Abstract:
A differential Shack-Hartmann curvature sensor for measuring a curvature including principal curvatures and directions of a wavefront. The curvature sensor includes a Shack-Hartmann sensor with an input beam and an optical element to split said input beam into three output beams traveling in different directions. Three lenslet arrays in each of the three beam paths produce corresponding Hartmann grids. A shearing device shears two of the three Hartmann grids in two perpendicular directions a differential difference. A measuring device measures the Hartmann grid coordinates generated by said three beams to determine various curvatures of the wavefront at each of the Hartmann grid points.

Description:
This application claims the benefit of priority to U.S. Provisional Patent Application No. 60/692,884 filed on Jun. 22, 2005. 

   FIELD OF THE INVENTION 
   This invention relates to sensors and, in particular, to methods, systems, apparatus and devices for a differential Shack-Hartmann curvature sensor to measure the differentials of wave front slope, i.e. wave front curvatures, to determine the wave front shape for use in active/adaptive optics, optical testing, opthalmology, telescope image analysis and atmosphere and random media characterizations. 
   BACKGROUND AND PRIOR ART 
   The Shack-Hartmann sensor is one of the most popular wavefront sensors presently available. The sensor measures the slope data of the wavefront by comparing the coordinates of Hartmann grid points from the measurement beam with coordinates from the reference beam.  FIG. 1  is a schematic layout of a prior art Shack-Hartmann sensor. Hartmann invented the Hartmann test in 1900, and Roland Shack improved this technique by introducing a lenslet array in 1971. The reference beam was generated by a pin-hole light source and injected into this system as ideal wavefront to calibrate the systematic error. In  FIG. 1 , reference beam  105  is represented by a dashed line, and the continuous line represents the measurement beam  120 . The wavefront is first collimated by the collimation lens then passes through a lenslet array  110 . The Hartmann screen divides the wavefront into many sub-apertures, then the micro-lens array  110  focus the wavefront of each sub-aperture into a group of Hartmann grid points. Comparing the coordinate differences between the measured wavefront and the reference wavefront, the wavefront slopes can be calculated according to the following formula: 
                         ∂   W       ∂   x       ⁢     |   i       =         x   i   mea     -     x   i   ref       f       ⁢     
     ⁢             ∂   W       ∂   y       ⁢     |   i       =         y   i   mea     -     y   i   ref       f       ,             (   1   )               
where (x i   ref ,y i   ref ) (i=1, 2, . . . , m, m=t×t is the total number of grid points) is the Hartmann grid coordinates of the reference beam, (x i   mea ,y i   mea ) is the Hartmann grid coordinates of the measurement beam, and f is the focal length of the lenslet array.
 
   Compared to the Hartmann test, the Shack-Hartmann wavefront sensor provides improved photon efficiency because the position of the focal spot is proportional to the average wavefront slope over each sub-aperture and the position is independent of higher-order aberrations and intensity profile variations. The Shack-Hartmann sensor is a parallel wavefront sensor operating in real time. It has application in active/adaptive optics, optical testing, opthalmology, telescope image analysis and atmosphere and random media characterizations. 
   However, the usage of the Shack-Hartmann sensor is restrictive because it requires an external reference, or a reference beam, and is sensitive to vibration, tilt and whole body movement. Therefore, a need exists for a sensor that provides the benefits of the Shack-Hartmann sensor without the limitations. 
   SUMMARY OF THE INVENTION 
   A primary objective of the invention is to provide a new method, system, apparatus and device for measuring differentials of the wave front slopes to determine the wave front curvature without the use of an external reference or reference light source after calibration. 
   A secondary objective of the invention is to provide a new method, system, apparatus and device measuring differentials of the wavefront slopes to determine the wavefront curvature independent of vibration, tilt or whole body movement so it is useful for measurements on a moving stage. 
   A third objective of the invention is to provide a new method, system, apparatus and device for a differential Shack-Hartmann curvature sensor that is scale tunable by varying differential values. 
   A fourth objective of the invention is to provide a new method, system, apparatus and device for a differential Shack-Hartmann curvature sensor that can be applied in optical testing with vibration. 
   A fifth objective of the invention is to provide a new method, system, apparatus and device for a differential Shack-Hartmann curvature sensor without moving parts for increased reliability. 
   A sixth objective of the invention is to provide a new method, system, apparatus and device for a differential Shack-Hartmann curvature sensor for use in optical testing, active and adaptive optics, shape-extraction in bio-optics and opthalmology, such as corneal measurement. 
   A seventh objective of the invention is to provide a new method, system, apparatus and device for a differential Shack-Hartmann curvature sensor for measuring the normal curvatures and the twist curvature terms. 
   An eighth objective of the invention is to provide a new method, system, apparatus and device for a differential Shack-Hartmann curvature sensor for measuring the wave front principal curvatures and directions. 
   A first preferred embodiment of the invention provides a method, system and apparatus for providing a differential Shack-Hartmann curvature sensor for measuring the local curvatures of a wavefront. The differential Shack-Hartmann curvature sensor includes a Shack-Hartmann sensor with an output beam and an optical element to split said output beam into three beams traveling in three different directions. Three lenslet arrays mounted in the paths of the three beams generate three corresponding Hartmann grids. A shearing device makes two of the three lenslet arrays shear a differential difference in two perpendicular directions, respectively, comparing to the third lenslet array. A measuring device measures the Hartmann grid coordinates generated by said three lenslet arrays to determine the curvature of the wavefront at each of said Hartmann grid point. 
   In the second and third embodiment, wavefront curvature is measured by using a Shack-Hartmann sensing system that has three output beams. Two of the three output beams are sheared in two perpendicular directions comparing to the third one with their corresponding two lenslet arrays conjugated to the third one, and the Hartmann grids are differentially displaced. Differentials of plural wavefront slopes at said plural Hartmann grid points of the said third lenslet array are measured, and the curvatures of the wavefront at the plural Hartmann grid points are obtained. 
   Further objects and advantages of this invention will be apparent from the following detailed description of preferred embodiments, which are illustrated schematically in the accompanying drawings. 

   
     BRIEF DESCRIPTION OF THE FIGURES 
       FIG. 1  shows a prior art Hartmann screen and lenslet array and their coordinate differences. 
       FIG. 2  shows an example of a prior art Hartmann grid. 
       FIG. 3  shows a Shack-Hartmann grid with displacements in the x and the y directions. 
       FIG. 4   a  shows a direct implementation layout of the differential Shack-Hartmann curvature sensor according to an embodiment of the present invention. 
       FIG. 4   b  shows another direct implementation layout of the differential Shack-Hartmann curvature sensor according to another embodiment of the present invention. 
       FIG. 4   c  shows yet another direct implementation of the differential Shack-Hartmann curvature sensor according to another embodiment of the present invention. 
       FIG. 5  shows a calibration block diagram according to the present invention. 
       FIG. 6  shows an experimental system for the differential Shack-Hartmann curvature sensor according to the first embodiment of the present invention. 
       FIG. 7   a  shows the image space of Roddier&#39;s curvature sensing technique of the prior art. 
       FIG. 7   b  shows the object space of Roddier&#39;s curvature sensing corresponding to the image space of  FIG. 7   a.    
   

   DESCRIPTION OF THE PREFERRED EMBODIMENTS 
   Before explaining the disclosed embodiments of the present invention in details it is to be understood that the invention is not limited in its application to the details of the particular arrangements shown since the invention is capable of other embodiments. Also, the terminology used herein is for the purpose of description and not of limitation. 
   The following is a list of the reference numbers used in the drawings and the detailed specification to identify components: 
   
     
       
             
             
           
         
             
                 
             
           
           
             
               100 
               Shack-Hartmann sensor 
             
             
               105 
               reference beam 
             
             
               110 
               Hartmann screen and lenslet 
             
             
                 
               array 
             
             
               120 
               measurement beam 
             
             
               200 
               Hartmann grid 
             
             
               310 
               Hartmann grid 
             
             
               320 
               horizontal displacement (s x ) 
             
             
               330 
               vertical displacement (s y ) 
             
             
               400 
               differential Shack-Hartmann 
             
             
                 
               curvature sensor 
             
             
               402 
               first lenslet array 
             
             
               404 
               second lenslet array 
             
             
               406 
               third lenslet array 
             
             
               410 
               beam splitters 
             
             
               412 
               first camera 
             
             
               414 
               second camera 
             
             
               416 
               third camera 
             
             
               417 
               lenslet shearing device (x) 
             
             
               418 
               lenslet shearing device (y) 
             
             
               610 
               first light source 
             
             
               620 
               second light source 
             
             
                 
             
           
        
       
     
   
   The method, system, apparatus and device of the present invention provides a Shack-Hartmann curvature sensor that shears the Shack-Hartmann grid, horizontally and vertically, as shown in  FIG. 3  to measure the wavefront curvatures at each Hartmann grid point. The curvature is the derivative of slopes. The differential Shack-Hartmann curvature sensor of the present invention is based on the prior art Shack-Hartmann sensor and the Hartmann grid shown in  FIGS. 1 and 2 , respectively. By measuring the differentials of the wavefront slopes at each Hartmann grid point, the wavefront local curvatures (normal curvatures and the twist term curvature) are determined, from which the principal curvatures and directions of the wave front are computed. 
   A. Normal Curvature Measurements 
   Differential displacements are made in the x and y directions to obtain slope differentials of the wavefront, the wavefront curvatures. The curvature sensor of the present invention is implemented as a Shack-Hartmann sensor with three output channels for achieving the Hartmann grid shearing in two perpendicular directions. The slope is measured in the three channels as well as the curvature of the wavefront at each Hartmann grid point. 
   The Hartmann grid is shifted a lateral differential distance in the x and in the y direction, which are suggested in this invention to adopt 1/10 to ½ of the pitch size of the Hartmann grid according to each specific application. As shown in  FIG. 3 , for example, the slopes at each point are measured before and after the differential shift to obtain the slope differentials, the curvature in the x direction and in the y direction can be obtained by applying 
                           c   x     ⁡     (   i   )       =       ⁢             ∂   2     ⁢   W       ∂     x   2         ⁢     |   i       =         1     s   x       ⁢     (         ∂   W       ∂   x       ⁢     |     i   ′       ⁢     -       ∂   W       ∂   x         ⁢     |   i       )       =         1   f     ⁢     (         x   i   ′mea     -     x   i   mea         s   x       )       -       c     0   ,   x       ⁡     (   i   )                               c   y     ⁡     (   i   )       =       ⁢             ∂   2     ⁢   W       ∂     y   2         ⁢     |   i       =         1     s   y       ⁢     (         ∂   W       ∂   y       ⁢     |     i   ′       ⁢     -       ∂   W       ∂   y         ⁢     |   i       )       =         1   f     ⁢     (         y   i   ″mea     -     y   i   mea         s   y       )       -       c     0   ,   y       ⁡     (   i   )               ,                 (   2   )               
where c 0,x (i) and c 0,y (i) are obtained by
 
                           c     0   ,   x       ⁡     (   i   )       =       ⁢       1   f     ⁢     (         x   i   ′Ref     -     x   i   Ref         s   x       )                         c     0   ,   y       ⁡     (   i   )       =       ⁢       1   f     ⁢     (         y   i   ″Ref     -     y   i   Ref         s   y       )         ,                 (   3   )               
and in this paper s x  and s y  are the differential shifts in the x- and y-directions, respectively. Theoretically c 0,x (i) and c 0,y (i) are “1/f”, because the lenslet array is moved a lateral distance s x  in x-direction, for example, the reference Shack-Hartmann grid moves exactly the same distance s x  accordingly, therefore, x i ′ Ref −x i   Ref =s x  and c 0,x (i)=1/f; similarly, c 0,y (i)=1/f. However, in practice c 0,x (i) and c 0,y (i) are not “1/f”, because the coordinates are measured by different CCD cameras, so they belong to different coordinate system and therefore c 0,x (i) and c 0,y (i) are obtained by calibration.
 
   The slope differentials in the cross directions, referred to as the corresponding twist curvatures, are given by 
                           c   xy     ⁡     (   i   )       =       ⁢             ∂   2     ⁢   W         ∂   x     ⁢     ∂   y         ⁢     |   i       =         1     s   y       ⁢     (         ∂   W       ∂   x       ⁢     |     i   ′       ⁢     -       ∂   W       ∂   x         ⁢     |   i       )       =         1   f     ⁢     (         x   i   ″mea     -     x   i   mea         s   x       )       -       c     0   ,   xy       ⁡     (   i   )                               c   yx     ⁡     (   i   )       =       ⁢             ∂   2     ⁢   W         ∂   y     ⁢     ∂   x         ⁢     |   i       =         1     s   x       ⁢     (         ∂   W       ∂   y       ⁢     |     i   ′       ⁢     -       ∂   W       ∂   y         ⁢     |   i       )       =         1   f     ⁢     (         y   i   ′mea     -     y   i   mea         s   x       )       -       c     0   ,   yx       ⁡     (   i   )               ,                 (   4   )               
where c 0,yx (i) and c 0,xy (i) are constants given by
 
                           c     0   ,   yx       ⁡     (   i   )       =       ⁢       1   f     ⁢     (         y   i   ′Ref     -     y   i   Ref         s   x       )                       c     0   ,   xy       ⁡     (   i   )       =       ⁢       1   f     ⁢       (         x   i   ″Ref     -     x   i   Ref         s   y       )     .                     (   5   )               
In Equations (2) and (4), (x i   mea ,y i   mea ), (x i ′ mea ,y i ′ mea ) and (x i   n     mea   ,y i   n     mea   ) (i=1, 2, . . . m) are the coordinates of the measured original Hartmann grid points, the measured sheared Hartmann grid points in the x-direction and in y-direction, respectively. When a differential shift s x  is made in x-direction, the y-coordinates remains unchanged, so c 0,yx (i)=0; similarly, c 0,xy (i)=0. However, in practice, these two constants may not be zero as the coordinates are measured by three CCD cameras, and they are usually obtained by calibration.
 
   If the wavefront travels in the z direction, and the beam is split into three parts traveling in three different directions, one beam travels in the z-direction, another beam travels in the x-direction, and the third beam travels in the y-direction. For example, three lenslet arrays  402 ,  404  and  406 , are placed in the three beams such that  402  is in the z-direction beam,  404  is in the x-direction beam, and  406  is in the y-direction beam as shown in  FIG. 4   a . Lenslet  404  is conjugated to  402 , and  406  is conjugated to  402 , separately. Using a micro-screw, lenslet array  404  is moved a differential distance in the x direction, and using a second micro-screw, lenslet array  406  is moved a differential distance value in the y direction. Thus, the lenslet  404  and lenslet  402  are shearing each other in the x direction, and lenslet  406  and lenslet  402  are shearing each other in the y direction. By measuring the coordinates of Hartmann grid points generated by the three beams, and applying Equations (2) and (4), the wavefront curvatures in the x and y directions and the twist terms are obtained. In an example, the Hartmann grid coordinates are measured by placing a charge-coupled device (CCD) camera in the path of each of the three beams for recording the Hartmann grid points. 
   Examples of alternative direct implementation layouts are shown in  FIGS. 4   b  and  4   c . As shown in  FIG. 4   b , optical parallel plates  410  are used as shearing elements to make differential displacements between the lenslet arrays. In the embodiment shown in  FIG. 4   c , the optical parallel plates  410  also act as beam splitters. 
   The differential Shack-Hartmann curvature sensor  400  of the present invention uses three lenslet arrays  402 ,  404 ,  406  in three channels and three CCD cameras  412 ,  414 ,  416  to record the coordinates of the each Hartmann grid points as shown in  FIG. 4   a . Since the lenslet  402 ,  404 , and  406  are recorded by different CCD cameras, a reference light beam is required to calibrate the system to obtain the unknown constants in Equation (2) and Equation (4). 
   The calibration diagram is shown in  FIG. 5 . First, it is necessary to make sure that lenslets  404  and  406  are conjugate to lenslet  402  and that the shearing differential values in the x and y directions are s x  and s y . A reference beam is introduced by an ideal flat mirror, and the coordinates of the grid points of the three Hartmann grids are recorded for use as references to compute the constant values c 0,x (i), c 0,y (i), c 0,yx (i) and c 0,xy (i) with Equations (3) and (5). 
   B. Principal Curvature Computations 
   The normal curvature is the change of the surface normal in an osculating plane, and the principal curvatures of a non-umbilical point are the maximum and minimum values of normal curvatures, say κ 1  and κ 2 , in two perpendicular directions. Regarding a local patch of surface, the principal curvatures are invariants, which are insensitive to the surface orientation. In order to evaluate the principal curvatures, it is assumed that the neighborhood of a Hartmann grid point is represented by a “Monge patch” of the form:
 
 X=x{right arrow over (e)}   1   +y{right arrow over (e)}   2   +W ( x,y ) {right arrow over (e)}   3 ,  (6)
 
where ({right arrow over (e)} 1 , {right arrow over (e)} 2 , {right arrow over (e)} 3 ) is an orthogonal frame in Euclidean 3-space. Then the second fundamental form has a matrix to describe the local surface shape as
 
                   II   =     (             ω   ^     1   13             ω   ^     1   23                 ω   ^     2   13             ω   ^     2   23           )       ,           (   7   )               
where {circumflex over (ω)} j   i3  (i=1,2; j=1,2.) defines the component in {right arrow over (e)} i  of the turning rate of the normal as the frame moves across the given point along {right arrow over (e)} j . For a wave front traveling in the z-direction ({right arrow over (e)} 3 ), W(x,y) is the “height” as a function of x and y in the pupil plane. Then at each “Monge patch”, the matrix II becomes
 
                   II   =     (             c   x     ⁡     (   i   )               c   yx     ⁡     (   i   )                   c   xy     ⁡     (   i   )               c   y     ⁡     (   i   )             )       ,     i   =   1     ,   2   ,     ⋯   ⁢           ⁢   m     ,           (   8   )               
where the diagonal terms c x (i) and c y (i) are the wave front normal curvatures in the x-direction and in the y-directions, i.e.
 
                 c   x     ⁡     (   i   )       =             ∂   2     ⁢   W       ∂     x   2         ⁢     (   i   )     ⁢           ⁢   and   ⁢           ⁢       c   y     ⁡     (   i   )         =           ∂   2     ⁢   W       ∂     y   2         ⁢     (   i   )           ;         
the off-diagonal terms c xy (i) and c yx (i) are the corresponding twist curvature terms, i.e.
 
               c   xy     ⁡     (   i   )       =             ∂   2     ⁢   w         ∂   x     ⁢     ∂   y         ⁢     (   i   )     ⁢           ⁢   and   ⁢           ⁢       c   yx     ⁡     (   i   )         =           ∂   2     ⁢   w         ∂   y     ⁢     ∂   x         ⁢       (   i   )     .               
It is assumed that c xy (i)=c yx (i).
 
   The determinant of matrix II, denoted as K, is known as the Gaussian curvature. The trace of the matrix II, denoted as 2H, is known as the mean curvature. Both Gaussian curvature and mean curvature are algebraic invariants, which do not change with rotation of the orthogonal frame ({right arrow over (e)} 1 , {right arrow over (e)} 2 , {right arrow over (e)} 3 ) about the normal. 
   By diagonalizing the matrix II to rotate the orthogonal frame about {right arrow over (e)} 3 , the off-diagonal terms disappear, and a new matrix II′ is obtained by
 
II′=P T IIP,  (9)
 
where P is an orthogonal matrix defined by
 
                   P   =     [           cos   ⁢           ⁢   θ             -   sin     ⁢           ⁢   θ               sin   ⁢           ⁢   θ           cos   ⁢           ⁢   θ           ]       ,           (   10   )               
where angle θ is defined as the frame rotation angle. The new matrix II′ is a diagonal matrix, which is
 
                     II   ′     =     (             κ   1     ⁡     (   i   )           0           0           κ   2     ⁡     (   i   )             )       ,           (   11   )               
where κ 1 (i) and κ 2 (i) (κ 1 (i)&gt;κ 2 (i)) are the eigenvalues of the matrix II, also known as the first and second principal curvatures that define the maximum and minimum normal curvatures at a given point patch, and i=1, 2, . . . , m, where m is the total number of Shack-Hartmann grid points. Plug the Equations (8), (10) and (11) into Equation (9), to obtain the principal curvatures κ 1  and κ 2  at each grid point as
 
                       κ     1   ,   2       ⁡     (   i   )       =           c   x     ⁡     (   i   )       +         c   y     ⁡     (   i   )       ±           (         c   x     ⁡     (   i   )       -       c   y     ⁡     (   i   )         )     2     +     4   ⁢         c   xy     ⁡     (   i   )       2               2       ,           (   12   )               
and the rotation angle θ is the angle between the first principal curvature and the x-direction, which can be obtained by
 
                   θ   ⁡     (   i   )       =       1   2     ⁢         tan     -   1       ⁡     (       2   ⁢       c   xy     ⁡     (   i   )               c   x     ⁡     (   i   )       -       c   y     ⁡     (   i   )           )       .               (   13   )               
The principal curvatures can also be computed by evaluating the eigenvalues of matrix II with its characteristic equation as
   det (κ I−II )=0,  (14) 
the result is the same as Equation (12). The rotation angle θ can also be computed with Euler&#39;s formula (1760) by
 
                     cos   ⁢           ⁢     2   ⁡     [     θ   ⁡     (   i   )       ]         =         2   ⁢       c   x     ⁡     (   i   )         -     2   ⁢     H   ⁡     (   i   )                 κ   1     ⁡     (   i   )       -       κ   2     ⁡     (   i   )             ,           (   15   )               
where H is the mean curvature given by
   H ( i )=( c   x ( i )+ c   y ( i ))/2,  (16) 
Then the angle θ is given by
 
                   θ   ⁡     (   i   )       =       1   2     ⁢         cos     -   1       ⁡     (           c   x     ⁡     (   i   )       -       c   y     ⁡     (   i   )               κ   1     ⁡     (   i   )       -       κ   2     ⁡     (   i   )           )       .               (   17   )               
Apply Equation (12) into Equation (17), to obtain
 
                     θ   ⁡     (   i   )       =       1   2     ⁢       cos     -   1       ⁡     (                c   x     ⁡     (   i   )       -       c   y     ⁡     (   i   )                      (         c   x     ⁡     (   i   )       -       c   y     ⁡     (   i   )         )     2     +     4   ⁢         c   xy     ⁡     (   i   )       2             )           ,           (   18   )               
which is equivalent to Equation (13).
 
C. Comparisons with the Previous Arts
 
   With the system calibrated, the discrepancies between the image de-magnifying systems are cancelled, the discrepancies between the two arms in the cube beam splitter are cancelled, and the aberrations in the collimator and the cube prisms are also cancelled. Besides the error from the ideal flat mirror used for introducing the reference beam, which can be very small, and the nominal errors in applying the shearing differential values in Equations (2) and (3), the remaining error sources are the discrepancies between lenslets  402 ,  404  and  406 , which are negligible for high quality micro-lenslet arrays. 
   As shown in the configuration in  FIG. 6 , a point light source  610  is used to generate the beam for measurement and the point light source  620  is used to generate the beam for calibration. After calibration, no reference light beam is necessary. 
   The following description compares the differential Shack-Hartmann curvature sensor with a prior art curvature sensor. In 1988, Francois Roddier proposed a method to measure the local curvature of the wavefront surface by measuring the difference in illumination of the two planes before and after the focal point as shown in  FIG. 7 . The Roddier&#39; curvature sensor is based on the irradiance Poisson equation, which is derived from Teague&#39;s irradiance transport equation. Roddier obtained 
                       I   1     -     I   2           I   1     +     I   2         =       (           ∂   W       ∂   n       ⁢     δ   c       -     P   ⁢       ∇   2     ⁢   W         )     ⁢   Δ   ⁢           ⁢   z             (   5   )               
where Δz is the distance from the pupil plane of the defocused plane P 1  or P 2  viewed from the object space. A plane at a distance from the focus is conjugated to a plane at a distance Δz from the pupil. Roddier proved that
 
                   Δ   ⁢           ⁢   z     =       f   ⁡     (     f   -   l     )       l             (   5   )               
So Roddier obtained the well-known equation
 
   
     
       
         
           
             
               
                 
                   
                     
                       I 
                       1 
                     
                     - 
                     
                       I 
                       2 
                     
                   
                   
                     
                       I 
                       1 
                     
                     + 
                     
                       I 
                       2 
                     
                   
                 
                 = 
                 
                   
                     
                       f 
                       ⁡ 
                       
                         ( 
                         
                           f 
                           - 
                           l 
                         
                         ) 
                       
                     
                     l 
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         
                           
                             ∂ 
                             W 
                           
                           
                             ∂ 
                             n 
                           
                         
                         ⁢ 
                         
                           δ 
                           c 
                         
                       
                       - 
                       
                         P 
                         ⁢ 
                         
                           
                             ∇ 
                             2 
                           
                           ⁢ 
                           W 
                         
                       
                     
                     ) 
                   
                 
               
             
             
               
                 ( 
                 6 
                 ) 
               
             
           
         
       
     
   
   Curvature sensing is a technique used typically in adaptive optics to measure the Laplacian of the wavefront by subtracting intensity profiles from an exact distance before and after the focus of a lens. The idea of the Roddier&#39;s curvature sensor is that the normalized differential intensity change along the optical axis provides the information of the local Laplacian curvature of the wavefront. 
   For adaptive optics systems, the image before and after the focus is usually switched mechanically, making the systems quite noisy during operation. In contrast, the Differential Shack-Hartmann curvature sensor contains no moving parts making it more reliable. 
   Interferometry is a technique to make the wavefront to interfere with itself or an ideal wavefront. It is especially good for measuring high spatial frequency aberrations and low amplitude aberrations. But air motion and mechanical vibrations make obtaining an image with an interferometer difficult, especially for testing large optics. Sophisticated software is necessary to extract meaningful and accurate information from interferograms. Commercial interferometers are typically expensive, where a high quality Shack-Hartmann wavefront sensor is typically much less expensive. 
   Foucault knife-edge testing involves moving a knife-edge through the focus of a beam and observing the intensity pattern on a screen. Like interferometry, knife-edge testing allows high-spatial frequency aberrations to be observed. But it requires very accurate alignment of the knife-edge to the beam focus, and it is qualitative test. 
   In summary, the Differential Shack-Hartmann Curvature sensor shares the important features of the Shack-Hartmann sensor, such as it is a real-time wavefront measurement, measurements are inherently two-dimensional and parallel, it is independent of higher-order aberrations and intensity profile variations, has good photon efficiency and is good for the all wavelength bands. 
   The Differential Shack-Hartmann Curvature Sensor also provides some unique features such as eliminating the need for external references after calibration, the sensor is independent of vibrations, tilt and whole body movements, which makes it a good choice for measurements with moving objects, and is scale tunable by changing differential values. 
   While the invention has been described, disclosed, illustrated and shown in various terms of certain embodiments or modifications which it has presumed in practice, the scope of the invention is not intended to be, nor should it be deemed to be, limited thereby and such other modifications or embodiments as may be suggested by the teachings herein are particularly reserved especially as they fall within the breadth and scope of the claims here appended.