PATENT ABSTRACT
Disclosed is a surface sensing apparatus, one embodiment having a source of coherent radiation capable of out-putting wavelength emissions to create a first illumination state to illuminate a surface and create a first speckle pattern, an emission deviation facility capable of influencing the emission to illuminate the surface and create a second illumination state and a second speckle pattern, and a sensor capable of sensing a representation of the first and a second speckle intensity from the first and second speckle pattern. Also disclosed are methods of sensing properties of the surface, one embodiment comprising the steps of illuminating the surface having a first surface state with the source of coherent radiation emission, sensing a first speckle intensity from the surface, influencing a relationship of the surface to the emission to create a second surface state and sensing a second speckle intensity from the surface at the second surface state.

PATENT DESCRIPTION
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application claims the benefit of U.S. Patent Application No. 61/115,923, filed on Nov. 18, 2008, entitled “DIMENSIONAL PROBE,” and U.S. Patent Application No. 61/070,352, filed on Mar. 22, 2008, entitled “DIMENSIONAL PROBE,” the entire contents of which are incorporated herein by reference. 
     
    
     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
       [0002]    Not Applicable 
       REFERENCE TO SEQUENCE LISTING, A TABLE, OR A COMPUTER PROGRAM LISTING COMPACT DISC APPENDIX 
       [0003]    Not Applicable 
       BACKGROUND 
       [0004]    This invention relates to the sensing of properties of a surface, more particularly, embodiments of the present invention relate to the precision measurement of the location, orientation, and motion of surfaces and to the formation of three-dimensional images and dimensional data of complex objects. 
       BRIEF SUMMARY OF THE INVENTION 
       [0005]    It is an object of one embodiment of the present invention to provide a surface sensing apparatus having at least one source of coherent radiation capable of outputting at least one wavelength emission to create a first illumination state and to illuminate a surface to create a first speckle pattern, an emission deviation facility capable of influencing the emission to illuminate the surface to create a second illumination state and a second speckle pattern, and at least one sensor capable of sensing a representation of the first and a second speckle intensity from the first and second speckle pattern. 
         [0006]    It is another object of one embodiment of the present invention to provide the surface sensing apparatus further comprising a waist creating assembly capable of influencing the emission to create at least one illumination waist location and the emission deviation facility is further capable of influencing the at least one illumination waist location to create the first and second speckle pattern. 
         [0007]    It is a further object of one embodiment of the present invention to provide the surface sensing apparatus wherein the emission deviation facility further comprises the at least one source of coherent radiation being capable of emitting a first and a second wavelength emission to create a first and second illumination state and create the first and second speckle pattern. 
         [0008]    It is yet another object of one embodiment of the present invention to provide the surface sensing apparatus wherein the emission deviation facility further comprises a waist offset creating assembly capable of creating a waist offset and changing the waist offset to create the first and second speckle pattern. 
         [0009]    It is another object of one embodiment of the present invention to provide the surface sensing apparatus wherein the emission deviation facility further comprises a waist offset creating assembly capable of creating a waist offset and changing the waist offset to create the first and second speckle pattern and a deflector assembly capable of altering a direction of the illumination state. In some embodiments of the present invention, the emission deviation facility further comprises a waist range offset assembly capable of creating a waist range offset and changing the waist range offset to create the first and second speckle pattern. 
         [0010]    It is yet another object of one embodiment of the present invention to provide the surface sensing apparatus further comprising a processor in communication with a memory and the at least one sensor, the processor capable of receiving a representation of the first and second speckle pattern, and the memory containing machine instructions capable of determining a speckle shift from the speckle patterns whereby the processor can calculate at least one system parameter. 
         [0011]    It is an object of one embodiment of the present invention to provide a surface sensing apparatus, having at least one source of coherent radiation capable of outputting at least one wavelength emission to illuminate a surface, a relative position of the surface and the emission being variable to create a first surface state and second surface state, each surface state having a speckle pattern, and at least one sensor capable of sensing a representation of a first and a second speckle intensity from the first and second speckle pattern. 
         [0012]    It is another object of embodiments of the present invention to provide the surface sensing apparatus further comprising elements such as, but not limited to, a surface deviation facility capable of varying the relative position of the surface and the emission, a range change assembly capable of varying the relative position of the surface and the emission by a range, or a lateral translation assembly capable of varying the relative position of the surface and the emission by a lateral translation. 
         [0013]    It is an object of one embodiment of the present invention to provide a method of sensing properties of a surface, and the method comprising the steps of illuminating a surface with at least one source of coherent radiation emission with a first illumination state, sensing a first speckle intensity from the surface from the first illumination state, influencing the emission to create a second illumination state, sensing a second speckle intensity from the surface from the second illumination state, and determining a speckle shift between the first and second speckle intensity. 
         [0014]    It is another object of one embodiment of the present invention to provide the method of sensing properties of a surface further comprising the step of calculating at least one surface parameter using the speckle shift. 
         [0015]    It is yet another object of one embodiment of the present invention to provide the method of sensing properties of a surface wherein the first and second illumination state have a first and second waist, the difference between the first and second waist comprising a waist offset, and the step of calculating the at least one surface parameter comprises calculating a surface height using the speckle shift and the waist offset. 
         [0016]    It is yet another object of one embodiment of the present invention to provide the method of sensing properties of a surface wherein each illumination state having a direction and the difference between the first and second illumination direction is an illumination direction offset, and the step of calculating the surface height further comprises using the speckle shift, the waist offset and the illumination direction offset. 
         [0017]    It is another object of one embodiment of the present invention to provide the method of sensing properties of a surface wherein the step of illuminating the surface with at least one source of coherent radiation comprises at least two emissions, each emission having a unique wavelength, the difference between each wavelength is a wavelength change, and the step of calculating the system parameters comprises calculating a surface orientation of the surface using the speckle shift and the wavelength change. 
         [0018]    It is an object of one embodiment of the present invention to provide a method of sensing properties of a surface, the method comprising the steps of illuminating a surface with at least one source of coherent radiation emission, the surface having a first surface state, sensing a first speckle intensity from the surface at the first surface state, influencing a relationship of the surface to the emission to create a second surface state, sensing a second speckle intensity from the surface at the second surface state, and determining a speckle shift between the first and second surface state. 
         [0019]    It is another object of one embodiment of the present invention to provide the method of sensing properties of a surface further comprising calculating at least one surface parameter using the speckle shift. 
         [0020]    It is a further object of one embodiment of the present invention to provide the method of sensing properties of a surface wherein the step of calculating surface parameters further comprises utilizing a processor in communication with a memory and the at least one sensor, the processor receiving a representation of the first and second speckle pattern, and the processor determining a speckle shift from the speckle patterns utilizing the memory containing machine instructions capable of determining a speckle shift from the speckle patterns and at least one system parameter. 
         [0021]    It is yet another object of one embodiment of the present invention to provide the method of sensing properties of a surface further comprising the step of illuminating the surface with at least one source of coherent radiation emission being performed at a first surface range, the step of influencing the relationship of the surface to the emission to create a second surface state comprises moving the relationship of the surface and the emission to create a second surface range, and calculating a first surface height and a surface orientation using the speckle shift. 
         [0022]    It is still another object of one embodiment of the present invention to provide the method of sensing properties of a surface wherein the step of illuminating the surface with at least one source of coherent radiation emission is performed at a first surface orientation, the step of influencing the relationship of the surface to the emission to create a second surface state comprises moving the relationship of the surface and the emission to create a second surface orientation, and further comprising calculating a change between the first and second surface orientation using the speckle shift. 
         [0023]    It is yet another object of one embodiment of the present invention to provide the method of sensing properties of a surface further comprising the step of illuminating the surface with at least one source of coherent radiation emission is performed at a first lateral location, the step of influencing the relationship of the surface to the emission to create a second surface state comprises moving a relationship of the surface and the emission to create a second lateral location, and calculating a change between the first and second lateral location using the speckle shift. 
         [0024]    Embodiments of the present invention can be useful as a stand-alone probe, as an element of a measurement system, or as a component of high-speed scanning system. As an example and not for limitation, embodiments of the present invention may find application in areas including industrial inspection, dimensional metrology, assembly, alignment, machine vision, robot vision, and three-dimensional imaging. 
     
    
     
       BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS 
         [0025]    In order that the manner in which the above-recited and other advantages and features of the invention are obtained, a more particular description of the invention briefly described above will be rendered by reference to specific embodiments thereof which are illustrated in the appended drawings. Understanding that these drawings depict only typical embodiments of the invention and are not therefore to be considered to be limiting of its scope, the invention will be described and explained with additional specificity and detail through the use of the accompanying drawings in which: 
           [0026]      FIG. 1 . Is a diagram illustrating a system state and sensor modalities for one embodiment of the invention. 
           [0027]      FIG. 2 . Is a diagram illustrating two system states for one embodiment of the invention. 
           [0028]      FIG. 3 . Is a diagram illustrating one embodiment of an overlap between two illumination states. 
           [0029]      FIG. 4 . Is a block diagram showing one embodiment of the steps for varying system states and calculating system parameters. 
           [0030]      FIG. 5 . Is a block diagram illustrating example measurement modalities for one embodiment of the invention. 
           [0031]      FIG. 6 . Is an illustration of multiple embodiments of an emission deviation facility. 
           [0032]      FIG. 7 . Is an illustration of additional embodiments of an emission deviation facility. 
           [0033]      FIG. 8 . Is an illustration of multiple embodiments of receiver modalities. 
           [0034]      FIG. 9 . Is an illustration of multiple detector embodiments. 
           [0035]      FIG. 10 . Is an illustration of one embodiment of a system for measuring surface profiles. 
           [0036]      FIG. 11 . Is an illustration of one embodiment of a measurement system. 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0037]    A surface sensing system will now be described in detail with reference to the accompanying drawings. It will be appreciated that, while the following description focuses on an optical surface sensing system that probes surface position and orientation at a single surface element, the systems and methods disclosed herein have wide applicability and can be used for the determination of three-dimensional surface shape and object orientation of complex objects. For example, the surface sensing system described herein may be readily employed as a scanning system to rapidly determine dimensions, geometry, location, and orientation of a wide range of objects having surfaces that scatter radiation. The methods are not limited to the optical wavelength range and apply generally to electromagnetic and acoustic waves. The invention provides the capability of performing high-precision measurements at low cost without the shadowing and surface-access issues inherent in triangulation-based approaches. For example, the narrow illumination beam utilized in this invention allows for the measurement of high-aspect-ratio features of an object such as deep holes. In addition, the invention provides for enhanced range resolution and precision compared with time-of-flight approaches that measure the transit time of radiation scattered from the surface of an object. The invention applies to a wide range of measurement scales ranging from microscopic applications to laser-radar applications. Notwithstanding the specific example embodiments set forth below, all such variations and modifications that would be envisioned by one of ordinary skill in the art are intended to fall within the scope of this disclosure. 
         [0038]    The phemonology utilized by this invention is described with reference to  FIG. 1 , which shows surface element  110  in relation to Cartesian coordinate system  105 . Surface element  110  intersects the z axis at location  114  lying on plane  116  having z coordinate  117 . The direction of surface normal unit vector  112  is denoted in coordinate system  105  through spherical-polar angles (θ s , φ s ) as 
         [0000]        s=i  sin θ s  cos φ s   +j  sin θ s  sin cos φ s   +k  cos θ s .  (1)
 
         [0000]    The height profile of surface element  110  is represented by 
         [0000]        z   s   =z   0   +αx+βy,   (2)
 
         [0000]      where 
         [0000]      α=−tan θ s  cos φ s .  (3)
 
         [0000]      β=−tan θ s  sin φ s  
 
         [0039]    Coherent illumination beam  120  with wavelength λ and beam axis  123  propagates in the direction opposite to unit vector  122  as denoted in coordinate system  105  through spherical-polar angles (θ b ,φ b ) as 
         [0000]        b=i  sin θ b  cos φ b   +j  sin θ b  sin cos φ b   +k  cos θ b .  (4)
 
         [0040]    In one embodiment, coherent illumination beam  120  is substantially a Gaussian laser beam. Other beam types, however, may also be utilized and will exhibit the same general behavior as described by the following theoretical analysis of Gaussian beams. Coherent illumination beam  120  has a beam waist located at waist position  124 . The complex amplitude of a Gaussian beam propagating along the negative ζ axis of a Cartesian coordinate system (ξ,η,ζ) with waist position  124  located at the origin of coordinate system (ξ,η,ζ) is represented by 
         [0000]    
       
         
           
             
               
                 
                   
                     
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         [0000]    where the Rayleigh range z r  is related to the 1/e 2  intensity radius w 0  at waist position  124  by 
         [0000]        z   r   =πw   0   2 /λ,  (6)
 
         [0000]    and the beam intensity profile is expressed as 
         [0000]        I=|A   0 | 2 exp[−2(ξ 2 +η 2 )/ w   2 ],  (7)
 
         [0000]    where w is the 1/e 2  intensity radius point ζ along the beam axis, which is given by 
         [0000]        w=w   0 √{square root over (1+ζ 2   /z   r   2 )}.  (8)
 
         [0000]    Within the paraxial approximation, the beam waist radius and the Rayleigh range are related to the f/# that represents the convergence rate of the beam by 
         [0000]    
       
         
           
             
               
                 
                   
                     
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         [0041]    The complex amplitude impinging on surface point  118  lying on surface element  110  at lateral coordinates (x, y) is obtained through the coordinate transformation, 
         [0000]    
       
         
           
             
               
                 
                   
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         [0000]    that transforms the coordinate system in which the Gaussian beam is defined in Eq. (5) into coordinate system  105 . Calculation of the quantity 
         [0000]      ρ 2 =ξ 2 +η 2   (12)
 
         [0000]    using Eq. (11) yields 
         [0000]      ρ 2 =[( x−x   b )sin φ b −( y−y   b )cos φ b ] 2 +[( x−x   b )cos θ b  cos φ b +( y−y   b )cos θ b  sin φ b −( z   0   −z   b   +αx+βy )sin θ b ] 2 .  (13)
 
         [0000]      Likewise 
         [0000]      ζ=( x−x   b )sin θ b  cos φ b +( y−y   b )sin θ b  sin φ b +( z   0   +z   b   +αx+βy )cos θ b .  (14)
 
         [0000]    Substitution of Eqs. (13) and (14) into Eq. (5) yields the complex amplitude of the Gaussian beam that impinges on the arbitrary surface point  118  lying on surface element  110 : 
         [0000]    
       
         
           
             
               
                 
                   
                     
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         [0042]    The scattered complex amplitude v P   + (x, y) emanating from point  118  is represented by multiplication of the incident complex amplitude at surface point  118  by the complex function g(x, y): 
         [0000]        v   P   + ( x,y )= v   P ( x,y ) g ( x,y ).  (16)
 
         [0000]    The complex function g(x, y) represents the phase change and reflection loss introduced by scattering from surface point  118 . Fresnel propagation of v P   + (x, y) to plane  134 , located at z=L, yields 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
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         [0000]    is the Fresnel propagation kernel. 
         [0043]    In  FIG. 1  option (A)  151 , detector  140  is placed at plane  134  to detect the speckle intensity at plane  134 . The intensity is found by taking the magnitude squared of the complex amplitude, which yields 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
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                                     x 
                                     ′ 
                                   
                                 
                                  
                                 
                                    
                                   
                                     y 
                                     ′ 
                                   
                                 
                               
                             
                           
                            
                         
                         2 
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   where 
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       φ 
                       L 
                     
                      
                     
                       ( 
                       
                         
                           x 
                           ′ 
                         
                         , 
                         
                           y 
                           ′ 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         - 
                         
                           
                             2 
                              
                             π 
                           
                           λ 
                         
                       
                        
                       
                         ( 
                         
                           
                             α 
                              
                             
                                 
                             
                              
                             
                               x 
                               ′ 
                             
                           
                           + 
                           
                             β 
                              
                             
                                 
                             
                              
                             
                               y 
                               ′ 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         π 
                         λ 
                       
                        
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   x 
                                   - 
                                   
                                     x 
                                     ′ 
                                   
                                 
                                 ) 
                               
                               2 
                             
                             + 
                             
                               
                                 ( 
                                 
                                   y 
                                   - 
                                   
                                     y 
                                     ′ 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                           
                             L 
                             - 
                             
                               z 
                               0 
                             
                             - 
                             
                               α 
                                
                               
                                   
                               
                                
                               
                                 x 
                                 ′ 
                               
                             
                             - 
                             
                               β 
                                
                               
                                   
                               
                                
                               
                                 y 
                                 ′ 
                               
                             
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
         [0044]    In  FIG. 1  option (B)  152  and option (C)  153 , detector  140  is placed at distance d  142  behind plane  134 . In option (C)  153 , distance d  142  is the focal length f  144  of lens  130 . Within the Fresnel approximation, the complex amplitude leaving the lens is the product of the complex amplitude v L  arriving at plane  134  and the complex quadratic phase function representing the transmission function of lens  130  having focal length f  144 . The complex amplitude at distance d  142  to the right of lens  130  is obtained by Fresnel propagation over the distance d  142 : 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           v 
                           d 
                         
                          
                         
                           ( 
                           
                             x 
                             , 
                             y 
                           
                           ) 
                         
                       
                       = 
                       
                         
                           ∫ 
                           
                             - 
                             ∞ 
                           
                           ∞ 
                         
                          
                         
                           
                             ∫ 
                             
                               - 
                               ∞ 
                             
                             ∞ 
                           
                            
                           
                             
                               
                                 v 
                                 L 
                               
                                
                               
                                 ( 
                                 
                                   
                                     x 
                                     ′ 
                                   
                                   , 
                                   
                                     y 
                                     ′ 
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               exp 
                               [ 
                               
                                 
                                   - 
                                   π 
                                 
                                  
                                 
                                   
                                     
                                       x 
                                       ′2 
                                     
                                     + 
                                     
                                       y 
                                       ′2 
                                     
                                   
                                   
                                     λ 
                                      
                                     
                                         
                                     
                                      
                                     f 
                                   
                                 
                               
                               ] 
                             
                              
                             
                               h 
                                
                               
                                 ( 
                                 
                                   
                                     x 
                                     - 
                                     
                                       x 
                                       ′ 
                                     
                                   
                                   , 
                                   
                                     
                                       y 
                                       - 
                                       
                                         y 
                                         ′ 
                                       
                                     
                                     ; 
                                     d 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     ) 
                   
                    
                   
                      
                     
                       x 
                       ′ 
                     
                   
                    
                   
                     
                        
                       
                         y 
                         ′ 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The resulting intensity at detector  140  is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         I 
                         d 
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           y 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         1 
                         
                           
                             λ 
                             2 
                           
                            
                           
                             d 
                             2 
                           
                         
                       
                        
                       
                         
                            
                           
                             
                               ∫ 
                               
                                 - 
                                 ∞ 
                               
                               ∞ 
                             
                              
                             
                               
                                 ∫ 
                                 
                                   - 
                                   ∞ 
                                 
                                 ∞ 
                               
                                
                               
                                 
                                   g 
                                    
                                   
                                     ( 
                                     
                                       
                                         x 
                                         ′ 
                                       
                                       , 
                                       
                                         y 
                                         ′ 
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     v 
                                     P 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         x 
                                         ′ 
                                       
                                       , 
                                       
                                         y 
                                         ′ 
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     exp 
                                      
                                     
                                       [ 
                                       
                                         
                                           φ 
                                           d 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               x 
                                               ′ 
                                             
                                             , 
                                             
                                               y 
                                               ′ 
                                             
                                           
                                           ) 
                                         
                                       
                                       ] 
                                     
                                   
                                   / 
                                   
                                     Q 
                                      
                                     
                                       ( 
                                       
                                         
                                           x 
                                           ′ 
                                         
                                         , 
                                         
                                           y 
                                           ′ 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                  
                                 
                                    
                                   
                                     x 
                                     ′ 
                                   
                                 
                                  
                                 
                                    
                                   
                                     y 
                                     ′ 
                                   
                                 
                               
                             
                           
                            
                         
                         2 
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   where 
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       Q 
                        
                       
                         ( 
                         
                           
                             x 
                             ′ 
                           
                           , 
                           
                             y 
                             ′ 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       1 
                       + 
                       
                         
                           ( 
                           
                             
                               1 
                               d 
                             
                             - 
                             
                               1 
                               f 
                             
                           
                           ) 
                         
                          
                         
                           ( 
                           
                             L 
                             - 
                             
                               z 
                               0 
                             
                             - 
                             
                               α 
                                
                               
                                   
                               
                                
                               
                                 x 
                                 ′ 
                               
                             
                             - 
                             
                               β 
                                
                               
                                   
                               
                                
                               
                                 y 
                                 ′ 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       φ 
                       d 
                     
                      
                     
                       ( 
                       
                         
                           x 
                           ′ 
                         
                         , 
                         
                           y 
                           ′ 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     - 
                     
                       
                         
                           
                             2 
                              
                             π 
                           
                           λ 
                         
                         [ 
                         
                           
                             α 
                              
                             
                                 
                             
                              
                             
                               x 
                               ′ 
                             
                           
                           + 
                           
                             β 
                              
                             
                                 
                             
                              
                             
                               y 
                               ′ 
                             
                           
                           + 
                           
                             
                               
                                 xx 
                                 ′ 
                               
                               + 
                               
                                 yy 
                                 ′ 
                               
                             
                             
                               
                                 Q 
                                  
                                 
                                   ( 
                                   
                                     
                                       x 
                                       ′ 
                                     
                                     , 
                                     
                                       y 
                                       ′ 
                                     
                                   
                                   ) 
                                 
                               
                                
                               d 
                             
                           
                           + 
                           
                             
                               
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             x 
                                             ′2 
                                           
                                           + 
                                           
                                             y 
                                             ′2 
                                           
                                         
                                         ) 
                                       
                                        
                                       
                                         ( 
                                         
                                           d 
                                           - 
                                           f 
                                         
                                         ) 
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           x 
                                           2 
                                         
                                         + 
                                         
                                           y 
                                           2 
                                         
                                       
                                       ) 
                                     
                                      
                                     
                                       ( 
                                       
                                         L 
                                         - 
                                         f 
                                         - 
                                         
                                           z 
                                           0 
                                         
                                         - 
                                         
                                           α 
                                            
                                           
                                               
                                           
                                            
                                           
                                             x 
                                             ′ 
                                           
                                         
                                         - 
                                         
                                           β 
                                            
                                           
                                               
                                           
                                            
                                           
                                             y 
                                             ′ 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             
                               2 
                                
                               
                                 Q 
                                  
                                 
                                   ( 
                                   
                                     
                                       x 
                                       ′ 
                                     
                                     , 
                                     
                                       y 
                                       ′ 
                                     
                                   
                                   ) 
                                 
                               
                                
                               fd 
                             
                           
                         
                         ] 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Note that for option (C)  153  that Q reduces to unity so that Eqs. (22)-(24) take on a particularly simple form with the phase term being reduced to a linear function of x′ and y′. Option (C)  153  shows one embodiment where detector  140  is at the Fourier transform plane of lens  130 . Equations (19)-(20) and (22)-(24) with the additional equations they reference embody the phenomenology which is exploited in the invention. 
         [0045]      FIG. 2  describes two system states. System state  1  is comprised of illumination state  1  and surface state  1 . System state  2  is comprised of illumination state  2  and surface state  2 . As used throughout this description, an illumination state is defined as the properties or parameters of an emission of coherent radiation as they illuminate a surface. Examples of these properties or parameters include but are not limited to the direction of illumination, lateral position of the emission beam waist, offset range of the emission beam waist, the wavelength of the emission/illumination beam and other properties as shown in Table 1. As used throughout this description, a surface state is defined as the properties or parameters of a surface or object. Examples of these properties or parameters include but are not limited to the position and orientation of the surface being sensed, including lateral translations and other properties as shown in Table 1. As used throughout this description, an emission is one or more wavelengths of coherent radiation emitted from a coherent radiation source such as but not limited to a laser. Examples of illumination states and emissions are also described in U.S. Patent Publication No. US20080154524A1 entitled “METHOD AND APPARATUS FOR REMOTE SENSING OF OBJECTS UTILIZING RADIATION SPECKLE” which is herein incorporated by reference in its entirety. 
         [0046]    Generally, and not for limitation, parameters associated with the two system states in  FIG. 2  are described in Table 1. 
         [0000]                                                                            TABLE 1                       Parameter Name   State 1   State 2                                    System   Illumination   Wavelength   λ 1     λ 2                 Direction   (θ b1 , φ b1 )   (θ b2 , φ b2 )               Waist offset   (x b1 , y b1 ,)   (x b2 , y b2 ,)               Waist range offset   z b1     z b2             Surface   Height   z 01     z 02                 Lateral translation   (0, 0)   (Δx, Δy)               Orientation   (α 1 , β 1 ) or   (α 2 , β 2 ) or                   (θ s1 , φ s1 )   (θ s2 , φ s2 )               Orientation change   (0, 0)   (Δα, Δβ)            Speckle intensity   Location of tracked   (x 1 , y 1 )   (x 2 , y 2 )           speckle           Speckle shift   (0, 0)   (s x , s y )                    
Illumination state  1  is represented by coherent illumination beam  120  propagating along beam axis  123  in the direction opposite to unit vector  122 . Coherent illumination beam  120  has beam waist position  124  and a first wavelength. Surface state  1  is represented by the location and orientation of surface element  110 . Illumination state  2  is represented by coherent illumination beam  220  propagating along beam axis  223  in the direction opposite to unit vector  222 . Coherent illumination beam  220  has beam waist position  224  and a second wavelength. Surface state  2  is represented by surface element  110  being translated and rotated into the position represented by repositioned surface element  210 . Repositioned surface element  210  has surface normal unit vector  212  and intersects the z axis at intersection location  214  lying on plane  216 . Point  250  represents a lateral translation of the center of surface element  110  to the center of repositioned surface element  210 . Illumination state  1  and illumination state  2  may be the same or they may comprise parameters that are the same and parameters that are different between state  1  and state  2 . For example, wavelength  1  may equal wavelength  2 . Likewise, surface state  1  and surface state  2  may be the same or they may have parameters that are the same and parameters that are different between state  1  and state  2 . For example, there may be no lateral translation or orientation change between surface state  1  and surface state  2 . In addition, there may be no relative motion between surface state  1  and surface state  2  so that repositioned surface element  210  coincides exactly with surface element  110 . System state  1  and system state  2  differ when at least one system parameter, represented by illumination parameters and surface parameters in Table 1, differ. Coherent illumination beam  120  and coherent illumination beam  220  may either be derived from the same coherent radiation source, or they may be derived from different coherent radiation sources. It is not necessary for coherent illumination beam  120  and coherent illumination beam  220  to be coherent with respect to each other. System state  1  produces speckle intensity pattern  1  at detector  140 . System state  2  produces speckle intensity pattern  2  at detector  140 . If coherent illumination beam  1  and coherent illumination beam  2  have a high degree of overlap at surface element  110  and repositioned surface element  210  so that substantially the same scattering cells are illuminated in system state  1  and system state  2 , then speckle pattern  1  and speckle pattern  2  are highly correlated. This correlation may manifest itself as speckle pattern  1  and speckle pattern  2  being substantially identical except for a lateral shift in the speckle pattern at the location of detector  140 . If a feature in speckle pattern  1  is located at position (x 1 , y 1 ) at the location of detector  140  and this feature moves to position (x 2 , y 2 ) in speckle pattern  2  at the location of detector  140 , then the shift in the speckle pattern is denoted by
 
         [0000]        s   x   =x   2   −x   1   (25)
 
         [0000]    
       
      
       s 
       y 
       =y 
       2 
       −y 
       1  
      
     
         [0047]      FIG. 3  describes one embodiment of the invention where beam axis  123  and beam axis  223  intersect at or near the z axis of coordinate system  105 . In  FIG. 3  the intersection point  360  of beam axis  123  and beam axis  223  lies in plane  362  located at z=z c . Lateral offset  378  is the distance between beam center points  374  and  376  lying on plane  372  located at coordinate z. The condition for beam axis  123  or beam axis  223  to intersect on the z axis at z=z c  is 
         [0000]        x   b +( z   c   −z   b )tan θ b  cos φ b =0.  (26)
 
         [0000]        y   b +( z   c   −z   b )tan θ b  sin φ b =0
 
         [0000]    For illumination directions with small angles, θ b &lt;&lt;1, 
         [0000]        x   b +( z   c   −z   b )θ b  cos φ b =0.  (27)
 
         [0000]        y   b +( z   c   −z   b )θ b  sin φ b =0
 
         [0000]    Substantially satisfying Eq. (26) or Eq. (27) for both beam axis  123  and beam axis  223  produces a location in measurement space near z=z c  on the z axis around which measurements are not significantly degraded by decorrelation between speckle pattern  1  and speckle pattern  2 . The location of z c  can be chosen to suit the application. A high degree of beam overlap can be achieved during a measurement by coordinating the waist offset and the illumination direction between system state  1  and system state  2  to substantially maintain the relations given by Eq (26) or Eq. (27). 
         [0048]    The degree of overlap at the surface is a function of both beam displacement  378  and beam waist w at the surface element. For surface states with equal heights, z 0 =z 0     1   =z 0     2   , and no lateral translation and for illumination states with symmetric waist offsets and equal waist range offsets (x b , y b , z b )=(x b     2   , y b     2   , z b     2   )=(−x b     1   , −y b     1   , z b     1   ) and for small angles θ b =θ b     1   =θ b     2   &lt;&lt;1 and azimuth angles related by φ b     2   −φ b     1   =π, the beam displacement  378  is given by 
         [0000]      Δ r   c =2θ b   |z−z   c |=2 r   b |1− z/z   c |,  (28)
 
         [0000]      where 
         [0000]        r   b =√{square root over ( x   b   2   +y   b   2 )}.  (29)
 
         [0000]    The beam waist radius at z is 
         [0000]        w=w   0 √{square root over (1+( z−z   b ) 2   /z   r   2 )}.  (30)
 
         [0000]    An indication of the degree of overlap for surface elements with height z 0  is given by 
         [0000]    
       
         
           
             
               
                 
                   γ 
                   = 
                   
                     
                       1 
                       - 
                       
                         
                           Δ 
                            
                           
                               
                           
                            
                           
                             r 
                             c 
                           
                         
                         
                           2 
                            
                           w 
                         
                       
                     
                     = 
                     
                       1 
                       - 
                       
                         
                           r 
                           b 
                         
                          
                         
                           
                             
                                
                               
                                 1 
                                 - 
                                 
                                   
                                     z 
                                     0 
                                   
                                   / 
                                   
                                     z 
                                     c 
                                   
                                 
                               
                                
                             
                             
                               
                                 w 
                                 0 
                               
                                
                               
                                 
                                   1 
                                   + 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             z 
                                             0 
                                           
                                           - 
                                           
                                             z 
                                             b 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                     / 
                                     
                                       z 
                                       r 
                                       2 
                                     
                                   
                                 
                               
                             
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   31 
                   ) 
                 
               
             
           
         
       
     
         [0000]    When γ=1, the beam axes intersect at the surface element and speckle correlation is high. When γ=0, the 1/e 2  intensity points of the beams approximately touch, and there is very little beam overlap and a correspondingly low speckle correlation. For fixed z c  there is a range of z values for which γ is high enough to provide good measurements. It is possible to achieve high γ values over long ranges. For example, if z b =0, r b =50 μm, z c =5 mm, and f/#=10, then γ≧0.9 over the range of 3.3 mm≦z≦10.0 mm. Furthermore, γ≧0.8 for z≧2.5 mm. If θ b =0 so that there is no beam tilt with respect to the z axis, i.e., z c =∞, then γ≧0.8 for |z|≧5.0 mm and γ≧0.9 for |z|≧10.0 mm. These results for γ are insensitive to wavelength and hold from the ultraviolet to the far infrared. 
         [0049]      FIG. 4  describes a method for determining a change in at least one system parameter between system state  1  and system state  2  from measurements of the speckle shift resulting from the change between system state  1  and system state  2 . In step  410  surface state  1  is illuminated with illumination state  1 . In step  420  speckle pattern  1  is measured at least 1 point. In step  430  surface state  2  is illuminated with illumination state  2 . In step  440  speckle pattern  2  is measured at least 1 point. In step  450 , the speckle shift between speckle pattern  1  and speckle pattern  2  is determined. In step  460  at least one system parameter is calculated in response to the change between system state  1  and system state  2 . 
         [0050]    Measurement of speckle intensity can occur in any of numerous modes. For example, in one embodiment detector  140  is a two-dimensional detector array comprised of at least three detector elements or pixels. Detector element need not lie on a plane. In one embodiment of a two-dimensional detector array the pixels are closely packed and contiguous. In another embodiment detector  140  is a linear array comprised of at least two elements arranged in substantially a straight line. In one embodiment of a linear array, the array is aligned to substantially coincide with the direction of speckle shift. In another embodiment detector  140  is comprised of at least two separated detection elements. In one embodiment of at least two separated detection elements, said detection elements are positioned to be substantially aligned with the direction of speckle shift. In one embodiment of said substantially aligned detector elements, a time history of speckle intensity over a multiple set of system states is produced for each of said detector elements. In one embodiment of said time history, the set of system states is a continuous history produced by a smooth variation of at least one system parameter. 
         [0051]    Step  450  of determining the speckle shift can be accomplished through any of many algorithms known by those skilled in the art. For example, speckle shift can be obtained through calculating a cross correlation of two sets of speckle intensity data. The location of the peak of the cross correlation provides the speckle shift. Enhanced precision is achieved through sub-pixel interpolation algorithms. Another example of a method for determining speckle shift is to utilize optical flow algorithms known to those skilled in the art. Yet another method for determining speckle shift is to compare the speckle intensity time histories produced by a set of system states and to determine the time delay associated with features in the speckle pattern to move between detector elements. In one embodiment the determination of speckle shift is performed by a processor in communication with a memory and containing machine instructions capable of determining a speckle shift. 
         [0052]    Step  460  of calculating at least one system parameter is performed by relating speckle shift to changes in at least one system parameter through system calibration or through measurement equations derived from the analysis of speckle intensity as related to Eq. (19) or Eq. (22). System calibration may be achieved, for example, by compensating measurement equations for system errors or inaccuracies in the model from which the measurement equations are derived. 
         [0053]    A generalized measurement equation is obtained by continuing the analysis that led to Eq. (19) and Eq. (22). Equations (19) and (22) indicate that the speckle pattern intensity can be formulated as the magnitude squared of a Fourier transform. The argument of the Fourier transform contains the spatial coordinates (x, y) of the speckle intensity at detector  140 . The speckle shift resulting from transitioning from system state  1  to system state  2  is determined by keeping the arguments of the Fourier transform constant. The argument of the Fourier transform consists of all complex exponential phase terms in Eq. (19) or Eq. (22) that are linear in terms of the variables of integration x′ and y′. As long as the remainder of the integrand, excluding g(x′, y′), varies slowly with respect to x′ and y′ and γ is high, the integral behaves approximately as the Fourier transform of g, weighted by the magnitude of the incident beam. Consequently, speckle pattern  1  and speckle pattern  2  are highly correlated and related by a simple shift. The behavior of the speckle shift is determined by writing the integrand as a magnitude multiplied by a complex exponential phase component. The phase is then expanded in a power series to obtain an approximation of the linear phase terms. 
         [0054]    The phase function φ d  of Eq. (24) is expanded in a power series as 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         φ 
                         d 
                       
                        
                       
                         ( 
                         
                           
                             x 
                             ′ 
                           
                           , 
                           
                             y 
                             ′ 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         φ 
                         
                           d 
                            
                           
                               
                           
                            
                           0 
                         
                       
                       + 
                       
                         
                           φ 
                           dx 
                         
                          
                         
                           x 
                           ′ 
                         
                       
                       + 
                       
                         
                           φ 
                           dy 
                         
                          
                         
                           y 
                           ′ 
                         
                       
                       + 
                       … 
                     
                   
                    
                   
                       
                   
                   , 
                   
                     
 
                   
                    
                   where 
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       φ 
                       dx 
                     
                     = 
                     
                       - 
                       
                         
                           
                             2 
                              
                             π 
                           
                           λ 
                         
                         [ 
                         
                           
                             x 
                             
                               f 
                               d 
                             
                           
                           + 
                           
                             α 
                             ( 
                             
                               1 
                               - 
                               
                                 
                                   
                                     x 
                                     2 
                                   
                                   + 
                                   
                                     y 
                                     2 
                                   
                                 
                                 
                                   2 
                                    
                                   
                                     f 
                                     d 
                                   
                                 
                               
                             
                             ) 
                           
                         
                         ] 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         φ 
                         dy 
                       
                       = 
                       
                         - 
                         
                           
                             
                               2 
                                
                               π 
                             
                             λ 
                           
                           [ 
                           
                             
                               y 
                               
                                 f 
                                 d 
                               
                             
                             + 
                             
                               α 
                               ( 
                               
                                 1 
                                 - 
                                 
                                   
                                     
                                       x 
                                       2 
                                     
                                     + 
                                     
                                       y 
                                       2 
                                     
                                   
                                   
                                     2 
                                      
                                     
                                       f 
                                       d 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                           ] 
                         
                       
                     
                     , 
                     
                       
 
                     
                      
                     and 
                   
                 
               
               
                 
                   ( 
                   33 
                   ) 
                 
               
             
             
               
                 
                   
                     f 
                     d 
                   
                   = 
                   
                     d 
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           
                             d 
                             / 
                             f 
                           
                         
                         ) 
                       
                        
                       
                         
                           ( 
                           
                             L 
                             - 
                             
                               z 
                               0 
                             
                           
                           ) 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The same results are applicable to the expansion of φ L  of Eq. (20) if we set d=0. Equations (33) and (34) are therefore used in the general case. Option (C)  153  of  FIG. 1  reduces to f d =f and is preferred because the expansion φ d  in linear terms is exact and because f d  does not depend on the surface height z 0 . 
         [0055]    The complex amplitude incident on point  118  is expressed in terms of a magnitude and a phase component as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         v 
                         P 
                       
                        
                       
                         ( 
                         
                           
                             x 
                             ′ 
                           
                           , 
                           
                             y 
                             ′ 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                          
                         
                           A 
                           0 
                         
                          
                       
                        
                       
                         
                           z 
                           r 
                         
                         
                           
                             
                               ζ 
                               2 
                             
                             + 
                             
                               z 
                               r 
                               2 
                             
                           
                         
                       
                        
                       
                         exp 
                         [ 
                         
                           
                             - 
                             
                               π 
                               λ 
                             
                           
                            
                           
                             ( 
                             
                               
                                 
                                   ρ 
                                   2 
                                 
                                  
                                 
                                   z 
                                   r 
                                 
                               
                               
                                 
                                   ζ 
                                   2 
                                 
                                 + 
                                 
                                   z 
                                   r 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                         ] 
                       
                        
                       
                         exp 
                          
                         
                           [ 
                           
                             
                               φ 
                               P 
                             
                              
                             
                               ( 
                               
                                 
                                   x 
                                   ′ 
                                 
                                 , 
                                 
                                   y 
                                   ′ 
                                 
                               
                               ) 
                             
                           
                           ] 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                       
                   
                    
                   where 
                 
               
               
                 
                   ( 
                   35 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         φ 
                         P 
                       
                        
                       
                         ( 
                         
                           
                             x 
                             ′ 
                           
                           , 
                           
                             y 
                             ′ 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         tan 
                         ( 
                         
                           ζ 
                           
                             z 
                             r 
                           
                         
                         ) 
                       
                       - 
                       
                         
                           π 
                           λ 
                         
                          
                         
                           
                             
                               ρ 
                               2 
                             
                              
                             ζ 
                           
                           
                             
                               ζ 
                               2 
                             
                             + 
                             
                               z 
                               r 
                               2 
                             
                           
                         
                       
                       - 
                       
                         
                           
                             2 
                              
                             π 
                           
                           λ 
                         
                          
                         
                           ζ 
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   36 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Expansion of Eq. (36) in a power series yields 
         [0000]      φ P ( x′,y ′)=φ P0 +φ Px   x′+φ   Py   y′+ . . . ,   (37)
 
         [0000]    where the linear coefficients are 
         [0000]    
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         φ 
                         Px 
                       
                       = 
                       
                         
                           
                             
                               2 
                                
                               π 
                             
                             λ 
                           
                            
                           
                             
                               1 
                               
                                 
                                   ζ 
                                   0 
                                   2 
                                 
                                 + 
                                 
                                   z 
                                   r 
                                   2 
                                 
                               
                             
                             [ 
                             
                               
                                 
                                   - 
                                   
                                     χ 
                                     x 
                                   
                                 
                                  
                                 
                                   ζ 
                                   0 
                                 
                               
                               + 
                               
                                 
                                   
                                     ζ 
                                     x 
                                   
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         ρ 
                                         0 
                                         2 
                                       
                                        
                                       
                                         
                                           
                                             ζ 
                                             0 
                                             2 
                                           
                                           - 
                                           
                                             z 
                                             r 
                                             2 
                                           
                                         
                                         
                                           
                                             ζ 
                                             0 
                                             2 
                                           
                                           + 
                                           
                                             z 
                                             r 
                                             2 
                                           
                                         
                                       
                                     
                                     + 
                                     
                                       
                                         λ 
                                          
                                         
                                             
                                         
                                          
                                         
                                           z 
                                           r 
                                         
                                       
                                       π 
                                     
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                         
                         - 
                         
                           
                             
                               2 
                                
                               π 
                             
                             λ 
                           
                            
                           
                             ζ 
                             x 
                           
                         
                       
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       
                         
                           φ 
                           Py 
                         
                         = 
                         
                           
                             
                               
                                 2 
                                  
                                 π 
                               
                               λ 
                             
                              
                             
                               
                                 1 
                                 
                                   
                                     ζ 
                                     0 
                                     2 
                                   
                                   + 
                                   
                                     z 
                                     r 
                                     2 
                                   
                                 
                               
                               [ 
                               
                                 
                                   
                                     - 
                                     
                                       χ 
                                       y 
                                     
                                   
                                    
                                   
                                     ζ 
                                     0 
                                   
                                 
                                 + 
                                 
                                   
                                     
                                       ζ 
                                       y 
                                     
                                     2 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           ρ 
                                           0 
                                           2 
                                         
                                          
                                         
                                           
                                             
                                               ζ 
                                               0 
                                               2 
                                             
                                             - 
                                             
                                               z 
                                               r 
                                               2 
                                             
                                           
                                           
                                             
                                               ζ 
                                               0 
                                               2 
                                             
                                             + 
                                             
                                               z 
                                               r 
                                               2 
                                             
                                           
                                         
                                       
                                       + 
                                       
                                         
                                           λ 
                                            
                                           
                                               
                                           
                                            
                                           
                                             z 
                                             r 
                                           
                                         
                                         π 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ] 
                             
                           
                           - 
                           
                             
                               
                                 2 
                                  
                                 π 
                               
                               λ 
                             
                              
                             
                               ζ 
                               y 
                             
                           
                         
                       
                       , 
                       
                         
 
                       
                        
                       
                           
                       
                        
                       and 
                     
                   
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             
                                 
                             
                              
                             
                               
                                 
                                   ζ 
                                   0 
                                 
                                 = 
                                 
                                   
                                     
                                       ( 
                                       
                                         z 
                                         - 
                                         
                                           z 
                                           b 
                                         
                                       
                                       ) 
                                     
                                      
                                     cos 
                                      
                                     
                                         
                                     
                                      
                                     
                                       θ 
                                       b 
                                     
                                   
                                   - 
                                   
                                     
                                       ( 
                                       
                                         
                                           
                                             x 
                                             b 
                                           
                                            
                                           cos 
                                            
                                           
                                               
                                           
                                            
                                           
                                             φ 
                                             b 
                                           
                                         
                                         + 
                                         
                                           
                                             y 
                                             b 
                                           
                                            
                                           sin 
                                            
                                           
                                               
                                           
                                            
                                           
                                             φ 
                                             b 
                                           
                                         
                                       
                                       ) 
                                     
                                      
                                     sin 
                                      
                                     
                                         
                                     
                                      
                                     
                                       θ 
                                       b 
                                     
                                   
                                 
                               
                                
                               
                                 
 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ζ 
                                   x 
                                 
                                 = 
                                 
                                   
                                     α 
                                      
                                     
                                         
                                     
                                      
                                     cos 
                                      
                                     
                                         
                                     
                                      
                                     
                                       θ 
                                       b 
                                     
                                   
                                   + 
                                   
                                     sin 
                                      
                                     
                                         
                                     
                                      
                                     
                                       θ 
                                       b 
                                     
                                      
                                     cos 
                                      
                                     
                                         
                                     
                                      
                                     
                                       φ 
                                       b 
                                     
                                   
                                 
                               
                                
                               
                                 
 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ζ 
                                   y 
                                 
                                 = 
                                 
                                   
                                     β 
                                      
                                     
                                         
                                     
                                      
                                     cos 
                                      
                                     
                                         
                                     
                                      
                                     
                                       θ 
                                       b 
                                     
                                   
                                   + 
                                   
                                     sin 
                                      
                                     
                                         
                                     
                                      
                                     
                                       θ 
                                       b 
                                     
                                      
                                     sin 
                                      
                                     
                                         
                                     
                                      
                                     
                                       φ 
                                       b 
                                     
                                   
                                 
                               
                                
                               
                                 
 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ρ 
                                   0 
                                   2 
                                 
                                 = 
                                 
                                   
                                     
                                         
                                       
                                         
                                           
                                               
                                             
                                               
                                                 
                                                   
                                                     ( 
                                                     
                                                       
                                                         
                                                           
                                                             
                                                             
                                                             y 
                                                             b 
                                                             
                                                              
                                                             cos 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             φ 
                                                             b 
                                                             
                                                             
                                                             - 
                                                           
                                                         
                                                       
                                                       
                                                         
                                                           
                                                             
                                                             x 
                                                             b 
                                                             
                                                              
                                                             sin 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             φ 
                                                             b 
                                                             
                                                           
                                                         
                                                       
                                                     
                                                     ) 
                                                   
                                                   2 
                                                 
                                                 + 
                                                 
                                                   
                                                     
                                                       [ 
                                                       
                                                         
                                                           
                                                             
                                                             
                                                             
                                                             ( 
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             x 
                                                             b 
                                                             
                                                              
                                                             cos 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             φ 
                                                             b 
                                                             
                                                             
                                                             + 
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             y 
                                                             b 
                                                             
                                                              
                                                             sin 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             φ 
                                                             b 
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             ) 
                                                             
                                                              
                                                             cos 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             θ 
                                                             b 
                                                             
                                                             
                                                             + 
                                                             
                                                           
                                                         
                                                         
                                                           
                                                             
                                                             
                                                             ( 
                                                             
                                                             z 
                                                             - 
                                                             
                                                             z 
                                                             b 
                                                             
                                                             
                                                             ) 
                                                             
                                                              
                                                             sin 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             θ 
                                                             b 
                                                             
                                                             
                                                           
                                                         
                                                       
                                                       ] 
                                                     
                                                     2 
                                                   
                                                    
                                                   
                                                     
 
                                                   
                                                    
                                                   
                                                     χ 
                                                     x 
                                                   
                                                 
                                               
                                               = 
                                               
                                                 
                                                   
                                                     - 
                                                     
                                                       x 
                                                       b 
                                                     
                                                   
                                                    
                                                   
                                                     sin 
                                                     2 
                                                   
                                                    
                                                   
                                                     φ 
                                                     b 
                                                   
                                                 
                                                 + 
                                                 
                                                   
                                                     y 
                                                     b 
                                                   
                                                    
                                                   sin 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   
                                                     θ 
                                                     b 
                                                   
                                                    
                                                   sin 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   
                                                     φ 
                                                     b 
                                                   
                                                 
                                               
                                             
                                               
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 
                                                   
                                                     
                                                       αcosθ 
                                                       b 
                                                     
                                                     + 
                                                   
                                                 
                                               
                                               
                                                 
                                                   
                                                     sin 
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     
                                                       θ 
                                                       b 
                                                     
                                                      
                                                     cos 
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     
                                                       φ 
                                                       b 
                                                     
                                                   
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                         + 
                                       
                                         
                                     
                                      
                                     
                                       [ 
                                       
                                         
                                           
                                             
                                               
                                                 
                                                   ( 
                                                   
                                                     z 
                                                     - 
                                                     
                                                       z 
                                                       b 
                                                     
                                                   
                                                   ) 
                                                 
                                                  
                                                 sin 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   θ 
                                                   b 
                                                 
                                               
                                               + 
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 x 
                                                 b 
                                               
                                                
                                               cos 
                                                
                                               
                                                   
                                               
                                                
                                               
                                                 θ 
                                                 b 
                                               
                                                
                                               cos 
                                                
                                               
                                                   
                                               
                                                
                                               
                                                 φ 
                                                 b 
                                               
                                             
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           
                                             
                                               α 
                                                
                                               
                                                   
                                               
                                                
                                               sin 
                                                
                                               
                                                   
                                               
                                                
                                               
                                                 θ 
                                                 b 
                                               
                                             
                                             - 
                                           
                                         
                                       
                                       
                                         
                                           
                                             cos 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               θ 
                                               b 
                                             
                                              
                                             cos 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               φ 
                                               b 
                                             
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                                
                               
                                 
 
                               
                                
                               
                                 
                                   χ 
                                   y 
                                 
                                 = 
                                 
                                   
                                     
                                       - 
                                       
                                         y 
                                         b 
                                       
                                     
                                      
                                     
                                       cos 
                                       2 
                                     
                                      
                                     
                                       φ 
                                       b 
                                     
                                   
                                   + 
                                   
                                     
                                       x 
                                       b 
                                     
                                      
                                     sin 
                                      
                                     
                                         
                                     
                                      
                                     
                                       θ 
                                       b 
                                     
                                      
                                     cos 
                                      
                                     
                                         
                                     
                                      
                                     
                                       φ 
                                       b 
                                     
                                   
                                 
                               
                             
                               
                           
                            
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       β 
                                        
                                       
                                           
                                       
                                        
                                       cos 
                                        
                                       
                                           
                                       
                                        
                                       
                                         θ 
                                         b 
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     sin 
                                      
                                     
                                         
                                     
                                      
                                     
                                       θ 
                                       b 
                                     
                                      
                                     sin 
                                      
                                     
                                         
                                     
                                      
                                     
                                       φ 
                                       b 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                         + 
                       
                         
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               
                                 
                                   ( 
                                   
                                     z 
                                     - 
                                     
                                       z 
                                       b 
                                     
                                   
                                   ) 
                                 
                                  
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   b 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 y 
                                 b 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 b 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 φ 
                                 b 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                    
                   
                     
                       ( 
                       
                         
                           
                             
                               
                                 βsinθ 
                                 b 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 b 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 φ 
                                 b 
                               
                             
                           
                         
                       
                       ) 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
           
         
       
     
         [0000]    In one embodiment, the illumination parameters θ b , x b , and y b  are small and Eq. (38) reduces to: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       φ 
                       Px 
                     
                     = 
                     
                       
                         
                           2 
                            
                           π 
                         
                         λ 
                       
                        
                       
                         ( 
                         
                           
                             
                               
                                 
                                   x 
                                   b 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       z 
                                       0 
                                     
                                     - 
                                     
                                       z 
                                       b 
                                     
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   z 
                                   r 
                                   2 
                                 
                                  
                                 
                                   θ 
                                   b 
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   φ 
                                   b 
                                 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     α 
                                     + 
                                     
                                       
                                         θ 
                                         b 
                                       
                                        
                                       cos 
                                        
                                       
                                           
                                       
                                        
                                       
                                         φ 
                                         b 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   
                                     λ 
                                      
                                     
                                         
                                     
                                      
                                     
                                       z 
                                       r 
                                     
                                   
                                   
                                     2 
                                      
                                     π 
                                   
                                 
                               
                             
                             
                               
                                 
                                   ( 
                                   
                                     
                                       z 
                                       0 
                                     
                                     - 
                                     
                                       z 
                                       b 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               + 
                               
                                 z 
                                 r 
                                 2 
                               
                             
                           
                           - 
                           α 
                         
                         ) 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       φ 
                       Py 
                     
                     = 
                     
                       
                         
                           2 
                            
                           π 
                         
                         λ 
                       
                        
                       
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     y 
                                     b 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         z 
                                         0 
                                       
                                       - 
                                       
                                         z 
                                         b 
                                       
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   
                                     z 
                                     r 
                                     2 
                                   
                                    
                                   
                                     θ 
                                     b 
                                   
                                    
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   
                                     φ 
                                     b 
                                   
                                 
                                 + 
                                 
                                   
                                     ( 
                                     
                                       β 
                                       + 
                                       
                                         
                                           θ 
                                           b 
                                         
                                          
                                         sin 
                                          
                                         
                                             
                                         
                                          
                                         
                                           φ 
                                           b 
                                         
                                       
                                     
                                     ) 
                                   
                                    
                                   
                                     
                                       λ 
                                        
                                       
                                           
                                       
                                        
                                       
                                         z 
                                         r 
                                       
                                     
                                     
                                       2 
                                        
                                       π 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         z 
                                         0 
                                       
                                       - 
                                       
                                         z 
                                         b 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 + 
                                 
                                   z 
                                   r 
                                   2 
                                 
                               
                             
                             - 
                             α 
                           
                           ) 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   40 
                   ) 
                 
               
             
           
         
       
     
         [0056]    The measurement equation for relating speckle shift to system parameters for a specific operation mode is obtained by requiring that the total combined linear phase remain constant between system state  1  and system state  2 : 
         [0000]      φ Px     1   +φ dx     1   =φ Px     2   +φ dx     2   .  (41)
 
         [0000]      φ Py     1   +φ dy     1   =φ Py     2   +φ dy     2    
 
         [0057]    In reference to  FIG. 5 , six specific operation modes are described in detail. These modes correspond to the specific parameters that are changed in step  430  to achieve system state  2  and to the specific system parameter or parameters are being calculated in step  460 . For the purpose of the following illustrative example calculations pertaining to  FIG. 5 , it is assumed that measurements at detector  140  are symmetric about the origin: 
         [0000]      ( x   2   ,y   2 )=(− x   1   ,−y   1 )=( s   x /2 ,s   y /2),  (42)
 
         [0000]    and unless otherwise stated, that θ b =0 and z b =0. (It is not necessary to number the subscripts when the values of the parameter are the same for both state  1  and state  2 .) These assumptions are for illustration only and do not reduce the generality of the results. 
         [0058]    In  FIG. 5  option (A) step  430  corresponds to the step of changing the waist offset  510  while keeping all other parameters constant. Step  460  corresponds to calculating the height of the surface element. For the purpose of the calculation, let the waist offset be in the x direction and symmetric about the origin: 
         [0000]      ( x   b     2     ,y   b     2   )=(− x   b     1     ,y   b     1   )=( b/ 2,0),  (43)
 
         [0000]    where the total combined waist offset shift between state  1  and state  2  is 
         [0000]        b=x   b     2     −x   b     1   .  (44)
 
         [0000]    The resulting measurement equation is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       s 
                       x 
                     
                     = 
                     
                       
                         - 
                         
                           bf 
                           d 
                         
                       
                        
                       
                         
                           z 
                           0 
                         
                         
                           
                             z 
                             0 
                             2 
                           
                           + 
                           
                             z 
                             r 
                             2 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   45 
                   ) 
                 
               
             
           
         
       
     
         [0000]    which when solved for height yields 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       z 
                       0 
                     
                     = 
                     
                       - 
                       
                         
                           
                             b 
                              
                             
                               ( 
                               
                                 d 
                                 + 
                                 
                                   κ 
                                    
                                   
                                       
                                   
                                    
                                   L 
                                 
                               
                               ) 
                             
                           
                           ± 
                           
                             
                               
                                 
                                   
                                     b 
                                     2 
                                   
                                    
                                   
                                     ( 
                                     
                                       d 
                                       + 
                                       
                                         κ 
                                          
                                         
                                             
                                         
                                          
                                         L 
                                       
                                     
                                     ) 
                                   
                                 
                                 2 
                               
                               - 
                               
                                 4 
                                  
                                 
                                   
                                     s 
                                     x 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         s 
                                         x 
                                       
                                       - 
                                       
                                         κ 
                                          
                                         
                                             
                                         
                                          
                                         b 
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   z 
                                   r 
                                   2 
                                 
                               
                             
                           
                         
                         
                           2 
                            
                           
                             ( 
                             
                               
                                 s 
                                 x 
                               
                               - 
                               
                                 κ 
                                  
                                 
                                     
                                 
                                  
                                 b 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   where 
                 
               
               
                 
                   ( 
                   46 
                   ) 
                 
               
             
             
               
                 
                   κ 
                   = 
                   
                     1 
                     - 
                     
                       d 
                       / 
                       
                         f 
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   47 
                   ) 
                 
               
             
           
         
       
     
         [0059]    In the embodiment of detection option  153 , κ=0, and Eq. (46) reduces to 
         [0000]    
       
         
           
             
               
                 
                   
                     z 
                     0 
                   
                   = 
                   
                     - 
                     
                       
                         
                           bf 
                           ± 
                           
                             
                               
                                 
                                   b 
                                   2 
                                 
                                  
                                 
                                   f 
                                   2 
                                 
                               
                               - 
                               
                                 4 
                                  
                                 
                                   s 
                                   x 
                                   2 
                                 
                                  
                                 
                                   z 
                                   r 
                                   2 
                                 
                               
                             
                           
                         
                         
                           2 
                            
                           
                             s 
                             x 
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   48 
                   ) 
                 
               
             
           
         
       
     
         [0000]    When the height is large with respect to the Rayleigh range, z r &lt;&lt;|z 0 |, and Eq. (46) reduces to 
         [0000]    
       
         
           
             
               
                 
                   
                     z 
                     0 
                   
                   = 
                   
                     - 
                     
                       
                         
                           b 
                            
                           
                             ( 
                             
                               d 
                               + 
                               
                                 κ 
                                  
                                 
                                     
                                 
                                  
                                 L 
                               
                             
                             ) 
                           
                         
                         
                           
                             s 
                             x 
                           
                           - 
                           
                             κ 
                              
                             
                                 
                             
                              
                             b 
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   49 
                   ) 
                 
               
             
           
         
       
     
         [0000]    When both conditions κ=0 and z r &lt;&lt;|z 0 | are satisfied, Eq. (46) takes the particularly simple form 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       z 
                       0 
                     
                     = 
                     
                       - 
                       
                         bf 
                         
                           s 
                           x 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   50 
                   ) 
                 
               
             
           
         
       
     
         [0000]    so that there is an inverse proportionality between height and speckle shift. The relation between height and speckle shift is linear when the magnitude of the height is small with respect to the Rayleigh range |z 0 |&lt;&lt;z r , and κ=0: 
         [0000]    
       
         
           
             
               
                 
                   
                     z 
                     0 
                   
                   = 
                   
                     - 
                     
                       
                         
                           
                             s 
                             x 
                           
                            
                           
                             z 
                             r 
                             2 
                           
                         
                         bf 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   51 
                   ) 
                 
               
             
           
         
       
     
         [0060]    In  FIG. 5  option (B), step  430  corresponds to a coordinated change of waist offset and illumination direction  510  and step  460  corresponds to calculation of height  525 . In this operation mode, the illumination direction is changed according to Eq. (27) to maintain a high value of γ so that measurement noise due to speckle pattern decorrelation is minimized. Therefore, in this embodiment θ b  is not assumed to be zero. It is shown explicitly in the equations, but should obey the relation 
         [0000]      θ b   =−x   b   /z   c   (52)
 
         [0000]    in order to satisfy Eq. (27) at position z c . The azimuth angles are φ b     1   =0 and φ b     2   =π so that Eq. (27) can be satisfied for the waist offset being in the x direction. The measurement equation for option (B) of  FIG. 5  is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       s 
                       x 
                     
                     = 
                     
                       
                         - 
                         
                           [ 
                           
                             d 
                             + 
                             
                               κ 
                                
                               
                                 ( 
                                 
                                   L 
                                   - 
                                   
                                     z 
                                     0 
                                   
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                        
                       
                         
                           
                             bz 
                             0 
                           
                           + 
                           ɛ 
                         
                         
                           
                             z 
                             0 
                             2 
                           
                           + 
                           
                             z 
                             r 
                             2 
                           
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   where 
                 
               
               
                 
                   ( 
                   53 
                   ) 
                 
               
             
             
               
                 
                   ɛ 
                   = 
                   
                     
                       θ 
                       b 
                     
                      
                     
                       
                         
                           z 
                           r 
                         
                          
                         
                           ( 
                           
                             
                               2 
                                
                               
                                 z 
                                 r 
                               
                             
                             - 
                             
                               λ 
                               / 
                               π 
                             
                           
                           ) 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   54 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The solution of Eq. (53) for height is: 
         [0000]    
       
         
           
             
               
                 
                   
                     z 
                     0 
                   
                   = 
                   
                     - 
                     
                       
                         
                           bd 
                           + 
                           
                             
                               κ 
                                
                               
                                 ( 
                                 
                                   bL 
                                   - 
                                   ɛ 
                                 
                                 ) 
                               
                             
                             ± 
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           [ 
                                           
                                             bd 
                                             + 
                                             
                                               κ 
                                                
                                               
                                                 ( 
                                                 
                                                   bL 
                                                   - 
                                                   ɛ 
                                                 
                                                 ) 
                                               
                                             
                                           
                                           ] 
                                         
                                         2 
                                       
                                       - 
                                     
                                   
                                 
                                 
                                   
                                     
                                       4 
                                        
                                       
                                         
                                           ( 
                                           
                                             
                                               s 
                                               x 
                                             
                                             - 
                                             
                                               κ 
                                                
                                               
                                                   
                                               
                                                
                                               b 
                                             
                                           
                                           ) 
                                         
                                          
                                         
                                           [ 
                                           
                                             
                                               
                                                 s 
                                                 x 
                                               
                                                
                                               
                                                 z 
                                                 r 
                                                 2 
                                               
                                             
                                             + 
                                             
                                               ɛ 
                                                
                                               
                                                 ( 
                                                 
                                                   d 
                                                   + 
                                                   
                                                     κ 
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     L 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                         
                           2 
                            
                           
                             ( 
                             
                               
                                 s 
                                 x 
                               
                               - 
                               
                                 κ 
                                  
                                 
                                     
                                 
                                  
                                 b 
                               
                             
                             ) 
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   55 
                   ) 
                 
               
             
           
         
       
     
         [0000]    In detection option  153  of  FIG. 1 , κ=0, and Eq. (55) simplifies to 
         [0000]    
       
         
           
             
               
                 
                   
                     z 
                     0 
                   
                   = 
                   
                     - 
                     
                       
                         
                           bf 
                           ± 
                           
                             
                               
                                 
                                   b 
                                   2 
                                 
                                  
                                 
                                   f 
                                   2 
                                 
                               
                               - 
                               
                                 4 
                                  
                                 
                                   
                                     s 
                                     x 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           s 
                                           x 
                                         
                                          
                                         
                                           z 
                                           r 
                                           2 
                                         
                                       
                                       + 
                                       
                                         ɛ 
                                          
                                         
                                             
                                         
                                          
                                         f 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         
                           2 
                            
                           
                             s 
                             x 
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   56 
                   ) 
                 
               
             
           
         
       
     
         [0061]    In  FIG. 5  option (C), step  430  corresponds to a change of wavelength  530  and step  460  corresponds to calculating surface orientation  535 . In this operation mode the waist offset is zero in all directions. For a small change Δv in optical frequency v, the orientation is related to the speckle shift through 
         [0000]    
       
         
           
             
               
                 
                   
                     α 
                     = 
                     
                       
                         - 
                         
                           ν 
                           Δν 
                         
                       
                        
                       
                         
                           s 
                           x 
                         
                         
                           2 
                            
                           
                             f 
                             d 
                           
                         
                       
                        
                       
                         1 
                         
                           1 
                           - 
                           
                             
                               
                                 s 
                                 x 
                                 2 
                               
                               + 
                               
                                 s 
                                 y 
                                 2 
                               
                             
                             
                               16 
                                
                               
                                 f 
                                 d 
                                 2 
                               
                             
                           
                           + 
                           
                             
                               
                                 λ 
                                  
                                 
                                     
                                 
                                  
                                 
                                   z 
                                   r 
                                 
                               
                               
                                 4 
                                  
                                 π 
                               
                             
                              
                             
                               
                                 
                                   z 
                                   0 
                                   2 
                                 
                                 - 
                                 
                                   z 
                                   r 
                                   2 
                                 
                               
                               
                                 
                                   ( 
                                   
                                     
                                       z 
                                       0 
                                       2 
                                     
                                     + 
                                     
                                       z 
                                       r 
                                       2 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     β 
                     = 
                     
                       
                         - 
                         
                           ν 
                           Δν 
                         
                       
                        
                       
                         
                           s 
                           y 
                         
                         
                           2 
                            
                           
                             f 
                             d 
                           
                         
                       
                        
                       
                         
                           1 
                           
                             1 
                             - 
                             
                               
                                 
                                   s 
                                   x 
                                   2 
                                 
                                 + 
                                 
                                   s 
                                   y 
                                   2 
                                 
                               
                               
                                 16 
                                  
                                 
                                   f 
                                   d 
                                   2 
                                 
                               
                             
                             + 
                             
                               
                                 
                                   λ 
                                    
                                   
                                       
                                   
                                    
                                   
                                     z 
                                     r 
                                   
                                 
                                 
                                   4 
                                    
                                   π 
                                 
                               
                                
                               
                                 
                                   
                                     z 
                                     0 
                                     2 
                                   
                                   - 
                                   
                                     z 
                                     r 
                                     2 
                                   
                                 
                                 
                                   
                                     ( 
                                     
                                       
                                         z 
                                         0 
                                         2 
                                       
                                       + 
                                       
                                         z 
                                         r 
                                         2 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                               
                             
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   57 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The azimuth angle of the surface orientation is given simply as 
         [0000]      θ s =tan −1 ( s   y   /s   x ).  (58)
 
         [0000]    Equation (57) is approximated well by the simplified expression 
         [0000]    
       
         
           
             
               
                 
                   
                     α 
                     = 
                     
                       
                         - 
                         
                           ν 
                           Δν 
                         
                       
                        
                       
                         
                           s 
                           x 
                         
                         
                           2 
                            
                           
                             f 
                             d 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     β 
                     = 
                     
                       
                         - 
                         
                           ν 
                           Δν 
                         
                       
                        
                       
                         
                           s 
                           y 
                         
                         
                           2 
                            
                           
                             f 
                             d 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   59 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Because the second and third terms in the denominator of the third factor are very small with respect to unity. 
         [0062]    In  FIG. 5  option (D), step  430  corresponds to a change of waist range offset  540  and step  460  corresponds to initially estimating height  543  and calculating surface orientation  545 . To illustrate this operation mode, let z b     1   =0, z b     2   =Δz b , and κ=0. The surface orientation is related to the speckle shift through 
         [0000]    
       
         
           
             
               
                 
                   
                     α 
                     = 
                     
                       
                         s 
                         x 
                       
                        
                       
                         
                           2 
                            
                           π 
                         
                         
                           λ 
                            
                           
                               
                           
                            
                           f 
                         
                       
                        
                       
                         
                           
                             z 
                             0 
                             2 
                           
                           + 
                           
                             z 
                             r 
                             2 
                           
                         
                         
                           z 
                           r 
                         
                       
                        
                       
                         
                           
                             
                               ( 
                               
                                 
                                   z 
                                   0 
                                 
                                 - 
                                 
                                   Δ 
                                    
                                   
                                       
                                   
                                    
                                   
                                     z 
                                     b 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                           + 
                           
                             z 
                             r 
                             2 
                           
                         
                         
                           Δ 
                            
                           
                               
                           
                            
                           
                             
                               z 
                               b 
                             
                              
                             
                               ( 
                               
                                 
                                   2 
                                    
                                   
                                     z 
                                     0 
                                   
                                 
                                 - 
                                 
                                   Δ 
                                    
                                   
                                       
                                   
                                    
                                   
                                     z 
                                     b 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     β 
                     = 
                     
                       
                         s 
                         y 
                       
                        
                       
                         
                           2 
                            
                           π 
                         
                         
                           λ 
                            
                           
                               
                           
                            
                           f 
                         
                       
                        
                       
                         
                           
                             z 
                             0 
                             2 
                           
                           + 
                           
                             z 
                             r 
                             2 
                           
                         
                         
                           z 
                           r 
                         
                       
                        
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     z 
                                     0 
                                   
                                   - 
                                   
                                     Δ 
                                      
                                     
                                         
                                     
                                      
                                     
                                       z 
                                       b 
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                             + 
                             
                               z 
                               r 
                               2 
                             
                           
                           
                             Δ 
                              
                             
                                 
                             
                              
                             
                               
                                 z 
                                 b 
                               
                                
                               
                                 ( 
                                 
                                   
                                     2 
                                      
                                     
                                       z 
                                       0 
                                     
                                   
                                   - 
                                   
                                     Δ 
                                      
                                     
                                         
                                     
                                      
                                     
                                       z 
                                       b 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   60 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Since Eq. (60) has a high degree of dependence on the height, z 0 , an initial estimate of the height is needed to determine the proportionality constant. However, Eq. (58) applies here as well and the azimuth angle of the surface orientation is easily obtained without the necessity of estimating the height. One application of this option is to align a surface perpendicular to a beam. The knowledge of φ s  provides feedback as to which direction to tilt the surface. The rate of speckle shift provides feedback as to how much additional tilt is required to achieve alignment. 
         [0063]    In  FIG. 5  option (E), step  430  corresponds to changing the surface orientation  550  and step  460  corresponds to calculating the change in surface orientation  555 . This operation mode provides a method for sensing how much an object is tilting and for sensing the dynamics of motion. Let the surface orientation before and after the tilt be given by 
         [0000]      (α 1 ,β 1 )=(α,β).  (61)
 
         [0000]      (α 2 ,β 2 )=(α+Δα,β+Δβ)
 
         [0000]    The change in orientation is related to the speckle shift through 
         [0000]    
       
         
           
             
               
                 
                   
                     Δα 
                     = 
                     
                       
                         - 
                         
                           
                             s 
                             x 
                           
                           
                             2 
                              
                             
                               f 
                               d 
                             
                           
                         
                       
                        
                       
                         1 
                         
                           1 
                           - 
                           
                             
                               
                                 s 
                                 x 
                                 2 
                               
                               + 
                               
                                 s 
                                 y 
                                 2 
                               
                             
                             
                               16 
                                
                               
                                 f 
                                 d 
                                 2 
                               
                             
                           
                           - 
                           
                             
                               λ 
                               
                                 4 
                                  
                                 π 
                               
                             
                              
                             
                               
                                 z 
                                 r 
                               
                               
                                 
                                   z 
                                   0 
                                   2 
                                 
                                 + 
                                 
                                   z 
                                   r 
                                   2 
                                 
                               
                             
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     Δβ 
                     = 
                     
                       
                         - 
                         
                           
                             s 
                             y 
                           
                           
                             2 
                              
                             
                               f 
                               d 
                             
                           
                         
                       
                        
                       
                         
                           1 
                           
                             1 
                             - 
                             
                               
                                 
                                   s 
                                   x 
                                   2 
                                 
                                 + 
                                 
                                   s 
                                   y 
                                   2 
                                 
                               
                               
                                 16 
                                  
                                 
                                   f 
                                   d 
                                   2 
                                 
                               
                             
                             - 
                             
                               
                                 λ 
                                 
                                   4 
                                    
                                   π 
                                 
                               
                                
                               
                                 
                                   z 
                                   r 
                                 
                                 
                                   
                                     z 
                                     0 
                                     2 
                                   
                                   + 
                                   
                                     z 
                                     r 
                                     2 
                                   
                                 
                               
                             
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   62 
                   ) 
                 
               
             
           
         
       
     
         [0000]    These expressions are approximated well by 
         [0000]    
       
         
           
             
               
                 
                   
                     Δα 
                     = 
                     
                       - 
                       
                         
                           s 
                           x 
                         
                         
                           2 
                            
                           
                             f 
                             d 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     Δβ 
                     = 
                     
                       - 
                       
                         
                           
                             s 
                             y 
                           
                           
                             2 
                              
                             
                               f 
                               d 
                             
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   63 
                   ) 
                 
               
             
           
         
       
     
         [0064]    In  FIG. 5  option (F), step  430  corresponds to a relative lateral translation  560  between the illumination beam and the surface and step  460  corresponds to calculating the relative lateral translation  560  or the velocity associated with this translation by taking into account the time between measurement of speckle pattern  1  and speckle pattern  2 . The measurement equation associated with option (F) is simply a rearrangement of Eq. (45) for option (A) where b in Eq. (45) is interpreted as Δx and the effective waist offset is achieved by motion of the surface rather than changing the waist offset. There is a similar equation for offset in the y direction. Thus 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       Δ 
                        
                       
                           
                       
                        
                       x 
                     
                     = 
                     
                       
                         - 
                         
                           s 
                           x 
                         
                       
                        
                       
                         
                           
                             z 
                             0 
                             2 
                           
                           + 
                           
                             z 
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         [0000]    And both the magnitude and direction of the lateral translation can be determined. The direction of motion is obtained from 
         [0000]      φ m =tan −1 (Δ y/Δx )=tan −1 ( s   y   /s   x )  (65)
 
         [0000]    Determination of the magnitude of the lateral translation requires knowledge of the height, which may be known, estimated, or obtained through an auxiliary measurement. 
         [0065]      FIG. 6  describes six embodiments of an emission deviation facility that is capable of producing illumination state  1  and illumination state  2 . The emission deviation facility in  FIG. 6A  comprises a coherent radiation source  604  producing an expanding beam emanating from point  606  and incident on lens  610 . In one embodiment coherent radiation source  604  is a laser coupled into a single-mode fiber and point  606  is the tip of the fiber. Assembly  612  holds point  606  and lens  610  in a fixed position relative to each other. Assembly  612  may undergo translation  614  to move waist position  124  to produce illumination states  1  and  2 .  FIG. 6B  is an emission deviation facility comprising the additional component of an optical window  620  that may be moved into the path of the non-collimated laser beam to influence waist range offset  625 .  FIG. 6C  is an emission deviation facility that achieves a coordinated change of waist offset and illumination direction through a deflector assembly to produce region  639  of high beam overlap at plane  638 . Plane  638  is the optical conjugate plane of plane  636  where beams emanating from point  606  and point  607  have a high degree of overlap. Coherent radiation sources  604  and  605  may be derived from coherent radiation source  632  through optical switch  634  or they may be derived from different coherent radiation sources not connected to optical switch  634 .  FIG. 6D  is an emission deviation facility that achieves a coordinated change of waist offset and illumination direction by the insertion of a deflector assembly comprised of optical wedge  641  at the plane corresponding to the optical conjugate of plane  638 . Optical wedge  641  may rotate about the optical axis to a second position  642  to produce a second illumination state that has a high degree of overlap with illumination state  1 .  FIG. 6E  is an emission deviation facility that achieves a coordinated change of waist offset and illumination direction by reflecting the illumination beam with a deflector assembly comprised of mirror  655  that tilts between position  651  and position  652  to achieve a high degree of beam overlap at plane  638 .  FIG. 6F  is an emission deviation facility that produces a first illumination state with wavelength  1  and a second illumination state with wavelength  2 . Coherent radiation source  661  emits radiation at wavelength  1  and coherent radiation source  662  emits radiation at wavelength  2 . Combiner  663  combines radiation sources  661  and  662  to produce radiation source  604 . Combining may consist of switching between sources or providing both wavelengths simultaneously. 
         [0066]      FIG. 7A  is an emission deviation facility capable of producing illumination state  1  and illumination state  2  by moving radiation source  604 , which produces point  606 , to new location  605 , which produces point  607 . This emission deviation facility may also scan the beam waist to produce measurements at various locations on a surface having multiple surface elements.  FIG. 7B  is an emission deviation facility that produces highly repeatable waist range offsets  124 ,  224 , and  724  by sequentially placing optical windows  712 ,  742 , and  772  of different thicknesses into the path of the non collimated beam. In one embodiment, windows  124 ,  224 , and  724  are contained in assembly  735  which rotates different windows into place to achieve the desired waist range offset. 
         [0067]      FIG. 8A  shows an optical receiver assembly in relation to an emission deviation facility. Beam splitter  802  allows outgoing radiation to pass and reflects incoming radiation into the receiver leg. In the embodiment  153 , lens  130  focuses  812  each set of collimated rays  810  of the incoming signal onto single point  814  on detector  140 .  FIG. 8B  shows an optical receiver comprising detector  140  placed to receive incoming radiation without passing through other optics. Optional addition detectors  840  may be placed at various locations around the lens.  FIG. 8C  shows an optical receiver in relation to an emission deviation facility where lens  610  provides the function of lens  130 . In one embodiment detector  140  is located one focal length behind lens  610 . Additional detectors  840  may be placed around the outgoing beam emanating from point  606 .  FIG. 8D  shows a further embodiment of an optical receiver assembly in relation to an emission deviation facility. Minor  804  reflects incoming radiation into lens  130  and detector  140 . In one embodiment detector  140  is a focal length f  144  behind lens  130 . 
         [0068]      FIG. 9A  describes an embodiment of detector  140  consisting of a two-dimensional array of closely packed pixels. Speckle shift  906  corresponds to the shift between feature  902  of speckle pattern  1  and the same feature  904  of speckle pattern  2 .  FIG. 9B  shows a detector consisting of a linear detector array  940  of closely packed pixels. Speckle shift  906  lies substantially along the axis of linear detector array  940 .  FIG. 9C  describes an embodiment of detector  140  consisting of at least two separated detectors  925  and  926  recording time histories  927  and  928  of the speckle intensity at locations separated by the distance D  906 . Speckle feature  904  detected by detector  926  is observed by detector  925  at a later time Δt  956  corresponding to the time for speckle feature  904  to move the distance D between detectors  926  and  925 . In one embodiment detector  925  and  926  are fiber coupled through lenses  921  and  922 , which focus incoming radiation onto fiber tips  923  and  924 . 
         [0069]      FIG. 10A  describes a measurement system for producing a surface profile of measurement object  1050 . Coherent radiation source point  1006  is imaged by lens  1010  substantially onto the surface of mirror  1020 . Mirror  1020  reflects outgoing radiation into lens  610 . Lens  610  produces beam waist  124  in the vicinity of surface element  110 . Measurement object  1050  is connected to translation stage  1034 , which translates object  1050  laterally. Encoder  1034  produces a pulse train  1036  that triggers detector  140  to take images corresponding to precise increments of translation of object  1050 . The effective value of the waist offset b between system state  1  and system state  2  is the known translation increment between measurements as obtained by encoder  1034 . The height is measured at each location and the height profile of object  1050  is calculated by processor  1040 . In one embodiment, the processor  1040  is a component in a specially programmed computer having the processor in communication with a memory capable of storing and retrieving machine instructions to be executed by the processor. The specially programmed computer being able to communicate the results of executing the machine instructions. In one embodiment detector  140  is located focal length f  144  behind lens  130 . In a further embodiment detailed in  FIG. 10B , mirror  1020  pivots about the image of coherent radiation source point  1006 . A coordinated change of waist offset and illumination direction is achieved at a nominal location z c  by tilting the mirror in response to encoder pulses  1036 . The tilt of the mirror oscillates between successive trigger pulses such that illumination state  1  and illumination state  2  produce a coordinated change of waist offset and illumination direction. The amplitude of oscillation can be adjusted to produce an area of high overlap at a desired range of height values. 
         [0070]      FIG. 11  describes a measurement system that automatically produces a coordinated change of waist offset and illumination direction based on the concept of  FIG. 6E . Motor  1150  spins tilted mirror  1151  such that the beam waist follows a circular trajectory while producing a region of high overlap at a plane that is the optical conjugate of the plane where the focused beam strikes the tilted mirror. Motor  1150  is equipped with rotary encoder  1134 . Detector  140  is triggered by encoder pulses  1134  to takes images repeatably at precise points of rotation. In the example of  FIG. 11  there are six images per revolution of the mirror. A given speckle feature thus rotates through positions  1171  through  1176  on detector  140 . Speckle intensity patterns that occur from pulses that occur at diametrically opposed points of rotation are cross correlated. Cross correlation pattern composite  1180  shows the locations of correlation peaks corresponding to each cross correlation pair. The distance of each cross correlation peak from the center is a measurement of the speckle shift. The magnitude of these speckle shifts is averaged to produce an enhanced measurement. Alternatively, data is only taken at two diametrically opposed locations and processed to form an estimate of the speckle shift. The height value is calculated by processor  1040 . 
         [0071]    With respect to the above description then, it is to be realized that the optimum dimensional relationships for the parts of the invention, to include variations in size, materials, shape, form, function and manner of operation, assembly and use, are deemed readily apparent and obvious to one skilled in the art, and all equivalent relationships to those illustrated in the drawings and described in the specification are intended to be encompassed by the present invention. 
         [0072]    Therefore, the foregoing is considered as illustrative only of the principles of the invention. Further, since numerous modifications and changes will readily occur to those skilled in the art, it is not desired to limit the invention to the exact construction and operation shown and described, and accordingly, all suitable modifications and equivalents may be resorted to, falling within the scope of the invention. Although this invention has been described in the above forms with a certain degree of particularity, it is understood that the present disclosure has been made only by way of example and numerous changes in the details of construction and combination and arrangement of parts may be resorted to without departing from the spirit and scope of the invention.