Abstract

In this paper the conjugate differential method is proposed to measure the absolute surface shape of the flat mirror using a phase-shifting interferometer. The conjugate differential method is derived from the differential method, which extracts absolute phase differences by introducing the slight transverse shifts of the optic. It employs the measurement schemes making transverse shifts on the orthogonally bilateral symmetry positions. So the measurement procedures have been changed into four-step tests to get the phase difference map instead of three-step tests for the differential method. The precision of the slope approximation is enhanced by reducing couplings between multi-step tests, and the reliability of the measurements can be improved. Several differential wavefront reconstruction methods, such as Fourier transform, Zernike polynomial fitting and Hudgin model method, can be applied to reconstruct the absolute surface shape from the differencing phase maps in four different simulation environment. They were also used to reconstruct the absolute surface shape with the conjugate differential method in the experiment. Our method accords with the classical three-flat test better than the traditional differential method, where the deviation of RMS value between the conjugate differential method and the three-flat test is less than 0.3 nm.

© 2015 Optical Society of America

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References

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2014 (1)

2013 (2)

Z. Han, L. Chen, T. Wulan, and R. Zhu, “The absolute flatness measurements of two aluminum coated mirrors based on the skip flat test,” Optik (Stuttg.) 124(19), 3781–3785 (2013).
[Crossref]

W. Song, F. Wu, X. Hou, W. Zhao, and Y. Wan, “Absolute measurement of flats with the method of shift-rotation,” Opt. Rev. 20(5), 374–377 (2013).
[Crossref]

2012 (2)

2011 (1)

2010 (1)

2009 (1)

2008 (1)

2006 (2)

G. M. Dai, “Comparison of wavefront reconstructions with Zernike polynomials and Fourier transforms,” J. Refract. Surg. 22(9), 943–948 (2006).
[PubMed]

U. Griesmann, “Three-flat test solutions based on simple mirror symmetry,” Appl. Opt. 45(23), 5856–5865 (2006).
[Crossref] [PubMed]

2005 (1)

M. Vannoni and G. Molesini, “Validation of absolute planarity reference plates with a liquid mirror,” Metrologia 42(5), 389–393 (2005).
[Crossref]

2001 (2)

1996 (1)

1992 (1)

C. Ai and J. C. Wyant, “Absolute testing of flatness decomposed to even and odd functions,” Proc. SPIE 1776, 73–83 (1992).
[Crossref]

1984 (2)

B. S. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 23(4), 379–383 (1984).
[Crossref]

P. B. Keenan, “Pseudo-shear interferometry,” Proc. SPIE 123(4), 2–9 (1984).

1980 (1)

1968 (1)

1967 (1)

1893 (1)

L. Rayleigh, “Interference bands and their applications,” Nature 48(1235), 212–214 (1893).
[Crossref]

Ai, C.

C. Ai and J. C. Wyant, “Absolute testing of flatness decomposed to even and odd functions,” Proc. SPIE 1776, 73–83 (1992).
[Crossref]

Bloemhof, E. E.

Bon, P.

Bünnagel, R.

Chen, L.

Z. Han, L. Chen, T. Wulan, and R. Zhu, “The absolute flatness measurements of two aluminum coated mirrors based on the skip flat test,” Optik (Stuttg.) 124(19), 3781–3785 (2013).
[Crossref]

C. Xu, L. Chen, and J. Yin, “Method for absolute flatness measurement of optical surfaces,” Appl. Opt. 48(13), 2536–2541 (2009).
[Crossref] [PubMed]

Dai, G. M.

G. M. Dai, “Comparison of wavefront reconstructions with Zernike polynomials and Fourier transforms,” J. Refract. Surg. 22(9), 943–948 (2006).
[PubMed]

Evans, C. J.

Freischlad, K. R.

Fritz, B. S.

B. S. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 23(4), 379–383 (1984).
[Crossref]

Gao, Z.

Griesmann, U.

Han, Z.

Z. Han, L. Chen, T. Wulan, and R. Zhu, “The absolute flatness measurements of two aluminum coated mirrors based on the skip flat test,” Optik (Stuttg.) 124(19), 3781–3785 (2013).
[Crossref]

Hou, X.

W. Song, F. Wu, X. Hou, W. Zhao, and Y. Wan, “Absolute measurement of flats with the method of shift-rotation,” Opt. Rev. 20(5), 374–377 (2013).
[Crossref]

Keenan, P. B.

P. B. Keenan, “Pseudo-shear interferometry,” Proc. SPIE 123(4), 2–9 (1984).

Kestner, R. N.

Küchel, M. F.

M. F. Küchel, “A new approach to solve the three flat problem,” Optik (Stuttg.) 112(9), 381–391 (2001).
[Crossref]

Ma, J.

Miao, E.

Molesini, G.

M. Vannoni and G. Molesini, “Validation of absolute planarity reference plates with a liquid mirror,” Metrologia 42(5), 389–393 (2005).
[Crossref]

Monneret, S.

Nam, J.

Oehring, H. A.

Osten, W.

Pruss, C.

Rayleigh, L.

L. Rayleigh, “Interference bands and their applications,” Nature 48(1235), 212–214 (1893).
[Crossref]

Rubinstein, J.

Schulz, G.

Schwider, J.

Song, W.

W. Song, F. Wu, X. Hou, W. Zhao, and Y. Wan, “Absolute measurement of flats with the method of shift-rotation,” Opt. Rev. 20(5), 374–377 (2013).
[Crossref]

Southwell, W. H.

Steiner, K.

Su, D.

Sui, Y.

Vannoni, M.

M. Vannoni and G. Molesini, “Validation of absolute planarity reference plates with a liquid mirror,” Metrologia 42(5), 389–393 (2005).
[Crossref]

Wan, Y.

W. Song, F. Wu, X. Hou, W. Zhao, and Y. Wan, “Absolute measurement of flats with the method of shift-rotation,” Opt. Rev. 20(5), 374–377 (2013).
[Crossref]

Wattellier, B.

Wu, F.

W. Song, F. Wu, X. Hou, W. Zhao, and Y. Wan, “Absolute measurement of flats with the method of shift-rotation,” Opt. Rev. 20(5), 374–377 (2013).
[Crossref]

Wulan, T.

Z. Han, L. Chen, T. Wulan, and R. Zhu, “The absolute flatness measurements of two aluminum coated mirrors based on the skip flat test,” Optik (Stuttg.) 124(19), 3781–3785 (2013).
[Crossref]

Wyant, J. C.

C. Ai and J. C. Wyant, “Absolute testing of flatness decomposed to even and odd functions,” Proc. SPIE 1776, 73–83 (1992).
[Crossref]

Xu, C.

Yang, H.

Yin, J.

Yuan, C.

Zhao, W.

W. Song, F. Wu, X. Hou, W. Zhao, and Y. Wan, “Absolute measurement of flats with the method of shift-rotation,” Opt. Rev. 20(5), 374–377 (2013).
[Crossref]

Zhu, R.

Z. Han, L. Chen, T. Wulan, and R. Zhu, “The absolute flatness measurements of two aluminum coated mirrors based on the skip flat test,” Optik (Stuttg.) 124(19), 3781–3785 (2013).
[Crossref]

J. Ma, C. Pruss, R. Zhu, Z. Gao, C. Yuan, and W. Osten, “An absolute test for axicon surfaces,” Opt. Lett. 36(11), 2005–2007 (2011).
[Crossref] [PubMed]

Appl. Opt. (8)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Refract. Surg. (1)

G. M. Dai, “Comparison of wavefront reconstructions with Zernike polynomials and Fourier transforms,” J. Refract. Surg. 22(9), 943–948 (2006).
[PubMed]

Metrologia (1)

M. Vannoni and G. Molesini, “Validation of absolute planarity reference plates with a liquid mirror,” Metrologia 42(5), 389–393 (2005).
[Crossref]

Nature (1)

L. Rayleigh, “Interference bands and their applications,” Nature 48(1235), 212–214 (1893).
[Crossref]

Opt. Eng. (1)

B. S. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 23(4), 379–383 (1984).
[Crossref]

Opt. Lett. (3)

Opt. Rev. (1)

W. Song, F. Wu, X. Hou, W. Zhao, and Y. Wan, “Absolute measurement of flats with the method of shift-rotation,” Opt. Rev. 20(5), 374–377 (2013).
[Crossref]

Optik (Stuttg.) (2)

M. F. Küchel, “A new approach to solve the three flat problem,” Optik (Stuttg.) 112(9), 381–391 (2001).
[Crossref]

Z. Han, L. Chen, T. Wulan, and R. Zhu, “The absolute flatness measurements of two aluminum coated mirrors based on the skip flat test,” Optik (Stuttg.) 124(19), 3781–3785 (2013).
[Crossref]

Proc. SPIE (2)

P. B. Keenan, “Pseudo-shear interferometry,” Proc. SPIE 123(4), 2–9 (1984).

C. Ai and J. C. Wyant, “Absolute testing of flatness decomposed to even and odd functions,” Proc. SPIE 1776, 73–83 (1992).
[Crossref]

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Figures (13)

Fig. 1
Fig. 1 (a) Differential method and (b) conjugate differential method experimental process schematic diagram.
Fig. 2
Fig. 2 The ideal (a) test surface and (b) reference surface (units: nm)
Fig. 3
Fig. 3 Reconstructed surface by conjugate differential method with (a) Fourier transform method, (b) Zernike Fitting method and (c) Hudgin model method. (d), (e) and (f) show the residual errors corresponding to these three wavefront reconstruction methods (units: nm).
Fig. 4
Fig. 4 Reconstructed surface by differential method with (a) Fourier transform method, (b) Zernike Fitting method and (c) Hudgin model method. (d), (e) and (f) show the residual errors corresponding to these three wavefront reconstruction methods (units: nm).
Fig. 5
Fig. 5 Residual errors of (a) Fourier transform method, (b) Zernike Fitting method and (c) Hudgin model method reconstructed by conjugate differential method with 2 nm noise (units: nm)
Fig. 6
Fig. 6 Residual errors trend curves with different random noise.
Fig. 7
Fig. 7 Residual errors of (a) Fourier transform method, (b) Zernike Fitting method and (c) Hudgin model method reconstructed by conjugate differential method with 0.02 mm translation along x direction (units: nm).
Fig. 8
Fig. 8 Residual errors trend curves with different translation error along x direction.
Fig. 9
Fig. 9 Residual errors of (a) Fourier transform method, (b) Zernike Fitting method and (c) Hudgin model method reconstructed by conjugate differential method with 0.4° rotation about z axes (units: nm).
Fig. 10
Fig. 10 Residual errors trend curves with different rotation deviation about z axes.
Fig. 11
Fig. 11 Measurement results at five positions (the values showed in the textbox with types of PV/RMS and units of nm).
Fig. 12
Fig. 12 The absolute surface shape reconstructed by conjugate differential method with (a) Fourier transform method, (b) Zernike Fitting method and (c) Hudgin model method (units: nm).
Fig. 13
Fig. 13 Surface shape result at x = 0 by conjugate differential method, differential method and traditional three-flat method.

Tables (2)

Tables Icon

Table 1 Wavefront reconstruction errors of the test surface (Ideal PV = 118.5nm and RMS = 15.2nm, the values showed with types of PV/RMS unit: nm)

Tables Icon

Table 2 PV and RMS of the surface shape at x = 0

Equations (5)

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Φ ( x , y ) = ϕ t e s t ( x , y ) + ϕ r e f e r e n c e ( x , y ) + ϕ noise ( x , y ) + ϕ adjustment ( x , y )
Φ ( x , y ) = ϕ t e s t ( x , y ) + ϕ r e f e r e n c e ( x , y )
Φ ( x δ , y ) = ϕ t e s t ( x δ , y ) + ϕ r e f e r e n c e ( x , y ) Φ ( x + δ , y ) = ϕ t e s t ( x + δ , y ) + ϕ r e f e r e n c e ( x , y ) Φ ( x , y + δ ) = ϕ t e s t ( x , y + δ ) + ϕ r e f e r e n c e ( x , y ) Φ ( x , y δ ) = ϕ t e s t ( x , y δ ) + ϕ r e f e r e n c e ( x , y )
Φ ( x + δ , y ) Φ ( x δ , y ) = ϕ t e s t ( x + δ , y ) ϕ t e s t ( x δ , y ) Φ ( x , y + δ ) Φ ( x , y δ ) = ϕ t e s t ( x , y + δ ) ϕ t e s t ( x , y δ )
d w x ( x , y ) = ϕ t e s t ( x + δ , y ) ϕ t e s t ( x δ , y ) 2 δ = Φ ( x + δ , y ) Φ ( x δ , y ) 2 δ d w y ( x , y ) = ϕ t e s t ( x , y + δ ) ϕ t e s t ( x , y δ ) 2 δ = Φ ( x , y + δ ) Φ ( x , y δ ) 2 δ

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