Abstract

Interferograms obtained with ordinary interferometers, such as the Fizeau interferometer or the Twyman–Green interferometer, show the contour maps of a wave front under test. On the other hand, lateral shearing interferograms show the difference between a wave front under test and a sheared wave front, that is, the inclination of the wave front. Therefore the shape of the wave front under test is reconstructed by means of analyzing the difference. To reconstruct the wave front, many methods have been proposed. An integration method is usually used to reconstruct the wave front under test rapidly. However, this method has two disadvantages: The analysis accuracy of the method is low, and part of the wave front cannot be measured. To overcome these two problems, a new, to our knowledge, integration method, improved by use of polynomials, is proposed. The validity of the proposed method is evaluated by computer simulations. In the simulations the analysis accuracy achieved by the proposed method is compared with the analysis accuracy of the ordinary integration method and that of the method proposed by Rimmer and Wyant. The results of the simulations show that the analysis accuracy of the newly proposed method is better than that of the integration method and that of the Rimmer–Wyant method.

© 2000 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  4. W. Shen, M. W. Chang, D. S. Wan, “Zernike polynomial fitting of lateral shearing interferometry,” Opt. Eng. 36, 905–913 (1997).
    [CrossRef]
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    [CrossRef]
  6. H. Schreiber, J. Schwider, “Lateral shearing interferometer based on two Ronchi phase gratings in series,” Appl. Opt. 36, 5321–5324 (1997).
    [CrossRef] [PubMed]
  7. M. P. Rimmer, J. C. Wyant, “Evaluation of large aberration using a lateral-shear interferometer having variable shear,” Appl. Opt. 14, 142–150 (1975).
    [CrossRef] [PubMed]
  8. T. Yatagai, T. Kanou, “Aspherical surface testing with shearing interferometer using the fringe scanning detection method,” Opt. Eng. 23, 357–360 (1997).
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    [CrossRef]
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    [CrossRef]
  11. D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, New York, 1992), pp. 461–466.

1999 (1)

1998 (1)

T. Nomura, K. Kamiya, S. Okuda, H. Miyashiro, K. Yoshikawa, H. Tashiro, “Shape measurements of mirror surfaces with a lateral-shearing interferometer during machine running,” Precis. Eng. 22, 185–189 (1998).
[CrossRef]

1997 (3)

H. Schreiber, J. Schwider, “Lateral shearing interferometer based on two Ronchi phase gratings in series,” Appl. Opt. 36, 5321–5324 (1997).
[CrossRef] [PubMed]

T. Yatagai, T. Kanou, “Aspherical surface testing with shearing interferometer using the fringe scanning detection method,” Opt. Eng. 23, 357–360 (1997).

W. Shen, M. W. Chang, D. S. Wan, “Zernike polynomial fitting of lateral shearing interferometry,” Opt. Eng. 36, 905–913 (1997).
[CrossRef]

1996 (2)

1978 (1)

F. M. Dickey, T. M. Harder, “Shearing plate optical alignment,” Opt. Eng. 17, 295–298 (1978).
[CrossRef]

1975 (1)

1974 (1)

Chang, M. W.

W. Shen, M. W. Chang, D. S. Wan, “Zernike polynomial fitting of lateral shearing interferometry,” Opt. Eng. 36, 905–913 (1997).
[CrossRef]

Dickey, F. M.

F. M. Dickey, T. M. Harder, “Shearing plate optical alignment,” Opt. Eng. 17, 295–298 (1978).
[CrossRef]

Elster, C.

Harbers, G.

Harder, T. M.

F. M. Dickey, T. M. Harder, “Shearing plate optical alignment,” Opt. Eng. 17, 295–298 (1978).
[CrossRef]

Kamiya, K.

T. Nomura, K. Kamiya, S. Okuda, H. Miyashiro, K. Yoshikawa, H. Tashiro, “Shape measurements of mirror surfaces with a lateral-shearing interferometer during machine running,” Precis. Eng. 22, 185–189 (1998).
[CrossRef]

Kanou, T.

T. Yatagai, T. Kanou, “Aspherical surface testing with shearing interferometer using the fringe scanning detection method,” Opt. Eng. 23, 357–360 (1997).

Kunst, P. J.

Leibbrandt, G. W. R.

Malacara, D.

Marroquin, J. L.

Miyashiro, H.

T. Nomura, K. Kamiya, S. Okuda, H. Miyashiro, K. Yoshikawa, H. Tashiro, “Shape measurements of mirror surfaces with a lateral-shearing interferometer during machine running,” Precis. Eng. 22, 185–189 (1998).
[CrossRef]

Nomura, T.

T. Nomura, K. Kamiya, S. Okuda, H. Miyashiro, K. Yoshikawa, H. Tashiro, “Shape measurements of mirror surfaces with a lateral-shearing interferometer during machine running,” Precis. Eng. 22, 185–189 (1998).
[CrossRef]

Okuda, S.

T. Nomura, K. Kamiya, S. Okuda, H. Miyashiro, K. Yoshikawa, H. Tashiro, “Shape measurements of mirror surfaces with a lateral-shearing interferometer during machine running,” Precis. Eng. 22, 185–189 (1998).
[CrossRef]

Rimmer, M. P.

Schreiber, H.

Schwider, J.

Servin, M.

Shen, W.

W. Shen, M. W. Chang, D. S. Wan, “Zernike polynomial fitting of lateral shearing interferometry,” Opt. Eng. 36, 905–913 (1997).
[CrossRef]

Smartt, R. N.

Tashiro, H.

T. Nomura, K. Kamiya, S. Okuda, H. Miyashiro, K. Yoshikawa, H. Tashiro, “Shape measurements of mirror surfaces with a lateral-shearing interferometer during machine running,” Precis. Eng. 22, 185–189 (1998).
[CrossRef]

Wan, D. S.

W. Shen, M. W. Chang, D. S. Wan, “Zernike polynomial fitting of lateral shearing interferometry,” Opt. Eng. 36, 905–913 (1997).
[CrossRef]

Weingartner, I.

Wyant, J. C.

Yatagai, T.

T. Yatagai, T. Kanou, “Aspherical surface testing with shearing interferometer using the fringe scanning detection method,” Opt. Eng. 23, 357–360 (1997).

Yoshikawa, K.

T. Nomura, K. Kamiya, S. Okuda, H. Miyashiro, K. Yoshikawa, H. Tashiro, “Shape measurements of mirror surfaces with a lateral-shearing interferometer during machine running,” Precis. Eng. 22, 185–189 (1998).
[CrossRef]

Appl. Opt. (6)

Opt. Eng. (3)

T. Yatagai, T. Kanou, “Aspherical surface testing with shearing interferometer using the fringe scanning detection method,” Opt. Eng. 23, 357–360 (1997).

F. M. Dickey, T. M. Harder, “Shearing plate optical alignment,” Opt. Eng. 17, 295–298 (1978).
[CrossRef]

W. Shen, M. W. Chang, D. S. Wan, “Zernike polynomial fitting of lateral shearing interferometry,” Opt. Eng. 36, 905–913 (1997).
[CrossRef]

Precis. Eng. (1)

T. Nomura, K. Kamiya, S. Okuda, H. Miyashiro, K. Yoshikawa, H. Tashiro, “Shape measurements of mirror surfaces with a lateral-shearing interferometer during machine running,” Precis. Eng. 22, 185–189 (1998).
[CrossRef]

Other (1)

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, New York, 1992), pp. 461–466.

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Figures (13)

Fig. 1
Fig. 1

Lateral shearing interferometer.

Fig. 2
Fig. 2

Lateral shearing interferogram.

Fig. 3
Fig. 3

Wave front under test and two sheared wave fronts.

Fig. 4
Fig. 4

Measuring area: (a) interference area of the lateral shearing interferogram; (b) area used for analysis in lateral shearing interferograms; (c) area reconstructed by integration method.

Fig. 5
Fig. 5

Result of measurement with the Fizeau interferometer: (a) Fizeau interferogram; (b) three-dimensional plot of the shape error.

Fig. 6
Fig. 6

Shape reconstructed with Zernike coefficients.

Fig. 7
Fig. 7

Analysis error in shape reconstructed with the integration method.

Fig. 8
Fig. 8

Analysis error in shape reconstructed with the proposed method.

Fig. 9
Fig. 9

Comparison of the integration method and proposed method: (a) relationship between shear ratio and relative error of PV; (b) relationship between shear ratio and relative error of rms.

Fig. 10
Fig. 10

Analysis error in shape reconstructed with the Rimmer–Wyant method.

Fig. 11
Fig. 11

Analysis error in shape reconstructed with the proposed method.

Fig. 12
Fig. 12

Comparison of the Rimmer–Wyant and proposed methods: (a) relationship between the shear ratio and the relative error of PV; (b) relationship between the shear ratio and the relative error of the rms.

Fig. 13
Fig. 13

Condition number of the matrices used.

Tables (3)

Tables Icon

Table 1 Zernike Polynomials in Cartesian Coordinates to as High as the Fourth Order

Tables Icon

Table 2 Polynomials Obtained by Integration Method

Tables Icon

Table 3 Coefficients of Zernike Polynomials

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

Wx, y=i=1L AiUix, y,
ΔWxx, y, s=Wx+s/2, y-Wx-s/2, y=i=1L AiΔUixx, y, s,
ΔUixx, y, s=Uix+s/2, y-Uix-s/2, y.
ΔWyx, y, s=Wx, y+s/2-Wx, y-s/2=i=1L AiΔUiyx, y, s,
ΔUiyx, y, s=Uix, y+s/2-Uix, y-s/2.
Wx, y, s=i=1L AiUix, y, s,
Uix, y, s=1sx0x ΔUiξξ, y, sdξ+y0y ΔUiηx0, η, sdη.
X2+Y21,
X=x/1-s,  Y=y/1-s.
WX, Y, s=i=1L AiUiX, Y, s.
UiX, Y, s=m=1L amiUmX, Y,
WX, Y, s=i=1L AiUiX, Y, s=i=1LAim=1L amiUmX, Y=i=1L AiUiX, Y,
Ai=Aim=1L aim.
c=Mb,
c=A1A2AL, b=A1A2AL, M=a11a12a1La21a22a2LaL1aL2aLL.
b=M-1c.

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