Abstract:
A system for providing vision contains an aberrometer, a wavefront sensor, and a transfer optical system. The aberrometer is configured to measure a received wavefront. The aberrometer includes a wavefront sensor and a transfer optical system for transferring an input wavefront so as to the provide the received wavefront at or near the wavefront sensor. The system also includes a processor in communication with the aberrometer, a readable memory, and instructions located within the memory. The readable memory contains one or more system error parameters and instructions for calculating the input wavefront based on the received wavefront and the one or more system error parameters.

Description:
RELATED APPLICATION 
       [0001]    This application is a continuation application of, and claims prior to, U.S. patent application Ser. No. 12/263,433, filed Oct. 31, 2008, which claims priority under 35 U.S.C §119(e) to provisional application No. 60/984,355, filed on Oct. 31, 2008 and provisional application No. 61/028,890, filed on Feb. 14, 2008, the entire contents of each of which applications are hereby incorporated by reference in their entirety for all purposes as if fully set forth herein. 
     
    
     BACKGROUND OF THE INVENTION 
       [0002]    1. Field of the Invention 
         [0003]    The present invention relates generally to systems and software for determining aberrations, and more specifically to systems and software for determining aberrations of an input wavefront by correcting for, compensating for, or reducing system errors. 
         [0004]    2. Description of the Related Art 
         [0005]    Wavefront imaging system may be generally considered to be systems for transferring optical information contained in an object or input wavefront from an object space to an image space containing an image or output wavefront generally located at an image plane. A wavefront sensor, containing a detector or detector array such as a CCD or CMOS detector array, is generally placed at or near an image plane within the image space in order to provide one or more electronic or digital images of the image wavefront. The wavefront sensor may include additional optics disposed at or near the image space to further transfer or process the image wavefront. For example, a lenslet array may be disposed at or near the image plane in order to sample various portions the image wavefront. In some cases, a recording media such as a holographic plate and a reference beam may be provided to record the wavefront for later reconstruction. In any case, the recorded information in the image plane or space is generally used to provide information contained in the original object wavefront. 
         [0006]    The transfer of optical information by the wavefront imaging system to the wavefront sensor is, however, generally imperfect. At a minimum, the effects of diffraction, which arise from the use of a finite system aperture, will limit the ability of the wavefront imaging system to transfer information contained in the object space to the conjugate image space. In addition, the wavefront imaging system will produce other errors or aberrations during the transfer process, for example, aberrations produced by imperfect optical elements within the system and/or misalignment between these optical elements. 
         [0007]    In prior art wavefront imaging systems, aberrations or errors of the imaging system are typically not accounted for, which can lead to errors in accurately determining aberrations of an input wavefront, which has passed through an aberrated imaging system, based on measurements of an output wavefront. Rather, the imaging system is designed so as to minimize system optical aberrations, and is carefully aligned to reduce alignment errors. However, the inventor have found that even relatively small system aberrations or misalignment can lead to large errors in calculating the magnitude of higher order aberration terms of the original input wavefront. 
         [0008]    In some cases, the errors or aberrations produced by the optical imaging system may depend, not only on the system aberrations themselves, but also on the aberrations contained in the original input wavefront. 
         [0009]    Accordingly, better methods and systems are needed to provide more accurate calculation of the aberrations contained in an input wavefront after it has passed through a wavefront imaging system containing system aberrations and misalignments. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0010]    Embodiments of the present invention may be better understood from the following detailed description when read in conjunction with the accompanying drawings. Such embodiments, which are for illustrative purposes only, depict novel and non-obvious aspects of the invention. The drawings include the following figures: 
           [0011]      FIG. 1  is a schematic drawing of a wavefront sensor according to an embodiment of the present invention. 
           [0012]      FIG. 2  is a schematic drawing of the image or wavefront transfer system for the wavefront sensor shown in  FIG. 1 . 
           [0013]      FIG. 3  is system for performing a corneal ablation surgery according to an embodiment of the present invention. 
       
    
    
     DETAILED DESCRIPTION 
       [0014]    Embodiments of the present invention are generally directed to systems and software for measuring or determining aberrations of an input wavefront based on an output wavefront produced when the input wavefront passes through an imaging system that produces an output wavefront that is received by a wavefront sensor. The measurement or determination of the input wavefront is provided by correcting for, compensating for, or reducing system errors and/or input wavefront errors that may alter the error produced by the system. Embodiments of the present invention will be illustrated using aberrometer systems used in the area of ophthalmic measurement and correction. However, it will be understood that embodiments of the present invention may be utilized in other optical applications where measurement of a wavefront or image may be affected by transfer of the wavefront or image through a transfer optical system. 
         [0015]    Referring to  FIG. 1 , in certain embodiments of the present invention, an aberrometer or wavefront measurement system  10  is configured to provide a characterization of an input or object wavefront W of a subject eye  20  containing a cornea  22  and a retina  25 . The wavefront measurement system  10  comprises an illumination optical system  30  that includes an illumination source  32  and may include one or more lenses  35  or other optical elements that are configured to direct light from the light source  32  to the eye  20  and to preferably focus the light onto the retina  25 . The wavefront measurement system  10  also includes a computer or processor  36  that is used to control various components of the system  10 , to collect input data from the system  10 , and to make calculations regarding aberrations of the wavefronts W and/or W′, aberrations of the system  10 , and/or aberrations of the eye  20 . The wavefront measurement system  10  also comprises an optical relay or image relay system  40  that is configured to receive and relay an input wavefront W from the eye  20 . The input wavefront W is transformed by the relay system or telescope  40  into a wavefront W′ that is received by a wavefront sensor  38 . 
         [0016]    In certain embodiments, the optical relay system  40  comprises a first lens  42  and a second lens  45 . As the input wavefront W passes through the first lens  42  of relay system  40 , light is generally directed or focused onto an internal focal plane. This light then passes through the second lens  45  to form the wavefront W′, which is received by the wavefront sensor  38 . Some embodiments, at least one of the lenses  42 ,  45  may be replaced or supplemented by another optical element, for example, a diffractive optical element (DOE) or mirror. In general, aberrations may be introduced into the wavefront W′ that were not present in the input wavefront W, for example, by misalignment between the lenses  42 ,  45  in translation (dx, dy, dz) or rotation (dθx, dθy, dθz). Alternatively or additionally, aberrations may be introduced due to inherent optical characteristics of the optical relay system  40 , for example, spherical aberrations introduced through the use of spherical lenses. 
         [0017]    In the illustrated embodiment, the wavefront sensor  38  is a Shack-Hartmann wavefront sensor comprising a lenslet array  55  and a detector  60 , which may be a CCD, CMOS, or similar type detector comprising an array of pixel elements. Alternatively, other types of wavefront sensors may be used, for example, an interferometer or phase diversity sensor arrangement. In the illustrated embodiment shown in  FIG. 2 , the test object is an eye  70  which produces the wavefront W at an exit pupil  75 . The wavefront W is produced by focused light reflected by the eyes retina that passes through the eye. In such embodiments, a mathematical relationship may be built of the wavefront aberrations in between the entrance pupil of the eye and the exit pupil of the relay system using Fourier polynomials and Zernike polynomials. In certain embodiments, the wavefront W also includes aberrations as it passes from the cornea of the eye to the pupil  75 , which may be included in the aberrations calculated for the transferred wavefront W′. In other embodiments, aberrations introduced into the wavefront sensor  50  may also be included in the wavefront analysis. 
         [0018]    In some embodiments, more than one relay system may be disposed between the test object and the wavefront sensor  50 , in which case an analysis according to embodiments of the present invention may be used to calculate aberrations introduced into the wavefront from the test object. 
         [0019]    In certain embodiments, the computer or processor  36  contains memory including instructions for calculating aberrations of the wavefront W′, for example, by representing the wavefront W′ as a polynomial, such as a Zernike polynomial or Fourier polynomial, for example as taught in U.S. Pat. Nos. 6,609,793, 6,830,332, which are incorporated by reference in their entirety. The computer  36  may be a desktop or portable computer. Alternatively, the computer  36  may be incorporated into an electronic circuit board or chip containing on-board memory or in communications with separate external memory. 
         [0020]    In some embodiments, a memory for the computer  36  contains value for certain parameters. For example, the memory may contain one or more system error parameters. For example, the system error parameters may include errors, aberrations, or misalignment information regarding the wavefront measurement system  10  and/or the optical relay system  40 . The system error parameters may include coefficients of a polynomial equation representing an error or aberration, such as Zernike polynomial, a Taylor polynomial, or the like. 
         [0021]    Referring to  FIG. 3 , in certain embodiments, a corneal refractive surgical system  100  includes a laser for performing a corneal refractive surgery such as a LASIK or PRK procedure, for example, as disclosed in U.S. Pat. No. 6,964,659, which is herein incorporated by reference in its entirety. The surgical system  100  is provided with a treatment profile for a cornea based on the wavefront W′ that is calculated based on the wavefront W and corrections or compensations for the transfer system  40  and/or cross-terms of the transfer system and the input wavefront W in accordance to embodiments discussed herein, or the like. 
         [0022]    In some embodiments, a method or formulas are used in wavefront measurements using a confocal system from one point (pin hole or small aperture) to another conjugated point (pin hole or small aperture) instead of from one plane to another plane. In other embodiments, a method may be applied for any other aerometer or device to measure accurately the aberrations of human eye or other optical system that uses a relay system before the wavefront sensor. 
         [0023]    In certain embodiments, the aberrations introduced into the object wavefront W may reduced or eliminated by adjusted, supplementing, or replacing optical elements of the relay system  40  or some other optical element of the measurement system  10 . In some embodiments, the final design of a wavefront measurement system or relay system  10  may be adjusted to reduce or eliminate the sensitivity to misalignment of certain optical elements of the system. 
         [0024]    In some embodiments, a wavefront propagation theory may be used to derive mathematical expressions of the wavefront propagation through the optical relay system  40 . The basis for the wavefront propagation may be based on: 
         [0025]    Fresnel-Kirchhoff theory 
         [0026]    Distribution and phase function 
         [0027]    Fourier optics 
         [0000]    In such embodiments, the wavefront transfer function or matrix may be given by: 
         [0000]        W ′( x,y )=WTF( x,y )* W ( x,y )  (1) 
         [0000]      [ W ′]=[WTF]*[ W]   (2) 
         [0028]    W, WTF, W′ may use sets of polynomials 
         [0000]    Additionally or alternatively, ray tracing simulations may be used to derive mathematical expressions of the wavefront propagation through the optical relay system  40 , as discussed in further detail below. 
         [0029]    The Fresnel-Kirchhoff&#39;s approximate formula may include the following relations: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
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         [0030]    Embodiments of the present invention generally relate to optical imaging systems and methods of determining the image errors produced by such optical imaging systems. Embodiments of the present invention discussed below are for illustrative purposes and are not intended to limit the scope of the present invention. Aspects of the present invention are discussed below in conjunction with the “Slides” referenced therein and include with this disclosure. 
         [0031]    As discussed above, it is typically assumed that the output at an image plane of an optical system is given as: 
         [0000]      Output=Input+System  (6) 
         [0000]    This assumes that the input and the system errors or aberrations may be independently assessed, and that, accordingly, there is no “cross-talk” among the input, system, and pupil diffraction. However, the inventors have found that such cross-talk can exist. For example, aberrations in the term “Input” in Equation (6) may affect the magnitude of the term “System” In such embodiment, the output may be calculated, for example, according to the relation: 
         [0000]      Output=Input+System+Input*System  (6′) 
         [0000]    If either Input or System errors are zero or very close to zero, then the cross-term, Input*System, is zero, Equation (6) is valid. 
         [0032]    The inventor have found that in cases where either the input or the system have no error or aberrations, there is no cross-talk and the above relationship is valid. However, the inventors have also found that in some situations, cross-talk can actually have a significant affect on the output, particularly in terms of higher order aberration (e.g., above 4th order). Accordingly, a more accurate assessment of the output may be provided by: 
         [0000]      Output=Input+System Error+ f (input, system)+ g (input, pupil)+ h (system, pupil)  (7) 
         [0000]    Accordingly, the following effort functions were defined: 
         [0000]      Error Function 1=Output−Input  (8) 
         [0033]    Error Function 2=(output−input−system error)/input Referring to Equations (9) Referring to Equations (3) through (5) above, the inventors used Fresnel-Kirchhoff&#39;s approximate formula to provide a more accurate analysis of an optical imaging system output that accounts for cross-talk. The need for such an approach has been illustrated by the inventors using the following method  200 :
   1. At input plane, provide values in Zernike coefficients.   2. At lens  42  and lens  45  (see  FIG. 2 ), input aberrations (e.g., 0.1 microns spherical aberration)   3. Using a ray tracing program, decenter lens  42  relative to lens  45  (e.g., 0.1 mm to 0.5 mm) and introduce relative tilt (2 to 5 degrees) and defocused (e.g., 0.1 mm);   4. Calculate the wavefront at an output plane by wavefront maps and Zernike coefficients;   5. Calculate an error function using the wavefront maps and fit to Zernike polynomials.   6. Repeat 1-5 with lens  42  shifted left by 10 mm.   7. Repeat 1-5 with zero input (2) OR zero system error (zero all in 4) will make the error function 2 zero.   
 
         [0041]    Accordingly, and by way of example, the relay system shown in  FIG. 2  was used to demonstrate the effects of cross-talk in producing significant output error. It will be appreciated that this specific example is not meant to limit the scope of the present invention, but is give as a way of demonstrating the importance of using an analysis according to embodiments of the present invention in the design of optical systems, for example in the field of ophthalmic diagnostic. 
         [0042]    In steps 1-3 of method  200  above, various errors are introduced into the input and system shown in  FIG. 2 . Input errors are shown in terms of Zernike coefficients in the second column of TABLE 1 below. The column labeled “OUTPUT” in TABLE 1 shows the resulting output after the system errors are subtracted out. Thus, the error in percent introduced by cross-talk is shown in the column in TABLE 1 labeled “Error Function 2”. As can be seen, these errors are significant for many of the Zernike terms. In TABLE 2, an additional error is introduce by way of displacement of the object plan by 10 millimeters. This amount of displacement error is typical in aberrometers used to measure aberrations of a subjects eye or cornea. 
         [0000]    
       
         
               
               
               
               
             
               
               
               
               
             
           
               
                 TABLE 1 
               
               
                   
               
               
                 INPUT 
                 CONJUGATED 
                 OUTPUT 
                 Error Function 2 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 Z 1 
                  0.73:1 
                 1.185371 
                   
               
               
                 Z 2 
                 −0.10:4{circumflex over ( )}(1/2) (p) * COS (A) 
                 −0.100265 
               
               
                 Z 3 
                 −0.10:4{circumflex over ( )}(1/2) (p) * SIN (A) 
                 −0.309667 
               
               
                   
               
               
                 Z 4 
                  0.32:3{circumflex over ( )}(1/2) (2p{circumflex over ( )}2 − 1) 
                 0.331621 
                 
                   
                             
                     
                         
                         
                     
                   
                 
               
               
                   
               
               
                 Z 5 
                 −0.10:6{circumflex over ( )}(1/2) (p{circumflex over ( )}2) * SIN (2A) 
                 −0.095975 
                 −4.02% 
               
               
                 Z 6 
                 −0.10:6{circumflex over ( )}(1/2) (p{circumflex over ( )}2) * COS (2A) 
                 −0.096498 
                 −3.50% 
               
               
                 Z 7 
                 −0.10:8{circumflex over ( )}(1/2) (3p{circumflex over ( )}3 − 2p) * SIN (A) 
                 −0.094691 
                 −5.31% 
               
               
                 Z 8 
                 −0.10:8{circumflex over ( )}(1/2) (3p{circumflex over ( )}3 − 2p) * COS (A) 
                 −0.093219 
                 −6.78% 
               
               
                 Z 9 
                 −0.10:8{circumflex over ( )}(1/2) (p{circumflex over ( )}3) * SIN (3A) 
                 −0.095711 
                 −4.29% 
               
               
                 Z 10 
                 −0.10:8{circumflex over ( )}(1/2) (p{circumflex over ( )}3) * COS (3A) 
                 −0.097180 
                 −2.82% 
               
               
                   
               
               
                 Z 11 
                 −0.10:5{circumflex over ( )}(1/2) (6p{circumflex over ( )}4 − 6p{circumflex over ( )}2 + 1) 
                 −0.288831 
                 
                   
                             
                     
                         
                         
                     
                   
                 
               
               
                   
               
               
                 Z 12 
                 −0.05:10{circumflex over ( )}(1/2) (4p{circumflex over ( )}4 − 3p{circumflex over ( )}2) * COS (2A) 
                 −0.046635 
                 −6.73% 
               
               
                 Z 13 
                 −0.05:10{circumflex over ( )}(1/2) (4p{circumflex over ( )}4 − 3p{circumflex over ( )}2) * SIN (2A) 
                 −0.046091 
                 −7.82% 
               
               
                 Z 14 
                 −0.05:10{circumflex over ( )}(1/2) (p{circumflex over ( )}4) * COS (4A) 
                 −0.048774 
                 −2.45% 
               
               
                   
               
               
                 Z 15 
                 −0.05:10{circumflex over ( )}(1/2) (p{circumflex over ( )}4) * SIN (4A) 
                 −0.046940 
                 
                   
                             
                     
                         
                         
                     
                   
                 
               
               
                   
               
               
                 Z 16 
                 −0.05:12{circumflex over ( )}(1/2) (10p{circumflex over ( )}5 − 12p{circumflex over ( )}3 + 3p) * COS (A) 
                 −0.043968 
                 
                   
                             
                     
                         
                         
                     
                   
                 
               
               
                   
               
               
                 Z 17 
                 −0.03:12{circumflex over ( )}(1/2) (10p{circumflex over ( )}5 − 12p{circumflex over ( )}3 + 3p) * SIN (A) 
                 −0.024911 
                 
                   
                             
                     
                         
                         
                     
                   
                 
               
               
                   
               
               
                 Z 18 
                 −0.03:12{circumflex over ( )}(1/2) (5p{circumflex over ( )}5 − 4p{circumflex over ( )}3) * COS (3A) 
                 −0.027555 
                 
                   
                             
                     
                         
                         
                     
                   
                 
               
               
                   
               
               
                 Z 19 
                 −0.03:12{circumflex over ( )}(1/2) (5p{circumflex over ( )}5 − 4p{circumflex over ( )}3) * SIN (3A) 
                 −0.026558 
                 
                   
                             
                     
                         
                         
                     
                   
                 
               
               
                   
               
               
                 Z 20 
                 −0.03:12{circumflex over ( )}(1/2) (p{circumflex over ( )}5) * COS (5A) 
                 −0.028616 
                 
                   
                             
                     
                         
                         
                     
                   
                 
               
               
                   
               
               
                 Z 21 
                 −0.03:12{circumflex over ( )}(1/2) (p{circumflex over ( )}5) * SIN (5A) 
                 −0.027833 
                 
                   
                             
                     
                         
                         
                     
                   
                 
               
               
                   
               
               
                 Z 22 
                 −0.03:7{circumflex over ( )}(1/2) (20p{circumflex over ( )}6 − 30p{circumflex over ( )}4 + 12p{circumflex over ( )}2 − 1) 
                 −0.021155 
                 
                   
                             
                     
                         
                         
                     
                   
                 
               
               
                   
               
               
                 Z 23 
                 −0.01:14{circumflex over ( )}(1/2)A) 
                 −0.007083 
                 
                   
                             
                     
                         
                         
                     
                   
                 
               
               
                   
               
               
                 Z 24 
                 −0.01:14{circumflex over ( )}(1/2)A) 
                 −0.007631 
                 
                   
                             
                     
                         
                         
                     
                   
                 
               
               
                   
               
               
                 Z 25 
                 −0.01:14{circumflex over ( )}(1/2) (6p{circumflex over ( )}6 − 5p{circumflex over ( )}4) * SIN (4A) 
                 −0.008299 
                 
                   
                             
                     
                         
                         
                     
                   
                 
               
               
                   
               
               
                 Z 26 
                 −0.01:14{circumflex over ( )}(1/2) (6p{circumflex over ( )}6 − 5p{circumflex over ( )}4) * COS (4A) 
                 −0.009175 
                 
                   
                             
                     
                         
                         
                     
                   
                 
               
               
                   
               
               
                 Z 27 
                 −0.01:14{circumflex over ( )}(1/2) (p{circumflex over ( )}6) * SIN (6A) 
                 −0.008496 
                 
                   
                             
                     
                         
                         
                     
                   
                 
               
               
                   
               
               
                 Z 28 
                 −0.01:14{circumflex over ( )}(1/2) (p{circumflex over ( )}6) * COS (A) 
                 −0.009279 
                 
                   
                             
                     
                         
                         
                     
                   
                 
               
               
                   
               
               
                 Z 29 
                 −0.01:16{circumflex over ( )}(1/2) SIN (A) 
                 −0.005762 
                 
                   
                             
                     
                         
                         
                     
                   
                 
               
               
                   
               
               
                 Z 30 
                 −0.01:16{circumflex over ( )}(1/2) COS (A) 
                 −0.005568 
                 
                   
                             
                     
                         
                         
                     
                   
                 
               
               
                   
               
               
                 Z 31 
                 −0.01:16{circumflex over ( )}(1/2)3A) 
                 −0.007889 
                 
                   
                             
                     
                         
                         
                     
                   
                 
               
               
                   
               
               
                 Z 32 
                 −0.01:16{circumflex over ( )}(1/2)3A) 
                 −0.008411 
                 
                   
                             
                     
                         
                         
                     
                   
                 
               
               
                   
               
               
                 Z 33 
                 −0.01:16{circumflex over ( )}(1/2) (7p{circumflex over ( )}7 − 6p{circumflex over ( )}5) * SIN (5A) 
                 −0.009393 
                 
                   
                             
                     
                         
                         
                     
                   
                 
               
               
                   
               
               
                 Z 34 
                 −0.01:16{circumflex over ( )}(1/2) (7p{circumflex over ( )}7 − 6p{circumflex over ( )}5) * COS (5A) 
                 −0.008911 
                 
                   
                             
                     
                         
                         
                     
                   
                 
               
               
                   
               
               
                 Z 35 
                 −0.01:16{circumflex over ( )}(1/2) (p{circumflex over ( )}7) * SIN (7A) 
                 −0.009280 
                 
                   
                             
                     
                         
                         
                     
                   
                 
               
               
                   
               
               
                 Z 36 
                 −0.01:16{circumflex over ( )}(1/2) (p{circumflex over ( )}7) * COS (7A) 
                 −0.009105 
                 
                   
                             
                     
                         
                         
                     
                   
                 
               
               
                   
               
               
                 Z 37 
                 −0.01:9{circumflex over ( )}(1/2) + 1) 
                 −0.004645 
                 
                   
                             
                     
                         
                         
                     
                   
                 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
               
               
               
             
               
               
               
               
             
           
               
                 TABLE 2 
               
               
                   
               
               
                 INPUT 
                 NON-CONJUGATED                          
 
                 OUTPUT 
                 Error Function 2 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 Z 1 
                  0.73:1 
                 1.188192 
                   
               
               
                 Z 2 
                 −0.10:4{circumflex over ( )}(1/2) (p) * COS (A) 
                 −0.099986 
               
               
                 Z 3 
                 −0.10:4{circumflex over ( )}(1/2) (p) * SIN (A) 
                 −0.308426 
               
               
                   
               
               
                 Z 4 
                  0.32:3{circumflex over ( )}(1/2) (2p{circumflex over ( )}2 − 1) 
                 0.334590 
                 
                   
                             
                     
                         
                         
                     
                   
                 
               
               
                   
               
               
                 Z 5 
                 −0.10:6{circumflex over ( )}(1/2) (p{circumflex over ( )}2) * SIN (2A) 
                 −0.095868 
                 −4.13% 
               
               
                 Z 6 
                 −0.10:6{circumflex over ( )}(1/2) (p{circumflex over ( )}2) * COS (2A) 
                 −0.096638 
                 −3.36% 
               
               
                 Z 7 
                 −0.10:8{circumflex over ( )}(1/2) (3p{circumflex over ( )}3 − 2p) * SIN (A) 
                 −0.093921 
                 −6.08% 
               
               
                 Z 8 
                 −0.10:8{circumflex over ( )}(1/2) (3p{circumflex over ( )}3 − 2p) * COS (A) 
                 −0.093020 
                 −6.98% 
               
               
                 Z 9 
                 −0.10:8{circumflex over ( )}(1/2) (p{circumflex over ( )}3) * SIN (3A) 
                 −0.096921 
                 −4.08% 
               
               
                 Z 10 
                 −0.10:8{circumflex over ( )}(1/2) (p{circumflex over ( )}3) * COS (3A) 
                 −0.097297 
                 −2.70% 
               
               
                   
               
               
                 Z 11 
                 −0.10:5{circumflex over ( )}(1/2) (6p{circumflex over ( )}4 − 6p{circumflex over ( )}2 + 1) 
                 −0.286254 
                 
                   
                             
                     
                         
                         
                     
                   
                 
               
               
                   
               
               
                 Z 12 
                 −0.06:10{circumflex over ( )}(1/2) (4p{circumflex over ( )}4 − 3p{circumflex over ( )}2) * COS (2A) 
                 −0.046927 
                 −6.15% 
               
               
                 Z 13 
                 −0.06:10{circumflex over ( )}(1/2) (4p{circumflex over ( )}4 − 3p{circumflex over ( )}2) * SIN (2A) 
                 −0.046072 
                 −7.86% 
               
               
                 Z 14 
                 −0.06:10{circumflex over ( )}(1/2) (p{circumflex over ( )}4) * COS (4A) 
                 −0.048841 
                 −2.32% 
               
               
                 Z 15 
                 −0.06:10{circumflex over ( )}(1/2) (p{circumflex over ( )}4) * SIN (4A) 
                 −0.047399 
                 −5.20% 
               
               
                 Z 16 
                 −0.06:12{circumflex over ( )}(1/2) (10p{circumflex over ( )}5 − 12p{circumflex over ( )}3 + 3p) * COS (A) 
                 −0.043978 
                 −12.04% 
               
               
                 Z 17 
                 −0.03:12{circumflex over ( )}(1/2) (10p{circumflex over ( )}5 − 12p{circumflex over ( )}3 + 3p) * SIN (A) 
                 −0.024479 
                 −18.40% 
               
               
                 Z 18 
                 −0.03:12{circumflex over ( )}(1/2) (5p{circumflex over ( )}5 − 4p{circumflex over ( )}3) * COS (3A) 
                 −0.027777 
                 −7.41% 
               
               
                 Z 19 
                 −0.03:12{circumflex over ( )}(1/2) (5p{circumflex over ( )}5 − 4p{circumflex over ( )}3) * SIN (3A) 
                 −0.026923 
                 −10.26% 
               
               
                 Z 20 
                 −0.03:12{circumflex over ( )}(1/2) (p{circumflex over ( )}5) * COS (5A) 
                 −0.028486 
                 −5.05% 
               
               
                 Z 21 
                 −0.03:12{circumflex over ( )}(1/2) (p{circumflex over ( )}5) * SIN (5A) 
                 −0.028431 
                 −5.23% 
               
               
                 Z 22 
                 −0.03:7{circumflex over ( )}(1/2) (20p{circumflex over ( )}6 − 30p{circumflex over ( )}4 + 12p{circumflex over ( )}2 − 1) 
                 −0.019803 
                 −26.17% 
               
               
                 Z 23 
                 −0.01:14{circumflex over ( )}(1/2)A) 
                 −0.007201 
                 −27.99% 
               
               
                 Z 24 
                 −0.01:14{circumflex over ( )}(1/2)A) 
                 −0.008088 
                 −19.12% 
               
               
                 Z 25 
                 −0.01:14{circumflex over ( )}(1/2) (6p{circumflex over ( )}6 − 5p{circumflex over ( )}4) * SIN (4A) 
                 −0.008896 
                 −11.04% 
               
               
                 Z 26 
                 −0.01:14{circumflex over ( )}(1/2) (6p{circumflex over ( )}6 − 5p{circumflex over ( )}4) * COS (4A) 
                 −0.009299 
                 −7.01% 
               
               
                 Z 27 
                 −0.01:14{circumflex over ( )}(1/2) (p{circumflex over ( )}6) * SIN (6A) 
                 −0.009083 
                 −9.17% 
               
               
                 Z 28 
                 −0.01:14{circumflex over ( )}(1/2) (p{circumflex over ( )}6) * COS (6A) 
                 −0.009026 
                 −9.74% 
               
               
                 Z 29 
                 −0.01:16{circumflex over ( )}(1/2) SIN (A) 
                 −0.005442 
                 −45.58% 
               
               
                 Z 30 
                 −0.01:16{circumflex over ( )}(1/2) COS (A) 
                 −0.005801 
                 −41.99% 
               
               
                 Z 31 
                 −0.01:16{circumflex over ( )}(1/2)3A) 
                 −0.008378 
                 −16.22% 
               
               
                 Z 32 
                 −0.01:16{circumflex over ( )}(1/2)3A) 
                 −0.008716 
                 −12.84% 
               
               
                 Z 33 
                 −0.01:16{circumflex over ( )}(1/2) (7p{circumflex over ( )}7 − 6p{circumflex over ( )}5) * SIN (5A) 
                 −0.010059 
                 0.59% 
               
               
                 Z 34 
                 −0.01:16{circumflex over ( )}(1/2) (7p{circumflex over ( )}7 − 6p{circumflex over ( )}5) * COS (5A) 
                 −0.008808 
                 −11.92% 
               
               
                 Z 35 
                 −0.01:16{circumflex over ( )}(1/2) (p{circumflex over ( )}7) * SIN (7A) 
                 −0.009793 
                 −2.07% 
               
               
                 Z 36 
                 −0.01:16{circumflex over ( )}(1/2) (p{circumflex over ( )}7) * COS (7A) 
                 −0.008750 
                 −12.50% 
               
               
                 Z 37 
                 −0.01:9{circumflex over ( )}(1/2) +1) 
                 −0.004604 
                 −53.24% 
               
               
                   
               
             
          
         
       
     
         [0043]    By way of summary, it was concluded that:
   1. A real relay system can introduce an input-dependent error due to the crosstalk between the input, the system, and the pupil diffraction to the output wavefront   2. Error could be introduced if the measured wavefront plane is not conjugated to the pupil plane because of changes in the wavefront during propagation   3. The quality of the relay optics and the alignment of the relay system are critical.   4. Errors measuring lower order aberrations of the input wavefront may be relatively low, while those of higher order aberrations may be relatively high, as summarized in the following table:   
 
         [0000]    
       
         
               
               
               
             
           
               
                   
               
               
                 Not ignorable 
                 1 
                 2 (system RMS &lt; λ/4) 
               
               
                   
               
             
             
               
                 Low order 
                 Determined by the 
                 Up to 5% 
               
               
                 (up to 4th order) 
                 system 
               
               
                 High order 
                 Determined by the 
                 Up to 50% 
               
               
                 (above 4th order) 
                 system 
               
               
                   
               
             
          
         
       
     
         [0048]    The above presents a description of the best mode contemplated of carrying out the present invention, and of the manner and process of making and using it, in such full, clear, concise, and exact terms as to enable any person skilled in the art to which it pertains to make and use this invention. This invention is, however, susceptible to modifications and alternate constructions from that discussed above which are fully equivalent. Consequently, it is not the intention to limit this invention to the particular embodiments disclosed. On the contrary, the intention is to cover modifications and alternate constructions coming within the spirit and scope of the invention as generally expressed by the following claims, which particularly point out and distinctly claim the subject matter of the invention.