Patent Publication Number: US-2022229288-A1

Title: Ladar Transmitter with Ellipsoidal Reimager

Description:
CROSS-REFERENCE AND PRIORITY CLAIM TO RELATED PATENT APPLICATIONS 
     This patent application is a continuation of U.S. patent application Ser. No. 16/865,687, filed May 4, 2020, and entitled “Ladar Transmitter with Ellipsoidal Reimager”, now U.S. Pat. No. ______, which is a continuation of U.S. patent application Ser. No. 16/051,707, filed Aug. 1, 2018 and entitled “Ladar Transmitter with Ellipsoidal Reimager”, now U.S. Pat. No. 10,642,029, which is a continuation of U.S. patent application Ser. No. 15/644,242, filed Jul. 7, 2017 and entitled “Ladar Transmitter with Optical Field Splitter/Inverter”, now U.S. Pat. No. 10,042,159, which is (1) a continuation-in-part of U.S. patent application Ser. No. 15/431,065, filed Feb. 13, 2017 and entitled “Ladar Transmitter with Optical Field Splitter/Inverter for Improved Gaze on Scan Area Portions”, which claims priority to (i) U.S. provisional patent application 62/297,126, filed Feb. 18, 2016 and entitled “Ladar Transmitter with Resonant Scan Optical Field Splitter/Inverter” and (ii) U.S. provisional patent application 62/439,378, filed Dec. 27, 2016 and entitled “Ladar Transmitter with Improved Gaze on Scan Area Portions”, and (2) a continuation-in-part of U.S. patent application Ser. No. 15/431,096, filed Feb. 13, 2017 and entitled “Ladar Transmitter with Induced Phase Drift for Improved Gaze on Scan Area Portions”, which claims priority to (i) U.S. provisional patent application 62/297,126, filed Feb. 18, 2016 and entitled “Ladar Transmitter with Resonant Scan Optical Field Splitter/Inverter” and (ii) U.S. provisional patent application 62/439,378, filed Dec. 27, 2016 and entitled “Ladar Transmitter with Improved Gaze on Scan Area Portions”, the entire disclosures of each of which are incorporated herein by reference. 
    
    
     INTRODUCTION 
     It is believed that there are great needs in the art for improved computer vision technology, particularly in an area such as automobile computer vision. However, these needs are not limited to the automobile computer vision market as the desire for improved computer vision technology is ubiquitous across a wide variety of fields, including but not limited to autonomous platform vision (e.g., autonomous vehicles for air, land (including underground), water (including underwater), and space, such as autonomous land-based vehicles, autonomous aerial vehicles, etc.), surveillance (e.g., border security, aerial drone monitoring, etc.), mapping (e.g., mapping of sub-surface tunnels, mapping via aerial drones, etc.), target recognition applications, remote sensing, safety alerting (e.g., for drivers), and the like). 
     As used herein, the term “ladar” refers to and encompasses any of laser radar, laser detection and ranging, and light detection and ranging (“lidar”). Ladar is a technology widely used in connection with computer vision. Ladar systems share the high resolution and intuitive feel of passive optic sensors with the depth information (ranging) of a radar system. In an exemplary ladar system, a transmitter that includes a laser source transmits a laser output such as a ladar pulse into a nearby environment. Then, a ladar receiver will receive a reflection of this laser output from an object in the nearby environment, and the ladar receiver will process the received reflection to determine a distance to such an object (range information). Based on this range information, a clearer understanding of the environment&#39;s geometry can be obtained by a host processor wishing to compute things such as path planning in obstacle avoidance scenarios, way point determination, etc. However, conventional ladar solutions for computer vision problems suffer from high cost, large size, large weight, and large power requirements as well as large data bandwidth use. The best example of this being vehicle autonomy. These complicating factors have largely limited their effective use to costly applications that require only short ranges of vision, narrow fields-of-view and/or slow revisit rates. 
     For example, ladar systems are known in the art where a ladar transmitter illuminates a large number of range points simultaneously. Flash ladar is an example of such a system. However, these conventional systems are believed to suffer from a number of shortcomings. For example, flash ladar systems require a very high energy per pulse laser, which is not only costly but can also be an eye hazard. Furthermore, the read-out integrated circuits for flash ladar systems are typically quite noisy. Also, the wide field-of-view signal-to-noise ratio (SNR) for flash ladar systems is typically very low, which results in short ranges, thereby detracting from their usefulness. 
     In an effort to satisfy the needs in the art for improved ladar-based computer vision technology, the inventor has disclosed a number of embodiments for methods and systems that apply scanning ladar transmission concepts in new and innovative ways, as described in U.S. patent application Ser. No. 62/038,065, filed Aug. 15, 2014 and U.S. Pat. App. Pubs. 2016/0047895, 2016/0047896, 2016/0047897, 2016/0047898, 2016/0047899, 2016/0047903, and 2016/0047900, the entire disclosures of each of which are incorporated herein by reference. 
     The inventor believes that there are needs in the art for further improvements on how scanning ladar transmitters can be designed to optimize their gaze on regions of interest in the environment. While radars have been highly optimized with scheduling methods to dwell (gaze) where gaze is needed when gaze is needed, conventional ladar systems today do not share this dwell optimality. This is because ladar suffer from the very thing that makes them attractive: their resolution. 
     This is because, while even the world&#39;s largest radars have thousands of beam choices upon which to dwell, even a small automotive ladar system fitting in the palm of the hand routinely has 100,000+ or even millions of choices for dwell. This leads to two general design choices for ladar engineers: (i) mechanically step from dwell to dwell, or (ii) use resonant mirrors that rapidly scan through the scene. Design approach (i) is precise and adaptable but is extremely slow in environments where there are large numbers of interrogation cells present. Design approach (ii) has historically been non-adaptable. To improve upon these conventional design approaches, the inventors disclose techniques by which one can reduce the disadvantages of resonant mirror-based ladar while achieving much of the acuity and specificity that historically has only been available to mechanical stepping techniques and without losing the speed advantages of resonant scanning mirrors. 
     In example embodiments, the inventors disclose designs for a field splitter/inverter that can be used in combination with high speed scans such as Lissajous scans to improve the gaze of a ladar transmitter on desired scan areas (such as a center region of a scan area). The inventors further disclose how an induced periodic phase drift can be incorporated into a Lissajous scan to further improve gaze by reducing scan gaps in desired regions, while doing so in a manner that can ensure low amplitude and zero impact on periodicity, thereby ensuring beam quality and preserving mensuration. 
     In example embodiments, the inventors also disclose a compact beam scanner assembly that includes an ellipsoidal conjugate reflector reimaging mirror. The ellipsoidal mirror can be positioned optically between first and second scannable mirrors. A lens can be positioned optically upstream from the first scannable mirror. Such an arrangement can provide (among other benefits) a compact beam scanner design where the two scannable mirrors are equally sized and placed closely together within the assembly. Moreover, reimaging can be especially useful when used in combination with field inversion, since it limits the additional upscope headroom needed for an inverter lens. 
     These and other features and advantages of the present invention will be described hereinafter to those having ordinary skill in the art. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIGS. 1A  and B illustrate example embodiments of ladar transmitter/receiver systems. 
         FIG. 2A  depicts an example embodiment of a ladar transmitter. 
         FIG. 2B  shows a scan area defined by two scannable mirrors for an example scanning ladar transmitter. 
         FIG. 2C  shows a top view of the scan arrangement defined by  FIG. 2B . 
         FIG. 3  shows a beam scanner that includes a field splitter/inverter in accordance with an example embodiment. 
         FIG. 4  shows a top view of the beam scanner arrangement defined by  FIG. 3 . 
         FIG. 5  shows how the field splitter/inverter operates to split and invert the scan pattern of the mirrors across the scan area. 
         FIG. 6A  shows a perspective view of a field splitter/inverter in accordance with another example embodiment. 
         FIGS. 6B and 6C  show additional views of the field splitter/inverter of  FIG. 6A . 
         FIG. 6D  shows an example embodiment of a hinged field splitter/inverter that exhibits an adjustable mirror pitch angle. 
         FIG. 7  shows an example ray tracing with respect to beam paths in the field splitter/inverter of  FIGS. 6A-C . 
         FIG. 8  shows an example of how a split/inverted scan area can be overlapped. 
         FIG. 9  is a table that exhibits performance results for different use cases of a ladar transmitter. 
         FIG. 10A  depicts an example of a standard Lissajous scan pattern. 
         FIG. 10B  depicts an example of a split/inverted Lissajous scan pattern. 
         FIG. 10C  depicts the revisit performance of the Lissajous scan pattern of  FIG. 10A . 
         FIG. 10D  depicts a plot of revisit performance for an example of a split/inverted Lissajous scan revisit when a field inverter is employed to provide inversion along elevation. 
         FIGS. 11A  and B depict where scan gaps exist in connection with the scan patterns of  FIGS. 10A and 10B  respectively. 
         FIG. 12  depicts an example process flow for inducing a periodic phase drift into a Lissajous scan pattern. 
         FIGS. 13A  and B depict scan gaps projected onto a ground plane for example use cases of a non-inverted Lissajous scan and an inverted Lissajous scan respectively. 
         FIGS. 14A and 14B  show example inverted Lissajous scan patterns with induced phase drift and no induced phase drift respectively. 
         FIG. 15  shows an example embodiment of an ellipsoidal conjugate reflector (ECR) reimaging system. 
         FIG. 16  displays the scanned field for a conventional scanner as configured without ray fan aligned reimaging optics. 
         FIG. 17  displays an embodiment of a reimager with geometry chosen per our disclosed design formula. Note that the ray fan directed downward from the reimager lies in a plane containing the center of the second scanning mirror. 
         FIG. 18  displays the elimination of distortions in the scanned output field made possible by using an ellipsoidal reflector in a manner consistent with an example embodiment. 
         FIG. 19  is a side-looking schematic of an example embodiment of an ECR 2D scanner. 
         FIG. 20  shows the use of paired “kissing” mirrors as a virtual, low cost, source of field inversion. 
     
    
    
     DETAILED DESCRIPTION OF EXAMPLE EMBODIMENTS 
       FIG. 1A  illustrates an example embodiment of a ladar transmitter/receiver system  100 . The system  100  includes a ladar transmitter  102  and a ladar receiver  104 , each in communication with system interface and control  106 . The ladar transmitter  102  is configured to transmit a plurality of ladar pulses  108  toward a plurality of range points  110  (for ease of illustration, a single such range point  108  is shown in  FIG. 1A ). Ladar receiver  104  receives a reflection  112  of this ladar pulse from the range point  110 . Ladar receiver  104  is configured to receive and process the reflected ladar pulse  112  to support a determination of range point distance [depth] and intensity information. In addition the receiver  104  determines spatial position information [in horizontal and vertical orientation relative to the transmission plane] by any combination of (i) prior knowledge of transmit pulse timing, and (ii) multiple detectors to determine arrival angles. 
     In example embodiments, the ladar transmitter  102  can take the form of a ladar transmitter that includes scanning mirrors. Furthermore, in an example embodiment, the ladar transmitter  102  uses a range point down selection algorithm to support pre-scan compression (which can be referred herein to as “compressive sensing”), as shown by  FIG. 1B . Such an embodiment may also include an environmental sensing system  120  that provides environmental scene data to the ladar transmitter  102  to support the range point down selection. Specifically, the control instructions will instruct a laser source when to fire, and will instruct the transmitter mirrors where to point. Example embodiments of such ladar transmitter designs can be found in U.S. patent application Ser. No. 62/038,065, filed Aug. 15, 2014 and U.S. Pat. App. Pubs. 2016/0047895, 2016/0047896, 2016/0047897, 2016/0047898, 2016/0047899, 2016/0047903, and 2016/0047900, the entire disclosures of each of which are incorporated herein by reference. Through the use of pre-scan compression, such a ladar transmitter can better manage bandwidth through intelligent range point target selection. An example embodiment of ladar receiver  104  can be found in U.S. patent application Ser. No. 62/297,126, filed Feb. 18, 2016. While these referenced and incorporated patent applications describe example embodiments for the ladar transmitter  102  and ladar receiver  104 , it should be understood that practitioners may choose to implement the ladar transmitter  102  and/or ladar receiver  104  differently than as disclosed in these referenced and incorporated patent applications. 
       FIG. 2A  depicts an example embodiment for a ladar transmitter  104  as disclosed by the above-referenced and incorporated patent applications. The ladar transmitter  104  can include a laser source  200  in optical alignment with laser optics  202 , a beam scanner  204 , and transmission optics  206 . These components can be housed in a packaging that provides a suitable shape footprint for use in a desired application. For example, for embodiments where the laser source  200  is a fiber laser or fiber-coupled laser, the laser optics  202 , the beam scanner  204 , and any receiver components can be housed together in a first packaging that does not include the laser source  200 . The laser source  200  can be housed in a second packaging, and a fiber can be used to connect the first packaging with the second packaging. Such an arrangement permits the first packaging to be smaller and more compact due to the absence of the laser source  200 . Moreover, because the laser source  200  can be positioned remotely from the first packaging via the fiber connection, such an arrangement provides a practitioner with greater flexibility regarding the footprint of the system. 
     Based on the control information transmitter control instructions, such as a shot list  212  generated by system control  106 , a beam scanner controller  208  can be configured to control the nature of scanning performed by the beam scanner  204  as well as control the firing of the laser source  200 . A closed loop feedback system  210  can be employed with respect to the beam scanner  204  and the beam scanner controller  208  so that the scan position of the beam scanner  204  can be finely controlled, as explained in the above-referenced and incorporated patent applications. 
     The laser source  200  can be any of a number of laser types suitable for ladar pulse transmissions as described herein. 
     For example, the laser source  200  can be a pulsed fiber laser. The pulsed fiber laser can employ pulse durations of around 1-4 ns, and energy content of around 0.1-100 μJ/pulse. The repetition rate for the pulsed laser fiber can be in the kHz range (e.g., around 1-500 kHz). Furthermore, the pulsed fiber laser can employ single pulse schemes and/or multi-pulse schemes as described in the above-referenced and incorporated patent applications. However, it should be understood that other values for these laser characteristics could be used. For example, lower or higher energy pulses might be employed. As another example, the repetition rate could be higher, such as in the 10&#39;s of MHz range (although it is expected that such a high repetition rate would require the use of a relatively expensive laser source under current market pricing). 
     As another example, the laser source  200  can be a pulsed IR diode laser (with or without fiber coupling). The pulsed IR diode laser can employ pulse durations of around 1-4 ns, and energy content of around 0.01-10 μJ/pulse. The repetition rate for the pulsed IR diode fiber can be in the kHz or MHz range (e.g., around 1 kHz-5 MHz). Furthermore, the pulsed IR diode laser can employ single pulse schemes and/or multi-pulse schemes as described in the above-referenced and incorporated patent applications. 
     The laser optics  202  can include a telescope that functions to collimate the laser beam produced by the laser source  200 . Laser optics can be configured to provide a desired beam divergence and beam quality. As example, diode to mirror coupling optics, diode to fiber coupling optics, and fiber to mirror coupling optics can be employed depending upon the desires of a practitioner. 
     The beam scanner  204  is the component that provides the ladar transmitter  104  with scanning capabilities such that desired range points can be targeted with ladar pulses. The beam scanner receives an incoming ladar pulse from the laser source  200  (by way of laser optics  202 ) and directs this ladar pulse to a desired downrange location (such as a range point on the shot list) via reflections from movable mirrors. Mirror movement can be controlled by one or more driving voltage waveforms  216  received from the beam scanner controller  208 . Any of a number of configurations can be employed by the beam scanner  204 . For example, the beam scanner can include dual microelectromechanical systems (MEMS) mirrors, a MEMS mirror in combination with a spinning polygon mirror, or other arrangements. An example of suitable MEMS mirrors are single surface tip/tilt/piston MEMS mirrors. By way of further example, in an example dual MEMS mirror embodiment, a single surface tip MEMS mirror and a single surface tilt MEMS mirror can be used. However, it should be understood that arrays of these MEMS mirrors could also be employed. Also, the dual MEMS mirrors can be operated at any of a number of frequencies, examples of which are described in the above-referenced and incorporated patent applications, with additional examples being discussed below. As another example of other arrangements, a miniature galvanometer mirror can be used as a fast-axis scanning mirror. As another example, an acousto-optic deflector mirror can be used as a slow-axis scanning mirror. Furthermore, for an example embodiment that employs the spiral dynamic scan pattern discussed below, the mirrors can be resonating galvanometer mirrors. Such alternative mirrors can be obtained from any of a number of sources such as Electro-Optical Products Corporation of New York. As another example, a photonic beam steering device such as one available from Vescent Photonics of Colorado can be used as a slow-axis scanning mirror. As still another example, a phased array device such as the one being developed by the DARPA SWEEPER program could be used in place of the fast axis and/or slow axis mirrors. More recently, liquid crystal spatial light modulators, such as those offered by Boulder Nonlinear Systems and Beamco, can be considered for use. 
     Also, in an example embodiment where the beam scanner  204  includes dual mirrors, the beam scanner  204  may include relay imaging optics between the first and second mirrors, which would permit that two small fast axis mirrors be used (e.g., two small fast mirrors as opposed to one small fast mirror and one long slower mirror). 
     The transmission optics  206  are configured to transmit the ladar pulse as targeted by the beam scanner  204  to a desired location through an aperture. The transmission optics can have any of a number of configurations depending upon the desires of a practitioner. For example, the environmental sensing system  106  and the transmitter  104  can be combined optically into one path using a dichroic beam splitter as part of the transmission optics  206 . As another example, the transmission optics can include magnification optics as described in the above-referenced and incorporated patent applications or descoping [e.g., wide angle] optics. Further still, an alignment pickoff beam splitter can be included as part of the transmission optics  206 . 
     Field Splitting and Inversion to Optimize the Ladar Transmitter&#39;s Gaze on Desirable Regions within the Scan Area: 
     The beam scanner controller  208  can provide voltage waveforms  214  to the beam scanner  204  that will drive the mirrors of the beam scanner to a desired scan position pairing (e.g., scan angles). The voltage waveforms  214  will define a scan pattern for the targeting of the ladar transmitter  102  within a scan area. The firing commands  214  generated by the beam scanner controller  208  can be coordinated with the scan pattern so that the ladar transmitter  102  fires ladar pulses toward desired range points within the scan area. Example embodiments for the beam scanner controller  208  are described in the above-referenced and incorporated patent applications. 
       FIG. 2B  shows an example beam scanner arrangement where the positioning of mirrors  250  and  252  about rotational axes  258  and  260  respectively defines the targeting for the beam scanner within a scan area  254 . An incoming incident ladar pulse launched from laser source  200 , coming from direction  256 , will impact scanning mirror  250 , whereupon it is reflected toward scanning mirror  252 , whereupon it is reflected toward a range point within the scan area  254 . The positioning of the scanning mirrors  250  and  252  will control which horizontal and vertical positions within the scan area  254  are targeted, and the range [depth] for these positions will then be extracted from pulse compression processing within the receiver. 
     In an example embodiment, mirror  250  can control where the beam scanner is targeted along a first-axis  262  of the scan area  254 , and mirror  252  can control where the beam scanner is targeted along a second axis  264  of the scan area  104 . The first and second axes may be orthogonal to each other (e.g., a horizontal X-axis and a vertical Y-axis). Note that the second mirror,  252 , is shown in this embodiment to be larger than the first mirror  250 . This is because of the sweep in angles arising as the first mirror scans, three such positions are shown in the figure. It is desirable for compaction to introduce relay imaging optics between  250  and  252  which reduces the size of the second mirror. When one or both mirrors are scanning at a resonant frequency, the speed of the scan will be fast in the middle of the scan area and slower at the edges of the scan area. This characteristic is shown by  FIG. 2C , which provides a top view of the beam scanner arrangement. In the view of  FIG. 2C , as mirror  250  scans at resonance, the targeting of the beam scanner along the X-axis  262  of the scan area  254  will be faster in the middle of the scan area than it is on the edges. The same would hold true with respect to the Y-axis  264  of the scan area  254  (where the targeting will move faster in the middle of the scan area that is does on the edges). This high rate of speed in the middle of the scan area may pose efficiency problems when a practitioner wants to implement a scan pattern that includes a large number of range points within the middle of the scan area and/or includes interline skips/detours in the middle of the scan area. A description of interline skips and detours in connection with ladar transmitters can be found in the above-referenced and incorporated patent applications. This “fast in the middle” scan characteristic be particularly acute when both mirrors are scanned at resonance so as to achieve a Lissajous scan pattern (example embodiments of which are discussed below). Lissajous scan patterns allow for the mirrors to be scanned at high rates, and thus provides a fast moving scan pattern. However, with a “fast in the middle” scan characteristic, of which a Lissajous scan pattern is a prime example, the ladar transmitter may not have enough time to fire ladar pulses at all of the range points within the middle of the scan area without at least increasing the laser firing rate (which may decrease pulse energy) or including additional line repeats in the scan pattern. 
     In an effort to solve this problem, the inventor discloses the use of an optical field splitter/inverter (hereinafter “field inverter”) that is positioned optically downstream from the mirrors  250  and  252  and positioned to receive and re-direct ladar pulses that are outgoing from the mirrors so that the scan area is split and inverted in a manner that causes the fast portion of the scan to reside at the edges of the scan area and the slower portion of the scan to reside in the middle of the scan area. 
       FIG. 3  shows an example embodiment where the beam scanner for a scanning ladar transmitter includes a field splitter/inverter  300  that is positioned optically downstream from mirrors  250  and  252  to split and invert the scan area relative to the scan area  254  of  FIG. 2B . 
     For the purpose of brief explanation, in this example, the beam scanner includes mirrors  250  and  252  that may take the form of dual MEMS mirrors. However, it should be understood that other mirrors could be used for the first and/or second mirrors (e.g., galvo-meter mirrors). First mirror  250  is positioned to receive an incident ladar pulse. Mirror  250  will reflect this ladar pulse to the second scanning mirror  252 . It should be understood that this reflection can be a direct reflection or an indirect reflection whereby the beam reflected from mirror  250  passes through relay imaging optics such as a unity magnification telescope on its way to mirror  252 . Mirror  252  is positioned to receive the reflected laser pulse from mirror  250  and further reflect this laser pulse onto the field inverter mirror  300 . The reflection off this mirror can then passed through a telescope/descope to produce the outgoing laser pulse  108  which then travels to a designated horizontal/vertical location within a scan area  302  corresponding to the range point a shot list that is being targeted by the beam scanner. 
     The first and second mirrors  250  and  252  are controllably rotatable around their respective axes of rotation  258  and  260  as discussed above. Thus, mirror  250  will be rotatable to control the position of the ladar pulse within the scan area  302  along the scan area&#39;s X-axis, while mirror  252  will be rotatable to control the position of the ladar pulse within the scan area  302  along the scan area&#39;s Y-axis. Thus, the combined positioning of mirrors  250  and  252  along their respective axes of rotation when the ladar pulse strikes each mirror will be effective to direct the launched ladar pulse  108  to the desired location within the scan area  302 . In an example embodiment, the X-axis mirror  250  is scanned at resonance. However, it should be understood that the Y-axis mirror  252  could also be scanned at resonance. Further still, any combination of mirrors  250  and  252  could be scanned at resonance. 
     It should also be understood that one of the axes can serve as a “fast axis” and the other axis serving as a “slow axis” to reflect the relative scan rates of mirrors  250  and  252 . For example, the X-axis can serve as the fast axis, and the Y-axis could serve as the slow axis. It should be understood by a practitioner that the designation of the fast axis as the X-axis and the slow axis as the Y-axis is arbitrary as a 90 degree turn in position for the system would render the X-axis as the slow axis and the Y-axis as the fast axis. Furthermore, in an example embodiment, the fast axis mirror is smaller than the slow axis mirror in terms of mirror area and is also positioned upstream from the slow axis mirror (that is, the fast axis mirror receives the ladar pulse and reflects it to the slow axis mirror for transmission toward the targeted range point). However, this configuration could be changed for other embodiments. For example, while making the slow axis mirror larger than the fast axis mirror provides a benefit in terms of permitting a larger scan area, for embodiments where a decrease in the size of the scan area is permissible, the slow axis mirror could be the same size or even smaller than the fast axis mirror. As another example, if the fast axis mirror were downstream from the slow axis mirror, reimaging optics, such as relay imaging optics, could be used between the two mirrors to support such an arrangement. 
       FIG. 3  shows an embodiment where an optical field splitter/inverter  300  is positioned to receive and redirect the ladar pulse reflected by mirror  252 . Thus, the field splitter/inverter  300  is optically downstream from mirrors  250  and  252 . The field splitter/inverter  300  may comprise mirrors or lenses that are arranged to both split the field of view for the transmitter and invert the split field of view. Note the figure assumes a mirror/reflector is used so the imaged scene  302  is behind the MEMs assembly to the far right. 
     In an example embodiment, the field inverter  300  may take the form of reflectors arranged in a W-shape as shown by  FIG. 3  (see also  FIG. 4 , which is a top view of the  FIG. 3  arrangement). Here, the laser pulse launched from the second mirror,  107 , is inverted and passed along to the scene to be imaged,  302 , in speed inverted fashion. The launched pulse  108  will now be slowly scanning where  107  is fast (and vice versa). 
       FIG. 4  is a top view of the beam scanner arrangement of  FIG. 3 , and it shows the effect of the field inverter  300  on the resultant field of view/scan area. As explained above, unlike the  FIG. 1  embodiment (where the beam scanner scans through the scan area in a manner that is faster in the middle and slower at the edges while mirror  250  scans at resonance (as reflected in the parenthetical shown by  FIG. 4  above mirror  102 )), the beam scanner for the embodiments of  FIGS. 3 and 4  scans through the scan area in manner that is slower in the middle and faster at the edges due to the field inverter  300 . The beam ray path for the beam scanner of  FIGS. 3 and 4  is shown via the arrows in  FIG. 4 , and it can be seen that the beams which would otherwise have landed in the middle of the scan area are instead redirected to the edge of the scan area  400 , the top view of  302 , via the field inverter. 
       FIG. 5  illustrates the effect of the field inverter  300  on the beam field. The top frame of  FIG. 5  shows a plot of the scan speed across the field when mirror  250  is scanning at resonance. As can be seen, the scan speed is faster in the middle of the field than it is at the edges. The field inverter  300  provides a splitting effect and an inverting effect. The middle frame of  FIG. 5  shows the split effect at line  500 , where the field is split into two halves. The bottom frame of  FIG. 5  shows the inversion effect that operates on each of the split fields (where each half defined by line  500  is inverted). For the example embodiment of  FIGS. 3 and 4 , the split line  500  corresponds to the central peak of the W-shape of the field inverter. The field inverter  300  can be optically positioned so that this central peak corresponding to split line  500  falls within the middle of the field of view. Due to the field inverter  300 , the beam scan path within each split field is inverted. This yields the plot of scan speed across the field as shown by the bottom frame of  FIG. 5 . As can be seen, the scan speed now becomes faster at the edges than it is in the middle of the field. This scan characteristic provides the ladar transmitter with more time to accommodate groupings of range points in the middle of a frame. Furthermore, when paired with a ladar transmitter that employs a shot list and dynamic scan pattern that includes interline skipping and/or detouring, this scan characteristic also provides the ladar transmitter with more time to accommodate the line skips and/or line detours that may occur in the middle of the frame. 
     It should be understood that a shot list employed by the ladar transmitter would employ a re-mapping of range points on the shot list to accommodate the split and inverted field of view. 
     In another example embodiment, the field inverter  300  may take the form a triangular prism  600  as shown by  FIG. 6A . The triangular prism  600  can define an inverted V-shape for reflection of light pulses as discussed below. The example triangular prism of  FIG. 6  is shown in perspective view and exhibits a shape that includes first triangular face  602  and second triangular face  604  at opposite ends of the prism  600 . First, second, and third sides  606 ,  608 , and  610  serve as the underside, right side, and left side respectively of the triangular prism  600  shown by  FIG. 6 , and these sides connect the triangular faces  602  and  604 . In the example of  FIG. 6A , the triangular faces  602  and  604  are oriented perpendicularly with respect to sides  606 ,  608 , and  610 , although this need not be the case. Triangular prism  600  can be formed from any material transparent to the incident laser wavelength with the active regions where bounce paths may occur treated with reflective material. Since empty space (air or vacuum) is transparent to ladar laser light, the prism can be hollow or dense whichever any desired manufacturing dictates. 
     The view of  FIG. 6A  is from a perspective above a laser feed  107 , with partially transparent mirrors to permit visualization of the prism interior.  FIG. 6B  provides a vertical cut view into the triangular prism  600  of  FIG. 6A , and  FIG. 6C  provides a depth cut view into the triangular prism  600  of  FIG. 6A . In this example, we can take the horizontal direction (azimuth) to be a fast axis scan which we seek to invert, and elevation can be step-scanned or resonant-scanned. The architecture of prism  600  shown by  FIG. 6  is invariant to this choice. 
     With reference to  FIG. 6A , a laser pulse  107  enters the prism  600  through injection region  620 . The precise site of entry  622  will vary as a result of the scanning by mirrors to define a scan pattern over a scan area. Accordingly, injection region  620  is shown in  FIG. 6A  as a box area to indicate such range of potential entry points. After entry into the prism  600 , the pulse  107  will travel linearly until it strikes a reflector mirror  624  on an interior portion of underside  606 , where reflector mirror  624  is offset in depth within prism  600 . The bullet points in  FIG. 6A  show sites of reflection (or “bounce”) within the prism  600  where the pulse transiting through  600  is reflected in a new direction. Each bounce site is labeled as I, II, or III based on an order of bounces for the pulse  107 . Thus, after striking reflector mirror  624 , the pulse strikes a reflector mirror on an interior portion of right side  608  (bounce site II), whereupon the pulse  107  is re-directed to strike a reflector mirror on an interior portion of left side  610  (bounce site III). 
     The laser injection angle at injection site  622  can be obliquely selected part way from vertical to horizontal. This allows the pulse  107  to bounce against both right mirror  608  (bounce site II) and left mirror  610  (bounce site III) and finally to exit the prism  600  through exit cut  626  and on to an exterior lens  670  (see  FIG. 6C ; e.g., an up/down scope) without encountering occlusions. We denote the side pitch angle for mirror  608  as z+. With reference to  FIG. 6B , the azimuth scan of the scanning mirrors  250  and  252  is mapped through bounce sites I-III to the angle f,  654 , formed between the two dotted lines at the right hand side of the figure. The far field angle S,  655 , is formed from the angles between the vertical dotted line at the right and the “flight trajectory” of the pulse  108  exiting the field inversion assembly. Behold from congruence arguments that S=f. As in the W embodiment of  FIGS. 3 and 4 , the pulse  108  is fast when  107  is slow and vice versa since the angles are reciprocally related. Our goal, if we wish to avoid a gap,  500 , is to set S so that the leftmost angle we can scan to is vertical i.e. S(min)=0. This ensures that the scan angle starts vertical and moves right; if it starts anywhere else the slit  500  will be large, otherwise it is negligibly small, and limited to creeping waves. Algebra reveals that 
     
       
         
           
             f 
             = 
             
               S 
               ≈ 
               
                 
                   4 
                   ⁢ 
                   
                     z 
                     + 
                   
                 
                 - 
                 
                   π 
                   4 
                 
                 - 
                 
                   ϕ 
                   . 
                 
               
             
           
         
       
     
     (see  654  text). When the scanning mirror azimuth scan is set to ℠=0 [where the trajectory of  107  is fastest], we see that the launched pulse  108  is sent to 
     
       
         
           
             
               4 
               ⁢ 
               
                 z 
                 + 
               
             
             - 
             
               
                 π 
                 4 
               
               . 
             
           
         
       
     
     Likewise when the scan is at its angle we obtain 
     
       
         
           
             S 
             ≈ 
             
               
                 4 
                 ⁢ 
                 
                   z 
                   + 
                 
               
               - 
               
                 π 
                 4 
               
               - 
               
                 
                   ϕ 
                   
                     { 
                     
                       ma 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       x 
                     
                     } 
                   
                 
                 . 
               
             
           
         
       
     
     Thus we obtain 
     
       
         
           
             
               ϕ 
               
                 { 
                 
                   ma 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   x 
                 
                 } 
               
             
             , 
             
               = 
               
                 
                   4 
                   ⁢ 
                   
                     z 
                     + 
                   
                 
                 - 
                 
                   π 
                   4 
                 
               
             
           
         
       
     
     as the desired mirror pitch angle. This completes the field inversion prism shape&#39;s exterior  602 - 610  as it relates to pre-inversion scan angle and scene azimuth positioning. It remains to discuss the prism internal structure and elevation scan. 
     Referring to  FIG. 6C , it can be shown by algebra that the choice of injection angle c [the vertical angle for the scanning mirror output pulse  107 ],  675 , will produce a desired far field elevation scan angle of this exact same value, that is c,  675 , equals the depression angle of the ray  670 . As the injection elevation angle c varies from c′ to c, it sweeps through angle e,  673 , the elevation swath of the scanning mirror. The bounce point I in  FIG. 6C  will then move along the prism base  674 . It should be clear that the length of this sweep determines the extent to which the mirror interior must be treated with a reflective surface. The entire prism apparatus can be scaled by an overall scale factor to account for the projective geometry. Specifically the pulse launch from the prism will travel along the vector t sin(c)[sin(4z−ϕ), cos(4z−ϕ), cot(c)]. We can see that we can treat the x,h analysis as a two dimensional problem and afterwards correct by a scale factor of sin(c). This observation is in fact what enabled us to decompose the problem into x,h then y,h in the first place, as done previously. 
     Also shown in  FIG. 6B  is a dotted line  660  traversing the injection site  622  at the horizontal axis of symmetry This dotted line  660  is not traveled by the laser, but is a virtual path that is able to replace the actual path for the first bounce site I in modeling. The virtue of so doing is the geometric mathematics then greatly simplifies. One can observe that the laser pulse  107  arriving from the scanning mirror will come from different locations by virtue of the different points on mirror  252  that are illuminated. The fact that  107  moves about complicates the math, and it is much easier to invoke symmetry and “pretend” the laser pulse originates from point  650  with different arrival angles. 
     We can now introduce some notation. We can denote the horizontal direction as x, the elevation as h (height), and the depth of the prism, i.e. the distance along the line connecting  602  and  604 , as y. With this notation,  FIG. 6A  shows the prism  600  with sides  602 , 604  in the x,h plane and the underside lies in the x,y plane. Next, we introduce subscripts l,r to denote the left and right side mirrors  610  and  608  respectively. This means that the mirror pitch angle can be represented as z where 
     
       
         
           
             
               z 
               ± 
             
             ≡ 
             
               
                 π 
                 4 
               
               ± 
               
                 z 
                 . 
               
             
           
         
       
     
     This notation simplifies a mathematical representation of prism  600 . Recall, the pitch angle of the V shape is z + , and the internal angle inside the V shape (which is the angular complement of z + ) is z − . The field inversion is defined via the mathematical dance which the laser angles undergo as the follow through these angles. Recall, the notation f,  654 , denotes the scan angle in the horizontal direction that we desire in the far field, which also happens to equal S,  655 . The notation c′ denotes the steepest downward injection angle (when the pulse  107  travels path  672 ) used in the elevation scan (as measured from the vertical axis on  FIG. 6A ). Recall that the notation c is the tunable far field elevation angle, ranging from zero to e,  673  (see  FIG. 6C ), as measured against the horizontal plane (see the horizontal axis in  FIG. 6C ). We introduced the notation is 4), the azimuth scan angle at a point in time that the second scanning mirror  252  output pulse furnishes at the laser injection site  622 . Recall it can be seen that the far field angle f can be expressed as f≈4z−ϕ. By symmetry, the same behavior will arise if we begin scanning right versus left, with left and right defined as shown in  FIG. 6B . Since ϕ has been inverted, the prism  600  achieves splitting and inversion. 
     Furthermore, to demonstrate that the prism  600  can be blockage-free, we can lift the scaling.  FIG. 6C  shows a depth formula for yr as an intermediate to calculate input/output invariance,  673 . This formula can be used to position an exterior descope/telescope lens  670  beyond the last bounce III as measured in depth. 
     If desired by a practitioner, a position sensor (not shown; e.g., a 4 quad position sensor) can be positioned near the second scanning mirror  252  to precisely determine the scan position of mirror  252 , which allows for calibrating out material defects beyond the above formula for f. This calibration can be achieved by adding a secondary wavelength into the pulse  108  and placement of a frequency/wavelength-selective mirror (e.g., a dichroic mirror  674 ) near the injection site  622 . This mirror can selectively reflect light at the secondary wavelength to the position sensor for accurate detection of the scan position for mirror  252  over time. A dichroic mirror can also be used to calibrate the scanning of mirror  250  if desired by a practitioner. 
     Also, Taylor series applied to the far field angle S will reveal that for all practical purposes: 
     
       
         
           
             
               δ 
               ⁢ 
               f 
             
             = 
             
               
                 
                   - 
                   δ 
                 
                 ⁢ 
                 ϕ 
               
               - 
               
                 δ 
                 ⁢ 
                 
                   
                     
                       ϕ 
                       3 
                     
                     2 
                   
                   . 
                 
               
             
           
         
       
     
     This formula can be used to construct the morphing of the far field pattern, i.e. obtaining a desired corrected angle S by adjusting the scanning of mirror  252  and mirror pitch angle z+. Any corrections beyond this point can be masked by calibrating out imperfections in the optical assembly for most and perhaps any diffraction-limited commercial systems. As noted above, such calibration can be achieved by the dichroic mirror  674 . By correcting time warping analytically, the range and sophistication of the time warping calibration that relates sinusoidal scan mirror control signals to the scan position in the far field can be greatly simplified. 
     Also, it should be understood that the right and left mirrors  608  and  610  can be selected to be one-way in their entirety or over injection region  620  to allow the injection region  620  to also be a potential bounce site. This repurposing of the same physical mirror real estate can lead to a more compact design, thereby saving cost, weight, and size for a practitioner. As a minimum for an example embodiment, we can require that the segment of the prism side-traced by the bottom of the V to point  622  in  FIG. 6B  must by two-way, as must be the mirror image on the other side of the V. The reason for this is that as ϕ gets near zero, we want a bounce back when moving from plate  624  towards  660 , while we also want an absence of a bounce at the bottom of the V when moving in the opposite direction. 
     Furthermore, ray tracing can be used to determine which regions of the prism interior might serve as bounce sites for laser pulses  108  during operation. The left half of  FIG. 7  shows the xl,yl (left side of plot) and xr,yr (right side of plot) ray tracing and region lased in the exit window  626  across the scan volume, shown in mm, for a 1 mm inter scanning mirror spacing and a V slot base to reflector dimension of 1 mm. The x-axis of  702  can be used to select the L-R mirror x-axis ( 610 ,  608 ) onset and the  704  x-axis as the mirrors  610 ,  608  y-axis onset. The y-axis of  706  can be used as the terminus in depth. As shown in the right half of  FIG. 7 , we can use the x,y position of  708  to be the beginning (inner terminus) of the exit window  626  and we can use the x,y position of  710  as the end (outer terminus) of the exit window  626 . We use the exit window embodiment here to be constant height so the bottom of the prism  606  is coplanar with exit window  626 , which is directly attached to an up/downscope. However, it should be noted, canted interfaces with the up-scope may be chosen as compaction might require, and the ray tracing can still be employed to determine requirements on prism dimensions and one-way mirror minimum areas. Thus far, we have defined the rectangular boundaries of mirrors  610 / 608 / 606 . The portion of the mirror cavity that never receives bounces I, II, or III can now be eliminated. This can be accomplished by computing the contours traced out by  702 ,  712 ,  706  and  704 ,  710 ,  714  and assigning mirror (i.e., reflective coating) to the attendant convex hull varying over all sets of angles. The mirror need only be one-way over the region determined by ray tracing x,r of the scanning mirror output over their scan angles. Through this process, mirror compaction can be achieved which may reduce manufacturing and raw material costs. 
     Overlapped Inversion/Splitting: 
     The field inverter  300  can also be configured to provide an inverted/split field of view that is overlapped. This configuration not only decouples mirror inversion with field look direction but also allows multiple mirror inversions per field look direction within a single axis scan. This overlapping effect allows for a longer gaze along a desired region such as a middle region or centerline of a field of view. Accordingly, it should be understood that the field inverter  300  need not split the field of view into two non-overlapping fields. To achieve overlapping, the mirror pitch angle z+ can be modified so that the launch angle of pulse  108 , i.e. angle S swings negative (i.e. left of vertical) at extremal values of the scan angle ϕ. 
     By creating an overlap, we exchange high elevation scan time for double revisit of the specified foviation zone, which allows a deeper look down that can be useful for avoiding near obstacles. 
     An example of this arrangement is shown by  FIG. 8 . In this example, we choose to field invert the scan on azimuth (the horizontal axis), with a selected scan volume of 45 degrees, 
     
       
         
           
             
               
                 A 
                 az 
               
               = 
               
                 π 
                 4 
               
             
             . 
           
         
       
     
     Without field inversion, the relation between the x-axis, the resonant scan angle for mirror  252 , and the vertical axis, the far field scan angle f would be identical—i.e., =A az . 
     With field inversion, we observe that as the resonant mirror scan angle goes from −22.5 degrees to zero, the fair field scan angle f ranges from zero to −22.5 degrees. This “flip” is expressed mathematically by the formula for f, where there is a shift (4z) and a negative sign relating the vertical to horizontal axis. The moment the mirror scan moves from 0− to 0+, the far field scan angle f flips from −22.5 degrees to 21 degrees. In the absence of overlap, this flip would end up at 22.5 degrees rather than 21 degrees. The difference between overlapped and non-overlapped scans can be defined and achieved in real-time with a hinge as discussed below. Since we begin the far field scan angle at 21 degrees, and we must swing through 22.5 degrees net in the scanning mirror, we end at −1.5 degrees in the other direction (as opposed to zero as we would for non-overlapped field inversion). It is this change that provides the beneficial effect of overlapping because we now scan through the horizon twice for each scan leg (where it is expected there will be a need for more range point detection). 
     The horizontal axis of  FIG. 8  is the angle to which the scanning mirror (e.g., MEMS mirror) is “looking”. Because the scanning mirror is slowest when it changes direction, we can see from the curve at the top of  FIG. 8  that the scanning mirror is moving fastest at the center (0 degrees). It is also moving slowest at the edges of the scan (in this example at −22.5 degrees and 22.5 degrees). As discussed above, this can be problematic for practitioners when the center region is the area where it is desirable to maximize gaze time (to increase how many range points are lased in the center region) (e.g., for automotive applications, the center region is where most accidents are likely to occur). The vertical axis in  FIG. 8  is the location in the far field where the ladar transmitter is gazing, i.e. the instantaneous field of view. In a traditional, non-inverted field of view, this gaze location would be equal to the horizontal axis. If zoom or upscope is added to the ladar transmitter, the relationship remains linearly proportional. This means the “bad” edge gaze of the scanning mirror is “inherited” in the ladar transmitter&#39;s gaze into the scene. Since the scanning mirror&#39;s scan position alone determines where in the scene that the instantaneous gaze is located, there is a curve that we can plot that represents the relationship connecting these two axes. While the example of  FIG. 8  is shown for only one resonant scanning mirror, the same idea works for 2D resonant mirror scanning but is more complex with respect to articulation, so for ease of illustration the single resonant scanning mirror example is shown. For standard non-inverted ladar transmitters, this relationship curve is simply a straight continuous line. However, with field inversion, as shown by  FIG. 8 , the curve is presented by two line segments with a “jump” reflected around the vertical axis. The end result is seen in the plot of the speed of the instantaneous field of view within the scene to be imaged at the far right of  FIG. 8 . Relative to the top plot of  FIG. 8 , the speed is inverted vis a vis the horizontal, MEMS axis. Indeed, we now stare a relatively longer time at 0 degrees (where speed is slow) and spend relatively less time at the scene edges (where speed is high). 
     With regard to overlapping the inverted view, we can choose to scan beyond zero at one or both edges of the mirror scan.  FIG. 8  shows an example where the right hand scan (and only the right hand scan in this example) is selected for overlap. One can see that the scene is inspected a 0 degrees and down to −1.5 degrees on the right hand scan, which increases the amount of time we gaze at the centerline of 0 degrees and environs via overlapping. 
     To control and adjust how much overlap is achieved, one or more controlled hinges  628  can be used to define the mirror pitch angle z+, as shown by  FIG. 6D . Hinges  628  allow for the mirror pitch angle of the prism  600  to be adjusted. A mirror pitch angle adjustment mechanism (such as a threaded knob arrangement  630  and  632  by which the mirrors  608  and  610  are pivoted at hinges  628  to adjust the mirror pitch angle) can be used to slide mirrors  608  and  610  along base  606 . This results in the ability to dynamically adjust the degree of backscan and or the Field of View in real-time, at sub-second rates. By adjusting the mirror pitch angle z+, the scan volume and overlap of the field inverter is dynamically controlled. Specifically, the midfield and far field angles S and f become tunable using a mechanical actuator to adjust the intersection of right and left mirrors  608  and  610  with the plane subtending the reflector  606 . 
     It should be understood that still other embodiments for a field inverter  600  could be used by a practitioner. For example, the reflector  606  could be replaced with a scanning mirror to add more compaction and shrink the spans shown in  FIG. 7  in exchange for potential occlusion from non-reflecting material at the boundary of  606 . Another example is to use two resonant mirrors designed to scan a left and right angular interval, with a potential overlap. The right hand edge of the left mirror and the left hand edge of the right mirror can then serve as field inverters. An example of this is shown by  FIG. 20 .  FIG. 20  shows a laser  2000  which is common to each of the scanning mirrors  2010  and  2012  (e.g., MEMS scanners in this embodiment). Shown is a single scan axis, the orthogonal axis is scanned through a field inverter, a standard scan, with a common mirror, or independent mirrors as deemed desirable by the practitioner. Laser output  2002  is fed into a beamsplitter  2004 . As examples, The splitter  2004  can be a WDM (wave division multiplexer), a MEMS switch, Pockel switch, etc. To the right of the splitter  2004 , we have a scan mirror  2010  with a scan sector shown by the dotted lines, beginning at directly forward  2014  to a far right hand direction  2018 . To the left of the splitter  2004  is the other scan mirror  2012  with a scan sector shown by the dotted lines, beginning at directly forward  2016  to a far left hand direction  2020 . Since the speed of scan is slowest at the edges of the scan, it is evident that arrangement of  FIG. 20  implements an inverted field. 
       FIG. 9  depicts a table that demonstrates the effect of increased gaze times that can be achieved via field inversion (non-overlapped) and overlapped field inversion. The table of  FIG. 9  is based on a 90 degree scan range, and it shows the results of measuring the time spent near the centerline (within 3 degrees of the centerline) over each scan for a standard operation (no field inversion), field inversion operation (non-overlapped), and an overlapped field inversion (with a 1 degree split field for 5 mrad beam divergence). As can be seen, field inversion and non-overlapped field inversion yield significant improvements in dwell time near the centerline. 
     Lissajous Scan Patterns with Induced Periodic Phase Drift: 
     A 2D laser scan pattern is called a Lissajous scan pattern if and only if the 2D beam scans sinusoidally (in time) along each axis. The phase of both azimuth and elevation can be arbitrary, but are fixed in a standard Lissajous scan. The Lissajous pattern is generally desirable when one wants the fastest possible scans. This is because the fastest mirrors are resonant mirrors which must be driven periodically and resonantly, and hence sinusoidally. Thus, in a Lissajous scan pattern, both mirrors  250  and  252  will be driven by sinusoidal signals. While both phases of these sinusoids are free, the difference between them impacts scan performance. The usual choice for the phase difference amongst practitioners is 90 degrees, which minimizes the maximum gap between where adjacent beams scan. 
     The Lissajous can pattern for a 2D resonant beam scanner can be expressed as: 
       [height( t ),azimuth( t )]=[ A   h  sin( ft ), A   az  sin(( f+ 1) t +ξ)]  (Eq1)
 
     The resonant frequencies f,f+1 differ by only 1 in many applications as this is well-known to minimize gap times. However, for ladar in a road driving context, other choices might be desired, due to the fact that azimuth rate of change and elevation rate of change differ for objects on the road surface. 
       FIG. 10A  depicts an example of a Lissajous scan pattern for a ladar transmitter when no field inverter  300  is employed.  FIG. 10C  depicts the revisit performance of the Lissajous scan pattern of  FIG. 10A . The axes in  FIGS. 10A and 10C  are an X-axis corresponding to horizon/azimuth and a Y-axis corresponding to elevation. In  FIG. 10C , the Z axis is the number of revisits per Lissajous cycle, with periodicities  99 , 10 , for overall Lissajous cycle of  990 . The scan range chosen here is 45 degrees in each axis. For physical fidelity uniform phase drift of half the beam divergence of 9 mrad was added. In addition a road surface was added, with lidar velocity of 30 m/s, with an above horizon slice of 4 m. Motion is included because the true revisit must take this variable into account. However, this component has only a minor effect. 
     The lines in  FIG. 10A  are a tracing that shows where the ladar transmitter is targeted as the mirrors scan. The white regions are the spacings or gaps that would exist between beams that are theoretically fired all along the scan lines. The surface mesh in  FIG. 10C  shows how the revisit rate varies as a function of azimuth and elevation. For example at the edges we obtain about 8 opportunities to revisit a point per cycle. This aspect of  FIG. 10C  shows the “downside” of a conventional Lissajous curves for ladar: the best revisit is at the edge of the scan volume, where it is least needed. Specifically, the plots of  FIGS. 10A and 10C  show how a standard Lissajous scan results in denser visits at the edges of the scan area relative to a center region of the scan area. That is,  FIGS. 10A and 10C  show that there are larger gaps and delays between the scan lines and opportunities in the center region of the plot than there are at the edges. Again, this is contradictory to most desired applications (such as most automotive applications) where denser visits are desired at the center region. 
     As mentioned above, a field inverter  300  such as prism  600  can be used to split and invert a scan pattern, which when applied to the Lissajous scan patterns of  FIGS. 10A and 10C  yields a scan pattern with increased gaze time near the center region, as shown by the example of  FIG. 10B .  FIG. 10B  depicts an example of a split/inverted Lissajous scan pattern when a field inverter  300  is employed to provide inversion along the horizon (i.e., the horizontal axis when the elevation is zero), and  FIG. 10B  shows that the density of visits by the ladar transmitter near the center line is greatly increased relative to that of  FIG. 10A . This difference is dramatic, as the gaps along the horizon are virtually eliminated. 
       FIG. 10D  depicts a plot of revisit performance for an example of a split/inverted Lissajous scan revisit when a field inverter  300  is employed to provide inversion along elevation (i.e., the vertical axis when the horizon is zero), as well a kissing mirror pattern in the azimuth direction.  FIG. 10D  shows that the density of visits by the ladar transmitter near the center line is greatly increased relative to that of  FIG. 10C . This difference is also dramatic, there is roughly 20 times more access to the horizon directly in front of the ladar-equipped vehicle. 
     Another way to measure the significance of these gaps is to determine which gaps exceed a defined threshold. This threshold can be defined based on beam divergence to account for the profile/diameter of a laser pulse  108  at an assumed distance with respect to a targeted range point. If the gap is larger than the threshold, this would represent a potential blindspot that could not be targeted by the ladar transmitter. However, if the gap is smaller than the threshold, such a small gap could be subsumed by a laser pulse  108  targeted nearby.  FIG. 11A  shows a series of dots which represent potential blindspots for the ladar transmitter when a Lissajous scan pattern such as that shown by  FIGS. 10A and 10C  is employed (with no field inverter  300 ), now with periodicity  48 , 49 . As can be seen, there are a relatively large amount of blindspots in the center region of the plot.  FIG. 11B  shows the corresponding blindspot plot for the inverted Lissajous scan pattern, such as  FIG. 10B  (once again, where each dot represents a potential blindspot) but now with periodicity  48 , 49 . As can be seen by  FIG. 11B , the combination of the field inverter  300  with the Lissajous scan pattern yields a ladar transmitter with a significant blindspot-free zone in the center region. 
     However,  FIG. 11B  shows that blindspots do exist in the split/inverted Lissajous scan pattern in regions outside the center region. This raises the question of how the system can be design to reduce the extent of such blindspots, ideally in a manner that allows the system to intelligently select desired regions for increased gaze that are not necessarily inside the center region. 
     As solution to this problem, the inventors disclose the use of an induced periodic phase drift in the Lissajous scan pattern. With this approach, the fixed phase that is common in Lissajous scan patterns is replaced with a time-varying drift. One (or both) of the scanning mirrors  250 / 252  is driven slightly off resonance by gently varying its phase. Accordingly, with this embodiment, we can represent the phase as (t) rather than as shown above by Equation 1 in the formulaic representation of the Lissajous scan pattern. This phase drift (t) is induced by having the beam scanner controller slowly vary a command signal provided to a driver for the subject scanning mirror (e.g., where the driver could be a motor for a stepped scan or a piston for a MEMS mirror). This command signal controls the mirror with respect to how it scans. In an example embodiment, the phase drift can be represented as: 
       ξ( t )=Σ i=1   M/2   A   i  sin( K   i   t+μ   i )  (Eq2)
 
     Accordingly, the Lissajous scan pattern as modified to include the induced periodic phase drift in both dimensions can be represented by: 
     
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           height 
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                         , 
                         
                           azimuth 
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                       
                       ] 
                     
                     drifted 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           A 
                           h 
                         
                         ⁢ 
                         
                           sin 
                           ⁡ 
                           
                             ( 
                             
                               ft 
                               + 
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   
                                     M 
                                     / 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     A 
                                     i 
                                   
                                   ⁢ 
                                   
                                     sin 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           
                                             K 
                                             i 
                                           
                                           ⁢ 
                                           t 
                                         
                                         + 
                                         
                                           μ 
                                           i 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                       , 
                       
                         
                           A 
                           az 
                         
                         ⁢ 
                         
                           sin 
                           ( 
                           
                             
                               
                                 ( 
                                 
                                   f 
                                   + 
                                   1 
                                 
                                 ) 
                               
                               ⁢ 
                               t 
                             
                             + 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   
                                     1 
                                     + 
                                     
                                       M 
                                       2 
                                     
                                   
                                 
                                 M 
                               
                               ⁢ 
                               
                                 
                                   A 
                                   i 
                                 
                                 ⁢ 
                                 
                                   sin 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         
                                           K 
                                           i 
                                         
                                         ⁢ 
                                         t 
                                       
                                       + 
                                       
                                         μ 
                                         i 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   Eq3 
                   ) 
                 
               
             
           
         
       
     
     In this expression, we denote M as the total number of phase frequency drift components across both mirrors. We can take all M drift components and apply them to one of the scanning mirrors, or as shown in the above formula, we can distribute the phase frequency drift components across both scanning mirrors. This distribution can be an even distribution or some other distribution. It should be understood that even with a single resonant mirror, a system can still benefit from the induced drift in terms of increasing gaze time in targeted regions, although we will describe the example embodiment in terms of two mirrors scanning at resonance. For a linearized approximation of Equation 3, associated with small amplitudes A i , we have: 
     
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           height 
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                         , 
                         
                           azimuth 
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                       
                       ] 
                     
                     drifted 
                   
                   ≈ 
                   
                     [ 
                     
                         
                     
                     ⁢ 
                     
                       
                         
                           A 
                           h 
                         
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                   ( 
                   Eq4 
                   ) 
                 
               
             
           
         
       
     
     For this example discussion, we can set M=4 to simplify the narrative, and we will ignore phase drift terms μ i  with the understanding that, for the example embodiment where a total least squares approach is used to find the desired drift, the terms μ i  behave just like the drift frequency terms K1, . . . during optimization and actuation. 
     It is also advantageous for the phase drift to be periodic with a harmonic sub-period of the initial pattern, i.e., where 
     
       
         
           
             
               
                 K 
                 i 
               
               f 
             
             , 
             
               
                 K 
                 
                   
                     M 
                     / 
                     2 
                   
                   + 
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                 + 
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     are rational and less than unit modulos. This assures that the revisit time is not reduced, thereby ensuring that the gap reductions are not achieved at the cost of a slower pattern revisit time. 
     With regarding to choosing the periodic drift frequencies Ki, there will be tradeoffs involved. If we choose periodic drift frequencies Ki that are too close to the Lissajous frequency f, then there will be very little effect on the pattern because the phase rate of change will blur into the frequency. Also, if the drift frequency is too low, the phase rate of change blurs into the fixed phase term of the Lissajous pattern. In the examples presented below, we describe an embodiment where the drift frequencies are set to lie at the midpoint prior to iteration. However, it should be understood that these are examples only and other values could be chosen. 
     In an example embodiment, an optimal phase for fixed drift frequency is determined as a solution to a total least squares problem. Total least squares (TLS) fits parameters when both dependent and independent variables are free, which will be the case here with the independent variable for TLS being time. The minimization employs a cost function. As an example, consider automotive ladar where we select a region in the ground plane as the area where we wish to remove gaps. 
       FIG. 12  discloses an example process flow for controlling how a scanning mirror is induced with a periodic phase drift. Driver  1250  for a scanning mirror includes an actuator which can periodically modulate the phase ξ using parameters A az , A h A i K i μ i , i=1, . . . , M while retaining the frequencies f,f+1. Step  1204  in  FIG. 12  describes frame-dependent feedback in the scan pattern for the scanning mirror centered on total least squares The use of singular value decomposition (SVD) assures that the  FIG. 12  process flow can be executed in real-time, specifically on the order of milliseconds. 
     At step  1200 , the process flow is initiated via selection of system parameters. As part of this, the minimum acceptable ground gap is defined. This can be set as an angle or a distance in meters. We chose the gap in meters for an example embodiment. We compute the gaps by looking at where the pulses can be fired from scan to scan, and measuring if the distance is larger than this amount, measured as an extent beyond the full width half maximum of beam divergence. We declare a gap, and use that gap in calculating phase drift when, as shown in  1202 - 1204 , the gaps are (i) larger than 1 m, and (ii) within the preferred scene (gaze region) Q. Examples of this are shown in  FIG. 13A  (for a non-inverted case) and  FIG. 13B  (for an inverted case), where f=49. The black regions in  FIGS. 13A and 13B  (e.g., see  1302 ) are the regions in the x,y axis which are further away than 1 meter by 1 meter from a laser pulse for all values of time t in Equation 1 above. In this example, the ground plane is used to determine gaps. However, in another embodiment, one could use the azimuth elevation plane for identifying gaps, in which case we would use the black cluster in  FIGS. 11A  and B rather than  FIGS. 13A  and B. 
     Next, the drift frequencies are selected (where, for M=4 in this example, 2 drift frequencies are used for each scan axis). We also select the Lissajous frequencies f,f+1, field of view (FOV) and the tolerable gap size. The Lissajous frequencies will be set by the transmitter control instructions,  103 . The FOV will be determined by the speed of the mirror scans as well as the desired region we wish the laser to inspect, coupled with the gaps. For example, if the maximum scan frequency is 10 Khz, and we scan across 100 degrees, with 1 microsecond pulse spacing, the gaps will be about three degrees, if we scan across 20 degrees at 10 microseconds the gaps will be 4 deg. For example, with  FIGS. 13A  and B, the FOV is +/−45 degrees in azimuth (as evidenced by the slope of black triangle  1304 ). Likewise, the vertical FOV ranges from the horizon down to 12 degrees, as evidenced by the small black rectangle  1306  near 0,0 assuming a 2 meter height monitoring of the ladar transmitter. 
     For the purposes of phase drift, the only part of the FOV that matters is the range below the horizon, so the upper limit of the elevation FOV beyond zero is immaterial (as indicated in  FIG. 13A  by no black dots being shown above 1600 feet and in  FIG. 13B  by no black dots being shown above 800 feet). 
     At step  1202 , we generate the standard fixed phase Lissajous pattern according to the defined parameters.  FIG. 13A  shows the gaps for a standard Lissajous pattern where f,f+1 is 49,50 and no inversion takes place, and  FIG. 13B  shows the gaps for a standard Lissajous pattern where f,f+1 is 49,50 and where inversion with a 1 degree overlap takes place.  FIGS. 11A  and B show the respective gaps in the vertical (laser line of sight) for the non-inverted and inverted space for a 3 mrad beam width, and  FIGS. 13A  and B show these same gaps projected onto the ground plane with a one meter gap tolerance. Region  1310  shows gaze region Q where gap reduction is desired. The gaze region enters into the processing stream in  FIG. 12  in section labeled  1202 . Only the gaps in this section are used in the subsequent algorithm stages, all other regions are ignored. In our example all the gaps are removed in  1310 , which is why we labeled them grey. Region  1310  can exhibit any shape, and it can be selected based on the environmental scene (see  120  in  FIG. 1B ). For example, consider a case where we have a side road that we wish to scan near its intersection of a ladar transmitter-equipped vehicle which is moving, along azimuth=0, from 0 to 800 feet where the intersection arises. The Q selection might be either data adaptive (we observe traffic moving along the region  1310  and seek to investigate) or it may be selected from a priori information (such as a road network map). The size of M will determine the rate of change for the Q selectin that is allowable in real-time. Preliminary work indicates that M=4 works quite nicely and easily converges in time to update at a millisecond rate. In our example, the projections are to a ground plane, but it should be understood that any projected surface will do, incorporating terrain elevation or other considerations that might replace ground planes with other topological manifolds. 
     The detected gaps that exceed the defined minimum accepted gap are grouped into a set of points as S. Thus, S will be a collection of points that represent the black dots  1302  in  FIG. 13A  (or  13 B). 
     Next, Equation 1 is used to find the times t that are yield coordinates that are closest to each point in S. These times can be denoted in the set T. 
     Step  1204  follows where we solve the TLS solution using the principal components from steps  1202  and  1204 . In the following description, we will restrict the embodiment to ground plane gaze, with field inversion and overlap. First, we linearize using Equation 4 with times t set based on the elements in T. The TLS solution will pick pairs A i , μ i . as well as updated time stamp T based on the linearized Taylor series representation of the drift. We can now substitute these values into Equation 3, and from this find the new (generally smaller) set of black dots that define a new set which we will denote as S_drift. Next, we recurse again, resolving TLS over time stamps, amplitudes, and drift phases and frequencies until a desired performance level is achieved. 
     As per step  1210 , if at any point S or S_drift is zero, we inject the phase drift controls corresponding to that S or S_drift into driver  1250 . There are two mutually exclusive and collectively exhaustive outcomes of this procedure. The first is that the set in S no longer shrinks (see step  1212 ), in which case we can update the Lissajous parameters [K,f] and repeat to test if performance improves (S,S_drift is reduced). Alternatively, S,S_drift reduces to a size that is deemed worthy of termination (or vanishes entirely). 
     For an example run, all of the black dots in region  1310  of  FIG. 13B  can be removed (i.e., S_drift is empty). The example run took only 1 Mflops, executable in 1 ms for 1 Gflop processor, which is fast enough to be updated on a single frame basis. The zoom of the field inverted and phase drifted overlapped pattern is shown by  FIG. 14A .  FIG. 14B  shows the corresponding non-phase drifted Lissajous pattern.  FIGS. 14A  and B zoom in near the horizon since that is where the main activity occurs, and an inspection of  FIGS. 14A  and B reveals clues as to how the induced phase drift improves performance and removes gaps. First, we note that in both cases the scan range region is shown in expanded view since the full view plot in  FIGS. 10A  and B is too coarse to garner insight. Next, we note that there is a lot of scanning slightly below the horizon, and this region has very tight spacing between laser scans. This is combined feature of both field inversion and overlapped coverage, common to both examples. In both cases we have chosen to tune the two mirror field inverter to place renewed emphasis on the elevation scan between about −0.3 degrees and −1.2 degrees in elevation. Inspection of  FIG. 14A  shows that within the dense scan region and the coarser scan region (above −0.2 degrees elevation and below 1.2 degrees elevation), the diamond-shaped gap regions are irregular, and more irregular than in  FIG. 14B . This is the gift that the  FIG. 12  process flow bestows by determining the best way to drift the phase so as to drive down the set S. Also, we note that the vertical stripes in  FIG. 14A  are more spaced apart than they are in  FIG. 14B . This is a result of the phase drift bifurcating the revisit pattern and replaced diamond gaps with tighter gaps. 
     Accordingly, it should be understood that the induced phase drift can be used to intelligently selection regions of the scan area for longer dwelling gaze. Moreover, when combined with a field inverter  300 , the use of mirrors scanning in a Lissajous pattern with induced phase drift is expected to provide significant performance improvements that allows for better interrogation of desirable regions in a scan area per frame. 
     Ellipsoidal Reimaging Mirror for Compact Beam Scanner Assembly: 
     The inventors recognize that there is also a desire in the art for compact beam scanner assemblies. For example, the inventors believe there is a growing interest in compact 2D scan mirrors for automotive and airborne ladar, biomedical imaging (i.e. endoscopy), virtual and augmented reality, and confocal active imaging. Scan mirrors, whether implemented as galvanometers, MEMS, or other mirrors, are often used in laser scanning systems due to the associated high scan speed and compact form factor. The fastest real scan rate and tilt angle is usually obtained by cascading a pair of in-plane and out-of-plane single axis (as opposed to dual axis) MEMS devices. The second mirror in the light path has a larger spot size than the first due to beam divergence. The inventors disclose a device which reimages the spot beam on the second mirror, thereby shrinking the required mirror size. Not only does this reduce the form factor of the scanner, it also increases scan speed, and/or maximum tilt angle, and therefore scan field of view, since mirror area is proportional to torque and scan speed. 
     In an example embodiment, two scan mirrors (e.g. MEMS mirrors) can be placed at the foci of an ellipsoid defined by an ellipsoidal reflector/mirror. A focusing lens (or mirror) can be positioned to condition the input beam prior to directing the beam onto the first scan mirror in order that the output beam can remain collimated. This is optically equivalent to placing an image of the first scan mirror at the location of the second scan mirror, a situation known as being optically conjugate. For this reason, the reflector assembly can be referred to as an elliptical conjugate reflector (ECR) assembly. In an example embodiment, only a relatively small portion of the complete ellipsoid will intercept light reflected from the first scan mirror, as determined by the angle of incidence of the light beam at the first scan mirror. This allows construction of the ECR using only the corresponding section of the ellipsoid. This in turn provides a ready mechanism for allowing both the incoming and outgoing light beams to enter and leave the assembly. 
     Analysis of the imaging properties of the ellipsoid shows that the angle of incidence at the first scan mirror can be chosen so that the reflected ray fan from the first scan mirror towards the reflecting surface of the ellipsoid interior is oriented so that the intersection of all the rays in the ensuing fan lie in a plane which also contains the center of the second scan mirror. We disclose a design formula that ensures this coplanar dependency, with or without a tilt offset on the scanners. A tilt offset allows for flexibility in the length, height, and width of the assembly, which has the benefit of increasing the trades available to a practitioner. 
     In addition to 2D scan applications, the ECR techniques disclosed herein offers improvements in any cascaded mirror assembly. Cascaded mirrors increase overall scan aperture, and the reimager disclosed herein renders these systems more compact as well. In contrast to prior art, the ECR solutions disclosed herein provide a more compact solution (see, for example, an embodiment that uses a single mirror for reimaging) without introducing artifacts into the scanned field. 
     A laser can be scanned with a pair of single axis mirrors. If the mirrors are attached to a solenoid, this is referred to as a galvanometric scanner. In many modern compact laser systems (which includes copy machines, bar code readers, and ladar systems), MEMS single chip devices are often used as the tilt mirrors to reduce size, weight, and cost, while increasing scan speed. Since it is desired that the mirrors freely articulate, and the light cone communicating between them be unoccluded, there are hard constraints on how close the distance between the articulating mirrors can be. Since the second scan mirror must be large enough to accommodate the entire range of angles induced by the first scan mirror, it is conventional that the second scan mirror in general be larger than the first scan mirror. This in turn reduces achievable maximum scan angle, or maximum achievable scan frequency, or both. Since both are important design parameters for practitioners of the laser arts, the inventors disclose in an example embodiment a design that allows a significant increase in scan volume by rendering a system with two scanning mirrors (such as MEMS devices) of small and equal size. The limitation on mirror size is a function of both laser beam waist and scan volume. Reimaging allows a MEMS device on the order of a few millimeters. To solve this problem in the art, the inventors disclose the use of an ellipsoidal reimaging reflector that is positioned optically between the first and second scan mirrors. Such a design can preserve the simplicity of planar MEMS mirrors as the scannable mirrors while also offering improved performance. Moreover, this ellipsoidal reflector can be the single reimaging mirror used by the system. 
       FIG. 15  shows an example embodiment of a design employing an ellipsoidal reflector  1510 . The reference H. Rehn, “Optical Properties of Elliptical Reflectors”, Opt. Eng. 43(7) 1480 (2004), the entire disclosure of which is incorporated herein by reference, provides additional details including optical properties associated with an ellipsoidal reflector. It should be understood that when reflector  1510  is referred to as an ellipsoidal reflector, this means that the reflector  1510  exhibits a curvature that corresponds to at least a portion of an ellipsoid shape. Thus, the ellipsoidal reflector  1510  preferably exhibits a shape and curvature corresponding to a section of a hollow ellipsoid. The example system shown by  FIG. 15  uses the ellipsoidal reflector  1510  in an offset configuration. Also, in an example embodiment, the specific ellipsoidal structure used for reflector  1510  can be a prolate spheroidal shape. Such ellipsoids have rotational symmetry about the major axis, and this structure allows physical separation of the two scan mirrors. 
     Consider an ellipsoid of revolution defined by the formula 
     
       
         
           
             
               
                 
                   
                     z 
                     2 
                   
                   
                     A 
                     2 
                   
                 
                 + 
                 
                   
                     r 
                     2 
                   
                   
                     B 
                     2 
                   
                 
               
               = 
               1 
             
             , 
           
         
       
     
     where r′=x 2 +y 2 . The projection of this into a plane is an ellipse  1550  with horizontal length of 2A (see  1520  in  FIG. 15  which identifies the length A), and vertical height 2B (see  1526  in  FIG. 15  which identifies the height B). The scan mirrors  1522  and  1524  are each set at a distance C (see  1518  in  FIG. 15 ) from the ellipse center  1526 . For these locations to be at the focal points of the ellipse, the value of C should be defined as C=√{square root over (A 2 −B 2 )}. 
     Upstream from the reflector  1510  we insert a lens  1502 , which focuses the light emitted from the source  1500 . As explained below, the ellipsoidal reflector  1510  and lens  1502  can serve jointly as an afocal lensing system. The shape and position of lens  1502  is chosen so that the focal point  1506  lies between the first scan mirror  1522  and the reflective surface of the ellipsoidal reflector  1510 . Recall, that by definition, the focal point  1506  represents the location where the spot size is at a minimum. The distance from  1506  to the location on the ellipsoidal mirror whereupon the light source projects we denote by F2 ( 1512 ). It is desirable that the light beam incident on the second scan mirror be collimated, in order that the output of the scan mirror  1522  is also collimated. Hence, the optimum location of the focal point  1506  as determined by the characteristics of the input beam  1500  and focusing element  1502  can be made to conform to the requirement that the distance  1512  is equal to the effective focal length F2 of the ellipsoid corresponding to ellipse  1550  defined by the shape and curvature of ellipsoidal reflector  1510  at the point of reflection from  1510 . 
     The angle α,  1516 , is the offset tilt of the first scan mirror  1522 . Note that as the tilt is varied on the 1 st  scan mirror  1522 , the angle of incidence (AOI)  1504  also varies. This does not constitute a requirement for using the system but offers additional flexibility to practitioners wishing to incorporate the system by decoupling the trajectory of the input light from subsequently described geometric requirements. We denote by the offset  1508  as the distance from the center  1526  of the ellipsoid projection  1550  to the center of the portion of the reflective surface of the ellipsoidal reflector  1510 . 
     If a point source is positioned at one of the two foci of a prolate spherical ellipsoid, then light will all arrive at the second focus without aberration, and the total path length for all light rays will be equal. Therefore, in principle one can direct a light beam onto the first scan mirror  1522  from any angle and it will reflect onto the second scan mirror  1524  as long as that second scan mirror  1524  is located at the second focus of the ellipsoid. 
     A more important factor influencing the beam input angle arises from the desire to optimize the characteristics of the field covered by the scan pattern of the output beam. This can be appreciated by considering the operation of an ideal two-mirror scanning system operating on optical rays with no intervening optics. In such a system, the accumulation of rays reflected for various tilt angles of the first mirror results in a set of reflected rays at various angles referred to here as a ray fan. It is desirable that all the rays in this fan lie in the same plane. This ray fan is then incident on a second mirror of sufficient extent that all of the rays in the fan can be accommodated. When this second mirror is scanned in a direction orthogonal to the first mirror, the resulting 2D output fan has the property that, when projected onto a plane perpendicular to the center ray, the 2D output fan forms a scan pattern in which the scan rows are linear and horizontal. The plane of incidence of each member of the ray fan emanating from the first mirror, when incident on the second mirror, will then be rotated to an extent determined by the magnitude of the scan angle imparted by the X mirror. This results in a small pincushion distortion in the X direction only, which is visible in  FIG. 16  as a deviation from the exact rectilinear pattern illustrated by the rectangular boundary. This distortion can be readily accommodated by either adjusting the amplitude of the X mirror scan for each Y position, or adjusting the laser pulse timing in the ladar system. 
     Consider the ray fan from the first scan mirror  1522  as it encounters the inside reflective surface of the ellipsoidal reflector  1510 , from which it reflects down onto the second scan mirror  1524 . For an example embodiment, in order for the scanner to operate in the same desirable fashion as the ideal mirror pair previously described, after reflection from the ellipsoidal reflector  1510 , the fan of rays now converging onto the center of the second scan mirror  1524  should all lie in the same plane. This can occur only for the case where the intersection of the center ray of the fan lies directly above the second scan mirror  1524 . 
       FIG. 17  illustrates a 3-dimensional view of the arrangement shown by  FIG. 15 , with a focus on the ray fan geometry. In  FIG. 17 , various elements of  FIG. 15  are again labeled ( 1500 ,  1502 ,  1522 ,  1524 ,  1510 ,  1514 ), and the 3D ellipsoid is now drawn as a wire mesh,  1700 .  FIG. 17  also adds the following labels:  1704  (for the image plane as presented to the environmental scene),  1706  (for the fan beam from the second scan mirror  1524 ,  1708  (for the fan beam reflected by ellipsoidal reflector  1510 ).  FIG. 17  also shows the plane  1702  which encompasses the fan beam  1708 . It is useful to note that, in the case of an arbitrary geometry, the beam reflecting off the ellipsoidal reflector  1510  onto the second scan mirror  1524  has a fan beam  1708  that is not planar. 
     For an example embodiment, making this fan beam  1708  planar places a requirement on the angle of reflection from the first scan mirror  1522 . This angle is abbreviated the CPA (see  1514  in  FIG. 15 ) for the coplanar angle. CPA is the angle subtended between the symmetry axis of the ellipsoid and the intersection of the ellipsoid with the perpendicular line passing through the center of the second scan mirror  1524 . CPA can be calculated from the values of A and C which serve to define the ellipsoid  1550 , using the following expression (shown as  1528  in  FIG. 15 ): 
     
       
         
           
             CPA 
             = 
             
               2 
               ⁢ 
               
                 
                   
                     tan 
                     
                       - 
                       1 
                     
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         A 
                         - 
                         C 
                       
                       
                         A 
                         + 
                         C 
                       
                     
                     ) 
                   
                 
                 . 
               
             
           
         
       
     
     To aid in ensuing design trades, we can add in an optional offset in tilt angle, α, to the 1 st  scan mirror  1522 . We then obtain a modified formula for the CPA, shown as  1528  in  FIG. 15  and re-created here for convenience: 
     
       
         
           
             CPA 
             = 
             
               
                 2 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     tan 
                     
                       - 
                       1 
                     
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         A 
                         - 
                         C 
                       
                       
                         A 
                         + 
                         C 
                       
                     
                     ) 
                   
                 
               
               - 
               
                 
                   α 
                   2 
                 
                 . 
               
             
           
         
       
     
     Note that CPA  1528  is no longer mathematically exact (as is in the first formula that did not include the addition of the optional offset tilt angle), but is rather an approximation sufficient for practical use. 
     Note that light does not interact with the ECR following reflection from the second scan mirror  1524 , so an offset angle can be imposed on the second scan mirror  1524  to facilitate exit of the scanned volume without prejudice to performance. 
     The magnification between a collimated input  1500  and a collimated exit beam ( FIG. 15 ) is given by the ratio M=F2/F1. This is a consequence of the lens equation as applied to cascaded optical systems. In practice, a practitioner may want this ratio to be near unity, to keep both scanning mirrors  1522  and  1524  equal in size. 
       FIGS. 16 and 17  show how the CPA constraint can impact the construction of useful ECR.  FIG. 16  shows an example of the field resulting from a 24 degree×20 degree (optical) scan of X and Y angles, respectively, when no attention is paid to ensuring the ECR is constructed and used with the CPA constraint. Note that  FIG. 16  shows strong curvature in both Y and X scan lines, making this pattern difficult to match with a rectilinear coordinate system, especially problematic for co-boresighting camera registration with passive optics.  FIG. 18  shows the same scan field operated with the ECR constrained to operate according to  1528 . In contrast to  FIG. 16 , the scan rows (constant Y angle) are linear, and the pincushion distortion along the X direction is equivalent to that seen in the ideal (albeit non reimaged and therefor non-compact) system with no intervening optics. Note that in this pattern distortions in the second (vertical) scan angle mathematically vanish for alpha=0. The residual distortion in the first (horizontal) scan direction includes a minor over-scan similar to a 1D pincushion distortion, and is easily compensated in post processing. 
       FIG. 19  shows the elegant form factor compaction we can obtain in an example embodiment. The two scan mirrors, viewed from the side, are tightly packed with millimeter scales that are eminently feasible for a nominal beam waist of order 100 um. Recall the direction of the first mirror scan in this example embodiment is out of the plane, i.e. towards the viewer, while the second mirror scans within the plane containing the image itself. For brevity and clarity labels are omitted in  FIG. 19 , but visible are the CPA angle, the scan mirror input and output rays, the input light beam source and input lens, and the ellipsoidal reflector. The 3D nature of this mirror is also visible in  FIG. 19 . 
     While the invention has been described above in relation to its example embodiments, various modifications may be made thereto that still fall within the invention&#39;s scope. Such modifications to the invention will be recognizable upon review of the teachings herein. For example, while Lissajous scan patterns are disclosed as being a prime example of scan patterns that can be enhanced via field inversion and induced periodic phase drift, it should be understood that spiral scan patterns can be similarly enhanced. Such patterns are often implemented as dampened Lissajous patterns where the amplitude is slowly modulated.