This paper develops some analytical approximations for the transient focal waveform produced at the second focus of a prolate-spheroidal reflector due to a pulse TEM wave launched from the first focus. This is extended to consider the spot size of the peak field near the second focus.
 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 ∞.
 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.
 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 (r1, θ1, ϕ1). These are related to cylindrical (Ψ1, ϕ1, z1) and Cartesian (x, y, z) coordinates as
The prolate spheroid is described by
The thin-cone electrical centers are described by the angle θc with
At the reflector we have (subscript p)
Given θc one can solve for zp and Ψp. One can also specify zp and compute Ψp and θc.
 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 = zp 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 Sa. It is this surface which will be used for integrating over the reflected TEM wave to find the fields at the second focus, 0.
 The stereographic projections in Appendix A can be used to calculate the fields. Let wave 1 have the form
Let the wave have V1 = ±V0 on the two cones.
 The stereographic projection of this wave is
The electrical center of the thin wire on this projection plane is
This is the prepulse orientation and magnitude (for step excitation). It lasts for a time (Appendix A)
after which the reflector signal arrives at the second focus.
 For the special case as in (6) and (7) for which the launcher intersects the reflector at the z = 0 symmetry plane we have
EpΔtp ≃ (time integral or “area” of prepulse).
3. Fields on Aperture Plane
 Now we consider the wave heading from the reflector toward the second focus. There is no set of conical conductors guiding the wave there. So we consider this second spherical TEM wave (Appendix A) on the aperture plane, which we can use in turn to find the fields at 0 (and other positions as well). For present purposes we take the aperture plane as z = zp, as in Figure 1. The reflected wave illuminates Sa, a disk of radius Ψp.
 Note that the reflector is truncated at the aperture plane. This is because the field from the wave-launcher reverses sign for the wave on the “other side” of the launching conductors. This is the same truncation used for typical reflector IRAs, based on a self-reciprocal geometry [Farr and Baum, 1995]. There are more sophisticated (nonplanar) truncation geometries which may be considered in the future.
 In Appendix A the reflected wave was related to the first wave by a double stereographic transformation. They are equal (except for a minus sign) on the stereographic projection plane (z = −a) for which
The second TEM wave takes the form
Note the factor giving an incoming (on 0) step-function wave.
 For the aperture integral we need the tangential electric field on Sa. First we have
For θ2 near zero we have
Comparing this to the prepulse amplitude in (18) we have (θ2 near 0)
For the special case (as in (20)) that the launcher intersects the reflector at the z = 0 symmetry plane we have (for θ2 near zero)
a fairly simple result.
 At the center of the aperture plane we have
For zp = 0 this reduces to
Wave number 2 is focused on 0. Without guiding conductors (25) cannot hold all the way as r2 → 0. So we are considering the fields on Sa for later integration. On the center of Sa we have the electric field E0 polarized in the x direction.
 In IRA-related calculations [Baum, 1991b, 1992] it has been seen that, for circular apertures, the field at the center is an important parameter. The boresight radiated field can be found by integrating the TEM field over the aperture, or by integrating a uniform field of the center value (including polarization) over the same circular aperture. Seen another way, one can expand the field in cylindrical coordinates and note that terms with cos(mϕ) and sin(mϕ) for m ≥ 2 integrate to zero (for observation field points on the z axis). (There is no m = 0 term.) This is basically a symmetry result.
 Similarly here, let us consider a uniform field on the (Ψ0, ϕ0) projection plane. This corresponds to a potential function
for a uniform field E0 polarized in the x direction. Matching this field to the second wave at
For zp = 0 this becomes
Now convert V′2 on the projection plane to (θ2, ϕ2) coordinates as
giving an electric field
On the aperture plane we have
The tangential electric field is then
Converting to Cartesian coordinates we need only the x component (due to symmetry) as
4. Fields at Second Focus
 We are now in a position to evaluate the fields at 0 by integrals over the fields on Sa. A previous paper [Baum, 1987] has developed the formulae. From Baum [1987, equation (3.3)] we have (in time domain for step excitation)
Using cylindrical coordinates on Sa we can evaluate the integrals. Considering first the ϕ variable we have
Next we have
Note that Es has units V/m, while Eδ has units Vs/m corresponding to the time integral (or area) of the δ function.
5. Some Design Considerations
 Summarizing, we have
One can compute c from zp (or Ψp) in (5). For the special symmetric case of zp = 0, merely replace Ψp = b, giving
A first observation concerns the common factor V0(πfg). For large fields one needs large voltage and low wave-launcher impedance. Note that an extra factor of increase in the fields is obtained by going to a 4-arm feed in the usual sense of an IRA with arms at 45° [Baum, 1991a], and a little more can be achieved by arms at about 60° from the horizontal (the x = 0 plane) as projected on a constant-z plane [Baum, 1998; Tyo, 1999]. This is just another common factor with which we can deal separately.
 The δ-function part of the field does not have infinite amplitude when the incident step-like wave has a nonzero risetime. For simplicity let us imagine that the δ-function is replaced by
i.e., a gate function of width tδ and height tδ−1. This makes the impulsive part of the field have amplitude Eδtδ−1, showing the importance of a small pulse width. For convenience define
We will want this to be large, which argues for large a.
 Considering the simpler case zp = 0 we have
If we want the step from the reflector (the post pulse if you like) to cancel the prepulse step then one would like b to approach a, but this has other consequences. Next compare the main (δ) pulse amplitude to the prepulse amplitude giving
The factor z0/a is less than one, but it needs to be large to give a large main pulse. This is further aided by a large T implying large a − z0 and small tδ. So, intermediate values of z0 are called for, i.e., at the maximum of z0[a − z0] which is
6. Spot Size
 Given that the impulse has some small width tδ, the maximum fields will exist in some small region around 0. So let us estimate a pulse width of this impulsive part for positions near 0. For the δ-function pulse we have the wave from every position on Sa arriving at exactly the same time at 0. We can then estimate a pulse width near 0 by the dispersion in the arrival times from all parts of Sa at the observation point.
 For this purpose, consider the lengths of the raypaths in Figure 2 indicating the maximum time differences from the edges and center of Sa to the observer. For an observer at z0 + Δz on the z axis we have
Then we have
For the special case of zp = 0 we have
Large z0/a (small b/a) minimizes this.
 For an observer radially displaced ΔΨ from 0 we have
Then we have
For the special case of zp = 0 we have
Small b/a also minimizes this.
 Comparing these pulse widths to tδ, the width at 0, we can note that the pulse widths in the z direction are tδ + tz, and in the Ψ direction tδ + tΨ. The physical spot size is then given by (counting width in both directions from 0)
given by (48) and (50). For the special case of zp = 0 we have
Of course, these are just rough estimates. Detailed waveform calculations will give more accurate results, including actual waveforms instead of bounds on pulse widths.
7. Concluding Remarks
 This modest study has found some analytic approximations useful for estimating the focal waveforms and focal spot size for the prolate-spheroidal IRA. One may think of this as analogous to the first of the papers dealing with the IRA using a paraboloidal reflector [Baum, 1989]. Considering the sophisticated design papers which followed the introduction of the IRA concept, there is much yet to be done for the prolate-spheroidal version.
 Here we have found some especially simple results for the case of reflector truncation at zp = 0. More detailed numerical treatment of the results for zp ≠ 0 may lead to yet more insight into an optimal design.
Appendix A:: Prolate Spheroidal Scatterer for Spherical TEM Waves
 Previous papers [Farr and Baum, 1992a, 1992b; Baum and Farr, 1992] have established that inhomogeneous TEM waves in a uniform, isotropic medium (e.g., free space) are exactly transformed by stereographic projection into second such waves in the case of paraboloidal and hyperboloidal scatterers, provided the incident wave is centered on an appropriate focal point (including infinity) of these quadric surfaces. One spherical or planar TEM wave is then transformed into another with an exact matching of the boundary conditions on the (perfectly conducting) reflector. This gives exact solutions of the Maxwell equations, valid up until some time related to a signal arriving at the observer from some truncation of the reflector, or from some waveguiding structure used to guide the incident wave (i.e., conical or cylindrical perfectly conducting transmission lines). These “early-time” exact solutions (valid up to some “clear time” are examples of partial geometric symmetries as discussed by Baum .
 Keeping with bodies of revolution, which give focal points, another quadric surface to consider is the prolate spheroid, a special case of an ellipsoid. In this case both focal points are inside the volume enclosed by the surface Sp. So our consideration is to launch an inhomogeneous plane wave from one focus, and reflect it toward the second.
A2. Matching Spherical TEM Waves
 In spherical coordinates (r1, 1, 1) centered on −0, with θ1 = 0 pointing along the negative z axis (toward the stereo graphic-projection plane), we have an outward propagating (from −0) inhomogeneous transient TEM wave as
This is guided by two or more perfectly conducting cones (with apices at −0) toward the reflector (see Figure A1).
 The stereographic transformation relating spherical TEM waves to cylindrical TEM waves takes the form
 The projection plane is here taken as z = −a, tangential to the reflecting surface. Note that for purposes of this transform the projection plane extends to ∞, corresponding to θ1 = π. Note that (Ψ0, ϕ0) corresponds to a point on the projection plane. In this projection V1 satisfies the Laplace equation in cylindrical (Ψ0, ϕ0) coordinates.
 Here we are imagining a wave launched to the “left,” to be reflected on S′p [Baum, 2005a] (the portion of Sp on which a reflector is built). The portion to the “right,” around the target location, is assumed not used for the reflector, allowing access to the target vicinity. However, the symmetry of the geometry allows one to interchange the roles of source and target.
 Let us consider a second spherical TEM wave, centered (incoming) on 0 of the form
The projection formula for this wave is
The formulas in (A1) also apply, substituting subscripts 2 for 1.
 Now let
on the projection plane. Since V1 satisfies the Laplace equation there, so does V2. Going through the projection formula (A4), then V2(θ2, ϕ2) satisfies the spherical Laplace equation ((A1) using 2 subscripts). This double stereographic transformation is of the same general form as in the work by Farr and Baum [1995, section IV] with a few sign changes. Here we have a diverging wave reflected into a converging wave. Note that the waveforms are the same f(t) for these two waves.
 We merely need now that V2 + V1 = 0 (or its tangential derivative, i.e., tangential E-field) on the reflector. On the reflector we have, due to the stereographic transforms
The two waves match in time as well on the reflector. Differentiating the potential (net zero) on the reflection gives zero tangential electric field, the required boundary condition. This gives an exact solution of the Maxwell equations for times (clear times) before scattering from feed arms, and Sp truncation to S′p is seen by the observer. Such clear time is observer-position-dependent. For analytical convenience we can take the time domain waveform as a step function
applying to both transmitted and reflected waves.
 The feed arms also fit into the spherical Laplace equation. Their electrical “centers” have been considered in the case of impulse-radiating antennas (IRAs) [Farr and Baum, 1992a, 1995], allowing for placement which in some sense is optimal. Consider that these intersect Sp at some zp. The double stereographic transform then has “image” feed arms pointing to 0 from the intersection at zp. This leads to an interesting symmetry concept by setting
This makes z = 0 a symmetry plane between the wave launching side (z < 0) and the wave receiving side (z > 0). In practice (inverse) feed arms are not included on the receiving side (at least not down to the focal point at 0).
A3. Separation of Two TEM Waves Incident on Target
 The foregoing is limited in that it includes the wave reflected from S′p. There is also a direct wave from − 0 to 0. This is a step function in the simple case (27). It arrives at a time
It is analogous to the prepulse from a reflector IRA. The reflected wave arrives at 0 at a time
 The length of the prepulse is then
provided there are no guiding arms for the reflected waves to interfere with this.
 Assuming that the prepulse is a negative Ex, corresponding to a positive potential on the upper feed arms (positive x in Figure A1), the reflected pulse reaching toward 0 has a positive sign. The waveform, however, is quite different. In the work by Baum  the case of the guiding arms is considered by use of an aperture integral. The field at 0 has a delta-function pact and a step-function part. A detailed treatment of this may appear in the future.
 This work was sponsored in part by the Air Force Office of Scientific Research.