Abstract:
System and method for estimating and correcting an aberration of an optical system. The method includes capturing a first plurality of images on a first plurality of planes. The first plurality of images is formed by at least the optical system. Additionally, the method includes processing at least information associated with the first plurality of images, and determining a first auxiliary function based upon at least the information associated with the first plurality of images. The first auxiliary function represents a first aberration of the optical system. Moreover, the method includes adjusting the optical system based upon at least information associated with the first auxiliary function.

Description:
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     BACKGROUND OF THE INVENTION 
     The present invention relates generally to optics. More particularly, the invention provides techniques for correcting optical aberrations. Merely by way of example, the invention has been applied to optical mirrors, but it would be recognized that the invention has a much broader range of applicability. 
     Optical system has been widely used for detecting images of various targets. The optical system usually introduces discrepancies to the images. The discrepancies including phase errors result from various sources, such as aberrations associated with individual segments of optical system including optical mirrors and discrepancies between input and output of optical system. These errors often need to be estimated and corrected in order to improve image quality. For example, a space telescope such as the James Webb Space Telescope may have large phase errors after its deployment, and these aberrations often need to be corrected with the telescope remaining in space. 
     In order to correct the optical aberrations, a Green&#39;s function approach has been proposed. This method derives the transport of intensity equation and solves for the auxiliary function. In other words, the Green&#39;s function approach uses known phase or phase gradient at the boundary of optical aperture of the optical system and determines the phase map of the entire optical aperture. Applied to an astronomical telescope, this method measures irradiance on either side of telescope focus and radial gradient of wavefront at the edge of telescope aperture. Irradiance measurements do not need to be performed on planes symmetrically located on either side of telescope focus. Consequently, a Poisson equation is solved to obtain the wavefront error in the interior of the telescope aperture. 
     When the wavefront error of an aperture is large, the Green&#39;s function approach usually cannot effectively sample the entire optical aperture. Instead, the optical aperture is usually divided into several sub-apertures, and phases within each sub-aperture are measured. Phase errors in each sub-aperture are then determined and corrected. Afterwards, sizes of sub-apertures are increased, and phase errors within enlarged sub-apertures are further corrected. Through iterations, phase errors within the aperture become so small that the entire aperture may be sampled. This iterative sub-aperture approach requires additional masks and setups, and may even require several iterative corrections at each sub-aperture size. Hence this method is costly and time consuming. 
     In addition, the above method sometimes uses curvature-based wavefront sensing. This sensing technique requires information about radial derivative of phase at the boundary of optical aperture. For large mirrors with several segments, a large number of boundary radial derivatives need to be determined. Hence this method may be cumbersome. 
       FIG. 1  is a simplified diagram illustrating technique for phase error correction. The correction method includes at least five processes: secondary mirror alignment process  110 , coarse tilt adjustment process  120 , coarse petal figuring process  130 , inter-petal phasing process  140 , tilt/figure refinement process  150 , and full aperture figuring process  160 . Inter-petal phasing process  140  and tilt/figure refinement process  150  may be performed iteratively. As shown in  FIG. 1 , processes  110 ,  120 ,  130 , and  140  use different pupil plane masks  112 ,  122 ,  132 ,  142 , and  152  respectively. In addition, processes  110 ,  130 ,  150 , and  160  use additional hardware. For example, process  110  uses Phase Diverse Phase Retrieval (“PDPR”) plates  114 , process  130  uses fine steering mirror  134 , process  150  uses PDPR plates  154  and fine steering mirror  155 , and process  160  uses PDPR plates  164 . At secondary mirror alignment process  110 , point source functions (“PSFs”) in focal plane and defocus planes are measured, and sharpness maximization and PDPR analysis are performed. At coarse tilt adjustment process  120 , PSFs for each petal is measured, and centroid analysis is performed. At coarse petal figuring process  130 , PSFs for each sub-aperture is measured, and analysis based on PSF maximization algorithm is performed. At inter-petal phasing process  140 , grism fringes are measured, and fringe analysis is performed. At tilt/figure refinement process  150 , PSFs for each petal in focal plane and defocus planes are measured, and centroid analysis and PDPR analysis are performed. At full aperture figuring process  160 , PSFs for entire aperture in focal plane and defocus planes are measured, and PDPR analysis is performed. 
     Hence it is desirable to simplify and improve phase correction technique. 
     BRIEF SUMMARY OF THE INVENTION 
     The present invention relates generally to optics. More particularly, the invention provides techniques for correcting optical aberrations. Merely by way of example, the invention has been applied to optical mirrors, but it would be recognized that the invention has a much broader range of applicability. 
     According to one embodiment of the present invention, a method for estimating and correcting an aberration of an optical system includes capturing a first plurality of images on a first plurality of planes. The first plurality of images is formed by at least the optical system. Additionally, the method includes processing at least information associated with the first plurality of images, and determining a first auxiliary function based upon at least the information associated with the first plurality of images. The first auxiliary function represents a first aberration of the optical system. Moreover, the method includes adjusting the optical system based upon at least information associated with the first auxiliary function. 
     According to another embodiment of the present invention, a method for estimating and correcting an aberration of an optical system includes capturing a first plurality of images on a first plurality of planes. The first plurality of images is formed by at least the optical system. Additionally, the method includes processing at least information associated with the first plurality of images, and determining a first auxiliary function based upon at least the information associated with the first plurality of images. The first auxiliary function represents a first aberration of the optical system. Moreover, the method includes adjusting the optical system based upon at least information associated with the first auxiliary function. The capturing, the processing, the determining, and the adjusting are free from dividing an aperture of the optical system into a plurality of sub-apertures, estimating an aberration for each sub-aperture, or reducing the aberration for each sub-aperture. 
     According to yet another embodiment of the present invention, a method for estimating and correcting an aberration of an optical system includes capturing a plurality of images on a plurality of planes. The plurality of images is formed by at least the optical system. Additionally, the method includes measuring a plurality of intensities for each of the plurality of images. The plurality of intensities corresponds to a plurality of locations on each of the plurality of planes respectively. Moreover, the method includes obtaining a plurality of derivatives of intensity with respect to an optical axis of the optical system using at least information associated with the plurality of intensities. The plurality of derivatives corresponds to the plurality of locations on a focal plane of the optical system. Also, the method includes determining a first auxiliary function based upon at least information associated with the plurality of derivatives. The first auxiliary function represents an aberration of the optical system. 
     According to yet another embodiment of the present invention, a system for estimating and correcting an aberration of an optical system includes a testing system, a control system connected to the testing system, and an adjustment system connected to the testing system and to the control system. The testing system and the control system are configured to capture a plurality of images on a plurality of planes. The plurality of images is formed by at least the optical system. The control system is configured to process at least information associated with the plurality of images and determine an auxiliary function based upon at least the information associated with the plurality of images. The first auxiliary function represents a first aberration of the optical system. The adjustment system and the control system are configured to adjust the optical system based upon at least information associated with the auxiliary function. 
     The techniques of the present invention have numerous advantages. Certain embodiments of the present invention can sense and correct aberrations on the entire aperture of an optical system without dividing the aperture into sub-apertures. The amount of time required for aberration reduction may be shortened. Some embodiments of the present invention work for segmented apertures. Certain embodiments of the present invention can improve aberration reduction by iterations. The iterative process alleviates convergence problem encountered by conventional techniques. Some embodiments of the present invention can simplify hardware requirements for aberration reduction, such as hardware requirements for coarse alignments of large telescopes. Certain embodiments of the present invention do not use the Pseudo-Hartmann mask, which is often used by conventional techniques for coarse figuring. Conventional techniques for coarse figuring often require Pseudo-Hartmann masks, each of which is made up of sets of several multi-faceted prisms. Fabrication of the masks is difficult, time consuming and costly. Therefore, certain embodiments of the present invention can lower the cost and shorten the time for aberration reduction. Some embodiments of the present invention usually can be implemented with minimum computation time. 
     Depending upon the embodiment under consideration, one or more of these benefits may be achieved. These benefits and various additional objects, features and advantages of the present invention can be fully appreciated with reference to the detailed description and accompanying drawings that follow. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a simplified diagram illustrating technique for phase error correction. 
         FIG. 2  is a simplified block diagram for correcting optical aberrations according to one embodiment of the present invention. 
         FIG. 3  illustrates a simplified geometry for process of intensity measurement according to one embodiment of the present invention. 
         FIGS. 4A through 4D  illustrate measured image intensities on different planes with aberrations on mirror surface. 
         FIGS. 5A through 5D  illustrate measured image intensities on different planes with other aberrations on mirror surface. 
         FIGS. 6A through 6D  illustrate measured image intensities on different planes with yet other aberrations on mirror surface. 
         FIG. 7  is a simplified system for estimation and correction of aberrations according to one embodiment of the present invention. 
         FIG. 8  is a simplified system for estimation and correction of aberrations according to another embodiment of the present invention. 
         FIG. 9  is a simplified system for estimation and correction of large aberrations according to yet another embodiment of the present invention. 
         FIGS. 10A through 10C  show simplified experimental results according to yet another embodiment of the present invention. 
         FIG. 11  shows actuator commands for each iteration of aberration reduction process. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     The present invention relates generally to optics. More particularly, the invention provides techniques for correcting optical aberrations. Merely by way of example, the invention has been applied to optical mirrors, but it would be recognized that the invention has a much broader range of applicability. 
       FIG. 2  is a simplified block diagram for correcting optical aberrations according to one embodiment of the present invention. This diagram is merely an example, which should not unduly limit the scope of the claims. One of ordinary skill in the art would recognize many variations, alternatives, and modifications. The method of correcting optical aberrations includes process  210  for intensity measurement, process  220  for derivative estimation, process  230  for aberration determination, process  240  for aberration reduction, and process  250  for additional measurement determination. Although the above has been shown using a selected sequence of processes, there can be many alternatives, modifications, and variations. For example, some of the processes may be expanded and/or combined. For example, process  220  of derivative estimation and process  230  of aberration determination may be combined. Other processes may be inserted to those noted above. For example, conventional phase diversity process for aberration reduction may be used in combination with processes  210 ,  220 ,  230 ,  240 , and  250 . Depending upon the embodiment, the specific sequences of steps may be interchanged with others replaced. Process  240  for aberration reduction is optional and may be skipped under certain conditions. Further details of these processes are found throughout the present specification and more particularly below. 
     At process  210  of intensity measurement, optical images are formed on various planes and image intensities are measured. The planes may be located on either side of focal plane or optionally coincide with the focal plane. If aberrations of the optical system do not change image intensities on a certain plane, image intensities on this plane do not need to be measured. The skipped plane may be the focal plane or a defocus plane of the optical system. The optical system may be a telescope, a mirror, or any system with an optical aperture. Measured image intensities describe intensity as a function of location on respective planes. 
       FIG. 3  illustrates a simplified geometry for process  210  of intensity measurement according to one embodiment of the present invention. One of ordinary skill in the art would recognize many variations, alternatives, and modifications. As shown in  FIG. 3 , optical system  310  with optical aberrations has an optical axis z. Focal plane  320  of optical system  310  is located at z equal to zero. Positions of defocus planes are measured by z values. Z value is larger than zero for defocus planes, such as plane  330 , located on the right side of focal plane  320 , as shown in  FIG. 3 . Similarly, z value is smaller than zero for defocus planes, such as plane  340 , located to the left side of focal plane  320 , as shown in  FIG. 3 . Locations on each plane are measured by x and y values. Hence measured intensities depend on x, y, and z. More specifically, measured intensities may include I measure (x,y,z 1 ), I measure (x,y,z 2 ), . . . , I measure (x,y,z n ), . . . , I measure (x,y,z N ), where N is a positive integer representing the number of different planes on which image intensities are measured. For example, N may be equal to 2, 3, 100, or any other positive integer. 
     At process  220  of derivative estimation, the derivative of measured intensities taken along the z axis at z equal to zero is estimated as shown below. 
     
       
         
           
             
               
                 
                   
                     D 
                     ⁡ 
                     
                       ( 
                       
                         x 
                         , 
                         y 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ∂ 
                         
                           I 
                           ⁡ 
                           
                             ( 
                             
                               x 
                               , 
                               y 
                               , 
                               z 
                             
                             ) 
                           
                         
                       
                       
                         ∂ 
                         z 
                       
                     
                     ⁢ 
                     
                       ❘ 
                       
                         z 
                         = 
                         0 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1 
                   
                   ) 
                 
               
             
           
         
       
     
     where I(x,y,z) is image intensity as a function of x, y, and z. D(x,y) is the derivative of intensity taken along the z axis at z equal to zero. z equal to zero corresponds to location of the focal plane, so D(x,y) is effectively the derivative of I(x,y,z) along the z direction on the focal plane. 
     D(x,y) may be estimated with various methods. For example, D(x,y) may be obtained if I(x,y,z) is obtained within at least the vicinity of the focal plane, i.e., −a&lt;z&lt;b, where a and b is larger than or equal to zero. Preferably a and b are both larger than zero. I(x,y,z) may be estimated by fitting measured intensities on various planes to a function. The measured intensities includes I measure (x,y,z 1 ), I measure (x,y,z 2 ), . . . , I measure (x,y,z n ), I measure (x,y,z N ). The function that can describe I(x,y,z) in the vicinity of the focal plane may include at least 
                 ∑     m   =   0     M     ⁢         a   m     ⁡     (     x   ,   y     )       ⁢     z   m         ,         
where M is an arbitrary positive integer. a m (x,y) varies with x and y but is independent of z. For example,
 
when  M= 1 , I ( x,y,z )= a   0 ( x,y )+ a   1 ( x,y )× z   (Equation 2)
 
when  M= 2 , I ( x,y,z )= a   0 ( x,y )+ a   1 ( x,y )× z+a   2 ( x,y )× z   2   (Equation 3)
 
when  M= 3 , I ( x,y,z )= a   0 ( x,y )+ a   1 ( x,y )× z+a   2 ( x,y )× z   2   +a   3 ( x,y )× z   3   (Equation 4)
 
     Magnitude of M determines the minimum number of different planes on which image intensities need to be measured at process  210  of intensity measurement. N usually needs to be larger than M. Regardless of magnitude of M, a m (x,y) is usually estimated with measured intensities such as I measure (x,y,z), I measure (x,y,z 2 ), . . . , I measure (x,y,z n ), . . . , I measure (x,y,z N ). 
     Coefficients of a fitting function I(x,y,z) may be estimated by the least square fit method. I(x,y,z) may be 
               ∑     m   =   0     M     ⁢         a   m     ⁡     (     x   ,   y     )       ⁢     z   m             
or any other function. For example,
 
               ∑     m   =   0     M     ⁢         a   m     ⁡     (     x   ,   y     )       ⁢     z   m             
has coefficients a m (x,y), where 0≦m≦M. The least square fit method assesses closeness of the fitting function I(x,y,z) to measured intensities as follows.
 
     
       
         
           
             
               
                 
                   
                     χ 
                     2 
                   
                   = 
                   
                     
                       
                         
                           ∫ 
                           ∫ 
                         
                         ImagingArea 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             N 
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       I 
                                       measure 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         x 
                                         , 
                                         y 
                                         , 
                                         
                                           z 
                                           i 
                                         
                                       
                                       ) 
                                     
                                   
                                   - 
                                   
                                     I 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         x 
                                         , 
                                         y 
                                         , 
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                                       ) 
                                     
                                   
                                 
                                 
                                   
                                     I 
                                     measure 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       x 
                                       , 
                                       y 
                                       , 
                                       
                                         z 
                                         i 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                         
                         ] 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       x 
                     
                     ⁢ 
                     
                       ⅆ 
                       y 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     5 
                   
                   ) 
                 
               
             
           
         
       
     
     where ImagingArea covers the area on a plane where any respective one of I measure (x,y,z 1 ), I measure (x,y,z 2 ), . . . , I measure (x,y,z n ), . . . , I measure (x,y,z N ) is captured. By minimizing χ 2 , the least square fit method finds values of coefficients, such as a m (x,y) for 
     
       
         
           
             
               ∑ 
               
                 m 
                 = 
                 0 
               
               M 
             
             ⁢ 
             
               
                 
                   a 
                   m 
                 
                 ⁡ 
                 
                   ( 
                   
                     x 
                     , 
                     y 
                   
                   ) 
                 
               
               ⁢ 
               
                 
                   z 
                   m 
                 
                 . 
               
             
           
         
       
     
     In addition, the least square fit method may also be used to compare capabilities of various fitting functions to describe measured intensities. For each fitting function, its coefficients may be determined by minimizing χ 2 . The resulting χ 2  minimums for different fitting functions may be different. The fitting function with the smallest χ 2  minimum usually provides the best fit to the measured intensities, and may be chosen to calculate D(x,y) according to Equation 1. 
     At process  230  of aberration determination, the aberration of the optical system is obtained. The aberration is described by a function called Ψ(x,y,z) at z equal to zero. Ψ(x,y,z) is called auxiliary function. Ψ(x,y,0) can be calculated as follows: 
     
       
         
           
             
               
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         π 
                       
                       λ 
                     
                     ⁢ 
                     
                       D 
                       ⁡ 
                       
                         ( 
                         
                           x 
                           , 
                           y 
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       - 
                       
                         
                           ∇ 
                           2 
                         
                         ⁢ 
                         
                           Ψ 
                           ⁡ 
                           
                             ( 
                             
                               x 
                               , 
                               y 
                               , 
                               0 
                             
                             ) 
                           
                         
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     where 
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     6 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     ∇ 
                     2 
                   
                   ⁢ 
                   
                     = 
                     
                       
                         
                           ∂ 
                           2 
                         
                         
                           ∂ 
                           
                             x 
                             2 
                           
                         
                       
                       + 
                       
                         
                           ∂ 
                           2 
                         
                         
                           ∂ 
                           
                             y 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     7 
                   
                   ) 
                 
               
             
           
         
       
     
     At process  240  of aberration reduction, the optical system is adjusted in order to reduce aberrations. The adjustment may be performed with various methods. For example, surface of an optical mirror may be polished. Also, surface of an optical mirror may be adjusted with actuators. Actuators may be placed on the backside of the mirror. In order to use actuators to reduce aberrations on optical mirror, the relationship between settings of actuators and aberrations, also called influence function, needs to be determined. The influence function may be obtained by measuring influence function data and fitting the measured data to an influence function. The fitting process may use the least square fit method or any other fitting method. The influence function may take the form of various functions. In addition, measurements of influence function data and fitting of the influence function may be performed before process  240 , during process  240 , or combination thereof. Further, process  240  may be skipped if process  230  of aberration determination shows that aberrations are sufficiently small. 
     At process  250  of additional measurement determination, the need for any additional intensity measurement is determined. For example, if process  230  has determined that aberrations are small or if process  240  has been skipped, no additional measurement may be needed. Other factors may also affect the need for additional intensity measurement, such as time, cost, and performance requirement. If process  250  determines an additional measurement is needed, processes  210 ,  220 ,  230 , and  240  may be performed. As discussed above, process  240  may be skipped. 
     In order to effectively reduce aberrations through iterations of processes  210 ,  220 ,  230 , and  240 , process  210  of intensity measurement may be performed on different sets of planes at different iterations. For example, I measure (x,y,z 1 ), I measure (x,y,z 2 ), . . . , I measure (x,y,z n ), . . . , I measure (x,y,z N ) may be measured on planes having greater distances from the focal plane than respective planes from the focal plane at the previous performance of process  210 . Therefore z 1 , z 2 , . . . , z n , . . . , z N  at a subsequent iteration may be larger than z 1 , z 2 , . . . , z n , . . . , z N  for previous performance of process  210  respectively. Alternatively, the subsequent iteration may use z 1 , z 2 , . . . , z n , . . . , z N  all which are the same as those used for previous measurement respectively. Subsequent iteration may use z 1 , z 2 , . . . , z n , . . . , z N  some of which are the same as and rest of which are different from those used for previous measurement respectively. Subsequent iteration may measure intensities on the same number of planes as previous performance of process  210 . Subsequent iteration may measure intensities on different number of planes than previous performance of process  210 . 
       FIGS. 4A through 4D  illustrate measured image intensities on different planes with aberrations on mirror surface. The measurements are merely examples, which should not unduly limit the scope of the present invention. One of ordinary skill in the art would recognize many variations, alternatives, and modifications. As shown in  FIG. 4(A) , direct measurements by Michelson interferometer shows that mirror surface  410  has a valley and a bump in or around center region  412 . These aberrations create bright area  422  and dark area  424  on image  420  that is captured on a plane located farther away from mirror surface  410  than the focal plane from mirror surface  410  by 11 mm, as shown in  FIG. 4B . Hence the image plane has a z value of 11 mm as defined in  FIG. 3 . In  FIG. 4C , image  430  is captured on a plane having a z value of 20 mm. Bright area  432  and dark area  434  indicates the existence of aberrations on mirror surface  410 . Similarly, image  440  is captured on a plane having a z value of −20 mm. Dark area  442  and bright area  444  indicate the existence of aberrations on mirror surface  410 . By comparison, images  430  and  440  are captured on planes symmetrically located on opposite sides of the focal plane. Bright area  432  is located in roughly the same location as dark area  442 ; dark area  434  is located in roughly the same location as bright area  444 . In addition, both images  430  and  440  are captured on planes further away from the focal plane than image  420  from the focal plane. Consequently, areas  432 ,  434 ,  442 , and  444  have generally bigger sizes and stronger contrasts than areas  422  and  424 . Hence images captured on planes further away from the focal plane usually reflects aberrations on mirror surface more sensitively than images captured on planes closer to the focal plane. 
       FIGS. 5A through 5D  illustrate measured image intensities on different planes with other aberrations on mirror surface. The measurements are merely examples, which should not unduly limit the scope of the present invention. One of ordinary skill in the art would recognize many variations, alternatives, and modifications. As shown in  FIG. 5A , measured Michelson fringes show that mirror surface  510  has a bump and a valley in or around center area  512 . But the bump and valley in  FIG. 5A  are not as severe as those in  FIG. 4A . Consequently, the low bump or the shallow valley does not create strong intensity variations on the plane at z equal to 11 mm, as shown in image  520  of  FIG. 5B . In contrast, image  530  captured at z equal to 20 mm has bright area  532  and dark area  534 , as shown in  FIG. 5C . Similarly,  FIG. 5D  shows dark area  542  and bright area  544  on image  540  captured at z equal −20 mm. 
       FIGS. 6A through 6D  illustrate measured image intensities on different planes with yet other aberrations on mirror surface. The measurements are merely examples, which should not unduly limit the scope of the present invention. One of ordinary skill in the art would recognize many variations, alternatives, and modifications. As shown in  FIG. 6A , measured Michelson fringes show that mirror surface  610  has a bump and a valley in or around center area  612 . But the bump and valley in  FIG. 6A  are not as severe as those in  FIGS. 4A and 5A . As shown in  FIGS. 5B ,  5 C, and  5 D, images  620 ,  630 , and  640  are captured at z equal to 11 mm, 20 mm, and 40 mm respectively. For image  640 , there appear bright area  642  and dark area  644  reflecting aberrations on mirror surface  610 . Also, image  640  shows locations of actuators attached to the back of mirror surface  610 , such as locations  646  and  648 . 
       FIG. 7  is a simplified system for estimation and correction of aberrations according to one embodiment of the present invention. The system is merely an example, which should not unduly limit the scope of the present invention. One of ordinary skill in the art would recognize many variations, alternatives, and modifications. System  700  includes light source  710 , lens  720 , beam splitter  730 , lens  740 , mirror  750 , and image detector  760 . Although the above has been shown using systems  710  through  760 , there can be many alternatives, modifications, and variations. For example, some of the systems may be expanded and/or combined. Lens  720  may be expanded to several lenses. Also, lens  740  may be expanded to several lenses. Other systems may be inserted to those noted above. Depending upon the embodiment, the specific systems may be replaced. For example, mirror  750  may be replaced by a telescope or other system with optical aperture. Further details of these systems are found throughout the present specification and more particularly below. 
     As shown in  FIG. 7 , light source  710  is a point light source such as a laser source combined with a pin-hole or a fiber-optic, and is placed at the focal point of lens  720 . Light source  710  generates radiation with substantially spherical wavefront. The radiation is converted into collimated beam  722  by lens  720 . Collimated beam  722  travels to beam splitter  730  and is partially reflected to form collimated beam  732 . Beam  732  travels to lens  740  and is converted into beam  744 . Lens  740  focuses beam  744  to focal point  742 , which is also the center of curvature for mirror  750 . Beam  744  travels to focal point  742  and then spreads out to reach mirror  750 . Mirror  750  reflects beam  744  to form beam  752  and focuses beam  752  at focal point  742 . Passing through focal point  742 , beam  752  is then collimated by lens  740  and reaches beam splitter  730 . Beam  752  partially passes through beam splitter  730  and then forms images on planes located either at focal plane  762  of mirror  750  or on either side of focal plane  762 . The images, including their intensities, are captured by image detector  760 . Image detectors  760  may be any detecting device that can measure intensities of images. System  700  may be used to perform method for estimation and correction of aberrations of mirror  750  including process  210  for intensity measurement as shown in  FIG. 2 . For example, an aberration includes a hill and a valley on the aperture. The vertical distance between the top of the hill and the bottom of the valley is at least one wavelength of radiation from the light source  710 . 
       FIG. 8  is a simplified system for estimation and correction of aberrations according to another embodiment of the present invention. The system is merely an example, which should not unduly limit the scope of the present invention. One of ordinary skill in the art would recognize many variations, alternatives, and modifications. 
       FIG. 9  is a simplified system for estimation and correction of large aberrations according to yet another embodiment of the present invention. The system is merely an example, which should not unduly limit the scope of the present invention. One of ordinary skill in the art would recognize many variations, alternatives, and modifications. System  900  includes testing system  910 , control system  920 , and adjustment system  930 . Although the above has been shown using systems  910 ,  920 , and  930 , there can be many alternatives, modifications, and variations. For example, some of the systems may be expanded and/or combined. For example, testing system  910  and testing  930  may be combined. Other systems may be inserted to those noted above. For example, system for performing conventional phase diversity process may be added. Depending upon the embodiment, the specific systems may be replaced. Further details of these systems are found throughout the present specification and more particularly below. 
     Testing system  910  may have some or all components of system  700  as described in  FIG. 7 . Control system  920  may be a personal computer, a server, a customized processor, or any other system. Control system  920  may perform process  220  for derivative estimation, process  230  for aberration determination, and process  250  of additional measurement determination as described in  FIG. 2 . In addition, control system  920  and testing system  910  may perform process  210  of intensity measurement. Adjustment system  930  may include polishing system, actuators, or combination thereof. For example, actuators may be placed on the backside of mirror  750  if testing system  910  has at least some components of system  700 . Optical adjustment system  930  and control system  920  may perform process  240  for aberration reduction. 
     In addition, control system  920  may include code that automatically directs testing system  910 , control system  920 , and adjustment system  930  to perform the inventive process  210  for intensity measurement, process  220  for derivative estimation, process  230  for aberration determination, process  240  for aberration reduction, and process  250  for additional measurement determination. The computer code may be implemented in Matlab, C++, or any other computer language. 
       FIGS. 10A through 10C  show simplified experimental results according to yet another embodiment of the present invention. The experiment is merely an example, which should not unduly limit the scope of the present invention. One of ordinary skill in the art would recognize many variations, alternatives, and modifications. In the experiment, mirror surface is measured by Michelson interferometer before any aberration reduction process and after each aberration reduction process in order to examine effectiveness of method and system of the present invention. As shown in  FIG. 10A , mirror surface has certain aberrations. After the first aberration reduction process is performed according to the present invention, the measured Michelson fringes show reduced aberrations on the mirror surface, as shown in  FIG. 10B . After four iterations of aberration reduction processes, the aberrations on the mirror surface are almost eliminated, as shown in  FIG. 10C . 
       FIG. 11  shows actuator commands for each iteration of aberration reduction process as described in  FIGS. 10A through 10C . The actuator commands are merely examples, which should not unduly limit the scope of the present invention. One of ordinary skill in the art would recognize many variations, alternatives, and modifications. For each actuator, the magnitude of correction usually decreases with number of iteration. In the meantime, the magnitudes of optical aberrations also decrease with number of iteration, as shown in  FIGS. 10 through 10C . 
     The techniques of the present invention have numerous advantages. Certain embodiments of the present invention can sense and correct aberrations on the entire aperture of an optical system without dividing the aperture into sub-apertures. The amount of time required for aberration reduction may be shortened. Some embodiments of the present invention work for segmented apertures. Certain embodiments of the present invention can improve aberration reduction by iterations. The iterative process alleviates convergence problem encountered by conventional techniques. Some embodiments of the present invention can simplify hardware requirements for aberration reduction, such as hardware requirements for coarse alignments of large telescopes. Certain embodiments of the present invention do not use the Pseudo-Hartmann mask, which is often used by conventional techniques for coarse figuring. Conventional techniques for coarse figuring often require two Pseudo-Hartmann masks, each of which is made up of sets of several multi-faceted prisms. Fabrication of the masks is difficult, time consuming and costly. Therefore, certain embodiments of the present invention can lower the cost and shorten the preparation time for aberration reduction. Some embodiments of the present invention usually can be implemented with minimum computation time. 
     It is understood the examples and embodiments described herein are for illustrative purposes only and that various modifications or changes in light thereof will be suggested to persons skilled in the art and are to be included within the spirit and purview of this application and scope of the appended claims.