A variety of synthetic-aperture ladar (SAL) imaging techniques are investigated on a table-top laboratory setup using an ultra-broad bandwidth (>3 THz) actively linearized chirp laser centered at 1.55 microns. Stripmap and spotlight mode demonstrations of SAL in monstatic and bistatic geometries are presented. Interferometric SAL for 3D topographical relief imaging is demonstrated highlighting the coherent properties of the SAL imaging technique.
©2012 Optical Society of America
Synthetic Aperture (SA) imaging has long been a valuable tool in radar allowing high spatial resolution without the need to utilize inconveniently large real apertures. In the optical domain, SA techniques have only more recently been successfully demonstrated including: traditional range and cross-range synthetic-aperture ladar (SAL) in direct analogy to the most common form of synthetic-aperture radar (SAR) [1,2], holographic aperture ladar (HAL) to improve the resolutions of 2D transverse imaging , and laser optical feedback imaging to improve 2D and 3D confocal imaging . A major hurdle in the direct application of common frequency modulated continuous wave (FMCW) SAR techniques to the optical domain has been the lack of linear frequency chirped lasers sources that are both broadly tunable and sufficiently coherent. The use of active electronic feedback to stabilize the frequency sweep of tunable laser sources is one solution, has led to the demonstration of the highest range resolution for free-space time-of-flight ladar ranging , and is used here to accomplish table-top-scale laboratory demonstrations of SAL based on FMCW ladar.
Synthetic aperture radar has a significant history, and many operational modes, applications, and geometries have been utilized or demonstrated including: stripmap and spotlight collection modes, interferometric imaging for 3D topographic relief or coherent change detection, and advanced configurations such as bistatic or multistatic geometries [6,7]. Exploring synthetic aperture techniques with SAL is important to determining the potential utility of SAL for different applications including long-range imaging systems  and 3D surface profiling at a large standoff. We report here laboratory (table-top) demonstrations of advanced SAL imaging modes, including what we believe are the first reported demonstrations of interferometric SAL (IFSAL) based 3D topographic relief imaging, spotlight mode SAL imaging, and bistatic SAL imaging. These demonstrations highlight some of the benefits and drawbacks of each method with respect to particular applications.
The synthetic aperture radar or ladar technique produces images in the dimensions of range (direction of illumination beam) and cross-range (a.k.a along track, along the direction of aperture motion). To provide range diversity a planar scene must be observed at a slant angle to the surface (i.e. one cannot take SAR\SAL images from directly overhead). In this work, as in most modern SAR systems, the 1D range profile measurements are collected via the coherent chirped FMCW technique, which results in a Fourier representation of the scene. This Fourier space range profile of the scene is then recorded from a number of equally spaced positions along a 1D track, the synthetic aperture, generating an angularly diverse data set known as the phase history data (PHD). The PHD represents the scene in the radial Fourier dimension (range) and azimuth Fourier dimension (cross-range). If the illumination is perpendicular to the synthetic aperture (SA), the SA positions are regularly spaced, and the geometrical angles are all small, a complex SAL image can be formed, or compressed, by a simple 2D Fast Fourier Transform (FFT) of the PHD. The SAL image pixel sizes in each dimension are set by the temporal and spatial Fourier relations leading to a dimensions of δR = c/(2B) in range where c is the speed of light and B is the chirp bandwidth, and δCR = λ/(2θSA) in cross-range where the range is R, θSA is the angle subtended by the SA. The actual resolution of the image depends on the successful coherent image compression in both dimensions and any intentional or unavoidable Fourier windowing (e.g. in stripmap mode the beam scans across the scene limiting the angular bandwidth to the divergence angle of the illumination). Generally, the active stabilization of the chirped laser source is sufficient to achieve the Fourier limited resolution in range, while the cross-range resolution depends greatly on the effectiveness of SA compression.
Practical difficulties that complicate SA imaging include: 1) If the SA subtends large angles, the PHD must be interpolated to a Cartesian grid before compression with FFT’s, referred to in the SAR community as polar formatting [6,7,9]. Polar formatting algorithms are designed to compensate for the deterministic migration through resolution cells of stationary points in the scene as determined by the geometry of the SA collection and the position in the scene. While polar formatting is a necessity to most modern wide aperture SAR systems, it has been less important to SAL due the smaller diffraction angles and simpler geometries (e.g. broadside stripmap) attempted to date. 2) Motion errors that occur during the SA collection must be compensated, ideally to less than the carrier wavelength. In modern SAR systems, GPS based navigation followed by data driven algorithms such as Phase Gradient Autofocus (PGA) for final focusing has found to be sufficient for phase errors that occur due to motion [6,7,10]. Despite the much smaller wavelength used for SAL, we have found that good mechanical control combined with the PGA algorithm is sufficient in most cases to correct the motion and phase errors that occur during the SA collection.
The experimental setup used for the SAL and IFSAL imaging demonstrations is shown in Fig. 1 . It begins with the actively stabilized source described in , which provides > 3.6 THz bandwidth linear chirps centered at 1550 nm at a chirp rate of 5 THz/s of which we utilize a 600 ms portion for the images presented here. This results in a 3 THz bandwidth and corresponding 49.8 μm range pixels. A hydrogen cyanide cell (10 Torr, HCN) provides an absolute frequency reference for the chirps. A polarization maintaining (PM) Erbium Doped Fiber Amplifier (EDFA) amplifies the chirp to a level of ~200 mW, from which the Local Oscillator (LO) and signal paths are generated. A fiber circulator is used to separate the transmission from the returned target signal. In the stripmap mode and interferometric images (Fig. 2 and Fig. 3(a) ) the transmit\receive aperture is the face of the single mode fiber (10.5 μm mode field diameter) yielding a spot diameter of approximately 40 cm for a target 2 meters away. In the spotlight and bistatic SAL demonstrations (Fig. 3(c), 4 , and 5 ), a larger real aperture was approximated by re-imaging the fiber mode to a larger size using an aspheric lens (OFR 7 mm focal length) to collimate followed by a focusing lens (plano-convex 35 mm focal length) resulting in an aperture of approximately 50 μm. In stripmap mode, aperture optics are mounted on a precision linear ball screw stage (THK KR46) driven by a computer controlled stepper motor (Zaber NM Series) to implement the aperture motion. Spotlight mode utilizes this and additional hardware for implementation (see Fig. 3(d)).
At each position of the stage, the return signal is optical heterodyne detected by coherently beating with the LO path and using an auto-balanced detector to achieve near-shot-noise-limited detection followed by digitization with a 14 bit ADC (NI5122) at a sample of rate of 1 to 5 Ms/s. The transmission through HCN frequency reference is digitized simultaneously. It is used to measure and remove the pulse-to-pulse frequency jitter of the chirp by shifting the time window saved in the PHD. This ensures that the data from each SA position span the same optical frequencies. A raised-cosine window is then applied to the time domain signal before being range compressed by an FFT. The positive frequency portion of the compressed complex range profile around the scene (typically about 2048 points or 10 cm of range) is then saved as a column in a two dimensional array. This process is then repeated at the next position of the stage for the full number of steps in the SA. The result is a complex valued matrix constituting the range compressed, but cross-range uncompressed SAL data.
The steps size in the demonstrations presented here were chosen to be about the diameter of the real aperture. Traditionally in stripmap mode SA imaging the step size is half the diameter of the real aperture. Due to the inverse relationship between size of the real aperture and the diffraction limited size of beam on the target, a step size of half the diameter of the real aperture ensures that the Nyquist sampling condition is met, which avoids aliasing issues related to objects at the edge of the scene aliasing down into the scene. While we have observed this effect, in the demonstrations presented here the targets were small enough that we could relax this constraint. Being able to increase the size of the steps and lower their number while achieving the same resolution was a useful tradeoff due to the long time it took to collect the data. Each aperture step takes about 1 second due to the > 600 ms duration chirp and to ensure the motion of the stage settles. This long sampling period leads to collection times on the order 40 minutes to collect data from the 2048 positions that make up most of the SA’s presented here. Smaller steps would have mainly produced more black space on either side of the target in the SAL image, while increasing the already long collection time.
At this point a brief note about the photometric budget is in order. Due to the small apertures (10 μm and 50 μm) relatively small irradiance is incident on the target and very little power is collected. In the case where the fiber is used as the real aperture the irradiance on the target ~2 m away using 200 mW of transmit power is approximately 1x1011 photons/s in a resolution pixel size of 100 μm x 100 μm. Assuming diffuse scattering into a 2π solid angle the fiber will collect about 1 photon per resolution cell for the 600 ms chirp at each position during SA collection. This rough photometric analysis, and the corresponding image (see Fig. 3(a)) is a strong testament to the near-single-photon sensitivity of coherent balanced heterodyne detection and the robustness of the PGA algorithm , which can effectively correct the phase errors and focus the cross-range dimension of the image even with very low raw signal-to-noise ratios (peaks of 5 to 10dB above shot-noise for an individual range profile in the dragonfly image of Fig. 3(a)) and thereby providing additional coherent integration gain due the SA image formation.
2. Interferometric SAL
Like SAR, SAL is coherent imaging method, which should enable the use of interferometric imaging functions such as coherent change detection and topographic relief mapping [6,7]. We demonstrate here IFSAL for the application of 3D topographic relief imaging. In 3D relief imaging two stripmap SAL images are taken from two equal length and parallel SA tracks that are displaced from each other in a direction orthogonal to the range and cross-range dimensions. In our setup the range and SA directions are in a horizontal plane so the displacement was made in the vertical direction. This displacement introduces phase differences in the images that depend on the 3D surface profile of the object. The origin of these phase differences can be modeled as differences in the geometrical path length. We have found it useful to imagine a bisection plane as a reference, which we refer to as the Zero Phase Plane (ZPP), in which all points in the ZPP are equidistant to either SA track. This ZPP contains the line parallel to and equidistant between the two SA tracks and the point at scene center. The phase difference in the two SAL images of a scattering point at height h above this ZPP is determined geometrically as:
For the IFSAL demonstration shown in Fig. 2, two stripmap SAL images were taken of a scene consisting of the Lincoln side of a U.S. penny painted white to present a uniformly diffuse target. The individual SAL images were taken sequentially using the single mode fiber as the real aperture with 10 μm step spacing and 2048 steps as the SA. For the 137 cm range, a separation of 1.9 mm between the SA tracks in the vertical direction was chosen so that the 100 μm features on the target caused a measurable (but not modulo 2π ambiguous) phase change (see Eq. (1)).
Achieving phase correlation between two SAL images is difficult due to the requirement of sub-wavelength effective stability, which is exacerbated by the long data acquisition times of our current SAL demonstration setup. In addition to motion errors, changes in the index of air in the ranging path, and relative length fluctuations in the signal and LO fiber paths can lead to large (many wavelengths) phase fluctuations. Fortunately, as long as the motion errors are less than a resolution cell (i.e. less than range resolution) the Phase Gradient Autofocus algorithm  works well for most images to correct the phase error that occurs between SA positions. The PGA algorithm corrects the phase errors by opportunistically exploiting bright points in the scene to produce an average estimate of the common mode phase change between aperture positions. While PGA is very robust for focusing most images, the PGA was not relied upon in this IFSAL demonstration as the white paint that provided uniform reflectivity and constant SNR for the phase measurement made it a poor candidate for PGA causing the final PGA corrected image to have noticeable blurring. In addition, because the PGA only tries to make bright points brighter, we speculated that it might possibly de-correlate the fairly uniform speckle pattern of the two images by arbitrarily focusing on different random speckle cells. Therefore to ensure image phase stability, a 2.5 mm diameter cat-eye n = 2 index of refraction ball retro-reflector was placed in the scene as a bright point phase reference (similar to the use of a guide star in adaptive optics) to which every range profile was motion and phase corrected. However, given a different target that contains physical bright points the PGA alone may be adequate for interferometric processing.
In addition to phase correction, the IFSAL processing requires good image registration and 2-D phase unwrapping. Image registration is required to ensure the speckle phase is correlated (i.e. when the interferogram between the two images is constructed the pixels correspond to the same patch on the actual scene and therefore the phase difference is related only to the change in perspective). Image registration was performed via intensity correlation, although complex phase correlation may improve the registration . After registration, the interferogram is formed as, , where M1,2 are the individual complex SAL images from the two aperture tracks. Next the interferogram was smoothed by an 8x8 phase preserving filter to make speckled the interference fringes more suitable for unwrapping  at the expense of lowering the image resolution. Alternatively, one could use other means to achieve speckle averaging such as averaging over interferogram pairs taken from slightly different perspectives or chirp frequencies. Figure 2(b) shows the interferogram after filtering but before phase unwrapping, the main source of the fringes is the “flat earth” 45 degree orientation plane of the penny used to provide range diversity. These fringes are removed by subtracting a linear phase to better reveal the penny surface topography information after unwrapping, Fig. 2(c). Two dimensional phase unwrapping is an ill-defined problem and many 2D unwrapping functions have been developed. While a basic Least Squares based algorithm  was adequate, we found MATLAB code available online  that implements a Network Flow based unwrapping algorithm  producing superior results.
This demonstration strongly points to usefulness of IFSAL. While the profile of Lincoln is unobservable in the magnitude SAL image of Fig. 2(a), the phase information yields the topographical relief revealing Lincoln’s profile. Additionally, the “flat earth” fringes can be used to indicate the overall orientation of the target\scene, which can help in determining 3D surface normals and interpreting layover effects of 3D targets.
3. Comparison of stripmap, spotlight, and bistatic
Spotlight mode SA collection is a popular technique in the SAR community . In this mode the transmitted beam is steered to illuminate the same patch of the scene during SA collection. This compares to stripmap mode where the beam is scanned across the scene during SA collection. The benefits of spotlight mode over the stripmap mode accrue from the decoupling of the maximum SA size (limited by the beam size on target in stripmap mode), which sets the cross-range resolution, from the size of the real aperture, which determines the light collection efficiency. As a consequence, one can choose to improve the cross-range resolution for a given real aperture size, or one can use a larger real aperture at a fixed resolution cross-range resolution to greatly improve the photon budget, albeit at the expense of the field-of-view in the image. Figures 3(a) and 3(c) illustrates the signal improvement possible with spotlight mode collection. The stripmap image (a) used the 10.5 μm end face of a single mode fiber as an aperture to image the dragonfly specimen target at 2 m range, for a 40 cm spot on target. The spotlight mode image Fig. 3(c) used an increased aperture size of 50 μm generating an ~8 cm spot size on target, increasing the transmitted irradiance on the target by a factor of 14 dB. Additionally the area of the receiver is increased by another factor of 14 dB, for a 28 dB total increase in the received signal power from the target (assuming the target is small compared to either beam size). This increase in received power aids in the effectiveness of post-processing algorithms such as PGA, and also improves the image contrast. Each synthetic aperture collection consisted of 2048 steps (10 μm steps for 2.048 cm SA length in Fig. 3(a), 50 μm steps for 10.24 cm SA length in Fig. 3(c)). While the larger SA in the spotlight image leads to modestly better resolution (although in this case the 40 cm beam on target for the stripmap image did not limit the size of the SA), the improved contrast due to the increased return power greatly improves the image quality.
For the spotlight mode collection additional mechanics were implemented to keep the beam directed onto the target during SAL data collection. The required kinetic motion was accomplished by placing the transmit\receive optics on top of a constraint arm that was allowed to rotate about a pivot centered at the target using a rotation stage as a bearing. The other end of the constraint arm was actuated by the precision linear ball screw stage using a second orthogonal spring loaded linear stage and a rotation stage to allow the constraint beam stretch to help convert the linear motion to a rotational motion, see Fig. 3(d). This set of mechanics causes the aperture to sweep out an arcing motion making it very similar to inverse SAR\SAL imaging of a rotating target. After performing the demonstrations presented here, we realized that moving the spring loaded stage to the pivot point near the target would make the aperture motion similar to modern spotlight mode SAR geometries where the aperture track remains straight and only the beam is steered. However, except for differences in the polar formatting post processing (see below) the imaging results should be largely the same.
The spotlight geometry described in Fig. 3(d), where the synthetic aperture forms an arc, required further formatting before application of PGA for phase correction. Unlike in the stripmap mode or standard spotlight, here the center of the scene does not change in range while away from the scene center individual scattering points begin to migrate between resolution cells in a circular fashion. A rough estimate of the requirement for polar formatting is that a point in the scene migrates more than a pixel during the SA formation. In this spotlight case, the 0.0314 radian half angle subtended by the SA causes a point 1 cm in cross-range from the center to migrate ~500 μm in range, which is much greater than the resolution of a pixel. A full polar formatting based on the collection geometry should be best [6,7,9], however we found that compensation in the range dimension using Chirp-Z Transform Polar Format (CZT-PF) techniques outlined by  followed by PGA in cross-range to correct phase errors was sufficient to improve image formation at least in the central portion of the scene. In many cases we also applied PGA to the range dimension to correct possible range ghosting (see below). For the image in Fig. 3(c) this extra step resulted in little image improvement. Further polar formatting in both the cross-range and range dimensions was attempted, however we were unable provide improvement due our poor estimation of geometric parameters such as the exact center of rotation in the scene. We are working to improve implementation of the polar formatting.
As an additional demonstration of spotlight SAL, and one that more clearly shows the improvement with the range only CZT-PF and dual PGA processing, we imaged an USAF 1951 resolution target (see Fig. 4) consisting of a lithographic negative pattern of chrome on a 1/8 inch glass substrate (i.e. the lines were clear glass and the spacing was chrome). The target was placed at a 45 degrees slant angle away from transmit\receiver. This target provided very little return signal by itself due to the specular reflection off the chrome being oriented away from the receiver, so a bright white piece of paper was placed behind the pattern on the backside of the glass to provide a stronger diffuse reflection. In the initial image (Fig. 4(a)), there appears to be a ghost image in the range dimension. This was caused by multiple reflections off of the backside of glass substrate and the backside of the chrome pattern creating a weak layover image that is shifted to longer range. It was this issue that led us to first apply the PGA in the range dimension in addition to the CZT-PF and PGA in cross-range. As one can see in Fig. 4(b), the processing greatly improves the range and cross-range resolution, showing a cross-range resolution of 8 lines per millimeter or 125 micron resolution (Group 3 No. 2 is resolved in cross-range). This is about 6 times worse than the resolution set by the 50 micron x 1048 steps = 5.12 cm SA length at 1.5 m, but is a factor of 300 improvement over the diffraction limit of the ~50 μm real aperture. Due to the ghosting issue we were not able to clearly demonstrate the full range resolution of the FMCW system with this USAF resolution test target. Furthermore, while we observed improvement in the range resolution in Fig. 4(b) (1 lines/mm to 2.24 lines/mm) by applying the PGA in the range dimension, it is hard to quantify how much was due to correcting the ghosting and how much was due to the polar formatting.
Finally, we present a simple demonstration of bistatic SAL configuration utilizing a moving spotlight mode transmit aperture and a stationary off axis receive aperture. The stationary receiver was located at similar range to the transmit location but offset in cross-range by about 15 cm (see Fig. 3(d)). The receiver used a pair of imaging lenses to provide a real aperture of about 50 μm and collected the scattered light into a single mode fiber that was then mixed with an LO copy of the chirp using a 2x2 PM fiber splitter\coupler. The transmitter moved 2048 steps for a total SA of 10.24 cm. The target was a PC memory module oriented with its roughly planar face at a slant angle of 45 degrees. Post processing closely matched that of the monostatic spotlight case (i.e. CZT-PF in range and dual PGA) resulting in the image shown in Fig. 5. With a stationary receiver the angle subtended by the effective SA is roughly halved as the effective SA is the vector sum of the transmitter and receiver positions. The central IC chip, which is physically 13 mm wide, is 630 pixels wide in the image giving a cross-range pixel size of approximately 20 μm. This agrees with twice the monostatic resolution δCR = 1.55 μm/(2 x 10.24 cm/1.4 m) = 10 μm.
Another interesting thing to observe in this image is that while the receiver was oriented to the right of the transmitter, the horizontal edges of the chip are slanted up and to the left, which is the exact opposite to what one would observe in normal 2D transverse imaging. This effect is due to the range imaging as the light scattering from the upper left corner of the chip travels a longer distance than the light scattered from the upper right. The perspective of SAL images look as if they are taken from a perspective perpendicular to the beam and SA directions (e.g. if the beam is transmitted along the table surface to the target the SAL image looks like it was taken from above the target and not from the transmit\receiver position). The significant blurring away from the center of the scene indicates that the polar formatting could be greatly improved. The ability to use bistatic, including multistatic, geometries increases the design space for SA imaging systems to achieve strategic and application specific objectives [15,16]. For example, multiple bistatic perspectives could support tomographic processing or simultaneous collection to improve image resolution or imaging rates.
In this work, we have demonstrated table-top SAL imaging in a number of configurations. A comparison of stripmap, spotlight, and bistatic SAL image collection geometries was shown. By allowing flexibility in the size of the real aperture for a given desired spatial resolution, the spotlight mode geometry was shown to exhibit better light collection at the expense of the field of view of the scene. Through the use of optical fibers for flexible delivery of the optical chirp for transmission, collection of the scattered light, and combining it with the LO, we have demonstrated a spotlight mode bistatic geometry and highlighted the change in cross-range resolution and other aspects of SAL imaging. The phase gradient autofocus, with limited polar formatting in the case of spotlight mode imaging, was sufficient to form high quality SAL images even in the presence of low signal levels and relatively large phase errors due to motion and drifts of the optical fibers used in the setup. We also demonstrated the coherent aspect of SAL imaging by using interferometric SAL imaging to perform 3D surface relief imaging of the Lincoln side of a U.S penny.
These SAL demonstrations were enabled by use of an actively linearized THz chirped source, which provides the necessary range resolution for table-top sized SAL demonstrations and greatly reduces the post-processing requirements to make high quality SAL images. The SAL images shown here exceed the diffraction limit of the real apertures used for light collection by a factor of 100 to 1000, however improved application of polar formatting could increase this factor. In all, we have demonstrated that the full range of imaging techniques and algorithms used in SAR can be almost directly translated to SAL. With faster chirped sources and motion control to decrease the long data collection times in the present setup, SAL has the potential for use in a variety of precision imaging applications.
The above demonstrations owe much to the foundation set by working with our collaborator Bridger Photonics Inc. in the area of ultra-high resolution FMCW ladar and synthetic aperture ladar imaging. Portions of this collaborative work, including more high quality short range stripmap mode imaging and long range SAL imaging demonstrations, will be published jointly with Randy R. Reibel, Peter A. Roos, B. Kaylor, T. Berg, C. Keith, and N. Greenfield of Bridger Photonics Inc. We would like thank them for these collaborations and for providing us with assistance on the hardware and software used to collect the SAL data presented here. This material is based upon work supported by the National Science Foundation under Grant No. CMMI-1031211.
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