## 1. Introduction

[2] Recent research [*Schoenbach et al.*, 2006] has shown that it is possible to kill cancer (melanoma, a skin cancer) by the application of fast, high-amplitude electric field pulses. This has been done by the insertion of electrodes near the tumor, with direct contact from a high-voltage (hundreds of kV) pulse generator. It has been suggested that it would be desirable to be able to apply fast, high-electric-field pulses without direct contact for this biological application, i.e., to radiate them from some kind of antenna. Given experience with impulse radiating antennas (IRAs) [*Baum et al.*, 1999; C. E. Baum, E. G. Farr, and D. V. Giri, John Kraus Antenna Award of the IEEE Antennas and Propagation Society, 2006], it seemed logical to see if this could be modified in some way to focus a pulse in the near field instead of at ∞.

[3] Two previous papers [*Baum*, 2005a, 2005b] have discussed the basic concept of using one or more prolate spheroidal reflectors to focus a fast-transient electromagnetic pulse on some targets of interest. Appendix A shows that an inhomogeneous spherical TEM wave launched on guiding conical conductors from one focus is converted by a double stereographic transformation to a second (reflected) inhomogeneous spherical TEM wave propagating toward the second focus. Both waves have the same temporal waveforms before other scattering (from feed arms, etc.) can reach the observer.

[4] The present paper is concerned with analytical calculations of the waveform at the second focus. For this purpose let us consider the geometry in Figure 1. Let there be two thin perfectly conducting cone wave launchers with electrical centers lying in the *xz* plane. With respect to the negative *z* axis they are oriented at

in the wave-launching spherical system (*r*_{1}, θ_{1}, ϕ_{1}). These are related to cylindrical (Ψ_{1}, ϕ_{1}, *z*_{1}) and Cartesian (*x*, *y*, *z*) coordinates as

The prolate spheroid is described by

The thin-cone electrical centers are described by the angle θ_{c} with

At the reflector we have (subscript *p*)

Given θ_{c} one can solve for *z*_{p} and Ψ_{p}. One can also specify *z*_{p} and compute Ψ_{p} and θ_{c}.

[5] As a special case one may choose the symmetry plane for which

This simplifies the equations to

For the present let us truncate the reflector at the *z* = *z*_{p} plane. The portion used is *S*′_{p}, to the left. This is consistent with the traditional truncation of paraboloidal reflector impulse radiating antennas (IRAs). More sophisticated truncation contours can be considered, but are beyond the scope of this paper. For later use the truncation plane will be taken as an aperture plane. The portion of this plane inside the prolate sphere is designated *S*_{a}. It is this surface which will be used for integrating over the reflected TEM wave to find the fields at the second focus, _{0}.