An array of radars is developed as a standoff sensor for use in seismic/elastic mine detection systems. The array consists of N radar sensors which operate independently to sense the displacement of the surface of Earth due to elastic waves propagating in Earth. Each of the sensors consists of a lens-focused, conical, corrugated, horn antenna and a homodyne radar. The focused antenna allows the sensor to have greater standoff than with the previous unfocused antenna while maintaining the spatial resolution required for a mine detection system. By using an array of N sensors instead of a single sensor, the scan rate of the array is improved by a factor of N. A theoretical model for the focused antenna is developed, and an array of two radars is developed and used to validate the theoretical model. The array is tested in the experimental model for the seismic mine detection system. Results from the experimental model are presented.
 Recent research has shown that seismic/elastic detection techniques show great promise for the detection of buried land mines. However, significant technical problems must be overcome to make a practical mine detection system that uses these waves. The technical problems include (1) the measurement of elastic waves with sufficient accuracy and spatial resolution to detect small mines while achieving a measurement speed consistent with operational requirements, (2) the generation of soilborne elastic waves in a robust and repeatable manner over all types of terrain, and (3) the integration of sources, sensors, and signal processing in a way that will result in the most effective system. The development of an appropriate sensor to sense the seismic waves is one of the biggest technical challenges. Various sensors can and have been used: laser doppler vibrometers [Sabatier et al., 1986; Sabatier and Xiang, 2000a, 2000b, 2001a, 2001b; Xiang and Sabatier, 2003], radar [Scott et al., 1998, 2001b; Stewart, 1960], ultrasonic sensors [Codron, 2000; Martin et al., 2002], air-acoustic displacement sensors [Cook and Wormser, 1973; Larson et al., 2000], ground contacting probes such as accelerometers [BBN Technologies, 1992], etc. Of the noncontact sensors only the radar has been shown to be able to function effectively with a significant layer of vegetation on the surface of the ground [Scott et al., 2001a]. In this paper, the development of an array of radar sensors to measure the elastic waves in the ground is described.
 In earlier papers, the results from a single radar sensor that used an open ended waveguide as an antenna have been presented. The system uses a radar-based displacement sensor for the measurement of seismic displacements. The radar system performed well, but had two significant issues for a practical mine detection system; the antenna for this radar had to be placed within a few centimeters from the surface to get the required spatial resolution, and the scanning speed was very slow. Both of these issues are addressed in this paper. A focused antenna is developed which increases the standoff distance from a couple of centimeters to 20 cm while maintaining sufficient spatial resolution. Preliminary results for the antenna were presented earlier [Scott et al., 2001a]; in this paper, the theoretical model, design, and performance of the antenna have been improved. To improve the scanning speed, an array of N independent radar sensors is developed, allowing the displacements of the surface to be measured at N positions simultaneously which results in a factor of N increase in scanning speed.
 A schematic diagram of a two-element array incorporated in a seismic system is shown in Figure 1. Here an elastic wave transducer is placed on the ground and induces an elastic wave into Earth. The wave causes both the surface of Earth and the mine to be displaced. The motion of the mine is quite different from the surrounding soil because the mechanical properties of the mine are quite different from those of the soil. The displacement of the surface of Earth when the mine is present is different from when it is not present because of waves scattered from the mine. The array of radars detects these displacements and thus the mine.
 The theoretical model and design of the antenna are presented in section 2. Theoretical and measured patterns for the antennae are presented in section 3. These results show that the antenna functions as designed when a matching layer is placed on the lens. In sections 4, the results for an array of two radars that was developed and tested in the experimental model for the seismic mine detection systems. These results show the viability of the array concept.
2. Focused Antenna
 A lens-focused conical corrugated horn has been developed for use in the mine detection system. A diagram of the antenna is shown in Figure 2. It consists of a waveguide tuner, a waveguide transformer, a conical corrugated horn, and a dielectric bifocal lens. The lens and the waveguide transformer are connected by the conical metal horn with a corrugated interior. The antenna is fed from a standard rectangular waveguide, and a waveguide transformer is used to transform the rectangular waveguide to the circular waveguide. The cross section of the transformer changes gradually from a rectangular shape to a circular one. A waveguide-type tuner is used in order to match the antenna by achieving the standing wave ratio (SWR) of the antenna desired for the measurement. The waveguide-type tuner consists of a standard rectangular waveguide and three screws inserted into it. By adjusting the length of the tuning screw, a match can be obtained. The antenna is monostatic so that the antenna functions as both a transmitter and a receiver.
 A theoretical model has been developed for the antenna. The model is shown in Figure 3a. The dielectric lens with a refraction index, n, and a diameter, D, has two focal lengths, F1 and F2 (inner and outer focal lengths). The diameter is confined by the semiflare angle, θe, of the horn. The inner focal point is the same as the phase center of the horn. The inner and outer surfaces of the lens are hyperbolic curves. The lens profile is a function of the diameter, the focal length, and the refraction index and is rotationally symmetric. Given those parameters with ϕ = 0° (y = 0), the geometry of the inner surface of the lens can be defined as [Lo and Lee, 1993]
where a1 = (n − 1)F1. The geometry of the outer surface of the lens can be given in a similar manner.
 The inner surface of the lens is illuminated by a spherical wave front emanating from the horn. The horn is excited by the TE11 mode propagating in the circular waveguide. The TE11 mode is converted into the hybrid HE11 mode by the discontinuity between the circular waveguide and the horn and facilitated by the corrugations in the horn [Lo and Lee, 1993]. The electric field on the inner surface of the lens can then be described in terms of the geometric parameters by:
The field on the outer surface of the lens is found by ray tracing. The refraction of the ray at the lens surface is obtained using ray tracing. Figure 3b illustrates the refraction of the ray ignoring reflections from the surfaces of the lens. The lens with a refraction index, n, is placed in the air. Incident fields, i and i, in the direction of i emanate from the source at point A, and are obtained from equation (3). The incident fields are partially reflected and transmitted at point B. The transmitted fields, 1t and 1t, propagate through the lens in the direction of 1t. Then 1t and 1t at the location just before C can be derived from geometrical optics:
where k2 and η2 are the wave number and the intrinsic impedance in the lens, respectively, DF1 is the divergence factor of the surface Ω1, and is the path length between B and C.
 The second refraction occurs at the interface of Ω2. In a similar manner, the transmitted fields, 2t and 2t, at the interface of Ω3 can be obtained from
where k1 and η1 are the wave number and the intrinsic impedance in the free space, respectively, DF2 is the divergence factor of the surface Ω2, and δ is the path length between C and D. δ is infinitesimal resulting in DF2 ≃ 1 and e− ≃ 1. Divergence factors are obtained using the procedure described in the literature [Lee et al., 1982]. T1 and T2 are the Fresnel transmission coefficients at the points B and C, respectively. These coefficients are divided into two components, a parallel and a perpendicular component to the plane of incident, and can be found in the literature [Hecht, 1990].
 Once the fields on the outer surface of the lens are found by ray tracing, the outer surface of the lens can be replaced by an imaginary surface that contains the equivalent electric and magnetic surface current densities (s and s) by using the equivalence theorem. The imaginary surface is presumed to be very close to the outer surface of the lens (Ω3 in Figure 3b). The equivalent electric and magnetic surface current densities can be determined from the electric and magnetic fields on the imaginary surface:
From these, the radiation pattern of the lens can be found. The electric and magnetic fields at position on the observation plane may be represented by the following current-field direct relationships [Silver, 1949]:
where Ψ = er and ∇′ = . Here x′, y′, and z′ are defined by the coordinate system of the source. The radiated pattern (one-way pattern) of the antenna is then given by ∣()∣.
 Rather than the one-way pattern of the antenna, it is the two-way pattern involving the reflection from a scatterer back to the antenna that is of interest for the spot size of the antenna when the antenna is used with the radar to measure surface displacements. This is because radar is monostatic, and the antenna is used for both transmitting and receiving. The two-way pattern can be easily obtained by introducing a scatterer in the presence of the antenna [Collin, 1985]. Figure 4 is a diagram showing a receiving antenna with a scatterer. Here Iin is an input current to the antenna, and () and () are the fields at position radiated by the antenna without the scatterer. When the scatterer exists, the antenna receives the fields scattered back from the scatterer. Let the scatterer be a sphere with radius a ≪ λ0, and the sphere is assumed to be located at position . Here the incident wave is assumed to be a plane wave so that the E and H fields are uniform over the sphere. Then, using the reciprocity theorem, the received open-circuit voltage of the antenna in the presence of the scatterer can be expressed as [Collin, 1985]
which is equivalent to the radar backscattering from the sphere. If the sphere is assumed to be a perfect conductor, ε = ∞, and μ = μo, resulting in
Let the sphere be infinitesimal and be scanned over an arbitrary plane placed h apart from the lens. Then the two-way pattern of the antenna is obtained from equation (14). The shape of the pattern is defined by only nonconstant term, (). Thus the two-way pattern can be approximated as α∣()∣2, where α is a constant that depends on the size of the scatterer and the details of how the antenna is fed.
 Using the theoretical model described above, a parametric study has been conducted to design a bifocal lens providing appropriate electrical characteristics such as focused spot size and sidelobe levels. A corrugated horn with specific dimensions has been selected. Some important predetermined parameters for the horn and the lens are shown in Table 1. With those parameters, the E plane two-way patterns of the antenna have been calculated as a function of the inner focal length, F1, and the outer focal length, F2, of the lens. From these two-way patterns, the 3 dB spot size and the sidelobe level are obtained for different values of F1 and F2. Here the 3 dB spot size is obtained by measuring the width of the main beam in these patterns 3 dB down from the maximum, and the sidelobe level is obtained by normalizing the total energy confined in the sidelobe level by the total energy confined in the main beam. The boundary between the main beam and the sidelobe level is taken at the position of ±10 cm from the center of the main beam. For the antenna, the contour graphs of the 3 dB spot size and the sidelobe level as a function of F1 and F2 are shown in Figure 5. The spot size is obtained at a standoff distance of 20 cm. The spot size indicated on each contour is shown in centimeters, and the sidelobe level is dimensionless. As shown in Figure 5a, the spot size decreases in the direction of the arrow, with the smallest spot size appearing on the upper left. However, along the gray line (arrow), the spot size does not change significantly. The sidelobe level behaves in the opposite manner to the spot size. The sidelobe level decreases in the direction of the arrow. It is seen that there is a tradeoff between the spot size and the sidelobe level. Consequently, an appropriate combination of two foci of the lens must be selected. These graphs have been used as a tool to design a bifocal lens to be manufactured. Two foci have been selected at the design point indicated in Figure 5.
Table 1. Predetermined Parameters for the Lens and the Horn
Diameter of the horn aperture, D
Phase center of the horn from the aperture
Semiflare angle of the horn, θe
Diameter of the lens, D
Dielectric constant of the lens, εr
Operating frequency, f
Polarization of the excited wave
Height of the antenna, h
 Two types of the lenses have been designed and manufactured: two surface-matched lenses and two surface unmatched lenses. These types of lenses are shown in Figure 6. Focal lengths of 20 cm and 30 cm and the diameter of 20 cm are selected for both types of the lenses. A surface unmatched lens has been made of Rexolite 1422 material which has a dielectric constant of 2.53 and a loss tangent of approximately 0.001 at 8 GHz. This material maintains a dielectric constant of 2.53 through 500 GHz with a low dissipation factor. For this reason, the Rexolite material is commonly used for microwave lenses and antennae. The surfaces of unmatched lenses are relatively easy to fabricate because the geometries of the surfaces of the lenses are simple hyperbolic curves and are rotationally symmetric. Thus a CNC lathe was used to fabricate the lens. A second surface unmatched lens has been manufactured using a different material. For this lens, a “Rapid Prototyping Machine” using stereolithography epoxy (SLA) has been used to fabricate a prototype. The SLA machine is available at the School of Mechanical Engineering at Georgia Tech. The SLA machine can make any shaped prototype regardless of the complexity of the geometry. However, the machine can only be used with certain materials, most typical SL 7510. This material belongs to the chemical family of epoxy resin and acrylate ester blend. The electrical properties of this material, such as a dielectric constant and a loss tangent, were measured by placing a sample of the material in a waveguide and measuring the transmission coefficient through the sample. It was found that the dielectric constant, ε′, and the loss tangent, ε″/ε′, are approximately 2.94 and 0.03 at 8 GHz, respectively. Note that SL 7510 material is significantly more lossy than the Rexolite material.
 For the surface-matched lenses, a quarter wavelength matching layer is usually used to match the surface of the lens, and several techniques of simulating a quarter wave matching have been reported: corrugated surfaces, arrays of dielectric cylinders, and arrays of holes in the surface of the lens [Morita and Cohn, 1956]. For this research, corrugated surfaces with horizontal corrugations are chosen. These corrugations are perpendicular to the polarization of the excited E field. Details of the theory and the design of the corrugations can be found in the literature [Morita and Cohn, 1956; Du and Scheer, 1976]. Because of the complexities of the corrugation geometries, this lens would be costly and time consuming to manufacture using typical machining practices. For this reason, rapid prototyping machines such as a fused deposition modeling (FDM) machine and a SLA machine have been used to fabricate the prototype.
 Two surface-matched lenses have been designed and fabricated. One of these lenses is made of the polycarbonate material using the FDM machine. The electrical properties of the material, such as a dielectric constant and a loss tangent, were measured by placing a sample of the material in a waveguide and measuring the transmission coefficient through the sample. It has been found that the dielectric constant and the loss tangent are approximately 2.58 and 0.004 at 8 GHz, respectively. Note that the measured values of the dielectric constant for the polycarbonate sample made using the FDM machine is lower than that published for bulk polycarbonate. This is believed to be due to small amounts of air trapped within the material because of the way the sample is fabricated in the FDM machine. It is not known how consistent the dielectric properties of this material are as a function of location within the lens. The polycarbonate material has been chosen because this material has a very low loss factor; its loss factor is comparable to that of Rexolite 1422. A second surface-matched lens was made of SL 7510 material using a SLA machine.
3. Measuring Two-Way Patterns of the Focused Antenna
 In order to measure the two-way pattern of the focused antenna, an experiment has been conducted using the method of modulated scatterers [Cullen and Parr, 1955; Harrington, 1962; Richmond, 1955]. Figure 7 shows the measurement setup to measure the two-way antenna pattern. The setup consists of an antenna under test (AUT), a radar, a modulating scatterer, and an audio frequency signal source. The scatterer is independently modulated using an audio frequency generated by a modulating source signal. The field incident on a small scatterer is then modulated and back-scattered. The back-scattered signal is then picked up by the AUT and coherently detected by the receiver. The received signal represents the two-way pattern of the AUT because the modulated and back-scattered signal depends on transmission from the transmitter to the scatterer, and then from the scatterer to the receiver.
 A half wavelength dipole has been designed and built as the scatterer. The dipole scatterer is shown in Figure 7. It consists of a printed dipole on a dielectric substrate, a diode switch, and a feeding/decoupling circuit. A microwave PIN switch diode is placed between the arms of the dipole. Chip resistors are used to limit the current driving the diode and as part of a filter. The two resistors are connected to a chip capacitor forming a low pass filter. The filter provides microwave frequency isolation between the diode and the feed lines. This scatterer is implemented to measure the two-way pattern of the AUT. The measurement is conducted by placing the small dipole in the center of a large anechoic surface. The dipole is driven by an audio frequency of 1 KHz with a peak to peak amplitude of 20 volts. The radar output is recorded at this frequency using 2-s integration time to build dynamic range above the noise floor. The AUT is then scanned linearly across the surface. The anechoic treatment assures that the measurement could not be contaminated by a signal or by multiple-reflected signals from the ground or from the scatterer.
 Using the modulated scatterer, two-way patterns of the antenna shown in Figure 2 with the different lenses have been measured. The antenna is placed 20 cm (h = 20 cm) above the modulated dipole scatterer.
Figure 8 shows the two-way patterns measured with four different frequencies for the surface unmatched lens made of Rexolite 1422. For all of frequencies, the E and H plane patterns are seen to be quite different from each other; the sidelobe levels of the H plane pattern are wider and higher than those of the E plane pattern. It was also found that the sidelobe levels are much higher and wider than predicted theoretically. Furthermore, the sidelobe levels change more than expected with changes in frequency.
 These anomalies for the sidelobe levels are thought to happen because of an unwanted resonance within the lens; the lens is seen to act as a waveguide resonator that allows resonant fields to flow along the lateral direction inside the lens. The resonance is believed to happen mainly because of the impedance mismatch between the lens and the air. The impedance mismatch of the lens allows multiple reflections at lens interfaces. Multiple reflections from the interfaces degrade the antenna performance by increasing sidelobe levels.
Figure 9 shows graphs of the two-way patterns measured for the surface unmatched lens made of SL 7510 material. When compared to Figure 8, the sidelobe level behaves in a quite different manner; the sidelobe levels are lower and narrower than those in Figure 8. The reason for this is believed to be the increased loss in the lens. The increased loss damps the resonance and decreases its effects.
 Two-way patterns have been measured for the surface-matched lenses made of polycarbonate and SL 7510. Figures 10 and 11 show the two-way patterns measured for the surface-matched lens made of polycarbonate and of SL 7510, respectively. When compared to Figures 8 and 9, for both lenses, the substantial reduction of the sidelobe levels is shown to be obtained with the surface-matched lenses. This is due to the surface matching layer, yielding the reduction of the multiple reflections within the lenses.
 From these resulting patterns in Figures 10 and 11, it is seen that the lens made of SL 7510 performs slightly better than the one made of polycarbonate. The reason for this is that both the loss in the material and the surface matching layer contribute to the reduction of the resonance within the lens made of SL 7510. However, the maximum power received using the lenses made of the low loss materials is found to be greater by approximately 3 dB than that received using the lenses made of SL 7510. Therefore there is a tradeoff between the maximum power received and the performance of the lens.
 For comparisons, the measured and calculated patterns are plotted together in Figures 12 and 13. The comparisons are shown for the surface-matched lenses made of polycarbonate and SL 7510. The calculated patterns are obtained from the theoretical model described above ignoring the reflections from the lens surfaces and the surface matching layer. In general, the agreement is seen to be good for both planes; the first sidelobe levels and the overall shapes of the patterns are in good agreement. However, for the polycarbonate lens, slight disagreement appears; the first sidelobe levels are slightly wider than the theoretical results. This disagreement is believed to mainly present because either the dielectric constant of a sample of polycarbonate is not sufficiently accurate, or the resonance has not been completely eliminated due to the low loss factor as discussed earlier.
4. Laboratory Experimental Model
 In the mine detection technique being developed at Georgia Tech [Scott et al., 2001a, 2001b], elastic waves are generated in the ground by an electrodynamic transducer in direct contact with the soil surface. The surface displacements are measured using a noncontacting radar sensor in a synthetic array by scanning the radar antenna over the scan region. Interactions of the elastic waves with buried land mines are used to detect the presence and location of the mines; resonant oscillations of the land mine and soil system provide a characteristic signature of the land mine, which is used as a detection cue. Previous system tests in the existing experimental model [Scott et al., 2001b] have used either a waveguide antenna or a horn antenna for the radar. The use of these antennae requires that either the standoff distance must be limited to a few centimeters or beamforming is needed for sufficient resolution of surface displacements. The focused antenna described in this paper has been developed to allow greater standoff heights above the soil surface with similar measurement resolution without beamforming algorithms in the postprocessing. To test the land mine detection capabilities of the radar using the focused antenna in an array configuration with a standoff distance of 20 cm, a two-element radar array is assembled as shown in Figure 14 and tested in the laboratory experimental model. Each antenna has a surface-matched lens (one is polycarbonate, and the other is SL7510) and a separate radar source. Each source operates at a distinct frequency of 7.8 GHz and 7.9 GHz to minimize interference between adjacent radars. Data are acquired using only one radar at a time but both sources are operating throughout the testing to include any interactions between the radars, which are found to be negligible as expected. A TS-50 antipersonnel mine is buried 1 cm deep at a distance of 1 m from the elastic wave source at a location in the center of the 120 cm by 80 cm scan region for the measurements. Each of the radars in the array produces similar results when scanned over the minefield and successfully detects the presence of the land mine; data from the two scans are plotted as images on a 30 dB scale in Figure 15. The image processing algorithm used to form these images involves separation of the forward and reverse propagating waves to determine the reflected and scattered energy from buried objects. The energy in the reverse-propagating waves is then used as a weighting factor to improve the resolution of the image.
 A lens-focused, conical, corrugated, horn antenna was developed for use in a seismic/elastic mine detection system. The antenna significantly improved the performance of the system by improving the standoff distance by a factor of ten. To design the antenna, a theoretical model was developed and validated by comparison to an experiment. The results show that the surface of the lens must be properly matched for the antenna to function as designed. To demonstrate the viability of using an array of sensors to speed up the system, a two-element array of sensors was designed and built using the antenna. The array was tested in the experimental model for mine detection system and performed well.
 This work was supported in part by the U.S. Army Night Vision Electronic Systems Directorate, S and T Division, Countermine Technology Team, by the OSD MURI program by the U.S. Army Research Office under contract DAAH04-96-0448, and by the Office of Naval Research under contract N00014-01-1-0743. The authors would like to thank James S. Martin, Gregg D. Larson, and George S. McCall II for their help in fabricating the antenna and help in performing the experimental measurements.