[1] Impulse radiating antennas (IRAs) are designed to radiate very fast pulses in a narrow beam with low dispersion and high field amplitude. For this reason they have been used in a variety of applications. IRAs have been developed for use in the transient far-field region using parabolic reflectors. However, in this paper we focus in the near field region and develop the field waveform at the second focus of a prolate-spheroidal IRA. Certain skin cancers can be killed by the application of a high-amplitude electric field pulse. This can be accomplished by either inserting electrodes near the skin cancer or by applying fast, high-electric field pulses without direct contact. We investigate a new manifestation of an IRA, in which we use a prolate spheroid as a reflector instead of a parabolic reflector and focus in the near-field region instead of the far-field region. This technique minimizes skin damage associated with inserting electrodes near the tumor. Analytical and experimental behaviors for the focal waveforms of two and four-feed arm prolate-spheroidal IRAs are explored. With appropriate choice of the driving waveform we maximize the impulse field at the second focus. The focal waveform of a prolate-spheroidal IRA has been explained theoretically and verified experimentally.

[2] Fast high-amplitude electric field pulses can be used to kill skin cancer [Schoenbach et al., 2006]. This has been demonstrated by the insertion of electrodes near the tumor. Our motivation in this paper is to apply fast, high-electric field pulses without direct contact for the possibility of killing skin cancer. Here we concentrate on the electromagnetic, rather than the biological aspects of the problem.

[3] This paper is an extension of a previous paper [Baum, 2007]. IRAs have been developed for radiating in the transient far-field region [Baum et al., 1999]. Related experimental and numerical aspects of this problem can be found in Tyo and Buchenauer [2002] and Kim and Scott [2003]. In this paper we use a prolate spheroid as a reflector, launching an inhomogeneous plane wave from one focal point and reflecting it toward the second focal point.

2. Description of Geometry

[4] We choose a special case of the prolate-spheroidal IRA's geometric parameters as in Baum [2007] and it is illustrated in Figure 1, where the geometric parameters are

Our design uses either two or four TEM feed arms. The dimensions of these feed arms are determined by 400 Ω and 200 Ω pulse impedances (ϕ_{0} = 90°, 60°). In Baum [2007] the focal fields are calculated for a two-TEM feed-arm prolate spheroidal IRA. However, the analytical results can be simply extended to the four-arm case [Baum et al., 2004; Baretela and Tyo, 2003]. Figure 2 shows the TEM feed-arms geometry.

[5] We have an increase of 1.606 in the fields using the 60° TEM-feed-arm case as compared with the two-arm case [Altunc and Baum, 2006b]. In our design we used 60° feed-arm because the voltage gain is nearly maximum [Baretela and Tyo, 2003] and it was easy to construct this version.

3. Analytical Focal Waveform Calculations for Step Excitation

[6] The analytical focal fields were calculated in Baum [2007] and they are summarized here as

where E_{δ} and E_{s} are the impulse and step terms from the reflection from the prolate sphere and E_{p} is the magnitude of the prepulse wave from first focus (valid up to the time of aperture truncation). The detailed calculations for tan(θ_{c}/2) are presented in Altunc and Baum [2006a] which is a simpler result compared with the result in Baum [2007]. E is the prepulse term after the impulse (not included in Baum [2007] and is discussed in detail in Appendix A), θ_{c} is the angle of the feed arms with respect to the negative z axis, c is the speed of light in free space, and f_{g} is the transmission-line parameter

where Z_{c} and Z_{0} are the transmission line and medium wave impedances, respectively. The analytical focal waveforms for a ramp rising step excitation are from Altunc and Baum [2006a]. The excitation is a 1 Volt (V_{0} = .5 Volt) step, rising as a ramp function lasting 100 ps.

[7] We take the simple example case in (1) to illustrate the analytical waveform. One can calculate the analytical focal fields of a two-arm prolate-spheroidal IRA from (3) as

t_{δ} is the risetime of the ramp rising step excitation and the focal waveforms for this excitation is presented in Figure 3. We can easily extend this result for the four 60° TEM feed arm case by multiplying all the values by 1.606.

4. Analytical Focal Waveforms for Various Source Waveforms Driving a Prolate-Spheroidal IRA

[8] We consider the time domain characteristics of some analytic source waveforms used for determining the waveform characteristic of a prolate-spheroidal IRA at the second focus. This is an analytical calculation of a prolate-spheroidal IRA that is based on Baum [2007, 1998], Farr [1993], and Altunc and Baum [2006b]. The analytical waveforms for the two-TEM-feed-arm and 60° four-TEM-feed-arm cases at the second focus are calculated. We analyze the analytical focal waveform behavior for two different source waveforms.

4.1. Double Exponential Excitation (DEE)

[9] Let us use the commonly used waveform, which is the difference between two exponentials multiplied by a unit step function instead of a unit step function as

where t_{δ} is the risetime and t_{d} is the decay time constant. The peak of the waveform is given by (2.14) in Baum [1976] as

where t_{max} is the time when the maximum occurs and can be found by taking the derivative of (7)

[10] From a step excitation u(t) and DEE response of a prolate-spheroidal IRA at the second focal point are composed of three parts, the prepulse, impulse and postpulse, as indicated in Figure 5.

[11] One can see from Figures 5a and 5b that if we use a DEE instead of a step excitation we have a decrease in the amplitude of the prepulse, an increase in the amplitude of the impulse and the postpulse goes to zero. By modifying the excitation waveform we require less energy, but obtain a larger impulse amplitude.

[12] The impulse part of the double exponential excitation is

and the peak value is E_{δ}(α − β), where t_{2} = 4.2 ns is the time when the impulse, arrives at the second focus. Finally, we obtain response waveforms from (9) in Figure 5a,

where E_{pδ} is the value of E_{p} at the time the impulse starts and t_{1} = 2.5 ns is the time that prepulse arrives at the second focus.

5. Experiments

[13] Experiments were performed using two-arm and 60° four-arm prolate-spheroidal IRAs and these results are compared with analytical results in Baum [2007]. This section presents a summary of the experimental setup and the dimensions of these experiments are based on Baum [2007] and Altunc and Baum [2006a]. We use maximum t_{mr} (based on maximum rate of rise) as t_{δ} to compare our experimental results with analytical results. For a step like f(t), the t_{mr} is

The experiment uses three components: a prolate-spheroidal reflector with feed arms, a sampling-oscilloscope, and a pulse generator. As seen from Figure 6, we use a Tektronix TDS 8000B Digital Sampling-Oscilloscope with a Tektronix 80E04 sampling head to measure the waveform at the second focal point. A Picosecond Pulse Labs pulser with a PSPL 4050 RPH fast pulser head generator is used for excitation. Figure 7 shows the 60° four-feed arm prolate-spheroidal IRA. The output of the step generator is a 45-ps risetime, 10 V amplitude. We have also used a 10 dB attenuator to decrease the voltage level for safety reasons. We use three types of probes to measure the field: B-Dot Probe (EG&G MGL S7(R), equivalent area Aeq = 1 × 10^{−4} m^{2} and a risetime <150 ps), slow D-Dot (Aeq = 1 × 10^{−4} m^{2}, risetime <150 ps) and fast D-Dot (Aeq = 2 × 10^{−5} m^{2}, risetime <15 ps) probes.

[14]Figures 8 and 9show that the results for the focal waveforms are close to one another but for the slow D-Dot sensor we do not have much oscillation in the postpulse since this probe has a slower frequency response compared with the other two. The measurements of the magnetic field are converted to an equivalent electric field as

for convenient comparison with the measured electric field. Here B is the magnetic flux density. The equivalent electric field gives the exact result for the prepulse because we have a TEM wave and E/H = η_{0} ≈ 377 Ω for free space. We calibrate our D-Dot data by comparing the prepulse term. Although we do not have TEM waves for the impulse, we calculate η = E/H. For the two-arm case η is observed to be 384 Ω and for the four-arm case η is observed to be 408 Ω. This shows that we do not have a purely TEM wave for the impulse. The B-Dot data is believed to be accurate to a few percent based on manufacturers calibration, and is also accurately calculable [Thomson and Luessen, 1986], at least below the upper bandwidth.

[15] One can see from Figure 10 that when we use the fast D-Dot sensor we have an oscillation that is present which may not be attributable to the different types of sensors that we are using. There will always be oscillations and aberrations in the signal observed in the fast D-dot probe. There are cable and connection non-uniformities, nonlinear effects in the sampler, sampling time errors, digitizing errors, etc. Most importantly, the generator signal is not pure and has some aberrations following the step voltage.

[16] The slow sensors are more sensitive than the fast D-Dot sensor, but they are not fast enough to obtain the minimal t_{mr} values. We observe larger t_{mr} values, which result in a decrease in the amplitude of the impulse part of the focal waveform. If we use the fast D-Dot sensor we obtain higher amplitudes in the impulse part, but we obtain larger differences between the amplitudes of the impulse part of the analytical and experimental focal waveforms. The average value of the analytical peak, experimental results, oscillation amplitude, t_{mr} and differences in experimental results compared with the analytical results are summarized in Table 1.

Table 1. Average Value of the Analytical Peak, Experimental Results, Oscillation Amplitude, t_{mr} and Difference in Experimental Results Compared to Analytical Results

Average Value of the Analytical Peak

Exp Results (V/m)

Oscillation (V/m)

tmr (ps)

Difference (%)

B-Dot 2 Arm

3.3

2.5

0.4

119

21.4

B-Dot 4 Arm

4.8

3.4

0.97

127

32.7

D-Dot 2 Arm slow

3.3

2.5

0.38

119

21.4

D-Dot 4 Arm slow

4.8

3.5

0.92

130

32.7

D-Dot 2 Arm fast

15

9.6

1.58

26.5

36.0

D-Dot 4 Arm fast

28.9

19.8

3.3

22

31.5

[17] There are several factors that can lead to differences in the analytical expressions and experiments. When the focal fields are calculated in Baum [2007], the aperture integral did not consider the feed arms and feed-arms' thicknesses. This can cause an error in the calculation of the impulse amplitude of the focal waveform. There are errors in the experiment for which one needs to account. We are at the limit of our measurement instrumentation and we have less accuracy because of the limitation of the probes and pulse generator.

[18] The geometric shape or alignment of the prolate-spheroidal reflector may also cause some errors. Any misshape of the reflector will lead to a broader focus and smaller amplitude. The prolate-spheroidal reflector was manufactured from fiberglass and the inside of the reflector is painted with copper conductive paint. We checked the reflection from the conductive paint on the reflector and calculated about 99% reflection using the transmission coefficient of the reflector; however, there might be some portions that do not reflect very well and this could cause some errors.

6. Numerical Simulation

[19] We compare our analytical, numerical and experimental focal waveforms for a two-arm prolate-spheroidal IRA in Figures 8 and 9. One can see by comparing the analytical, numerical and experimental focal waveforms that the prepulses agree very well. The analytical and numerical impulses' amplitudes agree as well. However, the experimental impulse amplitude is less than the others. It is also broader near the base. As discussed before, any misshape of the reflector may lead to this in the experiment or experimental inaccuracies also due to details of the excitation waveform.

7. Conclusion

[20] In this paper we have designed, constructed and tested a prolate-spheroidal IRA that was designed to assess this method's feasibility as part of a technique to kill skin cancer using pulsed electric fields. This work used analytical calculations, numerical simulations and experiments.

[21] The analytical behaviors of the focal waveforms of the two and four-feed arm prolate-spheroidal IRAs were calculated. The analytical waveform was illustrated and analyzed. These analytical calculations were for a two-arm prolate-spheroidal IRA. However, we have shown that these calculations can be easily extended to the 60° four TEM feed arm case by just multiplying the analytical values by 1.606.

[22] Finally, the time domain characteristics of some analytic source waveforms used for determining the waveform characteristic of a prolate-spheroidal IRA at the second focus were discussed. With appropriate choice of a driving waveform we can maximize the impulse field at the second focus.

[23] We performed several experiments at the UNM Transient Antenna Laboratory in order to compare our analytical and numerical results. Experiments with a two-arm and 60° four-arm prolate-spheroidal IRA were performed. Experimental, analytical and numerical results were compared. The small differences between these results were discussed and the differences were analyzed.

Appendix A:: Prepulse Term E After the Impulse

[24] What happens to the prepulse term after the impulse, i.e., after the truncation at the aperture boundary (Ψ = Ψ_{p}, or b for special case)? This was not treated in Baum [2007]. Before the aperture truncation the prepulse is given by E_{p} in (3).

[25] Let E_{pt} = tangential E field (x component) on S_{a} due to the prepulse wave. Then we have [Baum, 2007, 1987]

These are both integrals of the fields from the first focus on the aperture plane. After we see the edge of S_{a}, neglecting diffraction terms from this edge and approximating E_{pt} by the negative of the TEM prepulse wave out to this edge (for a positive parameter as in Baum [2007]) we have, for step-function excitation, a time-independent prepulse field on S_{a},

Next we require the static E_{p2}. As before, since we are confining ourselves to the z axis we can use a uniform field on the projection plane to give E_{p2} in the above integral. From (2.11) of Baum [2007] at r_{1} = z_{0} (aperture plane center)

This can be extended over S_{a} since, as we have seen before [Baum, 2007], for the z axis only the uniform field terms (on the projection plane) need be considered (by symmetry).

[26] In the end we obtain

Let us find the ratio of E_{p2}/E_{p} from (A4) and (3)