickness h of the disk must be a function of the distance to its centre, ρ. The associated source intensity Q, on the other hand, will be made a function of both ˆ ρ and the azimuth angle ϕ as follows: Q(ρ, ϕ) = Q(ρ) exp(−jmϕ). By applying the far-field argument |r − r0 | ≈ r − ur · r0 in the Green function G in (2) together with the thin-disk approximation, so that within the integration volume V the position vector r0 is always held to be transverse, i.e., r0 ≈ ρ0 , it follows that the far-field of an oscillating thin circular disk, having its centre at the origin of coordinates (see Fig. 1) can be expressed for constant far-field radial distance r as h

e−jkr ψ far-f (θ, ϕ) ≈ 4πr

=

Z2π Z2 Za ˆ 0 )ejkρ0 sin θ cos(ϕ0 −ϕ) ρ0 dρ0 dz 0 dϕ0 Q(ρ 0 −h 0 2

e−jkr m −jmϕ j e 2r

Za ˆ 0 )h(ρ0 )Jm (kρ0 sin θ)ρ0 dρ0 Q(ρ

(35)

0

where the integral definition of the Bessel p function Jm [30] has been employed. We also specify h(ρ) = h0 1 − (ρ/a)2 so that the disk matches p the shape of an oblate spheroid, whose semi-focal distance is f = a2 − (h0 /2)2 . For certain formally convenient choices of the ˆ functional form of the source intensity distribution Q(ρ) we will be able to evaluate the radiation pattern in closed form.

4.2

Radiation Patterns

The simplest radiation patterns are obtained for m = 0, generating fields which are rotationally symmetric around the z-axis. Accordingly, ˆ we assign Q(ρ)h(ρ) = qh0 , i.e.p a constant, which means that the source ˆ intensity must be Q(ρ) = q/ 1 − (ρ/a)2 . Then, by integrating the

12

formula [30]

d m (x Jm (kx)) = kxm Jm−1 (kx), dx Eq. (35) gives us the field ψ far-f (θ) ≈

e−jkr qh0 a J1 (ka sin θ) kr | 2 {zsin θ } exa 1 =F (θ)

(36)

(37)

where, by F exa 1 (θ), we denote our example 1 far-field radiation pattern. We remark that this expression would have resulted from any ˆ choice of the product Q(ρ)h(ρ) =constant, but since h(ρ) is fixed as it ˆ is, Q(ρ) is determined as well. With the idea of illustrating sources that generate similar far-field radiation patterns while having rather distinct internal source structures, it is worthwhile to also consider another kind of rotationally symmetric source distribution (m = 0), namely one that radiates a field quite similar to that of the previous example, F exa 1p (θ). The disˆ ˆ tribution Q(ρ) = q, or a constant and Q(ρ)h(ρ) = qh0 1 − (ρ/a)2 , turns out to be a distribution with the desired property. The relevant radiation pattern may be found by the aid of the following definite integral [35, 36] Z1 0

√ Pn ( 1 − x2 ) Γ((n + 1)/2) Jn+1/2 (k) √ √ √ J0 (kx)x dx = 2 Γ(n/2 + 1) 2 1−x k

(38)

valid for n even, where Pn is the Legendre polynomial and Γ the √ Gamma function [30]. In √fact, by rewriting 1 − x2 = (P0 (y) + 2P2 (y))/(3y), denoting y = 1 − x2 for short, it follows from equations (15), (35) and (38) that the corresponding far-field radiation pattern is F exa 2 (θ) ≈

qh0 a sin(ka sin θ) − ka sin θ cos(ka sin θ) 2 (ka)2 sin3 θ

(39)

which represents our second field pattern example. As will be shown in the numerical examples in Sect. 4.3, the two patterns F exa 1 (θ) and F exa 2 (θ) have similar features, despite the difference of the expressions (37) and (39), as well as that of the corresponding source functions. In the long wavelength limit, ka → 0, the patterns approach constants of magnitudes F exa 1 (θ) → (ka)2 /4 and F exa 2 (θ) → (ka)2 /6, i.e., their relation approaches 3:2. The similarity of the two patterns means that the corresponding minimum-energy sources must also be quite similar; thus the comparison of the difference

13

between the original source and the minimum-energy part (i.e., the essentially nonradiating part of the source) for the two source examples can reveal visible differences. As a third and last field example let us consider a higher-order radiation pattern such as a dipolar pattern, obtained by setting m = 1, and whose source and field display a sine/cosine-variation in ϕ. By ˆ assigning Q(ρ)h(ρ) = qh0 ρ/a, or a linear function of the distance from the centre of the disk, we obtain from (15), (35) and (36) for this, third example, the following radiation pattern: F exa 3 (θ, ϕ) ≈ jejϕ

4.3

qh0 a J2 (ka sin θ) . 2 sin θ

(40)

Sources of Different Size and Shape

In the following the three radiation patterns, their original sources and the corresponding minimum-energy sources, reconstructed from the far-field patterns, are examined under two scenarios, for which the electrical size and the shape of the source differ as follows. • I. A small disk in terms of wavelengths; ka = 0.51 and kf = 0.5. The source surface coordinate is ξ0 ≈ 0.2009975, corresponding to an ellipticity (axial-ratio) of 0.197. • II. A resonant size disk, with ka = 4.01 and kf = 4.0, and ξ0 ≈ 0.070755, the ellipticity being 0.071. Case I: First let us consider the two rotationally symmetric patterns F exa 1 and F exa 2 . The polar diagram of Fig. 2 shows the far-field intensity distribution in the xz-plane. Both patterns are omnidirectional, and the relation between their magnitudes is almost exactly 3 : 2, as in the theoretical limit derived in Sect. 4.2. The relevant generalized multipole moments of the spheroidal waves obtained from (17) are given below to serve as reference; F exa 1 corresponds to a00 ≈ −0.0126j, a02 ≈ −0.0000787j, and F exa 2 to a00 ≈ −0.00844j, a02 ≈ −0.0000908j, neglecting the multipoles for n ≥ 4, as their inclusion will only lead to increasing instability. Since the lowest order (n = 0) spheroidal mode is predominant in the long-wavelength limit of both F exa 1 and F exa 2 , it is not surprising to find that the quotient of the magnitudes of the respective a00 -coefficients is approximately 1.493, or almost 3 : 2. Upon specifying kf = 0.5 the matrix (29) evaluated for m = 0 becomes µ ¶ 0.0006885 7.654 · 10−6 † LL [m = 0] ≈ (41) 7.654 · 10−6 1.782 · 10−7

14

A singular value decomposition of LL† , such that LL† = USUT and UUT = I, yields ¶ ¶ µ µ 0.6886 0 0.9999 −0.0111 −3 S ≈ 10 U≈ (42) 0 0.00009 0.0111 0.9999 affirming the dominant role of the lowest order term in the expansion. Using LL† the filtered coefficients are found to be a ˜00 ≈ −25.65j, a ˜02 ≈ 659.2j for field example 1, and a ˜00 ≈ −12.62j, a ˜02 ≈ 32.48j for field example 2. The corresponding minimum energy solutions represented in Figs. 3 and 4, respectively, can be seen to mimic the features of the original source quite faithfully, except near the infinities at the rim of the spheroid. A more accurate description of these details would require higher order modes to be incorporated in the expansion. The reconstructed minimum energy solution of Fig. 4 shows an impressing resemblance with the original source. Thus, the source is almost entirely radiating, which turns out to be a general property of homogeneous low frequency sources [1, 2]. The dipolar radiation pattern of F exa 3 (θ, ϕ), expressed by (40), is represented in Fig. 5 (magnitude of the imaginary part). The generalized multipole moments are in this case a11 ≈ −0.00163j, a13 ≈ 7.33 · 10−7 j, neglecting moments of order n = 5 and higher. In Fig. 6 are represented the intensity of the original source distribution Q(ρ, ϕ)h(ρ), i.e., the “thickness compensated source intensity”, and the corresponding quantity reconstructed from the radiation pattern, when the source intensity Q has been evaluated in the xy-plane (ξ = 0). This quantity can actually be regarded as an approximation of the surface charge density corresponding to a tangential current distribution upon a circular disk. By comparing the two distributions, the recovered minimum energy source shows again good resemblance with the true source, indicating that the most part of the source is radiating. Case II: Fig. 7 displays the radiation patterns of the rotationally symmetric fields F exa 1 and F exa 2 when ka = 4.01. Here the two patterns have also similar features; sharp lobes appear at θ = 0±180◦ and small amplitudes at θ = ±90◦ . The generalized multipole coefficients for n = 0, 2, 4, 6 are in field example 1: a00 ≈ −0.1565j, a02 ≈ 0.4445j, a04 ≈ 0.03646j and a06 ≈ −0.000450j; and, correspondingly, in field example 2: a00 ≈ −0.12545j, a02 ≈ −0.02456j, a04 ≈ 0.02341j and a06 ≈ −0.000062j. The relevant matrix (29) for m = 0 and kf = 4.0 involving the modes n = 0, 2, 4, 6 can be expressed in a singular value decomposition

15

LL† = USUT as 

0.5611 0 0 0  0 0.0316 0 0  S≈ 0 0 0.00035 0 0 0 0 0.49 · 10−6  0.9911 −0.1327 0.0091 −0.0003  −0.1329 −0.9845 0.1145 −0.0039 U ≈  0.0063 0.1148 0.9926 −0.0383 0.0000 0.0005 0.0385 0.9993

   

(43)

   

(44)

whence it can be seen that the modes n = 0, 2, 4 are the predominant ones, and that the strongest coupling takes place between modes n = 1 and n = 3, as well as between modes n = 3 and n = 5. The filtered coefficients for the field example 1 are hence: a ˜00 ≈ 5.729j, a ˜02 ≈ 69.03j, a ˜04 ≈ 517.3j, and a ˜06 ≈ −7172j. In field example 2, correspondingly, a ˜00 ≈ 0.6062j, a ˜02 ≈ 11.54j, a ˜04 ≈ 119.3j, and a ˜06 ≈ −1675j. Figs. 8 and 9 show, respectively, the source intensity distributions and their corresponding minimum energy sources. Despite the differences in the original source distributions, the minimum energy sources are quite similar in consequence of the similarity of the radiated field in these two examples. It is also clear that in these examples the essentially nonradiating components are very marked relative to the preceding low frequency results (case I). Finally, we consider the third example, corresponding to radiation pattern F exa 3 (θ, ϕ) with ka = 4.01, which is represented in Fig. 10 (magnitude of the imaginary part), and whose most significant generalized multipole moments are a11 ≈ −0.3999j, a13 ≈ −0.00827j, and a15 ≈ 0.00102j. Fig. 11 represents the intensity of the “thickness compensated” source intensity of the original source and the corresponding quantity reconstructed from the radiation pattern, when the source intensity Q has been evaluated in the xy-plane (ξ = 0). The recovered minimum-energy source seems to have a greater likeness with the true source than in the two previous examples. This is attributed to the third source being less uniform (it has higher spatial frequency components) so that approximate representation via truncated wave modes in the support region is facilitated. In particular, all the examples above illustrate the fact that minimum-energy sources are waves truncated within the source support [1, 2]. Hence, objects that are intrinsically more wavelike can also be more accurately represented via such truncated wave modes.

16

5

Conclusion

The inverse source problem was investigated for scalar sources supported within a spheroidal volume. Our methodology provides an alternative approach to the more familiar spherical wave (multipole) representation, for the analytical and computational treatment of scatterers and antennas which due to conformal reasons can be best characterized using the spheroidal coordinate system. The present theory was developed for the case of scalar fields in order to facilitate interpretation; the key ingredients for a generalization to electromagnetic vector fields and sources have been outlined in [7] for elongated prolate sources. The present work focuses on oblates, however, and is expected to be of interest for certain remote sensing and antenna applications where polarization is not a primary concern. Furthermore, in choosing the scalar framework we have been able to explicitly carry out manipulations that for the respective vector formulation would have been rather cumbersome. This strategy has then facilitated analytical and computational corroboration of old and new results along with further examples which clarify the role of what we have termed “the essentially radiating and nonradiating parts” of a source as is applicable in practical settings including noise and other perturbations to the data. To illustrate the theory, numerical examples were given in which different radiation patterns (expressible in analytic form) generated by prescribed source distributions confined in a disk in the shape of a flat oblate spheroid were analyzed. Three example patterns—two rotationally symmetric and one three-dimensional—have been considered at two frequencies. The two rotationally symmetric examples, one corresponding to a source whose intensity displays a 1/d-type singularity at the disk’s edge, and another corresponding to a source that is uniform, were seen to produce almost identical radiation fields. Accordingly, the respective minimum-energy sources recovered from the far-field patterns were almost similar. Instead, a considerable difference emerges between the essentially nonradiating parts of each source. The third radiation pattern example given was a more complicated dipole-like field, relevant, e.g., to electromagnetic currents flowing across a circular disk, and the associated minimum-energy reconstruction was found to highly resemble the coarse features of the original source. Among other results, it was numerically verified within an essentially finite-dimensional data space context that the minimum-energy sources are truncated waves and that under low frequency conditions the sources are essentially almost purely radiating. However, they are so at the expense of much functional source energy and reactive energy relative to a larger source [25]. The addition of nonradiating components can actually reduce the reactive energy [1, 25] and hence such components are not to be seen as a negative ingredient in a source

17

but instead as an additional resource for antenna design flexibility. From the perspective of object reconstruction and imaging, on the other hand, nonradiating sources are a problem since they lead to nonuniqueness as we have been able to illustrate with two examples for which the generated fields are of the same general form, and yet the respective original sources are quite different functionally. As summarized earlier, in these cases, the respective minimum-energy source reconstructions were quite similar despite the original sources being also very different. Future work will consider the vector version of this and the preceding formulation in [7] when other solution constraints relevant to antenna theory and communications are considered. Of particular interest is the reactive power constraint pioneered in [25] for which a treatment within spheroidal coordinates may lead to tighter performance bounds (on both source functional energy and reactive power, the latter being “resources” that one must use for the radiation of a given, desirable field) than those derived for the formally simpler spherical volume case in [25]. Comparative analysis between the spheroidal and spherical source inversion cases and their respectively implied radiation performance bounds with given resources is also expected to be carried out within the body of the envisioned future work.

Acknowledgments E.A. Marengo acknowledges support from the USA Air Force Office of Scientific Research through Grant Number FA9550-06-01-0013, and his research is affiliated with the Center for Subsurface Sensing and Imaging Systems (CenSSIS), under the Engineering Research Centers Program of the USA National Science Foundation (award number EEC9986821.) J. C.-E. Sten thanks Northeastern University for supporting his recent invited academic stay at that institution during which this paper was finalized.

References [1] E.A. Marengo and R.W. Ziolkowski, “Nonradiating and minimum-energy sources and their fields: Generalized source inversion theory and applications”, IEEE Trans. Antennas Propag., AP-48, 1553-1562, 2000. [2] E.A. Marengo, A.J. Devaney and R.W. Ziolkowski, “Inverse source problem and minimum-energy sources”, J. Opt. Soc. Am. A, 17, 34-45, 2000.

18

[3] E.A. Marengo and A.J. Devaney, “The inverse source problem of electromagnetics: Linear inversion formulation and minimum energy solution”, IEEE Trans. Antennas Propagat., 47, 410-412, 1999. [4] N. Bleistein and J.K. Cohen, “Nonuniqueness in the inverse source problem in acoustics and electromagnetics”, J. Math. Phys., Vol. 18, 194-201, 1977. [5] E.A. Marengo, A.J. Devaney, and R.W. Ziolkowski, “New aspects of the inverse source problem with far field data”, J. Opt. Soc. Am. A, Vol. 16, 1612-1622, 1999. [6] A.J. Devaney and E. Wolf, “Multipole expansions and plane-wave representations of the electromagnetic field”, J. Math. Phys., Vol. 15, 234-244, 1974. [7] J.C.-E. Sten, “Reconstruction of electromagnetic minimum energy sources in a prolate spheroid”, Radio Sci., Vol. 39, RS2020, 2004. [8] A.J. Devaney, “Inverse source and scattering problems in ultrasonics”, IEEE Trans. Sonics Ultrason., Vol. 30, 355-364, 1983. [9] R.P. Porter and A.J. Devaney, “Holography and the inverse source problem”, J. Opt. Soc. Am., Vol. 72, 327-330, 1982. [10] S. Nordebo and M. Gustafsson, “Statistical signal analysis for the inverse source problem of electromagnetics”, IEEE Trans. Signal Processing, in press. [11] D.A.B. Miller, “Communicating with waves between volumes: evaluating orthogonal spatial channels and limits on coupling strengths”, Appl. Optics, Vol. 39, 1681-1699, 2000. [12] A.S.Y. Poon, R.W. Brodersen and D.N.C. Tse, “Degrees of freedom in multiple-antenna channels: A signal space approach”, IEEE Trans. Info. Theory, Vol. 51, 523-536, 2005. [13] A. Brancaccio, G. Leone and R. Pierri, “Information content of Born scattered fields: results in the circular cylindrical case”, J. Opt. Soc. Am. A, Vol. 15, 1909-1917, 1998. [14] R. Pierri and F. Soldovieri, “On the information content of the radiated fields in the near zone over bounded domains”, Inv. Probl., Vol. 14, 321-337, 1998. [15] L.W. Hanlen, A.J. Grant and R.A. Kennedy, “On capacity for single-frequency spatial channels”, pp. 446-451, ITW 2004 Conference Proceedings, San Antonio, Texas, USA, 2004. [16] J.C.-E. Sten, P.K. Koivisto, A. Hujanen, “Limitations for the radiation Q for small antennas enclosed in a spheroidal volume: axial ¨ Int. J. for Electronics and Comm., Vol. 55, polarisation”, AEU No. 3, 198-204, 2001.

19

[17] J.C.-E. Sten, “Radiation Q of a small antenna in an oblate spher¨ Int. J. for oidal volume: transverse-to-axis polarisation”, AEU Electronics and Comm., Vol. 57, No. 3, 201-205, 2003. [18] L.J. Chu, “Physical limitations of omni-directional antennas”, J. Appl. Phys., Vol. 19, 1163-1175, 1948. [19] R.F. Harrington, “Effect of antenna size on gain, bandwidth and efficiency”, J. Nat. Bur. Stand., Vol. 64D, 1-12, 1960. [20] R.E. Collin and S. Rothschild, “Evaluation of antenna Q”, IEEE Trans. Antennas Propag., Vol. AP-12, 23-27, 1964. [21] R. L. Fante, “Quality factor of general ideal antennas”, IEEE Trans. Antennas Propag., Vol. AP-17, 151-157, 1969. [22] J.D. Jackson, Classical Electrodynamics, third ed., John Wiley & Sons, New York, 1998. [23] F.G. Friedlander, “An inverse problem for radiation fields”, Proc. Lond. Math. Soc., Vol. 3, 551-576, 1973. [24] A.J. Devaney and E. Wolf, “Radiating and nonradiating classical current distributions and the fields they generate”, Phys. Rev. D., Vol. 8, 1044-1047, 1973. [25] E.A. Marengo, A.J. Devaney and F.K. Gruber, “Inverse source problem with reactive power constraint”, IEEE Trans. Ant. Propagat., Vol. AP-52, 1586-1595, 2004. [26] M. Bertero, “Linear inverse and ill-posed problems”, in P.W. Hawkes, ed., Advances in Electronics and Electron Physics, Vol. 75, Academic Press, New York, pp. 1-120, 1989. [27] E.A. Marengo and R.W. Ziolkowski, “Inverse source problem with regularity constraints: Normal solution and nonradiating source components”, Pure Appl. Optics: J. Optics A, Vol. 2, 179-187, 2000. [28] C. Flammer, Spheroidal Wave Functions, Stanford Univ. Press, Stanford CA, 1957. [29] L.-W. Li, X.-K. Kang and M.-S. Leong, Spheroidal Wave Functions in Electromagnetic Theory, J. Wiley, New York, 2002. [30] M. Abramowitz, and I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972. [31] G.B. Arfken and H.J. Weber, Mathematical Methods for Physicists, sixth ed., Elsevier Academic Press, Burlington, Massachusetts, 2005. [32] F. Soldovieri and R. Persico, “Reconstruction of an embedded slab from multifrequency scattered field data under the distorted Born approximation”, IEEE Trans. Antennas Propagat., Vol. 52, 2348-2356, 2004.

20

[33] L.-W. Li, M.-S. Leong, T.-S. Yeo, P.-S. Kooi, and K.-Y. Tan, “Computation of spheroidal harmonics with complex arguments: A review with an algorithm”, Phys. Rev. E, Vol. 58, 6792-6806, 1998. [34] E.A. Marengo, A.J. Devaney and E. Heyman, “Analysis and characterization of scalar, ultrawideband volume sources and the fields they radiate: Part II - Square pulse excitation”, IEEE Trans. Antennas Propag., Vol. AP-46, 243-250, 1998. [35] P. Wolfe, “Eigenfunctions of the integral equation for the potential of a charged disk”, J. Math. Phys., Vol. 12, No. 7, p. 1215-1218, 1971. [36] J. Boersma and E. Danicki, “On the solution of an integral equation arising in potential problems for circular and elliptic disks”, SIAM J. Appl. Math., Vol. 53, 931-941, 1993.

21

Figures and figure captions

z h/2 a x

-h/2

a y

Figure 1: Geometry of a flat spheroidal volume source.

22

radiation patterns at ka=0.51 90

0.3 60

120 Fexa 1 150

0.2 30

exa 2

F

0.1

180

0

210

330

240

300 270

Figure 2: Radiation patterns of a thin disk source as a function of θ when ka = 0.51. The solid line corresponds to field example 1 of Eq. (37) and the dashed line to field example 2 of Eq. (39). Magnitudes are normalized.

23

minimum energy part

3

3

2.5

2.5

2

2

magnitude

magnitude

original source

1.5

1.5

1

1

0.5

0.5

0.5 0 0.1 0 −0.1 z

0.5 0

0

0.1 0 −0.1

−0.5 x

z

0 −0.5 x

Figure 3: Cross-section view of the original source (left) of case I (ka = 0.51) of field example 1 and the corresponding minimum-energy part (right) recovered from the far-field.

24

minimum energy part

2

2

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2 magnitude

magnitude

original source

1 0.8

1 0.8

0.6

0.6

0.4

0.4

0.2 0 0.1 0 −0.1

0.2

0.5

0 0.1 0 −0.1

0 −0.5 x

0.5 0 −0.5 x

z

z

Figure 4: Cross-section view of the original source (left) of case I (ka = 0.51) for field example 2 and the corresponding minimum-energy part (right) recovered from the far-field. far−field pattern of ψexa 3 for ka=0.51

z

x y

Figure 5: 3D-radiation pattern of field example 3 (magnitude of imaginary part) when ka = 0.51. 25

original source

0.2

magnitude

0.1

0

−0.1

−0.2 1 0.5

1 0.5

0

0

−0.5

−0.5 −1

y

−1

x

minimum energy part

0.2 0.15

magnitude

0.1 0.05 0 −0.05 −0.1 −0.15 −0.2 1 1

0.5 0.5

0 0 −0.5 y

−0.5 −1

−1

x

Figure 6: Cross-section in the plane of the disk of Q(ρ, ϕ)h(ρ) of field example 3 for ka = 0.51: The original (above) and the reconstructed minimumenergy source (below).

26

radiation patterns at ka=4.01 90 2.5 60

120 2 1.5

30

150 1 0.5

exa 1

F

Fexa 2

180

0

210

330

240

300 270

Figure 7: Radiation patterns of the case II (ka = 4.01).

27

minimum energy part

12

6

10

4

8

2

6

0 magnitude

magnitude

original source

4 2

−2 −4

0

−6

−2

−8

−4

−10

4

4

2 −6 0.20 −0.2

2

−2

0.20 −0.2

−4 x

z

0

−12

0

−2 −4 x

z

Figure 8: Cross-section view of the original source (left) of case II (ka = 4.01) for field example 1 and the corresponding minimum-energy part (right) recovered from the far-field.

28

original source

4 2

3 2

magnitude

1.5 1 1

0 −1

0.5 −2 0 0.2 0 −0.2

−3

x

−4

z

minimum energy part

1

magnitude

0.5 4

0 3 −0.5

2 1

−1 0 −1.5 −1 −2 0.2 0 −0.2

−2 −3

x

−4

z

Figure 9: Cross-section view of the original source (uppermost) of case II (ka = 4.01) for field example 2 and the corresponding minimum-energy part (middle) recovered from the far-field.

29

z

far−field pattern of ψexa 3 for ka=4.01

x y

Figure 10: 3D-radiation pattern of F exa 3 (magnitude of imaginary part) when ka = 4.01.

30

original source

0.4 0.3

magnitude

0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 5 5 0 0

y

−5

−5

x

minimum energy part

0.4 0.3

magnitude

0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 5 5 0 0

y

−5

−5

x

Figure 11: Cross-section in the plane of the disk of the “thickness compensated source intensity”, Q(ρ, ϕ)h(ρ), for the field field example 3 with ka = 4.01: The original (above) and the reconstructed minimum energy source (below).

31