Abstract

We have proposed a new absolute method to test rotationally asymmetric surface deviation. Relying on the high accuracy of Zernike polynomial fitting with least-squares algorithm for the low-frequency component and preserving the high-frequency component with the averaging method, the new method can guarantee the high accuracy of the measurement result with fewer rotational measurements compared to the traditional multiangle averaging method. It realizes a balance between the accuracy and efficiency of the measurements. It has been verified by experiments; the root mean square (rms) of residual figure between the two methods is 0.6nm. Meanwhile, the new method can suppress environmental noise introduced in measurement results well.

© 2012 Optical Society of America

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References

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  1. B. S. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 23, 379–383 (1984).
  2. P. Hariharan, “Interferometric testing of optical surfaces: absolute measurements of flatness,” Opt. Eng. 36, 2478–2481 (1997).
    [CrossRef]
  3. V. Greco, R. Tronconi, C. Del Vecchio, M. Trivi, and G. Molesini, “Absolute measurement of planarity with Fritz’s method: uncertainty evaluation,” Appl. Opt. 38, 2018–2027 (1999).
    [CrossRef]
  4. S.-W. Kim and H.-G. Rhee, “Self-calibration of high frequency errors of test optics by arbitrary N-step rotation,” Int. J. Korean Soc. Precision Eng. 1, 115–123 (2000).
  5. X. Chen, C. Lei, and W. Tu-ya, “Rotational variant error removal of the reference flat,” Acta Meterol. Sin. 31, 219–222 (2010) (in Chinese).
  6. C. J. Evans and R. N. Kestner, “Test optics error removal,” Appl. Opt. 35, 1015–1021 (1996).
    [CrossRef]
  7. S. Weihong, W. Fan, and H. Xi, “Simulation analysis on absolute testing of spherical surface with shift-rotation method,” High Power Laser Particle Beams 23, 1–5 (2011) (in Chinese).
  8. H. Ichikawa and T. Yamamoto, “Apparatus and method for wavefront absolute calibration and method of synthesizing wavefronts,” U.S. patent 5,982,490 (9November1999).
  9. K. Otaki, T. Yamamoto, and Y. Fukuda, “Accuracy evaluation of the point diffraction interferometer for extreme ultraviolet lithography aspheric mirror,” J. Vac. Sci. Technol. B 20, 295–300 (2002).
    [CrossRef]
  10. G. Seitz and W. Otto, “Method for the interferometric measurement of non-rotationally symmetric wavefront errors,” U.S. patent 7,277,186 (2October2007).
  11. W. Otto and Aalen-Waldhausen, “Method for the interferometric measurement of non-rotationally symmetric wavefront errors,” U.S. patent 6,839,143 (4January2005).
  12. D. Malacara, Optical Shop Testing3rd ed. (Wiley, 2007).
  13. C. B. Moler, Numerical Computing with MATLAB (SIAM, 2004).
  14. L. A. Selberg, “Interferometer accuracy and precision,” Proc. SPIE 1400, 24–32 (1990).
    [CrossRef]

2011

S. Weihong, W. Fan, and H. Xi, “Simulation analysis on absolute testing of spherical surface with shift-rotation method,” High Power Laser Particle Beams 23, 1–5 (2011) (in Chinese).

2010

X. Chen, C. Lei, and W. Tu-ya, “Rotational variant error removal of the reference flat,” Acta Meterol. Sin. 31, 219–222 (2010) (in Chinese).

2002

K. Otaki, T. Yamamoto, and Y. Fukuda, “Accuracy evaluation of the point diffraction interferometer for extreme ultraviolet lithography aspheric mirror,” J. Vac. Sci. Technol. B 20, 295–300 (2002).
[CrossRef]

2000

S.-W. Kim and H.-G. Rhee, “Self-calibration of high frequency errors of test optics by arbitrary N-step rotation,” Int. J. Korean Soc. Precision Eng. 1, 115–123 (2000).

1999

1997

P. Hariharan, “Interferometric testing of optical surfaces: absolute measurements of flatness,” Opt. Eng. 36, 2478–2481 (1997).
[CrossRef]

1996

1990

L. A. Selberg, “Interferometer accuracy and precision,” Proc. SPIE 1400, 24–32 (1990).
[CrossRef]

1984

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

Aalen-Waldhausen,

W. Otto and Aalen-Waldhausen, “Method for the interferometric measurement of non-rotationally symmetric wavefront errors,” U.S. patent 6,839,143 (4January2005).

Chen, X.

X. Chen, C. Lei, and W. Tu-ya, “Rotational variant error removal of the reference flat,” Acta Meterol. Sin. 31, 219–222 (2010) (in Chinese).

Del Vecchio, C.

Evans, C. J.

Fan, W.

S. Weihong, W. Fan, and H. Xi, “Simulation analysis on absolute testing of spherical surface with shift-rotation method,” High Power Laser Particle Beams 23, 1–5 (2011) (in Chinese).

Fritz, B. S.

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

Fukuda, Y.

K. Otaki, T. Yamamoto, and Y. Fukuda, “Accuracy evaluation of the point diffraction interferometer for extreme ultraviolet lithography aspheric mirror,” J. Vac. Sci. Technol. B 20, 295–300 (2002).
[CrossRef]

Greco, V.

Hariharan, P.

P. Hariharan, “Interferometric testing of optical surfaces: absolute measurements of flatness,” Opt. Eng. 36, 2478–2481 (1997).
[CrossRef]

Ichikawa, H.

H. Ichikawa and T. Yamamoto, “Apparatus and method for wavefront absolute calibration and method of synthesizing wavefronts,” U.S. patent 5,982,490 (9November1999).

Kestner, R. N.

Kim, S.-W.

S.-W. Kim and H.-G. Rhee, “Self-calibration of high frequency errors of test optics by arbitrary N-step rotation,” Int. J. Korean Soc. Precision Eng. 1, 115–123 (2000).

Lei, C.

X. Chen, C. Lei, and W. Tu-ya, “Rotational variant error removal of the reference flat,” Acta Meterol. Sin. 31, 219–222 (2010) (in Chinese).

Malacara, D.

D. Malacara, Optical Shop Testing3rd ed. (Wiley, 2007).

Moler, C. B.

C. B. Moler, Numerical Computing with MATLAB (SIAM, 2004).

Molesini, G.

Otaki, K.

K. Otaki, T. Yamamoto, and Y. Fukuda, “Accuracy evaluation of the point diffraction interferometer for extreme ultraviolet lithography aspheric mirror,” J. Vac. Sci. Technol. B 20, 295–300 (2002).
[CrossRef]

Otto, W.

G. Seitz and W. Otto, “Method for the interferometric measurement of non-rotationally symmetric wavefront errors,” U.S. patent 7,277,186 (2October2007).

W. Otto and Aalen-Waldhausen, “Method for the interferometric measurement of non-rotationally symmetric wavefront errors,” U.S. patent 6,839,143 (4January2005).

Rhee, H.-G.

S.-W. Kim and H.-G. Rhee, “Self-calibration of high frequency errors of test optics by arbitrary N-step rotation,” Int. J. Korean Soc. Precision Eng. 1, 115–123 (2000).

Seitz, G.

G. Seitz and W. Otto, “Method for the interferometric measurement of non-rotationally symmetric wavefront errors,” U.S. patent 7,277,186 (2October2007).

Selberg, L. A.

L. A. Selberg, “Interferometer accuracy and precision,” Proc. SPIE 1400, 24–32 (1990).
[CrossRef]

Trivi, M.

Tronconi, R.

Tu-ya, W.

X. Chen, C. Lei, and W. Tu-ya, “Rotational variant error removal of the reference flat,” Acta Meterol. Sin. 31, 219–222 (2010) (in Chinese).

Weihong, S.

S. Weihong, W. Fan, and H. Xi, “Simulation analysis on absolute testing of spherical surface with shift-rotation method,” High Power Laser Particle Beams 23, 1–5 (2011) (in Chinese).

Xi, H.

S. Weihong, W. Fan, and H. Xi, “Simulation analysis on absolute testing of spherical surface with shift-rotation method,” High Power Laser Particle Beams 23, 1–5 (2011) (in Chinese).

Yamamoto, T.

K. Otaki, T. Yamamoto, and Y. Fukuda, “Accuracy evaluation of the point diffraction interferometer for extreme ultraviolet lithography aspheric mirror,” J. Vac. Sci. Technol. B 20, 295–300 (2002).
[CrossRef]

H. Ichikawa and T. Yamamoto, “Apparatus and method for wavefront absolute calibration and method of synthesizing wavefronts,” U.S. patent 5,982,490 (9November1999).

Acta Meterol. Sin.

X. Chen, C. Lei, and W. Tu-ya, “Rotational variant error removal of the reference flat,” Acta Meterol. Sin. 31, 219–222 (2010) (in Chinese).

Appl. Opt.

High Power Laser Particle Beams

S. Weihong, W. Fan, and H. Xi, “Simulation analysis on absolute testing of spherical surface with shift-rotation method,” High Power Laser Particle Beams 23, 1–5 (2011) (in Chinese).

Int. J. Korean Soc. Precision Eng.

S.-W. Kim and H.-G. Rhee, “Self-calibration of high frequency errors of test optics by arbitrary N-step rotation,” Int. J. Korean Soc. Precision Eng. 1, 115–123 (2000).

J. Vac. Sci. Technol. B

K. Otaki, T. Yamamoto, and Y. Fukuda, “Accuracy evaluation of the point diffraction interferometer for extreme ultraviolet lithography aspheric mirror,” J. Vac. Sci. Technol. B 20, 295–300 (2002).
[CrossRef]

Opt. Eng.

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

P. Hariharan, “Interferometric testing of optical surfaces: absolute measurements of flatness,” Opt. Eng. 36, 2478–2481 (1997).
[CrossRef]

Proc. SPIE

L. A. Selberg, “Interferometer accuracy and precision,” Proc. SPIE 1400, 24–32 (1990).
[CrossRef]

Other

G. Seitz and W. Otto, “Method for the interferometric measurement of non-rotationally symmetric wavefront errors,” U.S. patent 7,277,186 (2October2007).

W. Otto and Aalen-Waldhausen, “Method for the interferometric measurement of non-rotationally symmetric wavefront errors,” U.S. patent 6,839,143 (4January2005).

D. Malacara, Optical Shop Testing3rd ed. (Wiley, 2007).

C. B. Moler, Numerical Computing with MATLAB (SIAM, 2004).

H. Ichikawa and T. Yamamoto, “Apparatus and method for wavefront absolute calibration and method of synthesizing wavefronts,” U.S. patent 5,982,490 (9November1999).

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Figures (12)

Fig. 1.
Fig. 1.

Calculated result of traditional multiangle averaging, PV=16.90nm, rms=2.71nm.

Fig. 2.
Fig. 2.

Calculated result of the proposed method (N=2+1, θ0=60°), PV=16.32nm, rms=2.41nm.

Fig. 3.
Fig. 3.

Calculated result of the proposed method (N=3+1, θ0=60°), PV=17.84nm, rms=2.76nm.

Fig. 4.
Fig. 4.

Residual figure between Fig. 1 and Fig. 2, PV=18.61, rms=1.03nm.

Fig. 5.
Fig. 5.

Residual figure between Fig. 1 and Fig. 3, PV=19.49nm, rms=0.64nm.

Fig. 6.
Fig. 6.

Comparison of the first 36 terms Zernike decomposition of the absolute results shown in Fig. 1, Fig. 2, and Fig. 3, respectively.

Fig. 7.
Fig. 7.

Differences of the first 36 terms Zernike decomposition shown in Fig. 6.

Fig. 8.
Fig. 8.

Original spherical surface deviation with 992×992 data array, PV=83.59nm, rms=7.01nm.

Fig. 9.
Fig. 9.

666 terms of Zernike polynomials expression of surface deviation shown in Fig. 8, PV=52.80nm, rms=6.97nm.

Fig. 10.
Fig. 10.

Residual figure between Fig. 8 and Fig. 9, PV=42.22nm, rms=0.73nm.

Fig. 11.
Fig. 11.

RMS of the residual figure changing with the order of Zernike polynomials for fitting (j=(n+1)(n+2)/2, j, number of Zernike terms; n, order of Zernike polynomials).

Fig. 12.
Fig. 12.

PSD plots of the absolute results shown in Fig. 1, Fig. 2, and Fig. 3, respectively.

Tables (1)

Tables Icon

Table 1. Coordinate of Center and Data Points

Equations (8)

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

Tφj(ρ,θ)=Wφj(ρ,θ)+V(ρ,θ)φj=j×360°/N,j=0,1,2,,N1,
Tave(ρ,θ)=1Ni=0N1Tφj(ρ,θ)=V(ρ,θ)+Wsym(ρ)+WkNθ(ρ,θ),
WkNθ(ρ,θ)=iqi·(k,n(1)k·(N+1)·RnkN(ρ)·(ankNcoskNθ+ankNsinkNθ)),
Wasy(ρ,θ)=Tφj(ρ,θ)Tave(ρ,θ)=Wφj(ρ,θ)Wsys(ρ)WkNθ(ρ,θ).
T(ρ,θ+θ0)=W(ρ,θ+θ0)+V(ρ,θ).
T(ρ,θ+θ0)T(ρ,θ)=W(ρ,θ+θ0)W(ρ,θ)=Wasy(ρ,θ+θ0)Wasy(ρ,θ),
Wasy(ρ,θ)=i,n,mCi,nmcosmθ+Ci,nmsinmθ=WkNθ(ρ,θ)+Wmθ(ρ,θ),mkN,
Wasy(ρ,θ)=Wasy(ρ,θ)+WkNθ(ρ,θ).

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