Abstract

We propose a more accurate and efficient reconstruction method used in testing large aspheric surfaces with annular subaperture interferometry. By the introduction of the Zernike annular polynomials that are orthogonal over the annular region, the method proposed here eliminates the coupling problem in the earlier reconstruction algorithm based on Zernike circle polynomials. Because of the complexity of recurrence definition of Zernike annular polynomials, a general symbol representation of that in a computing program is established. The program implementation for the method is provided in detail. The performance of the reconstruction algorithm is evaluated in some pertinent cases, such as different random noise levels, different subaperture configurations, and misalignments.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. J. Kim, " Polynomial fit of interferograms," Appl. Opt. 21, 4521-4525 (1982).
    [Crossref] [PubMed]
  2. J. G. Thunen and O. Y. Kwon, " Full aperture testing with subaperture test optics," in Wavefront Sensing, N.Bareket and C.L.Koliopoulos, eds., Proc. SPIE 351, 19-27 (1982).
  3. W. W. Chow and G. N. Lawrence, " Method for subaperture testing interferogram reduction," Opt. Lett. 8, 468- 470 (1983).
    [Crossref] [PubMed]
  4. J. E. Negro, "Subaperture optical system testing," Appl. Opt. 23, 1921-1930 (1984).
    [Crossref] [PubMed]
  5. S. C. Jensen, W. W. Chow, and G. N. Lawrence, "Subaperture testing approaches: A comparison," Appl. Opt. 23, 740-745 (1984).
    [Crossref] [PubMed]
  6. C. R. De Hainaut and A. Erteza, "Numerical processing of dynamic subaperture testing measurements," Appl. Opt. 25, 503-509 (1986).
    [Crossref] [PubMed]
  7. M. A. Schmucker and J. Schmit, "Selection process for sequentially combing multiple sets of overlapping surface profile interferometric data to produce a continuous composite map," U.S. patent 5,991,461 ( 23 November 1999).
  8. M. Otsubo, K. Okada, and J. Tsujiuchi, "Measurement of large plane surface shapes by connecting small-aperture interferograms," Opt. Eng. 33, 608-613 (1994).
    [Crossref]
  9. M. Bray, "Stitching interferometer for large plano optics using a standard interferometer," in Optical Manufacturing and Testing II, H.PhilipStahl, ed., Proc. SPIE 3134, 39-50 (1997).
  10. G. N. Lawrence and R. D. Day, "Interferometric characterization of full spheres: Data reduction techniques," Appl. Opt. 26, 4875-4882 (1987).
    [Crossref] [PubMed]
  11. P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, "Stitching interferometry: A flexible solution for surface metrology," Opt. Photon. News 14, 38-43 (2003).
    [Crossref]
  12. P. E. Murphy, J. Fleig, G. Forbes, and M. Tricard, "High precision metrology of demes and aspheric optics," in Window and Dome Technologies and Materials IX, R.W.Tustison, ed., Proc. SPIE 5786, 112-121 (2005).
  13. T. Hänsel, A. Nickel, and A. Schindler, "Stitching interferometry of aspherical surfaces," in Optical Metrology Roadmap for the Semiconductor, Optical, and Data Storage Industries II, A.Duparré and B.Singh, eds., Proc. SPIE 4449, 265-273 (2001).
  14. Y. M. Liu, G. Lawrence, and C. Koliopoulos, "Subaperture testing of aspheres with annular zones," Appl. Opt. 27, 4504-4513 (1988).
    [Crossref] [PubMed]
  15. M. Melozzi, L. Pezzati, and A. Mazzoni, "Testing aspheric surfaces using multiple annular interferograms," Opt. Eng. 32, 1073-1079 (1993).
    [Crossref]
  16. D. Malacara, M. Servin, A. Morales, and Z. Malacara, " Aspherical wavefront testing with several defocusing steps," in International Conference on Optical Fabrication and Testing, T.Kasai, ed., Proc. SPIE 2576, 190-192 (1995).
  17. M. J. Tronolone, J. F. Fleig, C. Huang, and J. H. Bruning, "Method of testing aspherical optical surfaces with an interferometer," U.S. patent 5,416,586 ( 16 May 1995).
  18. F. Granados-Agustín, J. F. Escobar-Romero, and A. Cornejo-Rodríguez, "Testing parabolic surfaces with annular subaperture interferograsm," Opt. Rev. 11, 82-86 (2004).
    [Crossref]
  19. J. C. Wyant and K. Creath, "Basic wavefront aberration theory for optical metrology," in Applied Optics and Optical Engineering Series (Academic, 1992), Vol. XI, p. 28.
  20. V. N. Mahajan, "Zernike annular polynomials for imaging systems with annular pupils," J. Opt. Soc. Am. 71 , 75-85 (1981).
    [Crossref]
  21. J. Y. Wang and D. E. Silva, "Wave-front interpretation with Zernike polynomials," Appl. Opt. 19, 1510-1518 (1980).
    [Crossref] [PubMed]
  22. The mathematica software was developed by the Wolfram Research, Inc., http://www.wolfram.com.
  23. A. F. Slomba and L. Montagnino, "Subaperture testing for mid-frequency figure control on large aspheric mirror," in Precision Surface Metrology, J.C.Wyant, ed., Proc. SPIE 429, 114-118 (1983).
  24. W. Swantner and W. W. Chow, "Gram-Schmidt orthonormalization of Zernike polynomials for general aperture shapes," Appl. Opt. 33, 1832-1837 (1994).
    [Crossref] [PubMed]
  25. H. Liu, Q. Hao, Q. Zhu, D. Sha, and C. Zhang, "A novel aspheric surface testing method using part-compensating lens," in Optical Design and Testing II, Y.Wang, Z.Weng, S.Ye, and J.M.Sasián, eds., Proc. SPIE 5638, 324-329 (2004).

2005 (1)

P. E. Murphy, J. Fleig, G. Forbes, and M. Tricard, "High precision metrology of demes and aspheric optics," in Window and Dome Technologies and Materials IX, R.W.Tustison, ed., Proc. SPIE 5786, 112-121 (2005).

2004 (2)

F. Granados-Agustín, J. F. Escobar-Romero, and A. Cornejo-Rodríguez, "Testing parabolic surfaces with annular subaperture interferograsm," Opt. Rev. 11, 82-86 (2004).
[Crossref]

H. Liu, Q. Hao, Q. Zhu, D. Sha, and C. Zhang, "A novel aspheric surface testing method using part-compensating lens," in Optical Design and Testing II, Y.Wang, Z.Weng, S.Ye, and J.M.Sasián, eds., Proc. SPIE 5638, 324-329 (2004).

2003 (1)

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, "Stitching interferometry: A flexible solution for surface metrology," Opt. Photon. News 14, 38-43 (2003).
[Crossref]

2001 (1)

T. Hänsel, A. Nickel, and A. Schindler, "Stitching interferometry of aspherical surfaces," in Optical Metrology Roadmap for the Semiconductor, Optical, and Data Storage Industries II, A.Duparré and B.Singh, eds., Proc. SPIE 4449, 265-273 (2001).

1999 (1)

M. A. Schmucker and J. Schmit, "Selection process for sequentially combing multiple sets of overlapping surface profile interferometric data to produce a continuous composite map," U.S. patent 5,991,461 ( 23 November 1999).

1997 (1)

M. Bray, "Stitching interferometer for large plano optics using a standard interferometer," in Optical Manufacturing and Testing II, H.PhilipStahl, ed., Proc. SPIE 3134, 39-50 (1997).

1995 (2)

D. Malacara, M. Servin, A. Morales, and Z. Malacara, " Aspherical wavefront testing with several defocusing 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," U.S. patent 5,416,586 ( 16 May 1995).

1994 (2)

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

W. Swantner and W. W. Chow, "Gram-Schmidt orthonormalization of Zernike polynomials for general aperture shapes," Appl. Opt. 33, 1832-1837 (1994).
[Crossref] [PubMed]

1993 (1)

M. Melozzi, L. Pezzati, and A. Mazzoni, "Testing aspheric surfaces using multiple annular interferograms," Opt. Eng. 32, 1073-1079 (1993).
[Crossref]

1992 (1)

J. C. Wyant and K. Creath, "Basic wavefront aberration theory for optical metrology," in Applied Optics and Optical Engineering Series (Academic, 1992), Vol. XI, p. 28.

1988 (1)

1987 (1)

1986 (1)

1984 (2)

1983 (2)

W. W. Chow and G. N. Lawrence, " Method for subaperture testing interferogram reduction," Opt. Lett. 8, 468- 470 (1983).
[Crossref] [PubMed]

A. F. Slomba and L. Montagnino, "Subaperture testing for mid-frequency figure control on large aspheric mirror," in Precision Surface Metrology, J.C.Wyant, ed., Proc. SPIE 429, 114-118 (1983).

1982 (2)

C. J. Kim, " Polynomial fit of interferograms," Appl. Opt. 21, 4521-4525 (1982).
[Crossref] [PubMed]

J. G. Thunen and O. Y. Kwon, " Full aperture testing with subaperture test optics," in Wavefront Sensing, N.Bareket and C.L.Koliopoulos, eds., Proc. SPIE 351, 19-27 (1982).

1981 (1)

V. N. Mahajan, "Zernike annular polynomials for imaging systems with annular pupils," J. Opt. Soc. Am. 71 , 75-85 (1981).
[Crossref]

1980 (1)

Bray, M.

M. Bray, "Stitching interferometer for large plano optics using a standard interferometer," in Optical Manufacturing and Testing II, H.PhilipStahl, ed., Proc. SPIE 3134, 39-50 (1997).

Bruning, J. H.

M. J. Tronolone, J. F. Fleig, C. Huang, and J. H. Bruning, "Method of testing aspherical optical surfaces with an interferometer," U.S. patent 5,416,586 ( 16 May 1995).

Chow, W. W.

Cornejo-Rodríguez, A.

F. Granados-Agustín, J. F. Escobar-Romero, and A. Cornejo-Rodríguez, "Testing parabolic surfaces with annular subaperture interferograsm," Opt. Rev. 11, 82-86 (2004).
[Crossref]

Creath, K.

J. C. Wyant and K. Creath, "Basic wavefront aberration theory for optical metrology," in Applied Optics and Optical Engineering Series (Academic, 1992), Vol. XI, p. 28.

Day, R. D.

De Hainaut, C. R.

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, 38-43 (2003).
[Crossref]

Erteza, A.

Escobar-Romero, J. F.

F. Granados-Agustín, J. F. Escobar-Romero, and A. Cornejo-Rodríguez, "Testing parabolic surfaces with annular subaperture interferograsm," Opt. Rev. 11, 82-86 (2004).
[Crossref]

Fleig, J.

P. E. Murphy, J. Fleig, G. Forbes, and M. Tricard, "High precision metrology of demes and aspheric optics," in Window and Dome Technologies and Materials IX, R.W.Tustison, ed., Proc. SPIE 5786, 112-121 (2005).

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, "Stitching interferometry: A flexible solution for surface metrology," Opt. Photon. News 14, 38-43 (2003).
[Crossref]

Fleig, J. F.

M. J. Tronolone, J. F. Fleig, C. Huang, and J. H. Bruning, "Method of testing aspherical optical surfaces with an interferometer," U.S. patent 5,416,586 ( 16 May 1995).

Forbes, G.

P. E. Murphy, J. Fleig, G. Forbes, and M. Tricard, "High precision metrology of demes and aspheric optics," in Window and Dome Technologies and Materials IX, R.W.Tustison, ed., Proc. SPIE 5786, 112-121 (2005).

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, "Stitching interferometry: A flexible solution for surface metrology," Opt. Photon. News 14, 38-43 (2003).
[Crossref]

Granados-Agustín, F.

F. Granados-Agustín, J. F. Escobar-Romero, and A. Cornejo-Rodríguez, "Testing parabolic surfaces with annular subaperture interferograsm," Opt. Rev. 11, 82-86 (2004).
[Crossref]

Hänsel, T.

T. Hänsel, A. Nickel, and A. Schindler, "Stitching interferometry of aspherical surfaces," in Optical Metrology Roadmap for the Semiconductor, Optical, and Data Storage Industries II, A.Duparré and B.Singh, eds., Proc. SPIE 4449, 265-273 (2001).

Hao, Q.

H. Liu, Q. Hao, Q. Zhu, D. Sha, and C. Zhang, "A novel aspheric surface testing method using part-compensating lens," in Optical Design and Testing II, Y.Wang, Z.Weng, S.Ye, and J.M.Sasián, eds., Proc. SPIE 5638, 324-329 (2004).

Huang, C.

M. J. Tronolone, J. F. Fleig, C. Huang, and J. H. Bruning, "Method of testing aspherical optical surfaces with an interferometer," U.S. patent 5,416,586 ( 16 May 1995).

Jensen, S. C.

Kim, C. J.

Koliopoulos, C.

Kwon, O. Y.

J. G. Thunen and O. Y. Kwon, " Full aperture testing with subaperture test optics," in Wavefront Sensing, N.Bareket and C.L.Koliopoulos, eds., Proc. SPIE 351, 19-27 (1982).

Lawrence, G.

Lawrence, G. N.

Liu, H.

H. Liu, Q. Hao, Q. Zhu, D. Sha, and C. Zhang, "A novel aspheric surface testing method using part-compensating lens," in Optical Design and Testing II, Y.Wang, Z.Weng, S.Ye, and J.M.Sasián, eds., Proc. SPIE 5638, 324-329 (2004).

Liu, Y. M.

Mahajan, V. N.

V. N. Mahajan, "Zernike annular polynomials for imaging systems with annular pupils," J. Opt. Soc. Am. 71 , 75-85 (1981).
[Crossref]

Malacara, D.

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

Malacara, Z.

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

Mazzoni, A.

M. Melozzi, L. Pezzati, and A. Mazzoni, "Testing aspheric surfaces using multiple annular interferograms," Opt. Eng. 32, 1073-1079 (1993).
[Crossref]

Melozzi, M.

M. Melozzi, L. Pezzati, and A. Mazzoni, "Testing aspheric surfaces using multiple annular interferograms," Opt. Eng. 32, 1073-1079 (1993).
[Crossref]

Montagnino, L.

A. F. Slomba and L. Montagnino, "Subaperture testing for mid-frequency figure control on large aspheric mirror," in Precision Surface Metrology, J.C.Wyant, ed., Proc. SPIE 429, 114-118 (1983).

Morales, A.

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

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, 38-43 (2003).
[Crossref]

Murphy, P. E.

P. E. Murphy, J. Fleig, G. Forbes, and M. Tricard, "High precision metrology of demes and aspheric optics," in Window and Dome Technologies and Materials IX, R.W.Tustison, ed., Proc. SPIE 5786, 112-121 (2005).

Negro, J. E.

Nickel, A.

T. Hänsel, A. Nickel, and A. Schindler, "Stitching interferometry of aspherical surfaces," in Optical Metrology Roadmap for the Semiconductor, Optical, and Data Storage Industries II, A.Duparré and B.Singh, eds., Proc. SPIE 4449, 265-273 (2001).

Okada, K.

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

Pezzati, L.

M. Melozzi, L. Pezzati, and A. Mazzoni, "Testing aspheric surfaces using multiple annular interferograms," Opt. Eng. 32, 1073-1079 (1993).
[Crossref]

Schindler, A.

T. Hänsel, A. Nickel, and A. Schindler, "Stitching interferometry of aspherical surfaces," in Optical Metrology Roadmap for the Semiconductor, Optical, and Data Storage Industries II, A.Duparré and B.Singh, eds., Proc. SPIE 4449, 265-273 (2001).

Schmit, J.

M. A. Schmucker and J. Schmit, "Selection process for sequentially combing multiple sets of overlapping surface profile interferometric data to produce a continuous composite map," U.S. patent 5,991,461 ( 23 November 1999).

Schmucker, M. A.

M. A. Schmucker and J. Schmit, "Selection process for sequentially combing multiple sets of overlapping surface profile interferometric data to produce a continuous composite map," U.S. patent 5,991,461 ( 23 November 1999).

Servin, M.

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

Sha, D.

H. Liu, Q. Hao, Q. Zhu, D. Sha, and C. Zhang, "A novel aspheric surface testing method using part-compensating lens," in Optical Design and Testing II, Y.Wang, Z.Weng, S.Ye, and J.M.Sasián, eds., Proc. SPIE 5638, 324-329 (2004).

Silva, D. E.

Slomba, A. F.

A. F. Slomba and L. Montagnino, "Subaperture testing for mid-frequency figure control on large aspheric mirror," in Precision Surface Metrology, J.C.Wyant, ed., Proc. SPIE 429, 114-118 (1983).

Swantner, W.

Thunen, J. G.

J. G. Thunen and O. Y. Kwon, " Full aperture testing with subaperture test optics," in Wavefront Sensing, N.Bareket and C.L.Koliopoulos, eds., Proc. SPIE 351, 19-27 (1982).

Tricard, M.

P. E. Murphy, J. Fleig, G. Forbes, and M. Tricard, "High precision metrology of demes and aspheric optics," in Window and Dome Technologies and Materials IX, R.W.Tustison, ed., Proc. SPIE 5786, 112-121 (2005).

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, "Stitching interferometry: A flexible solution for surface metrology," Opt. Photon. News 14, 38-43 (2003).
[Crossref]

Tronolone, M. J.

M. J. Tronolone, J. F. Fleig, C. Huang, and J. H. Bruning, "Method of testing aspherical optical surfaces with an interferometer," U.S. patent 5,416,586 ( 16 May 1995).

Tsujiuchi, J.

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

Wang, J. Y.

Wyant, J. C.

J. C. Wyant and K. Creath, "Basic wavefront aberration theory for optical metrology," in Applied Optics and Optical Engineering Series (Academic, 1992), Vol. XI, p. 28.

Zhang, C.

H. Liu, Q. Hao, Q. Zhu, D. Sha, and C. Zhang, "A novel aspheric surface testing method using part-compensating lens," in Optical Design and Testing II, Y.Wang, Z.Weng, S.Ye, and J.M.Sasián, eds., Proc. SPIE 5638, 324-329 (2004).

Zhu, Q.

H. Liu, Q. Hao, Q. Zhu, D. Sha, and C. Zhang, "A novel aspheric surface testing method using part-compensating lens," in Optical Design and Testing II, Y.Wang, Z.Weng, S.Ye, and J.M.Sasián, eds., Proc. SPIE 5638, 324-329 (2004).

Appl. Opt. (8)

J. Opt. Soc. Am. 71 (1)

V. N. Mahajan, "Zernike annular polynomials for imaging systems with annular pupils," J. Opt. Soc. Am. 71 , 75-85 (1981).
[Crossref]

Opt. Eng. (2)

M. Melozzi, L. Pezzati, and A. Mazzoni, "Testing aspheric surfaces using multiple annular interferograms," Opt. Eng. 32, 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, 608-613 (1994).
[Crossref]

Opt. Lett. (1)

Opt. Photon. News (1)

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, "Stitching interferometry: A flexible solution for surface metrology," Opt. Photon. News 14, 38-43 (2003).
[Crossref]

Opt. Rev. (1)

F. Granados-Agustín, J. F. Escobar-Romero, and A. Cornejo-Rodríguez, "Testing parabolic surfaces with annular subaperture interferograsm," Opt. Rev. 11, 82-86 (2004).
[Crossref]

Other (11)

J. C. Wyant and K. Creath, "Basic wavefront aberration theory for optical metrology," in Applied Optics and Optical Engineering Series (Academic, 1992), Vol. XI, p. 28.

H. Liu, Q. Hao, Q. Zhu, D. Sha, and C. Zhang, "A novel aspheric surface testing method using part-compensating lens," in Optical Design and Testing II, Y.Wang, Z.Weng, S.Ye, and J.M.Sasián, eds., Proc. SPIE 5638, 324-329 (2004).

The mathematica software was developed by the Wolfram Research, Inc., http://www.wolfram.com.

A. F. Slomba and L. Montagnino, "Subaperture testing for mid-frequency figure control on large aspheric mirror," in Precision Surface Metrology, J.C.Wyant, ed., Proc. SPIE 429, 114-118 (1983).

P. E. Murphy, J. Fleig, G. Forbes, and M. Tricard, "High precision metrology of demes and aspheric optics," in Window and Dome Technologies and Materials IX, R.W.Tustison, ed., Proc. SPIE 5786, 112-121 (2005).

T. Hänsel, A. Nickel, and A. Schindler, "Stitching interferometry of aspherical surfaces," in Optical Metrology Roadmap for the Semiconductor, Optical, and Data Storage Industries II, A.Duparré and B.Singh, eds., Proc. SPIE 4449, 265-273 (2001).

D. Malacara, M. Servin, A. Morales, and Z. Malacara, " Aspherical wavefront testing with several defocusing 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," U.S. patent 5,416,586 ( 16 May 1995).

M. Bray, "Stitching interferometer for large plano optics using a standard interferometer," in Optical Manufacturing and Testing II, H.PhilipStahl, ed., Proc. SPIE 3134, 39-50 (1997).

J. G. Thunen and O. Y. Kwon, " Full aperture testing with subaperture test optics," in Wavefront Sensing, N.Bareket and C.L.Koliopoulos, eds., Proc. SPIE 351, 19-27 (1982).

M. A. Schmucker and J. Schmit, "Selection process for sequentially combing multiple sets of overlapping surface profile interferometric data to produce a continuous composite map," U.S. patent 5,991,461 ( 23 November 1999).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Geometry of the annular subaperture interferometric method.

Fig. 2
Fig. 2

Simulated interferograms from the aspheric surface tested with the spherical wavefront, showing the change of null fringe and evaluable subaperture area with the increase of d.

Fig. 3
Fig. 3

Extraction annular subaperture interferogram from (a) the oversampled interferogram with (b) the annular mask; (c) the evaluable subaperture interferogram.

Fig. 4
Fig. 4

Geometry of three complementary annular subapertures in world and pixel coordinates.

Fig. 5
Fig. 5

Partial listing of the mathematica program for calculating Zernike annular polynomials.

Fig. 6
Fig. 6

Results of simulation analysis (PV is peak to valley): (a) the original wavefront map, (b) isometric plot for a five-annular-subaperture wavefront, (c) reconstructed full-aperture wavefront map from data shown in (b) with random noise σ = 0.1, (d) reconstructed full-aperture wavefront map from data shown in (b) with random noise σ = 0.2, (e) the residual wavefront map between (c) and (a), (f) the residual wavefront map between (d) and (a).

Fig. 7
Fig. 7

Graphic representation of the matrix [T k ] obtained when the subaperture areas are as defined in Table.2 The 16-term Zernike annular polynomials are used in this simulation. The origin of graph is different from that of the matrix, and they have flip-vertical relation. The first column of matrix [T k ] corresponds to the fifth term of the full-aperture Zernike annular polynomials:(a) [T 1], (b) [T 2], (c) [T 3], (d) [T 4], (e) [T 5].

Fig. 8
Fig. 8

The relative rms wavefront error Δ versus the noise-to-signal ratio σ: (a) five results per each noise level are plotted, (b) the corresponding plot with error bar.

Fig. 9
Fig. 9

The layout of five subapertures within the full-aperture:(a) type I, (b) type II, (c) type III.

Fig. 10
Fig. 10

Profiles of defocuses with different coefficients.

Tables (5)

Tables Icon

Table 1 Orthogonal Zernike Annular Polynomials Zi (ρ, θ, ε) a

Tables Icon

Table 2 Area and Misalignment Coefficients of the Five Simulated Subapertures

Tables Icon

Table 3 Comparison of Annular Zernike Coefficients Reconstructed from Simulated Subaperture Data with Different Random Noise Levels

Tables Icon

Table 4 Comparison of Reconstructed Full-Aperture Wavefronts for Different Subaperture Configurations

Tables Icon

Table 5 Comparison of Reconstructed Full-Aperture Wavefronts for Different Subaperture Defocuses

Equations (70)

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

L = f + R 0 + d .
r in 1 = R in , r out1 = r in 2 , r out 2 = r in 3 , r out 3 = R out ,
r out 1 r in 2 , r out2 r in 3 , , r out K 1 r in K ,
m 1 = r in1 r in1 = r out 1 r out 1 , m 2 = r in 2 r in 2 = r out2 r out2 , , m k = r in k r in k = r out k r out k , , m K = r in K r in K = r out K r out K .
ε 1 = r in 1 r out 1 = r in 1 r out 1 , ε 2 = r in 2 r out 2 = r in2 r out2 , , ε k = r in k r out k = r in k r out k , , ε K = r in K r out K = r in K r out K ,
ε 0 = r in 1 r out K = R in R out .
( ρ 1 , θ ) { ε 1 ρ 1 1 , 0 θ 2 π } ,
( ρ 2 , θ ) { ε 2 ρ 2 1 , 0 θ 2 π } ,
( ρ k , θ ) { ε k ρ k 1 , 0 θ 2 π } ,
( ρ K , θ ) { ε K ρ K 1 , 0 θ 2 π } ,
( P 1 , Θ ) { k = 1 K ε k P 1 k = 2 K ε k , 0 Θ 2 π } ,
( P 2 , Θ ) { k = 2 K ε k P 2 k = 3 K ε k , 0 Θ 2 π } ,
( P k , Θ ) { k = k K ε k P k k = k + 1 K ε k , 0 Θ 2 π } ,
( P K , Θ ) { ε K P K 1 , 0 Θ 2 π } ,
P k = ρ k k = k + 1 K ε k , Θ = θ ,
ε 0 = k = 1 K ε k .
Z n m ( ρ , θ , ε ) = { R n m ( ρ , ε ) cos ( ) , m > 0 R n m ( ρ , ε ) sin ( m θ ) , m < 0 R n m ( ρ , ε ) , } m 0 , m = 0
R n     m ( ρ ) = s = 0 ( n | m | ) / 2 ( 1 ) s ( n s ) ! s ! ( n + m 2 s ) ! ( n m 2 s ) ! ρ n 2 s .
R 2 j 0 ( ρ , ε ) = R 2 j 0 [ ( ρ 2 ε 2 1 ε 2 ) 1 / 2 ] .
R 2 j + m m ( ρ , ε ) = [ 1 ε 2 2 ( 2 j + m + 1 ) h j     m ] 1 / 2 ρ m Q j     m ( ρ 2 ) ,
Q j     m ( ρ 2 ) =
2 ( 2 j + 2 m 1 ) ( j + m ) ( 1 ε 2 ) h j     m 1 Q j     m 1 ( 0 ) i = 0 j Q i     m 1 ( 0 ) Q i     m 1 ( ρ 2 ) h i     m 1 ,
h j m = 2 ( 2 j + 2 m 1 ) ( j + m ) ( 1 ε 2 ) Q j + 1 m 1 ( 0 ) Q j     m 1 ( 0 ) h j     m 1 ,
Q j 0 ( ρ 2 ) = R 2 j 0 ( ρ , ε ) ,
h j 0 = 1 ε 2 2 ( 2 j + 1 ) .
R 2 j + | m | m ( ρ , ε ) = R 2 j + | m | | m | ( ρ , ε ) .
R n     n ( ρ , ε ) = ρ n ( i = 0 n ε 2 i ) 1 / 2 .
n = 2 j + | m | .
p = ( n | m | ) 2 ,
q = | m |
ρ = x 2 + y 2 , θ = tan 1 ( y / x ) .
σ fit 2 = 1 N k n = 1 N k { w k ( x n , y n , ε k ) S k [ x n , y n ] } 2 ,
w k ( x , y , ε k ) = i = 1 L a k i Z k i ( x , y , ε k ) ,
A 1 = [ a 11 a 12 a 13 a 14 a 1 L ] ,
A 2 = [ a 21 a 22 a 23 a 24 a 2 L ] ,
A K = [ a K 1 a K 2 a K 3 a K 4 a K L ] .
A = [ A 1 A 2 A 3 A K ] T .
W ( P , Θ , ε 0 ) = k = 1 K i = 1 4 b k i Z k i ( ρ k , θ , ε k ) + i = 5 L B i Z i ( P , Θ , ε 0 ) ,
W ( P , Θ , ε 0 ) = k = 1 K i = 1 L a k i Z k i ( ρ k , θ , ε k ) ,
k = 1 K i = 1 L a k i Z k i ( ρ k , θ , ε k ) = k = 1 K i = 1 4 b k i Z k i ( ρ k , θ , ε k ) + i = 5 L B i Z i ( P , Θ , ε 0 ) .
k = 1 K i = 1 L a k i Z i ( ρ k , θ , ε k ) Z i ( ρ k , θ , ε k ) k = k = 1 K i = 1 4 b k i Z i ( ρ k , θ , ε k ) Z i ( ρ k , θ , ε k ) k + i = 5 L B i Z i ( P k , Θ , ε 0 ) Z i ( ρ k , θ , ε k ) k ,
Z i ( ρ k , θ , ε k ) Z i ( ρ k , θ , ε k ) k = 0 2 π ε k i Z i ( ρ , θ , ε k ) Z i ( ρ , θ , ε k ) ρ d ρ d θ ,
Z i ( P k , Θ , ε 0 ) Z i ( ρ k , θ , ε k ) k = 0 2 π ε k i Z i ( P k , Θ , ε k ) Z i ( ρ , θ , ε k ) ρ d ρ d θ .
Z i ( P k , Θ , ε 0 ) Z i ( ρ k , θ , ε k ) k =
0 2 π ε k 1 Z i ( ρ k = k + 1 K ε k , θ , ε 0 ) Z i ( ρ , θ , ε k ) ρ d ρ d θ .
[ M ] K L × K L A K L × 1 = [ [ N ] K L × 4 K [ T ] K L × ( L 4 ) ] B ( 4 K + L 4 ) × 1 .
[ M ] = [ [ M 1 ] 0 0 0 0 [ M 2 ] 0 0 0 0 [ M K 1 ] 0 0 0 0 [ M K ] ] .
[ M k ] = [ # 0 0 0 0 # 0 0 0 0 # 0 0 0 0 # ] ,
[ N ] = [ [ N 1 ] 0 0 0 0 [ N 2 ] 0 0 0 0 [ N K 1 ] 0 0 0 0 [ N K ] ] .
[ N k ] = [ # 0 0 0 0 # 0 0 0 0 # 0 0 0 0 # 0 0 0 0 ] .
[ T ] = [ [ T 1 ] [ T 2 ] [ T K ] ] ,
B = [ b 11 b 12 b 13 b 14 b 21 b 22 b 23 b 24 b K 1 b K 2 b K 3 b K 4 B 5 B 6 B L ] T .
[ G ] = [ [ N ] [ T ] ] .
G 1 = ( G T G ) 1 G T .
K L 4 K + L 4.
B = [ G ] 1 [ M ] A .
B = [ H ] A ,
g ( x ) = 1 ( 2 π σ 0 ) exp [ ( x μ ) 2 2 σ 0     2 ] ,
S = [ 1 N 1 i = 1 N ( x i x ¯ ) 2 ] 1 / 2 .
σ = σ 0 S .
Δ = [ n = 1 N i = 5 L ( B i B i ) 2 Z i     2 ( x n , y n , ε 0 ) n = 1 N i = 5 L B i     2 Z i     2 ( x n , y n , ε 0 ) ] 1 / 2 ,
P II 1 [ 0.1 , 0.3 ] , P II 2 [ 0.3 , 0.5 ] , P II 3 [ 0.5 , 0.7 ] , P II 4 [ 0.7 , 0.9 ] , P II 5 [ 0.9 , 1 ] ,
P III 1 [ 0.1 , 0.4 ] , P III 2 [ 0.4 , 0.65 ] , P III 3 [ 0.65 , 0.85 ] , P III 4 [ 0.85 , 0.95 ] , P III 5 [ 0.95 , 1 ] .
f ( x , y ) = 2 ( x 2 + y 2 ) 1.
f ( x ) = 2 x 2 1.

Metrics