Abstract

Abstract: A new method is proposed for testing a rotationally symmetric aspheric surface with several annular subapertures based on a Hartmann–Shack sensor. In consideration of the limited sampling of Hartmann–Shack subapertures in the matching annular subaperture, a new algorithm for whole-aperture wavefront reconstruction from annular subaperture Hartmann–Shack gradient data is established. The algorithm separates the tip, tilt, and defocus misalignments for each annular subaperture by introducing annular Zernike polynomials. The performance of the algorithm is evaluated for different annular subaperture configurations, and the sensitivity of the algorithm to the detector error of the wavefront gradient is analyzed. The algorithm is verified by the experimental results.

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  1. C. J. Kim, “Polynomial fit of interferograms,” Appl. Opt. 21(24), 4521–4525 (1982).
    [CrossRef] [PubMed]
  2. J. G. Thunen and O. Y. Kwon, “Full aperture testing with subaperture test optics,” Proc. SPIE 351, 19–27 (1982).
  3. W. W. Chow and G. N. Lawrence, “Method for subaperture testing interferogram reduction,” Opt. Lett. 8(9), 468–470 (1983).
    [CrossRef] [PubMed]
  4. T. W. Stuhlinger, “Subaperture optical testing: experimental verification,” Proc. SPIE 656, 118–127 (1986).
  5. Y.-M. Liu, G. N. Lawrance, and C. L. Koliopoulos, “Subaperture testing of aspheres with annular zones,” Appl. Opt. 27(21), 4504–4513 (1988).
    [CrossRef] [PubMed]
  6. X. Hou, F. Wu, L. Yang, S. B. Wu, and Q. Chen, “Full-aperture wavefront reconstruction from annular subaperture interferometric data by use of Zernike annular polynomials and a matrix method for testing large aspheric surfaces,” Appl. Opt. 45(15), 3442–3455 (2006).
    [CrossRef] [PubMed]
  7. X. Hou, F. Wu, L. Yang, and Q. Chen, “Experimental study on measurement of aspheric surface shape with complementary annular subaperture interferometric method,” Opt. Express 15(20), 12890–12899 (2007).
    [CrossRef] [PubMed]
  8. M. Melozzi, L. Pezzati, and A. Mazzoni, “Testing aspheric surfaces using multiple annular interferograms,” Opt. Eng. 32(5), 1073–1079 (1993).
    [CrossRef]
  9. D. Malacara, M. Servin, A. Morales, and Z. Maracara, “Aspherical wavefront testing with several defusing steps,” in International Conference on Optical Fabrication and Testing, T. Kasai,ed., Proc.SPIE 2576,190–192 (1995).
  10. M. J. Tronolone, J. F. Fleig, C. Huang, and J. H. Bruning, “Method of testing aspherical optical surfaces with an interferometer,” US Patent 5416586 (May.16, 1995).
  11. F. Granados-Agustín, J. F. Escobar-Romero, and A. Cornejo-Rodríguez, “Testing parabolic surfaces with annular subaperture interferograms,” Opt. Rev. 11, 82–86 (2004).
    [CrossRef]
  12. M. Otsubo, K. Okada, and J. Tsujiuchi, “Measurement of large plane surface shapes by connecting small-aperture interferograms,” Opt. Eng. 33(2), 608–613 (1994).
    [CrossRef]
  13. P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: a flexible solution for surface metrology,” Opt. Photon. News 14(5), 38–43 (2003).
    [CrossRef]
  14. P. Murphy, G. Devries, C. Brophy, and G. Forbes, “Stitching of near-nulled subaperture measurements,” US Patent 2009/0251702 A1 (October 8, 2009).
  15. S. Y. Chen, S. Y. Li, Y. F. Dai, L. Y. Ding, and S. Y. Zeng, “Experimental study on subaperture testing with iterative stitching algorithm,” Opt. Express 16(7), 4760–4765 (2008).
    [CrossRef] [PubMed]
  16. D. R. Neal, R. R. Rammage, D. J. Armstrong, W. T. Turner, and J. D. Mansell, “Apparatus and method for evaluating a target larger than a measuring aperture of a sensor,” US Patent 6184974 B1 (February 6, 2001).
  17. T. D. Raymond, D. R. Neal, D. M. Topa, and T. L. Schmitz, “High-speed noninterferometric nanotopographic characterization of Si wafer surfaces,” Proc. SPIE 4809, 208–216 (2002).
    [CrossRef]
  18. V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71(1), 75–85 (1981).
    [CrossRef]
  19. X. Hou, F. Wu, L. Yang, and Q. Chen, “Comparison of annular wavefront interpretation with Zernike circle polynomials and annular polynomials,” Appl. Opt. 45(35), 8893–8901 (2006).
    [CrossRef] [PubMed]

2008

2007

2006

2004

F. Granados-Agustín, J. F. Escobar-Romero, and A. Cornejo-Rodríguez, “Testing parabolic surfaces with annular subaperture interferograms,” Opt. Rev. 11, 82–86 (2004).
[CrossRef]

2003

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: a flexible solution for surface metrology,” Opt. Photon. News 14(5), 38–43 (2003).
[CrossRef]

2002

T. D. Raymond, D. R. Neal, D. M. Topa, and T. L. Schmitz, “High-speed noninterferometric nanotopographic characterization of Si wafer surfaces,” Proc. SPIE 4809, 208–216 (2002).
[CrossRef]

1994

M. Otsubo, K. Okada, and J. Tsujiuchi, “Measurement of large plane surface shapes by connecting small-aperture interferograms,” Opt. Eng. 33(2), 608–613 (1994).
[CrossRef]

1993

M. Melozzi, L. Pezzati, and A. Mazzoni, “Testing aspheric surfaces using multiple annular interferograms,” Opt. Eng. 32(5), 1073–1079 (1993).
[CrossRef]

1988

1986

T. W. Stuhlinger, “Subaperture optical testing: experimental verification,” Proc. SPIE 656, 118–127 (1986).

1983

1982

C. J. Kim, “Polynomial fit of interferograms,” Appl. Opt. 21(24), 4521–4525 (1982).
[CrossRef] [PubMed]

J. G. Thunen and O. Y. Kwon, “Full aperture testing with subaperture test optics,” Proc. SPIE 351, 19–27 (1982).

1981

Chen, Q.

Chen, S. Y.

Chow, W. W.

Cornejo-Rodríguez, A.

F. Granados-Agustín, J. F. Escobar-Romero, and A. Cornejo-Rodríguez, “Testing parabolic surfaces with annular subaperture interferograms,” Opt. Rev. 11, 82–86 (2004).
[CrossRef]

Dai, Y. F.

Ding, L. Y.

Dumas, P.

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: a flexible solution for surface metrology,” Opt. Photon. News 14(5), 38–43 (2003).
[CrossRef]

Escobar-Romero, J. F.

F. Granados-Agustín, J. F. Escobar-Romero, and A. Cornejo-Rodríguez, “Testing parabolic surfaces with annular subaperture interferograms,” Opt. Rev. 11, 82–86 (2004).
[CrossRef]

Fleig, J.

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: a flexible solution for surface metrology,” Opt. Photon. News 14(5), 38–43 (2003).
[CrossRef]

Forbes, G.

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: a flexible solution for surface metrology,” Opt. Photon. News 14(5), 38–43 (2003).
[CrossRef]

Granados-Agustín, F.

F. Granados-Agustín, J. F. Escobar-Romero, and A. Cornejo-Rodríguez, “Testing parabolic surfaces with annular subaperture interferograms,” Opt. Rev. 11, 82–86 (2004).
[CrossRef]

Hou, X.

Kim, C. J.

Koliopoulos, C. L.

Kwon, O. Y.

J. G. Thunen and O. Y. Kwon, “Full aperture testing with subaperture test optics,” Proc. SPIE 351, 19–27 (1982).

Lawrance, G. N.

Lawrence, G. N.

Li, S. Y.

Liu, Y.-M.

Mahajan, V. N.

Mazzoni, A.

M. Melozzi, L. Pezzati, and A. Mazzoni, “Testing aspheric surfaces using multiple annular interferograms,” Opt. Eng. 32(5), 1073–1079 (1993).
[CrossRef]

Melozzi, M.

M. Melozzi, L. Pezzati, and A. Mazzoni, “Testing aspheric surfaces using multiple annular interferograms,” Opt. Eng. 32(5), 1073–1079 (1993).
[CrossRef]

Murphy, P.

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: a flexible solution for surface metrology,” Opt. Photon. News 14(5), 38–43 (2003).
[CrossRef]

Neal, D. R.

T. D. Raymond, D. R. Neal, D. M. Topa, and T. L. Schmitz, “High-speed noninterferometric nanotopographic characterization of Si wafer surfaces,” Proc. SPIE 4809, 208–216 (2002).
[CrossRef]

Okada, K.

M. Otsubo, K. Okada, and J. Tsujiuchi, “Measurement of large plane surface shapes by connecting small-aperture interferograms,” Opt. Eng. 33(2), 608–613 (1994).
[CrossRef]

Otsubo, M.

M. Otsubo, K. Okada, and J. Tsujiuchi, “Measurement of large plane surface shapes by connecting small-aperture interferograms,” Opt. Eng. 33(2), 608–613 (1994).
[CrossRef]

Pezzati, L.

M. Melozzi, L. Pezzati, and A. Mazzoni, “Testing aspheric surfaces using multiple annular interferograms,” Opt. Eng. 32(5), 1073–1079 (1993).
[CrossRef]

Raymond, T. D.

T. D. Raymond, D. R. Neal, D. M. Topa, and T. L. Schmitz, “High-speed noninterferometric nanotopographic characterization of Si wafer surfaces,” Proc. SPIE 4809, 208–216 (2002).
[CrossRef]

Schmitz, T. L.

T. D. Raymond, D. R. Neal, D. M. Topa, and T. L. Schmitz, “High-speed noninterferometric nanotopographic characterization of Si wafer surfaces,” Proc. SPIE 4809, 208–216 (2002).
[CrossRef]

Stuhlinger, T. W.

T. W. Stuhlinger, “Subaperture optical testing: experimental verification,” Proc. SPIE 656, 118–127 (1986).

Thunen, J. G.

J. G. Thunen and O. Y. Kwon, “Full aperture testing with subaperture test optics,” Proc. SPIE 351, 19–27 (1982).

Topa, D. M.

T. D. Raymond, D. R. Neal, D. M. Topa, and T. L. Schmitz, “High-speed noninterferometric nanotopographic characterization of Si wafer surfaces,” Proc. SPIE 4809, 208–216 (2002).
[CrossRef]

Tricard, M.

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: a flexible solution for surface metrology,” Opt. Photon. News 14(5), 38–43 (2003).
[CrossRef]

Tsujiuchi, J.

M. Otsubo, K. Okada, and J. Tsujiuchi, “Measurement of large plane surface shapes by connecting small-aperture interferograms,” Opt. Eng. 33(2), 608–613 (1994).
[CrossRef]

Wu, F.

Wu, S. B.

Yang, L.

Zeng, S. Y.

Appl. Opt.

J. Opt. Soc. Am.

Opt. Eng.

M. Melozzi, L. Pezzati, and A. Mazzoni, “Testing aspheric surfaces using multiple annular interferograms,” Opt. Eng. 32(5), 1073–1079 (1993).
[CrossRef]

M. Otsubo, K. Okada, and J. Tsujiuchi, “Measurement of large plane surface shapes by connecting small-aperture interferograms,” Opt. Eng. 33(2), 608–613 (1994).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Photon. News

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: a flexible solution for surface metrology,” Opt. Photon. News 14(5), 38–43 (2003).
[CrossRef]

Opt. Rev.

F. Granados-Agustín, J. F. Escobar-Romero, and A. Cornejo-Rodríguez, “Testing parabolic surfaces with annular subaperture interferograms,” Opt. Rev. 11, 82–86 (2004).
[CrossRef]

Proc. SPIE

T. W. Stuhlinger, “Subaperture optical testing: experimental verification,” Proc. SPIE 656, 118–127 (1986).

T. D. Raymond, D. R. Neal, D. M. Topa, and T. L. Schmitz, “High-speed noninterferometric nanotopographic characterization of Si wafer surfaces,” Proc. SPIE 4809, 208–216 (2002).
[CrossRef]

J. G. Thunen and O. Y. Kwon, “Full aperture testing with subaperture test optics,” Proc. SPIE 351, 19–27 (1982).

Other

D. R. Neal, R. R. Rammage, D. J. Armstrong, W. T. Turner, and J. D. Mansell, “Apparatus and method for evaluating a target larger than a measuring aperture of a sensor,” US Patent 6184974 B1 (February 6, 2001).

P. Murphy, G. Devries, C. Brophy, and G. Forbes, “Stitching of near-nulled subaperture measurements,” US Patent 2009/0251702 A1 (October 8, 2009).

D. Malacara, M. Servin, A. Morales, and Z. Maracara, “Aspherical wavefront testing with several defusing steps,” in International Conference on Optical Fabrication and Testing, T. Kasai,ed., Proc.SPIE 2576,190–192 (1995).

M. J. Tronolone, J. F. Fleig, C. Huang, and J. H. Bruning, “Method of testing aspherical optical surfaces with an interferometer,” US Patent 5416586 (May.16, 1995).

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Figures (12)

Fig. 1
Fig. 1

Layout of three annular subapertures for a whole-aperture surface with central obstruction of ε 0 .

Fig. 2
Fig. 2

Three annular subaperture configurations, (a) Configuration 1, (b) Configuration 2, and (c) Configuration 3.

Fig. 3
Fig. 3

Simulation results. (a) Original wavefront, (b) isometric of the three annular subaperture wavefronts, (c) valid subaperture spot array of the first annular subaperture, (d) valid subaperture spot array of the second annular subaperture, (e) valid subaperture spot array of the third annular subaperture, (f) whole-aperture wavefront reconstructed from three annular subapertures gradient data with τ = 0.1 , and (g) residual wavefront between (f) and (a).

Fig. 4
Fig. 4

Schematic of annular subaperture measuring system based on Hartmann–Shack sensor, optical source (OS), relay-imaging telescope (RI), beam splitter (BS), charged coupled device (CCD), lens (L), converging lens (CL),lenslet array (LA).

Fig. 5
Fig. 5

(a) Image of the valid subaperture spots in the inner subaperture at the first defocusing position, (b) image of the valid subaperture spots in the outer subaperture at the second defocusing position, and (c) image of the valid subaperture spots of the whole-aperture wavefront at the vertex curvature center of the spherical surface.

Fig. 6
Fig. 6

Whole-aperture wavefront reconstructed from annular subaperture Hartmann–Shack gradient data ( PV  0 .736 λ, RMS 0 .151 λ, λ 0 .635 μm ).

Fig. 7
Fig. 7

Whole-aperture wavefront directly measured by the Hartmann–Shack sensor ( PV 0 .712 λ, 0 .148  λ, λ 0 .635 μm ).

Fig. 8
Fig. 8

Residual wavefront map between Figs. 6 and 7 ( PV 0 .073 λ , RMS 0 .011 λ , λ 0 .635 μm ).

Fig. 9
Fig. 9

(a) Experimental setup, (b) image of valid subaperture spots in the inner matching subaperture, and (c) image of valid subaperture spots in the outer matching subaperture.

Fig. 10
Fig. 10

Whole-aperture wavefront map reconstructed from annular subaperture Hartmann–Shack gradient data ( PV 5 .541 λ, RMS 1 .319 λ, λ 0 .635 μm ).

Fig. 11
Fig. 11

Whole-aperture wavefront map directly tested by infrared interferometer, ( PV 5 .698 λ, RMS 1 .310 λ, λ 0 .635 μm ).

Fig. 12
Fig. 12

Residual wavefront map between Figs. 10 and 11 ( PV 0 .248 λ, RMS 0 .028 λ, λ 0 .635 μm ).

Tables (2)

Tables Icon

Table 1 PV and RMS of Residual Wavefront for Different Configurations with Different Noise Levels

Tables Icon

Table 2 PV and RMS of Residual Wavefronts for Different Defocuses

Equations (7)

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ε 0 R 1 and  0 T 2 π ,
ε 0 / R o u t 1 r 1 1 and  0 t 1 2 π ,
{ 1 σ σ δ W δ X d X d Y = 1 σ j = 1 M i = 0 3 a i , j σ δ z i , j ( R j / j = j + 1 M ε j , T , ε j ) δ X d X d Y + 1 σ i = 4 L A i σ δ Z i ( R , T , ε 0 ) δ X d X d Y 1 σ σ δ W δ Y d X d Y = 1 σ j = 1 M i = 0 3 a i , j σ δ z i , j ( R j / j = j + 1 M ε j , T , ε j ) δ Y d X d Y + 1 σ i = 4 L A i σ δ Z i ( R , T , ε 0 ) δ Y d X d Y ,
{ G x k , j = j = 1 M i = 1 3 a i , j σ k , j X z i ( R j / j = j + 1 M ε j , T , ε j ) d X d Y δ k , j d X d Y + i = 4 L A i σ k , j X Z i ( R , T , ε 0 ) d X d Y σ k , j d X d Y G y k , j = j = 1 M i = 1 3 a i , j σ k , j Y z i ( R j / j = j + 1 M ε j , T , ε j ) d X d Y σ k , j d X d Y + i = 4 L A i σ k , j Y Z i ( R , T , ε 0 ) d X d Y σ k , j d X d Y ,
A A = [ a 1 , 1 , a 1 , 2 , a 1 , 3 , ... a M , 1 , a M , 2 , a M , 3 , A 4 , A 5 , ... A N ] T ,
T j = [ z x 1 j ( 1 ) z x 2 j ( 1 ) z x 3 j ( 1 ) z y 1 j ( 1 ) z y 2 j ( 1 ) z y 3 j ( 1 ) z x 1 j ( 2 ) z x 2 j ( 2 ) z x 3 j ( 2 ) z y 1 j ( 2 ) z y 2 j ( 2 ) z y 3 j ( 2 ) ... ... ... z x 1 j ( s j ) z x 2 j ( s j ) z x 3 j ( s j ) z y 1 j ( s j ) z y 2 j ( s j ) z y 3 j ( s j ) ] ,
τ = δ S ,

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