PATENT DOCUMENT

Publication Number: US-9835853-B1
Application Number: US-201414554078-A
Country: US
Kind Code: B1

Title: MEMS scanner with mirrors of different sizes

Abstract:
A scanning device includes a base and a gimbal, mounted within the base so as to rotate relative to the base about a first axis. A first mirror, which has a first area, is mounted within the gimbal so as to rotate about a second axis, which is perpendicular to the first axis. A second mirror, which has a second area that is at least twice the first area, is also mounted within the gimbal so as to rotate about a third axis, which is parallel to the second axis, in synchronization with the first mirror.

Claims:
The invention claimed is: 
     
       1. A scanning device, comprising:
 a base; 
 a gimbal, mounted within the base so as to rotate relative to the base about a first axis; 
 a first mirror, which has a first area and is mounted within the gimbal so as to rotate about a second axis, which is perpendicular to the first axis; and 
 a second mirror, which has a second area that is at least twice the first area, and which is mounted within the gimbal so as to rotate about a third axis, which is parallel to the second axis, in synchronization with the first mirror. 
 
     
     
       2. The device according to  claim 1 , wherein the scanner comprises a substrate, which is etched to define the base, the gimbal, and the first and second mirrors in a microelectromechanical systems (MEMS) process. 
     
     
       3. The device according to  claim 1 , wherein the second area is at least four times the first area. 
     
     
       4. The device according to  claim 1 , wherein the first and second mirrors are connected to the gimbal by respective hinges disposed along the second and third axes, respectively, and are configured and coupled together so that the first and second mirrors oscillate in mutual synchronization at a resonant frequency of oscillation about the respective hinges. 
     
     
       5. The device according to  claim 4 , wherein the respective hinges of the first and second mirrors have different, respective degrees of stiffness, which are chosen so that respective individual resonant frequencies of the first and second mirrors differ by no more than 10%. 
     
     
       6. The device according to  claim 4 , and comprising a drive, which is configured to drive the first and second mirrors to rotate about the respective hinges in a resonant mode, while driving the gimbal to rotate relative to the base in a non-resonant mode. 
     
     
       7. The device according to  claim 6 , wherein the drive is configured to apply respective drive signals to the first and second mirrors at the resonant frequency with different, respective amplitudes and phases. 
     
     
       8. The device according to  claim 4 , wherein rotations of the first and second mirrors are synchronized in frequency and phase while rotating with different amplitudes. 
     
     
       9. The device according to  claim 1 , and comprising:
 a transmitter, which is configured to emit a beam of light toward the first mirror, which reflects the beam so that the scanner scans the beam over a scene; and 
 a receiver, which is configured to receive, by reflection from the second mirror, the light reflected from the scene and to generate an output in response to the received light. 
 
     
     
       10. The device according to  claim 9 , wherein the beam emitted by the transmitter comprises pulses of light, and wherein the output generated by the receiver is indicative of a time of flight of the pulses to and from points in the scene. 
     
     
       11. The device according to  claim 9 , wherein the transmitter and the receiver are mounted on the gimbal so as to rotate with the gimbal about the first axis. 
     
     
       12. A method for scanning, comprising:
 mounting a gimbal within a base so as to rotate relative to the base about a first axis; 
 mounting a first mirror, which has a first area, within the gimbal so as to rotate about a second axis, which is perpendicular to the first axis; 
 mounting a second mirror, which has a second area that is at least twice the first area, within the gimbal so as to rotate about a third axis, which is parallel to the second axis; and 
 driving the first and second mirrors to rotate about the second and third axes, respectively, in mutual synchronization, while driving the gimbal to rotate about the first axis. 
 
     
     
       13. The method according to  claim 12 , wherein mounting the gimbal and the first and second mirrors comprises etching a substrate to define the base, the gimbal, and the first and second mirrors in a microelectromechanical systems (MEMS) process. 
     
     
       14. The method according to  claim 12 , wherein the second area is at least four times the first area. 
     
     
       15. The method according to  claim 12 , wherein mounting the first and second mirrors comprises connecting the first and second mirrors to the gimbal by respective hinges disposed along the second and third axes, respectively while coupling the first and second mirrors together so that the first and second mirrors oscillate in mutual synchronization at a resonant frequency of oscillation about the respective hinges. 
     
     
       16. The method according to  claim 15 , wherein driving the first and second mirrors comprises causing the first and second mirrors to rotate about the respective hinges in a resonant mode, while driving the gimbal comprises causing the gimbal to rotate relative to the base in a non-resonant mode. 
     
     
       17. The method according to  claim 16 , wherein driving the first and second mirrors comprises applying respective drive signals to the first and second mirrors at the resonant frequency with different, respective amplitudes and phases. 
     
     
       18. The method according to  claim 15 , wherein driving the first and second mirrors comprises causing the first and second mirrors to rotate in mutual synchronization of frequency and phase while rotating with different amplitudes. 
     
     
       19. The method according to  claim 12 , and comprising:
 driving a transmitter to emit a beam of light toward the first mirror, which reflects the beam so that as to scan the beam over a scene; and 
 receiving, by reflection from the second mirror, the light reflected from the scene and generating an output in response to the received light. 
 
     
     
       20. The method according to  claim 19 , and comprising mounting the transmitter and a receiver of the reflected light on the gimbal so as to rotate with the gimbal about the first axis.

Description:
FIELD OF THE INVENTION 
     The present invention relates generally to methods and devices for projection and capture of optical radiation, and particularly to compact optical scanners. 
     BACKGROUND 
     Various methods are known in the art for optical 3D mapping, i.e., generating a 3D profile of the surface of an object by processing an optical image of the object. This sort of 3D profile is also referred to as a 3D map, depth map or depth image, and 3D mapping is also referred to as depth mapping. 
     Some methods of 3D mapping use time-of-flight sensing. For example, U.S. Patent Application Publication 2013/0207970, whose disclosure is incorporated herein by reference, describes a scanning depth engine, which includes a transmitter, which emits a beam comprising pulses of light, and a scanner, which is configured to scan the beam, within a predefined scan range, over a scene. The scanner may comprise a micromirror produced using microelectromechanical system (MEMS) technology. A receiver receives the light reflected from the scene and generates an output indicative of the time of flight of the pulses to and from points in the scene. A processor is coupled to control the scanner and to process the output of the receiver so as to generate a 3D map of the scene. 
     Another example of a time-of-flight scanner using MEMS technology is the Lamda scanner module produced by the Fraunhofer Institute for Photonic Microsystems IPMS (Dresden, Germany). The Lamda module is constructed based on a segmented MEMS scanner device consisting of identical scanning mirror elements. A single scanning mirror of the collimated transmit beam oscillates parallel to a segmented scanning mirror device of the receiver optics. 
     PCT International Publication WO 2014/016794, whose disclosure is incorporated herein by reference, describes optical scanners with enhanced performance and capabilities. In a disclosed embodiment, optical apparatus includes a stator assembly, which includes a core containing an air gap and one or more coils including conductive wire wound on the core so as to cause the core to form a magnetic circuit through the air gap in response to an electrical current flowing in the conductive wire. A scanning mirror assembly includes a support structure, a base, which is mounted to rotate about a first axis relative to the support structure, and a mirror, which is mounted to rotate about a second axis relative to the base. At least one rotor includes one or more permanent magnets, which are fixed to the scanning mirror assembly and which are positioned in the air gap so as to move in response to the magnetic circuit. A driver is coupled to generate the electrical current in the one or more coils at one or more frequencies selected so that motion of the at least one rotor, in response to the magnetic circuit, causes the base to rotate about the first axis at a first frequency while causing the mirror to rotate about the second axis at a second frequency. 
     U.S. Patent Application Publication 2014/0153001, whose disclosure is incorporated herein by reference, describes an optical scanning device that includes a substrate, which is etched to define an array of two or more parallel micromirrors and a support surrounding the micromirrors. Respective spindles connect the micromirrors to the support, thereby defining respective parallel axes of rotation of the micromirrors relative to the support. One or more flexible coupling members are connected to the micromirrors so as to synchronize an oscillation of the micromirrors about the respective axes. 
     U.S. Pat. No. 7,952,781, whose disclosure is incorporated herein by reference, describes a method of scanning a light beam and a method of manufacturing a microelectromechanical system (MEMS), which can be incorporated in a scanning device. In a disclosed embodiment, a rotor assembly having at least one micromirror is formed with a permanent magnetic material mounted thereon, and a stator assembly has an arrangement of coils for applying a predetermined moment on the at least one micromirror. 
     SUMMARY 
     Embodiments of the present invention provide improved devices and methods for efficient scanning. 
     There is therefore provided, in accordance with an embodiment of the present invention, a scanning device, including a base and a gimbal, mounted within the base so as to rotate relative to the base about a first axis. A first mirror, which has a first area, is mounted within the gimbal so as to rotate about a second axis, which is perpendicular to the first axis. A second mirror, which has a second area that is at least twice the first area, is mounted within the gimbal so as to rotate about a third axis, which is parallel to the second axis, in synchronization with the first mirror. 
     In a disclosed embodiment, the scanner includes a substrate, which is etched to define the base, the gimbal, and the first and second mirrors in a microelectromechanical systems (MEMS) process. 
     In some embodiments, the second area is at least four times the first area. 
     Typically, the first and second mirrors are connected to the gimbal by respective hinges disposed along the second and third axes, respectively, and are configured and coupled together so that the first and second mirrors oscillate in mutual synchronization at a resonant frequency of oscillation about the respective hinges. In a disclosed embodiment, the respective hinges of the first and second mirrors have different, respective degrees of stiffness, which are chosen so that respective individual resonant frequencies of the first and second mirrors differ by no more than 10%. 
     In some embodiments, the device includes a drive, which is configured to drive the first and second mirrors to rotate about the respective hinges in a resonant mode, while driving the gimbal to rotate relative to the base in a non-resonant mode. The drive may be configured to apply respective drive signals to the first and second mirrors at the resonant frequency with different, respective amplitudes and phases. The rotations of the first and second mirrors are typically synchronized in frequency and phase while rotating with different amplitudes. 
     In some embodiments, the device includes a transmitter, which is configured to emit a beam of light toward the first mirror, which reflects the beam so that the scanner scans the beam over a scene. A receiver is configured to receive, by reflection from the second mirror, the light reflected from the scene and to generate an output in response to the received light. In one embodiment, the beam emitted by the transmitter includes pulses of light, and the output generated by the receiver is indicative of a time of flight of the pulses to and from points in the scene. Typically, the transmitter and the receiver are mounted on the gimbal so as to rotate with the gimbal about the first axis. 
     There is also provided, in accordance with an embodiment of the present invention, a method for scanning, which includes mounting a gimbal within a base so as to rotate relative to the base about a first axis. A first mirror, which has a first area, is mounted within the gimbal so as to rotate about a second axis, which is perpendicular to the first axis. A second mirror, which has a second area that is at least twice the first area, is mounted within the gimbal so as to rotate about a third axis, which is parallel to the second axis. The first and second mirrors are driven to rotate about the second and third axes, respectively, in mutual synchronization, while driving the gimbal to rotate about the first axis. 
     The present invention will be more fully understood from the following detailed description of the embodiments thereof, taken together with the drawings in which: 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a schematic, pictorial illustration of an optical scanning device, in accordance with an embodiment of the present invention; 
         FIGS. 2 and 3  are schematic top and side views, respectively, of the optical scanning device of  FIG. 1 , in accordance with an embodiment of the present invention; 
         FIG. 4  is a schematic, pictorial illustration of the optical scanning device of  FIG. 1  from another perspective, in accordance with an embodiment of the present invention; 
         FIG. 5  is a schematic top view of a scanning mirror array, in accordance with an embodiment of the present invention; and 
         FIGS. 6A and 6B  are schematic side views of parts of an optical scanning device, shown in two different, orthogonal planes, in accordance with an embodiment of the present invention. 
     
    
    
     DETAILED DESCRIPTION OF EMBODIMENTS 
     Overview 
     Scanning optical devices often face a tradeoff between angular scan range, size and precision. For example, in the sort of MEMS-based time-of-flight (ToF) optical scanners that are described above in the Background section, the precision of depth measurement is limited by the strength of the received optical signal. This signal, in turn, is a function of the size of the receive aperture of the scanner, which is determined by the size of the collection optics and scanning mirror that collect light reflected from the scanned scene and direct it onto the receiver (typically a high-speed photodetector). Thus, enlarging the collection lens and mirror will improve the signal/noise ratio for 3D mapping, but at the same time, of course, will increase the size of the device and, for a device of a given size, can limit the angular range over which it is able to scan. 
     Some embodiments of the present invention that are described herein address this problem with a compact scanner design, in which the collection optics and receiver are mounted on the same rotating gimbal as the scanning receive mirror. Consequently, the optics and receiver rotate about the gimbal axis together with the mirror, giving a larger effective aperture than scanners of comparable size in which the optics and receiver are stationary. In the disclosed embodiments, the scanner typically also comprises a beam transmitter (such as a laser diode with collimating optics) and a scanning transmit mirror, which are likewise mounted on the same gimbal as the collection optics, receive mirror and receiver. 
     Specifically, in these embodiments, a scanning device comprises a scanner, in which a gimbal is mounted within a base so as to rotate relative to the base about the gimbal axis, and a receive mirror is mounted within the gimbal so as to rotate about a mirror axis, perpendicular to the gimbal axis. A detector is mounted on the gimbal so as to rotate with the gimbal about the gimbal axis and receive light reflected from the receive mirror while the receive mirror rotates about the mirror axis. A collection lens is also mounted on the gimbal and likewise rotates with the gimbal while collecting the light, so as to focus the light onto the detector by reflection from the receive mirror while the receive mirror rotates about the mirror axis. The area from which the detector receives light is thus scanned over an area of interest by the combined rotations of the gimbal and the receive mirror. 
     This sort of gimbaled scanning device is particularly useful in the context of ToF sensing systems, as described above, but may alternatively be applied in other sorts of scanned receivers. The principles of the present invention can be used, for example, in other systems that involve scanned reception of radiation, including optical, radio-frequency, and ultrasonic radiation, in which media or objects are detected by means of reflected or otherwise scattered radiation, using spatially separated emitters and detectors, wherein a first scanned element (such as a mirror) directs emitted radiation onto the object while a second scanned element directs collected radiation onto the receiver. 
     In embodiments in which the scanning device also includes a transmitter, the scanner typically comprises a transmit mirror, which mounted within the gimbal so as to rotate about its own mirror axis, parallel to the axis of the receive mirror, in synchronization with the receive mirror. The transmitter is mounted on the gimbal so as to rotate with the gimbal about the gimbal axis and to transmit a beam of light toward the transmit mirror while the transmit mirror rotates about its mirror axis so as to scan the beam of light over a scene. The detector thus receives, via the collection lens and the receive mirror, the light that is reflected from each point in the scene that is illuminated by the transmitted beam. 
     When separate transmit and receive mirrors are used in this fashion, the rotations of the transmit and receive mirrors are synchronized in frequency and phase but may rotate with different amplitudes, depending on the optical configuration of the scanning device. Since the transmitted beam is collimated, it is generally unnecessary that the transmit mirror have an aperture that is as large as the receive mirror. On the other hand, reducing the size of the transmit mirror relative to the receive mirror can reduce the overall size and enhance the design flexibility of the scanning device. 
     Thus, in some embodiments of the present invention, in which transmit and receive mirrors are mounted to rotate together in synchronization in a gimbal, the area of the receive mirror is at least twice the area of the transmit mirror, and may be at least four times the area of the transmit mirror. Despite the differences in the sizes of the mirrors, the hinges connecting the transmit and receive mirrors to the gimbal may be configured so that the mirrors, considered in isolation from one another, have respective resonant frequencies of rotation about their hinges that are approximately equal, for example, within 10% of one another. 
     More precisely, the transmit and receive mirrors are coupled together so as to synchronize the frequency and phase of the oscillation, using a mechanical coupler of the sort described in the above-mentioned U.S. Patent Application Publication 2014/0153001, or other suitable means of coupling. In this configuration, the mirrors have common symmetrical and anti-symmetrical modes of oscillation, with respective eigenfrequencies, as explained further in the Appendix hereinbelow. (These system eigenfrequencies are in general different from the individual resonance frequencies referred to above.) To achieve the desired performance, the gimbal in the disclosed embodiments includes mechanical coupling elements that are designed to give the appropriate strength of coupling and quality factors of the oscillations. As noted earlier, the amplitudes of oscillation of the two mirrors may differ, depending on application requirements. 
     Although the transmit and receive mirrors are used in the embodiments described below for purposes of ToF depth mapping, gimbaled arrays of synchronized mirrors of different sizes may likewise be used in other applications, particularly in miniaturized, MEMS-based systems. These alternative applications are also considered to be within the scope of the present invention. 
     For example, an ultrasonic scanner for 3D mapping typically comprises a source of acoustic radiation (ultrasonic transducer), a detector (such as another transducer), and optics (such as an acoustic lens). The interrogated media in this case may be, for example, physiological tissues, with imaging contrast due to the difference in acoustic impedance between the different parts of the tissue. The same sort of imaging scanner described above could be used, mutatis mutandis, for 3D mapping of scattered acoustic waves. In this scanner, the mirrors have surfaces with high acoustic impedance, so that acoustic waves are reflected from the surface, and the second mirror oscillates synchronously with the first mirror. An acoustic lens in front of the second mirror may be mounted on the same gimbal, with the detector mounted in front of the second mirror and behind the lens. 
     A system of this sort can be very efficient in collecting backscattered acoustic waves from the tissue and can thus provide ultrasound images of better quality than conventional ultrasound systems, in which only a tiny portion of the backscattered radiation is actually collected. Because only a portion of the field of view is actually “seen” by the detector at any given time during the scan—the part of the field of view that receives radiated waves—the receiver effectively selects the reflected waves and filters them from background and multiply-reflected waves. Since the dimensions of the MEMS scanner are tiny (in the range of millimeters) this sort of 3D ultrasound scanner could even be inserted into catheter. 
     System Description 
     Reference is now made to  FIGS. 1-4 , which schematically illustrate an optical scanning device  20 , in accordance with an embodiment of the present invention.  FIG. 1  presents a pictorial overview of device  20 , while  FIGS. 2 and 3  are top and side views, respectively, and  FIG. 4  is a pictorial view of the device as seen from below. (Terms referring to orientation in the present description, such as “upward,” “above” and “below,” are used solely for the sake of convenience, in reference to the viewing perspective shown in the figures, as are the X, Y and Z axes identified in the figures. The positive Z-direction is arbitrarily taken, in this context, to be the upward direction. In practice, device  20  may operate in substantially any orientation.) 
     Device  20  can be particularly useful as a part of a ToF 3D mapping system or other depth-sensing (LIDAR) device, in conjunction with suitable control and drive circuits  26 , as are known in the art. (Details of circuits  26  are omitted from the figures, however, for the sake of simplicity.) Alternatively, device  20  may be adapted for use as a scanning optical transceiver in other applications. 
     Scanning device  20  is built around a scanner  22 , which is driven by a stator  24  under the control of circuits  26 . In the embodiment that is shown in the figures, stator  24  drives the moving elements of scanner  22  by means of a varying magnetic field, in a manner similar to the apparatus described in the above-mentioned PCT International Publication WO 2014/016794. Alternatively, scanner  22  may be driven, mutatis mutandis, by magnetic drives of other sorts or by electrostatic or other types of mechanisms, as are known in the art. 
     A transmitter  28 , mounted on scanner  22 , emits a beam comprising pulses of light, which are collimated and steered by transmission optics  30  and then directed by a turning mirror  32  toward a transmit mirror  34 . (The term “light,” in the context of the present description and in the claims, refers to optical radiation of any wavelength, including visible, infrared, and ultraviolet radiation.) Mirror  34  scans the transmitted beam over a scene within a predefined angular range, as determined by the scan limits of the mirror. To open a clear beam path from transmit mirror  34  to the scene, turning mirror  32  is held slightly off axis (relative to the X-axis) by a mount  66 . 
     As shown in  FIGS. 2 and 3 , transmitter  28  comprises a miniature emitter  70 , such as a pulsed laser diode, which emits the beam in the upward (Z) direction, followed by a reflector  72 , which turns the beam in a direction parallel to the X-Y plane into transmission optics  30 . These transmission optics comprise a collimating lens  62  and a prism  64 , which directs the beam from transmitter  28  toward off-axis turning mirror  32 . 
     Light reflected from the scene is collected by a collection lens  36 , which focuses the collected light via a receive mirror  38  onto a detector  40 , which serves as the receiver. Detector  40  typically comprises a high-speed optoelectronic sensor, such as an avalanche photodiode. (Alternatively, any other suitable sorts of emitting and sensing components may be used as the transmitter and receiver in device  20 .) For the sake of compactness and light weight, collection lens  36  may be a Fresnel lens, as shown in the figures. Alternatively, other sorts of collection optics may be used. 
     Transmit mirror  34  and receive mirror  38  are mounted together within a gimbal  42  and rotate relative to the gimbal about respective axes that are parallel to the Y-axis shown in the figures. Gimbal  42  is mounted to rotate about the X-axis, perpendicular to the Y-axis, relative to a base  46 , which is fixed to stator  24 . To drive the rotation of gimbal  42  about the X-axis, a pair of rotors  48 , typically comprising permanent magnets, are fixed to opposite ends of the gimbal. Rotors  48  are then positioned within respective air gaps between poles  50  of stator  24 , which protrude upward from a base  52  of stator  24 . Poles and base  24  (as well as poles  74 , described below) comprise magnetic cores, which are wound with electrical windings  54 , as described in the above-mentioned PCT International Publication WO 2014/016794. Control and drive circuits  26  drive electrical windings  54  with alternating electrical currents, which cause alternating magnetic fields to be generated between poles  50 . These alternating magnetic fields interact with the permanent magnetic fields of rotors  48  and thus cause rotation of the rotors, and hence of gimbal  42 , about the X-axis. 
     To drive the rotation of mirrors  34  and  38  about their respective axes relative to gimbal  42 , stator  24  comprises additional, internal poles  74 , similarly fixed to base  52  and wound with electrical windings  56 . Control and drive circuits  26  drive electrical windings  56  with alternating currents, typically at frequencies substantially higher than those driving windings  54 . The driving currents cause poles  74  to generate alternating magnetic fields above their respective tips, which are tapered to concentrate the fields in the area of mirrors  34  and  38 . These magnetic fields interact with miniature magnetic rotors  76 , comprising permanent magnets, which are mounted on the undersides of mirrors  34  and  38 . For strong interaction, the gap between poles  74  and rotors  76  is typically no more than a few hundred micrometers. 
     The drive frequencies of the currents in windings  56  are typically at or near the resonant frequencies of rotation of mirrors  34  and  38  about their respective axes. Rotation sensors (not shown in the figures), such as capacitive or strain-based sensors, as are known in the art, may be mounted in the vicinity of mirrors  34  and  38  in order to provide feedback to circuits  26  regarding the frequency, phase and amplitude of rotation of the mirrors. Circuits  26  may apply this feedback in adjusting the drive signals to coils  54  and  56  in order to achieve the desired mirror rotation parameters. 
     An upper frame  58  is mounted above gimbal  42  so as to rotate with the gimbal relative to base  46 . The optical and optoelectronic components of transmitter  28  and its associated transmission optics, as well as detector  40  and lens  36 , are mounted on frame  58  so as to rotate with gimbal  42 . In the particular example shown in the figures, mounting of frame  58  over gimbal  42  is conveniently achieved by fixing frame  58  to rotors  48 , which in turn are fixed to gimbal  42 . Alternatively, other modes of attachment of the transmitter, detector and associated optics above gimbal  42  will be apparent to those skilled in the art and are considered to be within the scope of the present invention. 
     A detector mount  60  attached to frame  58  holds detector  40  in a position between collection lens  36  and receive mirror  38 , at the focal point of the rays collected by the collection lens. Mount  60  may comprise narrow struts, as shown in  FIGS. 1 and 2 , in order to minimize interference with the incoming light captured by collection lens  36 . Positioning detector  40  and lens  36  in this fashion, with a folded beam path between them, provides a large collection aperture (which may have a numerical aperture greater than 0.5) with low overall height. For this purpose, the area of collection lens  36  is typically at least twice the area of receive mirror  38 , and may be even four times the area of the receive mirror. The distance between the collection lens and the receive mirror is roughly half the focal distance of the collection lens. Frame  58  typically comprises a suitable dielectric or semiconductor material, with conductive lines (not shown) deposited on its surface to convey electrical power from circuits  26  to emitter  70  and to convey signals output by detector  40  to circuits  26 . 
       FIG. 5  is a schematic top view of the scanning mirror array in device  20 , comprising transmit mirror  34  and receive mirror  38 , in accordance with an embodiment of the present invention. Mirrors  34  and  38  rotate about respective hinges  84  and  86  relative to gimbal  42 , while gimbal  42  rotates about hinges  80  relative to base  46 . Hinges  80  of gimbal  42  are contained between wings  82 , to which magnetic rotors  48  are fixed, as explained above. Hinges  84  and  86  (and hence the axes of rotation of mirrors  34  and  38 ) are parallel to one another, along the Y-axis in the figures. Hinges  80  are oriented so that the axis of rotation of gimbal  42 , shown as being oriented along the X-axis, is perpendicular to the mirror rotation axes. As noted earlier, scanner  22  may be made from a substrate, such as a semiconductor wafer, which is etched to define base  46 , gimbal  42 , transmit and receive mirrors  34 ,  38 , and the associated connecting elements in a MEMS process. (A reflective coating is deposited on the mirrors as a part of the process.) 
     The diameter of receive mirror  38  in the pictured implementation is twice that of transmit mirror  34 , meaning that the ratio of areas is 4:1. Typically, for the sake of optical efficiency, the area of the receive mirror is at least twice that of the transmit mirror. Although mirrors  34  and  38  in the pictured example are round, other mirror shapes (maintaining the desired ratio of respective mirror areas) may be used in alternative embodiments. Typically, for portable applications, the respective areas of mirrors  34  and  38  are in the range of 5 to 50 mm 2 , and the overall area of scanner  22  in the X-Y plane is on the order of 1 cm 2 . Alternatively, larger or even smaller scanners of this sort may be produced, depending on application requirements. 
     As noted earlier, mirrors  34  and  38  are driven by stator  24  to rotate at frequencies near their respective resonant frequencies of oscillation about hinges  84  and  86 . To compensate for the different dimensions and masses of transmit and receive mirrors  34  and  38 , respective hinges  84  and  86  may typically be made with different, respective degrees of stiffness, which are chosen so that the respective resonant frequencies of the two mirrors differ by no more than a predefined tolerance, for example 10%. For this purpose, as illustrated in  FIG. 5 , hinge  84  is typically considerably thinner, and therefore less stiff, than hinge  86 . 
     Although the actual resonant frequencies may be slightly different, the rotations of transmit mirror  34  and receive mirror  38  are coupled together by coupling belts  88 , etched from the silicon substrate, in order to synchronize their oscillations. Furthermore, although the drive signals applied to coils  56  of the respective stators of mirrors  34  and  38  have the same frequency, circuits  26  typically vary the relative amplitudes and possibly the phases of these drive signals in order to overcome the differences in mechanical resistance between the mirrors and provide the desired ratio of rotational amplitudes between the mirror angles. Typically, the coupling provided by belts  88 , along with appropriate adjustment of the drive signals provided by circuits  26 , synchronizes the rotations of the transmit and receive mirrors in frequency and phase, while the respective amplitudes of rotation may vary, as described further hereinbelow. Feedback from sensors in device may be applied, as noted above, in adjusting the drive signal parameters. 
     In contrast to the high-frequency, resonant rotations of mirrors  34  and  38 , stator  24  typically drives gimbal  42  to rotate relative to base  46  in a non-resonant mode, typically at a frequency substantially lower than the resonant frequency of the mirrors. Rotation of gimbal  42  on hinges  80  causes both of mirrors  34  and  38  to rotate by the same angle about the X-axis (along with transmitter  28 , collection lens  36 , and detector  40 ). The fast rotation of mirrors  34  and  38  about the Y-axis and the slower rotation of gimbal  42  about the X-axis may be coordinated so as to define a raster scan of the transmitted and received beams over an area of interest. Alternatively, the rotations of mirrors  34 ,  38  and gimbal  42  may be controlled to generate scan patterns of other sorts. 
     Belts  88  are stressed by the torsional forces exerted at the ends of hinges  84  and  86 , and thus couple the rotations of mirrors  34  and  38  by means of mechanical forces exerted by the belts on the hinges. Mechanical links of this sort within an array of mirrors are described, for example, in the above-mentioned U.S. Patent Application Publication 2014/0153001. Alternatively or additionally, the mirrors may be coupled together by means of a link exerted by electromagnetic force, which may operate without mechanical contact between the mirrors, as described, for example, in U.S. Provisional Patent Application 61/929,071, filed Jan. 19, 2014, whose disclosure is incorporated herein by reference. 
     Typically, a weak coupling force is sufficient to engender the desired synchronization between mirrors  34  and  38 , particularly when the mirrors are driven to scan at or near their resonant frequencies of rotation. The strength of mechanical coupling can be expressed in terms of a coupling strength coefficient γ, whose derivation is explained in detail in an Appendix below. For the mirror array shown in  FIG. 5 , for example, the inventors have estimated that γ= 1/36 will give good results in synchronizing mirrors  34  and  38 , at a resonant frequency of about 4670 Hz. 
       FIGS. 6A and 6B  are schematic side views of mirrors  34 ,  38  and associated optical components, shown in two different, orthogonal planes, in accordance with an embodiment of the present invention. The rotations of mirrors  34  and  38  are controlled and synchronized, as noted above, so that at each point in the scan, detector  40  receives light from the same area of the scene that is illuminated at that point by transmitter  28 .  FIG. 6A  shows the transmitted and received beam paths in the X-Z plane, in which each of mirrors  34  and  38  rotates about its own axis, while  FIG. 6B  shows the beam paths in the Y-Z plane, in which the mirrors and optical components rotate together on gimbal  42 . Although  FIG. 6B  shows the mirrors at the null point of rotation of the gimbal (with the gimbal parallel to the X-Y plane), the relations among the beam angles shown in  FIG. 6B  will remain the same regardless of gimbal rotation. 
     As illustrated particularly in  FIGS. 1 and 2 , turning mirror  32  and detector  40  are both offset in the X-Y plane relative to the centers of respective mirrors  34  and  38 , with an offset angle of a relative to the X-axis. Because the components of the transmitter and receiver rotate together with mirrors  34  and  38  about the X-axis, there is no need for relative rotational adjustment (and no means for providing such adjustment in scanner  22 ) between the transmit and receive beam paths about the X-axis. Thus, as shown in  FIG. 6B , the transmitted and received beams remain in mutual alignment throughout the scan, at the offset angle α relative to the normal of the respective mirrors, as shown in  FIG. 6B . 
     On the other hand, as shown in  FIG. 6A , the optical components of the transmitter and receiver do not rotate with mirrors  34  and  38  about the Y-axis. In this case, the relative distances of mirror  32  and detector  40  from the respective mirrors and the optical power of lens  36  create a relative magnification between the transmit and receive paths. In the present example, the magnification is taken to be M=2, although larger or smaller magnification factors may alternatively be used, depending on system design requirements and constraints. As a result of this magnification, when the transmit mirror  34  has rotated, as shown in  FIG. 6A , so that the transmitted beam is incident on the transmit at an angle φ (measured relative to the normal to the mirror, in the X-Z plane), it is necessary that receive mirror  38  rotate by an angle 2φ in order for detector  40  to capture the reflected light from the point in the scene on which the transmitted beam is incident. In other words, although mirrors  34  and  38  rotate with equal frequencies and phases, the amplitude of rotation of mirror is increased relative to that of mirror  34  by the magnification factor M=2. 
     In formal terms, the vector {right arrow over (u)} below gives the direction of the beam incident on transmit mirror  34  in the coordinate system of  FIGS. 6A and 6B : 
     
       
         
           
             
               
                 
                   
                     u 
                     → 
                   
                   = 
                   
                     ( 
                     
                       
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               α 
                               ) 
                             
                           
                         
                       
                       
                         
                           0 
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     Vector {right arrow over (n)} below gives the direction of the transmitted beam after reflection from transmit mirror  34  in this coordinate system, using the angles α and φ as shown in the figures: 
                     n   →     =     (             sin   ⁡     (     2   ⁢   φ     )       ⁢     cos   ⁡     (   α   )                   sin   ⁡     (   α   )                   cos   ⁡     (     2   ⁢   φ     )       ⁢     cos   ⁡     (   α   )               )             (   2   )               
The projection of the transmitted beam in the X-Z plane can be written as:
 
                       n   →     XZ     =       cos   ⁡     (   α   )       ⁢     (           sin   ⁡     (     2   ⁢   φ     )                 cos   ⁡     (     2   ⁢   φ     )             )               (   3   )               
This expression is equivalent to rotation by angle 2φ of the beam incident normally on the mirror plane.
 
     The distance to the scene is assumed to be much larger that the distance between the centers of mirrors  34  and  38 . The beam reflected from the scene thus returns at the same angle 2φ in the X-Z plane relative to the optical axis of the receiver optics (as defined by lens  36 , mirror  38  and detector  40 ). This beam is collected by lens  36 , and after reflecting from receive mirror  38  is focused onto detector  40  beneath the lens. The rotation angle of mirror  38  for this purpose is thus 2φ, twice the rotation (φ) for transmit mirror  34 . 
     In the Y-Z plane, as shown in  FIG. 6B , the beam is reflected from mirror  32  and is then incident on transmit mirror  34  at an angle α relative to the Z-axis. The beam is then reflected from mirror  34  by the same angle α, toward the target point in the scene. The reflected beam is likewise incident at angle α in the Y-Z plane relative to the optical axis of the collecting optical system. Lens  36  collects the beam, which then reflects from mirror  38  and is focused onto detector  40 , which is positioned slightly off-center in the Y-Z plane. More precisely, the distance between detector  40  and the optical axis is f sin(α), wherein f is the focal length of lens  36 . For example, if f=2 mm, and α=10 deg, the off-axis distance in detector position along the Y-axis is about 350 micrometers. 
     Although the figures described above show a particular optical design and layout of the components of scanning device  20 , the principles of the present invention may be applied in scanning devices of other designs. For example, scanner  22  may comprise mirrors and gimbals of different shapes, sizes, orientations and spacing from those shown in the figures, and may further comprise two or more parallel receive mirrors. As another example, transmitter may be positioned to transmit light to mirror  34  directly, without intervening reflector  32 . Alternative designs based on the principles set forth above will be apparent to those skilled in the art and are also considered to be within the scope of the present invention. 
     It will thus be appreciated that the embodiments described above are cited by way of example, and that the present invention is not limited to what has been particularly shown and described hereinabove. Rather, the scope of the present invention includes both combinations and subcombinations of the various features described hereinabove, as well as variations and modifications thereof which would occur to persons skilled in the art upon reading the foregoing description and which are not disclosed in the prior art. 
     APPENDIX—SYSTEM OF TWO COUPLED MICRO-MIRRORS 
     In terms of the mathematical framework for a system of two mechanically-coupled micro-mirrors  34  and  38 , we consider a model of two coupled oscillators. In this model, the two oscillators have two separate sources for actuation of vibrations that can be tuned in frequency, amplitude and phase. We will show that if the actuation of the system is done at the frequency corresponding to the symmetric mode of the two oscillators, there exists a solution in parametric space that will cause in-phase oscillation of the two micro-mirrors with any desired amplitude ratio between them. 
     1. Mathematical Model of Coupled Harmonic Oscillators 
     1.1 Equations of Motion 
     The equations of motion of two coupled oscillators are 
     
       
         
           
             
               
                 
                   
                     
                       
                         x 
                         ¨ 
                       
                       1 
                     
                     + 
                     
                       
                         
                           ω 
                           10 
                         
                         
                           Q 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           x 
                           . 
                         
                         1 
                       
                     
                     + 
                     
                       
                         ω 
                         10 
                         2 
                       
                       ⁢ 
                       
                         x 
                         1 
                       
                     
                   
                   = 
                   
                     
                       F 
                       1 
                     
                     + 
                     
                       
                         λ 
                         12 
                       
                       ⁢ 
                       
                         x 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     1 
                     ⁢ 
                     a 
                   
                   ) 
                 
               
             
           
         
       
     
                         x   ¨     2     +         ω   20       Q   2       ⁢       x   .     2       +       ω   20   2     ⁢     x   2         =       F   2     +       λ   21     ⁢     x   1                 (     1   ⁢   b     )               
wherein ω 10 , ω 20  are eigenfrequencies of the first and second oscillators respectively, and Q 1 , Q 2  and F 1  and F 2  are quality factors and driving forces for the first and second oscillator respectively. The dimensionless quantities λ 12  and λ 21  are coupling constants between the oscillators: λ 12  specifies the strength of the influence of the second oscillator on the first, and vice versa for λ 21 .
 
     The meaning of γ 12  and γ 21  can be illustrated by the following example. Suppose the strengths F 1 =0 and F 2 ≠0 are constant. Then the stationary solution of Eq. (1a) provides: 
                     λ   12     =       ω   10   2     ⁢       x   10       x   20                 (     2   ⁢   a     )               
and vice versa, for the case F 1 ≠0 and F 2 =0, in which the solution of Eq. (1b) is
 
                     λ   21     =       ω   20   2     ⁢       x   20   ′       x   10   ′                 (     2   ⁢   b     )               
Here the tags on x′ 10  and x′ 20  distinguish them from the solution with x 10  and x 20  of Eq. (2a) obtained for different initial conditions.
 
     Thus, the meaning of the coupling constants is the following: γ 12  is the ratio between the displacements x 10  and x 20  when a constant force is applied to the second oscillator, and γ 12  is the ratio between the displacements of the second and first oscillators when a constant force is applied to the first oscillator. 
     The mechanical principle of reciprocity requires that
 
λ 12 =λ 21   (3)
 
This principle follows from the Lagrangian of a system of two coupled oscillators:
 
                   L   =         1   2     ⁢       x   .     1   2       +       1   2     ⁢       x   .     1   2       -       1   2     ⁢     ω   10   2     ⁢     x   1   2       -       1   2     ⁢     ω   20   2     ⁢     x   2   2       -       1   2     ⁢     λ   12   ′     ⁢     x   1     ⁢     x   2       -       1   2     ⁢     λ   21   ′     ⁢     x   2     ⁢     x   1                 (   4   )               
From the Euler-Lagrange equation, it follows that:
 
     
       
         
           
             
               λ 
               12 
             
             = 
             
               
                 
                   1 
                   2 
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       λ 
                       12 
                       ′ 
                     
                     + 
                     
                       λ 
                       21 
                       ′ 
                     
                   
                   ) 
                 
               
               = 
               
                 λ 
                 21 
               
             
           
         
       
     
     The values ω 10 , ω 20  are eigenfrequencies of the two oscillators and not of the coupled system of the two oscillators. These eigenfrequencies ω 10  and ω 20  are obtained from solutions of the two harmonic oscillators below: 
     
       
         
           
             
               
                 
                   
                     
                       
                         x 
                         ¨ 
                       
                       1 
                     
                     + 
                     
                       
                         
                           ω 
                           10 
                         
                         
                           Q 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           x 
                           . 
                         
                         1 
                       
                     
                     + 
                     
                       
                         ω 
                         10 
                         2 
                       
                       ⁢ 
                       
                         x 
                         1 
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   
                     5 
                     ⁢ 
                     a 
                   
                   ) 
                 
               
             
           
         
       
     
     
       
         
           
             
               
                 
                   
                     
                       
                         x 
                         ¨ 
                       
                       2 
                     
                     + 
                     
                       
                         
                           ω 
                           20 
                         
                         
                           Q 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           x 
                           . 
                         
                         2 
                       
                     
                     + 
                     
                       
                         ω 
                         20 
                         2 
                       
                       ⁢ 
                       
                         x 
                         2 
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   
                     5 
                     ⁢ 
                     b 
                   
                   ) 
                 
               
             
           
         
       
     
     We now consider harmonic driving forces at the same driving frequency ω applied to the two oscillators. We can write equations for F 1  and F 2  as follows:
 
 F   1 =ω 0   2   A   1   e   iωt   (6a)
 
 F   2 =ω 0   2   A   2   e   iωt   (6b)
 
The normalized force amplitudes A 1  and A 2  are complex numbers that have units of [x 1,2 ], and ω 0   2  is defined as
 
     
       
         
           
             
               
                 
                   
                     ω 
                     0 
                     2 
                   
                   = 
                   
                     
                       
                         ω 
                         10 
                         2 
                       
                       + 
                       
                         ω 
                         20 
                         2 
                       
                     
                     2 
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     Substituting these expressions into Eqs. (1), we obtain the following set of equations for two coupled oscillators: 
     
       
         
           
             
               
                 
                   
                     
                       
                         x 
                         ¨ 
                       
                       1 
                     
                     + 
                     
                       
                         
                           ω 
                           10 
                         
                         
                           Q 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           x 
                           . 
                         
                         1 
                       
                     
                     + 
                     
                       
                         ω 
                         10 
                         2 
                       
                       ⁢ 
                       
                         x 
                         1 
                       
                     
                   
                   = 
                   
                     
                       
                         ω 
                         0 
                         2 
                       
                       ⁢ 
                       
                         A 
                         1 
                       
                       ⁢ 
                       
                         e 
                         
                           i 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ω 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           t 
                         
                       
                     
                     + 
                     
                       
                         ω 
                         0 
                         2 
                       
                       ⁢ 
                       γ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         x 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     8 
                     ⁢ 
                     a 
                   
                   ) 
                 
               
             
           
         
       
     
                         x   ¨     2     +         ω   20       Q   2       ⁢       x   .     2       +       ω   20   2     ⁢     x   2         =         ω   0   2     ⁢     A   2     ⁢     e     i   ⁢           ⁢   ω   ⁢           ⁢   t         +       ω   0   2     ⁢   γ   ⁢           ⁢     x   1                 (     8   ⁢   b     )               
wherein the normalized coupling constant γ is defined by the equation:
 
ω 0   2 γ=λ 12 =λ 21   (9)
 
1.2 Solution of the Equations for Coupled Harmonic Oscillators
 
     We seek a solution of the set of Eqs. (8) in the form:
 
 x   1   =x   10   e   iωt   (10a)
 
 x   2   =x   20   e   iωt   (10b)
 
wherein x 10 , x 20  are complex amplitudes of x 1 , x 2  respectively. Substituting Eqs. (10) into Eqs. (8) and dividing by the common factor e iωt  we get:
 
 x   10   D   1 (ω)−ω 0   2   γx   20 =ω 0   2   A   1   (11a)
 
 x   20   D   2 (ω)−ω 0   2   γx   10 =ω 0   2   A   2   (11b)
 
wherein
 
     
       
         
           
             
               
                 
                   
                     
                       D 
                       1 
                     
                     ⁡ 
                     
                       ( 
                       ω 
                       ) 
                     
                   
                   = 
                   
                     
                       ω 
                       10 
                       2 
                     
                     - 
                     
                       ω 
                       2 
                     
                     + 
                     
                       i 
                       ⁢ 
                       
                         
                           
                               
                           
                           ⁢ 
                           
                             ωω 
                             10 
                           
                         
                         
                           Q 
                           1 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     12 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     a 
                   
                   ) 
                 
               
             
           
         
       
     
     
       
         
           
             
               
                 
                   
                     
                       D 
                       2 
                     
                     ⁡ 
                     
                       ( 
                       ω 
                       ) 
                     
                   
                   = 
                   
                     
                       ω 
                       20 
                       2 
                     
                     - 
                     
                       ω 
                       2 
                     
                     + 
                     
                       i 
                       ⁢ 
                       
                         
                           
                               
                           
                           ⁢ 
                           
                             ωω 
                             20 
                           
                         
                         
                           Q 
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     12 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     b 
                   
                   ) 
                 
               
             
           
         
       
     
     The set of Eqs. (8) can be written as a single matrix equation: 
                       (             D   1     ⁡     (   ω   )               -     ω   0   2       ⁢   γ                 -     ω   0   2       ⁢   γ             D   2     ⁡     (   ω   )             )     ⁢     (           x   10               x   20           )       =     (             ω   0   2     ⁢     A   1                   ω   0   2     ⁢     A   2             )             (   13   )               
or
 
 Ux=A   (14)
 
wherein
 
     
       
         
           
             
               
                 
                   
                     U 
                     = 
                     
                       ( 
                       
                         
                           
                             
                               
                                 D 
                                 1 
                               
                               ⁡ 
                               
                                 ( 
                                 ω 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   ω 
                                   0 
                                   2 
                                 
                               
                               ⁢ 
                               γ 
                             
                           
                         
                         
                           
                             
                               
                                 - 
                                 
                                   ω 
                                   0 
                                   2 
                                 
                               
                               ⁢ 
                               γ 
                             
                           
                           
                             
                               
                                 D 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 ω 
                                 ) 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                   , 
                   
                     x 
                     = 
                     
                       ( 
                       
                         
                           
                             
                               x 
                               10 
                             
                           
                         
                         
                           
                             
                               x 
                               20 
                             
                           
                         
                       
                       ) 
                     
                   
                   , 
                   
                     A 
                     = 
                     
                       
                         ω 
                         0 
                         2 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               
                                 A 
                                 1 
                               
                             
                           
                           
                             
                               
                                 A 
                                 2 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     The solution of Eq. (14) is
 
 x=U   −1   A   (16)
 
or equivalently,
 
                     (           x   10               x   20           )     =         ω   0   2       det   ⁡     (   U   )         ⁢     (             D   2     ⁡     (   ω   )               ω   0   2     ⁢   γ                 ω   10   2     ⁢     γ   12     ⁢     ω   0   2     ⁢   γ             D   1     ⁡     (   ω   )             )     ⁢     (           A   1               A   2           )               (   17   )               
wherein
 
det( U )= D   1 (ω) D   2 (ω)−ω 0   4 γ 2   (18)
 
The solution may be written explicitly as:
 
     
       
         
           
             
               
                 
                   
                     x 
                     10 
                   
                   = 
                   
                     
                       
                         
                           ω 
                           0 
                           2 
                         
                         ⁢ 
                         
                           
                             D 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             ω 
                             ) 
                           
                         
                         ⁢ 
                         
                           A 
                           1 
                         
                       
                       + 
                       
                         
                           ω 
                           0 
                           4 
                         
                         ⁢ 
                         γ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           A 
                           2 
                         
                       
                     
                     
                       
                         
                           
                             D 
                             1 
                           
                           ⁡ 
                           
                             ( 
                             ω 
                             ) 
                           
                         
                         ⁢ 
                         
                           
                             D 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             ω 
                             ) 
                           
                         
                       
                       - 
                       
                         
                           ω 
                           0 
                           4 
                         
                         ⁢ 
                         
                           γ 
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     19 
                     ⁢ 
                     a 
                   
                   ) 
                 
               
             
           
         
       
     
                     x   20     =           ω   0   2     ⁢       D   1     ⁡     (   ω   )       ⁢     A   2       +       ω   0   4     ⁢   γ   ⁢           ⁢     A   1                 D   1     ⁡     (   ω   )       ⁢       D   2     ⁡     (   ω   )         -       ω   0   4     ⁢     γ   2                   (     19   ⁢   b     )               
The above equations solve the system of coupled harmonic oscillators expressed by Eqs. (8).
 
     From Eq. (19a) and Eq. (19b), it follows that the ratio of the amplitudes of the two oscillators is 
                   r   =         x   10       x   20       =             D   2     ⁡     (   ω   )       ⁢     A   1       +       ω   0   2     ⁢   γ   ⁢           ⁢     A   2                 D   1     ⁡     (   ω   )       ⁢     A   2       +       ω   0   2     ⁢   γ   ⁢           ⁢     A   1                     (   20   )               
Thus, at any given ω there exists a ratio A 1 /A 2  that provides the desired ratio r of amplitudes x 10 /x 20  at ω. From Eq. (20), it follows that A 1 /A 2  is
 
     
       
         
           
             
               
                 
                   
                     
                       A 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       1 
                     
                     
                       A 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       2 
                     
                   
                   = 
                   
                     
                       
                         
                           ω 
                           0 
                           2 
                         
                         ⁢ 
                         r 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         γ 
                       
                       - 
                       
                         
                           D 
                           1 
                         
                         ⁡ 
                         
                           ( 
                           ω 
                           ) 
                         
                       
                     
                     
                       
                         
                           ω 
                           0 
                           2 
                         
                         ⁢ 
                         γ 
                       
                       - 
                       
                         
                           rD 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           ω 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     By way of example, for the particular case of two in-phase oscillators at some maximum frequency ω 1  oscillating with the same amplitudes, r=1, and we get from Eq. (21) the ratio of driving force amplitudes that provides this synchronous movement of the oscillators: 
                       A   ⁢           ⁢   1       A   ⁢           ⁢   2       =           ω   0   2     ⁢   γ     -     D   1             ω   0   2     ⁢   γ     -     D   2                 (   22   )               
wherein D 1  and D 2  are computed using Eqs. (12) at frequency ω=ω 1 .

Metadata:
Filing Date: 20141126
Publication Date: 20171205
Grant Date: 20171205
Priority Date: 20141126
Inventors: SHPUNT ALEXANDER
AXELROD NOEL
ERLICH RAVIV
GERSON YUVAL
Assignee: APPLE INC
CPC Classifications: [{"code": "G01S17/89", "inventive": true, "first": false, "tree": "[]"}, {"code": "G01S17/42", "inventive": true, "first": false, "tree": "[]"}, {"code": "G02B26/085", "inventive": true, "first": false, "tree": "[]"}, {"code": "G02B26/105", "inventive": true, "first": true, "tree": "[]"}, {"code": "G01S7/4817", "inventive": true, "first": false, "tree": "[]"}, {"code": "G02B26/0833", "inventive": true, "first": false, "tree": "[]"}, {"code": "G02B26/105", "inventive": true, "first": false, "tree": "[]"}, {"code": "G01S17/89", "inventive": true, "first": false, "tree": "[]"}, {"code": "G02B26/0833", "inventive": true, "first": false, "tree": "[]"}, {"code": "G02B26/105", "inventive": true, "first": true, "tree": "[]"}, {"code": "G01S17/06", "inventive": true, "first": false, "tree": "[]"}, {"code": "G01S17/89", "inventive": true, "first": true, "tree": "[]"}, {"code": "G01S17/42", "inventive": true, "first": false, "tree": "[]"}, {"code": "G01S7/4817", "inventive": true, "first": false, "tree": "[]"}]
Family ID: 60452198