Abstract:
A method for measuring two-dimensional displacement using conjugate optics comprises the steps of emitting an incident beam onto a diffraction element to generate many firstly diffracted beams, selecting two axially symmetric beams of the same order of diffraction from the firstly diffracted beams, introducing corresponding sets of wavefront reconstruction optics to reflect the two selected beams back onto the same incident spot along the same optical paths and to generate many secondly diffracted beams, selecting two axially symmetric pairs of beams of the same order of diffraction from the secondly diffracted beams, forming two interference fringes by superposing the two selected pairs of beams via corresponding mirrors and interferometric optics, and obtaining two linearly independent displacements of the diffraction element relative to the rest of the optics by decoding the two interference fringes.

Description:
BACKGROUND OF THE INVENTION  
         [0001]    1. Field of the Invention  
           [0002]    The invention relates to a method for measuring two-dimensional displacement using conjugate optics, wherein the term, conjugate optics, implies the adoption of the wavefront reconstruction optics and the optics of the telecentric system.  
           [0003]    2. Description of the Related Art  
           [0004]    A pair of one-dimensional scales in cross-type configuration is ordinarily used to measure two-dimensional displacement. The setup of such a system involves a sequence of alignment steps, one of the most important of which is the alignment for orthogonality, which becomes more difficult as the precision requirement of the system increases. Good technical skills are needed to operate such a system, resulting in a high training cost and significant operator dependence. Recently, planar scales have been designed and developed for the same measuring purpose. This novel instrumentation concept avoids the difficulties of its predecessors.  
           [0005]    U.S. Pat. No. 5,204,524 discloses a design of planar scales. The design is rooted in the theory of geometrical optics and intrinsically constrains the precision and resolution of the system.  
           [0006]    U.S. Pat. No. 5,424,833 discloses another design of planar scales. In this design, the light beam is diffracted three times by diffraction gratings. The light intensity is thus greatly reduced, and so is the signal to noise ratio.  
           [0007]    Consequently, in order to build such a system, it is necessary to use better optical components and to ask for stricter assembly tolerances.  
           [0008]    U.S. Pat. No. 5,530,543 discloses another design of planar scales that utilizes a single diffraction grating. However, this design compensates for no misalignment, and the output signal is therefore fragile under motion.  
         SUMMARY OF THE INVENTION  
         [0009]    The submitted invention proposes a new design of planar scales to resolve the problems mentioned above.  
           [0010]    The design uses conjugate optics and is comprised of the following steps. A light source emits a light beam vertically onto a diffraction element; many firstly diffracted beams a regenerated, and at least two of them are reflected along the same optical paths back onto the said diffraction element via corresponding sets of wavefront reconstruction optics; many secondly diffracted beams are generated, of which at least two pairs are respectively superposed by corresponding sets of mirrors and interferometric optics to form interference fringes; two linearly independent displacements of the diffraction element are then obtained by decoding the interference fringes.  
           [0011]    The invention has the advantage of providing high tolerances to both the alignment of the optics and the manufacture of the diffraction element. Both the manufacturing and the assembling costs are reduced, and yield is increased. 
       
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0012]    The aforementioned and other objectives and features of the invention are clarified in the following description, with reference to the following drawings and the preferred embodiments.  
         [0013]    [0013]FIG. 1 schematically shows the first part of the optical design of the first embodiment of the invention.  
         [0014]    [0014]FIG. 2 schematically shows the rest of the optical design of the first embodiment of the invention.  
         [0015]    [0015]FIG. 3 schematically shows the first part of the optical design of the second embodiment of the invention.  
         [0016]    [0016]FIG. 4 schematically shows the rest of the optical design of the second embodiment of the invention.  
         [0017]    [0017]FIG. 5 schematically shows the diffraction element formed on a planar substrate.  
         [0018]    [0018]FIG. 6 schematically shows the diffraction element formed on a cylindrical substrate.  
         [0019]    [0019]FIG. 7 schematically shows the diffraction element formed on a spherical substrate.  
         [0020]    [0020]FIG. 8 schematically shows the diffraction element formed on a reflective substrate.  
         [0021]    [0021]FIG. 9 schematically shows the diffraction element formed on a transmitting substrate.  
         [0022]    FIGS.  10  to  12  schematically show many kinds of wavefront reconstruction optics. 
     
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS  
       [0023]    First Embodiment  
         [0024]    [0024]FIG. 1 schematically shows the first part of the optical design of the first embodiment of the invention. A beam,  901 , is either first emitted from a light source,  201 , before passing through a collimating lens,  301 , or directly emitted from a collimating light source,  201 . The collimated beam,  901 , is then incident onto a spot,  1011 , on a diffraction element,  101 , in an almost vertical direction. The collimated beam,  901 , is hereafter referred to as the first incident beam.  
         [0025]    As shown in FIG. 1, in the first embodiment of the invention, the diffraction element,  101 , is formed on a reflective planar substrate,  10 , and is a two-dimensional step-type diffraction grating with grating pitches of 4 μm in both directions and a depth of 0.2 μm. After being incident on the diffraction element,  101 , the first incident beam,  901 , is transferred into many firstly diffracted beams. The transfer function, which represents the functionality of the diffraction element,  101 , is given by,  
         [0026]    f(x,y)=[exp(i2πkx)+exp(−i2πkx)]×[exp(i 2πky )+exp(− i 2π ky )] 
         [0027]    in the X-Y plane, which is tangential to the incident spot,  1011 ; k is the propagation constant.  
         [0028]    [0028]FIG. 1 presents four of the firstly diffracted beams, namely  911 ,  912 ,  913 , and  914 . This set of four beams is axially symmetric and all the beams are of the same order of diffraction. Their projections on the X-Y plane are in the directions, (1,1), (−1,1), (−1,−1), and (1,−1), respectively. Now, as the diffraction element,  101 , moves relative to the rest of the optics along the X-Y plane, the four beams,  911 ,  912 ,  913 , and  914 , carry various phase shifts, namely +θx+θy, −θx+θy, −θx−θy, and +θx−θy, respectively, because of the Doppler Effect.  
         [0029]    Next, referring to FIG. 1, an axially symmetric pair of beams, such as  911  and  913 , is selected from the four beams. Furthermore, two sets of wavefront reconstruction optics,  41  and  43 , are introduced in a way that both beams are reflected back onto the incident spot,  1011 , along the same optical paths as their respective ones before reflection. In this embodiment, each set consists of a collimating lens and a mirror; that is,  41  consists of  311  and  401 , and  43  consists of  313  and  403 . In a separate embodiment, the other axially symmetric pair of beams,  912  and  914 , can be used instead of  911  and  913 .  
         [0030]    [0030]FIG. 1 reveals that the optical axes of the two collimating lenses,  311  and  313 , should be placed co-linearly with the optical paths of the two beams,  911  and  913 , respectively; the two mirrors,  401  and  403 , should be placed perpendicular to the optical axes of the two collimating lenses,  311  and  313 , respectively; the incident spot,  1011 , and the reflecting spot,  4011 , on the mirror,  401 , should respectively be located at the front and the back focal points of the collimating lens,  311 ; the incident spot,  1011 , and the reflecting spot,  4031 , on the mirror,  403 , should respectively be located at the front and the back focal points of the collimating lens,  313 . The two sets of wavefront reconstruction optics thus constitute the conjugate optics.  
         [0031]    [0031]FIG. 2 schematically shows the rest of the optical design of the first embodiment of the invention. After returning back onto the incident spot,  1011 , on the diffraction element,  101 , the two reflected beams are similarly transferred into many secondly diffracted beams. FIG. 2 presents four of the secondly diffracted beams, namely  921 ,  922 ,  923 , and  924 . This set of four beams is axially symmetric and all the beams are of the same order of diffraction. Their projections on the X-Y plane are in the directions, (1,0), (0,1), (−1,0), and (0,−1), respectively. Now, as the diffraction element,  101 , moves relative to the rest of the optics along the X-Y plane, the four beams,  921 ,  922 ,  923 , and  924 , further carry various phase shifts, namely −2θy,−2θx, +2θy, and +2θx, because of the Doppler Effect.  
         [0032]    [0032]FIG. 2 shows that the four selected secondly diffracted beams are grouped into two pairs. The first pair consists of  921  and  923 , and in order to form an interference fringe, the pair is superposed into the beam,  941 , by a pair of mirrors,  411  and  413 , and a piece of interferometric optics, such as a polarizing beam splitter,  501 . The interference fringe of the beam,  941 , can be used to determine the Y direction displacement because the two beams,  921  and  923 , carry phase shifts of −2θy and +2θy, respectively. Similarly, the second pair consists of  922  and  924 , and in order to form the other interference fringe, the pair is superposed into the other beam,  942 , by the other pair of mirrors,  412  and  414 , and the other piece of interferometric optics, such as a polarizing beam splitter,  502 . The interference fringe of the beam,  942 , can be used to determine the X direction displacement because the two beams,  922  and  924 , carry phase shifts of −2θx and +2θx, respectively.  
         [0033]    Second Embodiment  
         [0034]    [0034]FIG. 3 schematically shows the first part of the optical design of the second embodiment of the invention. A beam,  901 , is either first emitted from a light source,  201 , before passing through a collimating lens,  301 , or directly emitted from a collimating light source,  201 . The collimated beam,  901 , is then incident onto a spot,  1011 , on a diffraction element,  101 , in an almost vertical direction. The collimated beam,  901 , is hereafter referred to as the first incident beam.  
         [0035]    As shown in FIG. 3, in the second embodiment of the invention, the diffraction element,  101 , is formed on a reflective planar substrate,  10 , and is a two-dimensional step-type diffraction grating with grating pitches of 4 μm in both directions and a depth of 0.2 μm. After being incident on the diffraction element,  101 , the first incident beam,  901 , is transferred into many firstly diffracted beams. The transfer function, which represents the functionality of the diffraction element,  101 , is given by,  
         [0036]    f(x,y)=[exp(i2πkx)+exp(−i2πkx)]×[exp(i2πky)+exp(−i2πky)] 
         [0037]    in the X-Y plane, which is tangential to the incident spot,  1011 ; k is the propagation constant.  
         [0038]    [0038]FIG. 3 presents four of the firstly diffracted beams, namely  911 ,  912 ,  913 , and  914 . This set of four beams is axially symmetric and all the beams are of the same order of diffraction. Their projections on the X-Y plane are in the directions, (1,1), (−1,1), (−1,−1), and (1,−1), respectively. Now, as the diffraction element,  101 , moves relative to the rest of the optics along the X-Y plane, the four beams,  911 ,  912 ,  913 , and  914 , carry various phase shifts, namely +θx+θy, −θx+θy, −θx−θy, and +θx−θy, respectively, because of the Doppler Effect.  
         [0039]    Next, referring to FIG. 3, four sets of wavefront reconstruction optics,  41 ,  42 ,  43 , and  44 , are introduced in a way that the four beams are reflected back onto the incident spot,  1011 , along the same optical paths as their respective ones before reflection. In this embodiment, each set consists of a collimating lens and a mirror; that is,  41  consists of  311  and  401 ,  42  consists of  312  and  402 ,  43  consists of  313  and  403 , and  44  consists of  314  and  404 .  
         [0040]    [0040]FIG. 3 reveals that the optical axes of the four collimating lenses,  311 ,  312 ,  313 , and  314 , should be placed co-linearly with the optical paths of the four beams,  911 ,  912 ,  913 , and  914 , respectively; the four mirrors,  401 ,  402 ,  403 , and  404 , should be placed perpendicular to the optical axes of the four collimating lenses,  311 ,  312 ,  313 , and  314 , respectively; the incident spot,  1011 , and the reflecting spot,  4011 , on the mirror,  401 , should respectively be located at the front and the back focal points of the collimating lens,  311 ; the incident spot,  1011 , and the reflecting spot,  4021 , on the mirror,  402 , should respectively be located at the front and the back focal points of the collimating lens,  312 ; and so on. The four sets of wavefront reconstruction optics thus constitute the conjugate optics.  
         [0041]    [0041]FIG. 4 schematically shows the rest of the optical design of the second embodiment of the invention. After returning back onto the incident spot,  1011 , on the diffraction element,  101 , the four reflected beams are similarly transferred into many secondly diffracted beams. FIG. 4 presents four of the secondly diffracted beams, namely  921 ,  922 ,  923 , and  924 . This set of four beams is axially symmetric and all the beams are of the same order of diffraction. Their projections on the X-Y plane are in the directions, (1,0), (0,1), (−1,0), and (0,−1), respectively. Furthermore, each of the four beams is a superposition of two beams. For example,  921  is a superposition of two secondly diffracted beams transferred from  912  and  913 ;  922  is a superposition of two secondly diffracted beams transferred from  913  and  914 ; and so on. Now, as the diffraction element,  101 , moves relative to the rest of the optics along the X-Y plane, the four beams,  921 ,  922 ,  923 , and  924 , thereby carry interference fringes, because of the Doppler Effect, and can be used to determine either the Y direction displacement or the X direction displacement. For example,  921  and  923  can be used to determine the Y direction displacement, and similarly,  922  and  924  can be used to determine the X direction displacement.  
         [0042]    [0042]FIG. 5 schematically shows the diffraction element formed on a planar substrate. FIG. 6 schematically shows the diffraction element formed on a cylindrical substrate. FIG. 7 schematically shows the diffraction element formed on a spherical substrate. Referring to FIGS.  5  to  7 , the diffraction element,  101 , can be formed on a planar substrate,  10 , a cylindrical substrate,  20 , or a spherical substrate,  30 .  
         [0043]    [0043]FIG. 8 schematically shows the implementation of the present invention using a reflective substrate. As shown in the figure, the diffraction element,  101 , is formed on a reflective substrate,  10 , and therefore the diffraction element,  101 , is of the reflective type. FIG. 9 schematically shows the implementation of the present invention using a transmitting substrate. As shown in the figure, the diffraction element is formed on a transmitting substrate,  10 ′, and therefore the diffraction element,  101 , is of the transmitting type.  
         [0044]    FIGS.  10  to  12  schematically show various kinds of wavefront reconstruction optics, such as a module  41 , a GRIN lens  51 , or a corner cube  61 . The module  41  is composed of a collimating lens  311  and a mirror  401 ; the GRIN lens  51  has a reflecting film  51   b  on its end surface  51   a.    
         [0045]    The wavefront reconstruction optics nicely directs the light beam to be diffracted twice at a single optical spot. The term, conjugate optics, is adopted because the back and forth optical aberrations are automatically compensated for each other according to the Fourier transform pair. Consequently, the tolerances of the measuring system are improved.  
         [0046]    The light source,  201 , emits a light beam that is either coherent or partially coherent, and whose polarization can be linear, circular, or elliptical.  
         [0047]    Although the preferred embodiments of the present invention have been specified, it is to be understood that any modification apparent to those skilled in the art does not depart from the spirit of the present invention. The scope of the invention is therefore to be determined solely by the following claims.