PATENT DOCUMENT

Publication Number: US-10612910-B2
Application Number: US-201916283661-A
Country: US
Kind Code: B2

Title: Motion sensing by monitoring intensity of light redirected by an intensity pattern

Abstract:
Systems and techniques are described for measuring displacement of a moving mass by combining (i) information obtained from scanning, using a beam of light, an intensity pattern disposed on a surface of the mass, with (ii) information obtained when a coil interacts with a magnet attached to the moving mass.

Claims:
What is claimed is: 
     
       1. A displacement measuring system comprising:
 (i) a back electromotive force (bEMF) sensing system to acquire a first displacement signal that relates to a time dependence of a displacement of a mass, wherein the displacement is relative to a datum of the displacement measuring system; 
 (ii) an optical sensing system comprising 
 an intensity pattern that is coupled with the mass and comprises two or more tiles separated from each other by corresponding one or more tile borders, wherein the tile borders are at known locations relative to each other; 
 a light source that is at rest relative to the datum to illuminate the intensity pattern with a light beam, wherein multiple tile border crossings occur while the first displacement signal is being acquired, wherein a tile border crossing is said to occur when a tile border of the intensity pattern crosses through the light beam; and 
 a photodetector that is at rest relative to the datum to acquire an intensity signal corresponding to intensity of the light beam redirected to the photodetector from the intensity pattern, wherein the intensity signal is indicative of the tile border crossings; and 
 (iii) a processor to 
 spatially resolve the tile border crossings indicated by the intensity signal, at least in part, based on whether the first displacement signal increases or decreases at a time when a tile border crossing has occurred; and 
 determine the displacement of the mass based on the spatially resolved tile border crossings. 
 
     
     
       2. The system of  claim 1 , wherein the processor to
 determine a second displacement signal using the spatially resolved tile border crossings; and 
 determine the displacement of the mass by combining the first displacement signal and the second displacement signal. 
 
     
     
       3. The system of  claim 2 , wherein the processor to
 determine a scale factor equal to a ratio of a change in the second displacement signal over a predetermined time interval and a change in the first displacement signal over the predetermined time interval, 
 differentiate the first displacement signal, and 
 scale the differentiated first displacement signal based on the scale factor prior to the combining of the first displacement signal and the second displacement signal. 
 
     
     
       4. The system of  claim 3 , wherein the processor updates the scale factor when the first displacement signal over the predetermined time interval exceeds a threshold change. 
     
     
       5. The system of  claim 2 , wherein
 the bEMF sensing system to sample the first displacement signal using a first sampling frequency, and 
 the optical sensing system to sample the intensity signal using a second sampling frequency smaller than the first sampling frequency, thereby samples of the second displacement signal have the second sampling frequency. 
 
     
     
       6. The system of  claim 5 , wherein, to perform the combining of the first displacement signal and the second displacement signal, the processor to insert corresponding samples of the scaled differentiated first displacement signal between samples of the second displacement signal. 
     
     
       7. The system of  claim 1 , wherein
 each tile has a size larger than a beam spot formed by the light beam that illuminates the intensity pattern, and 
 each tile is configured to redirect to the photodetector light having an intensity different from an intensity of light redirected to the photodetector by any of its adjacent tiles. 
 
     
     
       8. The system of  claim 7 , wherein the intensity pattern is a binary intensity pattern in which each tile has only two adjacent tiles configured to redirect to the photodetector light having the same intensity. 
     
     
       9. The system of  claim 7 , wherein each tile is a hexagonal tile configured to redirect to the photodetector light having an intensity level that is one of (i) a minimum intensity level, (ii) a maximum intensity level, (iii) a first intermediate intensity level between the minimum intensity level and the maximum intensity level, and (iv) a second intermediate intensity level between the first intermediate intensity level and the maximum intensity level. 
     
     
       10. The system of  claim 9 , wherein
 the first displacement signal acquired by the bEMF sensing system represents the time dependence of a component of the displacement of the mass along a first direction, and 
 the processor to 
 (i) spatially resolve first tile border crossings indicated by the intensity signal based on whether the first displacement signal increases or decreases at a time when a first tile border crossing has occurred along the first direction, and (ii) determine the component of the displacement of the mass along the first direction based on the spatially resolved first tile border crossings; and 
 (iii) spatially resolve second tile border crossings indicated by the intensity signal based on changes between a pair of the minimum intensity level, the maximum intensity level, the first intermediate intensity level, and the second intermediate intensity level of redirected light that is captured by the photodetector when a second tile border crossing has occurred along a second direction orthogonal to the first direction, and 
 (iv) determine a component of the displacement of the mass along the second direction based on the spatially resolved second tile border crossings. 
 
     
     
       11. The system of  claim 9 , wherein
 the light source to concurrently illuminate three tiles of the intensity pattern that are adjacent to each other, one of the three adjacent tiles illuminated with the light beam, and the other two of the three adjacent tiles respectively illuminated with two reference light beams, the two reference light beams spaced apart from the light beam by a separation about equal to a separation between adjacent tiles, and 
 the light beam and the reference light beams to illuminate the three adjacent tiles with substantially equal intensities. 
 
     
     
       12. The system of  claim 11 , wherein the light source to concurrently illuminate the three adjacent tiles in a time multiplexed manner. 
     
     
       13. The system of  claim 11 , wherein
 the photodetector to further acquire reference signals corresponding to intensities of respective reference light beams redirected to the photodetector from the intensity pattern, and 
 the optical sensing system to sample the intensity signal using a second sampling frequency and the reference signals using a third sampling frequency smaller than the second sampling frequency. 
 
     
     
       14. The system of  claim 11 , wherein
 the processor to compare measured values and expected values of differences between intensity of the light beam redirected to the photodetector from one of the three adjacent tiles and respective ones of the other two of the three adjacent tiles respectively illuminated with two reference light beams, and 
 the light source to adjust the intensity of the light beam based on the compared differences. 
 
     
     
       15. The system of  claim 1 , wherein the photodetector comprises a photodiode. 
     
     
       16. The system of  claim 1 , wherein the photodetector comprises a threshold module to apply one or more threshold values to each intensity value of the light beam redirected to, and measured by, the photodetector to issue a corresponding expected value of the intensity value. 
     
     
       17. The system of  claim 16 , wherein the photodetector comprises a filter to adaptively determine the one or more threshold values. 
     
     
       18. The system of  claim 16 , wherein the one or more threshold values are predetermined. 
     
     
       19. The system of  claim 1 , wherein the light source comprises a vertical cavity surface emitting laser (VCSEL) to emit the light beam. 
     
     
       20. The system of  claim 1 , wherein the light source comprises
 a light emitting diode (LED) to emit probe light; and 
 beam-shaping optics to form the light beam. 
 
     
     
       21. The system of  claim 1 , wherein the intensity pattern is reflective to the light beam, and disposed on a surface of the mass. 
     
     
       22. The system of  claim 1 , wherein
 the intensity pattern is transparent to the light beam, and 
 the optical sensing system comprises an optical structure having a first surface and a second, opposing surface, the intensity pattern is disposed on the first surface of the optical structure, and the optical structure is attached to a surface of the mass adjacent the second surface of the optical structure. 
 
     
     
       23. The system of  claim 22 , wherein
 the optical structure comprises an array of micro-mirrors disposed between the first and second surfaces of the optical structure, and 
 the micro-mirrors of the array are oriented to redirect to the photodetector the light beam that impinges on the array of micro-mirrors after transmission through the intensity pattern. 
 
     
     
       24. The system of  claim 22 , wherein the optical structure comprises solid material that is transparent to the light beam. 
     
     
       25. The system of  claim 24 , wherein
 the optical sensing system comprises a diffusive film sandwiched between the second surface of the optical structure and the surface of the mass, and 
 the diffusive film is configured to redirect to the photodetector the light beam that impinges on the diffusive film after transmission through the intensity pattern. 
 
     
     
       26. The system of  claim 24 , wherein
 the second surface of the optical structure is spaced apart from the surface of the mass by an air gap, and 
 the second surface of the optical structure comprises facets arranged to reflect, via total internal reflection (TIR), to the photodetector, the light beam that impinges on the facets after transmission through the intensity pattern. 
 
     
     
       27. The system of  claim 24 , wherein
 the optical sensing system comprises a diffusive material sandwiched between the second surface of the optical structure and the surface of the mass, and 
 the second surface of the optical structure comprises facets arranged to diffusely reflect, to the photodetector, the light beam that impinges on the facets after transmission through the intensity pattern. 
 
     
     
       28. A haptic engine comprising the mass and the displacement measuring system recited in  claim 1 . 
     
     
       29. A computing device comprising the haptic engine of  claim 28 .

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a divisional application of and claims priority to U.S. patent application Ser. No. 15/674,260, filed Aug. 10, 2017, which claims priority to U.S. Provisional Application Ser. No. 62/396,010, filed Sep. 16, 2016, and to U.S. Provisional Application Ser. No. 62/396,030, filed Sep. 16, 2016, and to U.S. Provisional Application Ser. No. 62/396,022, filed Sep. 16, 2016. The disclosure of all related applications is incorporated herein by reference in their entirety. 
    
    
     TECHNICAL FIELD 
     The present disclosure is generally related to motion sensing. For example, aspects of the present disclosure are related to measuring displacement of a mass by using an array of beams for scanning a binary intensity pattern disposed on a surface of the mass. As another example, aspects of the present disclosure are related to measuring displacement of a moving mass by combining (i) information obtained from scanning, using a beam of light, an intensity pattern disposed on a surface of the mass, with (ii) information obtained when a coil interacts with a magnet attached to the moving mass. As yet another example, aspects of the present disclosure are related to measuring displacement of a mass by illuminating an intensity pattern disposed on a surface of the mass with an array of beams and monitoring intensity of each of the beams that is redirected by the intensity pattern. 
     BACKGROUND 
     A haptic engine (also referred to as a vibration module) is a linear resonant actuator that determines one of acceleration, velocity and displacement of a moving mass. Either one of electrical sensing or magnetic sensing can be conventionally used for measuring displacements of the mass moving in the haptic engine. An example of electrical sensing, that is referred to as back electromotive force (bEMF) sensing, is based on sensing current-voltage of a coil that interacts with a magnet attached to the moving mass. For certain applications, accuracy of an absolute value of displacement measured by bEMF sensing may be insufficient because the coil&#39;s resistance changes with temperature. An example of magnetic sensing, that is referred to as Hall sensing, is based on sensing Hall voltages using Hall sensors that interact with a magnet attached to the moving mass. A displacement measuring system based on conventional Hall sensing can be expensive to calibrate as the conventional Hall sensing uses a look-up-table calibration to linearize displacement sensitivity. Additionally, conventional Hall sensing can have displacement sensitivity dead-zones when a Z-offset between the Hall sensors and the magnet exceeds a small Z-offset threshold. Conventional Hall sensing is also susceptible to interference from external magnetic fields. 
     Some of the above issues are remedied if measuring displacement of a mass moving in a haptic engine is performed using optical sensing. For instance, a conventional optical system can be used in conjunction with conditioning electronics, for measuring displacements of a mass in a vibration module. Such a conventional optical system includes a light emitting diode (LED) module, a striped optical pattern attached to the moving mass, and a photodetector array module. In such a vibration module, position of the striped optical pattern can be determined relative to a beam provided by an LED module. The photodetector array module is used to image the striped optical pattern illuminated by the beam. Each of (i) a bias used to power the LED module and (ii) an output signal of the photodetector array module is conditioned by a signal processing module that is configured based on a conventional transceiver architecture. As part of the conventional transceiver architecture, the signal processing module operates essentially in class-A mode and, hence, it includes numerous analog circuits, e.g., op-amps, digital POTs, and trans-inductance amplifiers (TIAs). As such, the signal processing module, operated in such a conventional transceiver architecture, is power hungry, and hence it can be costly to operate. Further note that, because a photodetector array is used to image the striped optical pattern illuminated by a single LED, the size of the vibration module in the Z-direction tends to grow unnecessarily in order to facilitate optical focusing between the LED, the striped optical pattern, and the photodetector array. 
     SUMMARY 
     In this disclosure, technologies are described for measuring displacement of a mass by using an array of beams for scanning a binary intensity pattern disposed on a surface of the mass. The array of beams can be provided by an array of vertical cavity surface emitting lasers (VCSELs), and the binary intensity pattern includes at least an edge formed between two portions of the surface of the mass that have different reflectivities. In this manner, a displacement of the mass can be measured based on changes of reflected light intensity caused by a relative movement between the binary intensity pattern and the VCSEL array. Accuracy of the disclosed displacement measuring technique is determined by the geometry of the binary intensity pattern and the geometry of the VCSEL array. 
     Further in this disclosure, technologies are described for measuring displacement of a moving mass by combining (i) information obtained from scanning, using a beam of light, an intensity pattern disposed on a surface of the mass, with (ii) information obtained when a coil interacts with a magnet attached to the moving mass. The beam of light can be provided by a VCSEL. The intensity pattern includes two or more tiles supported by the mass, the tiles being configured to spatially modulate intensity of light redirected to a photodetector, as individual ones of the tiles are sequentially illuminated by the beam of light when the intensity pattern is displaced along with the mass relative to the beam of light. An intensity signal issued by the photodetector relates to the spatially modulated intensity of the redirected light. Additionally, a bEMF signal issued by the coil concurrently with the intensity signal relates to a spatially dependent bEMF induced in the coil due to the motion relative to the coil of the magnet that is attached to the mass. Information from the intensity signal is combined with information from the bEMF signal to determine both absolute value and direction of a displacement of the intensity pattern relative to the beam of light. Note that the displacement determined in this manner can be resolved at a scale smaller than what the size of the tiles of the intensity pattern would allow on its own. 
     Furthermore in this disclosure, technologies are described for measuring displacement of a mass by (i) illuminating an intensity pattern disposed on a surface of the mass with an array of beams, and (ii) monitoring intensity of each of the beams that is redirected by the intensity pattern. For instance, an array of VCSELs can be time multiplexed and used to scan an intensity pattern with respective beams emitted by the VCSELs of the array. In this manner, beams that have spatially modulated intensity are redirected by the intensity pattern to a single photo-diode operated in charge integration mode. A photo-diode signal relates to changes in intensity of each of the redirected beams and can be used to decode a corresponding motion vector of the intensity pattern that is moving along with the mass. 
     A first aspect of the disclosure can be implemented as a displacement measuring system that includes a vertical cavity surface emitting laser (VCSEL) array including two or more vertical cavity surface emitting lasers (VCSELs) distributed along a first direction; and an optical pattern supported by a mass, the optical pattern having two portions that form an edge oriented along a second direction that crosses the first direction, the two portions of the optical pattern having different reflectivities. Here, the VCSEL array is spaced apart from the optical pattern and arranged such that, during operation of the displacement measuring system, the VCSEL array illuminates the optical pattern, across the edge, with VCSEL light emitted by the VCSEL array. The displacement measuring system also includes a photodetector spaced apart from the optical pattern and arranged such that, during operation of the displacement measuring system, the photodetector integrates the VCSEL light that is redirected by the optical pattern to the photodetector and issues a photodetector signal from the integrated VCSEL light; and processing electronics to receive the photodetector signal and determine a displacement of the mass along the first direction based on a change in the photodetector signal caused by motion of the mass along a direction of motion that crosses the edge. 
     Implementations can include one or more of the following features. In some implementations, the two or more VCSELs of the VCSEL array can be arranged in a row parallel to the first direction and separated by a pitch. In some implementations, the two or more VCSELs of the VCSEL array can be arranged in two rows parallel to the first direction, each of the rows includes two or more VCSELs that are separated by a pitch, and the two rows are staggered relative to each other along the first direction by half the pitch and separated from each other by a separation. In either of these implementations, the pitch can be about a size of a beam spot of the VCSEL light impinging on the optical pattern. In some cases, the separation can be about a size of a beam spot of the VCSEL light impinging on the optical pattern. 
     Further in some cases, the two portions of the optical pattern can form a second edge orthogonal to the first edge; the VCSEL array can be further arranged such that, during operation of the displacement measuring system, the VCSEL array illuminates the optical pattern across the second edge with the VCSEL light. In this manner, the processing electronics can determine a second displacement of the mass along a direction orthogonal to the first direction based on a second change in the photodetector signal caused by motion of the mass along a second direction of motion that crosses the second edge. For example, a first of the two portions of the optical pattern can be shaped as a rectangle and is surrounded by the second portion, such that the first edge is a first side of the rectangle, and the second edge is a second side of the rectangle; additionally, a length of the second edge can be longer than a length of the VCSEL array, and a length of the first edge is longer than the separation between the two rows of the VCSEL array. 
     In some implementations, the system can include a second VCSEL array including two or more VCSELs distributed along the first direction. Here, the two portions of the optical pattern form a second edge parallel to the first edge; the second VCSEL array is spaced apart from the mass and arranged such that, during operation of the displacement measuring system, the second VCSEL array illuminates the optical pattern, across the second edge but not across the first edge, with VCSEL light emitted by the second VCSEL array; the photodetector further issues, during operation of the displacement measuring system, a second photodetector signal based on the VCSEL light emitted by the second VCSEL array that is redirected by the optical pattern to the photodetector. In this manner, the processing electronics can receive the second photodetector signal and determine the displacement of the mass along the first direction further based on a change in the second photodetector signal caused by the motion of the mass along the direction of motion. 
     In some cases, the processing electronics can determine a first ratio signal as a division of the first photodetector signal to the second photodetector signal, and a second ratio signal as a division of the second photodetector signal to the first photodetector signal; and determine the displacement of the mass along the first direction based on respective changes, caused by the motion of the mass along the direction of motion, in the first ratio signal and the second ratio signal. For example, the processing electronics can determine the displacement of the mass along the first direction based on the smaller of the first ratio signal and the second ratio signal. 
     In other cases, the first VCSEL array and the second VCSEL array can be spaced apart from the mass by the same separation. In this manner, the processing electronics can determine a change of the separation between the VCSEL arrays and the mass, based on a change in a common value of the first photodetector signal and second photodetector signal. 
     In some other cases, the first VCSEL array and the second VCSEL array can illuminate the optical pattern, during operation of the displacement measuring system, in a multiplexed manner. Additionally, the photodetector can issue the respective first photodetector signal and second photodetector signal in the same multiplexed manner. For example, the VCSELs of the first VCSEL array and the second VCSEL array can emit light of the same wavelength; and the VCSELs of the first VCSEL array illuminate the optical pattern when the VCSELs of the second VCSEL array do not, and the VCSELs of the second VCSEL array illuminate the optical pattern when the VCSELs of the first VCSEL array do not, in a time multiplexed manner. As another example, the VCSELs of the first VCSEL array can emit light of a first wavelength and the VCSELs of the second VCSEL array emit light of a second wavelength different from the first wavelength; and the VCSELs of the first VCSEL array and the second VCSEL array illuminate the optical pattern concurrently, in a wavelength multiplexed manner. Here, the photodetector can include a first sensor to output a first sensor signal, and a second sensor to output a second sensor signal. Additionally, the processing electronics can concurrently issue (i) the first photodetector signal based on a first combination of the first sensor signal and the second sensor signal, the first combination being selective of the first wavelength, and (ii) the second photodetector signal based on a second combination of the first sensor signal and the second sensor signal, the second combination being selective of the second wavelength. 
     In yet other cases, the first VCSEL array and the second VCSEL array can have the same length; a first of the two portions of the optical pattern is shaped as a strip bounded by the first edge and the second edge inside a second of the two portions; and a width of the strip is wider than the common length of the VCSEL arrays. 
     In some implementations, one of the two portions of the optical pattern can be reflective and the other one of the two portions of the optical pattern is absorptive. In some implementations, at least one of the two portions of the optical pattern can be printed using ink that absorbs IR light. In some implementations, one of the two portions of the optical pattern can be coated with a multilayer reflection coating, and the other one of the two portions of the optical pattern is coated with a multilayer anti-reflection coating. In some implementations, one of the two portions of the optical pattern can have a reflectivity that is at least twice as large as a reflectivity of the other one of the two portions of the optical pattern. In some implementations, the VCSEL light emitted by the VCSELs can have wavelengths in a range from 700 nm to 1100 nm. In some implementations, the second direction can be orthogonal to the first direction. In some implementations, the direction of motion can be parallel to the first direction. 
     Another aspect of the disclosure can be implemented as a 2D-displacement measuring system that includes two pairs of light-emitting element (LEE) arrays, each LEE array having two rows of light-emitting elements (LEEs), the rows of LEEs being parallel to a first direction, and each LEE being configured to output collimated light; an optical pattern supported by a mass, the optical pattern having two portions that form a rectangular edge, the rectangular edge having two sides parallel to the first direction, the two portions of the optical pattern having different reflectivities, where each LEE array illuminates the optical pattern, across a corresponding corner of the rectangular edge, with the collimated light output by the LEE array; a photodetector to separately integrate the collimated light output by the respective LEE arrays redirected by the optical pattern to the photodetector, and issue two pairs of photodetector signals from the separately integrated light output by the respective LEE arrays; and processing electronics to receive the photodetector signals and determine displacements of the mass along, and orthogonal to, the first direction based on changes in the corresponding photodetector signals caused by motion of the mass. 
     Another aspect of the disclosure can be implemented as an angular displacement measuring system that includes three pairs of light-emitting element (LEE) arrays, each LEE array having two rows of light-emitting elements (LEEs), the rows of LEEs within each pair of LEE arrays being parallel to each other, and the rows of LEEs from different pairs of LEE arrays forming an angle of 120° with each other; an optical pattern supported around the perimeter of a wheel, the optical pattern having two portions that form three rectangular edges, each rectangular edge having two sides parallel to the rows of LEEs when the rectangular edge is proximate to a pair of LEE arrays, the two portions of the optical pattern having different reflectivities, where each LEE array of the proximate pair illuminates the optical pattern, across a corresponding corner of the rectangular edge, with collimated light output by the LEE array; a photodetector to issue three pairs of photodetector signals based on the collimated light output by the respective pair of LEE arrays and redirected by the optical pattern to the photodetector, each pair of photodetector signals including a periodic photodetector signal; and processing electronics to receive the photodetector signals and determine (i) an angular displacement of the wheel based on changes in the periodic photodetector signals caused by rotation of the wheel about a rotation axis of the wheel, and (ii) a lateral displacement of the wheel based on changes in one or more of the remaining photodetector signals caused by translation of the wheel along the rotation axis. 
     Implementations of each of the 2D-displacement measuring system and the angular displacement measuring system can include one or more of the following features. In some implementations, each LEE can include a VCSEL. In some implementations, each LEE can include a light source configured to emit un-collimated light; and collimating optics optically coupled with the light source to collimate the emitted light. In some implementations, the photodetector can include a CMOS sensor array or a CCD sensor array. 
     Another aspect of the disclosure can be implemented as a haptic engine that includes the mass and any of the foregoing displacement measuring systems. In some implementations, a computing device can include the haptic engine. 
     A second aspect of the disclosure can be implemented as a displacement measuring system that includes (i) a back electromotive force (bEMF) sensing system to acquire a first displacement signal that relates to a time dependence of a displacement of a mass, where the displacement is relative to a datum of the displacement measuring system; (ii) an optical sensing system including an intensity pattern that is coupled with the mass and comprises two or more tiles separated from each other by corresponding one or more tile borders, where the tile borders are at known locations relative to each other; a light source that is at rest relative to the datum to illuminate the intensity pattern with a light beam, where multiple tile border crossings occur while the first displacement signal is being acquired, and where a tile border crossing is said to occur when a tile border of the intensity pattern crosses through the light beam; and a photodetector that is at rest relative to the datum to acquire an intensity signal corresponding to intensity of the light beam redirected to the photodetector from the intensity pattern, where the intensity signal is indicative of the tile border crossings; and (iii) a processor to spatially resolve the tile border crossings indicated by the intensity signal, at least in part, based on whether the first displacement signal increases or decreases at a time when a tile border crossing has occurred; and determine the displacement of the mass based on the spatially resolved tile border crossings. 
     Implementations can include one or more of the following features. In some implementations, the processor can determine a second displacement signal using the spatially resolved tile border crossings; and determine the displacement of the mass by combining the first displacement signal and the second displacement signal. In some cases, the processor can determine a scale factor equal to a ratio of a change in the second displacement signal over a predetermined time interval and a change in the first displacement signal over the predetermined time interval; differentiate the first displacement signal; and scale the differentiated first displacement signal based on the scale factor prior to the combining of the first displacement signal and the second displacement signal. Additionally, the processor can update the scale factor when the first displacement signal over the predetermined time interval exceeds a threshold change. In some cases, the bEMF sensing system can sample the first displacement signal using a first sampling frequency; and the optical sensing system can sample the intensity signal using a second sampling frequency smaller than the first sampling frequency, thereby samples of the second displacement signal have the second sampling frequency. In the latter cases, to perform the combining of the first displacement signal and the second displacement signal, the processor can insert corresponding samples of the scaled differentiated first displacement signal between samples of the second displacement signal. 
     In some implementations, each tile can have a size larger than a beam spot formed by the light beam that illuminates the intensity pattern, and each tile is configured to redirect to the photodetector light having an intensity different from an intensity of light redirected to the photodetector by any of its adjacent tiles. In some cases, the intensity pattern can be a binary intensity pattern in which each tile has only two adjacent tiles configured to redirect to the photodetector light having the same intensity. In other cases, each tile can be a hexagonal tile configured to redirect to the photodetector light having an intensity level that is one of (i) a minimum intensity level, (ii) a maximum intensity level, (iii) a first intermediate intensity level between the minimum intensity level and the maximum intensity level, and (iv) a second intermediate intensity level between the first intermediate intensity level and the maximum intensity level. 
     In the latter cases, the first displacement signal acquired by the bEMF sensing system represents the time dependence of a component of the displacement of the mass along a first direction. As such, the processor can (i) spatially resolve first tile border crossings indicated by the intensity signal based on whether the first displacement signal increases or decreases at a time when a first tile border crossing has occurred along the first direction, and (ii) determine the component of the displacement of the mass along the first direction based on the spatially resolved first tile border crossings. Further, the processor can (iii) spatially resolve second tile border crossings indicated by the intensity signal based on changes between a pair of the minimum intensity level, the maximum intensity level, the first intermediate intensity level, and the second intermediate intensity level of redirected light that is captured by the photodetector when a second tile border crossing has occurred along a second direction orthogonal to the first direction, and (iv) determine a component of the displacement of the mass along the second direction based on the spatially resolved second tile border crossings. Also in the latter cases, the light source can concurrently illuminate three tiles of the intensity pattern that are adjacent to each other, one of the three adjacent tiles illuminated with the probe beam, and the other two of the three adjacent tiles respectively illuminated with two reference light beams, the two reference light beams spaced apart from the probe beam by a separation about equal to a separation between adjacent tiles; and the probe beam and the reference light beams can illuminate the three adjacent tiles with substantially equal intensities. Here, the light source can concurrently illuminate the three adjacent tiles in a time multiplexed manner. Further here, the photodetector can acquire reference signals corresponding to intensities of respective reference light beams redirected to the photodetector from the intensity pattern, and the optical sensing system can sample the reference signals using a third sampling frequency smaller than the second sampling frequency. Also here, the processor can compare measured values and expected values of differences between intensity of the probe light beam redirected to the photodetector from one of the three adjacent tiles and respective ones of the other two of the three adjacent tiles respectively illuminated with two reference light beams, and the light source can adjust the intensity of the probe light beam based on the compared differences. 
     In some implementations, the photodetector can include a threshold module to apply one or more threshold values to each intensity value of the light beam redirected to, and measured by, the photodetector to issue a corresponding expected value of the intensity value. In some cases, the photodetector can include a filter to adaptively determine the one or more threshold values. In other cases, the one or more threshold values can be predetermined. 
     In some implementations, the photodetector can include a photodiode. In some implementations, the light source can include a vertical cavity surface emitting laser (VCSEL) to emit the probe beam. In some implementations, the light source can include a light emitting diode (LED) to emit probe light; and beam-shaping optics to form the probe beam. In some implementations, the intensity pattern can be reflective to the probe light beam, and disposed on a surface of the mass. 
     In some implementations, the intensity pattern can be transparent to the probe light beam, and the optical sensing system includes an optical structure having a first surface and a second, opposing surface, the intensity pattern is disposed on the first surface of the optical structure, and the optical structure is attached to a surface of the mass adjacent the second surface of the optical structure. In some cases, the optical structure can include an array of micro-mirrors disposed between the first and second surfaces of the optical structure, and the micro-mirrors of the array are oriented to redirect to the photodetector the light beam that impinges on the array of micro-mirrors after transmission through the intensity pattern. In some cases, the optical structure can include solid material that is transparent to the probe light beam. In the latter cases, the optical sensing system can include a diffusive film sandwiched between the second surface of the optical structure and the surface of the mass, and the diffusive film is configured to redirect to the photodetector the light beam that impinges on the diffusive film after transmission through the intensity pattern. Also in the latter cases, the second surface of the optical structure is spaced apart from the surface of the mass by an air gap, and the second surface of the optical structure can include facets arranged to reflect, via total internal reflection (TIR), to the photodetector, the light beam that impinges on the facets after transmission through the intensity pattern. Also in the latter cases, the optical sensing system can include a diffusive material sandwiched between the second surface of the optical structure and the surface of the mass, and the second surface of the optical structure can include facets arranged to diffusely reflect, to the photodetector, the light beam that impinges on the facets after transmission through the intensity pattern. 
     Another aspect of the disclosure can be implemented as a haptic engine that includes the mass and a displacement measuring system summarized above. In some implementations, a computing device can include the haptic engine. 
     A third aspect of the disclosure can be implemented as a method that includes a displacement measuring system that includes a vertical cavity surface emitting laser (VCSEL) array including two or more (N TOT ) vertical cavity surface emitting lasers (VCSELs) distributed along a first direction; an intensity pattern that is coupled with a mass and includes two or more tiles separated from each other by corresponding one or more tile borders, where the tile borders are at known locations relative to each other along the first direction, and where the VCSEL array is spaced apart from the intensity pattern and arranged such that, during operation of the displacement measuring system, the (N TOT ) VCSELs of the array illuminate respective locations of the intensity pattern across at least one of the tile borders; a photodetector spaced apart from the intensity pattern and arranged such that, during operation of the displacement measuring system, the photodetector to (i) capture beams redirected to the photodetector from the (N TOT ) illuminated locations of the intensity pattern, where each tile of the intensity pattern is configured to redirect to the photodetector light having an intensity different from an intensity of light redirected to the photodetector by its adjacent tiles, and (ii) issue a set of (N TOT ) intensity values corresponding to the respective captured beams; and a processor to determine (i) positions of the illuminated locations of the intensity pattern based on relative differences between the intensity values of the issued set, and (ii) a displacement of the mass along the first direction based on one or more changes of the intensity values of the set caused by motion of the mass along a direction of motion across the at least one of the tile borders. 
     Implementations can include one or more of the following features. In some implementations, the photodetector is a single photodiode, and the VCSELs of the VCSEL array are configured to illuminate the intensity pattern in a time multiplexed manner. In some implementations, the photodetector issues instances of the set of intensity values with a sampling frequency (f S ), and the processor can obtain an intensity signal as a sequence of the instances of the set of intensity values, the sequence having a frequency equal to the sampling frequency (f S ), and use the obtained intensity signal to determine the displacement of the mass along the first direction. 
     In some implementations, the VCSELs of the VCSEL array can be arranged in a row parallel to the first direction and separated by a pitch (δ) configured such that at least two of the beams emitted by the VCSEL array can concurrently illuminate, along the first direction, a single tile of the intensity pattern. In some cases, the intensity pattern can have a pattern period (P) along the first direction that is formed from (M≥2) different tiles, each tile of the pattern period being configured to redirect to the photodetector light having an associated intensity level from among (M) corresponding different intensity levels, the pattern period satisfying the condition P&gt;(N TOT −1)δ, and for a motion of the mass that causes a maximum velocity (v MAX ) of the intensity pattern, a sampling frequency (f S ) satisfies the condition f S &gt;2v MAX /[(N TOT −1)δ]. In some cases, the intensity pattern can have a pattern period (P) along the first direction that is formed from (M≥2) different tiles, each tile of the pattern period being configured to redirect to the photodetector light having an associated intensity level from among the (M) different intensity levels, the pattern period satisfying the condition P≤(N TOT −1)δ, and, for a motion of the mass that causes a maximum velocity (v MAX ) of the intensity pattern, a sampling frequency (f S ) satisfies the condition f S &gt;2v MAX /P. 
     In some implementations, the VCSEL array can include two or more VCSELs distributed along a second direction that crosses the first direction; the intensity pattern can include at least two or more tiles that form one or more tile borders across the first direction, and at least two or more tiles that form one or more tile borders across the second direction, the tile borders being at known locations relative to each other along the first and second directions, such that, during operation of the displacement measuring system, the VCSELs of the array illuminate respective locations of the intensity pattern, across at least one of the tile borders along the first direction and across at least another one of the tile borders along the second direction; and the processor can determine the displacement of the mass along the first direction based on one or more changes of the intensity values of the set caused by motion of the mass along a direction of motion across the at least one of the tile borders along the first direction and across the at least another one of the tile borders along the second direction. Here, the VCSELs of the VCSEL array can be arranged in a first row parallel to the first direction, the first row including (N X ) VCSELs separated by a first pitch (δ X ) configured such that at least two of the beams emitted by the VCSEL array can concurrently illuminate, along the first direction, a single tile of the intensity pattern, and a second row parallel to the second direction, the second row including (N Y ) VCSELs separated by a second pitch (δ Y ) configured such that at least two of the beams emitted by the VCSEL array can concurrently illuminate, along the second direction, a single tile of the intensity pattern. 
     In some cases of these implementations, the intensity pattern can have a first pattern period (P X ) along the first direction that is formed from (M X ≥2) different tiles, each tile of the first pattern period being configured to redirect to the photodetector light having an associated intensity level from among (M X ) corresponding different intensity levels, the first pattern period satisfying the condition P X &gt;(N X −1)δ X , and the intensity pattern can have a second pattern period (P Y ) along the second direction that is formed from (M Y ≥2) different tiles, each tile of the second pattern period being configured to redirect to the photodetector light having an associated intensity level from among (M Y ) corresponding different intensity levels, the second pattern period satisfying the condition P Y &gt;(N Y −1)δ Y . As such, for a motion of the mass that causes a maximum velocity (v MAX ) of the intensity pattern, a sampling frequency (f S ) satisfies the condition f S &gt;MAX{2v MAX,X /[(N X −1)δ X ], 2v MAX,Y /[(N Y −1)δ Y ]}. 
     In some cases of these implementations, the intensity pattern can have a first pattern period (P X ) along the first direction that is formed from (M X ≥2) different tiles, each tile of the first pattern period being configured to redirect to the photodetector light having an associated intensity level from among (M X ) corresponding different intensity levels, the first pattern period satisfying the condition P X ≤(N X −1)δ X , and the intensity pattern can have a second pattern period (P Y ) along the second direction that is formed from (M Y ≥2) different tiles, each tile of the second pattern period being configured to redirect to the photodetector light having an associated intensity level from among (M Y ) corresponding different intensity levels, the second pattern period satisfying the condition P Y ≤(N Y −1)δ Y . As such, for a motion of the mass that causes a maximum velocity (v MAX ) of the intensity pattern, a sampling frequency (f S ) satisfies the condition f S &gt;MAX{2v MAX,X /P X , 2v MAX,Y /P Y }. 
     In some cases of these implementations, the intensity pattern can have a first pattern period (P X ) along the first direction that is formed from (M X ≥2) different tiles, each tile of the first pattern period being configured to redirect to the photodetector light having an associated intensity level from among (M X ) corresponding different intensity levels, the first pattern period satisfying the condition P X &gt;(N X −1)δ X , and the intensity pattern can have a second pattern period (P Y ) along the second direction that is formed from (M Y ≥2) different tiles, each tile of the second pattern period being configured to redirect to the photodetector light having an associated intensity level from among (M Y ) corresponding different intensity levels, the second pattern period satisfying the condition P Y ≤(N Y −1)δ Y . As such, for a motion of the mass that causes a maximum velocity (v MAX ) of the intensity pattern, a sampling frequency (f S ) satisfies the condition f S &gt;MAX{2v MAX,X /[(N X −1)δ X ], 2v MAX,Y /P Y }. 
     In some cases of these implementations, the intensity pattern can have a first pattern period (P X ) along the first direction that is formed from (M X ≥2) different tiles, each tile of the first pattern period being configured to redirect to the photodetector light having an associated intensity level from among (M X ) corresponding different intensity levels, the first pattern period satisfying the condition P X ≤(N X −1)δ X , and the intensity pattern can have a second pattern period (P Y ) along the second direction that is formed from (M Y ≥2) different tiles, each tile of the second pattern period being configured to redirect to the photodetector light having an associated intensity level from among (M Y ) corresponding different intensity levels, the second pattern period satisfying the condition P Y &gt;(N Y −1)δ Y . As such, for a motion of the mass that causes a maximum velocity (v MAX ) of the intensity pattern, a sampling frequency (f S ) satisfies the condition f S &gt;MAX{2v MAX,X /P X , 2v MAX,Y /[(N Y −1)δ Y ]}. 
     Moreover, in these implementations, the processor can determine a total displacement of the mass along the second direction that corresponds to a maximum displacement of the mass along the first direction; determine an angular misalignment of the intensity pattern based on the total displacement of the mass along the second direction; and determine a scaling factor to scale the determined displacement of the mass along the first direction and the determined displacement of the mass along the second direction. 
     In some implementations, the intensity pattern can include a surface that is reflective to the beams emitted by the VCSEL array. In some implementations, the intensity pattern can include a surface that is transparent to the beams emitted by the VCSEL array and is spatially modulated by the tiles; and an array of redirecting micro-structures disposed between the first surface and the mass, the redirecting micro-structures of the array being oriented to redirect, by a folding angle, to the photodetector, the beams emitted by the VCSEL array that impinge on the array of redirecting micro-structures after at least one transmission through the first surface. In some cases, the system can include a mount including a surface onto which the VCSEL array and the photodetector are disposed side-by-side to each other, where the folding angle is an acute angle. In some cases, the system can include a mount including a first surface onto which the VCSEL array is disposed, and a second surface angled to the first surface, the photodetector being disposed on the second surface, where the folding angle is substantially a right angle. 
     In some implementations, to determine the positions of the illuminated locations of the intensity pattern, the processor can use the issued set of intensity values against a mapping of (A) sets of expected intensity values to (B) positions of illuminated locations of the intensity pattern. 
     Another aspect of the disclosure can be implemented as a displacement measuring system that includes a single light-emitting element (LEE); an intensity pattern that is coupled with a mass and includes three or more tiles separated from each other by corresponding one or more tile borders, where the tile borders are at known locations relative to each other along a first direction, and where the LEE is spaced apart from the intensity pattern and arranged such that, during operation of the displacement measuring system, the LEE illuminates a location of the intensity pattern; a single photodiode spaced apart from the intensity pattern and arranged such that, during operation of the displacement measuring system, the photodiode to (i) capture a beam redirected to the photodiode from the illuminated location of the intensity pattern, where each tile of the intensity pattern is configured to redirect to the photodiode light having an intensity different by (A) a first amount from an intensity of light redirected to the photodiode by one of its adjacent tiles, and (B) a second amount from another intensity of light redirected to the photodiode by another one of its adjacent tiles, and (ii) issue a single intensity value corresponding to the captured beam; and a processor to determine (i) a position of the illuminated location of the intensity pattern based on the issued intensity value, and (ii) a displacement of the mass along the first direction based on changes of the intensity value caused by motion of the mass along a direction of motion across at least one of the tile borders. 
     Implementations of the above-summarized measuring system can include one or more of the following features. In some implementations, the intensity pattern can have a pattern period (P) along the first direction that is formed from (M≥3) different tiles, each tile of the pattern period being configured to redirect to the photodiode light having an associated intensity level from among (M) corresponding different intensity levels. As such, for a motion of the mass that causes a maximum velocity (v MAX ) of the intensity pattern, the sampling frequency (f S ) satisfies the condition f S &gt;2v MAX /P, and the processor can obtain an intensity signal as a sequence of the intensity values, the sequence having a frequency equal to the sampling frequency (f S ). In some implementations, to determine the positions of the illuminated locations of the intensity pattern, the processor can use the issued intensity value against a mapping of (A) expected intensity values to (B) positions of illuminated locations of the intensity pattern. 
     Another aspect of the disclosure can be implemented as a haptic engine that includes the mass and one of the displacement measuring systems summarized above. 
     Another aspect of the disclosure can be implemented as an angular displacement measuring system that includes a light-emitting element (LEE) array including two or more (N TOT ) light-emitting elements (LEEs), each LEE being configured to output collimated light in the form of a beam; an intensity pattern that is disposed on a side wall surface of a wheel, the intensity pattern including tiles shaped as annulus sectors, the tiles separated from each other by corresponding one or more tile borders, where the tile borders are radially oriented at known angular locations relative to each other, and where the LEE array is spaced apart from the intensity pattern and arranged such that, during operation of the angular displacement measuring system, the (N TOT ) LEEs of the array output beams along a direction orthogonal to the side wall surface and illuminate respective locations of the intensity pattern across at least one of the tile borders; a photodetector spaced apart from the intensity pattern and arranged such that, during operation of the angular displacement measuring system, the photodetector (i) captures beams redirected along a radial direction through the rim surface of the wheel to the photodetector from the (N TOT ) illuminated locations of the intensity pattern, where each tile of the intensity pattern is configured to redirect to the photodetector light having an intensity different from an intensity of light redirected to the photodetector by its adjacent tiles, and (ii) issues a set of (N TOT ) intensity values corresponding to the respective captured beams; and a processor to determine (i) positions of the illuminated locations of the intensity pattern based on relative differences between the intensity values of the issued set, and (ii) an angular displacement of the wheel based on one or more changes of the intensity values of the set caused by rotation of the wheel across the at least one of the tile borders. 
     Another aspect of the disclosure can be implemented as an angular displacement measuring system that includes a light-emitting element (LEE) array including two or more (N TOT ) light-emitting elements (LEEs), each LEE being configured to output collimated light in the form of a beam; an intensity pattern that is disposed on the rim surface of a wheel, the intensity pattern includes tiles separated from each other by corresponding one or more tile borders, where the tile borders are oriented either along the length, or the width, of the rim at known locations relative to each other, and where the LEE array is spaced apart from the intensity pattern and arranged such that, during operation of the angular displacement measuring system, the (N TOT ) LEEs of the array output beams along a radial direction through the rim surface of the wheel and illuminate respective locations of the intensity pattern across at least one of the tile borders; a photodetector spaced apart from the intensity pattern and arranged such that, during operation of the angular displacement measuring system, the photodetector (i) captures beams redirected to the photodetector from the (N TOT ) illuminated locations of the intensity pattern, where each tile of the intensity pattern is configured to redirect to the photodetector light having an intensity different from an intensity of light redirected to the photodetector by its adjacent tiles, and (ii) issues a set of (N TOT ) intensity values corresponding to the respective captured beams; and a processor to determine (i) positions of the illuminated locations of the intensity pattern based on relative differences between the intensity values of the issued set, and (ii) an angular displacement, and a lateral displacement, of the wheel based on one or more changes of the intensity values of the set caused by rotation, and lateral translation, of the wheel across the at least one of the tile borders. 
     Implementations of the above-summarized angular displacement measuring systems can include one or more of the following features. In some implementations, the beams redirected from the (N TOT ) illuminated locations of the intensity pattern to the photodetector can be tilted by an acute angle relative the radial direction along which the LEEs of the array output the beams. In some implementations, the beams redirected from the (N TOT ) illuminated locations of the intensity pattern through a side wall surface of the wheel to the photodetector can be tilted by a substantially right angle relative the radial direction along which the LEEs of the array output the beams. 
     Another aspect of the disclosure can be implemented as a displacement measuring system that includes a light-emitting element (LEE) array including two or more (N TOT ) light-emitting elements (LEEs), each LEE being configured to output collimated light in the form of a beam; an intensity pattern that is disposed on a surface of an axle of a wheel, the intensity pattern includes tiles separated from each other by corresponding one or more tile borders, where the tile borders are oriented either around, or along, the axle at known locations relative to each other, and where the LEE array is spaced apart from the intensity pattern and arranged such that, during operation of the angular displacement measuring system, the (N TOT ) LEEs of the array output beams along a radial direction through the axle surface of the axle and illuminate respective locations of the intensity pattern across at least one of the tile borders; a photodetector spaced apart from the intensity pattern and arranged such that, during operation of the angular displacement measuring system, the photodetector (i) captures beams redirected from the (N TOT ) illuminated locations of the intensity pattern to the photodetector, where each tile of the intensity pattern is configured to redirect to the photodetector light having an intensity different from an intensity of light redirected to the photodetector by its adjacent tiles, where the beams redirected from the (N TOT ) illuminated locations of the intensity pattern to the photodetector are tilted by an acute angle relative the radial direction along which the LEEs of the array output the beams, and (ii) issues a set of (N TOT ) intensity values corresponding to the respective captured beams; and a processor to determine (i) positions of the illuminated locations of the intensity pattern based on relative differences between the intensity values of the issued set, and (ii) an angular displacement, and a lateral displacement, of the wheel based on one or more changes of the intensity values of the set caused by rotation, and lateral translation, of the wheel across the at least one of the tile borders. 
     Implementations of the above-summarized measuring systems can include one or more of the following features. In some implementations, each LEE can include a VCSEL. In some implementations, each LEE can include a light source configured to emit un-collimated light; and collimating optics optically coupled with the light source to collimate the emitted light. In some implementations, the photodetector is a single photodiode; and the LEEs of the LEE array are configured to illuminate the intensity pattern in a time multiplexed manner. 
     Another aspect of the disclosure can be implemented as a watch that includes one of the above-summarized angular displacement measuring systems or displacement measuring systems. 
     Another aspect of the disclosure can be implemented as a computing device that includes one or more of the above summarized haptic engine, angular displacement measuring systems, or displacement measuring system. 
     The above-disclosed technologies can result in one or more of the following potential advantages. For example, absolute positions of a moving mass, disposed in vibration modules, that are measured by the disclosed displacement measuring systems can be used to effectively control saliency and prevent noise and damage. As such, accurately measured displacement of the moving mass allows closed-loop control. The closed-loop control enables richer saliency vocabularies, compensation against aging degradation, and crash of, or damage to, the vibration modules. As another example, implementations of the disclosed displacement measuring systems used for 1D motion sensing can be extended to 2D motion sensing, where displacements ΔX and ΔY of the moving mass can be concurrently measured. 
     As yet another example, thickness along the Z-axis of a vibration module that uses the disclosed displacement measuring systems can be significantly reduced relative to a conventional vibration module, because the disclosed displacement measuring systems&#39; VCSEL-based optical source does not need focusing, so it can be placed at any arbitrary distance to the intensity pattern. As yet another example, a vibration module that uses the disclosed displacement measuring systems can be self-calibrated with the intensity pattern acting as displacement reference, so they do not need to be placed in a calibration tester like conventional displacement measuring systems. 
     As yet another example, the disclosed displacement measuring systems can be insensitive to Z-offset given by relative alignment/misalignment between the moving mass and the VCSEL array. As yet another example, the disclosed displacement measuring system can be insensitive to temperature change as it uses a ratiometric measurement technique. 
     As yet another example, the transceiver architecture of the disclosed displacement measuring systems is configured to operate in pulse width modulation (PWM) mode which uses a reduced number of analog components compared to the class-A mode in which the transceiver architecture used in a conventional vibration module is configured to operate. 
     Details of one or more implementations of the disclosed technologies are set forth in the accompanying drawings and the description below. Other features, aspects, descriptions and potential advantages will become apparent from the description, the drawings and the claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIGS. 1A-1C  show aspects of a displacement measuring system. 
         FIG. 1D  is an example of a timing diagram used by the displacement measuring system of  FIGS. 1A-1C . 
         FIGS. 1E-1F  shows aspects of a haptic system that uses the displacement measuring system of  FIGS. 1A-1C . 
         FIGS. 2A-2B  show aspects of a technique for measuring displacements of a binary intensity pattern. 
         FIGS. 3A-3C  show aspects of a technique for measuring displacements of a binary intensity pattern in a single-ended manner. 
         FIGS. 4A-4C  show aspects of show aspects of another technique for measuring displacements of a binary intensity pattern in a differential manner. 
         FIGS. 5A-5C  show aspects of a differential measurement used by the displacement measuring system of  FIGS. 1A-1C . 
         FIGS. 6A-6B  show aspects of a technique for measuring displacements of a binary intensity pattern in a combined single-ended and differential manner. 
         FIGS. 7A-7C  show aspects of an angular displacement measuring system. 
         FIGS. 8A-8B  show aspects of a technique for measuring angular displacement used by the angular displacement measuring system of  FIGS. 7A-7C . 
         FIGS. 9A-9L  show aspects of a displacement measuring system that includes an optical sensing system and a bEMF sensing system. 
         FIGS. 10A-10B  show aspects of an example of an interpolator module used by the displacement measuring system of  FIGS. 9A-9L . 
         FIGS. 11A-11B  show aspects of a haptic system that uses the displacement measuring system of  FIGS. 9A-9L . 
         FIGS. 12A-12I  show aspects of another optical sensing system to be used in conjunction with the displacement measuring system of  FIG. 9A . 
         FIGS. 13A-13C  show examples of optical structures of the displacement measuring system of  FIG. 9A  that are used for redirecting probe light that illuminates an intensity pattern to a photodetector. 
         FIGS. 14A-14C  show other examples of optical structures of the displacement measuring system of  FIG. 9A  that are used for redirecting probe light that illuminates an intensity pattern to a photodetector. 
         FIG. 15A  shows an example of displacement measuring system. 
         FIG. 15B  shows another example of displacement measuring system. 
         FIG. 15C  shows an example of a binary intensity pattern used to measure 1D displacements. 
         FIGS. 15D-15F  show aspects of operating displacement measuring systems. 
         FIGS. 16A-16B  show aspects of determining location on the binary intensity pattern of  FIG. 15C . 
         FIG. 17A  shows an example of a three-level intensity pattern used to measure 15D displacements. 
         FIGS. 17B-17C  show aspects of determining location on the three-level intensity pattern of  FIG. 17A . 
         FIGS. 18A-18B  show modifications to the displacement measuring system of  FIGS. 15A-15B . 
         FIG. 18C  shows an example of a four-level intensity pattern used to measure 2D displacements. 
         FIGS. 18D-18E  show aspects of determining location on the four-level intensity pattern of  FIG. 18C . 
         FIGS. 19A-19B  show aspects of a technique for addressing misalignment of intensity pattern relative to a light emitting element array. 
         FIGS. 20A-20B  show aspects of an angular displacement measuring system. 
         FIGS. 21A-21B  show aspects of another angular displacement measuring system. 
         FIGS. 22A-22B  show aspects of another displacement measuring system. 
     
    
    
     Certain illustrative aspects of the systems, apparatuses, and methods according to the disclosed technologies are described herein in connection with the following description and the accompanying figures. These aspects are, however, indicative of but a few of the various ways in which the principles of the disclosed technologies may be employed and the disclosed technologies are intended to include all such aspects and their equivalents. Other advantages and novel features of the disclosed technologies may become apparent from the following detailed description when considered in conjunction with the figures. 
     DETAILED DESCRIPTION 
       FIG. 1A  is a side view, e.g. in the (x,z) plane, of an example of a displacement measuring system  100 . The displacement measuring system  100  includes a mount  104 , a light source  106  supported by the mount, an optical pattern  118  disposed on a surface  135 XY of a mass  134  that is spaced apart from the light source, a photodetector  124  supported by the mount, and processing electronics  125  coupled with the photodetector. 
       FIG. 1B  is a plan view, e.g., in the (x,y) plane, of the optical pattern  118 . In this example, the optical pattern  118  has two portions  120 ,  122  that form a rectangular edge  121  with sides parallel to corresponding x-axis and y-axis. The first portion  120  (shown in white) has a first reflectivity R 1 , and the second portion  122  (shown in grey) has a second reflectivity R 2 , smaller than the first reflectivity. For example, the second reflectivity R 2  can be at most half the first reflectivity R 1 , e.g., R 2 =0.3R 1 , 0.1R 1 , 0.05R 1  or other fractions of R 1 . As light impinging on the first portion  120  reflects off it with a first intensity, and light impinging on the second portion  122  reflects off it with a second intensity smaller than the first intensity, the optical pattern is also referred to as a binary intensity pattern  118 . In some implementations, the first portion  120  is coated with a reflective film and the second portion  122  is coated with an absorptive film. In some implementations, the first portion  120  is coated with a multilayer reflection coating and the second portion  122  is coated with a multilayer anti-reflection coating. 
       FIG. 1C  is a plan view, e.g., in the (x,y) plane, of the components of the displacement measuring system  100  supported by the mount  104 . The light source  106  can include one or more light emitting element (LEE) arrays  110 . In the example illustrated in  FIG. 1C , the light source  106  includes LEE arrays  110 A,  110 B,  110 C,  110 D. 
     Each LEE array, e.g., LEE array  110 A, includes a driving board  112  and a plurality of light emitting elements (LEEs)  114 , such that the driving board concurrently powers the LEEs of the LEE array, during operation of the displacement measuring system  100 . Note that the LEEs  114  of an LEE array  110  can be arranged in one or more rows parallel to each other. In the example illustrated in  FIG. 1C , each LEE array, e.g., LEE array  110 A, includes rows  116 A,  116 B of LEEs  114  distributed along the x-axis. The LEEs  114  in a row  116 A or  116 B are separated by a pitch X. Moreover, the rows  116 A and  116 B are separated from each other along the y-axis by a separation δY, and are staggered relative to each other along the x-axis by half the pitch A. Further, a total size (e.g., length) along the x-axis of an LEE array  110  sets an upper bound MAX ΔX for a displacement along the x-axis that can be measured by the displacement measuring system  100 . Furthermore, a total size (e.g., width) along the y-axis of an LEE array  110  sets an upper bound MAX ΔY for a displacement along the y-axis that can be measured by the displacement measuring system  100 . Note that a range of translational motion for a mass  134  disposed inside a haptic engine, as described below in connection with  FIGS. 1E-1F , is less than 0.6 mm. 
     Each LEE  114  of the light source  106  is configured to output collimated light, such that the LEE illuminates the optical pattern  118  with a beam  115  that forms a well-defined (e.g., top-hat or Gaussian) beam spot on the optical pattern. In this manner, displacement measurements performed by the displacement measuring system  100  are insensitive to a separation Z-offset between the light source  106  and the optical pattern  118 . In some implementations, each LEE  114  includes a light emitting diode (LED) and a collimating optic (e.g., a lens, a compound parabolic concentrator, etc.) optically coupled with the LED. Such an LED emits un-collimated light (e.g., in accordance with a Lambertian distribution), and the collimating optic collimates the emitted light to issue collimated light. In other implementations, each LEE  114  includes a vertical cavity surface emitting laser (VCSEL) that emits collimated light. In this manner, when the LEEs are VCSELs, Z-offset between the light source  106  and the optical pattern  118  can be very short, e.g., in the range of 0.1 mm to 0.5 mm. Moreover, as light emitted by the VCSELs  114  can have wavelengths in a range from 700 nm to 1100 nm, the second portion  122  of the optical pattern  118  can be printed using ink that absorbs IR light. 
     Additionally, each LEE array  110  is arranged relative to the optical pattern  118  such that its LEEs  114  illuminate with collimated light, during operation of the displacement measuring system  100 , the optical pattern  118  across a corresponding corner of the rectangular edge  121 , as explained below in this specification. The optical pattern  118  redirects to the photodetector  124 , e.g., via reflection or scattering, at least some of the collimated light  115  from an LEE array  110  that impinges on the optical pattern. The photodetector  124  captures the redirected light  119  associated with the LEE array  110  and integrates it. In some implementations, the photodetector  124  can be a CMOS sensor array. In some implementations, the photodetector  124  can be a CCD sensor array. 
     Moreover, as a result of integrating the captured light, the photodetector  124  issues a photodetector signal relating to the collimated light output by an LEE array  110  that is redirected by the optical pattern  118  to the photodetector. In the example shown in  FIGS. 1A and 1C , the photodetector  124  issues a respective photodetector signal relating to the collimated light output by a corresponding one of the LEE arrays  110 A,  110 B,  110 C,  110 D, where the issued photodetector signals are multiplexed based on a multiplexing scheme used to illuminate the optical pattern  118 . 
     In some implementations, the LEEs  114  of the LEE arrays  110 A,  110 B,  110 C,  110 D emit light of different wavelengths, λ A , λ B , λ C , λ D . In this case, the LEEs  114  of the LEE arrays  110 A,  110 B,  110 C,  110 D illuminate the optical pattern  118  concurrently, using a wavelength multiplexing scheme. As such, the issued photodetector signals also are wavelength multiplexed. For example, the photodetector  124  can include a set of four sensors v 1 , v 2 , v 3 , v 4 , each sensor being covered with a filter of different cut-off wavelength to match the wavelengths λ A , λ B , λ C , λ D  of light emitted by the LEE arrays  110 A,  110 B,  110 C,  110 D. And the signals corresponding to at least some of the wavelengths λ A , λ B , λ C , λ D  of the emitted light can be isolated by combining signals output by multiple of the sensors. For example, the first sensor, v 1 , has a filter that passes wavelengths up to λ A  and outputs a first sensor signal V 1 , the second sensor, v 2 , has a filter that passes wavelengths up to λ B  and outputs a second sensor signal V 2 , the third sensor, v 3 , has a filter that passes wavelengths up to λ C  and outputs a third sensor signal V 3 , and the fourth sensor, v 4 , has a filter that passes wavelengths up to λ D  and outputs a fourth sensor signal V 4 , where λ A &lt;λ B &lt;λ C &lt;λ D . Then a first photodetector signal in the band of λ A  can be isolated as V 1 , a second photodetector signal in the band of λ B  can be isolated as (V 2 −V 1 ), a third photodetector signal in the band of can be isolated (V 3 −V 2 ), and a fourth photodetector signal in the band of λ D  can be isolated as (V 4 −V 3 ). As another example, the first sensor, v 1 , has a filter that stops wavelengths up to λ A  and outputs a first sensor signal V 1 , the second sensor, v 2 , has a filter that stops wavelengths up to λ B  and outputs a second sensor signal V 2 , the third sensor, v 3 , has a filter that stops wavelengths up to and outputs a third sensor signal V 3 , and the fourth sensor, v 4 , has a filter that stops wavelengths up to λ D  and outputs a fourth sensor signal V 4 , where λ A &lt;λ B &lt;λ C &lt;λ D . Then a first photodetector signal in the band of λ A  can be isolated as (V 1 −V 2 ), a second photodetector signal in the band of λ B  can be isolated as (V 2 −V 3 ), a third photodetector signal in the band of λ C  can be isolated as (V 3 −V 4 ), and a fourth photodetector signal in the band of λ D  can be isolated as V 4 . 
     In other implementations, the LEEs  114  of the LEE arrays  110 A,  110 B,  110 C,  110 D emit light of the same wavelength. In this case, the LEEs  114  of the LEE arrays  110 A,  110 B,  110 C,  110 D illuminate the optical pattern  118  concurrently, using a time multiplexing scheme.  FIG. 1D  is an example of a timing diagram used for time multiplexing the light redirected by the optical pattern  118  to the photodetector  124 . As such, light from the LEE array  110 A that is redirected by the optical pattern  118  reaches the photodetector  124  in accordance with timing gate  126 A; light from the LEE array  110 B that is redirected by the optical pattern reaches the photodetector in accordance with timing gate  126 B; light from the LEE array  110 C that is redirected by the optical pattern reaches the photodetector in accordance with timing gate  126 C; and light from the LEE array  110 D that is redirected by the optical pattern reaches the photodetector in accordance with timing gate  126 D. Note that the timing gates  126 A,  126 B,  126 C,  126 D are chosen such that light from the LEE array  110 A that is redirected by the optical pattern  118  reaches the photodetector  124  when light from the remaining LEE arrays  110 B,  110 C,  110 D is prevented from reaching the photodetector; light from the LEE array  110 B that is redirected by the optical pattern  118  reaches the photodetector when light from the remaining LEE arrays  110 A,  110 C,  110 D is prevented from reaching the photodetector; and so on. As such, the photodetector  124  is said to operate in “shutter mode” because it receives light from only one LEE array  110  at a time. For example, the photodetector  124  can use a timing gate  128  that is synchronized with the timing gates  126 A,  126 B,  126 C,  126 D to issue photodetector signals that are time multiplexed. 
     The processing electronics  125  receive one or more photodetector signals issued by the photodetector  124  and determine displacements of the mass  134  based on corresponding changes in the one or more photodetector signals caused by motion of the mass along, and orthogonal to, a direction of the rows  116  of LEEs  114  of the LEE arrays  110 . The specific displacements that are determined by the processing electronics  125  depend on positioning of the displacement measuring system  100  inside a haptic engine  130  that includes the mass  134 , as shown in FIGS.  1 E- 1 F. Note that the haptic engine  130  has a frame  132  that encompasses the mass  134  and the displacement measuring system  100 . 
       FIG. 1E  is a side view, in the (y,z) plane, of the haptic engine  130 , in which the optical pattern  118  of the displacement measuring system  100  is supported by the mass  134  on a surface  135 XY parallel to the (x,y) plane, and the mount  104  of the displacement measuring system is supported by the frame  132  on a face parallel to the (x,y) plane. Here, the rows  116  of LEEs  114  of the LEE arrays  110  are oriented along the x-axis. In this case, the processing electronics  125  determine a displacement ΔX and a displacement ΔY of the mass  134  based on corresponding changes in the one or more photodetector signals caused by motion of the mass along the x-axis (e.g., vibration in-and-out of page) and motion of the mass along the y-axis (e.g., vibration left-right on page). 
       FIG. 1F  is a side view, in the (y,z) plane, of the haptic engine  130 , in which the optical pattern  118  of the displacement measuring system  100  is supported by the mass  134  on a surface  135 XZ parallel to the (x,z) plane, and the mount  104  of the displacement measuring system is supported by the frame  132  on a face parallel to the (x,z) plane. Here, the rows  116  of LEEs  114  of the LEE arrays  110  are oriented along the x-axis. In this case, the processing electronics  125  determine a displacement ΔX and a displacement ΔZ of the mass  134  based on corresponding changes in the one or more photodetector signals caused by motion of the mass along the x-axis (e.g., vibration in-and-out of page) and motion of the mass along the z-axis (e.g., vibration up-down on page). 
     Prior to describing techniques for measuring displacements of the mass  134  using the displacement measuring system  100 , techniques for sensing motion of an optical pattern that is illuminated by one or more beams is described next. 
       FIG. 2A  is a plan view, in the (x,y) plane, of a binary intensity pattern  218  that includes a first portion  220  corresponding to white strips and a second portion  222  corresponding to grey strips, where the stripes extend along the y-axis. A beam spot  236  shows a location where a beam of collimated light (e.g., VCSEL light) illuminates the binary intensity pattern  218 . Note that in this example, a size of the beam spot  236  is about equal to a width of a strip along the x-axis. Here, the first portion  220  has a first reflectivity R 1 , and the second portion  222  has a second reflectivity R 2 , smaller than the first reflectivity. In this manner, as the binary intensity pattern  218  is translated along the x-axis through the beam spot  236 , the beam reflects off with a first intensity when it illuminates the first portion  220 , and a second intensity, smaller than the first intensity, when it illuminates the second portion  222 . A photodetector, to which the reflected beam is reflected, issues a photodetector signal  240  proportional to the intensity of the reflected beam. Translation of the binary intensity pattern  218  along the x-axis through the beam spot  236 , causes changes of the photodetector signal  240 .  FIG. 2B  shows changes of the photodetector signal  240  as a function of displacement ΔX of a datum of the binary intensity pattern  218  relative to the beam spot  236 . 
       FIGS. 3A-3C  show aspects relating to a displacement measurement technique that has a sensitivity free of dead-zones.  FIG. 3A  is a plan view, in the (x,y) plane, of a binary intensity pattern  318  that has two portions  320 ,  322  that form an edge  321  parallel to the y-axis. A plurality of beam spots  336  show locations where respective beams of collimated light (e.g., VCSEL light) illuminate the binary intensity pattern  318 . The beam spots  336  have equal sizes, so the associated beams that illuminate the binary intensity pattern  318  have the same intensity. The first portion  320  (shown in white) has a first reflectivity R 1 , and the second portion  322  (shown in grey) has a second reflectivity R 2 , smaller than the first reflectivity. In this manner, a beam associated with a beam spot located on the first portion  320  reflects off with a first intensity, and a beam associated with a beam spot located on the second portion  322  reflects off with a second intensity, smaller than the first intensity. 
     Half of the beam spots  336  are arranged in a first row  338 A and the other half of the beam spots are arranged in a second row  338 B, where the rows  338 A,  338 B are parallel to each other and the x-axis. An in-row pitch of the beam spots  336  within a row  338 A or  338 B is in a range of 1 to 1.5 of a beam spot size. The rows  338 A,  338 B are staggered with respect to each other along the x-axis by half the in-row pitch, and are separated from each other along the y-axis by an inter-row pitch in a range of 0.8 to 1 of the beam spot size. 
     A first group of beams associated with the beam spots  336  of the first row  338 A is referred to as a first macro-beam  338 A, and a second group of beams associated with the beam spots  336  of the second row  338 B is referred to as a second macro-beam  338 B. Note that the macro-beams  338 A,  338 B can be turned ON/OFF in a time multiplexed manner, i.e., when the first macro-beam  338 A illuminates the binary intensity pattern  318 , the second macro-beam  338 B does not do so; and when the second macro-beam  338 B illuminates the binary intensity pattern  318 , the first macro-beam  338 A does not do so. The binary intensity pattern  318  reflects the first macro-beam  338 A (or second macro-beam  338 B) to a photodetector. The photodetector captures the reflected first macro-beam  338 A (or reflected second macro-beam  338 B), and integrates it. As a result of integrating the captured light, the photodetector issues a first photodetector signal  342 A (or second photodetector signal  342 B) proportional to the intensity of the reflected first macro-beam  338 A (or reflected second macro-beam  338 B). In other words, the first photodetector signal  342 A (or second photodetector signal  342 B) is proportional to cumulative intensity of the reflected beams of the first macro-beam  338 A (or second macro-beam  338 B). 
     Translation of the binary intensity pattern  318  along the x-axis through the first macro-beam  338 A (or second macro-beam  338 B), causes changes of the first photodetector signal  342 A (or second photodetector signal  342 B).  FIG. 3B  shows changes of the first photodetector signal  342 A (or second photodetector signal  342 B) as a function of displacement ΔX of the edge  321  of the binary intensity pattern  318  relative to the first macro-beam  338 A (or second macro-beam  338 B). For instance, as beams of the first macro-beam  338 A (or second macro-beam  338 B) transition over the edge  321 , from the second portion  322  to the first portion  320 , the intensity of the reflected first macro-beam  338 A (or reflected second macro-beam  338 B) increases, and so does the first photodetector signal  342 A (or second photodetector signal  342 B), as shown in  FIG. 3B . In this manner, the maximum measurable displacement MAX ΔX of the edge  321  of the binary intensity pattern  318  relative to the first macro-beam  338 A (or second macro-beam  338 B) is substantially equal to the length of the first macro-beam  338 A (or second macro-beam  338 B) along the x-axis. 
       FIG. 3C  shows a gradient  344 A of the first photodetector signal  342 A (or gradient  344 B of the second photodetector signal  342 B). Note that each peak/valley pair of the gradient  344 A (or gradient  344 B) corresponds to a crossing of the edge  321  through another beam of the first macro-beam  338 A (or second macro-beam  338 B) as the binary intensity pattern  318  is translated along the x-axis. The resolution of this displacement measurement is given by a spot size of beam spot  336 . For instance, spot size for a VCSEL beam can be less than 10 μm. By counting, from the gradient  344 A of the first photodetector signal  342 A (or gradient  344 B of the second photodetector signal  342 B), the number of beam spot crossings over the edge  321 , one can determine an accurate value of the absolute displacement ΔX. This can be used as an optical ruler to calibrate absolute displacement for a known spot size of the beam spot  336 . 
     Note that dead-zones in the sensitivity of the foregoing displacement measurement occur when the edge  321  of the binary intensity pattern  318  is between beams of the first macro-beam  338 A (or second macro-beam  338 B), for values of the gradient  344 A of the first photodetector signal  342 A (or gradient  344 B of the second photodetector signal  342 B) equal to zero (ΔI/ΔX=0). To avoid the foregoing dead-zones, the multiplexed first photodetector signal  342 A and second photodetector signal  342 B can be combined together, e.g., by averaging or interpolating them together, to obtain a smooth photodetector signal. The combining together of the first photodetector signal  342 A and second photodetector signal  342 B can be can be accomplished by activating, at the same time, both the first macro-beam  338 A and the second macro-beam  338 B. In this manner, the measurement of the displacement ΔX is performed using a single macro-beam  350  that includes both rows  338 A and  338 B of beams, as explained below in this specification. 
       FIGS. 4A-4C  show aspects relating to a differential sensing mode of a displacement measurement technique.  FIG. 4A  is a plan view, in the (x,y) plane, of a binary intensity pattern  418  that has a first portion  420  (shown in white) and a second portion  422  (shown in grey), the two portions forming a first edge  421 A and a second edge  421 B that are parallel to each other and the y-axis. A plurality of beam spots  436  show locations where respective beams of collimated light (e.g., VCSEL light) illuminate the binary intensity pattern  418 . The beam spots  436  have equal sizes, so the associated beams that illuminate the binary intensity pattern  418  have the same intensity. The first portion  420  has a first reflectivity R 1 , and the second portion  422  has a second reflectivity R 2 , smaller than the first reflectivity. In this manner, a beam associated with a beam spot located on the first portion  420  reflects off with a first intensity, and a beam associated with a beam spot located on the second portion  422  reflects off with a second intensity, smaller than the first intensity. 
     A first group of beams associated with half of the beam spots  436  is referred to as a first macro-beam  450 A, and a second group of beams associated with the other half of the beam spots  436  is referred to as a second macro-beam  450 B. The beam spots  436  of the first macro-beam  450 A (or second macro-beam  450 B) are arranged in two rows  438 A,  438 B that are parallel to each other and the x-axis. An in-row pitch of the beam spots  436  within a row  438 A (or  438 B) is in a range of 1 to 1.5 of a beam spot size. The rows  438 A,  438 B are staggered with respect to each other along the x-axis by half the in-row pitch, and are separated from each other along the y-axis by an inter-row pitch in a range of 0.8 to 1 of the beam spot size. A length of the macro-beams  450 A,  450 B along the x-axis is about equal to a separation between the first edge  421 A and the second edge  421 B of the binary intensity pattern  418 . In this manner, for any displacement ΔX of the binary intensity pattern  418  relative to the macro-beams  450 A,  450 B that is smaller than a maximum measurable displacement MAX ΔX, if the first macro-beam  450 A illuminates the binary intensity pattern across the first edge  421 A, then the second macro-beam  450 B illuminates the binary intensity pattern across the second edge  421 B. 
     Note that the macro-beams  450 A,  450 B are turned ON/OFF in a time multiplexed manner, i.e., when the first macro-beam  450 A illuminates the binary intensity pattern  418 , the second macro-beam  450 B does not do so; and when the second macro-beam  450 B illuminates the binary intensity pattern  418 , the first macro-beam  450 A does not do so. The binary intensity pattern  418  reflects the first macro-beam  450 A and the second macro-beam  450 B to a photodetector. The photodetector sequentially captures (in a time multiplexed manner) the reflected first macro-beam  450 A and the reflected second macro-beam  450 B, and separately integrates them. As a result of integrating the captured light, the photodetector issues a first photodetector signal  452 A and a second photodetector signal  452 B respectively proportional to the intensity of the reflected first macro-beam  450 A and the intensity of the reflected second macro-beam  450 B. In other words, the first photodetector signal  452 A is proportional to cumulative intensity of the reflected beams of the first macro-beam  450 A, and the second photodetector signal  452 B is proportional to cumulative intensity of the reflected beams of the second macro-beam  450 B). 
     Translation of the binary intensity pattern  418  along the x-axis through the first macro-beam  450 A and the second macro-beam  450 B, causes changes of the first photodetector signal  452 A and changes of the second photodetector signal  452 B.  FIG. 4B  is a graph  451  which shows changes of the first photodetector signal  452 A and changes of the second photodetector signal  452 B as a function of displacement ΔX of a datum of the binary intensity pattern  418  relative to the first macro-beam  450 A and the second macro-beam  450 B. The datum of the binary intensity pattern  418  can be either one of the first edge  421 A or the second edge  421 B, for instance. Here, when the binary intensity pattern  418  is translated from left-to-right on page, the following occurs: (1) As beams of the first macro-beam  450 A transition over the first edge  421 A, from the first portion  420  to the second portion  422 , the intensity of the reflected first macro-beam  450 A decreases, and so does the first photodetector signal  452 A; and (2), as beams of the second macro-beam  450 B transition over the second edge  421 B, from the second portion  422  to the first portion  420 , the intensity of the reflected second macro-beam  450 B increases, and so does the second photodetector signal  452 B. Here, the first photodetector signal  452 A and the second photodetector signal  452 B have the same value, referred to as mid-point intensity, at a mid-point displacement ΔX MP . 
     In this manner, displacement ΔX of the binary intensity pattern  418  along the x-axis through the first macro-beam  450 A and the second macro-beam  450 B can be determined in a differential manner, as follows. A value of the first photodetector signal  452 A and a value of the second photodetector signal  452 B is measured for an unknown displacement ΔX. A difference between the measured value of the first photodetector signal  452 A and the measured value of the second photodetector signal  452 B is determined. Then, a value of the unknown displacement ΔX is uniquely obtained by mapping the determined difference value onto graph  451 . Note that when the determined difference value is positive, the mapping is performed for displacements that satisfy ΔX&lt;ΔX MP , and when the determined difference value is negative, the mapping is performed for displacements that satisfy ΔX&gt;ΔX MP . In some implementations, the foregoing differential displacement measurement can be performed by the processing electronics  125  of the displacement measuring system  100 . 
     Note that the mid-point intensity value depends on a Z-offset between a light source (e.g., a portion of the light source  106  of the displacement measuring system  100 ) that provides the macro-beams  450 A,  450 B and the binary intensity pattern  418  (e.g., the optical pattern  118  of the displacement measuring system  100 ). For example, the mid-point intensity value is large for a small Z-offset and small for a large Z-offset. In this manner, displacements ΔZ along the z-axis of the binary intensity pattern  418  relative the light source can be determined based on changes of the mid-point intensity value. 
     Sensitivities of the differential displacement measurement—described above in connection with  FIG. 4B —due to drift in absolute light intensity, caused by variations in temperature, Z-offset, VCSEL bias, etc., can be avoided by performing a displacement measurement in a ratiometric mode. A first ratio signal  452 A/ 452 B is obtained by dividing the first photodetector signal  452 A to the second photodetector signal  452 B, and a second ratio signal  452 B/ 452 A is obtained by dividing the second photodetector signal  452 B to the first photodetector signal  452 A.  FIG. 4C  is a graph  453  which shows changes of the first ratio signal  452 A/ 452 B and changes of the second ratio signal  452 B/ 452 A as a function of displacement ΔX of the datum of the binary intensity pattern  418  relative to the first macro-beam  450 A and the second macro-beam  450 B. Here, the first ratio signal  452 A/ 452 B and the first ratio signal  452 A/ 452 B are both equal to one at a mid-point displacement ΔXMP, where the latter corresponding to the mid-point intensity. 
     Displacement ΔX of the binary intensity pattern  418  along the x-axis through the first macro-beam  450 A and the second macro-beam  450 B can be determined in a ratiometric manner, as follows. A value of the first photodetector signal  452 A and a value of the second photodetector signal  452 B is measured for an unknown displacement ΔX. A first ratio of the measured value of the first photodetector signal  452 A to the measured value of the second photodetector signal  452 B, and a second ratio of the measured value of the second photodetector signal  452 B to the measured value of the first photodetector signal  452 A are obtained. A smaller of the obtained first ratio and second ratio is selected, and then, a value of the unknown displacement ΔX is determined by mapping, in graph  453 , the selected first ratio or second ratio onto the corresponding first ratio signal  452 A/ 452 B or second ratio signal  452 B/ 452 A. For the example illustrated in  FIG. 4C , the obtained second ratio is smaller than the obtained first ratio, so the obtained second ratio is mapped, in graph  453 , onto the second ratio signal  452 B/ 452 A, for displacements that satisfy ΔX&lt;ΔX MP . As another example, if the obtained first ratio were smaller than the obtained second ratio, then the obtained first ratio would be mapped, in graph  453 , onto the first ratio signal  452 A/ 452 B, for displacements that satisfy ΔX&gt;ΔX MP . Note that, by switching between the first ratio signal  452 A/ 452 B and the second ratio signal  452 B/ 452 A when the mid-point is crossed, divide-by-zero problem can be avoided. In other words, the second ratio signal  452 B/ 452 A is used for determining small displacements ΔX, and the first ratio signal  452 A/ 452 B is used for sensing large displacements ΔX. In some implementations, the foregoing ratiometric displacement measurement can be performed by the processing electronics  125  of the displacement measuring system  100 . 
       FIGS. 5A-5C  are used to explain techniques for measuring displacements of the mass  134  using the displacement measuring system  100  that was described above in connection with  FIGS. 1A-1D .  FIG. 5A  is a plan view, in the (x,y) plane, of the optical pattern  118  described above in connection with  FIG. 1B . Note that the rectangular edge  121  of the optical pattern  118  is formed from two edges  121 XA,  121 XB parallel to the y-axis, and two edges  121 YA,  121 YB parallel to the x-axis. A plurality of beam spots  136  show locations where respective beams of collimated light—output by LEEs  114  of light source  106 —illuminate the optical pattern  118 . The beam spots  136  have equal sizes, so the associated beams—output by LEEs  114  of light source  106 —that illuminate the optical pattern  118  have the same intensity. 
     A first group of beams—that form some of the beam spots  136  and are output by the LEEs  114  of the first LEE array  110 A—is referred to as a first macro-beam  150 A, a second group of beams—that form some other of the beam spots  136  and are output by the LEEs of the second LEE array  110 B—is referred to as a second macro-beam  150 B, a third group of beams—that form yet some other of the beam spots  136  and are output by the LEEs of the third LEE array  110 C—is referred to as a third macro-beam  150 C, and a fourth group of beams—that form yet some other of the beam spots  136  and are output by the LEEs of the fourth LEE array  110 D—is referred to as a fourth macro-beam  150 D. The beam spots  136  of the first macro-beam  150 A (or any other macro-beam  150 B,  150 C,  150 D) are arranged in two rows  138 A,  138 B that are parallel to each other and the x-axis. The beams that form the beam spots  136  of the rows  138 A,  138 B relating to any of the macro-beams  150 A,  150 B,  150 C or  150 D are output by the LEEs  114  of the corresponding rows  116 A,  166 B. 
     An in-row pitch of the beam spots  136  within a row  138 A (or  138 B) is in a range of 1 to 1.5 of a beam spot size and is related to the pitch δX of the LEEs  114  within a row  116 A (or  116 B) of LEEs  114 . The rows  138 A,  138 B of beam spots  136  are staggered with respect to each other along the x-axis by half the in-row pitch, as are the rows  116 A,  116 B of LEEs  114 . Additionally, the rows  138 A,  138 B of beam spots  136  are separated from each other along the y-axis by an inter-row pitch—in a range of 0.8 to 1 of the beam spot size—that is related to the separation δY between the rows  116 A,  116 B of LEEs  114 . 
     A length of a macro-beam  150  along the x-axis—which relates to a length of a corresponding LEE array  110 —is about equal to a separation between the edges  121 XA,  121 XB that are parallel to the y-axis. In this manner, for any displacement ΔX of the optical pattern  118  relative to the macro-beams  150 A,  150 B,  150 C or  150 D that is smaller than a maximum measurable displacement MAX ΔX, if the first macro-beam  150 A (or third macro-beam  150 C) illuminates the optical pattern across the first edge  121 XA parallel to the y-axis, then the second macro-beam  150 B (or fourth macro-beam  150 D) illuminates the binary intensity pattern across the second edge  121 XB parallel to the y-axis. Additionally, a width of a macro-beam  150  along the y-axis—which relates to a width of a corresponding LEE array  110 —is about equal to a separation between the edges  121 YA,  121 YB that are parallel to the x-axis. In this manner, for any displacement ΔY of the optical pattern  118  relative to the macro-beams  150 A,  150 B,  150 C or  150 D that is smaller than a maximum measurable displacement MAX ΔY, if the first macro-beam  150 A (or second macro-beam  150 B) illuminates the optical pattern across the first edge  121 YA parallel to the x-axis, then the third macro-beam  150 C (or fourth macro-beam  150 D) illuminates the binary intensity pattern across the second edge  121 YB parallel to the x-axis. 
     Note that the macro-beams  150 A,  150 B,  150 C,  150 D are turned ON/OFF in a time multiplexed manner based on the timing gates  126 A,  126 B,  126 C,  126 D shown in  FIG. 1D . In this manner, when the first macro-beam  150 A illuminates the optical pattern  118 , the macro-beams  150 B,  150 C,  150 D do not do so; when the second macro-beam  150 B illuminates the optical pattern  118 , the macro-beams  150 A,  150 C,  150 D do not do so; and so on. The optical pattern  118  reflects the macro-beams  150 A,  150 B,  150 C,  150 D to the photodetector  124 . The photodetector  124  sequentially captures (in a time multiplexed manner based on the timing gate  128  shown in  FIG. 1D ) the reflected macro-beams  150 A,  150 B,  150 C,  150 D, and separately integrates them. As a result of integrating the captured light, the photodetector  124  issues a first photodetector signal  552 A, a second photodetector signal  552 B, a third photodetector signal  552 C and a fourth photodetector signal  552 D respectively proportional to the intensity of the reflected first macro-beam  150 A, the intensity of the reflected second macro-beam  150 B, the intensity of the reflected third macro-beam  150 C and the intensity of the reflected fourth macro-beam  150 D. 
     Translation of the optical pattern  118  along the x-axis through the macro-beams  150 A,  150 B,  150 C,  150 D, causes changes of the first photodetector signal  552 A, changes of the second photodetector signal  552 B, changes of the third photodetector signal  552 C and changes of the fourth photodetector signal  552 D.  FIG. 5B  is a graph  551 X which shows changes of the first photodetector signal  552 A, changes of the second photodetector signal  552 B, changes of the third photodetector signal  552 C and changes of the fourth photodetector signal  552 D as a function of displacement ΔX of a datum of the optical pattern  118  relative to the macro-beams  150 A,  150 B,  150 C,  150 D. The datum of the optical pattern  418  can be either one of the corners of the rectangular edge  121 , for instance. Here, when the optical pattern  118  is translated from left-to-right on page without being translated up-or-down on page, the following occurs: (1) As beams of the first macro-beam  150 A and beams of the third macro-beam  150 C transition over the first edge  121 XA parallel to the x-axis, from the first portion  120  to the second portion  122 , the intensity of each of the reflected first macro-beam  150 A and the reflected third macro-beam  150 C decreases, and so does each of the respective first photodetector signal  552 A and the third photodetector signal  552 C; and (2) as beams of the second macro-beam  150 B and beams of the fourth macro-beam  150 D transition over the second edge  121 XB parallel to the y-axis, from the second portion  122  to the first portion  120 , the intensity of each of the reflected second macro-beam  150 B and the reflected fourth macro-beam  150 D increases, and so does each of the respective second photodetector signal  552 B and the fourth photodetector signal  552 D. 
     As such, displacement ΔX of the optical pattern  118  along the x-axis through the macro-beams  150 A,  150 B,  150 C,  150 D can be determined in any one or more of the following manners. For example, a differential displacement measurement along the x-axis using (i) first photodetector signal  552 A and (ii) second photodetector signal  552 B can be performed similarly to the differential displacement measurement along the x-axis using (i) the first photodetector signal  452 A and (ii) the second photodetector signal  452 B, as described above in connection with  FIG. 4B . As another example, a differential displacement measurement along the x-axis using (i) third photodetector signal  552 C and (ii) fourth photodetector signal  552 D can be performed similarly to the differential displacement measurement along the x-axis using (i) the first photodetector signal  452 A and (ii) the second photodetector signal  452 B, as described above in connection with  FIG. 4B . As yet another example, a differential displacement measurement along the x-axis using (i) a combination (e.g., an average, or interpolation) of first photodetector signal  552 A and third photodetector signal  552 C and (ii) a combination (e.g., an average, or interpolation) of second photodetector signal  552 B and fourth photodetector signal  552 D can be performed similarly to the differential displacement measurement along the x-axis using (i) the first photodetector signal  452 A and (ii) the second photodetector signal  452 B, as described above in connection with  FIG. 4B . 
     Moreover, translation of the optical pattern  118  along the y-axis through the macro-beams  150 A,  150 B,  150 C,  150 D, causes changes of the first photodetector signal  552 A, changes of the second photodetector signal  552 B, changes of the third photodetector signal  552 C and changes of the fourth photodetector signal  552 D.  FIG. 5C  is a graph  551 Y which shows changes of the first photodetector signal  552 A, changes of the second photodetector signal  552 B, changes of the third photodetector signal  552 C and changes of the fourth photodetector signal  552 D as a function of displacement ΔY of the datum of the optical pattern  118  relative to the macro-beams  150 A,  150 B,  150 C,  150 D. Here, when the optical pattern  118  is translated from bottom-to-top on page without being translated left-or-right on page, the following occurs: (1) As beams of the first macro-beam  150 A and beams of the second macro-beam  150 B transition over the first edge  121 YA parallel to the x-axis, from the second portion  122  to the first portion  120 , the intensity of each of the reflected first macro-beam  150 A and the reflected second macro-beam  150 B increases, and so does each of the respective first photodetector signal  552 A and the second photodetector signal  552 B; and (2) as beams of the third macro-beam  150 C and beams of the fourth macro-beam  150 D transition over the second edge  121 YB parallel to the x-axis, from the first portion  120  to the second portion  122 , the intensity of each of the reflected third macro-beam  150 C and the reflected fourth macro-beam  150 D decreases, and so does each of the respective third photodetector signal  552 C and the fourth photodetector signal  552 D. With the exception of aspect ratio, notice the symmetry between X and Y axes when comparing  FIGS. 5B and 5C . 
     As such, displacement ΔY of the optical pattern  118  along the y-axis through the macro-beams  150 A,  150 B,  150 C,  150 D can be determined in any one or more of the following manners. For example, a differential displacement measurement along the y-axis using (i) first photodetector signal  552 A and (ii) third photodetector signal  552 C can be performed similarly to the differential displacement measurement along the x-axis using (i) the first photodetector signal  452 A and (ii) the second photodetector signal  452 B, as described above in connection with  FIG. 4B . As another example, a differential displacement measurement along the y-axis using (i) second photodetector signal  552 B and (ii) fourth photodetector signal  552 D can be performed similarly to the differential displacement measurement along the x-axis using (i) the first photodetector signal  452 A and (ii) the second photodetector signal  452 B, as described above in connection with  FIG. 4B . As yet another example, a differential displacement measurement along the y-axis using (i) a combination (e.g., an average, or interpolation) of first photodetector signal  552 A and second photodetector signal  552 B and (ii) a combination (e.g., an average, or interpolation) of third photodetector signal  552 C and fourth photodetector signal  552 D can be performed similarly to the differential displacement measurement along the x-axis using (i) the first photodetector signal  452 A and (ii) the second photodetector signal  452 B, as described above in connection with  FIG. 4B . 
     Note that, when the macro-beams  150 A,  150 B,  150 C,  150 D are concurrently output by the respective LEE arrays  110 A,  110 B,  110 C,  110 D of the optical source  106 , i.e., without using time multiplexing, then a photodetector signal should have a constant value, regardless of whether the optical pattern  118  is at rest or it is translated in the (x,y) plane. As any deviation from this would be caused by irregularities (e.g., damage of, dust particles on, etc. a surface) of the optical pattern  118 , a self-health-check can be performed by translating the optical pattern  118  over its full range of motion, e.g., MAX ΔX, while the LEE arrays  110 A,  110 B,  110 C,  110 D are activated concurrently. 
     Moreover, a saturation point of the intensity of the light reflected by the optical pattern  118  depends on physical dimensions of the optical pattern and of LEE arrays  110 A,  110 B,  110 C,  110 D. For example, the saturation point (i.e., the maximum level of a photodetector signal) can be determined for (i) a given geometry and size of the optical pattern  118 , (ii) a given geometry and size of the LEE arrays  110 A,  110 B,  110 C,  110 D, (iii) a given arrangement of the optical pattern relative to the LEE arrays, e.g., Z-offset or another offset in the (x,y) plane, etc. Once determined, the saturation point can be used as a self-calibration reference. 
       FIGS. 6A-6B  show that when the area inside a haptic engine  130  and/or cost of the haptic engine is a restriction, the number of LEE arrays and an area of the optical pattern  118  can be halved if differential sensing is not needed in one of two directions. In the example shown in  FIG. 6A , a modification of the displacement measuring system  100  includes LEE arrays  110 A,  110 B and an optical pattern  618 A. The optical pattern  618 A corresponds to the top half of the optical pattern  118  described above in connection with  FIG. 1B  and includes two portions  620 A,  622 A that form an edge  621 AY parallel to the x-axis and two edges  621 AXA,  621 AXB parallel to the y-axis. The first LEE array  110 A provides a first macro-beam  650 A that crosses the first edge  621 AXA parallel to the y-axis and the edge  621 AY parallel to the x-axis; and the second LEE array  110 B provides a second macro-beam  650 B that crosses the second edge  621 AXB parallel to the y-axis and the edge  621 AY parallel to the x-axis. In this configuration, the modified displacement measuring system can be used to perform (i) a differential displacement measurement along the x-axis to accurately determine a value of a displacement ΔX, and (ii) a single-ended displacement measurement along the y-axis to only determine whether there is motion along the y-axis. Here, the differential displacement measurement along the x-axis based on the first macro-beam  650 A transitioning over the first edge  621 AXA parallel to the y-axis and the second macro-beam  650 B transitioning over the second edge  621 AXB parallel to the y-axis is performed similarly to the differential displacement measurement along the x-axis based on the first macro-beam  150 A transitioning over the first edge  121 XA parallel to the y-axis and the second macro-beam  150 B transitioning over the second edge  121 XB parallel to the y-axis, as described above in connection with  FIGS. 5A-5B . Additionally, the single-ended displacement measurement along the y-axis based on either the first macro-beam  650 A or the second macro-beam  650 B or both transitioning over the edge  121 AY is performed similarly to the single-ended displacement measurement along the x-axis based on either the first macro-beam  338 A or the second macro-beam  338 B or both transitioning over the edge  321 , as described above in connection with  FIGS. 3A-3B . 
     In the example shown in  FIG. 6B , another modification of the displacement measuring system  100  includes LEE arrays  110 B,  110 D and an optical pattern  618 B. The optical pattern  618 B corresponds to the right half of the optical pattern  118  described above in connection with  FIG. 1B  and includes two portions  620 B,  622 B that form an edge  621 BX parallel to the y-axis and two edges  621 BYA,  621 BYB parallel to the x-axis. The first LEE array  110 B provides a first macro-beam  650 B that crosses the first edge  621 BYA parallel to the x-axis and the edge  621 BX parallel to the y-axis; and the second LEE array  110 D provides a second macro-beam  650 D that crosses the second edge  621 BYB parallel to the x-axis and the edge  621 BX parallel to the y-axis. In this configuration, the modified displacement measuring system can be used to perform (i) a differential displacement measurement along the y-axis to accurately determine a value of a displacement ΔY, and (ii) a single-ended displacement measurement along the x-axis to only determine whether there is motion along the x-axis. Here, the differential displacement measurement along the y-axis based on the first macro-beam  650 B transitioning over the first edge  621 BYA parallel to the x-axis and the second macro-beam  650 D transitioning over the second edge  621 BYB parallel to the x-axis is performed similarly to the differential displacement measurement along the y-axis based on the second macro-beam  150 B transitioning over the first edge  121 YA parallel to the x-axis and the fourth macro-beam  150 D transitioning over the second edge  121 YB parallel to the x-axis, as described above in connection with  FIG. 5A  and  FIG. 5C . Additionally, the single-ended displacement measurement along the x-axis based on either the first macro-beam  650 B or the second macro-beam  650 D or both transitioning over the edge  121 BX is performed similarly to the single-ended displacement measurement along the x-axis based on either the first macro-beam  338 A or the second macro-beam  338 B or both transitioning over the edge  321 , as described above in connection with  FIGS. 3A-3B . 
     Additionally, the disclosed displacement measuring system  100  can be further modified to also determine angular displacements. Such an angular displacement measuring system can be used in conjunction with a crown of a watch device (e.g., a setting wheel that functions as a switch, as it translates sideways, and as a scroller, as it rotates around a center axis), a wheel of a pointing device, a rotating hinge for a screen of a laptop computer, etc. 
       FIG. 7A  is a side view, e.g. in the (x,z) plane, of an example of an angular displacement measuring system  700 . The angular displacement measuring system  700  includes a mount  704 , a light source  706  supported by the mount, an optical pattern  718  disposed on the edge (perimeter) surface of a wheel  734  such that optical pattern is spaced apart from the light source, a photodetector  724  supported by the mount, and processing electronics  725  coupled with the photodetector. 
     The optical pattern  718  wraps around the perimeter surface of the wheel  734 , as shown in  FIG. 7B . In this example, the optical pattern  718  has a background portion  722  (shown in grey) that wraps around the perimeter surface of the wheel  734 , and multiple rectangular portions  720 A,  720 B,  720 C, etc. (shown in white) distributed around the perimeter surface of the wheel that form corresponding rectangular edges  721 A,  721 B,  721 C, etc. Note that each rectangular edge  721  has two azimuthal sides parallel to a direction of rotation θ, and two transverse sides across the direction of rotation θ (i.e., parallel to the y-axis). Here, each of the rectangular portions  720 A,  720 B, etc. has a first reflectivity R 1 , and the background portion  722  has a second reflectivity R 2 , smaller than the first reflectivity. Moreover, in this example, a size of each rectangular portion  720  along the direction of rotation θ is substantially equal to a separation between adjacent rectangular portions. 
     In the above-noted uses of the angular displacement measuring system  700 , where a direction and magnitude of angular displacement is to be measured over multiple periods of the optical pattern  718 , a 3-phase architecture can be used as described below.  FIG. 7C  is a plan view, e.g., in the (x,y) plane, of the components of the angular displacement measuring system  700  supported by the mount  704 . In this example, the light source  706  includes a first pair of LEE arrays  710 A,  710 B, a second pair of LEE arrays  710 C,  710 D, a third pair of LEE arrays  710 E,  710 F, and the photodetector  724 . Each of the LEE arrays  710 A,  710 B,  710 C,  710 D,  710 E,  710 F and the photodetector  724  can be implemented, as described above in connection with  FIG. 1C , as an LEE array  110  and as the photodetector  124 , respectively. The LEE arrays  710 A,  710 B,  710 C,  710 D,  710 E,  710 F are configured to output light beams grouped in respective macro-beams  850 A,  850 B,  850 C,  850 D,  850 E,  850 F, such that the macro-beams illuminate, during operation of the angular displacement measuring system  700 , the optical pattern  718  across the rectangular edges  721 A,  721 B,  721 C, etc., as shown in  FIG. 8A . Note that in this example, a length of each macro-beam  850  is substantially equal to the size of each rectangular portion  720  along the direction of rotation θ. Moreover, the LEE arrays  710 A,  710 B,  710 C,  710 D,  710 E,  710 F output respective macro-beams  850 A,  850 B,  850 C,  850 D,  850 E,  850 F in a time multiplexed manner. 
     The optical pattern  718  reflects to the photodetector  724  the macro-beam output by each LEE array  710  that impinges on the optical pattern. Note that in this example, the macro-beams  850 B,  850 E emitted by the second pair of LEE arrays  710 B,  710 E are reflected to the photodetector  724  in a backward direction, such that a cross-section of the second pair of reflected macro-beams  850 B,  850 E is parallel to the surface of the photodetector  724 . However, because of the curved configuration of the optical pattern  718 , the macro-beams  850 A,  850 D (or  850 C,  850 F) emitted by the first pair of LEE arrays  710 A,  710 D (or third pair of LEE arrays  710 A,  710 D) are reflected to the photodetector  724  in a direction tilted relative to the backward direction, such that a cross-section of the first pair of reflected macro-beams  850 A,  850 D (or third pair of reflected macro-beams  850 C,  850 F) form an angle of +120° (or −120°) to the surface of the photodetector  724 . 
     The photodetector  724  captures the reflected macro-beam  850  associated with each LEE array  710  and integrates it. As a result of integrating the captured reflected macro-beam  850 , the photodetector  724  issues, in a time multiplexed manner, a respective photodetector signal relating to the associated macro-beam  850 . Note that the processing circuitry  725  can combine, e.g., average or interpolate together, the photodetector signals associated with the macro-beams of a pair of macro-beams to determine a combined photodetector signal associated with that pair of macro-beams. As such, a first combined photodetector signal  856 AD associated with the first pair of macro-beams  850 A,  850 D is determined by combining a first photodetector signal associated with the first macro-beam  850 A together with a fourth photodetector signal associated with the fourth macro-beam  850 D; a second combined photodetector signal  856 BE associated with the second pair of macro-beams  850 B,  850 E is determined by combining a second photodetector signal associated with the second macro-beam  850 B together with a fifth photodetector signal associated with the fifth macro-beam  850 E; and a third combined photodetector signal  856 CF associated with the third pair of macro-beams  850 C,  850 F is determined by combining a third photodetector signal associated with the third macro-beam  850 C together with a sixth photodetector signal associated with the sixth macro-beam  850 F. 
     Rotation of the of the optical pattern  718  along the direction of rotation θ through the first pair of macro-beams  850 A,  850 D, the second pair of macro-beams  850 B,  850 E, and the third pair of macro-beams  850 C,  850 F, causes changes of the respective combined photodetector signals  856 AD,  856 BE,  856 CF.  FIG. 8B  is a graph  8510  which shows changes of the first combined photodetector signal  856 AD, changes of the second combined photodetector signal  856 BE, and changes of the third combined photodetector signal  856 CF as a function of angular displacement Δθ of a datum of the optical pattern  718  relative to the first pair of macro-beams  850 A,  850 D, the second pair of macro-beams  850 B,  850 E, and the third pair of macro-beams  850 C,  850 F. The datum of the optical pattern  718  can be either one of the corners of either one of the rectangular edges  721 A,  721 B,  721 C, etc., for instance. Here, when the optical pattern  718  is rotated from left-to-right on page without being translated up-or-down on page, the following occurs: (1) As beams of the second pair of macro-beams  850 B,  850 E transition over the rectangular portions  720 A,  720 B,  720 C, etc., the intensity of the reflected second pair of macro-beams  850 B,  850 E forms corresponding peaks, and so does the second combined photodetector signal  856 BE; at the same time, beams of the first pair of macro-beams  850 A,  850 D and beams of the third pair of macro-beams  850 C,  850 F transition over the background portion  722  between corresponding adjacent rectangular portions  720 A,  720 B,  720 C, etc., so each of the intensity of the reflected first pair of macro-beams  850 A,  850 D and the intensity of the reflected third pair of macro-beams  850 C,  850 F forms corresponding valleys, and so does each of the first combined photodetector signal  856 AD and the third combined photodetector signal  856 CF; and (2) as beams of the second pair of macro-beams  850 B,  850 E transition over the background portion  722  between adjacent rectangular portions  720 A,  720 B,  720 C, etc., the intensity of the reflected second pair of macro-beams  850 B,  850 E forms corresponding valleys, and so does the second combined photodetector signal  856 BE; at the same time, beams of the first pair of macro-beams  850 A,  850 D and beams of the third pair of macro-beams  850 C,  850 F transition over the corresponding rectangular portions  720 A,  720 B,  720 C, etc., so each of the intensity of the reflected first pair of macro-beams  850 A,  850 D and the intensity of the reflected third pair of macro-beams  850 C,  850 F forms corresponding peaks, and so does each of the first combined photodetector signal  856 AD and the third combined photodetector signal  856 CF. 
     In this manner, angular displacement Δθ of the optical pattern  718  along the direction of rotation θ through the first pair of macro-beams  850 A,  850 D, the second pair of macro-beams  850 B,  850 E, and the third pair of macro-beams  850 C,  850 F can be determined in the following manner. A value of the first combined photodetector signal  856 AD, a value of the second combined photodetector signal  856 BE, and a value of the third combined photodetector signal  856 CF are measured for an unknown displacement ΔX. Then, a value of the unknown angular displacement Δθ is unambiguously obtained by mapping onto graph  451  the measured value of the first combined photodetector signal  856 AD, the measured value of the second combined photodetector signal  856 BE, and the measured value of the third combined photodetector signal  856 CF. 
     Moreover, the angular displacement measuring system  700  can be used to perform a differential displacement measurement along the y-axis to accurately determine a value of a displacement ΔY of the optical pattern  718 . Here, the differential displacement measurement along the y-axis based on the second macro-beam  850 B transitioning over a first azimuthal side of rectangular edges  721 A,  721 B,  721 C, etc., and the fifth macro-beam  650 E transitioning over the second azimuthal side of rectangular edges  721 A,  721 B,  721 C, etc., is performed similarly to the differential displacement measurement along the y-axis based on either the first macro-beam  150 A or the second macro-beam  150 B transitioning over the first edge  121 YA parallel to the x-axis, and the respective third macro-beam  150 C or fourth macro-beam  150 D transitioning over the second edge  121 YB parallel to the x-axis, as described above in connection with  FIG. 5A  and  FIG. 5C . 
     In some implementations, processing electronics  125  or  725  can be configured in analog electronics. Here, the analog electronics include one or more filters, subtractors, dividers, comparators, and other analog electronics components for performing operations described in this specification. In some implementations, processing electronics  125  or  725  can be configured as mixed signal circuitry that processes analog signals and digital signals. In some implementations, processing electronics  125  or  725  can be configured as one or more digital signal processors, e.g., ASIC, FPGA, CPU, etc. 
       FIG. 9A  is a side view, e.g. in the (x,z) plane, of an example of a displacement measuring system  900  configured to measure displacement of a mass  1164 . Here, the displacement measuring system  900  includes an optical sensing system  902 , a back electromotive force (bEMF) sensing system  930  and a processor  925  coupled with both the optical sensing system and the bEMF sensing system. 
     In this example, the optical sensing system  902  includes a mount  904 , a light source  906  supported by the mount, an intensity pattern  910  disposed on a surface  1165 XYA of the mass  1164  that is spaced apart from the light source, and a photodetector  920  supported by the mount and coupled with the processing electronics  925 . During operation of the optical sensing system  902 , the light source  906  illuminates the intensity pattern  910  with probe light  908 , and the intensity pattern redirects to the photodetector  920 , e.g., via reflection or scattering, at least some of the light impinging thereon. 
       FIG. 9B  is a plan view, e.g., in the (x,y) plane, of the intensity pattern  910  that includes a plurality of tiles  912  separated from each other by corresponding tile borders  913 . Note that each tile has a size larger than a beam spot  916  formed by the probe light  908  that illuminates the intensity pattern  910 . For instance, a size of each of the tiles  912  can be 1.1, 1.5, 2× of the size of the beam spot  916 . In some implementations, the light source  906  includes a VCSEL that emits collimated probe light that is delivered to the intensity pattern  910  as a probe beam (also referenced as  908 ). In some implementations, the light source  906  includes an LED and beam-shaping optics that are optically coupled with the LED. In this case, the LED emits un-collimated light, and the beam-shaping optics receive the un-collimated light and issue the probe light  908 , either as a probe beam or at least as focused probe light. In this manner, in either these implementations, a size of the beam spot  916  can be 20 μm, 10 μm, or smaller. 
     Moreover, each tile  912  of the intensity pattern  910  is configured to redirect to the photodetector  920  light having an intensity different from an intensity of light redirected to the photodetector by any of its adjacent tiles. In the example shown in  FIG. 9B , the intensity pattern  910  is a binary intensity pattern because each tile  912 A (or  912 B) has only two adjacent tiles  912 B (or  912 A) configured to redirect to the photodetector  920  light having the same intensity. As such, the binary intensity pattern  910  has tiles of first type  912 A and tiles of second type  912 B, where each tile of first type  912 A forms respective tile borders  913  with two adjacent tiles of second type  912 B, and each tile of second type  912 B forms respective tile borders  913  with two adjacent tiles of first type  912 A. A tile of first type  912 A is configured such that, when the probe beam  908  illuminates it, the redirected light  918  has a maximum intensity I MAX . Further, a tile of second type  912 B is configured such that, when the probe beam  908  illuminates it, the redirected light  918  has a minimum intensity I MIN , where I MIN &lt;I MAX . Here, the tile borders  913  are distributed at known locations relative to each other along the x-axis, so the binary intensity pattern  910  of the optical sensing system  902  can be used as part of the displacement measuring system  900  to measure displacement ΔX of the mass  1164  along the x-axis. An example of a four-level intensity pattern  1210  of the optical sensing system  902  that can be used as part of the displacement measuring system  900  to measure, as described below in connection with FIGS.  12 A- 12 I, displacement ΔX of the mass  1164  along the x-axis and displacement ΔY of the mass along the y-axis. 
     Referring again to  FIG. 9A , the photodetector  920  captures the redirected light  918  associated with the light source  906  and integrates it. In some implementations, the photodetector  920  can be implemented as a photodiode, e.g., a PIN photodiode. A result of integrating the captured light is a raw intensity value I(t) relating to the intensity of the light redirected by the intensity pattern  910  to the photodetector  920  at sampling time t. The photodetector  920  includes a threshold module that classifies the raw intensity value I(t) against a threshold value Th. In the example illustrated in  FIG. 9C , the threshold module of the photodetector  920  can set I(t)=I MIN  if I(t)≤Th, or I(t)=I MAX  if I(t)&gt;Th. In some implementations, the threshold value Th can be predetermined. In other implementations, the threshold value Th can be updated adaptively. For instance, a statistic&lt;I&gt; S  of the last K values of raw intensity that are smaller than the threshold value Th, and another statistic&lt;I&gt; L  of the last K values of raw intensity that are larger than the threshold value Th can be used to reset the threshold value to Th=(&lt;I&gt; L −&lt;I&gt; S )/2. Here, K≥2 and the statistic&lt;I&gt; can be an average, truncated average, median, maximum or minimum, for instance. In this manner, the photodetector  920  issues an intensity signal  922  that can have only two values {I MIN , I MAX } and is related to the intensity of the light redirected by the intensity pattern  910  to the photodetector, as the intensity pattern carried by the mass  1164  is displaced along the x-axis relative to the probe beam  908 . 
       FIG. 9C  further shows changes in the intensity signal  922  caused by multiple tile border crossings that occur as the intensity pattern  910  carried by the mass  1164  is displaced along the x-axis relative to the probe beam  908 . A tile border crossing is said to occur when the intensity pattern  910  is displaced, along a direction of motion that intersects the tile borders  913 , such that a tile border  913 , formed between a tile of first type  912 A and a tile of second type  912 B, crosses through the beam spot  916  associated with the static probe beam  908 . In this manner, the probe beam  908  illuminates the tile of first type  912 A (or the tile of second type  912 B) before the tile border crossing and illuminates the tile of second type  912 B (or the tile of first type  912 A) after the tile border crossing, such that the tile border crossing causes a predefined decrease “−ΔI” (or a predefined increase “+ΔI”) in the intensity of the redirected light  918  between I MAX  and I MIN . In the example illustrated in  FIG. 9C , the intensity signal  922  indicates that (i) tile border crossings from a tile of second type  912 B to a tile of first type  912 A have occurred at times t 1  and t 3 , where the tile border crossing times t 1 , t 3  are determined as the times when the intensity signal increases by +ΔI=I MAX −I MIN ; and (ii) tile border crossings from a tile of first type  912 A to a tile of second type  912 B have occurred at times t 2  and t 4 , where the tile border crossing times t 2 , t 4  are determined as the times when the intensity signal decreases by −ΔI=−(I MAX −I MIN ). Further in the example illustrated in  FIG. 9C , the intensity signal  922  indicates that the probe beam  906  has illuminated a tile of second type  912 B from the start time to time t 1 , between times t 2  and t 3  and after time t 4  until the final time; and a tile of first type  912 A between times t 1  and t 2 , and times t 3  and t 4 . 
     However, the foregoing information that can be extracted from the intensity signal  922  shown in  FIG. 9C  is insufficient for uniquely determining the displacement ΔX of the intensity pattern  910  carried by the mass  1164  along the x-axis relative to the probe beam  908 . For instance,  FIG. 9D  shows only a few from among many possible scans  926  of the intensity pattern  910  carried by the mass  1164  along the x-axis relative to the probe beam  908 . Here, a tile border crossing  928  is represented by an arrow over the tile border that is being crossed. Also note that a tilt is used to represent the border crossings  928  in  FIG. 9D  to suggest their sequence in time. Note that although each of the scans  926 A,  926 B,  926 C,  926 D includes four tile border crossings  928 , the scans have different starting and ending points, span different displacement ranges, as explained below. 
     For scan  926 A, the beam spot  916  starts at X 3  corresponding to the center of the third tile of the intensity pattern  910  and ends back at X 3 ; at t 1 , a first tile border crossing occurs from the third tile to the fourth tile, and then the beam spot  916  reaches X 4  corresponding to the center of the fourth tile; next at t 2 , a second tile border crossing occurs from the fourth tile to the third tile, and then the beam spot  916  reaches X 3  again; next at t 3 , a third tile border crossing occurs from the third tile to the fourth tile, and then the beam spot  916  reaches X 4  again; next at t 4 , a fourth tile border crossing occurs from the fourth tile to the third tile, and then the beam spot  916  returns back to X 3 . Thus, the largest displacement ΔX of the intensity pattern  910  relative to the probe beam  908  during scan  926 A is of order (X 4 −X 3 ), e.g., approximately the pitch of the intensity pattern. Here, the pitch of the intensity pattern  910  includes a tile of first type  912 A and a tile of second type  912 B. 
     For scan  926 B, the beam spot  916  starts at X 3  corresponding to the center of the third tile of the intensity pattern  910  and ends back at X 3 ; at t 1 , a first tile border crossing occurs from the third tile to the fourth tile, and then the beam spot  916  reaches X 4  corresponding to the center of the fourth tile; next at t 2 , a second tile border crossing occurs from the fourth tile to the fifth tile, and then the beam spot  916  reaches X 5  corresponding to the center of the fifth tile; next at t 3 , a third tile border crossing occurs from the fifth tile to the fourth tile, and then the beam spot  916  reaches X 4  again; next at t 4 , a fourth tile border crossing occurs from the fourth tile to the third tile, and then the beam spot  916  returns back to X 3 . Thus, the largest displacement ΔX of the intensity pattern  910  relative to the probe beam  908  during scan  926 B is of order (X 5 −X 3 ), e.g., approximately 1.5 times the pitch of the intensity pattern. 
     For scan  926 C, the beam spot  916  starts at X 3  corresponding to the center of the third tile of the intensity pattern  910  and ends at X 1  corresponding to the first tile of the intensity pattern; at t 1 , a first tile border crossing occurs from the third tile to the fourth tile, and then the beam spot  916  reaches X 4  corresponding to the center of the fourth tile; next at t 2 , a second tile border crossing occurs from the fourth tile to the third tile, and then the beam spot  916  reaches X 3  again; next at t 3 , a third tile border crossing occurs from the third tile to the second tile, and then the beam spot  916  reaches X 2  corresponding to the center of the second tile; next at t 4 , a fourth tile border crossing occurs from the second tile to the first tile, and then the beam spot  916  ends up at X 1  corresponding to the center of the first tile. Thus, the largest displacement ΔX of the intensity pattern  910  relative to the probe beam  908  during scan  926 C is of order (X 1 −X 4 ), e.g., approximately twice the pitch of the intensity pattern. 
     For scan  926 D, the beam spot  916  starts at X 1  corresponding to the center of the first tile of the intensity pattern  910  and ends at X 5  corresponding to the center of the fifth tile of the intensity pattern; at t 1 , a first tile border crossing occurs from the first tile to the second tile, and then the beam spot  916  reaches X 2  corresponding to the center of the second tile; next at t 2 , a second tile border crossing occurs from the second tile to the third tile, and then the beam spot  916  reaches X 3  corresponding to the center of the third tile; next at t 3 , a third tile border crossing occurs from the third tile to the fourth tile, and then the beam spot  916  reaches X 4  corresponding to the center of the fourth tile; next at t 4 , a fourth tile border crossing occurs from the fourth tile to the fifth tile, and then the beam spot  916  ends up at X 5 . Thus, the largest displacement ΔX of the intensity pattern  910  relative to the probe beam  908  during scan  926 D is of order (X 5 −X 1 ), e.g., 2.5 times the pitch of the intensity pattern. 
     In order to determine, e.g., from among the scans  926 A,  926 B,  926 C,  926 D shown in  FIG. 9D , the actual scan used by the optical sensing system  902  to acquire the intensity signal  922  shown in  FIG. 9C , the processor  925  has to spatially resolve the tile border crossings indicated by the intensity signal. A tile border crossing is said to be spatially resolved when the processor  925  specifies both the tile border at which the tile border crossing has occurred, and the direction in which the tile border crossing has occurred. In order to spatially resolve the tile border crossings indicated by the intensity signal  922 , the processor  925  uses additional information provided by the bEMF sensing system  930 . Note that in the example shown in  FIG. 9A , the bEMF sensing system  930  includes a board  932 , a coil  934  supported by the board, and a magnet  936  disposed on a surface  1165 XYB of the mass  1164  opposite the surface  1165 XYA on which the intensity pattern  910  is disposed. An output port  938  of the bEMF sensing system  930  is coupled internally with the coil  934  and externally with the processor  925 . The magnet  936  is arranged relative to the coil  934  to interact with it when the magnet moves along with mass  1164  relative to the coil. In the example shown in  FIG. 9A , the coil  934  is at rest relative to a datum of the displacement measuring system  900  (e.g., a position of the probe beam  908 ) and the longitudinal axis of the coil is parallel to the x-axis, as the magnet  936  can move inside the coil along the longitudinal axis of the coil. 
     In this manner, a bEMF signal v(t)  923  is induced in the coil  934  when the magnet  936  is displaced along the x-axis together with the mass  1164 . A value of the bEMF signal v(t)  923  is proportional to a magnitude of a velocity of the mass  1164  along the x-axis, such that if the value of the bEMF signal is zero, then the mass it at rest; also, if the bEMF signal v(t)  923  increases (or decreases), then the velocity of the mass increases (or decreases). Moreover, a sign of the bEMF signal v(t)  923  indicates a direction along the x-axis of the velocity of the mass  1164 , and, thus, a direction of the displacement ΔX. For example, if a value of the bEMF signal v(t)  923  is positive (or negative), then the mass is displaced forward (or backward) along the x-axis. Moreover, the bEMF sensing system  930  includes an integrator module that integrates over time the bEMF signal v(t)  923  induced in the coil  934  and obtains a first displacement signal ΔX bEMF (t)  940  associated with the displacement ΔX. In this manner, given reasonable analog-to-digital (ADC) resolution, the bEMF sensing system  930  issues, at its output port  938 , with good differential resolution, the first displacement signal ΔX bEMF (t)  940  associated with the displacement ΔX of the mass  1164  that has been acquired concurrently with the intensity signal  922 . In this manner, the processor  925  can extract additional information from the first displacement signal ΔX bEMF (t)  940  acquired by the bEMF sensing system  930 , and will use the extracted information to spatially resolve the tile border crossings indicated by the intensity signal  922 , as described below. 
       FIG. 9E  shows a first example of a bEMF signal v(t)  923 A induced in the coil  934  of the bEMF sensing system  930  over the same time duration when the intensity signal  922  shown in  FIG. 9C  was acquired by the optical sensing system  902 . Note that in this case, the bEMF signal v(t)  923 A indicates that the mass  1164  to which the magnet  936  is attached starts its motion at rest, then it moves in the positive direction of the x-axis over a first time interval T 1  until it comes to a first stop, then it moves in the negative direction of the x-axis over a second time interval T 2  until it comes to a second stop, then it moves again in the positive direction of the x-axis over a third time interval T 3  until it comes to a third stop, and then it moves again in the negative direction of the x-axis over a fourth time interval T 4  until it comes to a fourth and final stop.  FIG. 9G  shows a first displacement signal ΔX bEMF (t)  940 A obtained by the bEMF sensing system  930  by integrating the bEMF signal v(t)  923 A shown in  FIG. 9E . Note that in this first case, the first displacement signal ΔX bEMF (t)  940 A indicates that the mass  1164  is displaced forward from about X 3  (e.g., the center of the third tile) to about X 4  (e.g., the center of the fourth tile) over the first time interval T 1  and third time interval T 3 , and backward from about X 4  to about X 3  over the second time interval T 2  and fourth time interval T 4 . 
     At this point, the processor  925  can combine information extracted from the intensity signal  922  shown in  FIG. 9C  with information extracted from the bEMF signal v(t)  923 A shown in  FIG. 9E  and with information extracted from the first displacement signal ΔX bEMF (t)  940 A shown in  FIG. 9F  to obtain a second displacement signal ΔX OPT (t)  942 A shown in  FIG. 9G , in the following manner. The processor  925  uses the intensity signal  922  to determine the respective times t 1 , t 2 , t 3 , t 4  when tile border crossings have occurred, as explained above in connection with  FIG. 9C . Further, the processor  925  uses the bEMF signal v(t)  923 A to determine that, at t 1 , the first tile border crossing occurs in the forward x-axis direction because, during the first time interval T 1  which includes t 1 , the mass moves in the forward x-axis direction, as explained above in connection with  FIG. 9E ; at t 2 , the second tile border crossing occurs in the backward x-axis direction because, during the second time interval T 2  which includes t 2 , the mass moves in the backward x-axis direction; at t 3 , the third tile border crossing occurs in the forward x-axis direction because, during the third time interval T 3  which includes t 3 , the mass moves in the forward x-axis direction; and at t 4 , the fourth tile border crossing occurs in the backward x-axis direction because, during the fourth time interval T 4  which includes t 4 , the mass moves in the backward x-axis direction. As the foregoing tile border crossings alternate in direction, they must correspond to crossings of a single tile border. Furthermore, the first displacement signal ΔX bEMF (t)  940 A is used to determine that the single tile border that is crossed back and forth is the tile border  913 ( 3 , 4 ) which separates the third tile and the fourth tile, because the mass moves back and forth along the x-axis direction between the third tile and the fourth tile, as explained above in connection with  FIG. 9F . Note that the second displacement signal ΔX OPT (t)  942 A obtained in this manner is represented in  FIG. 9G  by filled diamonds with arrows. In this case, all the samples of the second displacement signal ΔX OPT (t)  942 A are plotted at a coordinate corresponding to the precise location of the tile border  913 ( 3 , 4 ), and direction of the respective arrows corresponds to the direction of the tile border crossings at respective times t 1 , t 2 , t 3 , t 4 . Further note that the signals shown in  FIGS. 9E, 9F and 9G  correspond to scan  926 A shown in  FIG. 9D . 
       FIG. 9I  shows a second example of a bEMF signal v(t)  923 B induced in the coil  934  of the bEMF sensing system  930  over the same time duration when the intensity signal  922  shown in  FIG. 9C  was acquired by the optical sensing system  902 . Note that in this case, the bEMF signal v(t)  923 B indicates that the mass  1164  to which the magnet  936  is attached starts its motion at rest, then it moves in the positive direction of the x-axis over a first time interval T 1  until it comes to a first stop, then it moves in the negative direction of the x-axis over a second time interval T 2  until it comes to a second and final stop.  FIG. 9J  shows a first displacement signal ΔX bEMF (t)  940 B obtained by the bEMF sensing system  930  by integrating the bEMF signal v(t)  923 B shown in  FIG. 9I . Note that in this second case, the first displacement signal ΔX bEMF (t)  940 B indicates that the mass  1164  is displaced forward from about X 3  (e.g., the center of the third tile) through X 4  (e.g., the center of the fourth tile) to about X 5  (e.g., the center of the fifth tile) over the first time interval T 1 , and backward from about X 5  to about X 3  over the second time interval T 2 . 
     At this point, the processor  925  can combine (i) information extracted from the intensity signal  922  shown in  FIG. 9C  with (ii) information extracted from the bEMF signal v(t)  923 B shown in  FIG. 9I  and with (iii) information extracted from the first displacement signal ΔX bEMF (t)  940 B shown in  FIG. 9J  to obtain a second displacement signal ΔX OPT (t)  942 B shown in  FIG. 9K , in the following manner. The processor  925  uses the intensity signal  922  to determine the respective times t 1 , t 2 , t 3 , t 4  when tile border crossings have occurred, as explained above in connection with  FIG. 9C . Further, the processor  925  uses the bEMF signal v(t)  923 B to determine that, at t 1  and at t 2 , the first tile border crossing and the second tile border crossing respectively occur in the forward x-axis direction because, during the first time interval T 1  which includes both t 1  and t 2 , the mass moves in the forward x-axis direction, as explained above in connection with  FIG. 9E ; and at t 3  and t 4 , the third tile border crossing and the fourth tile border crossing respectively occur in the backward x-axis direction because, during the second time interval T 2  which includes both t 3  and t 4 , the mass moves in the backward x-axis direction. As the first two of the foregoing tile border crossings have the same forward direction, they must correspond to respective crossings in the forward direction of both tile borders of a single tile, and as the last two of the foregoing tile border crossings have the same backward direction, they must correspond to respective crossings in the backward direction of both tile borders of the same tile. Furthermore, the first displacement signal ΔX bEMF (t)  940 B is used to determine that the single tile that is crossed back and forth is the fourth tile having the center at X 4  and sharing tile border  913 ( 3 , 4 ) with the third tile and tile border  913 ( 4 , 5 ) with the fifth tile, because the mass moves back and forth along the x-axis direction between the third tile and the fifth tile, as explained above in connection with  FIG. 9J . Note that the second displacement signal ΔX OPT (t)  942 B obtained in this manner is represented in  FIG. 9K  by filled diamonds with arrows. In this case, a first sample of the second displacement signal ΔX OPT (t)  942 B is plotted at a coordinate corresponding to a location of a first tile border  913 ( 3 , 4 ) of the fourth tile, and a second sample of the second displacement signal ΔX OPT (t)  942 B is plotted at a coordinate corresponding to a location of a second tile border  913 ( 4 , 5 ) of the fourth tile, and the forward direction of each of the arrows of the first two samples corresponds to the forward direction of the tile border crossings at respective times t 1 , t 2 . Further in this case, a third sample of the second displacement signal ΔX OPT (t)  942 B is plotted at the coordinate corresponding to the location of the second tile border  913 ( 4 , 5 ) of the fourth tile, and a fourth sample of the second displacement signal ΔX OPT (t)  942 A is plotted at the coordinate corresponding to the location of the first tile border  913 ( 3 , 4 ) of the fourth tile, and the backward direction of each of the arrows of the last two samples corresponds to the backward direction of the tile border crossings at respective times t 3 , t 4 . Note that the signals shown in  FIGS. 9I, 9J and 9K  correspond to scan  926 B shown in  FIG. 9D . 
     Moreover, the second displacement signal ΔX OPT (t)  942 A (or  942 B) determined based in part on the intensity signal  922  acquired by the optical sensing system  902  has more absolute accuracy than the first displacement signal ΔX bEMF (t)  940 A (or  940 B) acquired by the bEMF sensing system  930 , because the optical sensing system provides integrated non-linearity (INL) that is lower than the INL of the bEMF sensing system. However, note that as the tile borders  913  are distributed at known locations relative to each other along the x-axis, the spatial resolution of second displacement signal ΔX OPT (t)  942 A (or  942 B) is limited by a photomask lithography process. Hence, the first displacement signal ΔX bEMF (t)  940 A (or  940 B) acquired by the bEMF sensing system  930 , as shown in  FIG. 9F  (or  FIG. 9J ) may be available at a larger sampling rate than the sampling rate used by the processor  925  to determine the second displacement signal ΔX OPT (t)  942 A (or  942 B), as shown in  FIG. 9G  (or  FIG. 9K ). As such, the processor  925  can further determine a displacement signal ΔX(t)  944 A (or  944 B) by interpolating the second displacement signal ΔX OPT (t)  942 A (or  942 B) with appropriately scaled sampling values of the first displacement signal ΔX bEMF (t)  940 A (or  940 B).  FIG. 9H  shows a displacement signal ΔX(t)  944 A determined by the processor  925  by interpolating the second displacement signal ΔX OPT (t)  942 A with appropriately scaled sampling values of the first displacement signal ΔX bEMF (t)  940 A. And  FIG. 9L  shows a displacement signal ΔX(t)  944 B determined by the processor  925  by interpolating the second displacement signal ΔX OPT (t)  942 B with appropriately scaled sampling values of the first displacement signal ΔX bEMF (t)  940 B. 
       FIG. 10A  shows an example of an interpolator module  1045  for interpolating the second displacement signal ΔX OPT (t)  942  (examples of which are shown in either of  FIG. 9G or 9K ) using the first displacement signal ΔX bEMF (t)  940  (examples of which are shown in either of  FIG. 9F or 9J ) to obtain a displacement signal ΔX(t)  944  (examples of which are shown in  FIG. 9H or 9L ). A digitized version of the first displacement signal ΔX bEMF (t)  940  acquired by the bEMF sensing system  930  can be stored in a first buffer  1046 ; here, samples of the first displacement signal ΔX bEMF (t)  940  are denoted b[n] and are sampled at a first sampling rate. The second displacement signal ΔX OPT (t)  942 , determined by the processor  925  based at least on the intensity signal  922  acquired by the optical sensing system  902 , can be stored in a second buffer  1048 ; here, samples of the second displacement signal ΔX OPT (t)  942  are denoted p[m] and are sampled at a second sampling rate that can be smaller than the first sampling rate. The displacement signal ΔX(t)  944 , obtained by the interpolator module  1045 , can be stored in a third buffer  1059 ; here, samples of the displacement signal ΔX(t)  944  are sampled at the first sampling rate.  FIG. 10B  shows a portion of a displacement signal ΔX(t)  944  obtained, as described below, by the interpolator module  1045  and stored in the third buffer  1059 . 
     The interpolator module  1045  includes a subtractor  1050  linked to both the first and second buffers  1046 ,  1048 , and a differentiator  1055  linked to the first buffer  1046 . The interpolator module  1045  further includes a divider  1052  linked to the subtractor  1050 , and a filter  1054  linked to the divider. Furthermore, the interpolator module  1045  includes a multiplier  1056  linked to both the differentiator  1055  and the filter  1054 . Also, the interpolator module  1045  includes an adder  1051  linked to the second buffer  1048 . Additionally, the interpolator module  1045  includes a multiplexer  1058  linked to the second buffer  1048 , and an accumulator  1057  linked to both the multiplier  1056  and the multiplexer. 
     The interpolator module  1045  receives a sample p[m] of the second displacement signal ΔX OPT (t)  942  from the second buffer  1048  and passes it through to the multiplexer  1058  to be output as a non-interpolated term p[m] of the displacement signal ΔX(t)  944  to the third buffer  1059 . Then, the multiplexer  1058  is switched to output k sequential interpolated terms of the displacement signal ΔX(t)  944 , where k≥2, that are obtained in the following manner. 
     Prior to calculating the k interpolated terms, the subtractor  1050  determines a change b[n]−b[n−k] of the first displacement signal ΔX bEMF (t)  940  over k of its samples and a change p[m]−p[m−j] of the second displacement signal ΔX OPT (t)  942  over j of its samples, where j≥1, and the divider  1052  determines a scale factor C as a ratio of the foregoing changes. In some implementations, the filter  1054  filters the scale factor C and outputs a filtered scale factor C to increase stability of the interpolation. Also prior to calculating the k interpolated terms, the accumulator  1057  is initialized to zero. 
     To calculate the first interpolated term, the differentiator  1055  outputs a first change (b[n+1]−b[n]) of the first displacement signal ΔX bEMF (t)  940 . The multiplier  1056  scales the first change output by the differentiator  1055  to obtain C(b[n+1]−b[n]). As the accumulator  1057  has been initialized to zero, the first scaled change C(b[n+1]−b[n]) is simply passed through the accumulator to the adder  1051 . The adder  1051  adds the first output of the accumulator  1057  to the sample p[m] of the second displacement signal ΔX OPT (t)  942 , so the interpolator module  1045  can use the multiplexer  1058  to output the first interpolated term p[m]+C(b[n+1]−b[n]) to the third buffer  1059 . To calculate the second interpolated term, the differentiator  1055  outputs a second change (b[n+2]−b[n+1]) of the first displacement signal ΔX bEMF (t)  940 . The multiplier  1056  scales the second change output by the differentiator  1055  to obtain C(b[n+2]−b[n+1]). As the accumulator  1057  has held the first scaled change C(b[n+1]−b[n]), the accumulation thereof with the second scaled change C(b[n+2]−b[n+1]) results in C(b[n+2]−b[n]) which is output to the adder  1051 . The adder  1051  adds the second output of the accumulator  1057  to the sample p[m] of the second displacement signal ΔX OPT (t)  942 , so the interpolator module  1045  can use the multiplexer  1058  to output the second interpolated term p[m]+C(b[n+2]−b[n]) to the third buffer  1059 . And so, and so forth, such that the k th  interpolated term output by the interpolator module  1045  to the third buffer  1059  is p[m]+C(b[n+k−1]−b[n]). 
     Then, the multiplexer  1058  is switched to output the next non-interpolated term. Here, the interpolator module  1045  receives a sample p[m+1] of the second displacement signal ΔX OPT (t)  942  from the second buffer  1048  and passes it through to the multiplexer  1058  to be output as the next non-interpolated term p[m+1] of the displacement signal ΔX(t)  944  to the third buffer  1059 . The foregoing operations are then iterated as necessary. Note that a portion of a displacement signal ΔX(t)  944 , that has been obtained as described above, is plotted  FIG. 10B . 
     Moreover, background calibration of the interpolator module  1045  can be performed by using the intensity pattern  910  as absolute displacement ΔX reference, as described below. For instance, the filter  1054  can conditionally update the scale factor C after a change p[m]−p[m−j] of the second displacement signal ΔX OPT (t)  942  over j of its samples exceeds a certain displacement threshold. Setting a larger threshold (e.g., integrating over multiple tile border crossings) will improve calibration accuracy, especially if the second sampling rate of the second displacement signal ΔX OPT (t)  942  is low, but it would also reduce the update rate of the background calibration. In practice, the temperature coefficient of the coil  934  used to acquire the first displacement signal ΔX bEMF (t)  942  is on the order of seconds, while a motion frequency of the mass  1164  inside a haptic engine is usually &gt;50 Hz, so calibration rate is not expected to be a problem. 
     Note that the displacement signal ΔX(t)  944  determined by the processor  925 , when the displacement measuring system  900  is arranged relative the moving mass  1164  as shown in  FIG. 9A , is indicative of displacements ΔX of the mass along the x-axis. Other displacements can be determined by the processor  925  depending on positioning of the displacement measuring system  900  inside a haptic engine that includes the mass  1164 , or by using extended capability of the displacement measuring system.  FIGS. 11A-11B  show example implementations of a haptic engine  1160 . In each of these implementations the haptic engine  1160  has a frame  1162  that encompasses the mass  1164  and the displacement measuring system  900 . 
       FIG. 11A  is a side view, in the (y,z) plane, of a first implementation of the haptic engine  1160 . Here, the optical sensing system  902  of the displacement measuring system  900  is arranged such that the intensity pattern  910  is supported by the mass  1164  on a surface  1165 XYA parallel to the (x,y) plane, and the mount  904  is supported by the frame  1162  on a face parallel to the (x,y) plane. Here, the tile borders  913  of the intensity pattern  910  are oriented along the y-axis. Additionally, the bEMF sensing system  930  of the displacement measuring system  900  is arranged such that the magnet  936  is coupled with the mass  1164  at a surface  1165 XYB parallel to the (x,y) plane and opposite to the surface  1165 XYA, and the coil  934  is coupled with the frame  1162  at another face parallel to the (x,y) plane. Alternatively, the bEMF sensing system  930  can be arranged such that the magnet  936  is coupled with the mass  1164  at the same surface  1165 XYA on which the intensity pattern  910  is supported, and the coil  934  is coupled with the frame  1162  at the same face on which the mount  904  is supported. In either of these arrangements, the longitudinal axis of the coil  934  is oriented parallel to the x-axis. In this first implementation of the haptic engine  1160 , the processor  925  determines a displacement ΔX of the mass  1164  based on corresponding changes in the first displacement signal ΔX bEMF (t)  942  and the second displacement signal ΔX OPT (t)  942  caused by motion of the mass along the x-axis (e.g., vibration in-and-out of page). In other implementations, two instances of the bEMF sensing system  930  can be used in conjunction with one instance of the optical sensing system  902 . For instance, a first instance of the bEMF sensing system  930  can be arranged on surface  1165 XYA and a second instance of the bEMF sensing system  930  can be arranged on surface  1165 XYB. 
       FIG. 11B  is a side view, in the (y,z) plane, of a second implementation of the haptic engine  1160 . Here, the optical sensing system  902  of the displacement measuring system  900  is arranged such that the intensity pattern  910  is supported by the mass  1164  on a surface  1165 XZA parallel to the (x,z) plane, and the mount  904  is supported by the frame  1162  on a face parallel to the (x,z) plane. Here, the tile borders  913  of the intensity pattern  910  are oriented along the x-axis. Additionally, the bEMF sensing system  930  of the displacement measuring system  900  is arranged such that the magnet  936  is coupled with the mass  1164  at a surface  1165 XZB parallel to the (x,z) plane and opposite to the surface  1165 XZA, and the coil  934  is coupled with the frame  1162  at another face parallel to the (x,z) plane. Alternatively, the bEMF sensing system  930  can be arranged such that the magnet  936  is coupled with the mass  1164  at the same surface  1165 XZA on which the intensity pattern  910  is supported, and the coil  934  is coupled with the frame  1162  at the same face on which the mount  904  is supported. In either of these arrangements, the longitudinal axis of the coil  934  is oriented parallel to the z-axis. In this second implementation of the haptic engine  1160 , the processor  925  determines a displacement ΔZ of the mass  1164  based on corresponding changes in the first displacement signal ΔX bEMF (t)  942  and the second displacement signal ΔX OPT (t)  942  caused by motion of the mass along the z-axis (e.g., vibration up-and-down on page). In other implementations, two instances of the bEMF sensing system  930  can be used in conjunction with one instance of the optical sensing system  902 . For instance, a first instance of the bEMF sensing system  930  can be arranged on surface  1165 XZA and a second instance of the bEMF sensing system  930  can be arranged on surface  1165 XZB. 
     In order to also measure a displacement ΔY of the mass  1164  along the y-axis, e.g., corresponding to vibration left-and-right on page, concurrently with the displacement ΔX of the mass caused along the x-axis, e.g., corresponding to vibration in-and-out of page, an additional instance of the displacement measuring system  900  can be disposed adjacent to the instance of the displacement measuring system  900  shown in  FIG. 11A , for instance. In this case, the additional instance of the displacement measuring system  900  is arranged to have (i) the tile borders  913  of the intensity pattern  910  oriented along the x-axis, and (ii) the longitudinal axis of the coil  934  oriented parallel to the y-axis. 
     Alternatively, the optical sensing system  902  of the displacement measuring system  900  can be modified to allow for concurrently measuring the displacement ΔX of the mass along the x-axis, e.g., corresponding to vibration in-and-out of page, and the displacement ΔY of the mass along the y-axis, e.g., corresponding to vibration left-and-right on page. A first modification of the optical sensing system  902  includes replacement of the intensity pattern  910  with an intensity pattern that encodes 2D spatial information, e.g., in the (x,y) plane, as described below in connection with  FIGS. 12A-12C . Another modification of the optical sensing system  902  includes a modified configuration of the optical source  906  that provides, as described below in connection with  FIGS. 12A and 12D-12E , a pair of reference beams in addition to the probe beam  908 . 
       FIG. 12A  is a plan view, e.g., in the (x,y) plane, of an intensity pattern  1210  that includes a plurality of hexagonal-shaped tiles  1212 , each of which configured to redirect to the photodetector  920  light having an intensity different from an intensity of light redirected to the photodetector by any of its six adjacent tiles. In this example, the intensity pattern  1210  is a four-level intensity pattern that has tiles of first type  1212 A, tiles of second type  1212 B, tiles of third type  1212 A and tiles of fourth type  1212 D arranged to form a unit cell  1214 . Note that the honeycomb arrangement of the intensity pattern  1210  can be generated by translating the unit cell  1214  in any of the eight directions E, NE, N, NW, W, SW, S, SE. 
       FIG. 12B  shows that a tile of first type  1212 A is configured such that, when the probe beam  908  illuminates it, the redirected light  918  has an intensity I A  equal to maximum intensity I MAX , I A =I MAX . A tile of fourth type  1212 D is configured such that, when the probe beam  908  illuminates it, the redirected light  918  has an intensity I D  equal to minimum intensity I MIN , I D =I MIN . A tile of second type  1212 B is configured such that, when the probe beam  908  illuminates it, the redirected light  918  has a first intermediary intensity I B  that satisfies I MIN &lt;I B &lt;I MAX . And, a tile of third type  1212 C is configured such that, when the probe beam  908  illuminates it, the redirected light  918  has a second intermediary intensity I C  that satisfies I MIN &lt;I C &lt;I B . The four types of tiles can redirect light having any four arbitrary intensity levels. The above case is only an exemplary embodiment. As described above in connection with  FIG. 9C , a photodiode of the photodetector  920  integrates, at a sampling time t, the redirected light  918  to provide a raw intensity value I(t) a threshold module. In the example shown in  FIG. 12B , the threshold module of the photodetector  920  classifies the raw intensity value I(t) against threshold values Th AB , Th BC  and Th CD  in the following manner. If I(t)&gt;Th AB , then the threshold module of the photodetector  920  can set I(t)=I A . If I(t)≤Th CD , then the threshold module of the photodetector  920  can set I(t)=I D . If Th BC ≤I(t)&lt;Th AB , then the threshold module of the photodetector  920  can set I(t)=I B . If Th CD ≤I(t)&lt;Th BC , then the threshold module of the photodetector  920  can set I(t)=I C . In some implementations, the threshold values Th AB , Th BC  and Th CD  can be predetermined. In other implementations, the threshold values Th AB , Th BC  and Th CD  can be updated in an adaptive manner, as described below. In this manner, the photodetector  920  issues an intensity signal  1222  (shown in  FIG. 12F ) that can have only four values {I D =I MIN , I C , I B , I A =I MAX } and is related to the intensity of the light redirected by the intensity pattern  1210  to the photodetector, as the intensity pattern carried by the mass  1164  is displaced along the x-axis and the y-axis relative to the probe beam  908 . 
     Referring again to  FIG. 12A , the light source  906  provides, in addition to the probe beam  908  that forms the probe beam spot  916  as it illuminates a tile of the intensity pattern  1210 , a first reference beam that forms a first reference beam spot  1216 A and a second reference beam that forms a second reference beam spot  1216 B. Note that a separation between each of the probe beam spot  916 , the first reference beam spot  1216 A, and the second reference beam spot  1216 B matches the separation between adjacent tiles of the intensity pattern  1210 , such that the three beams provided by the light source  906  always illuminate tiles of different types. In some implementations, the light source  906  includes an array of three VCSELs that emit the probe beam  908  and the pair of the reference beams. In other implementations, the light source  906  includes an array of three LEDs that emit non-collimated light that are coupled with one or more beam forming optics to provide the probe beam  908  and the pair of the reference beams.  FIG. 12C  shows that redirected light associated with each of the first reference beam  1216 A and the second reference beam  1216 B can be sampled at slower rates relative to the sampling rate of redirected light  918  associated with the probe beam  908  to save power and bandwidth. Referring again to  FIG. 12B , the light source  906 &#39;s drive level and the photodetector  920 &#39;s ADC level can be adjusted to correspond to an appropriate one of the levels I A , I B , I C , I D  of the intensity of the redirected light, based on differences between the detected intensity levels corresponding to the probe beam  908  relative to the reference beams. 
     Moreover, in the example shown in  FIG. 12A , the first reference beam spot  1216 A and the second reference beam spot  1216 B are cast on adjacent tiles located NE and E, respectively, relative to the tile on which the probe beam spot  916  is cast. In this manner, once the processor  925  determines a type of a tile on which the probe beam spot  916  is currently located, a combination of a level of intensity of light redirected by the tile illuminated by the probe beam, a first level of intensity of light redirected by a first adjacent tile illuminated by the first reference beam, and a second level of intensity of light redirected by a second adjacent tile illuminated by the second reference beam can be only one from among combinations  1268  shown in  FIG. 12B . 
     In accordance with combination  1268 A, when the probe beam spot  916  is cast on a tile of first type  1212 A, the first and second reference beam spots  1216 A,  1216 B are cast on adjacent tiles of second type  1212 B and fourth type  1212 D located NE and E, respectively, relative to the tile of first type  1212 A. In accordance with combination  1268 B, when the probe beam spot  916  is cast on a tile of second type  1212 B, the first and second reference beam spots  1216 A,  1216 B are cast on adjacent tiles of first type  1212 A and third type  1212 C located NE and E, respectively, relative to the tile of second type  1212 B. In accordance with combination  1268 C, when the probe beam spot  916  is cast on a tile of third type  1212 C, the first and second reference beam spots  1216 A,  1216 B are cast on adjacent tiles of fourth type  1212 D and second type  1212 B located NE and E, respectively, relative to the tile of third type  1212 C. In accordance with combination  1268 D, when the probe beam spot  916  is cast on a tile of fourth type  1212 D, the first and second reference beam spots  1216 A,  1216 B are cast on adjacent tiles of third type  1212 C and first type  1212 A located NE and E, respectively, relative to the tile of fourth type  1212 D. 
     The combinations  1268 A,  1268 B,  1268 C and  1268 D of three signal levels issued by the photodetector  920  can be used to adaptively update the threshold values Th AB , Th BC , Th CD  in the following manner. For instance, the photodetector  920  can issue, for a sampling time t and the first combination  1268 A, a first set of raw intensity values {I 116 (t), I 416A (t), I 416B (t)} corresponding to the redirected probe beam  908 , the first reference beam, and the second reference beam, respectively, that will be classified as a first set of intensity values {I A , I B , I D }; for a sampling time t′ and the second combination  1268 B, a second set of raw intensity values {I 166 (t′), I 416A (t′), I 416B (t′)} corresponding to the redirected probe beam  908 , the first reference beam, and the second reference beam, respectively, that will be classified as a second set of intensity values {I B , I A , I C }; for a sampling time t″ and the third combination  1268 C, a third set of raw intensity values {I 116 (t″), I 416A (t″), I 416B (t″)} corresponding to the redirected probe beam  908 , the first reference beam, and the second reference beam, respectively, that will be classified as a third set of intensity values {I C , I D , I B }; for a sampling time t″ and the fourth combination  1268 D, a fourth set of raw intensity values {I 116 (t′″), I 416A (t′″), I 416B (t′″)} corresponding to the redirected probe beam  908 , the first reference beam, and the second reference beam, respectively, that will be classified as a fourth set of intensity values {I D , I C , I A }; and so on and so forth. 
     Moreover, the photodetector can include a digital low pass filter (LPF)  1270  configured to obtain a first statistic&lt;I&gt; A  of the raw intensity values that have been associated with the intensity value I A , where &lt;I&gt; A =&lt;{I 116 (t), I 416A (t′), I 416B (t′″), . . . }&gt;; a second statistic&lt;I&gt; B  of the raw intensity values that have been associated with the intensity value I B , where &lt;I&gt; B =&lt;{I 416A (t), I 116 (t′), I 416B (t″), . . . }&gt;; a third statistic&lt;I&gt; C  of the raw intensity values that have been associated with the intensity value I C , where &lt;I&gt; C =&lt;{I 416B (t′), I 116 (t″), I 416A (t′″), . . . }&gt;; and a fourth statistic&lt;I&gt; D  of the raw intensity values that have been associated with the intensity value I D , where &lt;I&gt; D =&lt;{I 416B (t), I 416A (t″), I 116 (t′″), . . . }&gt;. Here, the statistic denoted&lt;I&gt; X  can be a median, mean, truncated mean, maximum or minimum of a set of raw intensity values classified as the intensity level I X , where the subscript X is {A, B, C or D}. In this manner, the digital LPF can adaptively update the threshold Th AB  used by the threshold module to set the intensity values I A  or I B  as Th AB =(1/2)(&lt;I&gt; A −&lt;I&gt; B ); the threshold Th BC  used by the threshold module to set the intensity values I B  or I C  as Th BC =(1/2)(&lt;I&gt; B −&lt;I&gt; C ); and the threshold Th CD  used by the threshold module to set the intensity values I C  or I D  as Th CD =(1/2)(&lt;I&gt; C −&lt;I&gt; D ). The foregoing operations performed by the digital LPF of the photodetector  920  amount to a procedure for self-calibrating (also referred to as a procedure for background calibrating) the optical sensing system  902  that uses the intensity pattern  1210 . 
       FIG. 12D  shows that the four types of tiles of the intensity pattern  1210  form six possible tile borders  1213  across which the intensity of the redirected light  918  changes by a fraction of the intensity range I MAX −I MIN . In the first row of  FIG. 12D , the intensity of redirected light  918  changes by (1/3)(I MAX −I MIN ) for each of a tile border  1213 AB formed between a tile of first type  1212 A and a tile of second type  1212 B, a tile border  1213 BC formed between a tile of second type  1212 B and a tile of third type  1212 C, and a tile border  1213 CD formed between a tile of third type  1212 C and a tile of fourth type  1212 D. In the second row of  FIG. 12D , the intensity of redirected light  918  changes by (2/3)(I MAX −I MIN ) for each of a tile border  1213 AC formed between a tile of first type  1212 A and a tile of third type  1212 C, and a tile border  1213 BD formed between a tile of second type  1212 B and a tile of fourth type  1212 D. And, in the third row of  FIG. 12D , the intensity of redirected light  918  changes by (I MAX −I MIN ) for a tile border  1213 AD formed between a tile of first type  1212 A and a tile of fourth type  1212 D. 
     Referring again to  FIG. 12A , note that the tile borders of the intensity pattern  1210  are distributed at known locations relative to each other in the (x,y) plane, so this intensity pattern can be used, as part of the optical sensing system  902  of the displacement measuring system  900 , by the processor  925  to concurrently measure the mass  1164 &#39;s displacement ΔX along the x-axis and displacement ΔY along the y-axis. Note that during the motion of the mass  1164 , the intensity pattern  1210  moves (along with the mass) relative to a probe beam spot  916  corresponding to the probe beam  908  that illuminates the intensity pattern, one tile at a time. In this manner, multiple tile border crossings will occur as the intensity pattern  1210  is displaced in the (x,y) plane, where for each tile border crossing, a tile border formed between adjacent tiles of different type crosses through the beam spot  916  associated with the static probe beam  908 . 
       FIG. 12E  shows that for each type of tile, there are 6 possible tile border crossings, three out of which are ambiguously paired. When the probe beam  908  illuminates a tile of first type  1212 A, first tile border crossings  1228 A will occur when (i) the tile border  1213 AB crosses the probe beam, so the probe beam then illuminates either the tile of second type  1212 B that is disposed on the NE side of the tile of first type  1212 A or the tile of second type  1212 B that is disposed on the SW of the tile of first type  1212 A; (ii) the tile border  1213 AD crosses the probe beam, so the probe beam then illuminates either the tile of fourth type  1212 D that is disposed on the E side of the tile of first type  1212 A or the tile of fourth type  1212 D that is disposed on the W of the tile of first type  1212 A; and (iii) the tile border  1213 AC crosses the probe beam, so the probe beam then illuminates either the tile of third type  1212 C that is disposed on the SE side of the tile of first type  1212 A or the tile of third type  1212 C that is disposed on the NW of the tile of first type  1212 A. 
     Further, when the probe beam  908  illuminates a tile of fourth type  1212 D, fourth tile border crossings  1228 D will occur when (i) the tile border  1213 CD crosses the probe beam, so the probe beam then illuminates either the tile of third type  1212 C that is disposed on the NE side of the tile of fourth type  1212 D or the tile of third type  1212 C that is disposed on the SW of the tile of fourth type  1212 D; (ii) the tile border  1213 AD crosses the probe beam, so the probe beam then illuminates either the tile of first type  1212 A that is disposed on the E side of the tile of fourth type  1212 D or the tile of first type  1212 A that is disposed on the W of the tile of fourth type  1212 D; and (iii) the tile border  1213 BD crosses the probe beam, so the probe beam then illuminates either the tile of second type  1212 B that is disposed on the SE side of the tile of fourth type  1212 D or the tile of second type  1212 B that is disposed on the NW of the tile of fourth type  1212 D. 
     Furthermore, when the probe beam  908  illuminates a tile of second type  1212 B, second tile border crossings  1228 B will occur when (i) the tile border  1213 AB crosses the probe beam, so the probe beam then illuminates either the tile of first type  1212 A that is disposed on the NE side of the tile of second type  1212 B or the tile of first type  1212 A that is disposed on the SW of the tile of second type  1212 B; (ii) the tile border  1213 BC crosses the probe beam, so the probe beam then illuminates either the tile of third type  1212 C that is disposed on the E side of the tile of second type  1212 B or the tile of third type  1212 C that is disposed on the W of the tile of second type  1212 B; and (iii) the tile border  1213 BD crosses the probe beam, so the probe beam then illuminates either the tile of fourth type  1212 D that is disposed on the SE side of the tile of second type  1212 B or the tile of fourth type  1212 D that is disposed on the NW of the tile of second type  1212 B. 
     Also, when the probe beam  908  illuminates a tile of third type  1212 C, third tile border crossings  1228 C will occur when (i) the tile border  1213 CD crosses the probe beam, so the probe beam then illuminates either the tile of fourth type  1212 D that is disposed on the NE side of the tile of third type  1212 C or the tile of fourth type  1212 D that is disposed on the SW of the tile of third type  1212 C; (ii) the tile border  1213 BC crosses the probe beam, so the probe beam then illuminates either the tile of second type  1212 D that is disposed on the E side of the tile of third type  1212 C or the tile of second type  1212 B that is disposed on the W of the tile of third type  1212 C; and (iii) the tile border  1213 AC crosses the probe beam, so the probe beam then illuminates either the tile of first type  1212 A that is disposed on the SE side of the tile of third type  1212 C or the tile of first type  1212 A that is disposed on the NW of the tile of third type  1212 C. 
     Referring again to  FIG. 12A , a probe beam spot  916  corresponding to the probe beam  908  illuminates (i) a tile of first type  1212 A at a departure point (labeled “START”) when the motion of the mass  1164  begins, and (ii) a tile of fourth type  1212 D at an arrival point (labeled “END”) when the motion of the mass ends. During this motion of the mass, there have been successive tile border crossings at unknown points P C1 , P C2 , P C3 . The times when these tile border crossings occur and their exact location in the (x,y) plane is determined by the processor  925  in the following manner. 
     An intensity signal  1222  issued by the photodetector  920  relates to the intensity of the light redirected by the intensity pattern  1210  to the photodetector.  FIG. 12F  shows changes in the intensity signal  1222  caused by the multiple tile border crossings that occur as the intensity pattern  1210  carried by the mass  1164  is displaced relative to the probe beam spot  916  from the departure point to the arrival point. As described above in connection with  FIG. 9C , the tile border crossings occur at times t 1 , t 2 , t 3  corresponding to predefined changes of the intensity signal  1222 , e.g., corresponding ones the predefined changes shown in  FIG. 12D : ±(1/3)ΔI, ±(2/3)ΔI, or ±ΔI, where ΔI=(I MAX −I MIN ). 
     Additionally, a first displacement signal ΔX bEMF (t)  1240 X associated with the displacement ΔX of the mass  1164  along the x-axis is acquired by the bEMF sensing system  930  concurrently with the intensity signal  1222 .  FIG. 12G  shows the first displacement signal ΔX bEMF (t)  1240 X acquired, from a start time is to an end time t E , while the mass  1164  is displaced relative to the probe beam spot  916  from the departure point to the arrival point. As a slope of the first displacement signal ΔX bEMF (t)  1240 X is negative (due to a non-zero westward component of the mass  1164 &#39;s velocity) over the entire time interval (t E −t S ), the processor  925  determines that the direction of motion of the mass  1164  has a non-zero westward component over the entire time interval (t E −t S ). 
     Referring again to  FIG. 12F , the processor  925  determines that the first tile border crossing that occurs at t 1  causes a decrease of 2/3(I MAX −I MIN ) of the intensity signal  1222 . As such, the processor  925  determines, based on the second row of  FIG. 12D , that the first tile border crossing that occurs at t 1  is across either (i) a tile border  1213 AC, in which case the probe beam spot  916  transitions onto a tile of third type  1212 C; or (ii) a tile border  1213 BD, in which case the probe beam spot  916  transitions onto a tile of fourth type  1212 D. In the first case (i), the first reference spot  1216 A transitions onto a tile of fourth type  1212 D and the second reference spot  1216 B transitions onto a tile of second type  1212 B. In the second case (ii), the first reference spot  1216 A transitions onto a tile of third type  1212 C and the second reference spot  1216 B transitions onto a tile of first type  1212 A. In the example illustrated in  FIG. 12A , the difference between intensity of redirected light associated with the probe beam spot  916  and intensity of redirected light associated with the first reference spot  1216 A is −(1/3)(I MAX −I MIN ); and the difference between intensity of redirected light associated with the probe beam spot  916  and intensity of redirected light associated with the second reference spot  1216 B is +(1/3)(I MAX −I MIN ). This corresponds to case (i), and contradicts case (ii). In this manner, the processor  925  determines that, from the start time t S  to the time t 1  when the first tile border crossing occurs across a tile border  1213 AC, the probe beam spot  916  is cast on a tile of first type  1212 A; and, from the time t 1  to the time t 2  when a second tile border crossing occurs, the probe beam spot  916  is cast on a tile of third type  1212 C. Then, the processor  925  determines, based on  FIG. 12E , that the probe beam spot  916  is cast on either the tile of third type  1212 C that is disposed on the SE side of the tile of first type  1212 A or the tile of third type  1212 C that is disposed on the NW side of the tile of first type  1212 A. Here, the processor  925  uses the previous determination, made based on the first displacement signal ΔX bEMF (t)  1240 X, that the direction of motion of the mass  1164  has a non-zero westward component over the entire time interval (t E −t S ). 
     In this manner, the processor  925  determines that, from the start time t S  to the time t 1  (when the first tile border crossing occurs across a tile border  1213 AC), the probe beam spot  916  is cast on a tile of first type  1212 A; and, from the time t 1  to the time t 2  (when a second tile border crossing occurs), the probe beam spot  916  is cast on a tile of third type  1212 C that is disposed on the NW side of the tile of first type  1212 A. As such, by time t 1  when the first tile border crossing occurs, the mass  1164  has moved about one size of a tile westward and about one half of the size of the tile northward from the starting point. 
     Further, the processor  925  determines that the second tile border crossing that occurs at t 2  causes an increase of 1/3(I MAX −I MIN ) of the intensity signal  1222 . As such, the processor  925  determines, based on the first row of  FIG. 12D , that the second tile border crossing that occurs at t 2  is across a tile border  1213 BC, in which case the probe beam spot  916  transitions from the current tile of third type  1212 C onto a tile of second type  1212 B. Then, the processor  925  determines, based on  FIG. 12E , that the probe beam spot  916  is cast on either the tile of second type  1212 B that is disposed on the E side of the tile of third type  1212 C or the tile of second type  1212 B that is disposed on the W side of the tile of third type  1212 C. Here, the processor  925  uses the previous determination, made based on the first displacement signal ΔX bEMF (t)  1240 X, that the direction of motion of the mass  1164  has a non-zero westward component over the entire time interval (t E −t S ). 
     In this manner, the processor  925  determines that, from the time t 1  (when the first tile border crossing occurs across a tile border  1213 AC) to the time t 2  (when the second tile border crossing occurs across a tile border  1213 BC), the probe beam spot  916  is cast on a tile of third type  1212 C; and, from the time t 2  to the time t 3  (when a third tile border crossing occurs), the probe beam spot  916  is cast on a tile of second type  1212 B that is disposed on the W side of the tile of third type  1212 C. As such, by time t 2  when the second tile border crossing occurs, the mass  1164  has moved about two sizes of the tile westward and about one size of the tile northward from the starting point. 
     Furthermore, the processor  925  determines that the third tile border crossing that occurs at t 3  causes a decrease of 2/3(I MAX −I MIN ) of the intensity signal  1222 . As such, the processor  925  determines, based on the second row of  FIG. 12D , that the third tile border crossing that occurs at t 3  is across a tile border  1213 BD, in which case the probe beam spot  916  transitions from the current tile of second type  1212 B onto a tile of fourth type  1212 D. Then, the processor  925  determines, based on  FIG. 12E , that the probe beam spot  916  is cast on either the tile of fourth type  1212 D that is disposed on the SE side of the tile of second type  1212 B or the tile of fourth type  1212 D that is disposed on the NW side of the tile of second type  1212 B. Here, the processor  925  uses the previous determination, made based on the first displacement signal ΔX bEMF (t)  1240 X, that the direction of motion of the mass  1164  has a non-zero westward component over the entire time interval (t E −t S ). 
     In this manner, the processor  925  determines that, from the time t 2  (when the second tile border crossing occurs across a tile border  1213 BC) to the time t 3  (when the third tile border crossing occurs across a tile border  1213 BD), the probe beam spot  916  is cast on a tile of second type  1212 B; and, from the time t 3  to the end time t E , the probe beam spot  916  is cast on a tile of fourth type  1212 D that is disposed on the NW side of the tile of second type  1212 B. As such, by time t 3  when the third tile border crossing occurs, the mass  1164  has moved about three sizes of the tile westward and more than one and a half size of the tile northward from the starting point. 
     The processor  925  uses the foregoing information determined based on (i) the changes of the intensity signal  1222  and (ii) the slope of the first displacement signal ΔX bEMF (t)  1240 X to determine a second displacement signal ΔX OPT (t)  1242 X associated with the displacement ΔX of the mass  1164  along the x-axis, and a second displacement signal ΔY OPT (t)  1242 Y associated with the displacement ΔY of the mass  1164  along the y-axis.  FIG. 12H  shows the second displacement signal ΔY OPT (t)  1242 Y associated with the displacement ΔY of the mass  1164  along the y-axis corresponding to the trip in the (x,y) plane from the starting point to the ending point, as shown in  FIG. 12A . Note that, as the second displacement signal ΔY OPT (t)  1242 Y includes only the y-coordinates of the crossing points P C1 , P C2 , P C3 , a resolution of the second displacement signal ΔY OPT (t)  1242 Y is about 10 μm to 50 μm corresponding to a feature size of the intensity pattern  1210 . 
       FIG. 12I  shows (represented by filled diamonds) the second displacement signal ΔX OPT (t)  1242 X associated with the displacement ΔX of the mass  1164  along the x-axis corresponding to the trip in the (x,y) plane from the starting point to the ending point, as shown in  FIG. 12A . Note that, as the second displacement signal ΔX OPT (t)  1242 X includes only the x-coordinates of the crossing points P C1 , P C2 , P C3 , a resolution of the second displacement signal ΔX OPT (t)  1242 X is about 10 μm to 50 μm, corresponding to a feature size of the intensity pattern  1210 . 
       FIG. 12I  also shows a displacement signal ΔX(t)  1244 X associated with the displacement ΔX of the mass  1164  along the x-axis corresponding to the trip in the (x,y) plane from the starting point to the ending point, as shown in  FIG. 12A . Here, the processor  925  determines the displacement signal ΔX(t)  1244 X by interpolating the previously determined second displacement signal ΔX OPT (t)  1242 X with the first displacement signal ΔX bEMF (t)  1240 X, by using the interpolation technique described above in connection with  FIG. 10A . Note that, because the displacement signal ΔX(t)  1244 X includes, in addition to the x-coordinates of the crossing points P C1 , P C2 , P C3 , the x-components of the points between the starting point and the ending point of the trip as determined by the bEMF sensing system  930 , a resolution of the displacement signal ΔX(t)  1244 X is about 1 μm, corresponding to the resolution of the bEMF sensing system. 
     Referring again to  FIG. 9A , several optical structures can be used to redirect the probe beam  908  that illuminates the intensity pattern  910  to the photodetector  920 . In some implementations, these optical structures are configured to use a reflective intensity pattern  910 . In other implementations, these optical structures are configured to use a transmissive intensity pattern  910  in conjunction with one or more reflective elements. 
       FIG. 13A  shows an example of an optical structure  1310 A that includes a substrate  1374  having a first surface  1376  and a second surface  1378  opposing the first surface. The first surface  1376  is to be coupled with a surface  1165  of the mass  1164  that is spaced apart from and facing the light source  906  and the photodetector  920 . The second surface  1378  is configured with a structure that includes facets that are tilted relative to the probe beam  908 . Here, a reflective intensity pattern  910  is attached to the second surface  1378 , such that the probe beam  908  illuminates the reflective intensity pattern, one-tile-at-a-time. A tilt of the facets of the second surface  1378  is configured such that the reflective intensity pattern  910  redirects the probe beam  908  as a redirected beam  918  to the photodetector  920 , regardless of a displacement ΔX along the x-axis of the reflective intensity pattern relative to the probe beam. In this example, different tiles of the reflective intensity pattern  910  have different reflectivities, such that light redirected to the photodetector  920  from adjacent tiles has different intensities. 
       FIG. 13B  shows another example of an optical structure  1310 B that includes a micro-mirror array  1380  to be mounted on a surface  1165  of the mass  1164  that is spaced apart from and facing the light source  906  and the photodetector  920 . A transmissive intensity pattern  910  is coupled with the micro-mirror array  1380 , such that the latter is sandwiched between the surface  1165  of the mass  1164  and the transmissive intensity pattern  910 . As such, the probe beam  908  illuminates the transmissive intensity pattern  910 , one-tile-at-a-time. In this manner, the probe beam  908  is selectively transmitted through the transmissive intensity pattern  910  to reflectors  1382  of the micro-mirror array  1380 . The reflectors  1382  of the micro-mirror array  1380  are tilted relative to the transmitted probe beam  908  to reflect it as a redirected beam  918  to the photodetector  920 , regardless of a displacement ΔX along the x-axis of the transmissive intensity pattern  910  relative to the probe beam. In this example, different tiles of the transmissive intensity pattern  910  have different transmissivities, such that light redirected to the photodetector  920  from adjacent tiles has different intensities. 
       FIG. 13C  shows another example of an optical structure  1310 C that includes a substrate  1375  having a first surface  1376  and a second surface  1380  opposing the first surface. The substrate  1375  includes glass, plastic, or other material that is transparent to light provided by the light source  906 . The first surface  1376  is to be coupled with a surface  1165  of the mass  1164  that is spaced apart from and facing the light source  906  and the photodetector  920 . The optical structure  1310 C further includes a scattering layer  1382  sandwiched between the first surface  1376  and the surface  1165  of the mass  1164 . A transmissive intensity pattern  910  is disposed on the second surface  1380 , such that the probe beam  908  illuminates the transmissive intensity pattern, one-tile-at-a-time. In this manner, the probe beam  908  is selectively transmitted through the transmissive intensity pattern  910  and through the substrate  1375  to the scattering layer  1382 . A scattering structure of the scattering layer  1382  is configured such that the scattering layer scatters the transmitted probe beam  908  as a redirected beam  918  to the photodetector  920 , regardless of a displacement ΔX along the x-axis of the transmissive intensity pattern  910  relative to the probe beam. 
       FIG. 14A  shows another example of an optical structure  1410 A that includes a substrate  1375  having a first surface  1484  and a second surface  1380  opposing the first surface. The substrate  1375  includes glass, plastic, or other material that is transparent to light provided by the light source  906 . The optical structure  1410 A further includes a diffusive layer  1382  disposed on the first surface  1484  to render the first surface diffusely reflective. A transmissive intensity pattern  910  is disposed on the second surface  1380 , such that the probe beam  908  illuminates the transmissive intensity pattern, one-tile-at-a-time. In this manner, the probe beam  908  is selectively transmitted through the transmissive intensity pattern  910  and through the substrate  1375  to the diffusely reflective first surface  1484 . The first surface  1484  is further configured with a structure that includes facets that are tilted relative to the transmitted probe beam  908 . A tilt of the facets of the first surface  1484  is configured such that the first surface diffusely reflects the probe beam  908  as a redirected beam  918  to the photodetector  920 , regardless of a displacement ΔX along the x-axis of the reflective intensity pattern relative to the probe beam. 
       FIG. 14B  shows another example of an optical structure  1410 B that includes a substrate  1375  having a first surface  1486  and a second surface  1380  opposing the first surface. The substrate  1375  includes glass, plastic, or other material that is transparent to light provided by the light source  906 . A transmissive intensity pattern  910  is disposed on the second surface  1380 , such that the probe beam  908  illuminates the transmissive intensity pattern, one-tile-at-a-time. In this manner, the probe beam  908  is selectively transmitted through the transmissive intensity pattern  910  and through the substrate  1375  to the first surface  1486 . The first surface  1486  is further configured with a structure that includes facets that are tilted relative to the transmitted probe beam  908 . A tilt of the facets of the first surface  1486  is configured such that the first surface reflects via total internal reflection (TIR) the probe beam  908  as a redirected beam  918  to the photodetector  920 , regardless of a displacement ΔX along the x-axis of the reflective intensity pattern relative to the probe beam. 
       FIG. 14C  shows a technique for mounting the optical structure  1410 A or  1410 B to the moving mass  1164 . Mounting tabs  1488  can be set proud of the light redirecting structure to provide a gap  1490  from the moving mass  1164  of effective height Z GAP . The mounting tabs  1488  also set a datum to allow precision tolerance fitting, e.g., to determine a thickness of adhesive  1492  used to attach the optical structure  1410 A or  1410 B to the moving mass  1164 . When mounting the optical structure  1410 A, the gap  1490  can be an air gap or can be filed with a filler material. When mounting the optical structure  1410 B, the gap  1490  is an air gap. In this case, this air gap is important for maximizing the contrast of refractive index at the first surface  1486  to optimize optical performance. 
     In some implementations, a VCSEL, whether singular or in array form, can have an efficiency of 300 lm/W. In some implementations, the VCSEL(s) of the light source  906  can be operated using peak currents in the range of 1-10 mA, and peak voltages in the range of 1-2V, for a maximum power consumption of 1-10 mW. As this power would be consumed if the VCSEL(s) were run continuously, the actual power used the VCSEL(s) can 100-1000 times smaller, as typical VCSEL(s) operation is duty cycled. 
     As noted above in connection with  FIG. 9C  and  FIG. 12B , the photodetector  920  senses the change in light intensity and uses it to detect tile border crossings between tiles of the intensity pattern, e.g.,  910  and  1210 , as a result of their relative lateral motion ΔX and axial motion ΔY. In some implementations, threshold hysteresis can be used to reliably detect level transition. In such cases, in order to avoid false tile border crossings, a tile border crossing is determined while moving in a forward direction, then the same tile border crossing is determined again while moving in a backward direction. 
     In some implementations, processing electronics  925  can be configured as mixed signal circuitry that processes analog signals and digital signals. In some implementations, processing electronics  925  can be configured as one or more digital signal processors, e.g., ASIC, FPGA, CPU, etc. 
       FIG. 15A  is a side view, e.g. in the (x,z) plane, of an example of a displacement measuring system  1500 A configured to measure displacement of a mass  1534 . Here, a haptic engine  1530 A having a frame  1532 A encapsulates the mass  1534  and at least a portion of the displacement measuring system  1500 A. In this example, the displacement measuring system  1500 A includes a mount  1502 A that has a surface  1503 XY and is attached on an opposing surface to a surface of the frame  1532 A that is parallel to the (x,y) plane. The displacement measuring system  1500 A further includes a light emitting element (LEE) array  1504  disposed on the surface  1503 XY of the mount  1502 A, an intensity pattern  1510 A coupled with a surface  1535 XY of the mass  1534 , and a photodetector  1522  disposed on the same surface  1503 XY of the mount as the LEE array  1504 . A surface  1511  of the intensity pattern  1510 A is spaced apart from and faces both the LEE array  1504  and the photodetector  1522 . In this manner, during operation of the displacement measuring system  1500 A, the LEE array  1504  illuminates the surface  1511  of the intensity pattern  1510 A with N TOT  beams  1506 , and the intensity pattern redirects, to the photodetector  1522 , at least some of the light impinging on the illuminated surface, such that N TOT  redirected beams  1520 A form an acute angle relative the illuminating beams. 
       FIG. 15B  is a side view, e.g. in the (x,z) plane, of another example of a displacement measuring system  1500 B configured to measure displacement of the mass  1534 . Here, another haptic engine  1530 B having a frame  1532 B encapsulates the mass  1534  and at least a portion of the displacement measuring system  1500 B. In this example, the displacement measuring system  1500 B includes a mount  1502 B that has a surface  1503 XY and an angled surface  1503 YZ. In some cases, the angled surface  1530 XZ can be oriented to form a substantially right angle relative to the surface  1503 XY. A surface of the mount  1502 B opposing the surface  1503 XY is attached to a surface of the frame  1532 B that is parallel to the (x,y) plane, and another surface of the mount opposing the surface  1503 YZ is attached to another surface of the frame  1532 B that can be parallel to the (y,z) plane. The displacement measuring system  1500 B further includes the LEE array  1504  disposed on the surface  1503 XY of the mount  1502 B, an intensity pattern  1510 B coupled with a surface  1535 XY of the mass  1534 , and the photodetector  1522  disposed on the surface  1503 YZ of the mount. The intensity pattern  1510 B has the surface  1511  that is spaced apart from and faces the LEE array  1504 , and a surface  1516  that is spaced apart from and faces the photodetector  1522 . In this manner, during operation of the displacement measuring system  1500 B, the LEE array  1504  illuminates the surface  1511  of the intensity pattern  1510 B with the N TOT  beams  1506 , and the intensity pattern redirects, through the surface  1516  to the photodetector  1522 , at least some of the light impinging on the illuminated surface, such that N TOT  redirected beams  1520 B form a folding angle relative to the beams  1506 . In some cases, the folding angle is a substantially right angle. 
       FIG. 15C  is a plan view, e.g., in the (x,y) plane, of the surface  1511  of the intensity pattern  1510 A/ 1510 B. The intensity pattern  1510 A/ 1510 B includes a plurality of tiles  1512  separated from each other by corresponding tile borders  1513 . Each tile  1512  of the intensity pattern  1510 A/ 1510 B is configured to redirect to the photodetector  1522  light having an intensity different from an intensity of light redirected to the photodetector by any of its adjacent tiles. In the example shown in  FIG. 15C , the intensity pattern  1510 A/ 1510 B is a binary intensity pattern because each tile  1512 A (or  1512 B) has only two adjacent tiles  1512 B (or  1512 A) configured to redirect to the photodetector  1522  light having the same intensity. As such, the binary intensity pattern  1510 A/ 1510 B has M=2 tile types: tiles of first type  1512 A and tiles of second type  1512 B, where each tile of first type  1512 A forms respective tile borders  1513  with two adjacent tiles of second type  1512 B, and each tile of second type  1512 B forms respective tile borders  1513  with two adjacent tiles of first type  1512 A. A tile of first type  1512 A is configured such that, when one of the beams  1506  illuminates it, the corresponding one of the redirected beams  1520 A/ 1520 B has a maximum intensity I MAX . Further, a tile of second type  1512 B is configured such that, when one of the beams  1506  illuminates it, the corresponding one of the redirected beams  1520 A/ 1520 B has a minimum intensity I MIN , where I MIN &lt;I MAX . Here, the tile borders  1513  are distributed at known locations relative to each other along the x-axis, so the binary intensity pattern  1510 A/ 1510 B can be used as part of the displacement measuring system  1500 A/ 1500 B to measure displacement ΔX of the mass  1534  along the x-axis. An example of a four-level intensity pattern  1810  of a displacement measuring system  1800 , that has M=4 tile types, can be used to measure, as described below in connection with  FIGS. 18A-18E , displacement ΔX of the mass  1534  along the x-axis and displacement ΔY of the mass along the y-axis. 
     Referring again to  FIG. 15C , in some implementations, the intensity pattern  1510 A can be configured to have a reflective surface  1511  that selectively reflects, scatters or both the illuminating beams  1506  to the photodetector  1522  as redirected beams  1520 A. In such cases, a tile of first type  1512 A (shown in white) has a first reflectivity R 1 , and a tile of second type  1512 B (shown in grey) has a second reflectivity R 2 , smaller than the first reflectivity. For example, the second reflectivity R 2  can be at most half the first reflectivity R 1 , e.g., R 2 =0.3R 1 , 0.1R 1 , 0.05R 1  or other fractions of R 1 . For example, the tile of first type  1512 A can be coated with a reflective film and the tile of second type  1512 B can be coated with an absorptive film. As another example, the tile of first type  1512 A can be coated with a multilayer reflection coating and the tile of second type  1512 B can be coated with a multilayer anti-reflection coating. 
     In some implementations, the intensity pattern  1510 B can be configured as a combination of a transmissive surface  1511 , an array of micro-mirrors  1518  (or micro-prisms, or other redirecting micro-structures), and a transmissive surface  1516 . In this manner, the transmissive surface  1511  selectively transmits the illuminating beams  1506 , the array of micro-mirrors  1518  reflects, scatters or both the selectively transmitted beams, and the transmissive surface  1516  transmits the redirected beams  1520 B to the photodetector  1522 . In such cases, a tile of first type  1512 A (shown in white) has a first transmissivity T 1 , and a tile of second type  1512 B (shown in grey) has a second transmissivity T 2 , smaller than the first transmissivity. For example, the second transmissivity T 2  can be at most half the first transmissivity T 1 , e.g., T 2 =0.3T 1 , 0.1T 1 , 0.05T 1  or other fractions of T 1 . 
       FIG. 15C  also shows that the N TOT  beams  1506  illuminate an area  1514  of the surface  1511  of the intensity pattern  1510 A/ 1510 B with discrete beam spots  1508  separated by a known separation δ. The separation δ between adjacent beam spots determines the spatial resolution of the displacement measurements performed by the displacement measuring system  1500 A/ 1500 B. For the examples of displacement measurement systems described in this specification, the separation δ between adjacent beam spots can be as small as about 1 μm, and as large as a size along the x-axis of each tile  1512 . In addition, a combination of the separation δ between adjacent beam spots and a size along the x-axis of each tile  1512  determines a number N of beams from among the N TOT  beams  1506  provided by the LEE array  1504  that can concurrently illuminate the tile. In order for the displacement measuring system  1500 A/ 1500 B to resolve both direction and magnitude of the displacement ΔX of the mass  1534  along the x-axis, the number N of the N TOT  beams  1506  can be predetermined, based on the number M of tile types, in the following manner. In some implementations, if an intensity pattern has M≥2 tile types, then a number N from among the N TOT  beams  1506  that can concurrently illuminate each tile satisfies the conditions 2≤N≤N TOT . In the examples shown in  FIGS. 15A-15C , the LEE array  1504  provides N TOT =4 beams  1506 , and the size of each tile  1512  of the intensity pattern  1510 A/ 1510 B allows for N=2 from among the 4 beams  1506  to illuminate each tile. In other implementations, if an intensity pattern has M≥3 tile types, then a number N from among the N TOT  beams  1506  that can concurrently illuminate each tile satisfies the conditions 1≤N≤N TOT , e.g., as described below in connection with  FIG. 17A . 
     Referring again to  FIGS. 15A-15B , the displacement measuring system  1500 A/ 1500 B includes a controller system  1525  that is linked to the LEE array  1504  and the photodetector  1522 , and is configured to control the way the photodetector captures individual ones the N TOT  redirected beams  1520 A/ 1520 B associated with corresponding ones of the N TOT  illuminating beams  1506 .  FIG. 15D  shows that the controller system  1525  includes a pulse-width modulation (PWM) driver  1536  and a processor  1546 , and that, in this example, the LEE array  1504  includes N TOT ≥2 light emitting elements (LEEs)  1528  that are independently switchable. In this manner, the LEEs  1528  of the LEE array  1504  can be PWM-switched at maximum current, so no adjustable current level is needed for the LEE array  1504 . Further, the photodetector  1522  includes a single photo-diode  1538 , an integration capacitance  1540 , a switch  1542 , and an analog-to-digital converter (ADC)  1544 . 
     In some implementations, each LEE  1528  of the LEE array  1504  includes a respective VCSEL that emits, when switched ON by the PWM driver  1536 , one of the N TOT  beams  1506  that illuminate the intensity pattern  1510 A/ 1510 B. In some implementations, each LEE  1528  of the LEE array  1504  includes a respective LED and a beam-shaping optic (not shown in  FIGS. 15A-15B, 15D ) that is optically coupled with the LED. In this case, the respective LED emits, when switched ON by the PWM driver  1536 , un-collimated light, and the beam-shaping optic receives the un-collimated light and issues one of the N TOT  beams  1506  that illuminate the intensity pattern  1510 A/ 1510 B. In this manner, in either of these implementations, the beam spot  1508  over which the intensity pattern  1510 A/ 1510 B is illuminated by each of the beams  1506  has a size of 20 μm, 10 μm, or smaller. 
     Moreover, the processor  1546  can instruct the PWM driver  1536  to individually switch ON/OFF the LEEs  1528  of the LEE array  1504 , such that the beams  1506  successively illuminate the intensity pattern  1510 A/ 1510 B, on a one-beam-at-a-time basis. To do so, the PWM driver  1536  can use a switching gate  1524  like the one shown in  FIG. 15E . Here, the switching gate  1524  is a sequence of trains of pulses, each train including N TOT  pulses corresponding to the N TOT  LEEs  1528  of the LEE array  1504 . Each pulse has a pulse duration T A  during which a single corresponding LEE  1528  is ON while the other (N TOT −1) LEEs of the LEE array  1504  are OFF and the switch  1542  is open. The pulses are separated from each other by a reset duration T R , during which the switch  1542  is closed. Note that operation of the photodetector  1522  in charge integration mode, using the switching gate  1524  as described below, can beneficially provide exposure time control. Additionally, the trains are repeated in time with a period T S  that represents a sampling period, related to the sampling rate f S =1/T S . As such, a train of pulses can correspond to a sampling time t 0 , the next train of pulses corresponds to the next sampling time t 1 , the next train of pulses corresponds to the next sampling time t 2 , and so on. Moreover, the pulse duration T A  and the reset time T R  can be 10, 100, or 1000 time smaller than the sampling period T S . Note that the processor  1546  can adjust either the duty cycle of the LEE array  1504  or the integration capacitance  1540  of the photodetector  1522  or both to (i) control the dynamic range of the displacement measuring system  1500 A/ 1500 B; and (ii) ensure that the N TOT  amounts of charge sequentially accumulated on the integration capacitance are the same for the N TOT  beams  1520 A/ 1520 B sequentially redirected from a tile of the same type. 
     In this manner, consider a sampling time t. Here, for a time T A  corresponding to the first pulse, a first LEE  1528  is placed in an ON state, the other LEEs of the LEE array  1504  are maintained in an OFF state, and the switch  1542  is in an open state. As such, the first LEE  1528  illuminates with a first beam  1506  (e.g., the left-most one of the beams  1506  in  FIG. 15A  or  FIG. 15B ) a first location of the intensity pattern  1510 A/ 1510 B, while the photo-diode  1538  captures a first beam  1520 A (e.g., the left-most one of the beams  1520 A in  FIG. 15A ) or a first beam  1520 B (e.g., the top one of the beams  1520 B in  FIG. 15B ) redirected from the first illuminated location, so the integration capacitance  1540  accumulates a first charge corresponding to a first intensity of the first redirected beam, and the ADC  1544  issues a first value i 1 (t) for the accumulated first charge. Then, the first LEE  1528  is placed in an OFF state, and the switch  1542  is placed in a close state for a time T R  to reset the integration capacitance  1540 . Then, for another time T A  corresponding to the second pulse, a second LEE  1528  is placed in an ON state, the other LEEs of the LEE array  1504  are maintained in an OFF state, and the switch  1542  is in an open state. As such, the second LEE  1528  illuminates with a second beam  1506  (e.g., the second left-most one of the beams  1506  in  FIG. 15A  or  FIG. 15B ) a second location of the intensity pattern  1510 A/ 1510 B, while the photo-diode  1538  captures a second beam  1520 A (e.g., the second left-most one of the beams  1520 A in  FIG. 15A ) or a second beam  1520 B (e.g., the second top one of the beams  1520 B in  FIG. 15B ) redirected from the second illuminated location, so the integration capacitance  1540  accumulates a second charge corresponding to a second intensity of the second redirected beam, and the ADC  1544  issues a second value i 2 (t) for the accumulated second charge. Then, the second LEE  1528  is placed in an OFF state, and the switch  1542  is placed in a close state for a time T R  to reset the integration capacitance  1540 . And so on for the remaining (N TOT −2) pulses of the train associated with sampling time t. In this manner, by the end of the (N TOT ) th  pulse of the train associated with sampling time t, the photodetector  1522  has issued a set of “raw” intensity values {i 1 (t), . . . , i NTOT (t)} corresponding to intensities of the N TOT  redirected beams  1520 A/ 1520 B captured by the photodetector for sampling time t. 
     Note that the raw intensity values {i 1 (t), . . . , i NTOT (t)} of the set issued by the photodetector  1522  are processed by the processor  1546 , so they can take only M values corresponding to the M tile types of the intensity pattern. For the example shown in  FIG. 15C , the intensity pattern  1510 A/ 1510 B has M=2 tile types, so the processor  1546  obtains a set of intensity values {I 1 (t), . . . , I NTOT (t)} that can take only the values I MAX  or I MIN . In some implementations, the processor  1546  can perform a threshold based classification on the set of raw intensity values {i 1 (t), . . . , i NTOT (t)}. In this case, the processor  1546  determines a threshold Th=(max{i 1 (t), . . . , i NTOT (t)}+min{i 1 (t), . . . , i NTOT (t)})/2, and, for each of {I 1 (t), . . . , I NTOT (t)}, can set I j (t)=I MAX  if I j (t)≥Th, or I j (t)=I MIN  if I j (t)&lt;Th, where j=1 . . . N TOT . As such, the processor  1546  can then determine positions of the illuminated locations  1508  of the intensity pattern  1510 A/ 1510 B based only on relative differences between the intensity values {I 1 (t), . . . , I NTOT (t)} of the obtained set. Hence, displacement measurements performed by the displacement measuring system  1500 A/ 1500 B are insensitive to common mode drift of the LEE array  1504 , for instance. 
       FIG. 15F  shows such a set  1526 ( t   0 ) of intensity values {I MIN , I MAX , I MAX , I MIN } obtained by the processor  1546  based on the set of raw intensity values {i 1 , i 2 , i 3 , i 4 } issued by the photodetector  1522 , for sampling time t 0 , corresponding to the example shown in  FIG. 15C . This set  1526 ( t   0 ) of intensity values has been obtained by the processor  1546  because, in the example shown in  FIG. 15C , the surface  1511  of the binary intensity pattern  1510 A/ 1510 B is illuminated over an area  1514  by N TOT =4 beams  1506  provided by the LEE array  1504 , of which the two inner beams illuminate locations  1508  that are positioned on a tile of first type  1512 A, and the two outer beams illuminate locations  1508  that are positioned on respective adjacent tiles of second type  1512 B. In  FIG. 15F , the intensity values {I MIN , I MAX , I MAX , I MIN } of the set  1526 ( t   0 ) are represented, by open circles, as a function of position along the x-axis of the illuminated locations of the binary intensity pattern  1510 A/ 1510 B with which the intensity values are respectfully associated. Note that separation along the x-axis between adjacent illuminated locations is the known separation S. Further note that, in  FIG. 15F , the intensity values {I MIN , I MAX , I MAX , I MIN } are fit with a solid line. 
     Once the processor  1546  obtains, for sampling time t 0 , the set  1526 ( t   0 ) of intensity values {I MIN , I MAX , I MAX , I MIN } shown in  FIG. 15F , then the processor can determine, based on additional information, that the area  1514 , illuminated by the LEE array  1504  with the beams  1506 , spans, at t 0 , over the tile borders  1513 ( i −1,i),  1513 ( i,i +1) formed between the i th  tile  1512 A and respective adjacent (i−1) th , (i+1) th  tiles  1512 B. The determined tile borders  1513 ( i −1,i),  1513 ( i,i +1) can be represented, by the controller system  1525  as shown in  FIG. 15F , between the appropriate positions of the illuminated locations with which the intensity values {I MIN , I MAX , I MAX , I MIN } are respectfully associated. 
     If the intensity pattern  1510 A/ 1510 B were at rest relative to the beams  1506 , then instances of the set  1526 ( t   1 ),  1526 ( t   2 ), . . . of intensity values obtained by the processor  1546  at respective subsequent sampling times t 1 , t 2 , . . . would be constant in time, e.g., would be the same as the set  1526 ( t   0 ) obtained at sampling time t 0 . However, if the intensity pattern  1510 A/ 1510 B is displaced by displacement ΔX (or displacement ΔY or both) relative to the beams  1506 , then the instances of the set  1526 ( t   1 ),  1526 ( t   2 ), . . . of intensity values obtained by the processor  1546  at respective subsequent sampling times t 1 , t 2 , . . . will change in a particular manner, based on (i) a number of tile border crossings, and (ii) types of adjacent tiles associated with, and direction of, each of the tile border crossings. Here, a tile border crossing is said to occur when the intensity pattern  1510 A/ 1510 B is displaced, along the x-axis, such that a tile border  1513 , formed between a tile of first type  1512 A and a tile of second type  1512 B, crosses through the beam spot  1508  associated with a corresponding one of the beams  1506 . In this manner, the corresponding one of the beams  1506  illuminates the tile of first type  1512 A (or the tile of second type  1512 B) before the tile border crossing and illuminates the tile of second type  1512 B (or the tile of first type  1512 A) after the tile border crossing, such that the tile border crossing causes a change in the intensity of a corresponding one of the redirected beams  1520 A/ 1520 B between I MAX  and I MIN . 
     As such, the processor  1546  can determine magnitude and direction of a displacement ΔX of the mass  1534  along the x-axis, based on one or more changes of the intensity values {I 1 (t k ), . . . , I NTOT (t k )} of the set  1526 ( t   k ) obtained at sampling time t k  relative to the intensity values {I 1 (t k+1 ), I NTOT (t k+1 )} of the set  1526 ( t   k+1 ) obtained at a subsequent sampling time t k+1 , where the changes are caused by motion of the mass along the x-axis, across at least one of the tile borders  1513 . In this manner, all motion related spatial information for the mass  1534  is encoded in the intensity pattern  1510 A/ 1510 B. Determinations of the magnitude and direction of displacements ΔX of the mass  1534  along the x-axis are described below when the displacement measuring system  1500 A/ 1500 B uses the intensity pattern  1510 A/ 1510 B of  FIG. 15C , or another intensity pattern  1710  described in connection with  FIG. 17A . 
       FIG. 16A  shows an intensity signal that includes a sequence of sets  1526 ( t   0 ),  1626 E(t 1 ),  1626 E(t 2 ),  1626 E(t 3 ) of intensity values obtained by the processor  1546  at respective sampling times t 0 , t 1 , t 2 , t 3  as the intensity pattern  1510 A/ 1510 B moves relative to the beams  1506 , as the intensity pattern is carried in an eastward direction by the mass  1534 .  FIG. 16B  shows another intensity signal that includes a sequence of sets  1526 ( t   0 ),  1626 W(t 1 ),  1626 W(t 2 ),  1626 W(t 3 ) of intensity values obtained by the processor  1546  at respective sampling times t 0 , t 1 , t 2 , t 3  as the intensity pattern  1510 A/ 1510 B moves relative to the beams  1506 , as the intensity pattern is carried in a westward direction by the mass  1534 . Note that the starting point of each of the eastward and westward motions corresponds to a position of the area  1514  that includes the illuminated locations  1508 , shown in  FIG. 15C , when the processor  1546  obtains, at sampling time t 0 , the set  1526 ( t   0 ) of intensity values shown in  FIG. 15F  and reproduced in the top panel of each of  FIG. 16A  and  FIG. 16B . The intensity values of the sets  1626 E(t k ) in  FIG. 16A , and the intensity values of the sets  1626 W(t k ) in  FIG. 16B , are represented, by open circles, as a function of position along the x-axis of the illuminated locations of the binary intensity pattern  1510 A/ 1510 B with which the intensity values are respectfully associated. Here, separation along the x-axis between adjacent illuminated locations is the known separation δ. 
     Because the intensity pattern  1510 A/ 1510 B has a pattern period of two tiles, as it is formed from M=2 types of tiles  1512 A,  1512 B, and because a width of each tile allows for N=2 concurrently illuminated locations  1508 , there are only M×N=4 unique sets  1626 E or  1626 W of intensity values that can be obtained by the processor  1546 . In this manner, in  FIG. 16A , the next set  1626 E(t 4 ) will be equivalent to the set  1526 ( t   0 ), the next set  1626 E(t 5 ) will be equivalent to the set  1626 E(t 1 ), and so on a cyclical mod(M×N=4) basis. Similarly, in  FIG. 16B , the next set  1626 W(t 4 ) will be equivalent to the set  1526 ( t   0 ), the next set  1626 W(t 5 ) will be equivalent to the set  1626 W(t 1 ), and so on a cyclical mod(M×N=4) basis. 
     Also, a sampling frequency f S  is chosen such that, in  FIG. 16A , a tile border crossing in the eastward direction occurs between each consecutive pair of sampling points t k , t k+1 ; similarly, in  FIG. 16B , a tile border crossing in the westward direction occurs between each consecutive pair of sampling points t k , t k+1 . Note, however, that the set  1626 E(t 2 ) and the set  1626 W(t 2 ) obtained by the processor  1546  at sampling time t 2  are the same,  1626 E(t 2 )= 1626 W(t 2 ), so the processor cannot unambiguously determine whether a displacement ΔX of the mass  1534  along the x-axis is in the eastward direction or westward direction based only on the set issued at this sampling time. For this reason, the sampling rate f S  must be larger than 1/(t 2 −t 0 ). In general, to unambiguously determine both the magnitude and direction of the displacement ΔX, the sampling rate f S  is chosen to satisfy the condition f S &gt;2v MAX /[δ*(N TOT −1)], if a period P of the intensity pattern  1510 A/ 1510 B satisfies the condition P&gt;δ*(N TOT −1); or the sampling rate f S  is chosen to satisfy the condition f S &gt;2v MAX /P, if the period P of the intensity pattern satisfies the condition P≤δ*(N TOT −1). Here, v MAX  is the maximum velocity of the mass  1534 , and δ is the separation between the illuminated locations  1508 . Note that the above conditions hold only when δ&gt;0, which is always true for N TOT ≥2. For the example illustrated in  FIG. 15C , N TOT =4, and the period P of the intensity pattern  1510 A/ 1510 B, which is equal to a sum of the width of the first type tile  1512 A and the width of the second type tile  1512 B, is larger than δ*3. As such, in this example, the sampling frequency f S  satisfies f S &gt;2v MAX /(3*δ). 
     In some implementations, sets  1626 E(t k ) of intensity values shown in  FIG. 16A  and sets  1626 W(t k ) of intensity values shown in  FIG. 16B , for k=0, 1, 2, . . . , can be deemed as sets of expected intensity values corresponding to particular positions along the x-axis of illuminated locations of the intensity pattern  1510 A/ 1510 B. In such cases, a first mapping of the sets  1626 E(t k ) of expected intensity values shown in  FIG. 16A  to the particular positions along the x-axis of illuminated locations of the intensity pattern  1510 A/ 1510 B can be recorded in a data store, e.g., in a register, a lookup table, etc. Similarly, a second mapping of the sets  1626 W(t k ) of expected intensity values shown in  FIG. 16B  to the particular positions along the x-axis of illuminated locations of the intensity pattern  1510 A/ 1510 B can be recorded in the data store. In these implementations, the processor  1546  uses an obtained set of intensity values (e.g.,  1626 W(t 3 )) against the second mapping of the sets  1626 W(t k ) of expected intensity values to positions of illuminated locations of the intensity pattern  1510 A/ 1510 B (e.g., as shown in  FIG. 16B ), to determine a position on the intensity pattern of the area  1514  that includes the illuminated locations  1508  (e.g., two left beams illuminate a tile of first type  1512 A between tile borders  1513 ( i −3,i−2),  1513 ( i −2,i−1), and two right beams illuminate a tile of second type  1512 B between tile borders  1513 ( i −2,i−1),  1513 ( i −1,i), as shown in the fourth panel of  FIG. 16B ). 
     As noted above, even if the LEE array  1504  were replaced with a single LEE  1528 , the displacement measuring system  1500 A/ 1500 B would still be capable of resolving both direction and magnitude of the displacement ΔX of the mass  1534  along the x-axis, as long as the intensity patterns  1510 A/ 1510 B were replaced with an intensity pattern with M≥3 tile types.  FIG. 17A  is a plan view in the (x,y) plane of surface  1511  of such an intensity pattern  1710  that has M=3 tile types: tiles of first type  1712 A, tiles of second type  1712 B and tiles of third type  1712 C, where a tile of first type  1712 A forms a tile border  1713 AB with a tile of second type  1712 B and a tile border  1713 AC with a tile of third type  1712 C, and a tile of second type  1712 B forms a tile border  1713 BC with a tile of third type  1712 C. A tile of first type  1712 A is configured such that, when a single beam (e.g., only the left-most one from among the beams  1506  depicted in  FIG. 15A  or  FIG. 15B ) illuminates it, the single beam  1520 A (e.g., only the left-most one of the beams  1520 A depicted in  FIG. 15A ) or the single beam  1520 B (e.g., only the top one of the beams  1520 B depicted in  FIG. 15B ) has a maximum intensity I A . Further, a tile of third type  1712 C is configured such that, when the single beam  1506  illuminates it, the single redirected beam  1520 A/ 1520 B has a minimum intensity I C , where I C &lt;I A . Furthermore, a tile of second type  1712 B is configured such that, when the single beam  1506  illuminates it, the single redirected beam  1520 A/ 1520 B has an intermediate intensity I B , where I C &lt;I B &lt;I A . Here, the tile borders  1713 AB,  1713 AC,  1713 BC are distributed at known locations relative to each other along the x-axis, so the intensity pattern  1710  can be used as part of the displacement measuring system  1500 A/ 1500 B to measure displacement ΔX of the mass  1534  along the x-axis. 
     In the implementations of the displacement measuring system  1500 A/ 1500 B that uses a single LEE  1528  in conjunction with the intensity pattern  1710 , the switching gate  1524 , that is used by the PWM driver  1536  for timing acquisition of raw intensity values issued by the photodetector  1522 , is modified such that each train includes a single pulse, such that the modified timing gate is a sequence of pulses repeated in time with a period T S . Here, the pulses of the modified timing gate correspond to sampling times t 0 , t 1 , t 2 , and so on. In this manner, at a sampling time t, the LEE  1528  is placed in an ON state for a time T A  corresponding to the pulse length, and the switch  1542  is in an open state. As such, the LEE  1528  illuminates with a single beam a single location  1708  of the intensity pattern  1710 , while the photo-diode  1538  captures a beam redirected from the illuminated location  1708 , so the integration capacitance  1540  accumulates a charge corresponding to an intensity of the redirected beam  1520 A/ 1520 B, and the ADC  1544  issues a single value i(t) for the accumulated charge. Then, the single LEE  1528  is placed in an OFF state, and the switch  1542  is placed in a close state for a time T S  corresponding to the sampling period. In this manner, the photodetector  1522  has issued a single “raw” intensity value i(t) corresponding to the intensity of the single redirected beam  1520 A/ 1520 B captured by the photodetector for sampling time t. 
     The raw intensity value i(t) issued by the photodetector  1522  is processed by the processor  1546 , so it can take only M values corresponding to the M tile types of the intensity pattern. For the example shown in  FIG. 17A , the intensity pattern  1710  has M=3 tile types, so the processor  1546  obtains a single intensity value I(t) that can take only the values I A , I B  or I C . In this example, the processor  1546  can perform a threshold based classification on the raw intensity value i(t) using two thresholds Th AB  and Th BC . Here, the processor  1546  can set I(t)=I A  if I(t)≥Th AB , or I(t)=I C  if I(t)≤Th BC , or I(t)=I B  if Th BC &lt;I(t)&lt;Th AB . If the intensity pattern  1710  is displaced by displacement ΔX relative to the single beams  1506 , then intensity values I(t 0 ), I(t 1 ), I(t 2 ), . . . obtained by the processor  1546  at respective subsequent sampling times t 0 , t 1 , t 2 , . . . will change in a particular manner, based on (i) a number of tile border crossings, and (ii) types of adjacent tiles associated with, and direction of, each of the tile border crossings. As such, the processor  1546  can determine magnitude and direction of a displacement ΔX of the mass  1534  along the x-axis, based on a change of the intensity value I(t k ) obtained at sampling time t k  relative to the intensity value I(t k+1 ) obtained at a subsequent sampling time t k+1 , where the change is caused by motion of the mass along the x-axis, across at least one of the tile borders  1713 AB,  1713 AC or  1713 BC. In this manner, all motion related spatial information for the mass  1534  is encoded in the intensity pattern  1710 . Determinations of the magnitude and direction of displacements ΔX of the mass  1534  along the x-axis are described below in connection with  FIGS. 17B and 17C . 
       FIG. 17B  shows an intensity signal that includes a sequence of intensity values I(t 0 ), I E (t 1 ), I E (t 2 ) obtained by the processor  1546  at respective sampling times t 0 , t 1 , t 2  as the intensity pattern  1710  moves relative to the single beam  1506 , as the intensity pattern is carried in an eastward direction by the mass  1534 .  FIG. 17C  shows another intensity signal that includes a sequence of intensity values I(t 0 ), I W (t 1 ), I W (t 2 ) obtained by the processor  1546  at respective sampling times t 0 , t 1 , t 2  as the intensity pattern  1710  moves relative to the single beam  1506 , as the intensity pattern is carried in a westward direction by the mass  1534 . Note that the starting point of each of the eastward and westward motions corresponds to position of the illuminated location  1708 , shown in  FIG. 17A , when the processor  1543  issues, at sampling time t 0 , the intensity value I(t 0 )=I A  shown in the top panel of each of  FIG. 17B  and  FIG. 17C . The intensity values I E (t k ) in  FIG. 17B , and the intensity values I W (t k ) in  FIG. 17C  are represented, by open circles, as a function of positions along the x-axis of the illuminated location of the intensity pattern  1710  with which the intensity values are respectfully associated. Here, the tiles are counted, starting from 0 at the center tile, as tiles+1, +2, . . . going eastward, and as tiles−1, −2, . . . going westward. 
     Because the intensity pattern  1710  has a pattern period of three tiles, as it is formed from M=3 types of tiles  1712 A,  1712 B,  1712 C and because there is no more than N=1 illuminated location  1708  per tile, there are only M×N=3 unique intensity values I E (t k ) or I W (t k ) that can be obtained by the processor  1546 . In this manner, in  FIG. 17B , the next intensity value I E (t 3 ) will be equivalent to the intensity value I(t 0 ), the next intensity value I E (t 4 ) will be equivalent to the intensity value I E (t 1 ), and so on a cyclical mod(M×N=3) basis. Similarly, in  FIG. 17C , the next intensity value I W (t 3 ) will be equivalent to the intensity value I(t 0 ), the next intensity value I W (t 4 ) will be equivalent to the intensity value I W (t 1 ), and so on a cyclical mod(M×N=3) basis. 
     Also, a sampling frequency f S  is chosen such that, in  FIG. 17B , a tile border crossing in the eastward direction occurs between each consecutive pair of sampling points t k , t k+1 ; similarly, in  FIG. 17C , a tile border crossing in the westward direction occurs between each consecutive pair of sampling points t k , t k−1 . As such, in this case, to unambiguously determine both the magnitude and direction of the displacement ΔX, the sampling rate f S  is chosen to satisfy the condition f S &gt;2Mv MAX /P, where v MAX  is the maximum velocity of the mass  1534 , and P is the period of an intensity pattern with M types of tiles. For the example illustrated in  FIG. 17A  in which N TOT =1, the number of tile types is M=3, and the period P of the intensity pattern  1710  is equal to a sum of the width of the first type tile  1712 A, the width of the second type tile  1712 B, and the width of the third type tile  1712 C. As such, in this example, the sampling frequency f S  satisfies f S &gt;6v MAX /P. 
     In some implementations, intensity values I E (t k ) shown in  FIG. 17B  and intensity values I W (t k ) shown in  FIG. 17C , for k=0, 1, 2 can be deemed as expected intensity values corresponding to particular positions along the x-axis of the illuminated location of the intensity pattern  1710 . In such cases, a first mapping of expected intensity values I E (t k ) shown in  FIG. 17B  to the particular positions along the x-axis of an illuminated location of the intensity pattern  1710  can be recorded in a data store, e.g., in a register, a lookup table, etc. Similarly, a second mapping of expected intensity values I W (t k ) shown in  FIG. 17C  to the particular positions along the x-axis of an illuminated location of the intensity pattern  1710  can be recorded in the data store. In these implementations, the processor  1546  uses an obtained intensity value (e.g., I W (t 2 )) against the second mapping of the expected intensity values I W (t k ) to positions of an illuminated location of the intensity pattern  1710  (e.g., as shown in  FIG. 17C ), to determine a position on the intensity pattern of the illuminated location  1708  (e.g., the single beam  1506  illuminates a tile of third type  1712 C between the tile border  1713 (+1,+2) and the tile border  1713 (+2,+3), as shown in the third panel of  FIG. 17C ). 
     The intensity patterns  1510 A/ 1510 B,  1710  described above, having a 1D profile, can be used by the displacement measuring system  1500 A/ 1500 B to perform measurements of displacement ΔX along a single direction. Some of the components of the displacement measuring system  1500 A/ 1500 B can be modified, as described below, to allow for concurrently measuring the displacement ΔX of the mass along the x-axis, e.g., corresponding to vibration left-and-right on page, and the displacement ΔY of the mass along the y-axis, e.g., corresponding to vibration in-and-out of page. 
       FIG. 18A  is a zoomed-in view in the (x,z) plane of a portion of the displacement measuring system  1500 A/ 1500 B that shows a modified LEE array  1804  and a portion of a modified intensity pattern  1810  attached to the surface of the mass  1534  facing the LEE array. The LEE array  1804  is spaced apart from the intensity pattern  1810  and outputs N TOT  beams  1806  that illuminate the surface  1511  of the intensity pattern.  FIG. 18B  is a plan view in the (x,y) plane of the LEE array  1804 . The LEE array  1804  includes a board  1803  and LEEs  1828  distributed in rows along the x-axis, each row having N X  LEEs, and columns along the y-axis, each column having N Y  LEEs. The LEEs  1828  respectively provide the beams  1806  used to illuminate the intensity pattern  1810 . In this example, parameters of the LEE array  1804  are N TOT =4, N X =2, N Y =2. In some implementations, the LEEs  1828  can be the same as the LEEs  1528  described above in connection with  FIGS. 15A-15D , and can be time multiplexed using a timing gate like the timing  1524  shown in  FIG. 15E . 
       FIG. 18C  is a plan view in the (x,y) plane of surface  1511  of the modified intensity pattern  1810 . In this example, the intensity pattern  1810  has M=4 tile types: tiles of first type  1812 A, tiles of second type  1812 B, tiles of third type  1812 C and tiles of fourth type  1812 D. A tile of first type  1712 A is configured such that, when one of the beams  1806  illuminates it, the corresponding one of the redirected beams  1520 A/ 1520 B has a maximum intensity I A . Further, a tile of fourth type  1712 D is configured such that, when one of the beams  1806  illuminates it, the corresponding one of the redirected beams  1520 A/ 1520 B has a minimum intensity I D , where I D &lt;I A . Furthermore, a tile of second type  1812 B is configured such that, when one of the beams  1806  illuminates it, the corresponding one of the redirected beams  1520 A/ 1520 B has a first intermediate intensity I B . Also, a tile of third type  1812 C is configured such that, when one of the beams  1806  illuminates it, the corresponding one of the redirected beams  1520 A/ 1520 B has a second intermediate intensity I C . In general, the intensities I A , I D , I C , and I B  can have any values as long as they are different from each other. Also, the separation between the values of the intensities I A , I D , I C , and I B  also needs to be sufficiently large to ensure sufficient signal to noise ratio (SNR) for level detection. In the example illustrated in  FIG. 18C , the values of the intensities I A , I D , I C , and I B  satisfy the following conditions, I D &lt;I C &lt;I B &lt;I A , as an illustrative example. 
     Moreover, tiles of first type  1812 A and tiles of second type  1812 B are separated by tile borders  1813 AB, oriented along the x-axis; tiles of first type  1812  and tiles of third type  1812 C are separated by tile borders  1813 AC, oriented along the y-axis; tiles of second type  1812 B and tiles of fourth type  1812 D are separated by tile borders  1813 BD, oriented along the y-axis; and tiles of third type  1812 C and tiles of fourth type  1812 D are separated by tile borders  1813 CD, oriented along the x-axis. In this manner, the tiles are arranged such that each corner  1850  of each tile is a corner that is shared by the tile with three other tiles of different type. The tile borders  1813 AC,  1813 BD are distributed at known locations relative to each other along the x-axis, and the tile borders  1813 AB,  1813 CD are distributed at known locations relative to each other along the y-axis, so the intensity pattern  1810  can be used as part of the displacement measuring system  1500 A/ 1500 B to concurrently measure displacement ΔX of the mass  1534  along the x-axis and displacement ΔY of the mass along the y-axis. 
     Further in the example shown in  FIG. 18C , the LEE array  1804  illuminates an area  1814  of the intensity pattern  1810  with N TOT =4 beams  1806 , such that a separation between the illuminated locations  1808  is a first separation δ X  along the x-axis, and a second separation δ Y  along the y-axis. In some implementations, the first separation δ X  and the second separation δ Y  are equal, δ X =δ Y =δ. Furthermore, in the example shown in  FIG. 18C , a size along each of the x-axis and y-axis of each of tiles  1812 A,  1812 B,  1812 C,  1812 D can accommodate an N=2 concurrently illuminated locations  1808 . In this manner, the processor  1546  (in conjunction with the photodetector  1522  of the displacement measuring system  1500 A/ 1500 B) obtains, for each sampling time t k , a set of N TOT =4 intensity values corresponding to intensities of respective beams  1520 A/ 1520 B redirected from the area  1814  of the intensity pattern  1810  that includes the illuminated locations  1806 . 
     In some implementations, the surface  1511  of the intensity pattern  1810  can be configured as a reflective surface that selectively reflects, scatters or both the illuminating beams  1806  to the photodetector  1522  as redirected beams  1520 A, as described in detail in connection with  FIG. 15A  and  FIG. 15C . In other implementations, the intensity pattern  1810  can be configured as a combination of a transmissive surface  1511 , an array of micro-mirrors  1518  (or micro-prisms, or other redirecting micro-structures), and a transmissive surface  1516 , as described in detail in connection with  FIG. 15B  and  FIG. 15C . In this manner, the transmissive surface  1511  selectively transmits the illuminating beams  1806 , the array of micro-mirrors  1518  reflects, scatters or both the selectively transmitted beams, and the transmissive surface  1516  transmits the redirected beams  1520 B to the photodetector  1522 , as explained in connection with  FIG. 15B . 
     Referring again to the example shown in  FIG. 18C , at sampling time t 0 , an area  1814  of the intensity pattern  1810  is positioned across four tile borders  1813 AB,  1813 AC,  1813 CD,  1813 BD (or equivalently straddle a corner  1850 ), such that a corresponding one of the four illuminated locations  1808  lies on each one of the four adjacent tiles  1812 A,  1812 C,  1812 D,  1812 B that form the concurrently crossed tile borders. As such, a set  1826 ( t   0 ) obtained by the processor  1546  includes the intensity values {I A , I C , I D , I B } corresponding to quadrants I, II, III, IV, respectively, of area  1814 . The intensity pattern  1810  can be moved along with the mass  1534  relative to the four beams  1806 , based on an axial motion vector  1852 , along the x-axis or the y-axis. When the intensity pattern  1810  is moved eastward along the axial motion vector  1852  illustrated in  FIG. 18C , at sampling time t 1 , the area  1814  is positioned across a single tile border  1813 CD, such that a row of the four illuminated locations  1808  lies on each of the adjacent tiles  1812 C,  1812 D that form the crossed tile border. As such, a set  1826 E(t 1 ) obtained by the processor  1546  includes the intensity values {I C , I C , I D , I D } corresponding to quadrants I, II, III, IV, respectively, of area  1814 . When the intensity pattern  1810  is moved further eastward along axial motion vector  1852 , at sampling time t 2 , the area  1814  is positioned across four tile borders  1813 CD,  1813 AC,  1813 AB,  1813 BD, such that a corresponding one of the four illuminated locations  1808  lies on each one of the four adjacent tiles  1812 C,  1812 A,  1812 B,  1812 D that form the concurrently crossed tile borders. As such, a set  1826 E(t 2 ) obtained by the processor  1546  includes the intensity values {I C , I A , I B , I D } corresponding to quadrants I, II, III, IV, respectively, of area  1814 .  FIG. 18D  shows an intensity signal that includes the sequence of sets  1826 ( t   0 ),  1826 E(t 1 ),  1826 E(t 2 ) obtained by the processor  1546  at respective sampling times t 0 , t 1 , t 2  as the intensity pattern  1810  moves relative to the beams  1806 , as the intensity pattern  1810  is carried in the eastward direction by the mass  1534 . The intensity values of the sets  1826 E(t k ) in  FIG. 18D  are represented, as a grey-level inside circles, as a function of position in the (x,y) plane of the illuminated locations of the intensity pattern  1810  with which the intensity values are respectfully associated. Here, separation along the x-axis and the y-axis between adjacent illuminated locations is the known separation δ. 
     Referring again to  FIG. 18C , note that, at sampling time t 2 , the same set  1826 E(t 2 ) would be obtained by the processor  1546  if the intensity pattern  1810  were moved in a westward, opposing motion vector  1852 . As such, to unambiguously determine the direction of motion, the sampling frequency has to be larger than 1/(t 2 −t 0 ). 
     Referring again to  FIG. 18C , the intensity pattern  1810  can be moved along with the mass  1534  relative to the four beams  1806  based on a diagonal motion vector  1854 . When the intensity pattern  1810  is moved in a NE direction along the diagonal motion vector  1854  illustrated in  FIG. 18C , at sampling time t 1 , the area  1814  is inscribed in a single tile  1812 D, so it does not cross any tile border. As such, a set  1826 NE(t 1 ) obtained by the processor  1546  includes the intensity values {I D , I D , I D , I D } corresponding to quadrants I, II, III, IV, respectively, of area  1814 . When the intensity pattern  1810  is moved further in the NE direction along diagonal motion vector  1854 , at sampling time t 2 , the area  1814  is positioned across four tile borders  1813 CD,  1813 BD,  1813 AB,  1813 AC, such that a corresponding one of the four illuminated locations  1808  lies on each one of the four adjacent tiles  1812 D,  1812 B,  1812 A,  1812 C that form the concurrently crossed tile borders. As such, a set  1826 NE(t 2 ) obtained by the processor  1546  includes the intensity values {I D , I B , I A , I C } corresponding to quadrants I, II, III, IV, respectively, of area  1814 . 
     Note that, at sampling time t 2 , the same set  1826 E(t 2 ) of intensity values would be obtained by the processor  1546  if the intensity pattern  1810  were moved in westward, opposing axial motion vector  1852 . Also at sampling time t 2 , the same set  1826 NE(t 2 ) of intensity values would be issued by the photodetector  1522  if the intensity pattern  1810  were moved in a southwest direction, opposing diagonal motion vector  1854 . As such, to unambiguously determine the direction of motion, the sampling frequency has to satisfy the following condition: f S &gt;1/(t 2 −t 0 ). 
     As such,  FIG. 18E  shows a first group  1856  of unique sets  1826  of expected intensity values corresponding to particular positions in the (x,y) plane of illuminated locations of the intensity pattern  1810 , where each of the sets of the first group  1856  can be used by the processor  1546  to unambiguously determine the absolute value and direction of displacements ΔX and ΔY of the intensity pattern relative to the beams  1810 .  FIG. 18E  also shows a second group  1858  of non-unique sets  1826  of expected intensity values corresponding to particular positions in the (x,y) plane of illuminated locations of the intensity pattern  1810 . While the sets  1826  of intensity values of the second group  1858  can be used by the processor  1546  to determine the absolute value of displacements ΔX and ΔY of the intensity pattern relative to the beams  1810 , respective directions of the displacements ΔX and ΔY can be one of multiple possible directions. However, the sets  1826  of intensity values of the second group  1858  can be used for error correction, e.g., to detect motion aliasing when the mass  1534  supporting the intensity pattern  1810  is moving too fast or the sampling rate f S  is too slow. 
     In some implementations, at least the first group  1856  of unique sets  1826  of intensity values corresponding to particular positions in the (x,y) plane of illuminated locations of the intensity pattern  1810  can be recorded in a data store, e.g., in a register, a lookup table, etc. In these implementations, the processor  1546  uses an obtained set of intensity values (e.g.,  1826 E(t 1 )) against the recorded first group  1856  of the sets  1826  of expected intensity values corresponding to particular positions in the (x,y) plane of illuminated locations of the intensity pattern  1810  (e.g., as shown in  FIG. 18E ), to determine a position on the intensity pattern of the area  1814  that includes the illuminated locations  1808  (e.g., a row of two beams illuminate a tile of third type  1812 C and, across the tile border  1813 CD, a row of two beams illuminate a tile of fourth type  1812 D, as shown in the second panel of  FIG. 18D ). 
       FIG. 19A  shows an intensity pattern  1910  that is misaligned relative to the LEE array  1504  of the displacement measuring system  1500 A/ 1500 B, e.g., rotated relative to the x-axis and y-axis by a rotation angle θ. This misalignment can be compensated for by sweeping over the full ΔX MAX  motion range and measuring the corresponding displacement ΔY MEAS  over this range to estimate θ. Such a misalignment compensation can be performed at fabrication time, or in the field. For example, while a device carrying the displacement measuring system  1500 A/ 1500 B is charging, it can be verified whether there is a misalignment of the intensity pattern  1910  pattern to the LEE array  1504 . The processor  1546  determines the angular misalignment θ based on the known ΔX MAX  motion range and measured displacement ΔY MEAS . Once the angular misalignment θ is determined, the processor  1546  transforms the intensity pattern  1910  using a rotation matrix to determine a scaling factor [1/cos(θ)].  FIG. 19B  shows that a displacement ΔX measured by the displacement measuring system  1500 A/ 1500 B using the rotated intensity pattern  1910  will be scaled by the processor  1546 , based on the determined scaling factor, to output the “true” displacement as a scaled displacement ΔX S . A magnitude of scaled displacement ΔX S  is larger than a magnitude of the measured displacement ΔX based on the determined scaled factor: ΔX S =ΔX/cos(θ). 
     The displacement measuring system  1500 A/ 1500 B in conjunction with intensity patterns  1510 A/ 1510 B,  1710  and  1810  can be used to measure angular displacement in various rotational configurations, as described below. 
       FIG. 20A  is a side view, in the (y,z) plane, of a first example of an angular displacement measuring system  2000  configured to measure angular displacement Δθ of a wheel  2034 . Here, a device  2030  has a frame  2032  that encapsulates at least a portion of the angular displacement measuring system  2000  and a portion of the wheel  2034 . A remaining portion of the wheel  2034  protrudes outside of the frame  2032  through slot  2033 , for instance. In some implementations, the device  2030  is a watch, and the wheel  2034  is a setting/control crown. 
     In this example, the angular displacement measuring system  2000  is a modification of the displacement measuring system  1500 B described above in connection with  FIG. 15B . The angular displacement measuring system  2000  includes the mount  1502 B, the LEE array  1504 , and the photodetector  1522 , all of which described in detail above in connection with  FIGS. 15B-15E . Note that the N TOT  LEEs  1528  of the LEE array  1504  are distributed along the x-axis. 
     An intensity pattern  2010  of the angular displacement measuring system  2000  is a structure shaped like a wheel that is disposed co-axially with the wheel  2034  and is coupled with a side wall surface  2035 XY of the wheel  2034 . A side wall surface  2011  of the intensity pattern  2010  is spaced apart from and faces the LEE array  1504 , and a rim surface  2016  of the intensity pattern  2010  is spaced apart from and faces the photodetector  1522 . In this manner, during operation of the angular displacement measuring system  2000 , the LEE array  1504  illuminates a transmissive side wall surface  2011  of the intensity pattern  2010  with N TOT  beams  1506 , and the intensity pattern redirects, through its transmissive rim surface  2016  to the photodetector  1522 , at least some of the light impinging on the illuminated surface, such that N TOT  redirected beams  1520 B form a folding angle relative the illuminating beams. In some cases, the folding angle formed by the redirected beams  1520 B relative the illuminating beams  1506  can be a substantially right angle. In the example illustrated in  FIG. 20A , the intensity pattern  2010  includes an array of micro-mirrors  2018  (or micro-prisms, or other redirecting micro-structures) between the transmissive side wall surface  2011  and the transmissive rim surface  2016 . Here, the flat surface of the micro-mirrors of the array  2018  is oriented parallel to the x-axis. In this manner, the transmissive side wall surface  2011  selectively transmits the illuminating beams  1506 , the array of micro-mirrors  2018  reflects, scatters or both the selectively transmitted beams, and the transmissive rim surface  2016  transmits the redirected beams  1520 B to the photodetector  1522 . 
       FIG. 20B  is a plan view in the (x,y) plane of side wall surface  2011  of the intensity pattern  2010 . In this example, the intensity pattern  2010  has M=2 tile types: tiles of first type  2012 A and tiles of second type  2012 B separated from each other by tile borders  2013 . Note that the tiles  2012 A,  2012 B are shaped as annulus sectors. A tile of first type  2012 A is configured such that, when one of the beams  1506  illuminates it, the corresponding one of the redirected beams  1520 B has a maximum intensity I MAX . Further, a tile of second type  2012 B is configured such that, when one of the beams  1506  illuminates it, the corresponding one of the redirected beams  1520 B has a minimum intensity I MIN , where I MIN &lt;I MAX . Here, the tile borders  2013  are distributed at known angular locations relative to each other along the azimuthal θ-axis, so the intensity pattern  2010  can be used as part of the angular displacement measuring system  2000  to measure an angular displacement Δθ of the wheel  2034  along the azimuthal θ-axis. Further in the example shown in  FIGS. 20A-20B , the LEE array  1504  illuminates an area  1514  of side wall surface  2011  of the intensity pattern  2010  with N TOT  beams  1506 , and a separation between the illuminated locations  1508  is a known separation δ. Moreover, the size along the azimuthal θ-axis of each of tiles  2012 A,  2012 B can accommodate 1≤N≤N TOT  concurrently illuminated locations  1508 . 
     In this manner, the PWM driver  1536  of the angular displacement measuring system  2000  can use a switching gate, similar to the switching gate  1524  shown in  FIG. 15E , to time multiplex the N TOT  LEEs  1528  of the LEE array  1504 . In this manner, the processor  1546  (in conjunction with the photodetector  1522 ) obtains, for each sampling time t k , an associated set of N TOT  intensity values corresponding to intensities of respective beams  1520 B redirected by the intensity pattern  2010  via transmission through the side wall surface  2011  at the illuminated locations  1508 . As such, the processor  1546  of the angular displacement measuring system  2000  can determine, for each sampling time t k , positions along the azimuthal θ-axis of the illuminated locations  1508  of the side wall surface  2011  of the intensity pattern  2010  based on relative differences between the intensity values of the associated issued set. Then, in a manner similar to the manners described above in connection with  FIG. 16A-16B or 17B-17C  (e.g., by substituting ΔX with Δθ), the processor  1546  determines, for each sampling time t k , an angular displacement Δθ of the wheel  2034  based on one or more changes of the intensity values of the obtained set caused by rotation of the wheel that sweeps at least one of the tile borders  2013  through at least one of the illuminating beams  1506 . 
       FIG. 21A  is a side view, in the (y,z) plane, of a second example of an angular displacement measuring system  2100 A or a third example of an angular displacement measuring system  2100 B each configured to measure angular displacement Δθ of a wheel  2134 . Here, a device  2130  has a frame  2132  that encapsulates at least a portion of one of the angular displacement measuring systems  2100 A,  2100 B and a portion of the wheel  2134 . A remaining portion of the wheel  2134  protrudes outside of the frame  2132  through slot  2133 , for instance. In some implementations, the device  2130  is a watch, and the wheel  2134  is a setting/control crown. 
     In this example, the angular displacement measuring system  2100 A is a modification of the displacement measuring system  1500 A described above in connection with  FIG. 15A , and the angular displacement measuring system  2100 B is a modification of the displacement measuring system  1500 B described above in connection with  FIG. 15B . The angular displacement measuring system  2100 A/ 2100 B includes the mount  1502 A/ 1502 B, the LEE array  1504 , and the photodetector  1522 A/ 1522 B, all of which described in detail above in connection with  FIGS. 15A-15E . Note that the N TOT  LEEs  1528  of the LEE array  1504  are distributed along the x-axis. 
     An intensity pattern  2110  of the angular displacement measuring system  2100 A/ 2100 B is a structure shaped like a wheel that is disposed co-axially with the wheel  2134  and coupled with a side wall surface  2135 YZ of the wheel  2134 .  FIG. 21B  is a plan view in the (x,z) plane of the side wall surface  2116  of the intensity pattern  2110  that is distal from the side wall surface  2135 YZ of the wheel  2134 . In the example illustrated in  FIGS. 21A-21B , the intensity pattern  2110  includes an array of micro-mirrors  2118  (or micro-prisms, or other redirecting micro-structures) between a transmissive rim surface  2111  and a transmissive side wall surface  2116 . Here, the flat surface of the micro-mirrors of the array  2118  is oriented parallel to the y-axis. In some implementations, the transmissive rim surface  2111  is configured as the surface  1511  of the intensity pattern  1510 A/ 1510 B described above in connection with  FIG. 15C . In other implementations, the transmissive rim surface  2111  is configured as the surface  1511  of the intensity pattern  1710  described above in connection with  FIG. 17A . In some other implementations, the transmissive rim surface  2111  is configured as the surface  1511  of the intensity pattern  1810  described above in connection with  FIG. 18C . In either of these implementations, the tile borders  1513  or  1713  or the tile borders  1813  along the y-axis are distributed at known angular locations relative to each other along the azimuthal θ-axis, so the intensity pattern  2110  can be used as part of the angular displacement measuring system  2100 A/ 2100 B to measure an angular displacement Δθ of the wheel  2134  along the azimuthal θ-axis. 
     For the angular displacement measuring system  2100 A, the rim surface  2111  of the intensity pattern  2110  is spaced apart from and faces the LEE array  1504  and the photodetector  1522 A. In this manner, during operation of the angular displacement measuring system  2100 A, the LEE array  1504  illuminates the transmissive rim surface  2111  of the intensity pattern  2110  with N TOT  beams  1506 ; the transmissive rim surface  2111  selectively transmits the N TOT  beams  1506 ; the array of micro-mirrors  2118  reflects, scatters or both the selectively transmitted beams, such that N TOT  redirected beams  1520 A form an acute angle relative the selectively transmitted beams; and the rim surface  2111  transmits the redirected beams  1520 A to the photodetector  1522 A. In another implementation of the angular displacement measuring system  2100 A (not illustrated in  FIG. 21B ), the intensity pattern  2110  is configured to have a selectively reflective rim surface  2111  as described above in connection with  FIG. 15C ,  FIG. 17A  or  FIG. 18C . During operation of this implementation of the angular displacement measuring system  2100 A, the LEE array  1504  illuminates the reflective rim surface  2111  of the intensity pattern  2110  with N TOT  beams  1506 ; and the reflective rim surface  2111  selectively redirects (via reflection, scattering or both) the N TOT  beams  1506  to the photodetector  1522 A, such that the N TOT  redirected beams  1520 A form an acute angle relative the beams  1506 . 
     For the angular displacement measuring system  2100 B, the rim surface  2111  of the intensity pattern  2110  is spaced apart from and faces the LEE array  1504 , and the side wall surface  2116  of the intensity pattern  2110  is spaced apart from and faces the photodetector  1522 B. In this manner, during operation of the angular displacement measuring system  2100 B, the LEE array  1504  illuminates the transmissive rim surface  2111  of the intensity pattern  2110  with N TOT  beams  1506 ; the transmissive rim surface  2111  selectively transmits the N TOT  beams  1506 ; the array of micro-mirrors  2118  reflects, scatters or both the selectively transmitted beams, such that N TOT  redirected beams  1520 B form a folding angle relative the selectively transmitted beams (e.g., the folding angle formed by the redirected beams  1520 B relative the illuminating beams  1506  can be a substantially right angle); and the transmissive side wall surface  2116  transmits the redirected beams  1520 B to the photodetector  1522 B. 
     Further in the example shown in  FIGS. 21A-21B , the LEE array  1504  illuminates an area  1514  of the rim surface  2111  of the intensity pattern  2110  with N TOT  beams  1506 , and a separation between the illuminated locations  1508  is a known separation S. Moreover, the size along the azimuthal θ-axis of each of tiles  1512  or  1712  or  1812  can accommodate 1≤N≤N TOT  concurrently illuminated locations  1508 . 
     In this manner, the PWM driver  1536  of the angular displacement measuring system  2100 A/ 2100 B can use a switching gate, similar to the switching gate  1524  shown in  FIG. 15E , to time multiplex the N TOT  LEEs  1528  of the LEE array  1504 . In this manner, the processor  1546  (in conjunction with the photodetector  1522 A/ 1522 B) obtains, for each sampling time t k , an associated set of N TOT  intensity values corresponding to intensities of respective beams  1520 A/ 1520 B redirected by the intensity pattern  2110  via either reflection off a selectively reflective rim surface  2111  at the illuminated locations  1508 , or transmission through a selectively transmissive rim surface  2111  at the illuminated locations  1508 . As such, the processor  1546  of the angular displacement measuring system  2100 A/ 2100 B can determine, for each sampling time t k , positions along the azimuthal θ-axis of the illuminated locations  1508  of the rim surface  2111  of the intensity pattern  2110  based on relative differences between the intensity values of the associated issued set. Then, in a manner similar to the manners described above in connection with  FIGS. 16A-16B  or  FIGS. 17B-17C  or  FIGS. 18C-18E  (e.g., by substituting ΔX with Δθ), the processor  1546  determines, for each sampling time t k , an angular displacement Δθ of the wheel  2134  based on one or more changes of the intensity values of the obtained set caused by rotation of the wheel that sweeps at least one of the tile borders  1513  or  1713  or the tile borders  1813  along the y-axis through at least one of the illuminating beams  1506 . 
       FIG. 22A  is a side view, in the (y,z) plane, of another example of a displacement measuring system  2200  configured to concurrently measure angular displacement Δθ and axial displacement ΔY of a wheel  2235 . Here, a device  2230  has a frame  2232  that encapsulates at least a portion of the displacement measuring system  2200  and a portion of an axle  2234  on which the wheel  2235  is mounted. A remaining portion of the axle  2234  protrudes outside of the frame  2232  through an opening  2233  with bearing mechanism, for instance. In some implementations, the device  2230  is a watch, and the wheel  2235  is a setting/control crown. 
     In this example, the displacement measuring system  2200  is a modification of the displacement measuring system  1500 A described above in connection with  FIG. 15A  and  FIGS. 18A-18B . The displacement measuring system  2200  includes the mount  1502 A, the LEE array  1804 , and the photodetector  1522 , all of which described in detail above in connection with  FIGS. 15A-15E  and  FIGS. 18A-18B . Note that the N TOT  LEEs  1828  of the LEE array  1804  are distributed in N X  rows along the x-axis and N Y  rows along the y-axis. 
     An intensity pattern  2210  of the displacement measuring system  2200  is a structure shaped like a wheel that is disposed on a rim surface  2262  of the axle  2234 , i.e., co-axially with the wheel  2235 .  FIG. 22B  is a plan view in the (x,z) plane of one of the two side wall surfaces  2216  of the intensity pattern  2210 . In the example illustrated in  FIGS. 22A-22B , the intensity pattern  2210  includes an array of micro-mirrors  2218  (or micro-prisms, or other redirecting micro-structures) between a transmissive rim surface  2211  and the side wall surfaces  2216 . Here, the flat surface of the micro-mirrors of the array  2218  is oriented parallel to the y-axis. Further here, the transmissive rim surface  2211  is configured as the surface  1511  of the intensity pattern  1810  described above in connection with  FIG. 18C . As such, the tile borders  1813  parallel to the y-axis are distributed at known angular locations relative to each other along the azimuthal θ-axis, and the tile borders  1813  parallel to the x-axis are distributed at known axial locations relative to each other along the y-axis, so the intensity pattern  2210  can be used as part of the displacement measuring system  2200  to measure both an angular displacement Δθ of the wheel  2235  along the azimuthal θ-axis and an axial displacement ΔY of the wheel  2235  along the y-axis. 
     Moreover, the rim surface  2211  of the intensity pattern  2210  is spaced apart from and faces the LEE array  1804  and the photodetector  1522 . In this manner, during operation of the displacement measuring system  2200 , the LEE array  1804  illuminates the transmissive rim surface  2211  of the intensity pattern  2210  with N TOT  beams  1806 ; the transmissive rim surface  2211  selectively transmits the N TOT  beams  1806 ; the array of micro-mirrors  2218  reflects, scatters or both the selectively transmitted beams, such that N TOT  redirected beams  1520 A form an acute angle relative the selectively transmitted beams; and the rim surface  2211  transmits the redirected beams  1520 A to the photodetector  1522 A. In another implementation of the angular displacement measuring system  2200  (not illustrated in  FIG. 22B ), the intensity pattern  2210  is configured to have a selectively reflective rim surface  2211  as described above in connection with  FIG. 18C . During operation of this implementation of the angular displacement measuring system  2200 , the LEE array  1804  illuminates the reflective rim surface  2211  of the intensity pattern  2210  with N TOT  beams  1806 ; and the reflective rim surface  2211  selectively redirects (via reflection, scattering or both) the N TOT  beams  1806  to the photodetector  1522 , such that the N TOT  redirected beams  1520 A form an acute angle relative the beams  1806 . 
     Further in the example shown in  FIGS. 22A-22B , the LEE array  1804  illuminates an area  1814  of the rim surface  2211  of the intensity pattern  2210  with N TOT  beams  1806 , and separation along the azimuthal θ-axis and the y-axis between the illuminated locations  1808  are separations δ θ  and δ Y y. Moreover, the size along the azimuthal θ-axis and the size along the y-axis of each of tiles  1812  can accommodate 2≤N≤N X,Y  concurrently illuminated locations  1808 . 
     In this manner, the PWM driver  1536  of the displacement measuring system  2200  can use a switching gate, similar to the switching gate  1524  shown in  FIG. 15E , to time multiplex the N TOT  LEEs  1828  of the LEE array  1804 . In this manner, the processor  1546  (in conjunction with the photodetector  1522 ) obtains, for each sampling time t k , an associated set of N TOT  intensity values corresponding to intensities of respective beams  1520 A redirected by the intensity pattern  2210  via either reflection off a selectively reflective rim surface  2211  at the illuminated locations  1808 , or transmission through a selectively transmissive rim surface  2211  at the illuminated locations  1808 . As such, the processor  1546  of the displacement measuring system  2200  can determine, for each sampling time t k , positions along both the azimuthal θ-axis and axial y-axis of the illuminated locations  1808  of the rim surface  2211  of the intensity pattern  2210  based on relative differences between the intensity values of the associated obtained set. Then, in a manner similar to the manner described above in connection with  FIGS. 18C-18E  (e.g., by substituting ΔX with Δθ), the processor  1546  determines, for each sampling time t k , both an angular displacement Δθ and an axial displacement ΔY of the wheel  2235  based on one or more changes of the intensity values of the issued set caused by rotation of the wheel that sweeps at least one of the tile borders  1813  through at least one of the illuminating beams  1806 . 
     Each of the intensity pattern  1510 A/ 1510 B illustrated in  FIG. 15C , the intensity pattern  1710  illustrated in  FIG. 17A , and the intensity pattern  1810  illustrated in  FIG. 18C  can be fabricated using infra-red (IR) ink with varying thickness based on various photo-mask processes, for instance. In some implementations, multiple masks can be applied to produce the 1D 3-level intensity pattern  1710  as shown in  FIG. 17A , or the 2D 4-level intensity pattern  1810  shown in  FIG. 18C . Commercially available IR inks used in visible light pass-through filters have stop bands in the range of 800 nm. IR VCSELs that emit light in the range of 800-1100 nm are readily available. Processes for spin coating (for thickness and uniformity control) and photo mask (UV development) patterning, used to make CMOS image sensor color filter arrays (CFA), can be applied to the above-noted IR inks to fabricate the intensity patterns  1510 A/ 1510 B,  1710 , and  1810  with a spatial resolution of order ˜1 μm. Alternatively, in the case of cost-sensitive applications, inkjet printing can be used to fabricate the intensity patterns  1510 A/ 1510 B,  1710 , and  1810  with a spatial resolution of order ˜10 μm. 
     In some implementations, the controller system  1525  can be configured as mixed signal circuitry that processes analog signals and digital signals. In some implementations, the controller system  1525  can be configured as one or more digital signal processors, e.g., ASIC, FPGA, CPU, etc. In some implementations, at least portions of circuitry that powers and conditions the LEE array  1504 / 1804  can be combined with at least portions of circuitry illustrated in  FIG. 15D  that is associated with the photodetector  1522  in a single system on a chip (SOC), ASIC, FPGA, CPU, etc. 
     A few embodiments have been described in detail above, and various modifications are possible. The disclosed subject matter, including the functional operations described in this specification, can be implemented in electronic circuitry, computer hardware, firmware, software, or in combinations of them, such as the structural means disclosed in this specification and structural equivalents thereof, including system on chip (SoC) implementations, which can include one or more controllers and embedded code. 
     While this specification contains many specifics, these should not be construed as limitations on the scope of what may be claimed, but rather as descriptions of features that may be specific to particular embodiments. Certain features that are described in this specification in the context of separate embodiments can also be implemented in combination in a single embodiment. Conversely, various features that are described in the context of a single embodiment can also be implemented in multiple embodiments separately or in any suitable subcombination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claimed combination may be directed to a subcombination or variation of a subcombination. 
     Similarly, while operations are depicted in the drawings in a particular order, this should not be understood as requiring that such operations be performed in the particular order shown or in sequential order, or that all illustrated operations be performed, to achieve desirable results. In certain circumstances, multitasking and parallel processing may be advantageous. Moreover, the separation of various system components in the embodiments described above should not be understood as requiring such separation in all embodiments. 
     Other embodiments fall within the scope of the following claims.

Metadata:
Filing Date: 20190222
Publication Date: 20200407
Grant Date: 20200407
Priority Date: 20160916
Inventors: CHEN, DENIS G.
LIU, JUI HUA
CAI, Xingxing
Assignee: APPLE INC
CPC Classifications: [{"code": "G01D5/142", "inventive": false, "first": false, "tree": "[]"}, {"code": "G01D5/2013", "inventive": true, "first": false, "tree": "[]"}, {"code": "G01B11/002", "inventive": true, "first": false, "tree": "[]"}, {"code": "G01B11/002", "inventive": true, "first": false, "tree": "[]"}, {"code": "G01B11/14", "inventive": true, "first": true, "tree": "[]"}, {"code": "G01D5/30", "inventive": true, "first": false, "tree": "[]"}, {"code": "G01D5/30", "inventive": true, "first": false, "tree": "[]"}, {"code": "G01D5/142", "inventive": false, "first": false, "tree": "[]"}, {"code": "G01B11/002", "inventive": true, "first": false, "tree": "[]"}, {"code": "G01D5/2013", "inventive": true, "first": false, "tree": "[]"}, {"code": "G01B11/14", "inventive": true, "first": true, "tree": "[]"}, {"code": "G01D5/30", "inventive": true, "first": false, "tree": "[]"}, {"code": "G01D5/2013", "inventive": true, "first": true, "tree": "[]"}]
Family ID: 61618410