# Three-dimensional planar surface-current equivalence theorem with application to receiving antennas as linear differential operators

## Abstract

[1] A surface-current equivalence theorem that states that the electromagnetic fields outside a three-dimensional (3-D) source region can be generated (to any degree of accuracy) by electric and magnetic surface currents that lie in a single plane within the source region is proven by showing that the fields of each spherical electromagnetic multipole can be generated by delta-function electric and magnetic surface currents in an arbitrarily oriented plane of infinitesimal area located at the origin of the spherical multipole. The theorem can be used to justify integral equation methods based on equivalent surface currents to represent radiated and scattered fields. Application of the theorem to antennas reveals that the output of an arbitrary linear receiving antenna can be expressed in terms of just the transverse spatial derivatives of the transverse components of the incident electric and magnetic fields at a single point in space. The coefficients of the transverse linear differential operator are expressed in terms of the spherical multipole coefficients of the antenna's complex receiving pattern.

## 1. Introduction

[2] In the studies by Yaghjian and Hansen [1991] and Yaghjian [2001] a general equivalence theorem was derived that states that the electromagnetic fields outside the smallest circular cylinder that encloses the original sources on an infinitely long cylindrical scatterer can be generated (to any degree of accuracy) by electric and magnetic surface currents on a single, infinitely long planar strip oriented arbitrarily along an axis of the cylindrical scatterer and lying inside its source region. In the work of Yaghjian and Hansen [1991] this surface-current equivalence theorem was proven using surface-current representations for vector cylindrical wave expansions, and in the study by Yaghjian [2001] the proof was streamlined by combining the cylindrical wave expansions with the two-dimensional (2-D) Kottler-Franz formulas. The key features of the general theorem are that the planar strip can be confined to the original source region, and that both the equivalent electric and magnetic currents flow in the surface of the planar strip and not perpendicular to the planar surface. This equivalence theorem combines with a general even and odd decomposition of cylindrical electromagnetic fields to allow previous formulas [Shore and Yaghjian, 1988] for planar surface-current incremental length diffraction coefficients (ILDC's) to be used to derive ILDC's for arbitrary cylindrical scatterers in terms of their cylindrical far fields [Yaghjian and Hansen, 1991; Yaghjian, 2001; Shore and Yaghjian, 1993; Yaghjian et al., 1996; Shore and Yaghjian, 2001].

[3] In the present paper the method for proving the 2-D theorem in the work of Yaghjian [2001] is used to prove the analogous three-dimensional (3-D) surface-current equivalence theorem, which states that:

The electromagnetic fields outside the smallest sphere that encloses a 3-D source region can be generated (to any degree of accuracy) by electric and magnetic surface currents on a single planar surface oriented arbitrarily and located entirely inside the source region.

This 3-D theorem can be used to rigorously justify solution procedures such as the equivalent-current integral equation method for computing the far fields of antennas from measured near-field data [Petre and Sarkar, 1992], and the “adaptive integral method” for efficiently computing the scattering and radiation from electrically large bodies [Bleszynski et al., 1996]. For scalar wave equations, L. A. Kunyansky, and O. P. Bruno (A fast, high-order algorithm for the solution of surface scattering problems, 2, Theoretical considerations, submitted to Journal of Computational Physics, 2001) have recently derived a similar 3-D theorem and applied it to compute the scattering from multiwavelength surfaces.

[4] The proof of the theorem, which is given in section 2, is based on showing that the fields of each and every spherical electromagnetic multipole can be generated by delta-function electric and magnetic surface currents in an arbitrarily oriented plane of infinitesimal area located at the origin of the multipole. Explicit expressions for these delta-function electric and magnetic surface currents that generate the fields of spherical electromagnetic multipoles are derived in Appendix A, along with distributed surface currents that approximate the delta-function surface currents to any desired accuracy. Although these explicit expressions for the equivalent surface currents of spherical multipoles are not required to prove the surface current equivalence theorem, they are used in section 3 to derive a general linear operator representation for the output of a receiving antenna in terms of just the two transverse spatial derivatives of the transverse components of the incident electric and magnetic fields at a single point in space. The coefficients of the transverse linear differential operator are expressed in terms of the spherical multipole coefficients of the antenna's complex receiving pattern. This linear operator representation for the output of a receiving antenna is similar to previous representations used to derive probe-corrected spherical-wave transmission formulas [Yaghjian and Wittmann, 1985; Wittmann, 1992]. The previous representations were more restrictive, however, than those derived in the present paper in that the previous representations required derivatives with respect to all three spatial directions.

## 2. Proof of the 3-D Planar Surface-Current Equivalence Theorem

[5] Consider arbitrary electromagnetic sources with eiωt (ω > 0) time dependence located in a bounded volume of 3-D space. The spherical coordinates of an observation point r are given by (r, θ, ϕ) with respect to arbitrarily oriented xyz axes whose origin O can be, but need not be, chosen within the source region. The electric and magnetic fields radiated by the sources can be expanded in a complete set of outgoing spherical vector wave functions [Stratton, 1941, section 7.11]

In (1)(2) the harmonic time dependence has been suppressed, Z0 is the impedance of free space, and the spherical vector wave functions are defined as

where the hl(1)(kr) are spherical Hankel functions of the first kind (k = ω/c = 2π/λ, c being the speed of light in free space and λ the wavelength) and the Plm(cos θ) are associated Legendre polynomials. The Alm and Blm are the constant coefficients of the spherical magnetic and electric multipoles, respectively, and these coefficients are determined by the sources [Jackson, 1999, chap. 9]. In the notation of Jackson, we can write

where the vector spherical harmonics are defined here as

The normalization constants we are using for the mlm, nlm, and Xlm, however, are those of Stratton rather than Jackson.

[6] For L = ∞, the infinite series of multipoles in (1)(2) converge to the correct values of the fields for r > a, where a is the radius of the smallest sphere (centered at the origin O) that encloses the sources. The same infinite series diverges for (r, θ, ϕ) within the “minimum source region,” which lies inside the radius r = a [Yaghjian et al., 2000]. If, however, the series in (1)(2) is truncated to l = L < ∞, the resulting truncated series will converge for all r > 0. Moreover, for L > ka′, where a′ is the radial extent of the significant reactive fields, this truncated series will converge to virtually the correct values of the fields for r > a + λ [Yaghjian, 1996]. For nonresonant sources, a′ is equal to about a + λ. For resonant sources, a′ may be greater than a + λ. The error in the fields outside the original source region can be made arbitrarily small by choosing L large enough. We can therefore assume that L is chosen large enough so that outside the original source region (r > a) the fields (E, H) of the truncated series in (1)(2) are practically equal to the exact fields.

[7] The fields (E, H) given by the truncated series in (1)(2) are finite everywhere except at the origin O (r = 0) where the multipole sources reside. Surround these sources at r = 0 by a finite cylinder that lies entirely within the source region. Let the cylinder have an arbitrarily shaped cross-section and let the top and bottom of this cylindrical “pillbox” lie parallel to the xy plane and be separated by a height 2h with the top at z = +h and the bottom at z = −h, as shown in Figure 1. To this closed cylindrical surface denoted by S, apply the Kottler-Franz formulas, which are an alternative version of the Stratton-Chu formulas for expressing the fields outside S in terms of the tangential fields on S [Hansen and Yaghjian, 1999, section 2.3.8], to get

The vector is the outward unit normal to S and G(r, r′) is the free-space Green's function given by

[8] Letting h approach zero eliminates the integration around the side(s) of the cylindrical pillbox and allows (6)(7) to be rewritten as

where A is the projection of the top or bottom of the cylindrical pillbox onto the xy plane, ρ′ = x + y, and Ke(ρ) and Km(ρ) are equivalent electric and magnetic surface currents defined by

These surface currents are vector functions of ρ that are perpendicular to and thus lie in the xy plane. Because H(r) and E(r) are continuous functions for all r except at r = 0 where the fields produced by the spherical multipole sources at r = 0 have infinite singularities, (11)(12) show that these surface currents Ke(ρ) and Km(ρ) are zero for all ρ except for ρ = 0. Thus, Ke(ρ) and Km(ρ) are generalized functions of ρ. By letting E(r) and H(r) in (9)(10) be the fields of an arbitrary multipole, it is shown in Appendix A that the generalized functions Ke(ρ) and Km(ρ) consist of delta functions and their derivatives with respect to x and y. These delta functions are produced by the infinite singularities in the magnetic and electric fields across the spherical multipole sources at r = 0. It is unnecessary, however, to evaluate the Ke(ρ) and Km(ρ) explicitly in (9)(10) for an arbitrary multipole in order to immediately obtain from (9)(12) the desired general 3-D surface-current equivalence theorem stated in the Introduction. Nonetheless, explicit expressions for the delta-function electric and magnetic surface currents that generate the fields of each spherical multipole are given in Appendix A, along with distributed surface currents that approximate the delta-function surface currents to any desired accuracy. These explicit expressions in Appendix A for the surface currents that generate the fields of spherical multipoles are used in section 3 to represent a general linear receiving antenna as a linear differential operator that converts the transverse components of the incident field and its transverse spatial derivatives at a point in space to an output voltage.

[9] Although the plane in which the equivalent surface currents flow was chosen as the xy plane, the orientation of the coordinate system can be chosen arbitrarily. Thus, the orientation of the planar surface can be chosen arbitrarily. Also, the origin of the coordinate system and thus the planar area with the equivalent surface currents can be chosen outside the scatterer, as long as a larger radius of convergence can be tolerated. (Recall that the radius of convergence a is the minimum radius of the sphere, centered at the origin O of the coordinates, that encloses the original sources.) Consequently, there are any number of different multipole surface-current distributions, each of which will produce fields that agree to a given accuracy everywhere outside the smallest sphere centered on that multipole surface-current distribution and enclosing the original source distribution.

[10] The most important feature of this 3-D equivalence theorem is that no equivalent electric or magnetic currents perpendicular to the chosen xy plane, that is, no z-directed currents, are required to generate the electromagnetic fields outside the original source region (r > a). In other words, (11)(12) show that Ke(ρ) and Km(ρ) in (9)(10) have no z components.

[11] Even though this equivalence theorem has been derived for delta-function equivalent surface currents on planes with infinitesimally small areas A at r = 0, it is shown in Appendix A that the theorem also holds for finite equivalent surface currents on planes with extended areas inside the original source region (r < a). One can also see this heuristically as follows. Let r → ∞ in (9) or (10) to get the far field, and divide the resulting θ and ϕ components of the far field into even and odd parts with respect to the xy plane, so as to get a separate integral equation for Km(ρ) and Ke(ρ) on a plane with finite (nonzero) area within the original source region. One can, in principle, solve these two integral equations for Km(ρ) and Ke(ρ) of finite (nonzero) extent on the area A that will produce the given far-field pattern to any desired accuracy. These surface currents will then also produce accurate near fields for r > a. (In practice, numerical inaccuracies would limit the accurate near fields to outside r = a′, the extent of the significant reactive fields. Also, since the solution to the integral equations for a given accuracy exists but is not unique, one might have to resort to pseudo-inverse or comparable solution techniques in order to obtain an accurate numerical solution.) Neglecting the electric surface current, Petre and Sarkar [1992] have applied the integral equation (9) to solve for the equivalent magnetic surface current in the form of magnetic dipoles equally spaced in a truncated infinite plane in front of an antenna to determine the fields radiated by the antenna into the front hemisphere. Since their work was restricted to a single hemisphere, half of the unknowns, namely the equivalent electric surface current, could be ignored without strongly affecting the accuracy of the computed far fields for θ not too close to π/2 (the plane of the equivalent magnetic surface current).

## 3. Receiving Antenna as a Linear Differential Operator

[12] The surface-current equivalence theorem stated in the Introduction and proven in section 2 implies that the fields of any spherical electromagnetic multipole can be generated by surface currents in an infinitesimal planar area at the center of the multipole. Explicit expressions for the equivalent surface currents of spherical multipoles are derived in Appendix A. The theorem can be understood by first considering the magnetic dipole at the point O with dipole moment in an arbitrary direction that can be labeled as the direction. The fields outside of this magnetic dipole can be generated by a delta-function magnetic current flowing in the direction at O. However, the same fields outside the dipole can also be generated by a tiny circulating electric current in the xy plane normal to ; see (55) in Appendix A. Similarly, the fields of an electric dipole in the direction can be generated by either directed electric current or circulating magnetic current in the xy plane; see (56) in Appendix A. Moreover, the fields of electric and magnetic dipoles with dipole moments in the and directions can be generated by surface currents in the xy plane. Therefore, any combination of electric and magnetic dipoles at a point in space can be produced by surface currents in an arbitrarily oriented infinitesimal planar area at that point.

[13] Now the higher order spherical electromagnetic multipoles can be constructed from elementary electric and magnetic dipoles by a progressively higher order limiting procedure [Stratton, 1941, section 3.12; Yaghjian and Wittmann, 1985]. Consequently, every spherical electromagnetic multipole, not just dipoles, can be produced by electric and magnetic surface currents in an arbitrarily oriented infinitesimal planar area at the center of the multipole.

[14] The fields of an antenna can be expanded in a series of multipole fields, as in (1)(2), and we have shown that the fields of each of these multipoles can be produced by electric and magnetic surface currents composed of delta functions and their surface derivatives. Thus, it would seem likely, and indeed we shall prove in this section, that the output of a linear antenna, when it is used as a receiver, can be expressed in terms of surface components (transverse to a chosen direction) of the incident electric and magnetic fields at a point in space and the transverse derivatives of these field components at that point. In a previous paper [Yaghjian and Wittmann, 1985], a similar result was derived that required all three spatial derivatives, not just the transverse derivatives, of the incident fields.

[15] To begin the derivation of the linear operator representation of an antenna, express the output voltage Vp(r) at a reference plane S0 in the feed waveguide (terminated in a perfectly matched shielded load, that is, ΓL = 0) of a linear receiving antenna (probe), illuminated by given incident fields, as a two-dimensional integral of plane waves weighted by the scalar product of the receiving spectrum Rp() of the probe antenna and the transmitting spectrum T() of the source of the incident fields [Kerns, 1981; Yaghjian, 1982; Hansen and Yaghjian, 1999, chap. 6]:

where k = K + γ, = k/k, K = kx + ky, dK = dkxdky, and γ = (k2K2)1/2 is positive real or imaginary. The antenna's feed waveguide supports just one propagating mode, and the receiving spectrum Rp of the probe antenna and the transmitting spectrum T of the source are referenced to the origins O′ and O fixed with respect to the antenna and source, respectively. As can be seen in Figure 2, the output voltage of the antenna is a function of the position vector r measured from the origin O of the xyz coordinate system fixed to the source and the origin O′ fixed to the antenna. The plane wave transmission formula (13) is derived under the assumptions that the receiving antenna translates without rotation with respect to the transmitting source, that multiple interactions between the receiving antenna and the transmitting source are negligible, and that the receiving antenna remains on the positive side (right side in Figure 2) of the source. That is, the physical antenna and source region do not “encroach” upon each other with respect to the z direction. Both the spectra can be shown to be proportional to far-field functions, which have no radial components, so that they satisfy the relations

[16] The complete vectorial plane wave spectrum T() determines the incident electromagnetic fields radiated by the source to the right of the source region. Specifically,

[17] Because the receiving spectrum Rp() is proportional to the electric far-field pattern of the probe antenna (adjoint probe, if the probe is nonreciprocal) analytically continued to include the evanescent spectrum, it can be expanded in a series of vector spherical harmonics analytically continued to include the evanescent spectrum; see Jackson [1999, chap. 9], Yaghjian and Wittmann [1985], and (34)(35) in Appendix A of the present paper:

in which ClmM and ClmE are the coefficients of magnetic and electric vector spherical harmonics, Xlm() and × Xlm(), respectively, with Xlm() defined in (5). Inserting Xlm() from (47) of Appendix A into (18) and then the resulting Rp () into the plane wave transmission formula (13) yields (after interchanging the order of the integrations and summations)

where the PlmmM(K/k) and PlmeM(K/k) are two-dimensional polynomials in kx and ky defined by (41) and (42) of Appendix A in terms of the vector spherical harmonics. (The vector k corresponds to the vector k+ in Appendix A). By using simple vector identities and the relation (14), the integrands in (19) can be changed to

[18] Since the PlmmM(K/k) and PlmeM(K/k) are two-dimensional polynomials in kx and ky, they can be brought outside the integral signs in (20) by converting them to linear differential operators with K/k = kx/k + ky/k replaced by ∇xy/(ik) = ∂/(ikx) + ∂/(iky); see Appendix A. Then with the help of (16) and (17), (20) can be rewritten simply as

Use of (16) and (17) in (21) assumes that the point O′, to which the receiving spectrum Rp() of the receiving antenna is referenced, lies on the positive side (right side in Figure 2) of the source region. (Recall that the position vector r goes from O, the origin of the coordinate system fixed with respect to the source, to the point O′, which is fixed with respect to the receiving antenna; see Figure 2.) Because the orientation of the z axis of the coordinate system fixed in the source can be chosen arbitrarily as long as the receiving antenna translates without rotation with respect to the source, the linear differential operator representation in (21) holds for all r such that the minimum convex surface circumscribing the source region and the minimum convex surface circumscribing both the receiving antenna and its fixed reference point O′ do not intersect.

[19] Letting denote the summations of derivatives in (21) operating on × Einc(r) and × Hinc(r), such that

and

(21) can be written compactly as

or, equivalently

Equation (24) or (25) expresses the output voltage of any linear receiving antenna in terms of transverse (xy) components of the incident electric and magnetic fields and their transverse spatial derivatives at the point O′ to which the receiving spectrum is referenced. The transverse linear differential operators are defined by (22) and (23) in terms of the vector spherical multipole coefficients (ClmE, ClmM) of the receiving spectrum phase referenced to O′. In other words, if one computes or measures the complex vectorial receiving pattern of an antenna with respect to a point O′, one can compute the spherical mode coefficients of that far-field function with respect to that point, insert those multipole coefficients into (22) and (23) using the (PlmeM[∇xy/(ik)], PlmmM[∇xy/(ik)]) defined by (41) and (42) with (kx, ky) replaced by the transverse derivatives [∂/(ikx), ∂/(iky)], and determine from (24) or (25) the response of the receiving antenna in terms of transverse components of the external incident electric and magnetic fields at the point O′.

[20] As a simple example, consider a reciprocal antenna consisting of an elementary magnetic dipole at O′ with its dipole moment in the direction. Then all the modal coefficients of the receiving pattern of the antenna are zero except for C10M, and thus from (22)(23) and (54)

and

Substitution of from (26) and (27) into (25) gives the output voltage of this receiving magnetic dipole as

which rigorously confirms the commonly accepted result that the output voltage of an antenna having the complex vector receiving pattern of an elementary magnetic dipole located at a point r is proportional to the component of the incident magnetic field at that point in the direction of its dipole moment. If the reciprocal magnetic dipole antenna radiates the far-field of a dipole moment md when its feed waveguide is excited by a unit-amplitude incoming propagating mode, then one can show that C10M = iωmd/(4π). The analogous result for the electric dipole antenna can be found in the studies by Kerns [1981, p. 131] and Hansen and Yaghjian [1999, section 6.2.3].

[21] Finally, note that the output voltage in (24) or (25) can be expressed in terms of Einc(r) alone or Hinc(r) alone if derivatives with respect to the normal direction (z) are allowed in addition to the x and y derivatives. To see this, simply substitute from Maxwell's equations

or

into (24) or (25) to get

where are three-dimensional vector linear differential operators involving derivatives with respect to all three directions x, y, and z.

## Appendix A:: Electric and Magnetic Surface Currents That Produce the Fields of Spherical Electromagnetic Multipoles

[22] The Alm terms in (1)(2) are the coefficients of what are generally referred to as the spherical magnetic multipoles, and the Blm terms in (1)(2) are the coefficients of the spherical electric multipoles [Jackson, 1999, chap. 9]. Therefore, the delta-function electric and magnetic surface currents that produce the fields of the magnetic and electric multipoles can be expressed formally by inserting the appropriate mlm and nlm multipole functions into (11)(12). In particular, the electric and magnetic surface currents that produce the fields of the magnetic multipoles are given by

[23] To evaluate the surface currents in (32)(33), we can write the spherical vector wave functions in the form of (4)

and express them outside the source plane z = 0 as a Fourier transform [Wittmann, 1988, equations (46a) and (46b)]

where the + and − superscripts hold for z > 0 and z < 0, respectively. The Xlm(±) are the vector spherical harmonics defined in (5) and analytically continued to include the evanescent spectrum, and k± and ± are defined as

with γ = (k2K2)1/2 chosen positive real for the propagating spectra (K < k) and positive imaginary for the evanescent spectra (K > k). The vector K = kx + ky contains the integration variables kx and ky with dK = dkxdky. Since the divergences of the fields of the multipoles are zero outside the source region, the vector spherical harmonics obey the conditions ± · Xlm(±) = 0. From (34) we see that the Xlm() are proportional to the complex far-field patterns of the vector spherical wave functions mlm(). The need to introduce k± can be understood by noting that (x, y, z) corresponds to (kx, ky, γ) for z > 0 and to (kx, ky, −γ) for z < 0.

[24] Substitution of nlm and mlm from (37) and (36) into (32) and (33), respectively, then taking the limit as h → 0 gives

Writing out the vector spherical harmonics Xlm in terms of associated Legendre polynomials reveals that

and

are vectors whose rectangular components are two-dimensional polynomials in (kx/k, ky/k). (A straightforward way to prove these results is to express the Xlm defined by (5) in rectangular components and use the facts that Plm(−γ) = (−1)l+mPlm(γ) and is an even polynomial in γ.) Consequently, (39) and (40) can be rewritten as

The validity of (39) or (43) becomes apparent after realizing that the polynomial vector PlmeM(K/k) (divided by k2Z0il) defined in (41) is the difference between the plane wave spectra of the magnetic fields (of a magnetic multipole at r = 0) in the hemispheres z > 0 and z < 0. Likewise, the validity of (40) or (44) becomes apparent after realizing that the polynomial vector PlmmM(K/k) (divided by k2il) defined in (42) is the difference between the plane wave spectra of the electric fields (of a magnetic multipole at r = 0) in the hemispheres z > 0 and z < 0.

[25] By manipulating and combining (41) and (42), it can be shown that

and that the Xlm(±) can be expressed in terms of the PlmeM and PlmmM as

a relationship that will be used in section 3.

[26] Equations (43) and (44) can be recast in terms of delta functions and their derivatives with respect to x and y by noting that the polynomial spectral functions can be brought outside the integral signs if they are converted to linear differential operators with K/k = kx/k +ky/k replaced by ∇xy/(ik) = ∂/(ikx) + ∂/(iky). Then the equations (43) and (44) for the electric and magnetic surface currents that produce the fields of magnetic multipoles become simply

after using the delta-function orthonormality relation

[27] Similarly, the electric and magnetic surface currents that produce the fields of the electric multipoles are given by

[28] As an example, for magnetic and electric dipoles with dipole moments in the direction, (l = 1, m = 0) and the vector spherical harmonic is given from (5) as X10() = −(∂P1/∂θ) = sinθ = −y/r + x/r, so that X10(±) = (kx/kky/k). Thus, from (41) and (42) the vector polynomial functions for these dipoles are given by

or in operator form

which when substituted into (48)(49) and (51)(52) yield the delta-function surface currents in the xy plane that produce the fields of the directed magnetic and electric dipoles

where the primes on the delta functions indicate differentiation with respect to their arguments. The electric current in (55) and the magnetic current in (56) are delta-function representations of circulating currents in the xy plane that produce the fields of directed magnetic and electric dipoles, respectively.

[29] The equivalent surface currents in (48)(49) and (51)(52) are all delta functions and their surface derivatives located at a single point in space at r = 0. These singular surface currents become finite surface currents distributed over a finite (nonzero) area if the delta functions in (48)(49) and (51)(52) are replaced by ordinary functions that approximate delta functions. For example, if the delta function δ(ρ) = δ(x)δ(y) is replaced by the following Gaussian function

which approaches δ(ρ) as the constant g → ∞, the equivalent surface currents given in (48)(49) and (51)(52), with this finite function replacing δ(ρ) over a finite area, will produce fields outside the original source region to any degree of accuracy by choosing the constant g large enough. Using the Gaussian functions in (57) on the disk ρ < λ to approximate the delta functions in the surface currents (55)(56) of the directed magnetic and electric dipoles, then inserting these currents into the integrals (9)(10) for the fields, we recover the dipole fields to within a fractional accuracy of about k2/(4g2) for r greater than about a wavelength, that is, for r outside the reactive fields of the dipoles.

## Acknowledgments

[30] The author is grateful for the thoughtful review of the manuscript by R.W. Wittmann of the National Institute of Standards and Technology and for the support of the U.S. Air Force Office of Scientific Research.