# Patent

## Title
Method of reconstructing aspheric surface equations from measurements

## Abstract

Surface measurement data just provides the coordinates of an object surface without giving various parameters like the radius of curvature, conic constant, and deformation coefficients. In this paper, we propose a novel method for extracting the important parameters for the determination of unknown aspheric surface equations from the measurement of aspheric surfaces. The largest error between the original surface and the reconstructed surface in the theoretical case is shown to be about 8.6 nm. This fact implies that the new method is well suited for the reconstruction of unknown surface equations.

## Background
BACKGROUND OF INVENTION
Aspheric lens optics have received much attention in scientific, industrial, and military optical systems such as terahertz imaging, diagnostic systems, night-vision systems, and so on because they are able to remove spherical aberration and astigmatism for improving light gathering power and simplifying complex lens designs. For this reason, aspheric optics designs with parameters significantly affecting optical performance are treated as confidential information or are patented. Typically, aspheric surfaces fabricated according to surface designs are characterized by various evaluation methods. However, it is rare to reconstruct aspheric surface equations from surface measurements. In the present invention, a new method for reconstructing the aspheric surface equations from the measurement of unknown aspheric surfaces is disclosed.
SUMMARY OF INVENTION
A method of reconstructing aspheric surface equations in an information processing device from measurements, the method comprises steps for: obtaining zas(r) as r varies from ri to rf in steps of rstep by measuring a given aspheric surface that is represented by an aspheric surface equation; finding an error curve δze for each R value while changing R from Ri to Rf in steps of Rstep wherein K and deformation coefficients are set to 0, wherein the error curve δze is defined by a deviation of the aspheric surface equation from a quadratic term zs(r) of the aspheric surface equation; finding the deformation coefficients from δze using an inverse matrix method; replacing RN and aN with each R value and the deformation coefficients found from the δze in zN, a curve determined by the measured coordinates; checking if a shape error, δzd, is within a predetermined threshold value; and determining RN that is closest to the original R.
The aspheric surface equation may be given by




  
    
      z
      ⁡
      
        (
        r
        )
      
    
    =
    
      
        
          
            r
            2
          
          R
        
        
          1
          +
          
            
              1
              -
              
                
                  (
                  
                    1
                    +
                    K
                  
                  )
                
                ⁢
                
                  
                    (
                    
                      r
                      R
                    
                    )
                  
                  2
                
              
            
          
        
      
      +
      
        
          ∑
          
            i
            =
            2
          
        
        ⁢
        
          
            a
            
              2
              ⁢
              i
            
          
          ⁢
          
            r
            
              2
              ⁢
              i
            
          
        
      
    
  
  ,




where K is the conic constant, r is the lateral coordinate, R is the radius of curvature, and the a2i values are the deformation coefficients.

The zas(ri), the δs(r), the δze, the zN, and the δzd may be given by



  
    
      
        z
        as
      
      ⁡
      
        (
        r
        )
      
    
    =
    
      
        
          cr
          2
        
        
          1
          +
          
            
              1
              -
              
                
                  (
                  
                    1
                    +
                    K
                  
                  )
                
                ⁢
                
                  c
                  2
                
                ⁢
                
                  r
                  2
                
              
            
          
        
      
      +
      
        
          a
          4
        
        ⁢
        
          r
          4
        
      
      +
      
        
          a
          6
        
        ⁢
        
          r
          6
        
      
      +
      
        
          a
          8
        
        ⁢
        
          r
          8
        
      
      +
      
        
          a
          10
        
        ⁢
        
          r
          10
        
      
    
  




  
    
      
        z
        s
      
      ⁡
      
        (
        r
        )
      
    
    =
    
      
        cr
        2
      
      
        1
        +
        
          
            1
            -
            
              
                (
                
                  1
                  +
                  K
                
                )
              
              ⁢
              
                c
                2
              
              ⁢
              
                r
                2
              
            
          
        
      
    
  




  
    
      δ
      ⁢
      
        
      
      ⁢
      
        z
        e
      
    
    =
    
      
        z
        as
      
      -
      
        z
        s
      
    
  




  
    
      
        z
        N
      
      ⁡
      
        (
        r
        )
      
    
    =
    
      Re
      [
      
        
          
            
              c
              N
            
            ⁢
            
              r
              2
            
          
          
            1
            +
            
              
                1
                -
                
                  
                    (
                    
                      1
                      +
                      K
                    
                    )
                  
                  ⁢
                  
                    c
                    N
                    2
                  
                  ⁢
                  
                    r
                    2
                  
                
              
            
          
        
        +
        
          
            a
            
              N
              ⁢
              
                
              
              ⁢
              4
            
          
          ⁢
          
            r
            4
          
        
        +
        
          
            a
            
              N
              ⁢
              
                
              
              ⁢
              6
            
          
          ⁢
          
            r
            6
          
        
        +
        
          
            a
            
              N
              ⁢
              
                
              
              ⁢
              8
            
          
          ⁢
          
            r
            8
          
        
        +
        
          
            a
            
              N
              ⁢
              
                
              
              ⁢
              10
            
          
          ⁢
          
            r
            10
          
        
      
      ]
    
  




  
    
      δ
      ⁢
      
        
      
      ⁢
      
        z
        d
      
    
    =
    
      
      
        
          z
          as
        
        -
        
          z
          N
        
      
      
    
  



where c and cN are the curvatures, which are the reciprocals of R and RN; zN is the curve determined by the measured coordinate; aN4, . . . , aN10 are the deformation coefficients calculated with our method, and δzd is the shape error, which is the deviation from the original curve zas.

In an embodiment of the present invention, the ri may be 0, the rf 35 mm, and the rstep about 3.5 mm.
The Ri may be 20 mm, the Rf 120 mm, and the Rstep about 0.0001 mm.
The predetermined threshold value may be about 10 nm.
The method may further comprises steps for: setting the R and the RN with the determined values; finding an error curve δze for each K value while changing K from Ki to Kf in steps of Kstep; finding the deformation coefficients from δze using an inverse matrix method; replacing KN and aN with each K value and the deformation coefficients found from the δze in zN, a curve determined by the measured coordinates; checking if a shape error, δzd, is within a predetermined threshold value; and determining KN that is closest to the original K.
The Ki may be −1.0, the Kf 1.0, and the Kstep about 0.001.
The inverse matrix method comprises steps for: representing the coordinate data (ri, zas(ri)) by n+1 nth-order polynomial equations with polynomial coefficients, a0, a1, . . . , an; writing the n+1 nth-order polynomial equations in a form of matrix multiplication, Xa=b, where X is a variable matrix, a is a coefficient vector, and b is a function vector; and calculating the polynomial coefficients, a0, a1, . . . , an using a=X−1b.
The deformation coefficients may be obtained by




  
    [
    
      
        
          
            a
            4
          
        
      
      
        
          
            a
            6
          
        
      
      
        
          
            a
            8
          
        
      
      
        
          
            a
            10
          
        
      
    
    ]
  
  =
  
    
      
        
          [
          
            
              
                
                  
                    (
                    
                      r
                      0
                    
                    )
                  
                  4
                
              
              
                
                  
                    (
                    
                      r
                      0
                    
                    )
                  
                  6
                
              
              
                
                  
                    (
                    
                      r
                      0
                    
                    )
                  
                  8
                
              
              
                
                  
                    (
                    
                      r
                      0
                    
                    )
                  
                  10
                
              
            
            
              
                
                  
                    (
                    
                      r
                      1
                    
                    )
                  
                  4
                
              
              
                
                  
                    (
                    
                      r
                      1
                    
                    )
                  
                  6
                
              
              
                
                  
                    (
                    
                      r
                      1
                    
                    )
                  
                  8
                
              
              
                
                  
                    (
                    
                      r
                      1
                    
                    )
                  
                  10
                
              
            
            
              
                ⋮
              
              
                ⋮
              
              
                ⋮
              
              
                ⋮
              
            
            
              
                
                  
                    (
                    
                      r
                      10
                    
                    )
                  
                  4
                
              
              
                
                  
                    (
                    
                      r
                      10
                    
                    )
                  
                  6
                
              
              
                
                  
                    (
                    
                      r
                      10
                    
                    )
                  
                  8
                
              
              
                
                  
                    (
                    
                      r
                      10
                    
                    )
                  
                  10
                
              
            
          
          ]
        
        
          -
          1
        
      
      ⁡
      
        [
        
          
            
              
                δ
                ⁢
                
                  
                
                ⁢
                
                  z
                  
                    e
                    ⁢
                    
                      
                    
                    ⁢
                    0
                  
                
              
            
          
          
            
              
                δ
                ⁢
                
                  
                
                ⁢
                
                  z
                  
                    e
                    ⁢
                    
                      
                    
                    ⁢
                    1
                  
                
              
            
          
          
            
              ⋮
            
          
          
            
              
                δ
                ⁢
                
                  
                
                ⁢
                
                  z
                  
                    e
                    ⁢
                    
                      
                    
                    ⁢
                    10
                  
                
              
            
          
        
        ]
      
    
    .
  




The method according to the present invention provides a powerful method of reconstructing aspheric surface from measurements. And, further the method can be applied to aspheric lens optics easily.



BRIEF DESCRIPTION OF DRAWINGS
FIG. 1 shows a flowchart illustrating steps for finding R in a method of reconstructing aspheric surface;
FIG. 2 shows a flowchart illustrating steps for finding K in a method of reconstructing aspheric surface;
FIG. 3 shows three different curves illustrating an original aspheric curve, a spherical curve, and an error curve;
FIGS. 4(a) to 4(d) each show procedures for finding a radius of curvature;
FIG. 5(a) shows a photograph of a setup for measuring surface; and
FIG. 5(b) shows a aspheric surface profile reconstructed by using an inverse matrix method.



DETAILED DESCRIPTION OF INVENTION
FIGS. 1 and 2 show flowcharts illustrating methods of reconstructing of unknown aspheric surface equations from measurement of aspheric surfaces.
A method of reconstructing aspheric surface equations in an information processing device from measurements, the method comprising steps for: obtaining zas(r) as r varies from ri to rf in steps of rstep by measuring a given aspheric surface that is represented by an aspheric surface equation (S10); finding an error curve δze for each R value while changing R from Ri to Rf in steps of Rstep wherein K and deformation coefficients are set to 0, wherein the error curve δze is defined by a deviation of the aspheric surface equation from a quadratic term zs(r) of the aspheric surface equation (S20); finding the deformation coefficients from δze using an inverse matrix method (S30); replacing RN and aN with each R value and the deformation coefficients found from the δze in zN, a curve determined by the measured coordinates (S40); checking if a shape error, δzd, is within a predetermined threshold value (S50); and determining RN that is closest to the original R (S60).
The aspheric surface equation may be given by




  
    
      z
      ⁡
      
        (
        r
        )
      
    
    =
    
      
        
          
            r
            2
          
          R
        
        
          1
          +
          
            
              1
              -
              
                
                  (
                  
                    1
                    +
                    K
                  
                  )
                
                ⁢
                
                  
                    (
                    
                      r
                      R
                    
                    )
                  
                  2
                
              
            
          
        
      
      +
      
        
          ∑
          
            i
            =
            2
          
        
        ⁢
        
          
            a
            
              2
              ⁢
              i
            
          
          ⁢
          
            r
            
              2
              ⁢
              i
            
          
        
      
    
  
  ,




where K is the conic constant, r is the lateral coordinate, R is the radius of curvature, and the a2i, values are the deformation coefficients.

The zas(ri), the zs(r), the δze, the zN, and the δzd may be given by



  
    
      
        z
        as
      
      ⁡
      
        (
        r
        )
      
    
    =
    
      
        
          cr
          2
        
        
          1
          +
          
            
              1
              -
              
                
                  (
                  
                    1
                    +
                    K
                  
                  )
                
                ⁢
                
                  c
                  2
                
                ⁢
                
                  r
                  2
                
              
            
          
        
      
      +
      
        
          a
          4
        
        ⁢
        
          r
          4
        
      
      +
      
        
          a
          6
        
        ⁢
        
          r
          6
        
      
      +
      
        
          a
          8
        
        ⁢
        
          r
          8
        
      
      +
      
        
          a
          10
        
        ⁢
        
          r
          10
        
      
    
  




  
    
      
        z
        s
      
      ⁡
      
        (
        r
        )
      
    
    =
    
      
        cr
        2
      
      
        1
        +
        
          
            1
            -
            
              
                (
                
                  1
                  +
                  K
                
                )
              
              ⁢
              
                c
                2
              
              ⁢
              
                r
                2
              
            
          
        
      
    
  




  
    
      δ
      ⁢
      
        
      
      ⁢
      
        z
        e
      
    
    =
    
      
        z
        as
      
      -
      
        z
        s
      
    
  




  
    
      
        z
        N
      
      ⁡
      
        (
        r
        )
      
    
    =
    
      Re
      [
      
        
          
            
              c
              N
            
            ⁢
            
              r
              2
            
          
          
            1
            +
            
              
                1
                -
                
                  
                    (
                    
                      1
                      +
                      K
                    
                    )
                  
                  ⁢
                  
                    c
                    N
                    2
                  
                  ⁢
                  
                    r
                    2
                  
                
              
            
          
        
        +
        
          
            a
            
              N
              ⁢
              
                
              
              ⁢
              4
            
          
          ⁢
          
            r
            4
          
        
        +
        
          
            a
            
              N
              ⁢
              
                
              
              ⁢
              6
            
          
          ⁢
          
            r
            6
          
        
        +
        
          
            a
            
              N
              ⁢
              
                
              
              ⁢
              8
            
          
          ⁢
          
            r
            8
          
        
        +
        
          
            a
            
              N
              ⁢
              
                
              
              ⁢
              10
            
          
          ⁢
          
            r
            10
          
        
      
      ]
    
  




  
    
      δ
      ⁢
      
        
      
      ⁢
      
        z
        d
      
    
    =
    
      
      
        
          z
          as
        
        -
        
          z
          N
        
      
      
    
  



where c and cN are the curvatures, which are the reciprocals of R and RN; zN is the curve determined by the measured coordinate; aN4, . . . , aN10 are the deformation coefficients calculated with our method, and δzd is the shape error, which is the deviation from the original curve zas.

In an embodiment of the present invention, the ri may be 0, the rf 35 mm, and the rstep about 3.5 mm. The Ri may be 20 mm, the Rf 120 mm, and the Rstep about 0.0001 mm. The predetermined threshold value may be about 10 nm.
In addition to the above steps for finding R, in order to find K, the method may further comprises steps for: setting the R and the RN with the determined values (S70); finding an error curve δze for each K value while changing K from Ki to Kf in steps of Kstep (S80); finding the deformation coefficients from δze using an inverse matrix method (S90); replacing KN and aN with each K value and the deformation coefficients found from the δze in zN, a curve determined by the measured coordinates (S100); checking if a shape error, δzd, is within a predetermined threshold value (S110); and determining KN that is closest to the original K (S120).
The Ki may be −1.0, the Kf 1.0, and the Kstep about 0.001.
The inverse matrix method comprises steps for: representing the coordinate data (ri, zas(ri)) by n+1 nth-order polynomial equations with polynomial coefficients, a0, a1, . . . , an; writing the n+1 nth-order polynomial equations in a form of matrix multiplication, Xa=b, where X is a variable matrix, a is a coefficient vector, and b is a function vector; and calculating the polynomial coefficients, a0, a1, . . . , an using a=X−1b.
The deformation coefficients may be obtained by




  
    [
    
      
        
          
            a
            4
          
        
      
      
        
          
            a
            6
          
        
      
      
        
          
            a
            8
          
        
      
      
        
          
            a
            10
          
        
      
    
    ]
  
  =
  
    
      
        
          [
          
            
              
                
                  
                    (
                    
                      r
                      0
                    
                    )
                  
                  4
                
              
              
                
                  
                    (
                    
                      r
                      0
                    
                    )
                  
                  6
                
              
              
                
                  
                    (
                    
                      r
                      0
                    
                    )
                  
                  8
                
              
              
                
                  
                    (
                    
                      r
                      0
                    
                    )
                  
                  10
                
              
            
            
              
                
                  
                    (
                    
                      r
                      1
                    
                    )
                  
                  4
                
              
              
                
                  
                    (
                    
                      r
                      1
                    
                    )
                  
                  6
                
              
              
                
                  
                    (
                    
                      r
                      1
                    
                    )
                  
                  8
                
              
              
                
                  
                    (
                    
                      r
                      1
                    
                    )
                  
                  10
                
              
            
            
              
                ⋮
              
              
                ⋮
              
              
                ⋮
              
              
                ⋮
              
            
            
              
                
                  
                    (
                    
                      r
                      10
                    
                    )
                  
                  4
                
              
              
                
                  
                    (
                    
                      r
                      10
                    
                    )
                  
                  6
                
              
              
                
                  
                    (
                    
                      r
                      10
                    
                    )
                  
                  8
                
              
              
                
                  
                    (
                    
                      r
                      10
                    
                    )
                  
                  10
                
              
            
          
          ]
        
        
          -
          1
        
      
      ⁡
      
        [
        
          
            
              
                δ
                ⁢
                
                  
                
                ⁢
                
                  z
                  
                    e
                    ⁢
                    
                      
                    
                    ⁢
                    0
                  
                
              
            
          
          
            
              
                δ
                ⁢
                
                  
                
                ⁢
                
                  z
                  
                    e
                    ⁢
                    
                      
                    
                    ⁢
                    1
                  
                
              
            
          
          
            
              ⋮
            
          
          
            
              
                δ
                ⁢
                
                  
                
                ⁢
                
                  z
                  
                    e
                    ⁢
                    
                      
                    
                    ⁢
                    10
                  
                
              
            
          
        
        ]
      
    
    .
  




In general, arbitrary curves including two-dimensional shapes are given by data in the form of various coordinates, (x0, y0), (x1, y1), . . . , (xn, yn). This data can be represented by an nth-order polynomial equation, as follows:

y=a0+a1x+a2x2+ . . . +anxn  (1)

where a0, a1, . . . , an are the polynomial coefficients. Substituting each of the n+1 (x, y) pairs into Eq. (1) yields the following n+1 equations:





  
    
      
        
          
            
              a
              0
            
            +
            
              
                a
                1
              
              ⁢
              
                x
                0
              
            
            +
            
              
                
                  a
                  2
                
                ⁡
                
                  (
                  
                    x
                    0
                  
                  )
                
              
              2
            
            +
            ...
            +
            
              
                
                  a
                  n
                
                ⁡
                
                  (
                  
                    x
                    0
                  
                  )
                
              
              n
            
          
          =
          
            y
            0
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              a
              0
            
            +
            
              
                a
                1
              
              ⁢
              
                x
                1
              
            
            +
            
              
                
                  a
                  2
                
                ⁡
                
                  (
                  
                    x
                    1
                  
                  )
                
              
              2
            
            +
            ...
            +
            
              
                
                  a
                  n
                
                ⁡
                
                  (
                  
                    x
                    1
                  
                  )
                
              
              n
            
          
          =
          
            y
            1
          
        
        ⁢
        
          

        
        ⁢
        ⋮
        ⁢
        
          

        
        ⁢
        
          
            
              a
              0
            
            +
            
              
                a
                1
              
              ⁢
              
                x
                n
              
            
            +
            
              
                
                  a
                  2
                
                ⁡
                
                  (
                  
                    x
                    n
                  
                  )
                
              
              2
            
            +
            ...
            +
            
              
                
                  a
                  n
                
                ⁡
                
                  (
                  
                    x
                    n
                  
                  )
                
              
              n
            
          
          =
          
            y
            n
          
        
      
    
    
      
        (
        2
        )
      
    
  




Equation (2) is a system of nth-order equations having the unknown quantities a0, a1, . . . , an. Here, however, Eq. (2) can be considered a system of linear equations with respect to the ai coefficients (where i=0, . . . , n) because the (x, y) values can be determined through measurements. Therefore, the equation can be written in the form of matrix multiplication, Xa=b, where X is the variable matrix, a is the coefficient vector, and b is the function vector. The result, represented as a matrix calculation, is as follows:




  
    
      
        
          
            [
            
              
                
                  1
                
                
                  
                    x
                    0
                  
                
                
                  
                    
                      (
                      
                        x
                        0
                      
                      )
                    
                    2
                  
                
                
                  ...
                
                
                  
                    
                      (
                      
                        x
                        0
                      
                      )
                    
                    n
                  
                
              
              
                
                  1
                
                
                  
                    x
                    1
                  
                
                
                  
                    
                      (
                      
                        x
                        1
                      
                      )
                    
                    2
                  
                
                
                  ...
                
                
                  
                    
                      (
                      
                        x
                        1
                      
                      )
                    
                    n
                  
                
              
              
                
                  ⋮
                
                
                  ⋮
                
                
                  ⋮
                
                
                  ⋱
                
                
                  ⋮
                
              
              
                
                  1
                
                
                  
                    x
                    n
                  
                
                
                  
                    
                      (
                      
                        x
                        n
                      
                      )
                    
                    2
                  
                
                
                  ...
                
                
                  
                    
                      (
                      
                        x
                        n
                      
                      )
                    
                    n
                  
                
              
            
            ]
          
          ⁡
          
            [
            
              
                
                  
                    a
                    0
                  
                
              
              
                
                  
                    a
                    1
                  
                
              
              
                
                  ⋮
                
              
              
                
                  
                    a
                    n
                  
                
              
            
            ]
          
        
        =
        
          [
          
            
              
                
                  y
                  0
                
              
            
            
              
                
                  y
                  1
                
              
            
            
              
                ⋮
              
            
            
              
                
                  y
                  n
                
              
            
          
          ]
        
      
    
    
      
        (
        3
        )
      
    
  




We want to solve Eq. (3) for the elements of the vector a. So, a can be found by multiplying the inverse matrix X−1 of X at either side, as follows:

X−1Xa=X−1b

a=X−1b  (4)

In order for it be possible to calculate the inverse of the matrix X (that is, for X to be invertible), det(X) must not be equal to zero. In practice, if a is found by Eq. (4), the original equation can be reconstructed from the coefficients in the vector a. If we measure the curve given by a certain equation, the measured values will be y values for the corresponding x values. An important point here is that when we do not know the equation of the curve, the measured value will be y, which contains the coefficients multiplied by suitable powers of the corresponding x value. So, the coefficients become the elements of the vector a extracted from the calculation in Eq. (4). We refer to this method as the inverse matrix method.
In general, an aspheric surface equation7 is given by




  
    
      
        
          z
          ⁡
          
            (
            r
            )
          
        
        =
        
          
            
              
                r
                2
              
              R
            
            
              1
              +
              
                
                  1
                  -
                  
                    
                      (
                      
                        1
                        +
                        K
                      
                      )
                    
                    ⁢
                    
                      
                        (
                        
                          r
                          R
                        
                        )
                      
                      2
                    
                  
                
              
            
          
          +
          
            
              ∑
              
                i
                =
                2
              
            
            ⁢
            
              
                a
                
                  2
                  ⁢
                  i
                
              
              ⁢
              
                r
                
                  2
                  ⁢
                  i
                
              
            
          
        
      
    
    
      
        (
        5
        )
      
    
  




where K is the conic constant, r is the lateral coordinate, R is the radius of curvature, and the a2i values are the deformation coefficients. To verify the validity of the inverse matrix method, let us suppose an arbitrary aspheric equation zas, which has a radius of curvature at the vertex is R=89 mm, a diameter of φ=70 mm, a conic constant of K=0, and deformation coefficients of a4=−5×10−9, a6=−2.075×10−9, a8=7.65625×10−13, and a10=−1.67481×10−16. Now, we must reconstruct the aspheric surface equation from the coordinate data (r, zas) of Eq. (5) with the known parameters. When K=0 and a2i=0, Eq. (5) includes a spherical equation that can be considered as the basic form of the aspheric surface equation. So R, the radius of the circle, must be found. In order to find R, we must calculate the error curve δze(r)=zas−zs, where zas is the entire equation itself and zs is just the quadratic term in Eq. (5), neglecting the higher-order terms. FIG. 3 shows (a) the original aspheric curve zas, (b) the spherical curve zs, and (c) the error curve δze, which represents the difference between zas and zs. The deformation coefficients can be found from the error curve by using the inverse matrix method.

As r varies from 0 to 35 mm in steps of 3.5 mm, 11 data points are obtained from δze(r). Here, as there are 11 r values, 11 deformation coefficients are also obtained. Among these coefficients, we are interested in the 4th-, 6th-, 8th-, and 10th-order coefficients due to Eq. (5). Therefore, others coefficients, like the constant and the odd-order coefficients, are omitted in the process of matrix calculation of the error curve. This calculation of the error curve is also in the form of Xa=b like Eq. (3) and is given by Eq. (6). In Eq. (6), the X matrix is 11×4, and the a vector is 4×1, so the b vector must be 11×1. In order to find the coefficient vector a, both sides of the equation are multiplied by the inverse matrix X−1, and since X−1 is 4×11, the product of X−1 and b will be 4×1. Therefore, the matrix calculation works because the result equals a, as shown by Eq. (7).




  
    
      
        
          
            [
            
              
                
                  
                    
                      (
                      
                        r
                        0
                      
                      )
                    
                    4
                  
                
                
                  
                    
                      (
                      
                        r
                        0
                      
                      )
                    
                    6
                  
                
                
                  
                    
                      (
                      
                        r
                        0
                      
                      )
                    
                    8
                  
                
                
                  
                    
                      (
                      
                        r
                        0
                      
                      )
                    
                    10
                  
                
              
              
                
                  
                    
                      (
                      
                        r
                        1
                      
                      )
                    
                    4
                  
                
                
                  
                    
                      (
                      
                        r
                        1
                      
                      )
                    
                    6
                  
                
                
                  
                    
                      (
                      
                        r
                        1
                      
                      )
                    
                    8
                  
                
                
                  
                    
                      (
                      
                        r
                        1
                      
                      )
                    
                    10
                  
                
              
              
                
                  ⋮
                
                
                  ⋮
                
                
                  ⋮
                
                
                  ⋮
                
              
              
                
                  
                    
                      (
                      
                        r
                        10
                      
                      )
                    
                    4
                  
                
                
                  
                    
                      (
                      
                        r
                        10
                      
                      )
                    
                    6
                  
                
                
                  
                    
                      (
                      
                        r
                        10
                      
                      )
                    
                    8
                  
                
                
                  
                    
                      (
                      
                        r
                        10
                      
                      )
                    
                    10
                  
                
              
            
            ]
          
          ⁡
          
            [
            
              
                
                  
                    a
                    4
                  
                
              
              
                
                  
                    a
                    6
                  
                
              
              
                
                  
                    a
                    8
                  
                
              
              
                
                  
                    a
                    10
                  
                
              
            
            ]
          
        
        =
        
          [
          
            
              
                
                  δ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    z
                    
                      e
                      ⁢
                      
                        
                      
                      ⁢
                      0
                    
                  
                
              
            
            
              
                
                  δ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    z
                    
                      e
                      ⁢
                      
                        
                      
                      ⁢
                      1
                    
                  
                
              
            
            
              
                ⋮
              
            
            
              
                
                  δ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    z
                    
                      e
                      ⁢
                      
                        
                      
                      ⁢
                      10
                    
                  
                
              
            
          
          ]
        
      
    
    
      
        (
        6
        )
      
    
  
  
    
      
        
          [
          
            
              
                
                  a
                  4
                
              
            
            
              
                
                  a
                  6
                
              
            
            
              
                
                  a
                  8
                
              
            
            
              
                
                  a
                  10
                
              
            
          
          ]
        
        =
        
          
            
              [
              
                
                  
                    
                      
                        (
                        
                          r
                          0
                        
                        )
                      
                      4
                    
                  
                  
                    
                      
                        (
                        
                          r
                          0
                        
                        )
                      
                      6
                    
                  
                  
                    
                      
                        (
                        
                          r
                          0
                        
                        )
                      
                      8
                    
                  
                  
                    
                      
                        (
                        
                          r
                          0
                        
                        )
                      
                      10
                    
                  
                
                
                  
                    
                      
                        (
                        
                          r
                          1
                        
                        )
                      
                      4
                    
                  
                  
                    
                      
                        (
                        
                          r
                          1
                        
                        )
                      
                      6
                    
                  
                  
                    
                      
                        (
                        
                          r
                          1
                        
                        )
                      
                      8
                    
                  
                  
                    
                      
                        (
                        
                          r
                          1
                        
                        )
                      
                      10
                    
                  
                
                
                  
                    ⋮
                  
                  
                    ⋮
                  
                  
                    ⋮
                  
                  
                    ⋮
                  
                
                
                  
                    
                      
                        (
                        
                          r
                          10
                        
                        )
                      
                      4
                    
                  
                  
                    
                      
                        (
                        
                          r
                          10
                        
                        )
                      
                      6
                    
                  
                  
                    
                      
                        (
                        
                          r
                          10
                        
                        )
                      
                      8
                    
                  
                  
                    
                      
                        (
                        
                          r
                          10
                        
                        )
                      
                      10
                    
                  
                
              
              ]
            
            
              -
              1
            
          
          ⁡
          
            [
            
              
                
                  
                    δ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      z
                      
                        e
                        ⁢
                        
                          
                        
                        ⁢
                        0
                      
                    
                  
                
              
              
                
                  
                    δ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      z
                      
                        e
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                    
                  
                
              
              
                
                  ⋮
                
              
              
                
                  
                    δ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      z
                      
                        e
                        ⁢
                        
                          
                        
                        ⁢
                        10
                      
                    
                  
                
              
            
            ]
          
        
      
    
    
      
        (
        7
        )
      
    
  




In reality, we only know the coordinate data (r, zas) from the surface measurement and not R, K, and the a coefficient values. The following equations are given to find R, K, and the a coefficients.




  
    
      
        
          
            
              z
              as
            
            ⁡
            
              (
              r
              )
            
          
          =
          
            
              
                cr
                2
              
              
                1
                +
                
                  
                    1
                    -
                    
                      
                        (
                        
                          1
                          +
                          K
                        
                        )
                      
                      ⁢
                      
                        c
                        2
                      
                      ⁢
                      
                        r
                        2
                      
                    
                  
                
              
            
            +
            
              
                a
                4
              
              ⁢
              
                r
                4
              
            
            +
            
              
                a
                6
              
              ⁢
              
                r
                6
              
            
            +
            
              
                a
                8
              
              ⁢
              
                r
                8
              
            
            +
            
              
                a
                10
              
              ⁢
              
                r
                10
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              z
              s
            
            ⁡
            
              (
              r
              )
            
          
          =
          
            
              cr
              2
            
            
              1
              +
              
                
                  1
                  -
                  
                    
                      (
                      
                        1
                        +
                        K
                      
                      )
                    
                    ⁢
                    
                      c
                      2
                    
                    ⁢
                    
                      r
                      2
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            δ
            ⁢
            
              
            
            ⁢
            
              z
              e
            
          
          =
          
            
              z
              as
            
            -
            
              z
              s
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              z
              N
            
            ⁡
            
              (
              r
              )
            
          
          =
          
            Re
            [
            
              
                
                  
                    c
                    N
                  
                  ⁢
                  
                    r
                    2
                  
                
                
                  1
                  +
                  
                    
                      1
                      -
                      
                        
                          (
                          
                            1
                            +
                            K
                          
                          )
                        
                        ⁢
                        
                          c
                          N
                          2
                        
                        ⁢
                        
                          r
                          2
                        
                      
                    
                  
                
              
              +
              
                
                  a
                  
                    N
                    ⁢
                    
                      
                    
                    ⁢
                    4
                  
                
                ⁢
                
                  r
                  4
                
              
              +
              
                
                  a
                  
                    N
                    ⁢
                    
                      
                    
                    ⁢
                    6
                  
                
                ⁢
                
                  r
                  6
                
              
              +
              
                
                  a
                  
                    N
                    ⁢
                    
                      
                    
                    ⁢
                    8
                  
                
                ⁢
                
                  r
                  8
                
              
              +
              
                
                  a
                  
                    N
                    ⁢
                    
                      
                    
                    ⁢
                    10
                  
                
                ⁢
                
                  r
                  10
                
              
            
            ]
          
        
        ⁢
        
          

        
        ⁢
        
          
            δ
            ⁢
            
              
            
            ⁢
            
              z
              d
            
          
          =
          
            
            
              
                z
                as
              
              -
              
                z
                N
              
            
            
          
        
      
    
    
      
        (
        8
        )
      
    
  




where c and cN are the curvatures, which are the reciprocals of R and RN; zN is the curve determined by the measured coordinate; aN4, . . . , aN10 are the deformation coefficients calculated with our method, and δzd is the shape error, which is the deviation from the original curve zas. The procedure for finding the parameters is as follows:

(1) Obtain zas through measurements as r varies from 0 to 35 mm in steps of 3.5 mm.
(2) Change R from 20 to 120 mm in steps of 0.0001 mm, where K and a4, . . . , a10 are set to 0.
(3) Find the error curve δze for each R value.
(4) Find the deformation coefficients from δze using the inverse matrix method.
(5) In zN, replace RN and aN with each R value and the deformation coefficients found from δze.
(6) Check if δzd is within a desired threshold value.
This procedure was implemented in an information processing unit such as a computer, using a well-known MATLAB® program and was repeated iteratively until the δzd value fell below some threshold value. In this case, the threshold value was 1×10−5 mm (10 nm), and then the relevant RN was 89.0013 mm. FIGS. 4(a) to 4(d) each shows the procedure for finding the RN closest to the original R, more specifically, for finding the radius of curvature RN, where the red line is zas, the cyan line is zN, and the green line is zd, in which (a) RN=20.0000 mm, (b) RN=60.0000 mm, (c) RN=89.0013 mm, and (d) RN=120.0000 mm. As RN approaches the original R, δzd goes to 0. Here, the minimized δzd was obtained when RN=89.0013 mm as shown in FIG. 4(c). After that, RN increased further, and δzd increased again. Table 1 shows δzd for various r values when RN is 89.0013 mm. Even the largest δzd is no larger than about 9.77 nm at r=31.5 mm, so this result is quite satisfactory. One thing to keep in mind is that the coordinate data of zas will correspond to the measurement data of an unknown aspheric lens surface later, and it will become a reference used to find R.







TABLE 1







Minimized δzd values found for RN = 89.0013 mm









RN (mm)
r (mm)
δzd (mm)












89.0013
0
0.0000



3.5
9.2793 × 10−7



7
2.8688 × 10−6



10.5
3.8795 × 10−6



14
2.3828 × 10−6



17.5
1.2827 × 10−6



21
4.3100 × 10−6



24.5
2.8575 × 10−6



28
4.1517 × 10−6



31.5
9.7711 × 10−6



35
4.6779 × 10−6









The deformation coefficients extracted from the inverse matrix method (7) for RN=89.0013 mm are compared with the original one in Table 2. The extracted







TABLE 2







Comparison between original and extracted deformation


coefficients









Deformation coefficient
Original
Extracted





a4
    −5 × 10−9
 −4.465 × 10−9


a6
 −2.075 × 10−9
 −2.076 × 10−9


a8
 7.65625 × 10−13
 7.6632 × 10−13


 a10
−1.67481 × 10−16
−1.6763 × 10−16









where coefficients are very similar to the original ones. This suggests that the inverse matrix method is quite effective.

In the case where K=−0.8 in Eq. (5) rather than K=0, we were able to find R, K, and a using the inverse matrix method. In the same way, when K=0, R was found to be 88.998. In this case, the R value (88.998) differs from its value in the previous case (89.0013) due to the difference in K. Now, K can be also determined in a way that is similar to the way that R was found. To find K, we applied R=88.998 to Eq. (8) and changed K from −1 to 1 in steps of 0.001, and then examined the value of K that occurred for shape error values less than δzd=1×10−5 mm. The resultant K was −0.732, and at that time, the largest δzd was 8.6152×10−6 mm (about 8.6 nm). Finally, the deformation coefficients were a4=−1.7652×10−8, a6=−2.0746×10−9, a8=7.6537×10−13, and a10=−1.6752×10−16. Although the extracted K and the deformation coefficients are somewhat different than the original ones, the result is satisfactory because the largest shape error δzd is no larger than about 8.6 nm.
As a last step we applied the inverse matrix method to a real situation. A lens surface with unknown parameters was measured by a CONTURA G2® (Karl Zeiss®) which is a 3-dimensional measurement device. The parameters extracted from the measured coordinates using the inverse matrix method were R=121.33 mm, K=0, a4=1.4193×10−6, a6=−3.4706×10−9, a8=3.8349×10−12, and a10=−1.5313×10−15. Then the largest shape error δzd between the measured coordinate data and the data from the reconstructed equation was 2.328×10−3 mm (about 2.3 μm). In this case, a major cause of δzd being higher than that in the theoretical case may be the precision limitation of the measurement equipment, which has a repeatability precision of 1.8 μm, and the uncertainty of the measurement7, which is the doubt that exists about the result of any measurement. Accordingly, these limitations can be overcome if equipment with higher precision is used and if a large number of measurements are performed. FIG. 5 shows the measurement of the aspheric lens surface and the surface profile realized by using the reconstructed surface equation found from the inverse matrix method. The procedure for finding R, K, and the deformation coefficients of the unknown aspheric lens surface is shown with (a) photograph of the setup for the surface measurement and (b) the aspheric surface profile reconstructed by using the inverse matrix method.
In the above steps, all the programming codes, intermediate results, and necessary information for performing them may be stored in a memory device of the information processing device. In other words, the memory of the computer may be modified with the results of the steps. Of course, the performing of the steps is not limited to MATLAB®. It is just a choice out of many appropriate hardware or software means that can process the inventive steps.
In conclusion, a new method using the inverse matrix method is disclosed. From the results of this study, it may be concluded that the inverse matrix method is a powerful means for reconstructing the unknown equations of various aspheric surfaces, and it is possible for this technique to be applied to other fields.

## Claims

1. A method of reconstructing aspheric surface equations from measurements for aspheric lens fabrication, the method comprising steps for:
obtaining zas(r) as a lateral coordinate r varies from ri to rf in steps of rstep by measuring a given aspheric surface that is represented by an aspheric surface equation;
finding an error curve δze for each radius of curvature R value while changing R from Ri to Rf in steps of Rstep wherein a conic constant K and deformation coefficients are set to 0, wherein the error curve δze is defined by a deviation of the aspheric surface equation from a quadratic term zs(r) of the aspheric surface equation;
finding the deformation coefficients from δze using an inverse matrix method;
replacing RN and aN with each R value and the deformation coefficients found from the δze in zN, a curve determined by the measured coordinates;
checking if a shape error, δzd, is within a predetermined threshold value; and
determining RN that is closest to the original R,
whereby the aspheric surface equations are reconstructed to improve an optical function in such a device as an optical diagnostic computer system.

2. The method of claim 1, wherein the aspheric surface equation is given by




  
    
      z
      ⁡
      
        (
        r
        )
      
    
    =
    
      
        
          
            r
            2
          
          R
        
        
          1
          +
          
            
              1
              -
              
                
                  (
                  
                    1
                    +
                    K
                  
                  )
                
                ⁢
                
                  
                    (
                    
                      r
                      R
                    
                    )
                  
                  2
                
              
            
          
        
      
      +
      
        
          ∑
          
            i
            =
            2
          
        
        ⁢
        
          
            a
            
              2
              ⁢
              i
            
          
          ⁢
          
            r
            
              2
              ⁢
              i
            
          
        
      
    
  
  ,




where K is the conic constant, r is the lateral coordinate, R is the radius of curvature, and the a2i values are the deformation coefficients.

3. The method of claim 2, wherein the zas(ri), the zs(r), the δze, the zN, and the δzd are given by



  
    
      
        z
        as
      
      ⁡
      
        (
        r
        )
      
    
    =
    
      
        
          cr
          2
        
        
          1
          +
          
            
              1
              -
              
                
                  (
                  
                    1
                    +
                    K
                  
                  )
                
                ⁢
                
                  c
                  2
                
                ⁢
                
                  r
                  2
                
              
            
          
        
      
      +
      
        
          a
          4
        
        ⁢
        
          r
          4
        
      
      +
      
        
          a
          6
        
        ⁢
        
          r
          6
        
      
      +
      
        
          a
          8
        
        ⁢
        
          r
          8
        
      
      +
      
        
          a
          10
        
        ⁢
        
          r
          10
        
      
    
  




  
    
      
        z
        s
      
      ⁡
      
        (
        r
        )
      
    
    =
    
      
        cr
        2
      
      
        1
        +
        
          
            1
            -
            
              
                (
                
                  1
                  +
                  K
                
                )
              
              ⁢
              
                c
                2
              
              ⁢
              
                r
                2
              
            
          
        
      
    
  




  
    
      δ
      ⁢
      
        
      
      ⁢
      
        z
        e
      
    
    =
    
      
        z
        as
      
      -
      
        z
        s
      
    
  




  
    
      
        z
        N
      
      ⁡
      
        (
        r
        )
      
    
    =
    
      Re
      ⁡
      
        [
        
          
            
              
                c
                N
              
              ⁢
              
                r
                2
              
            
            
              1
              +
              
                
                  1
                  -
                  
                    
                      (
                      
                        1
                        +
                        K
                      
                      )
                    
                    ⁢
                    
                      c
                      N
                      2
                    
                    ⁢
                    
                      r
                      2
                    
                  
                
              
            
          
          +
          
            
              a
              
                N
                ⁢
                
                  
                
                ⁢
                4
              
            
            ⁢
            
              r
              4
            
          
          +
          
            
              a
              
                N
                ⁢
                
                  
                
                ⁢
                6
              
            
            ⁢
            
              r
              6
            
          
          +
          
            
              a
              
                N
                ⁢
                
                  
                
                ⁢
                8
              
            
            ⁢
            
              r
              8
            
          
          +
          
            
              a
              
                N
                ⁢
                
                  
                
                ⁢
                10
              
            
            ⁢
            
              r
              10
            
          
        
        ]
      
    
  




  
    
      δ
      ⁢
      
        
      
      ⁢
      
        z
        d
      
    
    =
    
      
      
        
          z
          as
        
        -
        
          z
          N
        
      
      
    
  



where c and cN are the curvatures, which are the reciprocals of R and RN; zN is the curve determined by the measured coordinate; aN4, . . . , aN10 are the deformation coefficients calculated with our method, and δzd is the shape error, which is the deviation from the original curve zas.

4. The method of claim 3, wherein the ri is 0, the rf is 35 mm, and the rstep is about 3.5 mm.

5. The method of claim 3, wherein the Ri is 20 mm, the Rf is 120 mm, and the Rstep is about 0.0001 mm.

6. The method of claim 3, wherein the predetermined threshold value is about 10 nm.

7. The method of claim 3, further comprising steps for:
setting the R and the RN with the determined values;
finding an error curve δze, for each K value while changing K from Ki to Kf in steps of Kstep;
finding the deformation coefficients from δze using an inverse matrix method;
replacing KN and aN with each K value and the deformation coefficients found from the δzze, in zN, a curve determined by the measured coordinates;
checking if a shape error, δzd, is within a predetermined threshold value; and
determining KN that is closest to the original K.

8. The method of claim 7, wherein the Ki is −1.0, the Kf is 1.0, and the Kstep is about 0.001.

9. The method of claim 3, wherein the inverse matrix method comprises steps for:
representing the coordinate data (ri, zas(ri)) by n+1 nth-order polynomial equations with polynomial coefficients, a0, a1, . . . , an;
writing the n+1 nth-order polynomial equations in a form of matrix multiplication, Xa =b, where X is a variable matrix, a is a coefficient vector, and b is a function vector; and
calculating the polynomial coefficients, a0, a1, . . . ,an using a=X−1b.

10. The method of claim 9, wherein the deformation coefficients are obtained by




  
    [
    
      
        
          
            a
            4
          
        
      
      
        
          
            a
            6
          
        
      
      
        
          
            a
            8
          
        
      
      
        
          
            a
            10
          
        
      
    
    ]
  
  =
  
    
      
        
          [
          
            
              
                
                  
                    (
                    
                      r
                      0
                    
                    )
                  
                  4
                
              
              
                
                  
                    (
                    
                      r
                      0
                    
                    )
                  
                  6
                
              
              
                
                  
                    (
                    
                      r
                      0
                    
                    )
                  
                  8
                
              
              
                
                  
                    (
                    
                      r
                      0
                    
                    )
                  
                  10
                
              
            
            
              
                
                  
                    (
                    
                      r
                      1
                    
                    )
                  
                  4
                
              
              
                
                  
                    (
                    
                      r
                      1
                    
                    )
                  
                  6
                
              
              
                
                  
                    (
                    
                      r
                      1
                    
                    )
                  
                  8
                
              
              
                
                  
                    (
                    
                      r
                      1
                    
                    )
                  
                  10
                
              
            
            
              
                ⋮
              
              
                ⋮
              
              
                ⋮
              
              
                ⋮
              
            
            
              
                
                  
                    (
                    
                      r
                      10
                    
                    )
                  
                  4
                
              
              
                
                  
                    (
                    
                      r
                      10
                    
                    )
                  
                  6
                
              
              
                
                  
                    (
                    
                      r
                      10
                    
                    )
                  
                  8
                
              
              
                
                  
                    (
                    
                      r
                      10
                    
                    )
                  
                  10
                
              
            
          
          ]
        
        
          -
          1
        
      
      ⁡
      
        [
        
          
            
              
                δ
                ⁢
                
                  
                
                ⁢
                
                  z
                  
                    e
                    ⁢
                    
                      
                    
                    ⁢
                    0
                  
                
              
            
          
          
            
              
                δ
                ⁢
                
                  
                
                ⁢
                
                  z
                  
                    e
                    ⁢
                    
                      
                    
                    ⁢
                    1
                  
                
              
            
          
          
            
              ⋮
            
          
          
            
              
                δ
                ⁢
                
                  
                
                ⁢
                
                  z
                  
                    e
                    ⁢
                    
                      
                    
                    ⁢
                    10
                  
                
              
            
          
        
        ]
      
    
    .