## 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.