Patent Publication Number: US-8525982-B2

Title: Refractive index distribution measuring method and refractive index distribution measuring apparatus

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to a refractive index distribution measuring method and a refractive index distribution measuring apparatus. 
     2. Description of the Related Art 
     Japanese Patent Laid-Open No. (“JP”) 01-316627 proposes a method for finding a refractive index distribution of an object by measuring its transmitted wavefront while the object is immersed in a medium (matching oil) which has approximately the same refractive index as that of the object. JP 02-008726 proposes a method for finding a refractive index distribution of an object by measuring its transmitted wavefront while the object is immersed in each of two types of matching oils, which has a slightly different refractive index from that of the object. 
     These methods disclosed in JPs 01-316627 and 02-008726 need matching oils each of which has approximately the same refractive index as that of the object. However, the matching oil having a high refractive index has a low transmittance, and a detector can output only a weak signal. Thus, the measuring precision of the object having a high refractive index is likely to lower. 
     SUMMARY OF THE INVENTION 
     The present invention provides a refractive index distribution measuring method and a refractive index distribution measuring apparatus which can highly precisely measure a refractive index distribution of an object. 
     A refractive index distribution measuring method according to the present invention includes the steps of measuring a transmitted wavefront of an object by arranging the object in a medium having a refractive index different from that of the object, and by introducing reference light into the object, and calculating a refractive index distribution of the object by using a measurement result of the transmitted wavefront. In a plurality of orientations of the object in the medium which are different from each other, the measuring step measures a first transmitted wavefront in a first medium having a first refractive index and a second transmitted wavefront in a second medium having a second refractive index different from the first refractive index. The calculating step obtains a refractive index distribution projected value of the object in each of the plurality of orientations by removing a shape component of the object utilizing measurement results of the first transmitted wavefront and the second transmitted wavefront and each transmitted wavefront of a reference object that has the same shape as that of the object and a specific refractive index distribution and is located in one of the first medium and the second medium with the same orientation as that of the object. The calculating step then calculates a three-dimensional refractive index distribution of the object based on a plurality of refractive index distribution projected values corresponding to the plurality of orientations. 
     Further features of the present invention will become apparent from the following description of exemplary embodiments with reference to the attached drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a block diagram of a refractive index distribution measuring apparatus according to a first embodiment. 
         FIG. 2  is a flowchart for illustrating a refractive index distribution measuring method according to the first embodiment. 
         FIGS. 3A-3B  are views for illustrating a coordinate system set in a reference object and an optical path of a ray in the refractive index distribution measuring apparatus according to the first embodiment. 
         FIGS. 4A-4B  are views for illustrating an inclination of the object according to the first embodiment. 
         FIG. 5  is a block diagram of a refractive index distribution measuring apparatus according to a second embodiment. 
         FIG. 6  is a flowchart for illustrating a refractive index distribution measuring method according to the second embodiment. 
     
    
    
     DESCRIPTION OF THE EMBODIMENTS 
     Referring now to the accompanying drawings, a description will now be given of embodiments of the present invention. 
     First Embodiment 
       FIG. 1  is a block diagram of a refractive index distribution measuring apparatus installed on a stabilizer  190 . The refractive index distribution measuring apparatus measures a transmitted wavefront of an object by immersing the object in each of two types of media, such as air and water, each of which has a refractive index different from that of the object and by introducing reference light from a light source into the object. The refractive index distribution measuring apparatus then calculates a refractive index distribution of the object utilizing a processor as a computer and a measurement result of the transmitted wavefront. This embodiment utilizes a Talbot interferometer as a measuring unit configured to measure the transmitted wavefront of the object arranged in the medium by utilizing the light from the light source. 
     The object  140  is an optical element, such as a lens. A container  130  houses a medium  1  such as air, or a medium  2  such as water. The refractive index of air or water is smaller than the refractive index of the object  140  by 0.01 or higher. 
     A laser beam  101  emitted from a laser light source  100 , such as a He—Ne laser, along an optical axis is diffracted when it passes through a pinhole (PH)  112  in a pinhole plate (optical member)  110 . The diffracted or reference light diffracted in the pinhole  112  is converted into convergent light  103  by a collimator lens (CL)  120 . 
     The convergent light  103  transmits the medium  1  or  2  and the object  140  in the container  130 . This embodiment assumes the object  140  to be a rotationally symmetrical lens around an axis. A diameter Φ of the pinhole  112  is so small that the diffracted light  102  can be regarded as an ideal spherical wave, and the diameter Φ is designed to satisfy the following expression using a numerical aperture NAO on the object side and a wavelength λ of the laser light source  100 : 
     
       
         
           
             
               
                 
                   Φ 
                   ≈ 
                   
                     λ 
                     
                       N 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       A 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       O 
                     
                   
                 
               
               
                 
                   Expression 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   1 
                 
               
             
           
         
       
     
     When λ is 600 nm and NAO is about 0.3, the diameter Φ of the pinhole  112  may be about 2 μm. 
     The laser beam that has transmitted the object  140  and air or water in the container  130  passes an orthogonal diffraction grating  170  as a two-dimensional diffraction grating, and is captured (measured) by an image-pickup element (CCD sensor or CMOS sensor)  180 . The orthogonal diffraction grating  170  and the image-pickup element  180  will be sometimes referred to as a “sensor” hereinafter. 
     When the numerical aperture (NA) of the object  140  on the image side is small and a (Talbot) distance Z between the diffraction grating  170  and the image-pickup element  180  satisfies the Talbot condition as represented by Equation 2, the spurious resolution of the diffraction grating  170  is obtained as an interference pattern on the image-pickup element  180 , where m is an integer except 0, d is a grating pitch of the diffraction grating  170 , Z 0  is a distance from the diffraction gratin  170  to the image plane of the object  140 . The grating pitch d is determined in accordance with a magnitude of the aberration of the object  140 . 
     
       
         
           
             
               
                 
                   
                     
                       
                         Z 
                         0 
                       
                       ⁢ 
                       Z 
                     
                     
                       
                         Z 
                         0 
                       
                       - 
                       Z 
                     
                   
                   = 
                   
                     
                       md 
                       2 
                     
                     λ 
                   
                 
               
               
                 
                   Expression 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   2 
                 
               
             
           
         
       
     
     The object  140  is configured rotatable around an axis perpendicular to the optical axis by the rotating unit  150 , and relatively movable in the optical axis direction by the parallel moving and decentering unit  160 . The rotating unit  150  serves as an adjuster configured to adjust an orientation of the object in the medium. The collimator lens  120 , the diffraction grating  170 , and the image-pickup element  180  are also configured to relatively move on a rail (not illustrated) installed parallel to the optical axis. 
       FIG. 2  is a flowchart for illustrating a refractive index distribution measuring method according to this embodiment, and “S” stands for the step. The refractive index distribution measuring method is executed as a computer program by the processor  200 , such as a microcomputer illustrated in  FIG. 1 . 
     As illustrated in  FIG. 1 , the medium  1  (air) as a first medium having a first refractive index is filled in the container  130  (S 10 ). 
     Next, a (first) wavefront aberration W 1  of the object  140  immersed in the medium  1  in the container  130  is measured in accordance with the step A (S 20 ). 
     The measurement result of the transmitted wavefront contains a refractive index distribution of the object, influence of the object shape, influence of the object shape error, and an offset by the measuring system. Among them, the influence of the object shape and the offset by the measuring system are calculated by simulation and removed from the measurement result of the transmitted wavefront. The step A finds the wavefront aberration W 1 , and obtains remaining information which contains the refractive index distribution of the object and the influence of the object shape error. 
     The step A initially determines an optical arrangement of each component, that is, intervals among the pinhole plate  110 , the collimator lens  120 , the container  130 , the diffraction grating  170 , and the image-pickup element  180  in the optical axis direction (S 201 ). The optical arrangement is to restrain the NA equal to or lower than about 0.3 and to make appropriate the light flux size on the image-pickup element  180  so as to obtain the spurious resolution of the diffraction grating  170  over the entire surface of the image-pickup element in the Talbot interferometer. The optical arrangement prevents the light fluxes that have passed different positions on the object  140  from converging on the same point on the image-pickup element  180  so as to correlate a position on the image-pickup element  180  with a position on the object  140  in the subsequent steps. 
     Next, each component is arranged in accordance with the determined optical arrangement and the object  140  is aligned with the sensor (S 202 ). The alignment is performed by a relative movement using the parallel moving and decentering unit  160  in  FIG. 1  and/or the relative movements on the rail (not illustrated). The object  140  illustrated in  FIG. 1  is a concave lens, but if the object  140  is a convex lens, the container  130  may be installed behind the condensing position of the collimator lens  120  (on the side of the diffraction grating  170 ) so as to make appropriate the light flux size on the image-pickup element  180 . 
     Next, a simulation wavefront W sim  of the transmitted wavefront is calculated by assuming an ideal refractive index distribution (i.e., specific refractive index distribution) that has no refractive index distribution (S 203 ). This embodiment refers to an object having a known shape (which is the same as that of the object in this embodiment) and a known specific refractive index distribution as a reference object, and its transmitted wavefront as a reference transmitted wavefront. In S 203 , each transmitted wavefront is obtained by arranging the reference object in each of the first medium and the second medium with the same orientation as that of the object. 
     The known refractive index distribution may be a designed value or a measured value. The simulation wavefront W sim  is found based on a relationship of Expression 3 at a coordinate (x, y) of the reference object:
 
 W   sim ( x,y )= OP   sim ( x,y )− OP   sim (0,0)
 
 OP   sim ( x,y )= L 1( x,y )+ L 2( x,y ) N   1   +L 3( x,y ) Ng+L 4( x,y ) N   1   +L 5( x,y )  Expression 3
 
     Here, L 1  to L 5  are geometric distances among the components along the ray  103  as illustrated in  FIG. 3B . The ray  103  schematically illustrates a ray that passes a point (x, y) in the reference object  141  illustrated in  FIG. 3A . N 1  denotes a refractive index of air, and Ng denotes an ideal refractive index of the reference object  141 . The reference object  141  is contemplated by replacing the refractive index distribution of the object  140  with a known value. In order to simplify the expression, a thickness of the wall of the container  130  is ignored. 
     Next, while the object  140  is immersed in the air, the (first) transmitted wavefront W m  of the object  140  in the first medium is measured (measuring step) (S 204 ). S 204  contains an acquisition of an image of the interference pattern by the image-pickup element  180  and an image restoration of the transmitted wavefront by the processor  200 . The image restoration of the transmitted wavefront (referred to as a “wavefront restoration” hereinafter) is performed by the fast Fourier transform (“FFT”) method. 
     The wavefront restoration by the FFT method is a method that separates the aberration from the carrier pattern by utilizing a characteristic in that the aberration disturbs the carrier pattern of the interference pattern. More specifically, the two-dimensional FFT is performed for the interference pattern, and the interference pattern is converted into the frequency map. Next, the inverse fast Fourier transform (iFFT) method is performed after only part near the carrier frequency is picked up in the frequency map and a coordinate transformation is performed so as to set the carrier frequency to the origin. Thereby, a phase term of the complex amplitude is found. The resultant phase map becomes the transmitted wavefront. 
     W m  are expressed as follows using L 1  to L 5 :
 
 W   m ( x,y )= OP   m ( x,y )− OP   m (0,0)
 
 OP   m ( x,y )= L 1( x,y )+ L 2( x,y ) N   1   +{L 3( x,y )+ dL}  N   ( x,y )+{ L 4( x,y )− dL}N   1   +L 5( x,y )  Expression 4
 
     Here, the N bar is a refractive index distribution projected value averaged in the optical path direction of the object  140  at the coordinate (x, y), and dL denotes a thickness error of the object  140  at the coordinate (x, y). 
     Next, the (first) wavefront aberration W 1  corresponding to a difference between the simulation wavefront W sim  and the transmitted wavefront W m  is found by utilizing the following expression (S 205 ). In order to simply the expression, the refractive index Ng is assumed to be equal to the refractive index N(0, 0) on the optical axis of the object  140 :
 
 W 1 =W   m   −W   sim   =L 3( x,y ){    N   ( x,y )− Ng}+dL ( x,y ){    N   ( x,y )− N   1   }−dL (0,0){ Ng−N   1 }  Expression 5
 
     S 203  is independent of S 202  or S 204 , and may be executed at any timing between S 201  to S 205 . 
     Next, while the medium  2  (water) as a second medium having a second refractive index is filled in the container  130 , the object  140  is installed in the container  130  (S 30 ). Next, in accordance with the above step A, the (second) wavefront aberration W 2  of the object  140  is measured (measuring step) (S 40 ), where N 2  denotes the refractive index of water. At this time, the measuring step in S 204  measures the (second) transmitted wavefront W m  of the object  140  in the second medium while the object  140  is immersed in water (S 204 ).
 
 W 2 =W   m   −W   sim   =L 3( x,y ){    N   ( x,y )− Ng}+dL ( x,y ){    N   ( x,y )− N   2   }−dL (0,0){ N   9   −N   2 }  Expression 6
 
     Next, the refractive index distribution projected value of the object  140  is calculated by removing the shape component dL of the object  140  from the wavefront aberration W 1  and the wavefront aberration W 2  (S 50 ). S 50  is the calculating step for obtaining the refractive index distribution projected vale containing information of the refractive index distribution of the object  140  by removing the influence of the shape error of the object using the two wavefront aberrations W 1  and W 2  of the object arranged at the same position. Here, an approximation of Expression 8 is used. 
     
       
         
           
             
               
                 
                   
                     
                       N 
                       _ 
                     
                     ⁡ 
                     
                       ( 
                       
                         x 
                         , 
                         y 
                       
                       ) 
                     
                   
                   = 
                   
                     Ng 
                     + 
                     
                       
                         1 
                         
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           3 
                           ⁢ 
                           
                             ( 
                             
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                               , 
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                               ( 
                               
                                 Ng 
                                 - 
                                 
                                   N 
                                   1 
                                 
                               
                               ) 
                             
                             ⁢ 
                             W 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                           
                           - 
                           
                             
                               ( 
                               
                                 Ng 
                                 - 
                                 
                                   N 
                                   2 
                                 
                               
                               ) 
                             
                             ⁢ 
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                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                         
                         
                           
                             N 
                             2 
                           
                           - 
                           
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                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Expression 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   7 
                 
               
             
             
               
                 
                   
                       
                   
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                         { 
                         
                           
                             
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                   Expression 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   8 
                 
               
             
           
         
       
     
     Thereby, a refractive index distribution projected value is found at a first object inclination relative to the object  140 , which is an inclination in a case in which the optical axis accords with a rotationally symmetrical axis of the object  140 . Since the refractive index distribution projected value is a refractive index averaged in the optical path direction of the light incident upon the object  140 , it is necessary to find the refractive index distribution projected value by introducing the light into the object  140  at an inclination different from the first object inclination so as to obtain three-dimensional refractive index distribution information. A description will now be given of this method. 
     In order to find the refractive index distribution projected values of the object  140  with a plurality of orientations of the object  140 , the object  140  is rotated and decentered (S 61 ). The measurement number of S 60  is different according to the refractive index distribution to be found. When it can be assumed that the object  140  has a rotationally symmetrical shape around an axis and the refractive index distribution is rotationally symmetrical around the same axis, the measurement number may be twice. For example, an orientation (first object inclination) in a case in which the optical axis is accorded with the rotationally symmetrical axis of the object  140 , and an orientation (second object inclination) in a case in which an optical axis is no accorded with the rotationally symmetrical axis of the object  140 . 
       FIG. 4A  illustrates the first object inclination and  FIG. 4B  illustrates the second object inclination in this embodiment. In the first object inclination, the incident ray  103  and the exit ray  104  for the object  140  are as illustrated in  FIG. 4A , and in the second object inclination, the incident ray  103  and the exit ray  105  are as illustrated in  FIG. 4B . 
     In order to precisely obtain three-dimensional refractive index distribution information by the small measurement number, a plurality of measurement orientations of the refractive index distribution projected value may be made greatly different from each other. In other words, the first object inclination and the second object inclination may be made greatly different from each other. The second object inclination may be adjusted so that the incident ray  103  can pass the ends of the first surface and the second surface of the object  140 , as illustrated in  FIG. 4B . The end of the first surface is a boundary between the cutting surface and the R surface of the first surface as the optical surface on the light incidence side of the object, and the end of the second surface is a boundary between the cutting surface and the R surface of the second surface as the optical surface on the light exit side of the object. 
     This embodiment parallel moves and decenters the object  140  relative to the optical axis direction and rotates the object  140  around an axis perpendicular to the optical axis in S 61  and arranges the object  140  at a position and angle as illustrated in  FIG. 4B . 
     After S 61 , S 10  to S 50  are again performed until the measurement number for calculating the refractive index distribution projected value reaches the designated number (which is twice in this embodiment) in S 60 . 
     When the measurement number reaches the designated number, the three-dimensional refractive index distribution is calculated from a plurality of obtained refractive index distribution projected values (S 70 ). S 70  is the calculating step for obtaining information of the three-dimensional refractive index distribution based on a plurality of refractive index distribution projected values corresponding to a plurality of different orientations of the object  140 . The three-dimensional refractive index distribution is calculated by determining the polynomial coefficients that express the three-dimensional refractive index distribution so that a plurality of calculated refractive index distribution projected values can be reproduced. 
     When the incident ray  103  is expressed by 100 rays, the refractive index distribution projected values are expressed as follows: 
     
       
         
           
             
               
                 
                   
                     
                       N 
                       _ 
                     
                     1 
                   
                   = 
                   
                     
                       
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                                 100 
                               
                             
                           
                         
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                                   101 
                                 
                               
                             
                             
                               
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                   9 
                 
               
             
           
         
       
     
     The N 1  bar and the N 2  bar are refractive index distribution projected values at the first and second object inclinations. In addition, assume that the three-dimensional refractive index distribution P to be found is expressed by twelve polynomial coefficients. 
     
       
         
           
             
               
                 
                   
                     
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                   ⁢ 
                   
                       
                   
                   ⁢ 
                   10 
                 
               
             
           
         
       
     
     When it is assumed that U is a refractive index distribution projected value using polynomial coefficients of Expression 10 as a unit amount, U can be expressed by the following expression: 
     
       
         
           
             
               
                 
                   U 
                   = 
                   
                     ( 
                     
                       
                         
                           
                             u 
                             
                               1 
                               , 
                               1 
                             
                           
                         
                         
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                             u 
                             
                               12 
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                   11 
                 
               
             
           
         
       
     
     Each coefficient of P can reproduce the plurality of found refractive index distribution projected values when P is determined so as to satisfy the following equate.
 
   N =UP   Expression 12
 
     When the least-squares method is used, Φ is defined as in Expression 13 and each coefficient of P is determined so that Φ 2  can be least.
 
Φ= UP−  N     Expression 13
 
     P can be directly found as in Expression 14 by finding U −1  when the eigenvalue decomposition is used.
 
 P=U   −1     N     Expression 14
 
     Alternatively, a combination method may be used which defines Φ as a value made by subtracting a right side value from a left side value in Expression 14 and determines each coefficient of P so that Φ 2  can be least or another known method may be used to find P. Thus, a refractive index distribution measuring method ends in this embodiment by finding a three-dimensional refractive index distribution P. 
     As discussed above, this embodiment measures two types of wavefront aberrations of the object utilizing two types of media and the reference light emitted from the light source, obtains the refractive index distribution projected values by removing the shape component of the object from the wavefront aberration, and acquires another refractive index distribution projected value by changing an angle of the object relative to the optical axis. The polynomial coefficients which express the three-dimensional refractive index distribution of the object are found based on a plurality of refractive index distribution projected values. Thereby, even when a refractive index of the object is high, an internal refractive index distribution of the object can be highly precisely measured by utilizing a medium having a refractive index that is lower than the refractive index of the object. 
     For simple description, this embodiment properly sets the number of rays that expresses the incident rays  103  and a polynomial that expresses the three-dimensional refractive index distribution. More specifically, the incident light is expressed as one ray, and the measurement designated number of the refractive index distribution projected value is set to m and the polynomial that expresses the refractive index distribution has n terms. Even in this case, P can be found by a method similar to Expression 12 by setting the refractive index distribution projected value N bar, the refractive index distribution P, and the refractive index distribution projected value U of a case where each coefficient of P has a unit amount as in Expression 15: 
     
       
         
           
             
               
                 
                   
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                     _ 
                   
                   = 
                   
                     
                       
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                                 1 
                               
                             
                           
                           
                             
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                       P 
                     
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                   Expression 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   15 
                 
               
             
           
         
       
     
     As in this embodiment, a large aberration caused by a refractive index difference between the object and the medium can be measured by using the Talbot interferometer for the measuring unit. The Talbot interferometer is one type of a lateral shearing interferometer configured to measure as an interference pattern a difference between the transmitted wavefront and its sheared transmitted wavefront. 
     The shearing interferometer is a measuring unit configured to find an amount corresponding to a gradient of a wavefront shape of the transmitted wavefront. A lateral shift amount of the transmitted wavefront is referred to as a shear amount, and a ratio of the shear amount to the diameter of the light is referred to as a shear ratio. By reducing a shear ratio, a large transmitted wavefront aberration can be measured as a small aberration (shear wavefront) that does not make dense the interference pattern. 
     In general, when the shear ratio is excessively small in the shearing interferometer, the shear wavefront is embedded in the noises and the precision deteriorates. Thus, the shear ratio may be 3 to 5% as large as the diameter of the pupil. However, this embodiment sets the shear ratio to 1.5% or smaller, e.g., about 0.4 to 0.9% so as to measure the transmitted wavefront having a large aberration with a small shear wavefront. 
     The shear ratio is defined as (λZ)/(dD) using a Talbot distance Z and a diameter D of the interference pattern data on the image-pickup element  180 , and is defined as (md)/D using Expression 2 and the diameter D 0  of the light flux on the diffraction grating  170 . Therefore, the shear ratio is proportional to the grating pitch of the diffraction grating  170 . From Expression 2, the pitch of the diffraction grating  170  affects the Talbot distance Z, and it is thus necessary to determine the pitch by considering the interference among the components in the measuring apparatus. For example, if it is assumed that D O  is about 10 to 20 mm when m=1, the grating pitch may be about 40 to 180 μm. 
     While this embodiment set two types of media to air and water, the medium is not limited as long as two types of media are different by 0.01 or higher. In addition, the two types of media may be made from the same material having different refractive indexes by changing its temperature. 
     While this embodiment discusses use of the Talbot interferometer, a lateral shearing interferometer, and a radial shearing interferometer, and another shearing interferometer different from the Talbot interferometer may be used. 
     Second Embodiment 
       FIG. 5  is a block diagram of a refractive index distribution measuring apparatus of the second embodiment. The refractive index distribution measuring apparatus of this embodiment finds a refractive index distribution by measuring the transmitted wavefront twice using two types of light sources and one type of medium M. The two types of light sources are, for example, a He—Ne laser (with a first wavelength of 633 nm) as a light source  100 A and a second harmonic of a YAG laser (with a second wavelength of 532 nm different from the first wavelength) as a light source  100 B. 
     The medium M has a refractive index different from that of the object  140 . For example, the refractive index of the medium is smaller than that of the object, and larger than that of air. One example of the medium M is water, and low refractive index oil having a refractive index of about 1.5 to about 1.8. 
     The pinhole plate  110  generates (reference) light having the ideal spherical wave using a laser beam emitted from the light source  100 A or  100 B. This light passes the object  140  similar to  FIG. 1 , and its transmitted wavefront is measured by a Shack-Hartman sensor  500  as a wavefront measuring sensor. The Shack-Hartman sensor  500  includes, in order from the light source along the optical path, a lens array  510  and an image pickup element  520 . 
     Similar to the first embodiment, the collimator lens  120 , the container  130 , and the sensor  500  are arranged on the rail (not illustrated) parallel to the optical axis. The light incident upon the object  140  can be converted into any one of the divergent light, collimated light, and the convergent light by moving these components on the rail. Thereby, the NA of the light flux incident upon the Shack-Hartman sensor  500  can be adjusted. 
     In comparison with the Talbot interferometer, the Shack-Hartman sensor requires the NA of the light flux incident upon the sensor to be strictly controlled but the alignment of the sensor  500  becomes easier because it is unnecessary to set an interval between the diffraction grating  170  and the CCD  160  to the Talbot distance. 
     The Shack-Hartman sensor  500  condenses the light incident upon the lens array  510  upon the CCD. When the inclined transmitted wavefront is incident upon the lens array  510 , a position of the condensing point shifts. Since the Shack-Hartman sensor  500  can convert an inclination of the transmitted wavefront into a positional shift of the condensing point and measure the positional shift, a wavefront having a large aberration can be measured. 
       FIG. 6  is a flowchart for illustrating a refractive index distribution measuring method of this embodiment, and “S” stands for the step. The refractive index distribution measuring method is executed as a computer program by the processor  200 , such as a microcomputer illustrated in  FIG. 5 . Most of the flow of  FIG. 6  is the same of the measurement flow in  FIG. 2 , and thus only the difference will be discussed. 
     Initially, light from the light source  100 A is introduced into the pinhole plate  110  (S 11 ), and the wavefront aberration W 1  using the first wavelength is measured (S 20 ). Next, light from the light source  100 B having a wavelength different from that of the light source  100 A is introduced into the pinhole plate  110  (S 31 ), and the wavefront aberration W 2  is measured (S 40 ). The wavefront aberrations obtained in these steps are expressed by the following expressions:
 
 W 1 =L 3( x,y ){   N T   HeNe ( x,y )− Ng   HeNe   }+dL ( x,y ){   N     HeNe ( x,y )− N   oilHeNe   }−dL (0,0){ Ng   HeNe   −N   oilHeNe }
 
 W 2 =L 3( x,y ){   N     YAG ( x,y )− Ng   YAG   }+dL ( x,y ){   N     YAG ( x,y )− N   oilYAG   }−dL (0,0){ Ng   YAG   −N   oilYAG }  Expression 16
 
     Here, the N HeNe  bar and the N YAG  bar are refractive index distribution projected values at a position (x, y) in the object for the first light source (He—Ne laser) and the second light source (YAG second harmonic), respectively. Ng HeNe  and Ng YAG  are ideal refractive indexes of the object (refractive indexes of the reference object) for respective light sources. N oilHeNe  and N oilYAG  are refractive indexes of the medium for the respective light sources. 
     The refractive index for the first light source and the refractive index for the second light source have the following approximation relationship: 
     
       
         
           
             
               
                 
                   
                     
                       
                         N 
                         _ 
                       
                       YAG 
                     
                     ⁡ 
                     
                       ( 
                       
                         x 
                         , 
                         y 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           Ng 
                           YAG 
                         
                         - 
                         1 
                       
                       
                         
                           Ng 
                           HeNe 
                         
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       
                         
                           N 
                           _ 
                         
                         HeNe 
                       
                       ⁡ 
                       
                         ( 
                         
                           x 
                           , 
                           y 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   Expression 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   17 
                 
               
             
           
         
       
     
     By utilizing Expressions 16 and 17, the refractive index distribution projected value can be found (S 51 ). 
     
       
         
           
             
               
                 
                   
                     
                       
                         N 
                         _ 
                       
                       HeNe 
                     
                     ⁡ 
                     
                       ( 
                       
                         x 
                         , 
                         y 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       Ng 
                       HeNe 
                     
                     + 
                     
                       
                         1 
                         
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           3 
                           ⁢ 
                           
                             ( 
                             
                               x 
                               , 
                               y 
                             
                             ) 
                           
                         
                       
                       × 
                       
                         
                           
                             
                               ( 
                               
                                 
                                   Ng 
                                   HeNe 
                                 
                                 - 
                                 
                                   N 
                                   oilHeNe 
                                 
                               
                               ) 
                             
                             ⁢ 
                             W 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                           
                           - 
                           
                             
                               ( 
                               
                                 
                                   Ng 
                                   YAG 
                                 
                                 - 
                                 
                                   N 
                                   oilYAG 
                                 
                               
                               ) 
                             
                             ⁢ 
                             W 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                         
                         
                           
                             
                               
                                 
                                   
                                     
                                       
                                         Ng 
                                         YAG 
                                       
                                       - 
                                       1 
                                     
                                     
                                       
                                         Ng 
                                         HeNe 
                                       
                                       - 
                                       1 
                                     
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         Ng 
                                         HeNe 
                                       
                                       - 
                                       
                                         N 
                                         oilHeNe 
                                       
                                     
                                     ) 
                                   
                                 
                                 - 
                               
                             
                           
                           
                             
                               
                                 ( 
                                 
                                   
                                     Ng 
                                     YAG 
                                   
                                   - 
                                   
                                     N 
                                     oilYAG 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Expression 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   18 
                 
               
             
           
         
       
     
     Thereafter, S 60 , S 61 , and S 70  follow and the measurement ends. 
     In Expression 18, when Ψ of Expression 20 is large, the errors of the measurement values W 1  and W 2  can be reduced. 
     
       
         
           
             
               
                 
                   Ψ 
                   = 
                   
                     
                       
                         
                           
                             Ng 
                             YAG 
                           
                           - 
                           1 
                         
                         
                           
                             Ng 
                             HeNe 
                           
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             Ng 
                             HeNe 
                           
                           - 
                           
                             N 
                             oilHeNe 
                           
                         
                         ) 
                       
                     
                     - 
                     
                       ( 
                       
                         
                           Ng 
                           YAG 
                         
                         - 
                         
                           N 
                           oilYAG 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   Expression 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   19 
                 
               
             
           
         
       
     
     For example, when the medium is air, N oil ≈0 is satisfied. Thus, Ψ≈0 is established and the measurement becomes unavailable. In addition, for example, when it is assumed that the refractive index of the object does not greatly change as the wavelength changes, Ng YAG ≈Ng HeNe  is established and thus Expression 19 can be expressed as Expression 20.
 
Ψ= N   oilYAG   −N   oilHeNe   Expression 20
 
     In this case, a medium having a large refractive index difference may be selected between the first light source and the second light source. In order to increase Ψ, the medium needs to be determined by considering the refractive index of the object. 
     The measuring apparatus of this embodiment may be one that can measure an amount corresponding to a gradient of a wavefront shape of the transmitted wavefront or an inclination of the ray, and that can detect the gradient or inclination as a measurable physical amount, even when the transmitted wavefront has a large aberration. Therefore, the measuring apparatus is not limited to the Shack-Hartman method and may use the Hartman method or the Ronchi test. 
     The result measured by the refractive index distribution measuring apparatuses or methods according to the first and second embodiments is applicable to the manufacturing method of the optical element. A method for manufacturing an optical element includes the steps of molding an optical element based on a designed optical element, measuring a shape of the molded optical element, evaluating the shape precision, and evaluating the optical performance of the optical element that satisfies the shape precision. The refractive index distribution measuring method of this embodiment is applicable to the step of evaluating the optical performance. When the evaluated optical performance does not satisfy the required specification, a correction amount of an optical surface of the optical element is calculated and the optical element is redesigned by using the result. When the evaluated optical performance satisfies the required specification, the optical element is mass-produced. 
     Since the manufacturing method of the optical element of this embodiment can highly precisely measure an internal refractive index distribution of the optical element, the optical element can be precisely mass-produced through molding even when the optical element is made of a high refractive index glass material. 
     While the present invention has been described with reference to exemplary embodiments, it is to be understood that the invention is not limited to the disclosed exemplary embodiments. The scope of the following claims is to be accorded the broadest interpretation so as to encompass all such modifications and equivalent structures and functions. 
     This application claims the benefit of Japanese Patent Application No. 2010-119636, filed May 25, 2010, which is hereby incorporated by reference herein in its entirety.