Abstract:
An improved method and apparatus are disclosed for monitoring relative movement between an incident beam of substantially coherent light and a grating on which the beam is directed, wherein light from the beam passing through the grating is sensed by a photodetector, and the cycles in the photodetector output signal are counted. The improvement comprises patterning the beam, transforming the patterned beam into a Fraunhofer diffraction pattern, and further diffracting the Fraunhofer diffraction pattern with the grating to cause an interference image, which is detected by the photodetector. Each two cycles in the output signal of the photodetector then correspond to a relative movement of the beam past one grating period.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     This invention relates to position monitoring systems, and more particularly, to systems for tracking relative movement between a light beam and a grating. 
     2. Description of the Prior Art 
     Systems for tracking relative movement between a light beam and a grating are well known. For example, a pattern generator in which a portion of a moving laser beam is directed onto a stationary grating and light from the beam passing through the grating is sensed by a photodetector is illustrated in FIG. 1 of an article by M. J. Cowan et al. entitled &#34;The Primary Pattern Generator Part I -- Optical Design,&#34; beginning on page 2033 in the issue of the Bell System Technical Journal dated November 1970. The grating in this example is alternatively called a code plate. The position of the laser beam is tracked by counting interruptions in the laser beam. A similar system is disclosed in copending application Ser. No. 466,313 of V. J. Zaleckas, filed May 2, 1974, and now U.S. Pat. No. 3,902,036 and assigned to the assignee of this invention, wherein a patterned laser beam is deflected by mirrors rotated by galvanometers. Portions of the pattern are laterally spaced so that one portion is blocked by a line in a grating when the other portion is passed by the grating. Using two photodetectors, two out-of-phase signals are then obtained from which the direction of relative motion between the beam and the grating can be determined. 
     Systems wherein a grating moves with respect to a stationary light beam are also known. For example, automated machine tools have been built having a grating mounted on a movable worktable and a light beam and a photodetector mounted on a stationary bed, for use in tracking the position of the bed. Again, the beam can be patterned and two photodetectors used to generate out-of-phase signals from which the direction of relative movement can be obtained. 
     The above systems depend on a light beam being either blocked or passed by the lines in the grating. The light beam must be small enough in diameter in the plane of the grating so that it can be blocked by a grating line, or passed by the space between grating lines. These lines may be spaced as closely as 20μm apart. Defects in a grating or dust on the grating can thus affect the passage of the light beam and cause inaccuracies in the recorded position. 
     The precision of the above systems is a function of the spacing between the grating lines, and can only be increased by reducing that spacing and reducing the diameter of the incident light beam. There are practical limits to the minimum spacing of grating lines, and these limits prevent more precise systems from being built using the above techniques. 
     What is desired for use in such systems as those described above is a system for tracking the relative positions of a light beam and a grating that is relatively insensitive to defects in the grating or dust on the grating, and that is capable of higher precision with a given prior art grating. 
     SUMMARY OF THE INVENTION 
     The invention relates to improved methods of monitoring relative movement between an incident beam of substantially coherent light and a grating on which the beam is directed, wherein light from the beam passing through the grating is detected by a photodetector and the cycles in the photodetector output signal are counted. In the improved methods, the incident beam is patterned into a number of elements, then transformed into a Fraunhofer diffraction pattern in the plane of the grating. The grating diffracts the transformed beam into various diffraction orders, which are resolved in an image plane. The spacing between the elements in the patterned beam is chosen so that the +1 diffraction order from one of the elements coincides with the -1 diffraction order from another of the elements to form an interference image in the image plane. The photodetector is positioned to detect the interference image. 
     Various embodiments are disclosed for employing the invention in both one- and two-dimensional systems, and for systems wherein the direction of relative movement between a light beam and a grating is also to be determined. Apparatus is disclosed for implementing the various embodiments of the invention. 
     These and other aspects of the invention will be apparent from the accompanying drawings and the Detailed Description. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a schematic representation of apparatus according to the invention for monitoring the position of a deflected light beam; 
     FIG. 2 illustrates an aperture plate for use in the apparatus of FIG. 1; 
     FIG. 3 is a diagram of images and a photodetector in an image plane in the apparatus of FIG. 1 having the aperture plate of FIG. 2; 
     FIG. 4 illustrates another aperture plate for use in the apparatus of FIG. 1; 
     FIG. 5 is a diagram of images and photodetectors in an image plane in the apparatus of FIG. 1 having the aperture plate of FIG. 4; 
     FIG. 6 is a graph of signals from the photodetectors of FIG. 5; 
     FIG. 7 is a schematic representation of another embodiment of apparatus according to the invention for monitoring the position of a deflected light beam; 
     FIG. 8 is a diagram of light patterns in the plane of a grating in the apparatus of FIG. 7; 
     FIGS. 9 and 10 are diagrams of images and photodetectors in image planes in the apparatus of FIG. 7; 
     FIG. 11 is a schematic representation of yet another embodiment of apparatus according to the invention for monitoring the position of a deflected light beam; 
     FIG. 12 illustrates an aperture plate for the apparatus of FIG. 11; 
     FIG. 13 is a diagram of a light pattern on a grating in the apparatus of FIG. 11; 
     FIGS. 14 and 15 are diagrams of images and photodetectors in image planes in the apparatus of FIG. 11; 
     FIG. 16 is a schematic representation of apparatus according to the invention for monitoring the position of a light beam that is deflected in two coordinates; 
     FIG. 17 illustrates an aperture plate for the apparatus of FIG. 16; 
     FIGS. 18, 19, 20, and 21 are diagrams of images and photodetectors in image planes in the apparatus of FIG. 17; 
     FIG. 22 is a schematic representation of apparatus according to the invention using a crossed-grid grating for monitoring the position of a light beam that is deflected in two coordinates; 
     FIG. 23 is a diagram of a light pattern on a crossed-grid grating in the apparatus of FIG. 22; 
     FIGS. 24 and 25 are diagrams of images and photodetectors in image planes in the apparatus of FIG. 22; 
     FIG. 26 is a schematic representation of another embodiment of apparatus according to the invention for monitoring the position of a light beam that is deflected in two coordinates; 
     FIG. 27 is a diagram of images and photodetectors in an image plane in the apparatus of FIG. 26; 
     FIG. 28 is a schematic representation of another embodiment of apparatus according to the invention using a crossed-grid grating for monitoring the position of a light beam that is deflected in two coordinates; 
     FIG. 29 is a diagram of images and photodetectors in an image plane in the apparatus of FIG. 28; 
     FIG. 30 is a schematic diagram of yet another embodiment of apparatus according to the invention for monitoring the position of a light beam that is deflected in two coordinates; 
     FIG. 31 illustrates an aperture plate for the apparatus of FIG. 30; 
     FIGS. 32 and 33 are diagrams of light patterns and photodetectors in image planes in the apparatus of FIG. 30; 
     FIG. 34 is a schematic representation of yet another embodiment of apparatus according to the invention using a crossed-grid grating for monitoring the position of a light beam that is deflected in two coordinates; 
     FIG. 35 is a diagram of light patterns on a grating in the apparatus of FIG. 34; 
     FIG. 36 is a diagram of images and photodetectors in an image plane in the apparatus of FIG. 34; and 
     FIG. 37 is an embodiment of apparatus according to the invention for monitoring the position of a moving grating with respect to a stationary light beam. 
    
    
     DETAILED DESCRIPTION 
     Like numerals will be used for like elements in the different figures of the drawing. 
     FIG. 1 is a schematic representation of exemplary apparatus according to the invention for monitoring the movement of a beam of substantially coherent light with respect to a grating. Beam 10 is directed through aperture plate 11, reflected from mirror 12, which is rotatable around axis 13, and directed through scan lens 14, and beam splitter 15, onto grating 16. Light passing through grating 16 is resolved by collecting lens 17 in image plane 24, where a portion of the image pattern thus formed is sensed by photodetector 18. 
     Beam 10 can be any substantially coherent light beam, for example, the beam from an HeNe laser having a wavelength of 0.6328μm. Beam 10 can also be a composite of more than one wavelength of light, as will be discussed below. 
     Mirror 12 is connected to an actuator (not shown), such as a galvanometer, which rotates the mirror around axis 13 to deflect beam 10 in response to a deflection signal applied to the actuator. Many other means for moving beam 10 with respect to grating 16 will be apparent to those skilled in the art. 
     Scan lens 14 is designed to direct the deflected beam into a path parallel to the axis of the lens, as indicated by dotted lines 10&#39; representing deflected beams. Such a lens, sometimes termed a telecentric lens, is well known in the art. 
     Beam splitter 15 reflects a portion 20 of beam 10 to utilizing means (not shown), for example, a photosensitive sheet in pattern generating apparatus. The remaining portion of beam 10 is used for monitoring the position of the beam and is directed onto grating 16. 
     Beam 10 can also be a composite beam, such as is described in the Zaleckas application noted above, wherein a relatively high-powered working beam having one wavelength and a relatively low-powered reference beam having another wavelength are directed along the same path through deflection apparatus, then split by a selective beam splitter to direct the working beam onto a workpiece and the reference beam onto a grating. Such a composite beam is useful for laser machining apparatus where a high-powered, intermittent working beam may be produced by a pulsed laser and a low-powered continuous reference beam may be used to monitor the positions of deflection mirrors, as in the apparatus disclosed by Zaleckas. If such a composite beam were to be used as beam 10 in the apparatus of this invention, then beam splitter 15 would be selective and beam portion 20 would be the intermittent, high-powered beam. 
     FIG. 2 illustrates aperture plate 11, which contains two apertures 22 and 23 for patterning the beam into two elements. These apertures are coded with vertical and horizontal shading, respectively, to aid in relating the apertures to their images in an image plane to be described subsequently. Such shading will be used as a convention throughout the disclosure. 
     In prior art systems wherein a grating is used as a code plate to block or pass a beam, the beam is focused in the plane of the grating to less than the width of a grating line. In the apparatus of the invention, scanning lens 14 and grating 16 are spaced so that scanning lens 14 transforms the patterned beam into a Fraunhofer diffraction pattern in the plane of grating 16 that covers a relatively larger area of the grating than a single grating line. See, for example, the pattern from beam 10 outlined in FIG. 13. Grating 16 diffracts this pattern into various diffraction orders. 
     According to the invention, the spacing between apertures 22 and 23 in aperture plate 11 is chosen so that the +1 and -1 diffraction orders from these apertures coincide when they are resolved in image plane 24 by collecting lens 17. FIG. 3 is a diagram of the principal images resolved in image plane 24 as a result of the interaction of aperture plate 11, grating 16 and lenses 14 and 17. The images remain stationary in the image plane; however, the intensities of the images change cyclically as beam 10 is moved with respect to grating 16. Image 25 is the zero-order from aperture 22; image 26 is the zero-order from aperture 23. Image 27 is an interference image formed by the +1 order from aperture 22 coinciding with the -1 order from aperture 23. Other images resulting from higher diffraction orders that are not important to the invention are not shown in the Figures. 
     Grating 16 consists of a grid of opaque lines separated by transparent spaces. In the prior art, such as described above, such gratings are used as code plates to block or pass a light beam. In the positioning apparatus according to the invention, however, grating 16 is used to diffract the Fraunhofer diffraction pattern from a transformed beam that has been patterned into a number of elements by apertures in an aperture plate. Several advantages are thus achieved. For example, the effects of grating defects having sizes comparable to the grid-line spacing are minimized; and, as will be shown, the output signal from a detector positioned to respond to interfering +1 and -1 diffraction orders from different apertures has twice the line grating frequency, therefore providing twice the resolution for a particular grating compared with its resolution when used as a code plate. 
     This last advantage can be demonstrated mathematically as follows: 
     For convenience, grating 16 is considered to have a cosine variation. The amplitude transmittance of grating 16, t g  (x), can then be written as 
     
         t.sub.g (x) = 1/2 + 1/2 cos 2π bx                       (1) 
    
     where b is the grating periodicity and x is the variable position of beam 10 with respect to the grating. Apertures 22 and 23 in aperture plate 11 are considered to be circular apertures having diameter c 1  and spaced a center-to-center distance d = 2λfb apart, where λ is the wavelength of the light in the beam and f is the focal length of scan lens 14. 
     We now examine the light amplitude of the +1 and -1 order elements of the image in plane 24. Light amplitude can be represented as Ae i .sup.φ. This amplitude will change as beam 21 is deflected across grating 16. A uniform grating will have a constant diffraction efficiency, and the amplitude term A will remain constant regardless of the position of beam 21. The phase term e i .sup.φ, however, will change during the scan. 
     The quantity φ is dependent on the position of beam 21 on grating 16. In particular, φ varies as 
     
         φ(x) = -2π bx for the -1 order element (2) 
    
     and 
     
         φ(x) = +2π bx for the +1 order element (3) 
    
     where b is the periodicity of the grating defined earlier. 
     When a +1 order from one aperture interferes with the -1 order from another aperture, the total light amplitude u will be the sum of the two interfering orders, viz. 
     
         u = Ae.sup.i2.sup.πbx + Ae.sup.-.sup.i2.sup.πbx      (4) 
    
     Using the identity 
     
         cos φ = (e .sup.i.sup.φ + e.sup..sup.-iφ) / 2  (5) 
    
     equation (4) can be rewritten as 
     
         u = 2A cos 2π bx                                        (6) 
    
     A photodetector will respond to the intensity of light impinging thereon. Intensity is the square of the absolute value of the amplitude of the light 
     
         I = |u|.sup.2                            (7) 
    
     thus 
     
         I = 4A.sup.2 cos.sup.2 2π bx.                           (8) 
    
     Using the identity 
     
         cos.sup.2 φ = (1 + cos 2φ)/2                       (9) 
    
     equation (8) can be rewritten 
     
         I = 2A.sup.2 (1 + cos 4π bx).                           (10) 
    
     Thus, the signal from photodetector 18 exhibits a cosinusoidal variation having twice the periodicity of grating 16 as beam 10 is moved with respect to grating 16. 
     It can be shown that substantially the same relationship is true if grating 16 has a square-wave variation, which is typically the case. 
     The ability of the above-described interference method to minimize signal errors caused by grating defects depends upon the size of beam 10 at the grating surface. The size of this beam can be related approximately to aperture size c and aperture spacing d by the expression 
     
         h = 1.22 (d/c) x.sub.g                                     (11) 
    
     where h is the pattern size at the grating and x g  = 1/b is the spacing of line pairs on the grating. Because c &lt; d, the size of the pattern h is larger than a grid line spacing x g . Thus, any defect having approximately the dimensions of a grid-line spacing does not affect the entire pattern as some of beam 10 is passed and diffracted properly by the grating. If the signal were entirely blocked or passed in the wrong location by a similar defect, as might be the case when the grating is used as a code plate, an error in interpretation of the signal could result. The fact that in the interference method of the invention the pattern size at the grating is larger than such a defect size eliminates this source of error. 
     To obtain information indicating the direction of movement of the beam with respect to the grating, two out-of-phase signals are required. Such signals and their interpretation are discussed for the prior art, wherein the grating is used as a code plate, in the Zaleckas application referenced above. By using aperture plate 27, shown in FIG. 4, instead of aperture plate 11, two differently phased signals can be produced. Aperture plate 27 comprises apertures 22 and 23 as in aperture plate 11, and also apertures 31 and 32. Aperture 31 is covered with phase plate 33, which retards the portion of beam 10 passing through aperture 31. Therefore, a distinct phase relationship exists among the portions of beam 10 passing through the various apertures. Preferably, phase plate 33 retards light passing through it by λ/4. 
     The principal images resolved in image plane 24 from aperture plate 27 are shown in FIG. 5. Again, images 25, 26, and 27 appear, and interference image 27 is detected by photodetector 18. Also images 34, 35, and 36 are present as a result of the portions of beam 10 passing through apertures 31 and 32 in aperture plate 27. Because of the presence of phase plate 33 over aperture 31, light in image 34 will be out of phase with respect to image 35. If phase plate 33 retards light by λ/4, the variation in intensity of interference image 36, which results from interference between +1 and -1 orders, respectively, from apertures 31 and 32, will be 90° out of phase with the variation in intensity of interference image 27. This can be shown mathematically as follows: 
     The total light amplitude in element 36 is the sum of the two interfering orders, viz. 
     
         u = -iAe .sup.i2.sup.πbx + Ae .sup..sup.-i2.sup.πbx. (12) 
    
     The first term in equation (12) includes the factor -i because the +1 order from aperture 31 is delayed λ/4 by phase plate 33. Since u is a complex number in equation (12), the intensity I = | u| 2  is found by multiplying u by its conjugate u*. Thus 
     
         I = u u* = A.sup.2 (-ie.sup.i2.sup.πbx + e.sup.-.sup.i2.sup.πbx)(ie.sup.-.sup.i2.sup.πbx + e.sup.i2.sup.πbx)                                      (13) 
    
     Multiplying and simplifying 
     
         I = A.sup.2[ 2 - i(e.sup.i4.sup.πbx - e.sup.-.sup.i4.sup.πbx)]. (14) 
    
     Using the identity 
     
         Sin φ = (e.sup.i.sup.φ - e.sup.-.sup.i.sup.φ)/2i (15) 
    
     equation (14) can be rewritten 
     
         I = 2A.sup.2 (1 + sin 4 πbx).                           (16) 
    
     Contrasting equation (16) with equation (10), it can be seen that the intensity of interference image 36 varies sinusoidally as the intensity of interference image 27 varies cosinusoidally. 
     FIG. 6 graphically shows the relationship between the signals from photodetectors 18 and 37. Wave 40 is a cosinusoidal signal as generated by photodetector 18, and wave 41 is a sinusoidal signal generated by photodetector 37. 
     A suitable λ/4 phase plate 33 for use with an HeNe laser beam having a wavelength λ of 0.6328μm can comprise an optical flat with a MgF coating 4163 Angstrom units thick in one quadrant of the flat. 
     As will be explained in more detail below, aperture plate 27 can also be used in systems wherein a beam is deflected in two dimensions. 
     A disadvantage of the direction-determining method described above is that it requires near-diffraction limited performance from scan lens 14 for the phase relationship between the photodetector signals to remain constant. This is because the differently phased portions of beam 10 pass through different portions of scan lens 14, and an aberration in the lens may not affect all portions of the beam equally. 
     FIG. 7 shows another embodiment of the invention that combines the interference method of the invention with a polarization-splitting technique that tends to overcome the disadvantage noted above, and ease the requirements on scan lens 14. In FIG. 7, the apparatus of FIG. 1 is shown modified by the addition of Wollaston prism 50, beam splitter 42, and polarizers 43 and 46. 
     Wollaston prism 50 divides beam 10, as patterned by aperture plate 11, into two orthogonally polarized, diverging portions 10A and 10B. By proper positioning of the Wollaston prism, the Fraunhofer diffraction patterns from portions 10A and 10B can be made to impinge on grating 16 so that they are separated by one-eighth cycle of the grating periodicity. Thus, the modulation of beam 10A caused by traversing beams 10A and 10B across grating 16 will be 90° out of phase with the modulation caused in beam 10B. Other apparatus, such as a beam splitter and polarizers, can be used instead of Wollaston prism 50 to divide beam 10 into orthogonally polarized, diverging portions. 
     FIG. 8 shows a portion of grating 16 and the vertically polarized beam 10A and horizontally polarized beam 10B. At the plane of grating 16, the areas of impingement of both beams 10A and 10B are Fraunhofer diffraction patterns because of the transforming action of scanning lens 14. However, it can be seen that the area of impingement of beam 10B is displaced to the right by an eighth cycle of the grating periodicity (one-quarter of a grating-line width) with respect to the area of impingement of beam 10A. 
     Diffraction orders from both beams 10A and 10B are split by beam splitter 42 so that the split beam portions fall on both vertical polarizer 43 and horizontal polarizer 46. Thus, the vertically polarized beam 10A passes through vertical polarizer 43 and is resolved in image plane 44; and the horizontally polarized beam 10B passes through horizontal polarizer 46 and is resolved in image plane 47. 
     FIG. 9 illustrates the principal images resolved in image plane 44 and FIG. 10 illustrates the principal images resolved in image plane 47. Both sets of images are similar to those shown previously in FIG. 3, that is, a zero-order image for each aperture in aperture plate 11 and interference images from the two apertures Photodetector 45 and photodetector 48 are shown in FIGS. 9 and 10 respectively positioned to sense the interference images. 
     Because of the offset of beams 10A and 10B with respect to the grating, as shown in FIG. 8, the intensity variations sensed by photodetectors 45 and 48 as beams 10A and 10B move across the grating are out of phase, thus providing the necessary directional information. 
     FIG. 11 shows apparatus for an embodiment of the invention wherein another method is used to form the polarized light signals. In the apparatus of FIG. 11, polarizer 51 and aperture plate 52 replace aperture plate 11 and Wollaston prism 50 shown in FIG. 7. Aperture plate 52, as shown in more detail in FIG. 12, includes apertures 53 and 54, and quarter-wave plate 55 covering aperture 54. Quarter-wave plate 55 is characterized by having different propagation speeds for different light polarizations, more specifically, a first light component aligned with a first axis of the quarter-wave plate will be retarded λ/4 with respect to a second light component aligned with a second axis of the quarter-wave plate. It will be clear that means for retarding the second component by other than λ/4 are possible, and are also contemplated by the invention. 
     Polarizer 51 linearly polarizes light in beam 10, and quarter-wave plate 55 is oriented with respect to the polarization of beam 10 to retard one component of this polarization in the portion of beam 10 passing through aperture 54. A set of components showing one particular orientation of quarter-wave plate 55 with respect to the polarization of beam 10 is shown in FIG. 12. Beam 10 is linearly polarized as represented by vector R, which can be resolved into vertical and horizontal components A and B. This polarization passes through aperture 53 unhindered. Quarter-wave plate 55 passes component A&#39; substantially without change, but delays component B&#39; by a quarter wavelength. This is indicated by the imaginary number symbol (-i). If components A&#39; and B&#39; are equal and orthogonal, light passing through quarter-wave plate 55 can be said to be circularly polarized. 
     If the source of beam 10 produces linearly polarized light, polarizer 51 is not needed. 
     Scanning lens 14 transforms patterned beam 10 into a Fraunhofer diffraction pattern in the plane of grating 16 that covers a number of grating lines and spaces, as shown in FIG. 13. All the information for the pattern passes through the same portion of lens 14, so aberrations in the lens are not as likely to disturb the phase relationships among different portions of the beam as in the embodiments shown in FIGS. 1 and 7. Thus, the embodiment of FIG. 11 is preferred in those applications where such phase relationships are critical. 
     FIG. 14 shows the principal images resolved in image plane 44 and FIG. 15 shows the principal images resolved in image plane 47. Detector 45 in image plane 44 detects the interference image caused by the coincidence of the +1 order from aperture 54 and the -1 order from aperture 53, both of which are vertically polarized and in phase; and detector 48 in image plane 47 detects the interference image caused by the coincidence of the +1 order from aperture 54 and the -1 order from aperture 53, which are both horizontally polarized, but since λ/4 plate 55 retards the horizontally polarized component of light passing through it, which are out of phase by λ/4. Thus, the light intensity signal detected by photodetector 45 can be expressed by equation (10), and the light intensity detected by photodetector 48 can be expressed by equation (16). 
     FIG. 16 shows another embodiment of the invention in which the apparatus of FIG. 11 has been expanded for use with a beam that can be deflected in two dimensions. For this purpose, an additional rotatable mirror 59 has been added, which is rotated around its axis 60 by a suitable actuator (not shown) in response to a deflection signal in a similar manner to the rotation of mirror 12. Also added is beam splitter 61, which splits the beam from scanning lens 14 into two portions, 10X and 10Y; portion 10X falling on grating 16X and thereafter being processed by collecting lens 17X, beam splitter 42X, polarizers 43X and 46X, and photodetectors 45X and 48X; and portion 10Y falling on grating 16Y and thereafter being processed by collecting lens 17Y, beam splitter 42Y, polarizers 43Y and 46Y, and photodetectors 45Y and 48Y. 
     Aperture plate 62, shown in FIG. 17, is used in the apparatus of FIG. 16. Aperture plate 62 includes apertures 53 and 54 and λ/4 plate 55, and additional aperture 63 spaced the same distance from aperture 54 as aperture 53; apertures 54 and 63 being on an axis orthogonal to the axis of apertures 53 and 54. 
     FIGS. 18 and 19 show the principal images in image planes 44X and 47X pertaining to horizontal deflection of beam 10; FIGS. 20 and 21 show the principal images in image planes 44Y and 47Y pertaining to vertical deflection of beam 10. 
     In FIGS. 18 and 19, the top rows of images in image planes 44X and 47X are analogous to the rows of images in image planes 44 and 47 in FIGS. 14 and 15, respectively; photodetectors 45X and 48X are positioned to detect the differently polarized versions of the interfering orders from apertures 53 and 54. The bottom row of images in each of image planes 44Y and 47Y stems from aperture 63, and are not used in image planes 44X and 47X. 
     In FIGS. 20 and 21, the left-hand columns of images in each of image planes 44Y and 47Y are analogous to the rows of images in image planes 44 and 47 in FIGS. 14 and 15, respectively; photodetectors 45Y and 48Y are positioned to detect the differently polarized versions of the interference images from apertures 54 and 63. If λ/4 plate 55 is oriented with respect to apertures 53 and 54 as in FIG. 12, so that the horizontal polarization is delayed by λ/4 and the vertical polarization is passed substantially without delay, the interfering orders detected by photodetector 45Y will be in phase, and the interfering orders detected by photodetector 48Y will be out of phase by λ/4, thus providing directional information for the vertical deflection of beam 10. The right-hand column of images in each of image planes 44Y and 47Y stems from aperture 53, and is not used. 
     FIG. 22 shows another embodiment of the invention for use with a beam that can be deflected in two dimensions, which includes a crossed-grid grating 66 in place of gratings 16X and 16Y shown in FIG. 16, and does not require the duplication of elements shown in the apparatus of FIG. 16, e.g., collecting lenses 17X and 17Y are replaced by collecting lens 17. A portion of crossed-grid grating 66 is shown in more detail in FIG. 23. The lines in crossed-grid grating 66 are shown to intersect at right angles; however, it will be clear that other intersection angles are possible, and are also contemplated by the invention. For best results, dimensions w, x, y, and z should be equal. 
     Crossed-grid grating 66 creates a more complex pattern of images in each of image planes 44 and 47 than did the simple gratings described above. The principal images in this pattern are shown for each image plane, respectively, in FIGS. 24 and 25. Five images for each aperture are shown, namely: a zero-order, +1 and -1 orders in the horizontal axis, and +1 and -1 orders in the vertical axis. For example, in FIG. 24, the five images from aperture 54 are zero-order 70; in the horizontal axis, -1 order 71 and +1 order 72 (interfering with the horizontal axis, -1 order from aperture 53); and in the vertical axis, -1 order 73 and +1 order 74 (interfering with the vertical axis, -1 order from aperture 63). Two photodetectors are placed in each image plane to sense the interference images: photodetectors 45X and 45Y in image plane 44, and photodetectors 48Y and 48X in image plane 47. The polarizations of the images are shown by arrows and horizontally polarized images stemming from aperture 54, and thus delayed by λ/4 plate 55, are identified by (-i). Again, other images that are not relevant to the invention, such as images from higher diffraction orders, and images resulting from interactions between diffraction orders from different axes, are not shown. 
     FIG. 26 shows the embodiment of the invention first shown in FIG. 1 expanded with two gratings for a beam movable in two dimensions. Note that aperture plate 27, shown in FIG. 4, is used. The image pattern in image plane 24 for horizontal deflection is identical to that shown in FIG. 5. The image pattern in image plane 69 for vertical deflection is shown in FIG. 27, which also shows the positioning of photodetectors 70 and 71. Note that orders out of phase by λ/4 interfere in the interference image sensed by photodetector 70 and in-phase orders interfere in the interference image sensed by photodetector 71 so that the light intensity signal from photodetector 71 can be expressed by equation (10) and the light intensity signal from photodetector 70 can be expressed by equation (16). 
     FIG. 28 shows the embodiment of the invention from FIG. 1 expanded with a crossed-grid grating for a beam movable in two dimensions. Again, aperture plate 27 is used. Note that all photodetectors shown in FIG. 26 are here grouped in single image plane 73. The image pattern in image plane 73, shown in FIG. 29 along with the placement of photodetectors 18, 37, 70, and 71 is believed to be self-explanatory in view of the previous descriptions. 
     FIG. 30 shows the embodiment of the invention first shown in FIG. 7 expanded with two gratings for a beam movable in two dimensions. Aperture plate 74, shown in FIG. 31, is used in this embodiment. Aperture plate 74 includes three apertures 53, 54, and 63. In this embodiment, Wollaston prism 50 must be carefully adjusted so that the two beams 10A and 10B are separated on both gratings 16X and 16Y by the proper amount to cause the pair of signals from photodetectors 45X and 48X and the pair of signals from photodetectors 45Y and 48Y each to be out of phase by 90° as beam 10 is deflected across the gratings. 
     FIG. 32 shows the pattern and the positioning of photodetector 45X in image plane 44X for the embodiment of FIG. 30. FIG. 33 shows the pattern and the positioning of photodetector 45Y in image plane 44X. All elements in image planes 44X and 44Y are vertically polarized. The images and photodetector positions in image planes 47X and 47Y are analogous to those in image planes 44X and 44Y, respectively, so these are not shown. The intensity variations of the images in planes 47X and 47Y will be out of phase with those in planes 44X and 44Y as beams 10A and 10B are moved across gratings 16X and 16Y because of the difference in positions of beams 10A and 10B in the planes of the gratings. 
     FIG. 34 shows the embodiment of the invention from FIG. 7 expanded with a crossed-grid grating 66 for a beam movable in two dimensions. Again, aperture plate 74 is used in the apparatus of FIG. 34. Wollaston prism 50 must be adjusted to position the Fraunhofer diffraction patterns of beams 10A and 10B in the plane of crossed-grid grating 66 so that the proper phase relationships result between the signals from photodetectors 45X and 48X and the signals from photodetectors 45Y and 48Y. The positioning of these patterns is shown in FIG. 35. Note in this figure that the diffraction pattern for beam 10B is offset the same distance downward as it is to the right with respect to the diffraction pattern for beam 10A. FIG. 36 shows the vertically polarized pattern and the positioning of the photodetectors in image plane 44. The horizontally polarized pattern and photodetector positions (not shown) in image plane 47 are identical to those in image plane 44. 
     FIG. 37 shows an embodiment of the invention for use in apparatus having a stationary light beam and a moving grating. Coherent light beam 10 passes through aperture plate 11, lens 80, grating 16, and collecting lens 81, and impinges on photodetector 18. Grating 16 is movable, for example, grating 16 could be attached to a movable portion of a numerically controlled machine tool. Lenses 80 and 81 perform similar functions to those of lenses 14 and 17, respectively, in the embodiments described above in that lens 80 transforms beam 10, as patterned by aperture plate 11, into a Fraunhofer diffraction pattern in the plane of grating 16, and lens 81 collects light passing through grating 16 to form an image in image plane 24. However, since beam 10 is not deflected with respect to the lenses, these need not be as large as in the previously described embodiments, hence, the requirements on their design are less severe. 
     Using aperture plate 11, as shown in FIG. 2, in the apparatus of FIG. 37 the image pattern of FIG. 3 will appear in image plane 24, and a signal having twice the periodicity of grating 16 will be generated by photodetector 18 when grating 16 is moved. It will be clear to those skilled in the art how the other embodiments of the invention described above could be adapted for use with a fixed light beam and a movable grating. 
     Clearly, other embodiments of the invention are possible for monitoring other methods of producing relative movement between a coherent light beam and a grating. 
     One skilled in the art may make changes and modifications to the embodiments of the invention disclosed herein, and may devise other embodiments, without departing from the spirit and scope of the invention.