Patent Publication Number: US-2006001886-A1

Title: Precision retroreflector positioning apparatus

Description:
STATEMENT AS TO RIGHTS TO INVENTIONS MADE UNDER FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT  
      Not Applicable  
     FIELD OF THE INVENTION  
      The present invention relates generally to positioning of optical elements within an optical system, and more particularly relates to the precise positioning of a retroreflector along one axis, with minimal axial deviations.  
     BACKGROUND OF THE INVENTION  
      In certain optical applications, it is desirable to precisely control the path distance between a light source and a reflective element. It is particularly desirable to precisely and repeatedly control the position of an optical element along a first axis relative to a fixed light source, without appreciably displacing the mirror along dimensions orthogonal to the first axis.  
      One specific optical application relating to high precision positioning involves the use of corner-cube retroreflectors. A corner-cube retroreflector is a prism or set of three first-surface mirrors each having three mutually perpendicular surfaces and a hypotenuse face, or effective aperture. Light entering through the effective aperture is reflected by each of the three surfaces, and emerges back through the effective aperture parallel to the entering beam.  
      As depicted in  FIG. 1 , a conventional laser heterodyne interferometer can be used as a metrology gauge to measure the distance L between corner-cube retroreflector  101  and corner-cube retroreflector  102 . Briefly, changes in the relative phase between signals received at reference photodiode  104  and measurement photodiode  105 , and reference photodiode  106  and measurement photodiode  107  are measured to calculate the optical path difference (“OPD”) between reference light beam  109  and a measurement light beam  110 . Through a series of known optical equations, computer  111  can calculate the change in distance between the corner-cube retroreflector  101  and corner-cube retroreflector  102  using the OPD. See Peter G. Halverson &amp; Robert E. Spero,  Signal Processing and Testing of Displacement Metrology Gauges with Picometer - Scale Cyclic Nonlinearity , J. of Optics A: Pure and Applied Optics, Vol. 4, No. 6 (November 2002).  
      Interferometric displacement gauges, such as the laser heterodyne interferometer discussed above, are typically susceptible to various errors, including but not limited to cyclic error, diffraction error, mispointing, thermal drift, laser drift, and errors introduced from other noise sources. As illustrated in  FIG. 2 , cyclic error is exhibited by known interferometric displacement gauges as a repeatable non-linearity when the distance measured is varied. This non-linearity is typically a sinusoidal deviation from expected measurements, when the distance between the corner-cube retroreflectors is adjusted linearly.  
      Reverting to  FIG. 1 , there are several sources of cyclic error in conventional laser heterodyne interferometer metrology gauges, including: 
          Frequency shifters &amp; RF leakage (Region A): After exiting laser  112 , the gauge&#39;s laser light is split into two paths, and the frequency is shifted to create two optical frequencies by acousto-optic modulator (“AOM”)  114  and AOM  115 . Mixing of the radiofrequency (“RF”) signals creates a predictable cyclic error.     Metrology head &amp; optical leaking (Region B): Leakage of the reference beam or measurement beam into unintended paths near metrology head  116  or metrology head  117  will cause a cyclic error.     Photodiode signal mixing (Region C): Electrical isolation is achieved by operating with photodiode preamps, filter and sine-to-square wave converters  119  to  122  on independent power supplies, and by preventing ground loops.     Timing signal mixing (Region D): The outputs of sine-to-square-wave converters  119  to  122  are inherently immune to cross-talk effects.     Phasemeter time-of-measurement error (Region E): The phasemeter measures the relative time of logic transitions signaling the zero-crossings of the photodiode signals. This creates ambiguity as to when phase measurements are made.        

      Since cyclic error is manifested as a periodic deviation from the linear ramp expected when constant velocity motion is applied to a fiducial (such as a corner-cube retroreflector), these periodic deviations can be used for detecting and measuring the cyclic error.  FIG. 3  is a simplified block diagram illustrating one such typical cyclic error measurement test bed.  
      During cyclic error measurement, corner cube retroreflector  301  is moved linearly along the Z-axis, or parallel to the axis defined by laser beam  302  of a known frequency, emitted from laser  303 . The displacement time history is measured by displacement measuring interferometer  304 , and virtual machine environment (“VME”) chassis  305  applies a Fourier transform to the output data to reveal the cyclic error at the frequency. Typically, corner-cube retroreflector  301  is positioned using a Z-axis coupler control loop, which includes lead-zirconate-titanate (“PZT”) actuator  306  connected to the backside surface of corner-cube retroreflector  301  via a coupler. Fold mirror  308  directs laser beam  302  from laser  303  to corner-cube retroreflector  301 , and a ramp generator (not depicted) transmits a signal to PZT actuator  306  to control the linear motion of corner-cube retroreflector  301 .  
      In order to avoid coupling or beam walk errors, the X-axis and Y-axis motion of the corner cube retroreflector must be minimized. If axial deviations occur during positioning of the corner cube retroreflector, displacement measurement errors will occur, and calculated cyclic error will be greater than actual cyclic errors. Typical couplering mechanisms exhibit deviations of straightness of motion and tilt, including undesirable pitch, yaw, and roll, resulting in Abbe error and Cosine error.  
      It is therefore considered highly desirable to provide an apparatus for precisely positioning or moving a reflective optical element, to repeatedly control the position of the reflective optical element along an axis defined by the light beam, while minimizing deviations orthogonal to that axis. In particular, it is desirable to provide an apparatus for positioning a corner-cube retroreflector along an axis parallel to a beam of light, while minimizing axial deviations.  
     SUMMARY OF THE INVENTION  
      Various optical applications require the precise movement or positioning of a retroreflector in one dimension, while minimizing axial deviations. Conventional positioning apparatus, however, typically exhibit off-axis motion, introducing errors. The present invention solves the foregoing problems by providing precise positioning of a retroreflector along one axis, with minimal axial deviations.  
      According to one aspect, the present invention is an apparatus for positioning a retroreflector. The apparatus includes a retroreflector, where the retroreflector further includes an effective aperture. The apparatus also includes a retroreflector mount for holding the retroreflector, where the retroreflector mount further includes a front end and a back end obverse to the front end, and where the effective aperture is exposed through an opening in the front end. Furthermore, the apparatus also includes a plurality of parallel radial flexures including a first radial flexure and a second radial flexure parallel to the first radial flexure, where the first radial flexure surrounds the front end, and where the second radial flexure surrounds the back end, and an actuator for positioning the retroreflector. The plurality of parallel radial flexures allow for one-axis movement of the retroreflector of ±2.5 millimeters perpendicular to the plurality of parallel radial flexures, with an axial deviation of less than 0.001 radians.  
      Since the precision retroreflector apparatus of the present invention includes a plurality of parallel radial flexures, an actuator which applies a force on the retroreflector mount at the center of the flexures effectuates a movement along one axis.  
      The first radial flexure is preferably comprised of steel, aluminum, Invar or titanium, and the actuator is preferably a PZT actuator, voice-coil actuator, piezo actuator, or linear motor. The apparatus further includes a coupler, where the coupler connects the actuator to the back end of the retroreflector mount. The stiffness of the flexures allows the retroreflector to be positioned over a 5 millimeter range (2.5 millimeters in each direction), without introducing axial deviations of greater than a milli-radian.  
      The first radial flexure is pinned and clamped to the retroreflector mount. The first radial flexure includes a notched cutout pattern, or a spiral cutout pattern. To its benefit, the apparatus according to the present invention uses a plurality of parallel radial flexures to provide repeatable, 1-axis range and resolution, with precision not available to conventional, “off the shelf” couplering mechanisms. As such, the apparatus can position an optical or non-optical object to a greater precision and repeatability than other conventional linear translation stages.  
      According to a second aspect, the present invention is a precision positioning apparatus, including a mount, where the mount further includes a front end and a back end obverse to the front end. The apparatus also includes a plurality of parallel radial flexures including a first radial flexure and a second radial flexure parallel to the first radial flexure, where the first radial flexure surrounds the front end, and where the second radial flexure surrounds the back end. The apparatus further includes an actuator for positioning the mount, where the plurality of parallel radial flexures allow for one-axis movement of the mount of ±2.5 millimeters perpendicular to the plurality of parallel radial flexures, with an axial deviation of less than 0.001 radians.  
      In the following description of the preferred embodiment, reference is made to the accompanying drawings that form a part thereof, and in which is shown by way of illustration a specific embodiment in which the invention may be practiced. It is to be understood that other embodiments may be utilized and changes may be made without departing from the scope of the present invention.  
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
      Referring now to the drawings in which like reference numbers represent corresponding parts throughout:  
       FIG. 1  depicts a block diagram of a typical laser heterodyne interferometer;  
       FIG. 2  is a chart illustrating cyclic error of a typical laser heterodyne interferometer, shown as a sinusoidal deviation from an expected measurement as the distance between fiducial elements is adjusted linearly;  
       FIG. 3  depicts a typical component layout for the detection and measurement of cyclic error;  
       FIG. 4  depicts a cross-sectional view of the apparatus for positioning a retroreflector, according to one embodiment of the present invention;  
       FIG. 5  illustrates a frontal view of the  FIG. 2  embodiment;  
       FIG. 6  shows a perspective view of the  FIG. 2  embodiment;  
       FIGS. 7A and 7B  illustrate a cross-sectional view of the apparatus for positioning a retroreflector, in a state where the retroreflector mount has been projected and retracted, respectively;  
       FIGS. 8 and 8 A depict a frontal view and a side view, respectively, of an example notched radial flexure used by the apparatus according to one embodiment of the present invention;  
       FIGS. 9 and 9 A depict a frontal view and a side view, respectively, of an example spiral-cut radial flexure used by the apparatus according to an alternate embodiment of the present invention;  
       FIG. 10  is a drawing of a cyclic error measurement test bed, including the apparatus for apparatus for positioning a retroreflector according to the  FIG. 2  embodiment of the present invention; and  
       FIG. 11  is a depiction of a 2-gauge test bed for detecting cyclic error, using the  FIG. 2  embodiment of the present invention.  
    
    
     DETAILED DESCRIPTION OF THE INVENTION  
      The present invention allows for the precision positioning of a retroreflector by an actuator, by surrounding a retroreflector mount with a plurality of parallel radial flexures.  
       FIG. 4  is a cross-sectional view of one embodiment of an apparatus for positioning a retroreflector, in accordance with the present invention.  FIG. 5  is a frontal view of the apparatus for positioning a retroreflector of  FIG. 4 , and  FIG. 6  is a perspective view of the same apparatus.  
      Briefly, the embodiment of the present invention illustrated in FIGS.  4  to  6  relates to an apparatus for positioning a retroreflector, where the apparatus includes a retroreflector, and where the retroreflector further comprises an effective aperture. The apparatus also includes a retroreflector mount for holding the retroreflector, where the retroreflector mount further includes a front end and a back end obverse to the front end, where the effective aperture is exposed through an opening in the front end. Additionally, the apparatus includes an actuator for positioning the retroreflector, and a plurality of parallel radial flexures including a first radial flexure and a second radial flexure parallel to the first radial flexure, where the first radial flexure surrounds the front end, and where the second radial flexure surrounds the back end. The plurality of parallel radial flexures allow for one-axis movement of the retroreflector of +2.5 millimeters perpendicular to the plurality of parallel radial flexures, with an axial deviation of less than 0.001 radians.  
      In more detail, apparatus  400  for positioning a retroreflector includes retroreflector  401 , where retroreflector  401  further includes effective aperture  402 . A retroreflector is a device which transmits light back to where it came from, regardless of the angle of incidence. As depicted in  FIG. 4 , light beam  404  enters effective aperture  402  and, due to the geometry of retroreflector  401 , light beam  404  is reflected back at the light source in a beam parallel to the incoming beam.  
      Apparatus  400  also includes retroreflector mount  405 , where retroreflector mount further includes front end  406  and back end  407  obverse to front end  406 . Effective aperture  402  is exposed through an opening in front end  406 . Retroreflector mount  405  is for holding retroreflector  401 , and allows for retroreflector  401  to be moved through one-axis motion with minimal axial deviations.  
      As illustrated in FIGS.  4  to  6 , apparatus  400  further includes actuator  409  and coupler  410  for positioning retroreflector  401 , by applying a positioning force to retroreflector mount  405 . Actuator  409  is a PZT actuator, although in alternate arrangements actuator  409  is another type of actuator known in the art to be effective for nanopositioning, such as a piezo actuator, voice coil actuator or a linear motor. The net force applied by actuator  409  is directed to the center of retroreflector mount  405 .  
      In an additional alternate aspect of the present invention, coupler  410  is omitted, and actuator  409  applies positioning forces to retroreflector mount  405  directly. Retroreflector mount  405  remains in a stable, neutral position when actuator  409  is not applying a positioning force.  
      Apparatus  400  includes a plurality of parallel radial flexures, including first radial flexure  411  and second radial flexure  412  parallel to first radial flexure  411 . First radial flexure  411  surrounds front end  406 , and second radial flexure  412  surrounds back end  407 . First radial flexure  411  and second radial flexure  412  are in physical communication with both retroreflector mount  405  and support  414 , and hold retroreflector mount  405  into place. The structure, composition and design of the radial flexures will be discussed in more detail in conjunction with the descriptions of  FIGS. 8 and 9 .  
      Support  414  and retroreflector mount  405  hold the plurality of parallel radial flexures in place by clamping. Specifically, as illustrated in  FIGS. 4 and 5 , a radial flexure is placed in between support  414  and outer clamp  415 , or retroreflector mount  405  and inner clamp  416 , and the flexure is clamped by securing outer clamp  415  or inner clamp  416  into place using bolts  517  ( FIG. 5 ). In an alternate, aspect of the invention, the plurality of parallel radial flexures is held in place by pinning and clamping.  
      Finally, apparatus  400  also includes shell  417  for protecting actuator  409  and coupler  410  from external influences, and frame  419 , upon which all the above described components are mounted.  
       FIGS. 7A and 7B  illustrate an enlarged, cross-sectional view of the apparatus for positioning a retroreflector, in a state where the retroreflector mount has been projected and retracted, respectively. Referring briefly to  FIG. 4 , actuator  409  pushes coupler  410 , and coupler  410  applies a one-axis force on retroreflector mount  405  perpendicular to second radial flexure  412 . As shown in  FIG. 7A , the force is directed to a location on back end  407  representing the radial center of second radial flexure  412 . In a similar manner, the same one-axis force is transmitted via retroreflector mount  405  to front end  406 , which moves perpendicular to first radial flexure  411  as well.  
      Retroreflector mount  405 , which holds retroreflector  401 , can be projected up to +2.5 millimeters from a neutral position, with an axial deviation of less than 0.001 radians. By projecting retroreflector  401 , apparatus  400  effectuates a shortened laser beam path length.  
      In  FIG. 7B , actuator  409  pulls coupler  410 , and coupler  410  applies a one-axis force on retroreflector mount  405  perpendicular to second radial flexure  412 , and in an obverse direction to the force applied in  FIG. 7A . The force applied by coupler  410  and actuator  409  is directed to a location on back end  407  representing the radial center of second radial flexure  412 . Similarly, the one-axis force is transmitted via retroreflector mount  405  to front end  406 , which moves perpendicular to first radial flexure  411 .  
      Retroreflector mount  405  and retroreflector  401  can be retracted up to −2.5 millimeters from the neutral position, with an axial deviation of less than 0.001 radians. By retracting retroreflector  401 , apparatus  400  effectuates a longer laser beam path length.  
       FIG. 8  depicts a frontal view of an example “notched” radial flexure used by the apparatus according to one embodiment of the present invention. The radial flexure  800  includes a series of 18 sets of overlapping notch-shaped cuts, including cutout  801 , around the periphery of flexure  800 , allowing flexure  800  to provide for one-axis motion orthogonal to the plane defined by flexure  800 , with minimal axial deviation.  
      The features of the notched radial flexure  800  are oriented on six discrete rings  802  to  807 . Ring  802 , the outermost ring, defines the outer perimeter of radial flexure  800 , and has a relative diameter of 8.50 units. The next smallest ring, ring  803 , defines a circle around which bolts and pins are inserted to hold outer clamp  415  onto support  414 , thereby clamping flexure  800  into place. Ring  803  has a relative diameter of 8.00 units.  
      Ring  804  and ring  805 , respectively, define the outer radius and the inner radius of the notch cuts. Ring  804  has a relative diameter of 7.50 units, and ring  805  has a relative diameter of 5.5 units. Ring  806  is similar in function to ring  803 , and defines a circle around which bolts and pins are inserted to hold inner clamp  416  onto retroreflector mount  405 , in order to clamp flexure  800  into place. Ring  806  has a relative diameter of 5.00 units. Finally, ring  807 , the smallest ring with a relative diameter of 4.30 units, defines the inner perimeter of radial flexure  800 .  
      Outer flexure portion  809  lies between ring  802  and ring  804 , and inner flexure portion  810  lies between ring  805  and ring  807 . As a result of the freedom of movement imparted by the presence of the overlapping notch-shaped cuts such as cutout  801 , inner flexure portion  810  can be can be displaced with respect to outer flexure portion  809 , in a direction orthogonal to the plane defined by flexure  800  up to 2.5 millimeters in each direction (for a total range of 5 millimeters), with an axial deviation of less than 0.001 radians.  
      Flexure  800  is comprised of Invar, although in an alternate aspect of the present invention flexure  800  is comprised of steel, aluminum, or titanium. Cutout  801  is formed by chemical etching, laser cutting, electro discharge machining (“EDM”) or other machining processes known in the art. Each cutout includes an arc-shaped portion, oriented on either ring  804  or ring  805 , and two straight-radial portions oriented relative to the center of the flexure, and each intersecting an obverse end of the arc-shaped portion.  
      A finite element model (“FEM”) study was developed using the flexure design depicted in  FIG. 8 , with flexures comprised of Invar, steel, and aluminum. Table 1 shows the results of the FEM, which compares total weight, active weight, axial spring constant (K axial ), lateral spring constant (K lateral ), lateral frequency (f lateral ), and axial frequency (f axial ) for each of four different configurations:  
               TABLE 1                          FEM Results                                                 Aluminum,           Steel, pinned   Steel, clamped   Invar, clamped   clamped                                                             Total weight   0.0957168   lb   0.0957168   lb   0.1009188   lb   0.033986   lb       Active weight   0.0478587   lb   0.0478587   lb   0.0504594   lb   0.016933   lb       K axial     126.34175   lb/in   505.155108   lb/in   365.843397   lb/in   176.5514   lb/in       K lateral     2.99709 E 5   lb/in   3.0455214 E 5   lb/in   2.19574 E 5   lb/in   1.05942 E 5   lb/in       ƒ axial     160.7434   Hz   321.4199   Hz   266.388   Hz   318.889   Hz       ƒ lateral     7.829   kHz   7.892   kHz   6.52617   kHz   7.811596   kHz                  
 
      While FIGS.  4  to  8  and their accompanying descriptions fully describe a specific embodiment of the present invention, various modifications, alternative constructions and equivalents may be used. For example, while the example embodiment illustrated in FIGS.  4  to  8  utilizes a pair of flexures having a specific notched cutout pattern, this pattern is not required by alternate embodiments of the present invention. In accordance with these alternate embodiments, other flexure designs or additional flexures can be used.  
      For instance,  FIG. 9  depicts a frontal view of an example of a spiral-cut radial flexure used by the apparatus according to an alternate embodiment of the present invention. Flexure  900  comprises a metal disk, in which a series of 3 spiral cutouts, including spiral cutout  901 , are formed, allowing flexure  900  to provide for one-axis motion orthogonal to the plane defined by flexure  900 , with minimal axial deviation.  
      The features of the radial flexure  900  are oriented on six discrete rings. Ring  902 , the outermost ring, defines the outer perimeter of radial flexure  900 , and has a relative diameter of 8.50 units. The next smallest ring, ring  903 , defines an circle around which bolts and pins are inserted to hold outer clamp  415  onto support  414 , thereby clamping flexure  900  into place. Ring  903  has a relative diameter of 8.00 units.  
      The start point and end point of cutout  901  lie on ring  904  and ring  905 , respectively. Ring  904  has a relative diameter of 7.50 units, and ring  905  has a relative diameter of 1.5 units. Ring  906  is similar in function to ring  902 , and defines a circle around which bolts and pins are inserted to hold inner clamp  416  onto retroreflector mount  405 , in order to clamp flexure  900  into place. Ring  906  has a relative diameter of 1.06 units. Finally, ring  907 , the smallest ring with a relative diameter of 0.62 units, defines the inner perimeter of radial flexure  900 .  
      Flexure  900  is comprised of Invar, although in an alternate aspect of the present invention flexure  900  is comprised of steel, aluminum or titanium. Cutout  901  is formed by chemical etching, laser cutting or other machining processes known to the art. Each cutout begins on ring  904 , and makes a spiral pattern, ending on ring  905 .  
      Outer flexure portion  909  lies between ring  902  and ring  904 , and inner flexure portion  910  lies between ring  905  and ring  907 . As a result of the freedom of movement imparted by the presence of the overlapping spiral-shaped cuts such as cutout  901 , inner flexure portion  910  can be can be displaced with respect to outer flexure portion  909 , in a direction orthogonal to the plane defined by flexure  900  more than 15 millimeters in each direction (for a total range of 30 millimeters), with a minimal axial deviation.  
      The spiral-shaped flexure design shown in  FIG. 9  offers the advantage of greater length of linear motion orthogonal to the plane defined by the flexures, as compared with the example notched flexure design of  FIG. 8 . However, displacement of inner flexure portion  910  along this axis of motion will be accompanied by some amount of rotation inner flexure portion  910  relative to outer flexure portion  909 . In certain applications, such as where retroreflector  901  is a symmetrical mirror, the rotation would be insignificant. In other applications, however, the rotation is beneficial, since the amount of rotation per displacement distance can be measured and controlled, by adjusting the shapes of the cutouts.  
       FIG. 10  is a block diagram depicting a cyclic error measurement test bed, including apparatus  400  for positioning a retroreflector according to the  FIG. 2  embodiment of the present invention. Specifically and as described above with respect to  FIG. 2 , apparatus  400  includes a retroreflector, where the retroreflector further comprises an effective aperture, and a retroreflector mount for holding the retroreflector, where the retroreflector mount further includes a front end and a back end obverse to the front end, where the effective aperture is exposed through an opening in the front end. Furthermore, apparatus  400  includes a plurality of parallel radial flexures including a first radial flexure and a second radial flexure parallel to the first radial flexure, where the first radial flexure surrounds the front end, and where the second radial flexure surrounds the back end. Moreover, the apparatus includes an actuator for positioning the retroreflector, where the plurality of parallel radial flexures allow for one-axis movement of the retroreflector of ±2.5 millimeters perpendicular to the plurality of parallel radial flexures, with an axial deviation of less than 0.001 radians.  
      In addition to apparatus  400 , as seen in  FIG. 10 , the cyclic error measurement test bed further includes laser  1001  for emitting laser beam  1002 . Laser beam  1002  reflects off of fold mirror  1004 , and laser beam  1002  is directed into the corner-cube retroreflector mounted on apparatus  400 . The corner cube retroreflector is moved linearly along the Z-axis, or parallel to the axis defined by laser beam  1002  of a known frequency. The displacement time history is measured by computing equipment  1005 , and a Fourier transform is applied to the output data to reveal the cyclic error at the frequency.  
       FIG. 11  shows a 2-gauge test bed for detecting cyclic error, using the  FIG. 2  embodiment of the present invention. The 2-gauge test bed is used for cyclic error testing, to verify a cyclic error sensitivity of less than or equal to 1 pm rms . The test bed can also be used to verify that thermal sensitivity and corner-cube translation fall within specified parameters. As seen in  FIG. 11 , 2-gauge test bed  1101  includes precision retroreflector positioning apparatus  1102  for controlling the path length of measurement light beam  1103  and reference light beam  1104  between precision retroreflector positioning apparatus  1102  and corner cube retroreflector  1105 . Metrology head  1106  and metrology head  1107  measure the OPD between reference light beam  1104  and a measurement light beam  1103 . The OPD is used to calculate the change in distance between the corner-cube retroreflector mounted on precision retroreflector positioning apparatus  1102  and corner-cube retroreflector  1105 .  
      With regard to cyclic error testing, the test bed illustrated in  FIG. 11  uses stable mounts and platforms, to reduce environmental interferences so that 1 pm can be observed at 100 Hz. The Z-axis coupler control loop utilizes an optical gauge and/or strain gauges, where the couplering axis is the Z-axis of the reference corner-cubes, and X-axis and Y-axis motion of the corner cubes is restricted to less than 1 μm, to avoid coupling with beam walk errors. An alignment of 1° is maintained between corner cubes with respect to the vertex-to-vertex axis while couplering. The couplering mechanism provides ±25 μm along the Z-axis and Z-axis tip and tilt. Coupler frequency is between 1 to 2 Hz, with a ±25 μm coupler amplitude. The coupler sweep requires a triangle configuration which is as linear as possible. Non-linearities should be identified and used to correct cyclic data.  
      While the embodiment of the invention illustrated in  FIG. 2  illustrates a corner cube reflecting element positioned at the center of flexures  411  and  412 , other uses for the apparatus are also contemplated by the present invention. In accordance with alternative embodiments, the linear distance of other types of reflecting structures along an axis relative to a light source can also be controlled.  
      In other embodiments, in the alternative, the apparatus depicted in  FIG. 4  can be used as a precision positioning apparatus without a retroreflector, for use in controlling the motion of parts in a precise fashion. This use of the present invention is particularly useful for optical lithography techniques, which are frequently employed in the fabrication of semiconductor devices. Owing to the extremely small size of features being fabricated during such lithographic processes, the position of the wafer relative to a light source must be determined with great precision. Therefore, alternative embodiments of the apparatus in accordance with the present invention control movement of a semiconductor wafer along an axis relative to a radiation source.  
      The invention has been described with particular illustrative embodiments. It is to be understood that the invention is not limited to the above-described embodiments and that various changes and modifications may be made by those of ordinary skill in the art without departing from the spirit and scope of the invention.