Abstract

An interferogram obtained by use of ordinary interferometers, such as Fizeau and Twyman-Green interferometers, will show a contour map of the wave front under test. A lateral-shearing interferogram, however, will show a contour map of the difference between the wave front under test and a sheared wave front, that is, a contour map of the derivative of the wave front under test. Therefore one can reconstruct the shape of the wave front under test by analyzing that difference. Many methods for reconstructing a wave front have been proposed. The Saunders method reconstructs a wave front; rapidly however the wave-front data are reconstructed only at intervals of the amount of shear along the direction of the shear. Therefore the method has low spatial resolution. A method for reconstructing a wave front that is based on the Saunders method and has high spatial resolution is proposed. The method analyzes the differences that are produced by shearing of the wave front under test in many directions. This method requires a large number of interferograms for reconstructing the wave front. Here the method is described, and its validity is confirmed by simulation.

© 2002 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2000 (2)

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 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)

1994 (1)

T. Nomura, K. Kamiya, H. Miyashiro, K. Yoshikawa, H. Tashiro, M. Suzuki, S. Ozono, F. Kobayashi, M. Usuki, “An instrument for manufacturing zone-plates by using a lathe,” Precis. Eng. 16, 290–295 (1994).
[CrossRef]

1993 (1)

T. Nomura, H. Miyashiro, K. Kamiya, K. Yoshikawa, H. Tashiro, M. Suzuki, M. Usuki, F. Kobayashi, “Shape measurement of workpiece surface with zone-plate interferometer during machine running,” Precis. Eng. 15, 86–92 (1993).
[CrossRef]

1978 (1)

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

1975 (1)

1974 (1)

1961 (1)

J. B. Saunders, “Measurement of wave fronts without a reference standard. 1. The wave-front-shearing interferometer,” J. Res. Natl. Bur. Stand. Sect. B 65, 239–244 (1961).
[CrossRef]

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]

Hatsuzo, H.

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]

T. Nomura, K. Kamiya, H. Miyashiro, K. Yoshikawa, H. Tashiro, M. Suzuki, S. Ozono, F. Kobayashi, M. Usuki, “An instrument for manufacturing zone-plates by using a lathe,” Precis. Eng. 16, 290–295 (1994).
[CrossRef]

T. Nomura, H. Miyashiro, K. Kamiya, K. Yoshikawa, H. Tashiro, M. Suzuki, M. Usuki, F. Kobayashi, “Shape measurement of workpiece surface with zone-plate interferometer during machine running,” Precis. Eng. 15, 86–92 (1993).
[CrossRef]

Kanou, T.

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

Kazuhide, K.

Kobayashi, F.

T. Nomura, K. Kamiya, H. Miyashiro, K. Yoshikawa, H. Tashiro, M. Suzuki, S. Ozono, F. Kobayashi, M. Usuki, “An instrument for manufacturing zone-plates by using a lathe,” Precis. Eng. 16, 290–295 (1994).
[CrossRef]

T. Nomura, H. Miyashiro, K. Kamiya, K. Yoshikawa, H. Tashiro, M. Suzuki, M. Usuki, F. Kobayashi, “Shape measurement of workpiece surface with zone-plate interferometer during machine running,” Precis. Eng. 15, 86–92 (1993).
[CrossRef]

Kunst, P. J.

Leibbrandt, G. W. R.

Malacara, D.

Marroquin, J. L.

Miyashiro, H.

S. Okuda, T. Nomura, K. Kazuhide, H. Miyashiro, H. Hatsuzo, K. Yoshikawa, “High precision analysis of lateral shearing interferogram using the integration method and polynomials,” Appl. Opt. 39, 5179–5186 (2000).
[CrossRef]

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]

T. Nomura, K. Kamiya, H. Miyashiro, K. Yoshikawa, H. Tashiro, M. Suzuki, S. Ozono, F. Kobayashi, M. Usuki, “An instrument for manufacturing zone-plates by using a lathe,” Precis. Eng. 16, 290–295 (1994).
[CrossRef]

T. Nomura, H. Miyashiro, K. Kamiya, K. Yoshikawa, H. Tashiro, M. Suzuki, M. Usuki, F. Kobayashi, “Shape measurement of workpiece surface with zone-plate interferometer during machine running,” Precis. Eng. 15, 86–92 (1993).
[CrossRef]

Nomura, T.

S. Okuda, T. Nomura, K. Kazuhide, H. Miyashiro, H. Hatsuzo, K. Yoshikawa, “High precision analysis of lateral shearing interferogram using the integration method and polynomials,” Appl. Opt. 39, 5179–5186 (2000).
[CrossRef]

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]

T. Nomura, K. Kamiya, H. Miyashiro, K. Yoshikawa, H. Tashiro, M. Suzuki, S. Ozono, F. Kobayashi, M. Usuki, “An instrument for manufacturing zone-plates by using a lathe,” Precis. Eng. 16, 290–295 (1994).
[CrossRef]

T. Nomura, H. Miyashiro, K. Kamiya, K. Yoshikawa, H. Tashiro, M. Suzuki, M. Usuki, F. Kobayashi, “Shape measurement of workpiece surface with zone-plate interferometer during machine running,” Precis. Eng. 15, 86–92 (1993).
[CrossRef]

Okuda, S.

S. Okuda, T. Nomura, K. Kazuhide, H. Miyashiro, H. Hatsuzo, K. Yoshikawa, “High precision analysis of lateral shearing interferogram using the integration method and polynomials,” Appl. Opt. 39, 5179–5186 (2000).
[CrossRef]

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]

Ozono, S.

T. Nomura, K. Kamiya, H. Miyashiro, K. Yoshikawa, H. Tashiro, M. Suzuki, S. Ozono, F. Kobayashi, M. Usuki, “An instrument for manufacturing zone-plates by using a lathe,” Precis. Eng. 16, 290–295 (1994).
[CrossRef]

Paez, G.

Rimmer, M. P.

Saunders, J. B.

J. B. Saunders, “Measurement of wave fronts without a reference standard. 1. The wave-front-shearing interferometer,” J. Res. Natl. Bur. Stand. Sect. B 65, 239–244 (1961).
[CrossRef]

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.

Strojnik, M.

Suzuki, M.

T. Nomura, K. Kamiya, H. Miyashiro, K. Yoshikawa, H. Tashiro, M. Suzuki, S. Ozono, F. Kobayashi, M. Usuki, “An instrument for manufacturing zone-plates by using a lathe,” Precis. Eng. 16, 290–295 (1994).
[CrossRef]

T. Nomura, H. Miyashiro, K. Kamiya, K. Yoshikawa, H. Tashiro, M. Suzuki, M. Usuki, F. Kobayashi, “Shape measurement of workpiece surface with zone-plate interferometer during machine running,” Precis. Eng. 15, 86–92 (1993).
[CrossRef]

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]

T. Nomura, K. Kamiya, H. Miyashiro, K. Yoshikawa, H. Tashiro, M. Suzuki, S. Ozono, F. Kobayashi, M. Usuki, “An instrument for manufacturing zone-plates by using a lathe,” Precis. Eng. 16, 290–295 (1994).
[CrossRef]

T. Nomura, H. Miyashiro, K. Kamiya, K. Yoshikawa, H. Tashiro, M. Suzuki, M. Usuki, F. Kobayashi, “Shape measurement of workpiece surface with zone-plate interferometer during machine running,” Precis. Eng. 15, 86–92 (1993).
[CrossRef]

Torales, G. G.

Usuki, M.

T. Nomura, K. Kamiya, H. Miyashiro, K. Yoshikawa, H. Tashiro, M. Suzuki, S. Ozono, F. Kobayashi, M. Usuki, “An instrument for manufacturing zone-plates by using a lathe,” Precis. Eng. 16, 290–295 (1994).
[CrossRef]

T. Nomura, H. Miyashiro, K. Kamiya, K. Yoshikawa, H. Tashiro, M. Suzuki, M. Usuki, F. Kobayashi, “Shape measurement of workpiece surface with zone-plate interferometer during machine running,” Precis. Eng. 15, 86–92 (1993).
[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]

Weingärtner, I.

Wyant, J. C.

Yatagai, T.

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

Yoshikawa, K.

S. Okuda, T. Nomura, K. Kazuhide, H. Miyashiro, H. Hatsuzo, K. Yoshikawa, “High precision analysis of lateral shearing interferogram using the integration method and polynomials,” Appl. Opt. 39, 5179–5186 (2000).
[CrossRef]

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]

T. Nomura, K. Kamiya, H. Miyashiro, K. Yoshikawa, H. Tashiro, M. Suzuki, S. Ozono, F. Kobayashi, M. Usuki, “An instrument for manufacturing zone-plates by using a lathe,” Precis. Eng. 16, 290–295 (1994).
[CrossRef]

T. Nomura, H. Miyashiro, K. Kamiya, K. Yoshikawa, H. Tashiro, M. Suzuki, M. Usuki, F. Kobayashi, “Shape measurement of workpiece surface with zone-plate interferometer during machine running,” Precis. Eng. 15, 86–92 (1993).
[CrossRef]

Appl. Opt. (8)

J. Res. Natl. Bur. Stand. Sect. B (1)

J. B. Saunders, “Measurement of wave fronts without a reference standard. 1. The wave-front-shearing interferometer,” J. Res. Natl. Bur. Stand. Sect. B 65, 239–244 (1961).
[CrossRef]

Opt. Eng. (3)

T. Yatagai, T. Kanou, “Aspherical surface testing with shearing interferometer using 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]

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

Precis. Eng. (3)

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]

T. Nomura, H. Miyashiro, K. Kamiya, K. Yoshikawa, H. Tashiro, M. Suzuki, M. Usuki, F. Kobayashi, “Shape measurement of workpiece surface with zone-plate interferometer during machine running,” Precis. Eng. 15, 86–92 (1993).
[CrossRef]

T. Nomura, K. Kamiya, H. Miyashiro, K. Yoshikawa, H. Tashiro, M. Suzuki, S. Ozono, F. Kobayashi, M. Usuki, “An instrument for manufacturing zone-plates by using a lathe,” Precis. Eng. 16, 290–295 (1994).
[CrossRef]

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Figures (19)

Fig. 1
Fig. 1

Lateral-shearing interferometer.

Fig. 2
Fig. 2

Difference within digitized image.

Fig. 3
Fig. 3

Lateral shear relative to rotation of the mirror.

Fig. 4
Fig. 4

Lateral shear in the 0° direction.

Fig. 5
Fig. 5

Lateral shear in the θ direction.

Fig. 6
Fig. 6

Three of the reconstructed points.

Fig. 7
Fig. 7

Reconstructed points for five directions.

Fig. 8
Fig. 8

Application of ΛW (x, y).

Fig. 9
Fig. 9

Application of ΔW 90°(x, y).

Fig. 10
Fig. 10

Set of scale lines.

Fig. 11
Fig. 11

Set of scale lines obtained with a different starting point from that for Fig. 10.

Fig. 12
Fig. 12

Reverse scale lines.

Fig. 13
Fig. 13

Scale lines placed on the reverse scale lines. (a) Placement of one set of scale lines; (b) placement of two sets of scale lines.

Fig. 14
Fig. 14

Shape under test. PV value, 1.87λ; rms value, 0.433λ.

Fig. 15
Fig. 15

Lateral-shearing interferogram.

Fig. 16
Fig. 16

Set of scale lines obtained with starting point (0, 0).

Fig. 17
Fig. 17

Reconstructed shape. PV value, 1.90λ; rms value, 0.433λ.

Fig. 18
Fig. 18

Analysis error between shape under test and reconstructed shape. PV value, 3.32 × 10-2λ; rms value, 7.32 × 10-3λ.

Fig. 19
Fig. 19

Shape reconstructed by the Saunders method. We have magnified the points to show their size more clearly. PV value, 1.85λ; rms value, 0.434λ.

Equations (6)

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

ΔW0°x, y=Wx-s, y-Wx, y.
ΔWθx, y=Wx-s cos θ, y-s sin θ-Wx, y,
d=s21-cos θ1/2.
Wx, y=a cos2πlx2+y2+bx2+y2cosmθ,
x2+y21,
θ=arctany/x,

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