Abstract:
A process is provided to detect an object within a defined region using standing longitudinal cavity mode waves. The process includes disposing first and second conductive lines substantially parallel to the axis, transmitting an electromagnetic signal through the first line at a set frequency, returning the transmitted signal through the second line, measuring power from a reflected signal through the first line, adjusting the set frequency based on the measured power; extracting an appropriate parameter from the reflected signal to obtain a reflected characteristic, comparing the reflected characteristic to an established characteristic that lacks the object to obtain a characteristic differential, and analyzing the characteristic differential to obtain a position of the object along the length. The first and second conductive lines have specified length and width that bound a defined region. The analyzing can be performed by Fourier transform across wave modes.

Description:
CROSS REFERENCE TO RELATED APPLICATION 
     The invention is a Division, claims priority to and incorporates by reference in its entirety U.S. patent application Ser. No. 11/890,098 filed Jul. 13, 2007, and issued Sep. 28, 2010 as U.S. Pat. No. 7,804,441. 
    
    
     STATEMENT OF GOVERNMENT INTEREST 
     The invention described was made in the performance of official duties by one or more employees of the Department of the Navy, and thus, the invention herein may be manufactured, used or licensed by or for the Government of the United States of America for governmental purposes without the payment of any royalties thereon or therefor. 
    
    
     BACKGROUND 
     The invention relates generally to methods and systems that use standing waves to detect a buried or submerged object in an optically opaque medium. In particular, the invention employs electromagnetic or sonic waves and analyzes their standing wave patterns to identify the presence and location of an otherwise hidden object having material characteristics that differ from the medium into which the object is embedded. 
     Military personnel may be assigned to patrol regions where hostile combatants operate. Concealed ordnance, e.g., landmine, improvised explosive device (IED), present a severe hazard of injury and/or death in such reconnaissance roles. The difficulty in identifying, locating and neutralizing such objects from a safe distance hampers efforts to pacify these territories by elevating risk to travelers (especially those on patrol) well above acceptable levels. 
     SUMMARY 
     Conventional techniques to detect unauthorized intrusion yield disadvantages addressed by various exemplary embodiments of the present invention. In particular, various exemplary embodiments provide a mechanism to detect such target objects with greater safety distance margins than conventionally available. 
     Various exemplary embodiments provide electromagnetic tomography implemented by spectral analysis in an excited Fabry Perot cavity to detect and locate enemy agents crossing security perimeters or buried enemy ordnance, such as improvised explosive devices (IEDs) within localized volumes such as the soil lying underneath roadways upon which Coalition military vehicles must travel on a regular basis. Variant embodiments employ radio frequency electromagnetic waves or sonic waves, depending on application. Microwaves represent an exemplary portion of the electromagnetic spectrum in conjunction with microwave frequency computer-aided tomography (MCAT) implemented by spectral analysis. 
     Various exemplary embodiments provide methods and systems to detect an object within a defined region using standing longitudinal cavity mode waves. The process includes disposing first and second conductive lines substantially parallel to the axis, transmitting an electromagnetic signal through the first line at a set frequency, returning the transmitted signal through the second line, measuring power from a reflected signal through the first line, adjusting the set frequency based on the measured power; extracting an appropriate parameter from the reflected signal to obtain a reflected characteristic, comparing the reflected characteristic to an established characteristic that lacks the object to obtain a characteristic differential, and analyzing the characteristic differential to obtain a position of the object along the length. The first and second conductive lines have specified length and width that bound a defined region. The analyzing can be performed by Fourier transform across wave modes. 
     The two lines constitute a parallel-conductor radio-frequency transmission line stub. The alternative embodiments further includes transmitting a swept radio frequency signal along the transmission line, measuring the frequency, amplitude, and width of the resonant mode peaks, comparing the frequencies, amplitudes and widths to those previously obtained without target objects in the vicinity of the transmission line, and Fourier transforming the difference thereby computed. 
     These procedures may include recording the received signal by suitable instruments to obtain a received frequency spectrum, comparing the received frequency spectrum to an established frequency spectrum without the object, and Fourier transforming the difference to compute an intelligence function containing the target location. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       These and various other features and aspects of various exemplary embodiments will be readily understood with reference to the following detailed description taken in conjunction with the accompanying drawings, in which like or similar numbers are used throughout, and in which: 
         FIG. 1  is a block diagram view of the system according to a reflector embodiment; 
         FIG. 2  is a flowchart view of the signal analysis process for detecting an object; 
         FIG. 3  is a signal mode plot from an acoustic aquarium test; 
         FIG. 4  is a frequency band plot of an electromagnetic microwave signal without a target in the region (representing the cavity); 
         FIG. 5  is a frequency band plot of an electromagnetic microwave signal with a target at the center of the region; 
         FIG. 6  is a position plot of an electromagnetic microwave signal with a wooden target at the center of the region; 
         FIG. 7  is a position plot of an electromagnetic microwave signal with a wooden target midway between the center and one end of the region; 
         FIG. 8  is a position plot of an electromagnetic microwave signal with a metal target quarter-way between the center and one end of the region; 
         FIG. 9  is a block diagram view of the system for a twin-lead embodiment; 
         FIG. 10  is an electric circuit diagram of a high impedance (open circuit) condition for the twin-lead embodiment; 
         FIG. 11  is an electric circuit diagram of a low impedance (closed circuit) condition for the twin-lead embodiment; 
         FIG. 12  is a mode plot of electric field vs. distance for the three lowest frequency transmission line modes for a closed or short circuit at the far end of the transmission line stub; 
         FIG. 13  is a mode plot of electric field vs. distance for the three lowest frequency transmission line modes for an open-circuit condition at the far end of the transmission line stub; 
         FIG. 14  is a frequency shift plot based on target presence; 
         FIG. 15  is a frequency shift plot as a function of modes; and 
         FIG. 16  is a transform plot from modes to position. 
     
    
    
     DETAILED DESCRIPTION 
     In the following detailed description of exemplary embodiments of the invention, reference is made to the accompanying drawings that form a part hereof, and in which is shown by way of illustration specific exemplary embodiments in which the invention may be practiced. These embodiments are described in sufficient detail to enable those skilled in the art to practice the invention. Other embodiments may be utilized, and logical, mechanical, and other changes may be made without departing from the spirit or scope of the present invention. The following detailed description is, therefore, not to be taken in a limiting sense, and the scope of the present invention is defined only by the appended claims. 
     Radar employs electromagnetic radiation to provide azimuth, elevation and range of an object. A transmitter projects radiation in the microwave or radio spectrum. An object within line-of-sight of the transmitter reflects a portion of the radiated energy, which proceeds as a traveling electromagnetic wave back, to a receiver, whereupon the radar system analyzes the reflected signal to obtain the desired information. However, signal attenuation of short radiation wavelengths, at high frequencies, limit signal propagation and detection distances. For low frequencies, the object might escape detection if smaller than the long radiation wavelengths having less attenuation. 
     The object possesses material properties that differ from the concealing medium. These properties influence real and/or imaginary portions of the refraction index at microwave frequencies, thereby indicating the object&#39;s presence. A defined area of the medium may be interrogated with standing waves that encompass a suitable range of wavelengths. The standing wave patterns contain integer numbers of nodes and antinodes, each of which is located at a known position along the cavity length. The object betrays its presence by selective absorption or scatter of the energy, rather than by reflection. The changes in energy, frequency and quality associated with standing wave patterns (from an empty medium) depend on the object&#39;s location. 
     The standing waves represent wavelengths related to the length of the area being searched for the object&#39;s presence and location. The search area may be designated as a “cavity” having characteristic length L for the distance between boundaries against which the waves reflect, such as a Fabry-Perot cavity. The wavelength λ=v/f, where f is frequency and v is the wave velocity for each longitudinal mode in the cavity has a quantized value that may be expressed as λ=2L/n, where n is a positive integer 1, 2, . . . N, and L is the cavity length. For an electromagnetic wave, this velocity corresponds to nearly the speed of light. For an acoustic wave, this velocity corresponds to the longitudinal speed of sound, or additionally the transverse speed of sound for a solid medium. 
     For the twin-lead application, the waves may be reflected at the boundaries of the search area to produce a node at the shorted end of the transmission line, an antinode at the open end of the line, and some integer number of nodes and antinodes along the length of the line, the exact numbers being proportional to the frequencies of each standing wave mode. For a shorted far-end of the transmission line (closed circuit), the transmission line length L must be an odd number multiple of a quarter wavelength λ for each mode. This can be expressed as L=(2n+1)λ/4, where n is a non-negative integer 0, 1, 2, . . . N. The object can perturb the standing wave either by shifting the frequency of the mode peaks, or by changing the amplitude of the power at the mode peaks. 
     Modern radar typically depends upon the reflection of very short microwave-frequency pulses by a target of interest. Unfortunately, the presence of microwave signals is recognized by many adversaries, some of whom have deployed buried or camouflaged ordnance with very small radar scattering cross sections and used it successfully against US forces in Iraq and Afghanistan. The problem of locating, identifying, and neutralizing enemy agents and ordnance (particularly buried IEDs) at a distance has been a vexing challenge for US military analysts and a major personal risk for Coalition warfighters. 
     Reflector Embodiment: Fortunately, recent developments in optoelectronic component technology may be applied fruitfully to address problems such as the detection of buried IEDs in arid Middle Eastern soil and the location of enemy agents crossing long security perimeters. An enemy agent or IED inherently contains substances that exhibit real and/or imaginary components of the index of refraction at audio, ultrasonic, radio, or microwave frequencies that do not precisely match those of the surrounding air in which an agent may traverse or else the soil within which an IED may be buried. 
     If such a target (operative or ordnance) can be interrogated with a set of standing radio waves that encompass a suitable range of wavelengths, the presence of the target causes absorption or scattering whose magnitude is dependent upon the precise location of the target with respect to the nodes and antinodes of each of the standing wave mode patterns. Thus the introduction of a target into a volume that has been illuminated with a set of standing waves having an initially-uniform distribution of power perturbs this distribution of power among the various wave modes. In addition, the frequencies and/or the quality of the standing wave modes may be perturbed by the presence of the target object. 
     This perturbation can be accurately measured with modern spectral analysis instrumentation, and the location of the target can then be extracted from this perturbation by an appropriate type of mathematical analysis. Furthermore, this method of detection is most sensitive to objects that are comparable to or smaller than the smallest wavelength used to interrogate them and is therefore not subject to the usual diffraction limit that affects conventional imaging technologies. 
     The detection algorithm (to be described subsequently) relies upon the differential absorption or scattering of a set of standing wave mode patterns and not upon the reflection of a narrow band of wavelengths as in traditional radar or imaging technology. This location technique can also circumvent countermeasures to conceal hostile operatives or ordnance by using stealth technologies that absorb electromagnetic radiation in the radar frequency bands. The standing wave pattern can be created by establishing a Fabry-Perot cavity in one or more physical dimensions and exciting a set of standing wave modes within this cavity by use of a low power microwave transmitter. The cavity is formed by placing two highly reflecting “mirror” surfaces at the ends of the desired region to be protected and orienting them so that they are parallel. 
     This ensures that waves with propagation directions parallel to the cavity axis may travel back and forth within the cavity and experience multiple reflections at the two mirror ends. In order to ensure that a large set of modes is excited within the cavity, the transmitter sweeps often over a sufficiently wide bandwidth to encompass the desired range of wavelengths. For the various embodiments presented herein, the instant disclosure proposes the exploration of a new technology that exploits the differential absorption and/or scattering of a set of standing waves to detect and locate accurately a concealed target and presents results from two preliminary trials of the technology. 
     Diagram:  FIG. 1  shows a block diagram  100  of the detection system. A generator  110  provides an electromagnetic wave signal to a transmitter  120 , which projects the signal through a transmit antennas  130  to be intercepted by a receive antenna  135 , with both antennas (e.g., transducers or horns) being embedded within a cavity  140 . The signal propagates within the cavity  140  through a medium (e.g., air, soil, water) along a longitudinal axis  145 . The cavity  140  is bounded at each end by corresponding reflectors  150 ,  155  (e.g., microwave mirrors). Frequencies of the signal received by the receive horn  135  are evaluated by a radio frequency (RF) signal analyzer  160  and further processed by a computer  170 , which may illustrate the resulting modes on a display monitor  175 . By monitoring the frequency modes, a change therein indicates the presence of a target  180 , shown as a buried IED. 
     The reflectors  150 ,  155  are oriented parallel to each other, being disposed at the longitudinal ends of the cavity  140 , which represents a region to be monitored. The longitudinal cavity axis  145  is formed by a line perpendicular to the reflectors  150 ,  155  that extends the distance length of the cavity  140 . These mirrors may be flat or parabolically curved to focus the reflected waves. 
     The transmitter  120  may transmit microwaves at low power, and these microwaves are frequency-swept by modulation generator  110  for sending to the transmit antenna  130 . This transmit antenna  130  would generally be implemented as an electromagnetic horn, possibly with some gain over an isotropic radiator to confine most of the radiation to the direction of the cavity axis  145 . 
     The power generated by the frequency-swept transmitter  120  is injected through the transducer  130  and into the cavity  140  as electromagnetic radiation. This power may be attenuated because some waves are emitted in directions that are not parallel to the cavity axis  145 . That fraction of the power in waves that run parallel to the cavity axis  145  travel back-and-forth between the two reflectors  150 ,  155  and resonate within the cavity  140  at all discrete frequencies for which the cavity length L is an exact integer multiple of a half-wavelength. 
     These standing waves interact with the target  180 , which will selectively absorb or scatter some of the cavity modes. As a result, the overall shape of the spectrum is perturbed by the presence of the target  180 . This perturbed spectrum can be detected by receive antenna  135  and analyzed by the spectrum analyzer  160 , which sends its information to the computer  170  through an appropriate interface (e.g., GPIB, USB) and processed by a suitable program. The computer  170  analyzes the spectral shape using an algorithm presented further herein. As a result of this analysis, the computer  170  extracts from the spectral shape an intelligence function that is displayed on the monitor  175 . The horizontal axis of this plot represents distance, and the vertical axis represents deviation from average cavity materials properties. The target  180  is thus represented by a peak in the monitor  175  that resembles the blip on a traditional radar range scan. 
     Process:  FIG. 2  illustrates a flowchart  200  that describes the analysis and procedure for receiving and processing the standing wave signals to detect an intrusive object within a defined region. An iterative step  210  provides data resulting from the measurement of the power, frequencies and/or quality (denoted as Q) for each of a suitably large number of longitudinal cavity modes. Previously acquired data under conditions absent any target are acquired and stored at step  220 . These results from step  210  are compared at step  230  with the corresponding data from step  220 . 
     The algebraic difference between these two data sets, computed at step  230 , are then subjected to the Discrete Fourier Transform at step  240 . The magnitude of the output data set from step  240  is computed in step  250  and either displayed for observation by a human operator and/or analyzed by the software for automatic detection of target peaks. After a suitable period of time, the process is repeated, and the program flow returns to step  210  to begin the analysis of the next set of data acquired by the receiver. 
     Algorithm: The mathematical details of a microwave frequency computer-aided tomography (MCAT) Algorithm used to process the received spectrum and extract therefrom the intelligence function that bears information on the location of targets of interest. This detection system relies upon the creation of a Fabry-Perot cavity that bounds the cavity  140  that defines the spatial region. This cavity  140  consists of two parallel mirrors  150 ,  155 , separated by a longitudinal distance L. 
     One of the well-known properties of such cavities is that standing waves can resonate in the region between the mirrors  150 ,  155  only at those discrete frequencies for which the cavity length L is an integral multiple of a half wavelength; 
                     L   =       m   ⁢           ⁢     λ   m       2       ,           (   1   )               
where λ m  is the wavelength of the longitudinal cavity mode for which m half wavelengths exactly fit in the region between the reflectors. For this example, the cavity length L is in the range of 10 km (kilometers). Specific field-deployable implementations of this invention may utilize cavity lengths ranging from tens of meters to tens of kilometers.
 
     Artisans of ordinary skill will recognize that the frequency separation δf between adjacent longitudinal modes is given by the expression: 
                       δ   ⁢           ⁢   f     =     c     2   ⁢           ⁢   nL         ,           (   2   )               
where c is the speed of light in vacuo, and n is the index of refraction. For n=2 as a reasonable value for soil in the arid climates of the Middle East at low microwave frequencies, the adjacent modal separation is in the neighborhood of 7.5 kHz (kilo-Hertz), a value easily resolved with modern microwave spectrum analyzers.
 
     This Fabry-Perot cavity can be excited by a low-GHz (giga-Hertz) frequency microwave transmitter  120  via the transmit horn (or other suitable antenna)  130  to carry the transmitter&#39;s signal in the cavity  140 . The transmit antenna  130  may preferably be disposed in close proximity to one of the two parallel reflectors  150 , both disposed within the soil of the region comprising the cavity  140 . For the transmitter  120  being frequency-swept by the generator  110  over a bandwidth of 130 MHz in a low GHz band, the resulting band-width may be sufficient to encompass about 17,300 longitudinal modes, each distinguished by the 7.5 kHz modal separation interval. 
     Assuming that signal losses associated with absorption in the medium (e.g., soil) are not excessive, these longitudinal modes lying within the frequency-swept bandwidth of the transmitter  120  may resonate within the cavity  140 . These cavity modes may be detected by the receive antenna  135  placed in the soil near the other reflector  155  and connected to the RF spectrum analyzer  160 . 
     Quantifying the energy present in each of these longitudinal modes can be achieved by a set of traditional first-order rate equations. These are similar to the rate equations for a semiconductor laser, but modified slightly to make them applicable to the geometry in  FIG. 1 . For S m  being the number of photons-per-cubic-meter associated with the m th  longitudinal mode (i.e., amplitude), α m  is the loss associated with the same mode, P t  being the RF power output of the microwave transmitter, and C m  being the probability-per-unit-time of exciting a photon in the m th  longitudinal mode, the rate of change in photon density for the m th  mode may be written as follows: 
     
       
         
           
             
               
                 
                   
                     
                       ⅆ 
                       
                         S 
                         m 
                       
                     
                     
                       ⅆ 
                       t 
                     
                   
                   = 
                   
                     
                       
                         C 
                         m 
                       
                       ⁢ 
                       
                         P 
                         t 
                       
                     
                     - 
                     
                       
                         c 
                         n 
                       
                       ⁢ 
                       
                         α 
                         m 
                       
                       ⁢ 
                       
                         
                           S 
                           m 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     The spectrum analyzer  160  may be used to measure the relative amplitude S m  of each longitudinal mode. To model signal loss as a function of wavelength, an analogy to first order laser gain may be applied. This first order laser gain is modulated by the first moment of the spatial gain with respect to the square of the longitudinal mode electric field distributions. See, e.g., Yamada, M. and Suematsu, Y., “Analysis of Gain Suppression in Undoped Semiconductor Lasers”,  J. Appl. Phys.,  52, 2653-2664, (1981). 
     This insight was applied for various exemplary embodiments in creating the software for the laser Defect Distribution Scan Algorithm. See, e.g., DeChiaro, L., Robinet, M., and Devoldere, P., “Effects of Drive Current Upon Defect Distribution Scan Features in Multi-longitudinal Mode Semiconductor Lasers”,  J. Lightwave Technology,  11, 2057-2065, (1993). The same principle may be extended to the presence of a target of interest such as a concealed enemy agent or a buried IED. Due to the various internal constituents, the presence of the target may create a sharply-localized perturbation in the real and/or the imaginary components of the index of refraction of the surrounding medium. 
     The physical size of the target  180  may be sufficiently small so as to exhibit a negligible scattering cross section at low microwave frequencies. Nonetheless, its presence yields a frequency-selective attenuation of some of the longitudinal modes, depending upon its exact longitudinal coordinate relative to the nodes and antinodes of the modal electric field distributions. Thus, the loss for the m th  longitudinal mode may be modeled as follows: 
                       α   m     =       α   0     +       ∫   0   L     ⁢       G   ⁡     (   z   )       ⁢              E   m     ⁡     (   z   )            2     ⁢     ⅆ   z             ,           (   4   )               
where α 0  is the constant background loss, G(z) is the spatially-dependent loss (i.e., intelligence function) represented by any perturbations of the cavity medium (including IEDs, buried rocks, subterranean tunnels, or foliage), and E m  (z) is the electric field distribution of the m th  mode.
 
     For the spectrum analyzer  160  able to measure values of photon density S m  for many excited cavity modes, eqns. (3) and (4) may be solved by a simple Fourier Transform method to obtain the intelligence function G(z). In steady state, all the time derivatives in the system for eqn. (3) vanish, and eqn. (3) may then be solved for the steady state spectrum to obtain: 
     
       
         
           
             
               
                 
                   
                     S 
                     m 
                   
                   = 
                   
                     
                       
                         
                           nC 
                           m 
                         
                         ⁢ 
                         
                           P 
                           t 
                         
                       
                       
                         c 
                         ⁡ 
                         
                           ( 
                           
                             
                               α 
                               0 
                             
                             + 
                             
                               
                                 ∫ 
                                 0 
                                 L 
                               
                               ⁢ 
                               
                                 
                                   G 
                                   ⁡ 
                                   
                                     ( 
                                     z 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                      
                                     
                                       
                                         E 
                                         m 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         z 
                                         ) 
                                       
                                     
                                      
                                   
                                   2 
                                 
                                 ⁢ 
                                 
                                   ⅆ 
                                   z 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     Next, the cavity losses may be postulated as not being exceedingly large. For such a condition, the modal electric field distribution can be approximated as follows: 
     
       
         
           
             
               
                 
                   
                     
                       E 
                       m 
                     
                     ⁡ 
                     
                       ( 
                       z 
                       ) 
                     
                   
                   = 
                   
                     sin 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           z 
                         
                         
                           λ 
                           m 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     Substituting eqn. (6) in the integral in the denominator of eqn. (5) and recalling the trigonometric identity sin 2 θ=(1−cos2θ)/2, eqn. (5) reduces to: 
                       S   m     =         nC   m     ⁢     P   t         c   ⁡     (       α   0     +       L   ⁢     G   _       2     +       1   2     ⁢         G   ^     +     ⁡     (   λ   )           )           ,           (   7   )               
where  G  is the spatial average of G(z) and Ĝ +  (λ) is the Inverse Discrete Fourier Transform of the intelligence function G(z), using the sine functions of the modal electric fields of eqn. (6) as Fourier basis functions for the series expansion.
 
     Because L and  G  are both constants, the second term in the denominator of eqn. (7) may be combined into the α 0  term with no loss of generality. Eqn. (7) can be solved to yield Ĝ(λ), and the intelligence function G(z) retrieved by a Fourier Transform as follows: 
                       G   ⁡     (   z   )       =     F   ⁢           ⁢   F   ⁢           ⁢     T   ⁡     [         2   ⁢           ⁢     nC   m     ⁢     P   t         cS   m       -     2   ⁢           ⁢     α   0         ]           ,           (   8   )               
where FFT designates the Discrete fast-Fourier Transform (FFT) operation.
 
     In order to implement eqn. (8) in software, the parameters remain defined as: S m  is the discrete spectrum measured by the analyzer, P t  is the average RF power supplied by the transmitter, C m  is the probability-per-unit-time of photon emission into the m th  longitudinal mode, n is the average index of refraction of the cavity medium, c is the speed of light in vacuo, and α 0  is the background loss of the cavity medium. 
     The C m  probabilities are given by the shape of the radiated emission spectrum of the continuous wave (CW) microwave transmitter and antenna combination. For a reasonably flat frequency response of this combination, the values of C m  should be constant over a bandwidth sufficiently large to encompass about 16,384 longitudinal modes. 
     The choice of a number of modes that is an integral power of two is made in order to permit the use of the fastest possible software implementation of the Cooley-Tukey Algorithm to perform the Fourier transformation and retrieve the intelligence function G(z) in minimal time. See, e.g., Cooley, J. W. and Tukey, J. W., “An Algorithm for the Machine Calculation of Complex Fourier Series”,  Mathematics of Computation,  19, 297-301, (1965). 
     In practice, the negative term on the right hand side of eqn. (8) should appear as a direct current (DC) level upon which the wavelength-dependent perturbations associated with the targets ride. This term can therefore be suppressed before the Fourier transform is done. Such practices may enhance the overall numerical sensitivity of the software. 
     Typically, a function is solved by transform of its inverse, as provided in eqn. (8). However, for results obtained for this intelligence function application, solving by FFT of the function may improve sensitivity for a small target. Thus, the intelligence function G(z) may be alternatively expressed as a Fourier transform of the difference between the photon density S m  for each mode m and a reference value S 0m  absent any target. This can be expressed as:
 
 G ( z )=FFT[ S   m   −S   0m ].  (9)
 
     Acoustic Tests: The MCAT Algorithm may be applied to a wide variety of wave phenomena. A preliminary feasibility test of the concept was therefore accomplished with a minimal investment by running a trial using ultrasonic acoustic water waves in a volume sufficiently small for demonstration purposes. The protected region and the cavity mirrors were both realized by an aquarium (i.e., fish tank) filled with tap water. The glass walls of the tank are very parallel and reflect sound waves. 
     Because the short dimension of the tank measured about 0.15 m (meter) and the speed of sound in fresh water is taken to be 1435 m/sec (meters-per-second), the modal separation may be computed from eqn. (2) to be 4.78 kHz. Thus, the demonstration used ultrasonic frequency equipment. A low-power function generator such as the Hewlett-Packard (HP) 8111A was chosen for the transmitter  120 , and an ICOM PCR1500 radio communications receiver was used as the spectrum analyzer  160 . (The PCR1500 is a computer-controlled receiver whose supplied software includes a spectrum analyzer mode and a data recorder function that uses the hard drive of the controlling computer to store the acquired spectra). Low cost piezoelectric transducers were used as the transmit and receive antennas  130 ,  135 . 
     The signal generator  110  was configured to generate square waves, and the frequency was adjusted to a value close to 4.78 kHz so that the harmonics would fall at the cavity mode frequencies. The PCR1500 was set to scan the frequency range from 50 kHz to 250 kHz and record the spectra measured with and without a target  180  in the cavity  140 . 
       FIG. 3  illustrates a plot  300  from these acoustic tests. This graph features frequency f up to 300 kHz as the abscissa  310  and signal strength in arbitrary units (au) as the ordinate  320 . The graph shows two example spectra: a solid line  330  representing measurements without a target, and a dash line  340  representing measurements in which the tank includes a ball-point-pen-tip inserted into the water near the tank center. Some of the modes show signal strength differences  350  between the respective measurements. A sharp spike near 250 kHz is caused by a spurious response with the receiver. 
     An example peak  360 , such as at 170 kHz illustrates several additional details. Its peak strength of ˜190 au may be used to determine a frequency resolution (i.e., peak width), denoted as Δf, that represents the frequency difference of the peak  360  at its half peak strength  370 . The frequency difference between the peak  360  and its nearest spectral neighbor represents free spectral range (FSR), quantified as v/4d. The FSR may be represented by a frequency difference  380  between modal peaks. (For electromagnetic waves, velocity v is analogous to the speed of light c.) The cavity  140  may be characterized by quality denoted as Q=f/Δf, and “finesse” quantified as FSR/Δf. 
     Compared with the empty cavity spectrum, the spectrum measured with the target in place shows a perturbed shape in which every other mode is enhanced, while the intervening modes are attenuated. The spectral deviation caused by the presence of the target  180  is thus described as a “strong-weak-strong-weak” pattern that is characteristic of a cavity with a perturbation (denoting the target) located at the cavity&#39;s mid-position. The period of the spectral deviation is the minimum meaningful value of two mode spacings. 
     Electromagnetic Tests: The observation of acceptable results from the acoustic test demonstrated the feasibility of the tomography concept. Subsequent trial would involve electromagnetic waves sent through air over a path length that is measured in meters rather than centimeters. Such tests were performed using the large anechoic chamber at Dahlgren with a cavity length of 10 m (meters). An Agilent E8251A signal generator was used as the transmitter. 
     The best results for these tests were obtained over the 4 GHz to 5 GHz frequency range, using microwave horns as the transmitting and receiving antennas  130 ,  135 . The transmitter  120  was set to cover the frequency band in steps of 1 MHz with a dwell time of 1 s (second) at each frequency. The reflectors  150 ,  155  were two rectangular pieces of sheet aluminum measuring about 0.6 m-by-1.0 m to act as microwave mirrors. Power levels were about +11 dBm at the transmitter output connector. 
     A 15 m length of coaxial cable was used between the transmitter output and the transmit antenna. In the 4 GHz to 5 GHz band, this cable exhibits an estimated 8 dB (deci-Bell) loss. Thus, the actual transmitter power at the antenna  130  is estimated to be about +3 dBm (deci-Bell-milliwatt). The transmit and receive antennas  130 ,  135  were placed in the cavity  140  about 15 cm (centimeters) from the reflectors  150 ,  155  and oriented so that the microwave beams were directed at the reflectors in order to minimize the signal sent directly from transmit to receive antennas. 
     Signal reception was implemented by an Agilent E4407B spectrum analyzer and was controlled by a legacy Dolch 486 computer equipped with an HP GPIB interface card. Applications programming was done in the ASYST programming language for reasons of convenience and cost. The E4407B was set to cover the 4 GHz to 5 GHz range. Resolution and video 3 dB bandwidths were each set to 100 kHz, and the time per scan was about 200 ms (milliseconds). The E4407B was programmed to identify the strongest peak in each scan and to report the frequency and strength to the computer  170 . 
       FIG. 4  shows a plot  400  of the spectrum measured in the absence of any target  180 . This graph features frequency between 4 GHz and 5 GHz as the abscissa  410  and received power in dBm as the ordinate  420 . A continuous line  430  represents the power for each frequency without a target within the anechoic chamber that defined the cavity  140 . The analyzer noise floor was about −72 dBm, and the spectrum consists of about 872 data points. 
     Received signal strength varied between −43.5 dBm and −51.5 dBm, providing a margin of about 20 dB above the analyzer noise floor. The plot  400  shows considerable structure on both large and small scales. The small scale structure of the line  430  consists of a series of small, regularly-spaced peaks that are separated by about 15.7 MHz on average. This agrees well with the value of 15 MHz that is predicted by eqn. (2) for an air cavity with a length of 10 m. These peaks are therefore the longitudinal mode peaks of the cavity that are necessary for the MCAT Algorithm to operate properly. 
     The large scale structure also shows evidence of periodic behavior with a period of about 430 MHz in frequency space. This structure is believed to originate in a physical process different from the cavity resonances that give rise to the longitudinal modes mentioned above. One possible candidate mechanism is resonances associated with the feed-horn antennas. These will invariably exhibit some frequency dependence in their radiation pattern, and variation of the main lobe size by a few dB from 4 GHz to 5 GHz is not unreasonable. 
     Another possibility, however, is that the very presence of the transmit and receive horns  130 ,  135  and the coaxial cable feed-lines in the cavity  140  can perturb the field configurations due to the scattering of the waves. Thus, the MCAT scans may provide information on any and all objects in the cavity whose real or imaginary parts of the index of refraction differ from those of air. Given the present physical design of our system, this specifically includes the two antennas and some fractional length of their coaxial feed-lines. Additionally, the peak-to-valley contrast of the modes varies between 1 dB and 2 dB can be observed from  FIG. 4 . Clearly, this contrast is preferably to be as large as possible in order to provide an improved level of sensitivity to a weakly absorbing or reflecting target. This contrast is related to cavity quality Q, related to resolution, and finesse, related to damping time. 
     The cavity quality Q may be increased by minimizing losses. This process would include spillover losses from the feed-horns (if the reflectors subtend a solid angle smaller than that of the main lobes), as well as diffraction losses and losses due to mirror misalignment and the finite conductivity of the reflector material. Careful engineering design can minimize most of these, and this portends subsequent opportunities to improve the performance and sensitivity of future systems. 
     Despite the modest modal contrast shown in  FIG. 4 , the introduction of a target  180  into the microwave beam resulted in a measurable perturbation of the power distribution among the longitudinal cavity modes.  FIG. 5  shows the spectrum that was measured after a wooden object (i.e., target  180 ) was placed in the cavity within the microwave beam. The portion of the target  180  that was intercepted by the microwaves consists of a single wooden 2×4 plank (i.e., stud) oriented vertically (i.e., upright and hence transverse to the ground surface) with the short dimension parallel to the cavity axis  145 . 
     This caused an overall drop of about 2 dB in the measured signal strength across the 4 GHz to 5 GHz band. More important, the presence of the target (in the anechoic chamber) also introduced a perturbation into the modal structure so that the modes exhibited the same strong-weak-strong-weak pattern of oscillations that is shown in the water wave graph of  FIG. 3  for the fish tank. This represents the spectral signature expected for a target  180  placed in the center of the cavity  140 . 
     In particular,  FIG. 5  shows the received spectrum as a plot  500  with the upright wooden 2×4 plank disposed in a 10 m cavity  140  at a position near the center at 5 m from either end. This graph features frequency between 4 GHz and 5 GHz as the abscissa  510  and received power in dBm as the ordinate  520 . A continuous line  530  represents the power for each frequency with the plank target near the center the anechoic chamber that defined the cavity  140 . In order to display this position information in a quantitative manner, the spectral perturbations caused by the target must be extracted. This can be accomplished by identifying the frequencies of the mode peaks in the unperturbed data of  FIG. 4 , examining the same frequencies in the  FIG. 5  data, subtracting away the unperturbed spectrum, and Fourier transforming the difference. The results are shown in  FIG. 6 . 
     In particular,  FIG. 6  shows the resulting MCAT scan of this configuration with the upright wooden 2×4 plank disposed at the cavity center. This graph features distance from the cavity&#39;s boundary edge in meters as the abscissa  610  and return strength in arbitrary units as the ordinate  620 . A continuous line  630  denotes the deviation from average cavity materials properties caused by the plank target as a function of position along the length of the cavity. The “radar return” from the target  180  denoted by an arrow  640  is shown at the right side of the plot  600 . This indicates an abrupt change in the materials properties intercepted by the microwave beam at the center of the cavity (i.e., the 5 m position of the 10 m cavity length). The large difference feature at the left side of the plot  600  is believed to be an artifact caused either by the antenna horn resonances or by the modal perturbations induced by the presence of the two horn antennas in the cavity. 
     Other target configurations were also investigated to ensure that the MCAT Algorithm faithfully responds to different target locations.  FIG. 7  shows a plot  700  of the MCAT scan for the same upright wooden 2×4 plank, but disposed about 3 m from the receiver reflector  155 . This graph features distance from the cavity&#39;s boundary edge in meters as the abscissa  710  and return strength as the ordinate  720 . A continuous line  730  denotes the deviation from average cavity materials properties caused by the plank target as a function of position along the length of the cavity. The target&#39;s location is denoted by an arrow  740  shown near the center of the plot  700   
       FIGS. 6 and 7  illustrate the sensitivity of the MCAT system to targets that are composed primarily of dielectric materials (e.g., wood). To deploy an MCAT network to detect buried IEDs or land mines, however, the system must demonstrate response to metallic objects also. In order to show this capability, a second target consisting of a vertically-oriented (i.e., upright) metal pipe supported at the bottom by a circular, metallic base was disposed in the cavity at a distance of 1.6 m from the receiver reflector  155 , and spectra were acquired. The resultant MCAT scan is shown in  FIG. 8 . 
     In particular,  FIG. 8  shows a plot  800  of the MCAT scan for the metal pipe and base, disposed about 1.6 m from the receiver reflector  155 . This graph features distance from the cavity&#39;s boundary edge in meters as the abscissa  810  and return strength as the ordinate  820 . A continuous line  830  denotes the deviation from average cavity materials properties caused by the plank target as a function of position along the length of the cavity. The target&#39;s location is denoted by an arrow  840 , shown near the 1.5 meter position along the cavity length in the plot  800 . 
     Application: The first application of the tomography algorithm described herein was developed for the location of atomic scale crystallographic defects lying within the active layers of semiconductor lasers. The cavity lengths of such low power communications lasers are typically in the neighborhood of 250 μm (micrometers). The successful application of this algorithm to the 15 cm fish tank required that the algorithm be scaled up by a factor of about six-hundred from the laser scales. The second step upward to our 10 m anechoic cavity involves a total scaling factor of about forty-thousand from the laser length scales. To incorporate the MCAT Algorithm on the battlefield would typically require an additional increase in scale by about an additional factor of a thousand, or three orders of magnitude. 
     This factor of a thousand can be accomplished in approximately three further steps. The first of these would involve a cavity length of 150 meters and will take place near the flight line in the Maginiot Open Air Test Site (MOATS) Facility at Dahlgren. Pending approvals by Base Operations, Safety, and the Flight Line organizations, the second step would involve a path of 1500 m and may be performed adjacent to the Dahlgren Flight Line. The final scale-up to a 10 km path and the transition to an underground path will need to be accomplished at an off-base facility such as Yuma, Ariz. where the testing of a 10 km MCAT will not interfere with other activities and the soil is sufficiently dry to approximate conditions in the Middle East. 
     The system described herein could be deployed as an automated network of sensors, spaced at regular intervals along a roadway of interest. For example, the straight line distance from Baghdad to Tikrit is about 159 km. If there were a road running directly between the cities and assuming that a separation of 10 km between neighboring stations yields acceptable sensitivity and spatial resolution in the Iraqi soil, then this road would be divided into sixteen segments, each of 10 km in length, and thus employ seventeen sensor stations, the first equipped with one transmitter and a mirror, and the last equipped with one receiver and a mirror. The fifteen stations along the roadway would each employ two reflectors, one transmitter, and one receiver. 
     The computers controlling each receiver would be programmed to command the local spectrum analyzer to measure an RF spectrum at periodic intervals such as every 5 minutes. Each spectrum would be uploaded to the local controlling computer, which would then process the data according to the algorithm, extract the intelligence function, and compare that with intelligence functions extracted from previous scans measured from the same volume of soil. 
     If the software determines that a significant, new peak suddenly appears in one of the sixteen straight road segments between Baghdad and Tikrit and remains stable for a predetermined period of time, the computer  170  would be programmed to automatically raise an alarm and relay the coordinate information of the new signal peak to Coalition personnel, who can then dispatch appropriate forces to take action as determined by the local commanding officer. 
     In summary, microwave frequency computer-aided tomography (MCAT) can be implemented by spectral analysis in an excited Fabry Perot cavity as a means of detecting and locating hostile agents crossing security perimeters or buried enemy ordnance such as IEDs within localized volumes such as the soil lying underneath roadways upon which Coalition military vehicles must travel on a regular basis. 
     This disclosure presents a description of the fundamental principles upon which such a detection system is based and contains the results of two trials of the system; the first implemented with ultrasonic acoustic waves in a 15 cm fish tank and the second with microwaves in a 10 m cavity housed within an anechoic chamber. We report the computation of MCAT scans containing peaks whose locations correspond accurately to the target positions for targets composed of wood and metal. These results indicate technical feasibility to extract from measured cavity spectra an intelligence function giving the locations of inhomogeneities lying within the cavity, particularly buried IEDs. 
     Twin-Lead Embodiment: In an alternative embodiment, the attenuation of the medium can be circumvented by stretching a pair of parallel wires along a distance L and across a width W to be interrogated. It is also possible to use existing local infrastructure such as overhead power lines to implement a sensor under the proper conditions. 
       FIG. 9  illustrates a block diagram  900  showing the twin-lead configuration. A signal oscillator  910  provides a tuned alternating current (AC) (AC) signal along the parallel conductor transmission line comprised by wires  920  and  930 . The wire leads  920 ,  930  stretch along the distance L and separated by the width W to a terminus  940  enclosing a target  950 . Depending on the configuration of the circuit, the terminus  940  may be open or closed. The characteristic impedance of the parallel conduction transmission line may be represented by the relation: 
                       Z   0     =     276   ⁢           ⁢       log   10     ⁡     (       2   ⁢   W     a     )           ,           (   10   )               
where Z 0  is the impedance in Ω(ohms), W the spacing width between the wires and a is wire diameter, both lengths in comparable units. As can be observed from eqn. (10), impedance increases with increasing width and decreasing wire diameter.
 
     For the terminus  940  being open, the system  900  represents a shunt or open loop, which can be schematically described as a parallel inductor-capacitor circuit. This shunt signal is reflected at the terminus  940 . For the terminus  940  being closed, the system  900  represents a short circuit at the terminus  940  so that impedance remains zero at all frequencies. This short signal is bounded at the terminus  940 , which may include a power meter, while the circuit at oscillator  910  remains open. 
       FIG. 10  shows an electrical schematic  1000  for the parallel inductor-capacitor circuit (also known as a “tank” circuit) that behaves as an open circuit or shunt. The oscillator  910 , represented by nodes  1010  are connected by a circuit  1020  that includes an inductor  1030  and a capacitor  1040  connected together in a parallel loop  1050 , thereby causing high impedance at resonance.  FIG. 11  shows an electrical schematic  1100  for the series inductor-capacitor circuit that behaves as a low impedance or near short circuit at resonance. The oscillator  910 , represented by nodes  1110  are connected by a circuit  1120  that includes an inductor  1130  and a capacitor  1140  connected together in series. This behaves as though the oscillator had been terminated with a series resonant inductor-capacitor circuit representing the transmission line stub. 
     The closed circuit, with the terminus  940  representing a closed switch, reflects waves as odd harmonics of the quarter wave-length defined by the distance L. As described previously, each closed circuit wavelength λ in the cavity has an harmonic expressed as L=(2n+1) λ/4, where n is a non-negative integer 0, 1, 2, . . . N. These harmonics are shown in  FIG. 12  as a plot  1200  for three lowest values of n. This graph features normalized distance between the oscillator  910  and the terminus  940  as the abscissa  1210  and wave amplitude as the ordinate  1220 . The plot  1200  shows harmonic cosine curves for n=0 as solid line  1230 , n=1 as dash line  1240  and n=2 as dot-dash line  1250 . Vertical lines are shown at about ⅓ and ⅝ of distance L at  1260  and  1270 , respectively. 
     As can be observed, the first line  1260  shows high (absolute) amplitudes for the first and third harmonic curves  1230 ,  1250 , but is negligible for the second harmonic curve  1240 ; and the second line  1270  shows high-to-moderate amplitudes (e.g., received power) for the first and second harmonic curves  1230 ,  1240 , but is negligible for the third harmonic curve  1250 . These differences can be exploited by a spectrum analyzer to determine amplitude differences induced by a target&#39;s presence. 
     In particular, a target located at ⅓ L may absorb the electromagnetic energy for the first and third wave modes (having non-zero magnitudes), but not the second mode (having substantially no magnitude). Alternately, for the target position at ⅝L may absorb the electromagnetic energy for the first and second modes, but not the third mode. Evaluation of additional modes enables intermediate positions within the modal region to be compared. The different levels of absorption depending on which modes are affected provides an indicator of the target&#39;s location. In this quarter wave example, the transmission line stub appears as a resonant circuit in which the capacitive impedance component is given by Z C =−j/(2πfC), where C is the capacitance per unit length of transmission line, f is frequency, and j=√−1 represents the imaginary coefficient. 
     Similarly, each open circuit wavelength λ in the cavity has an harmonic expressed as L=(n+1)λ/2, where n is a non-negative integer 0, 1, 2, . . . N. These harmonics are shown in  FIG. 13  as a plot  1300  for three lowest values of n. This graph features normalized distance between the oscillator  910  and the terminus  940  as the abscissa  1310  and wave amplitude as the ordinate  1320 . The plot  1300  shows harmonic cosine curves for n=0 as solid line  1330 , n=1 as dash line  1340  and n=2 as dot-dash line  1350 . Vertical lines are shown at about ⅛, ½ and ¾ of distance L at  1360 ,  1370  and  1380 , respectively. 
     As can be observed, the first line  1360  shows high-to-moderate (absolute) amplitudes for the first and second harmonic curves  1330 ,  1340 , but is negligible for the third harmonic curve  1350 . The second line  1370  shows high amplitude for the second harmonic curve  1340 , but is negligible for the first and third harmonic curves  1330  and  1350 . The third line  1380  shows high-to-moderate amplitudes for the first and third harmonic curves  1330 ,  1350 , but is negligible for the second harmonic curve  1340 . 
     In particular, a target located at ⅛L may absorb the electromagnetic energy for the first and second modes, but not the third mode. Alternately, for the target position at ½ L, the electromagnetic energy may be absorbed for the second mode, but not the first and third modes. Similarly, the target at ⅛L may absorb at the first and third modes, but not the second mode. These differences can be exploited by a spectrum analyzer to determine amplitude differences induced by a target&#39;s presence. The open circuit represents a high-impedance circuit with Z L =j2πfL. 
     A circuit&#39;s impedance is a complex quantity (i.e., having real and imaginary components) that can be more generally expressed by the relation 
                     Z   =       R   +     j   ⁡     (       2   ⁢   π   ⁢           ⁢   fL     -     1     2   ⁢   π   ⁢           ⁢   fC         )         =     R   +     Z   L     +     Z   C           ,           (   11   )               
where R is the real component resistance. At low frequencies, the Z C  term in eqn. (11) dominates so that the circuit appears capacitive. At high frequencies, the Z L  term in eqn. (11) dominates so that the circuit appears inductive. At the natural or resonant frequency f r ={2π√(LC)} −1  for the parallel circuit in  FIG. 10 , the Z C  and Z L  terms cancel, producing a pure resistance circuit having no imaginary component. The presence of a target in the vicinity of the transmission line sensor alters either the real and/or the imaginary components of the transmission line impedance. The changes so produced may vary as a periodic function of the mode number being measured. The exemplary embodiments provide for detection of a target despite its concealment by signal absorption.
 
     Resolution of the target&#39;s presence by the signal analyzer  160  may be determined by various methods, such as comparing amplitudes of peaks of signals with and without the target&#39;s influence  340 ,  330  for select harmonic frequencies, as described for  FIG. 3 . Alternatively, the signal analyzer  160  may determine a frequency offset for conditions of adequate quality and finesse.  FIG. 14  shows a plot  1400  with frequency f as the abscissa  1410  and wave amplitude as the ordinate  1420 . The plot  1400  features a nominal inverse cosine function wave  1430  and an offset inverse cosine function wave  1440  influenced by the target for a single mode. The nominal wave has peak amplitude at a first frequency  1450 , and the offset wave has peak amplitude at a second frequency  1460 , the two frequencies being offset by a frequency difference Δf shown as a gap  1470 . This offset can be determined for each mode over a series to characterize the absorption and reflection characteristics of the target. 
     The frequency difference Δf can then be plotted over a series of modes to thereby determine the pattern of shifts induced by the target. Information on the target location is contained within the pattern.  FIG. 15  shows a plot  1500  with mode numbers as the abscissa  1510  and frequency difference Δf as the ordinate  1520 . As can be observed, the value of the frequency shift is a periodic function of the mode number, and both positive and negative shifts are observed. This frequency shift is shown in this example to vary as a sinusoidal wave  1530  having positive maxima  1540  and negative minima  1550  at the extrema. The sinusoidal wave  1530  illustrates a continuous curve produced from a collection of discrete values for each mode. At the inflection points  1560 , the frequency shift reaches zero, reflecting modes with resonant frequencies unperturbed by the target. The slope  1570  at the inflection points  1560  provides a measure of the maximum rate of change in frequency shift with respect to transmission line mode number. 
     As the slope  1570  increases (i.e., becomes steeper), the target may be moving closer to the center of the cavity of the twin-lead configuration  900  along its longitudinal axis, or the target may be moving closer to the transmission line conductors  920 ,  930  themselves. A power meter that inductively senses the current through the transmission line  920  of the twin-lead circuit  900  may provide the output signal to be analyzed by a suitable instrument, such as a power meter or a spectrum analyzer. The modal curve has a wavelength or period  1580  that corresponds to the target&#39;s position with respect to the cavity ends. For a target about mid-way between the ends of the cavity, the corresponding wavelength is short, providing rapid peak-to-peak oscillations across the modes from one to the next. By contrast, for a target in proximity to one of the ends of the cavity, the corresponding wavelength or period is large, appearing as a gradual undulating curve across the modes. The wavelength or period of the oscillations in the mode plot therefore contains information about the physical location of the target with respect to the transmission line sensor. 
     In this transmission line implementation, resolving the target&#39;s position along the length of the sensor from the modal frequency differences can be accomplished by Fourier transform. An example technique for providing the frequency-to-position function translation is the discrete FFT.  FIG. 16  shows a plot  1600  illustrating relative position of the target or object (i.e., of the source of electromagnetic absorption and/or reflection) along the cavity&#39;s longitudinal axis  145  in correspondence to the frequency difference variation with modes. 
     The abscissa  1610  represents this relative position, and the ordinate  1620  represents deviations away from the average properties of the material surrounding the sensor (as lines  920  and  930  shown in  FIG. 9 ). Idealized (i.e., absent noise) above and below peaks  1630 ,  1640  are plotted from the Fourier transform of the modal frequency differences. (Both non-elevated levels adjacent either side of the peaks correspond to average properties of the material surrounding the target.) The upper peak  1630  represents the target&#39;s position near the center of the cavity. By contrast, the lower peak  1640  represents the target&#39;s position adjacent an edge of the cavity. 
     Those of ordinary skill in the art will recognize that although electromagnetic waves have been described with particularity, and even more specifically in conjunction with the microwave portion of the spectrum, application of these principles can be extended to other wave propagators, including acoustic signals through a material medium, and the frequencies utilized may vary from the low kilohertz well into the Gigahertz without departing from the spirit of the invention. 
     While certain features of the embodiments of the invention have been illustrated as described herein, many modifications, substitutions, changes and equivalents will now occur to those skilled in the art. It is, therefore, to be understood that the appended claims are intended to cover all such modifications and changes as fall within the true spirit of the embodiments.