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We present a two-dimensional, active, millimeter-wave, electronic beam scanner, with Doppler capabilities for stand-off imaging. The two-dimensional scan is achieved by mapping the millimeter wave spectrum to space using a pair of crossed gratings. The active transceiver and heterodyne quadrature detection allow the measurement of the relative phase between two consecutive measurements and the synthesis of the scene’s Doppler signature. The frame rate of the imager is currently limited by the sweep rate of the vector network analyzer which is used to drive the millimeter wave extenders. All of the beam steering components are passive and can be designed to operate at any wavelength. The system design, characterization and measurements are presented and further uses and improvements are suggested.
© 2014 Optical Society of America
1. Introduction
Millimeter waves are well-suited for concealed object imaging applications [1–3]. The wavelength is long enough to penetrate most man-made materials such as clothing, but short enough to provide the required resolution for detection of objects of interest with moderate size apertures [4]. State of the art stand-off millimeter and sub-millimeter wave imagers use only a single detector and optomechanical scanning systems [5–10], the latter limiting their frame rate. New applications of these imagers would be enabled if faster frame rates could be achieved. The advantages of higher frame rates include the ability to image larger areas more quickly, as well as rapidly changing scenes. The consensus among the millimeter and sub-millimeter wave community is that faster frame rates are possible through a mixture of multi-pixel detection and electronic rather than mechanical scanning [10, 11]. Millimeter and sub-millimeter wave detectors remain expensive and challenges exist for building inexpensive detector arrays to acquire simultaneous measurements. Electronic scanning methods in the adjacent K-band have proven successful [12, 13] and similar approaches are being investigated for millimeter waves. Some of these approaches are reflector arrays [11, 14] and transmission modulation phase masks [15]. These methods require active modulation of the antenna elements and are challenging to achieve with the current technology.
Imaging with a single or few detectors requires encoding high-dimensional features of an object onto an accessible lower dimensional manifold. For example, spatial features may be encoded temporally. Examples of this approach include mechanically scanned conjugate point incoherent and coherent millimeter and sub-millimeter wave imagers [5–10], Synthetic Aperture Radar (SAR) [16] and image plane coded aperture imagers [17]. Often, more than one measurement dimension exchange is necessary. Single detector spectroscopy systems encode spectral components into spatial locations, so that the spectral components may be discerned in time by the single detector [18]. A recent imaging system at optical wavelengths dubbed STEAM (serial time-encoded amplified microscopy), is an example where spectrum is traded for space and time is traded for spectrum [19]. In general, systems that can encode space into time and then rapidly sample the resulting signal can achieve very fast frame rates.
To achieve a feasible and practical imaging system, the correct encoding must be chosen for a given detector and class of objects to the imaged. For example, when spectrum and/or phase measurements are available from the detection process, range can be synthesized in SAR or space encoded into spectrum which is further encoded into time as in STEAM. The available single pixel millimeter wave technology spans the passive and active detection modes and incoherent and coherent measurement methods [20]. Active coherent systems provide frequency indexed field amplitude and phase and are suitable for systems that perform spectrum to space mapping.
In this paper we present an electronic scanning system that maps spectrum to space. We call it an echelle crossed grating beam scanner because of the two gratings that implement the beam steering. A scanned-frequency beam is projected onto a frequency-dependent point on a two-dimensional surface in space by a pair of diffraction gratings in the same manner as a dispersive spectrometer [18]. The spectrum to space map is also used successfully in STEAM at visible wavelengths but to our knowledge has never been demonstrated in the millimeter wave spectrum. Unlike the STEAM imager, the echelle crossed grating beam scanner is an active coherent system capable of phase measurements. The phase information and the time made available because of the fast electronic scanning of the spectrum is used to synthesize the Doppler of the scene. The Doppler can be integrated to obtain the third spatial dimension. The echelle crossed grating beam scanner is adaptable to other parts of the spectrum where spectrum indexed detection is possible.
2. System description and design
A sketch and photograph of the imager are shown in Fig. 1. A transceiver of W-band radiation emits a beam in a narrow cone through a 22 dB gain circular horn. This diverging beam is collimated by a polytetrafluoroethylene (PTFE) plano-convex lens. The collimated beam is incident on a pair of gratings, the first being an echelle grating that deflects the beam vertically, and the second a cross grating that deflects the beam horizontally. Both of these gratings are used in a nearly Littrow configuration where the diffracted beam almost overlaps the incident beam. The echelle grating diffracts a comb of frequencies separated by the free spectral range, which is determined by the grating period, to the same scattering angle. A cross grating of first order is used to separate frequencies within a comb. The combined effect of the two gratings is to direct the beam in a raster-scan pattern as the frequency is varied, with the echelle grating corresponding to the fast scan axis, and the cross grating the slow scan axis. The cross grating is slightly concave to focus the beam on a surface to be imaged. As the focus position of the beam is frequency dependent, a frequency present in the backscattered radiation indicates the presence of a scattering target at the position. The backscattered radiation follows the incident radiation path backwards to the transceiver to be detected. Using a transceiver multiplying chain (Virginia Diodes, Charlottesville, VA), the frequency map be swept from 75 GHz to 110 GHz spanning approximately 19 free spectral ranges of the designed echelle grating, which corresponds to 19 scan lines of raster scan. The multiplying chain is driven by a dual source vector network analyzer (VNA) (Agilent N5222A). With an intermediate frequency (IF) bandwidth of 200 kHz the VNA can measure 401 points across the band in approximately 35 ms. The sweep rate of the VNA limits the frame rate and Doppler range. To compare structure measured with millimeter-wave imaging with visible-light imaging, a Kinect camera (Microsoft Corporation, Redmond WA) simultaneously observes the same area as the echelle grating imager. By superimposing the millimeter wave and visible images we are able to provide context for the millimeter wave image and Doppler signature of the scene.
The gratings were designed using the grating equations and then the imaging system was optimized using the Zemax optical design software (Radiant Zemax, Redmond, WA). For a first order design the form factor of the echelle grating, the desired field of view (FOV) and the available spectrum/swept bandwidth, were used as design constraints and requirements. The equations for gratings having one-dimensional periodicity and off-plane incidence are illustrated in Fig. 2 and are given by [21]
where K = 2π/λ is the wave number, Ki is the incident wave vector, Km is the order m diffracted wave vector, d is the direction of the grating, n is the normal to the grating, s is along the direction of the grooves of the grating and L is the grating period.
Fig. 2 Illustration of the grating geometry for off-plane incidence and one-dimensional periodicity and the Zemax axis convention.
If in-plane incidence is assumed the grating equation becomes
where θi and θm are the angles that the incident wave and the diffracted wave of order m make with the normal to the grating n when the incident and diffracted waves are perpendicular to the vector s in Fig. 2. The convention used for the sign of the angles follows the sign of the dot product of the incident and diffracted vectors with the grating direction. The echelle grating is used in Littrow configuration so that θm = −θi, in this configuration the grating equation becomes All the equations necessary for the first order design can be derived using Eq. (3). To obtain the relationship between the incident angle, the grating order and angle of diffraction, we differentiate both sides of Eq. (3) with λ and L constant and divide the result by Eq. (3). The result is To obtain an equation for the modes, we rewrite Eq. (3) as and differentiate both sides with L and θ constant then substituting Eq. (5) in Eq. (6) we obtain Eqs. (3), (4) and (7) can be rewritten in terms of design parameters as where L sin θi is the grating depth as illustrated in Fig. 2, FOV is the field of view, the subscript c indicates center values.The available source and detector can be swept between 75 GHz and 110 GHz, the desired horizontal and vertical FOVs at the center mode are 15 degrees and the length of the echelle is desired to be less than one meter. After a few iterations selecting Δm = 19, results in a center mode number of mc ≈ −48. For the desired FOV = 15°, the incident angle is approximately θi ≈ 85.45°. The step depth for the center wavelength and mode is calculated as L sin θi ≈ 77.78mm, resulting in the grating period L ≈ 78.03mm, and step width of 6.19mm. If the length of the grating plate is one meter, approximately 12 steps resulting in a total aperture height of 75mm. A collimated beam spanning at least 75mm can be achieved with a 250mm diameter Teflon lens illuminated by a standard 22dB conical horn placed approximately 1m from the lens. Also having a larger lens allows greater cross grating illumination. The choice of design parameters is a compromise between the form factor of the grating, the field of view, the resolution in the vertical direction and the number of scan lines, Δm.
The cross grating is operated in first order Littrow configuration (m = −1, θm = −θi). Using the first order design equations for FOV = 15°, the incident angle is approximately θi ≈ 14.67°. The step depth for the center frequency and first order is L sin θi ≈ 1.62mm, the grating period is L = 6.4mm and the step width is 6.19mm. In addition to the FOV and form factor a standoff imaging distance of two meters was placed as a requirement. Beam focusing at two meters is achieved by adding a slight curvature to the crossed grating. The parameters from the first order design were entered in Zemax and the grating geometry and parameters were further optimized for throughput and wavefront error minimization. The prescription for the resulting design is summarized in Table 1.
Table 1. System prescription.
The Zemax design was exported to a CAD (STEP ISO 10303-21) file which was in turn used to produce Solidworks (Dassault Systemes Solidworks Corporation, Waltham MA) models of the lens and gratings. These in turn were sent to a machine shop for computer numerically controlled (CNC) manufacturing. The PTFE lens was machined on a CNC lathe and the two gratings were machined from 25.4 mm thick 6061 aluminum cast plates using a 3-axis CNC mill. Cast plates, rather than extruded plates, were used to prevent the bowing of the plates that would occur for extruded plates as these have internal stresses relieved by milling. Once manufactured, the grating and lens were mounted on an optical breadboard. The gratings sit on legs manufactured to length for the designed tilt angles while the lens is held in place by an adjustable mount for alignment.
3. System resolution
The resolution of the system was characterized by measuring the point spread functions (PSF) of the system across the field of view. A small corner reflector was scanned across the image plane. The corner reflector, 15 mm on a side, was rotated so that one of the mirror joints was turned to the 6 o’clock position when viewed from the crossed grating. The reflector was mounted on a two-axis linear stage using a 60 cm, half inch diameter, aluminum rod for scanning clearance. The region around the corner reflector was covered with anechoic foam and the rod was oriented to minimize its reflection towards the receiver. A 60 cm by 60 cm area was scanned at 2 mm steps as the entire band was measured using 401 points. A background measurement was also collected every 300 positions (corresponding to every column in the scan). The point spread function for 5 points in the FOV is shown in Fig. 3. The points correspond to the frequencies 81.13 GHz, 82.26 GHz, 91.27 GHz, 100.37 GHz, and 101.69 GHz.
The measured point spread functions were compared to the expected point spread functions from the Zemax design. The measured and simulated PSF cross sections are shown in Fig. 4. The simulated PSF as calculated from Zemax is the projected PSF from the source to the object plane only while the measured PSF is the monostatic PSF, from the source to the reflector and back. For this reason the measured PSF is compared to the simulated PSF raised to the second power. There is good agreement between the measured and simulated PSF cross sections, except for changes due to the apparent size of the reflector used to sample the PSF.
Fig. 4 Cross sections of the measured and simulated point spread functions. The measured PSFs are shown using the dotted line and the simulated PSFs are shown with the solid line. The frequencies from top to bottom correspond to measurements at 81.13 GHz, 82.26 GHz, 91.27 GHz, 100.37 GHz, and 101.69 GHz respectively.
The horizontal beam width at the center of the image measured using the full width half maximum is approximately 28 mm simulated and 30 mm measured, and the vertical beam width is 66 mm simulated and 87 mm measured. The width of the measured PSFs in relation to the simulated PSF varies across the field of view. This is due to the effective area of the corner reflector that was used as a sampling target. The vertical (y) PSFs for measurements at 81.3 GHz and 100.37 GHz which are located in the lower part of the FOV, are wider than the simulated PSFs while for the measurements at 82.26 GHz and 101.69 GHz, located in the upper part, they are only slightly wider. The vertical PSF for the measurement at 91.27 GHz, located in the center, is affected more than the measurements at the top of the FOV but less than the measurements at the bottom. The horizontal PSFs are less sensitive to the angle of the reflector.
The PSF is almost twice as large in the vertical direction than in the horizontal. This is the expected result of the effective vertical aperture dimension (approximately 70 mm) of the echelle compared to the horizontal dimension which is effectively controlled by the diameter of the PTFE lens (approximately 250 mm). The PSF is also wider for longer wavelengths as expected because of diffraction.
4. Frequency to space mapping
For proper image display and registration with the Kinect image, the frequency indexed measurements from the VNA are mapped to two-dimensional positions on the object plane. Using the model given by Eqs. (1) and the parameters from Table 1, a two-dimensional map is generated with a position corresponding to a measured frequency. First, the direction of the wave vector diffracted from the echelle grating is calculated using Eqs. (1) given the incident direction from the PTFE lens. This calculation is repeated for the cross-grating to find the direction of the beam to the image plane, using the incident wave vector calculated from the echelle grating. The equations were programmed in Matlab and the directions of 401 evenly spaced frequencies between 75 GHz and 110 GHz were calculated. The locations on the two-dimensional map are the positions at which rays diffracted by the cross grating with the calculated directions intersect the object plane. The locations are normalized in the horizontal and vertical directions prior to shifting and scaling necessary to overlay them on the visible image from the Kinect. A map of the locations of the frequencies is shown in Fig. 5. In the figure, some locations are annotated with the corresponding frequency, the color map represents the frequencies from 75 GHz to 110 GHz.
Fig. 5 Frequency to space map; the locations on the two-dimensional map are the positions at which rays from the origin with the calculated directions intersect the object plane. The locations are normalized in the horizontal and vertical directions prior to shifting and scaling necessary to overlay them on the visible image from the Kinect
The spectrum map is irregular and suggested the use of interpolation to form image estimates on a regular grid. We used an interpolation method suited for irregularly spaced points, based on using Voronoi diagrams of the scattered points to calculate weights [22] for the interpolation function. This “natural” interpolation method produces more symmetric point images for the scattering structure of the spectrum to space map than other methods that were examined, such as a bicubic interpolator. For this reason it was chosen for interpolating the spectrum map onto a regular grid of points.
5. Scene measurements
5.1. Imagery
Registration with the Kinect image was performed using a two step procedure. First, the parameters of the spectrum to space model were adjusted to match the measurement of the PSF at several locations within the FOV. Then, the location of a corner reflector at two corners of the image was determined with the both imager and the Kinect. These measurements were used to calculate the shift and scaling of the millimeter wave image with respect to the Kinect image. Measurements of a person standing in front of the imager were recorded with the imager and the Kinect. The overall frame rate was approximately 10 Hz, this included the Kinect measurement and data transfer, the 401 point VNA measurement and corresponding data transfer, the calculation of the interpolated image, and the image display. The IF bandwidth of the VNA was 20 kHz. The data acquisition, processing and display are performed with Matlab on a Windows7 PC with quad-core Intel i7 processor. If the IF bandwidth is increased to 100 kHz and the image display is turned off, the frame rate increases to 20 Hz. Additional frame rate improvement up to 25 Hz is achieved if the Kinect recording is also turned off.
Figure 6 shows visible and millimeter wave images of the same scene side-by-side. The images are frames from a recorded video. The millimeter-wave image, shown on a linear scale, has been embedded in the visible image for context. Part (a) shows the image of the front side of a person. The image is specular and the convex parts of the subject such as the legs and abdomen show strong returns. The belt buckle is also visible because of the many corner reflectors that it forms with the body and clothing. Some of the scatter from the clothing is visible and can be best noticed around the abdomen area. The arms are also visible. Part (b) shows the subject in a position similar to part (a) but now the subject has a metallic gun-like object holstered in his pants. Only part of the gun is visible in the Kinect image but in the millimeter wave image the entire gun is visible through the subject’s clothing. Part (c) shows the back side of the subject. Convex body parts and parts that are directed towards the receiver have strong returns. Part (d) is an image of the head and shoulders. The face, neck and chest are clearly visible. There is also a video ( Media 1) available on-line showing a ball being tossed to demonstrate the real-time motion capture ability without scanning.
Fig. 6 Millimeter wave images. (a) Front side of a person, (b) gun to the side, (c) back side, and (d) image of the face and shoulders.
The millimeter-wave images from this active imager are specular. The glints from specular reflections are common when the scatterer is smooth as compared to the scale of the illuminating wavelength. When a surface is oriented to reflect back towards the receiver, a signal results that may be orders of magnitude larger than a diffusely scattering signal. Specularity is a well-known phenomena for all active RF imaging systems [20], but may be mitigated by high frame rate imaging. For example, the lucky imaging technique [23] is one form of speckle imaging used in astronomy made possible using high speed and short exposure cameras. By taking a large number of very short exposure images, clear images may be acquired and combined as these become available when brief intervals of favorable atmospheric transmission occur. The echelle grating imager may be used similarly, by acquiring lucky glints of object features as the object moves and combining these glints into a more complete estimate of object structure.
The system resolution is wavelength, and therefore spatially, dependent as it was shown in Section 3. This wavelength dependence is expected to result in lower energy density at the target for the longer wavelengths (approximately two times less than the energy density for the shorter wavelengths), however this is not evident in the images in Fig. 6. The reason for this is the transceiver transmit power and dynamic range dependence on wavelength. At longer wavelengths the transceiver has higher power and dynamic range than it does at the shorter wavelengths. Another factor is the atmospheric attenuation which follows a similar trend. The mmw images in Fig. 6 show that for the particular implementation the losses in energy density due to PSF width are compensated by the power characteristics of the transceiver and atmospheric attenuation. For a proper system design, system resolution, transceiver power and scanning range, should be considered since the latter two compete with the former.
5.2. Doppler measurements
The radial velocity of objects can be measured by comparing the phase of two consecutive measurements [24]. If the target region that is being probed by the beam has moved less than a half wavelength between adjacent measurements, then the phase measurement is not aliased and the displacement of the target in time can be extracted (Δd = λΔθ/2π). To demonstrate the Doppler capabilities of the imager we placed a pendulum in the field of view and collected 500 consecutive measurements. The measured sweep rate of the measurement was approximately 35.2 ms and the pendulum arm was approximately 61 cm. The sweep rate is calculated by measuring the time it takes from triggering the VNA for acquisition of the 500 frames until the data is returned to the PC. The measured time includes the data transfer time which is anywhere between 3 ms and 4 ms for 401 points, as described in the VNA user manual.
The measurements were analyzed and the phase difference between consecutive measurements was calculated for frequencies with sufficient return magnitude to form reliable phase estimates. The phase was extracted from image points after the step of interpolating the frequencies onto a regular grid and registration with the visible image. Figure 7 (top) shows the phase difference of consecutive measurements. The Fourier transform of the Doppler is computed and shown in Fig. 7 (bottom). The period of oscillation as calculated from the measurement is 1.754 s. The period calculated from the small angle pendulum equation , where L is the length of the pendulum arm and g is the gravitational acceleration, is 1.567 seconds. The discrepancy is because of the incorrect value of the sweep rate used in the calculation of the Doppler frequency in Fig. 7. The measured and calculated Doppler periods coincide if the VNA sweep time is adjusted to 31.4 ms, corresponding to a data transfer time of 3.8 ms which is within the published Agilent specification.
Fig. 7 Result of the Doppler measurement. Phase differences of consecutive measurements as a function of time (top) and time spectrum of the doppler measurement (bottom).
5.3. Three dimensional point tracking
Another interesting use of the Doppler measurements is the tracking of a single target in three dimensions. To demonstrate this, we collected consecutive measurements as a corner reflector was moved in spiral motion towards the imager. The data was analyzed to locate the point on the 2-D map and the range was synthesized by integrating the Doppler at that point. The result is shown in Fig. 8. Part (a) shows the position along the three axes as a function of frame number. Part (b) shows the position in three-dimensions, the color map encodes the frame number. The x and y axes are plotted on the same axes scale as Fig. 3 and with respect to the origin of the linear stages used to sample the PSF. The range along the z axis is plotted in units of meters and is calculated as range = ∑λiΔθi/2π where the i indexes the frame number and λi and Δi are the wavelength and phase difference calculated for the ith frame. The phase was corrected for aliasing by adding 2π to all the negative phases. This is justified using the prior knowledge that the object was moving toward the imager during the measurements. The spiral shape of the motion is easily distinguished from the reconstruction. The spiral however is not smooth. In the range direction, this is due to aliasing and incorrect aliasing correction. In the cross range directions this is due to the maximum return always being detected at one of the 401 map points.
Fig. 8 Point tracking in three dimensional space. (a) The location in each dimension as a function of frame number, (b) 3-D trajectory, the color indexes the frame number.
6. Conclusions
The echelle crossed grating beam scanner electronically scans the object plane by sweeping the frequency of the transceiver. The increased speed enabled by this rapid scanning technique may be used to analyze the behavior of the target in time, such as its Doppler signature. The Doppler signature can in turn be used to obtain a third dimension from the scene – the range dimension. In the future, fast imaging frame rates could perhaps be used to perform lucky imaging to mitigate the problem of specularity which is a problem with all active radio frequency imaging. The frame rate of the imager is limited by the sweep rate of the source and ultimately by the total amount of signal power that is captured during the sweep. As the frame rate of the echelle grating scanner is currently limited by the sweep rate of the network analyzer, an oscillator is being constructed capable of sweeping the frequency band in less than a millisecond. This way the frame rate could be maximized so that the imaging speed is limited only by signal power. The echelle grating beam scanner may find applications at other wavelength bands where the convenience and speed of electronic scanning offsets the cost and bulk of the grating components. Furthermore, as the system is linear time-invariant, broadband radiation, such as available from terahertz sources generated by ultrafast laser pulses, may be used instead of coherent radiation as only the spectrum needs to be measured to obtain spatial structure. The dispersion introduced into the terahertz pulses by the echelle spectrometer can be engineered so that different frequencies are detected at different time intervals, so that a single detector could be used in this case as well. While millimeter-wave technology currently has limitations due to detector cost and scarcity, encoding of the fields may be used to maximize the utility of millimeter-wave imaging systems and therefore drive their further adoption.
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