Abstract

An innovative iterative search method called the synthetic phase-shifting (SPS) algorithm is proposed. This search algorithm is used for maximum-likelihood (ML) estimation of a wavefront that is described by a finite set of Zernike Fringe polynomials. In this paper, we estimate the coefficient, or parameter, values of the wavefront using a single interferogram obtained from a point-diffraction interferometer (PDI). In order to find the estimates, we first calculate the squared-difference between the measured and simulated interferograms. Under certain assumptions, this squared-difference image can be treated as an interferogram showing the phase difference between the true wavefront deviation and simulated wavefront deviation. The wavefront deviation is the difference between the reference and the test wavefronts. We calculate the phase difference using a traditional phase-shifting technique without physical phase-shifters. We present a detailed forward model for the PDI interferogram, including the effect of the finite size of a detector pixel. The algorithm was validated with computational studies and its performance and constraints are discussed. A prototype PDI was built and the algorithm was also experimentally validated. A large wavefront deviation was successfully estimated without using null optics or physical phase-shifters. The experimental result shows that the proposed algorithm has great potential to provide an accurate tool for non-null testing.

© 2013 Optical Society of America

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  1. V. Genberg, G. Michels, and K. B. Doyle, “Orthogonality of Zernike polynomials,” Proc. SPIE4771, 276–286 (2002).
    [CrossRef]
  2. D. Malacara, Optical Shop Testing (Wiley, 1978).
  3. J. E. Greivenkamp and R. O. Gappinger, “Design of a nonnull interferometer for aspheric wave fronts,” Appl. Opt.43(27), 5143–5151 (2004).
    [CrossRef] [PubMed]
  4. J. E. Greivenkamp, “Sub-Nyquist interferometry,” Appl. Opt.26(24), 5245–5258 (1987).
    [CrossRef] [PubMed]
  5. J. E. Greivenkamp, A. E. Lowman, and R. J. Palum, “Sub‐Nyquist interferometry: implementation and measurement capability,” Opt. Eng.35(10), 2962–2969 (1996).
    [CrossRef]
  6. H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).
  7. H. H. Barrett, C. Dainty, and D. Lara, “Maximum-likelihood methods in wavefront sensing: stochastic models and likelihood functions,” J. Opt. Soc. Am. A24(2), 391–414 (2007).
    [CrossRef] [PubMed]
  8. J. A. Sakamoto and H. H. Barrett, “Maximum-likelihood estimation of parameterized wavefronts from multifocal data,” Opt. Express20(14), 15928–15944 (2012).
    [CrossRef] [PubMed]
  9. J. A. Sakamoto, H. H. Barrett, and A. V. Goncharov, “Inverse optical design of the human eye using likelihood methods and wavefront sensing,” Opt. Express16(1), 304–314 (2008).
    [CrossRef] [PubMed]
  10. W. P. Linnik, “A simple interferometer for the investigation of optical systems,” C. R. Acad. Sci. URSS5, 208–210 (1933).
  11. R. N. Smartt and W. H. Steel, “Theory and Application of point-diffraction interferometers,” Jpn. J. Appl. Phys.14, 351–356 (1975).
  12. R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two‐dimensional phase unwrapping,” Radio Sci.23(4), 713–720 (1988).
    [CrossRef]
  13. D. Ruijters, B. M. ter Harr Romeny, and P. Suetens, "Efficient GPU-accelerated elastic image registration," in Proceedings Sixth IASTED international conference on biomedical engineering (BioMed), pp. 419-424 (2008).
  14. G. E. Sommargren, "Phase shifting diffraction interferometry for measuring extreme ultraviolet optics," No. UCRL-JC--123549, CONF-9604150--1, Lawrence Livermore National Lab., CA (1996).
  15. C. R. Mercer and K. Creath, “Liquid-crystal point-diffraction interferometer for wave-front measurements,” Appl. Opt.35(10), 1633–1642 (1996).
    [CrossRef] [PubMed]
  16. R. M. Neal and J. C. Wyant, “Polarization phase-shifting point-diffraction interferometer,” Appl. Opt.45(15), 3463–3476 (2006).
    [CrossRef] [PubMed]
  17. M. Paturzo, F. Pignatiello, S. Grilli, S. De Nicola, and P. Ferraro, “Phase-shifting point-diffraction interferometer developed by using the electro-optic effect in ferroelectric crystals,” Opt. Lett.31(24), 3597–3599 (2006).
    [CrossRef] [PubMed]
  18. J. E. Millerd, N. J. Brock, J. B. Hayes, and J. C. Wyant, “Instantaneous phase-shift point-diffraction interferometer,” Proc. SPIE5531, 264–272 (2004).
    [CrossRef]
  19. P. Su, J. Burge, R. A. Sprowl, and J. Sasian,"Maximum likelihood estimation as a general method of combining subaperture data for interferometric testing," Proc. SPIE 6342, 1X-1X-6 (2006).

2012

2008

2007

2006

2004

J. E. Millerd, N. J. Brock, J. B. Hayes, and J. C. Wyant, “Instantaneous phase-shift point-diffraction interferometer,” Proc. SPIE5531, 264–272 (2004).
[CrossRef]

J. E. Greivenkamp and R. O. Gappinger, “Design of a nonnull interferometer for aspheric wave fronts,” Appl. Opt.43(27), 5143–5151 (2004).
[CrossRef] [PubMed]

2002

V. Genberg, G. Michels, and K. B. Doyle, “Orthogonality of Zernike polynomials,” Proc. SPIE4771, 276–286 (2002).
[CrossRef]

1996

J. E. Greivenkamp, A. E. Lowman, and R. J. Palum, “Sub‐Nyquist interferometry: implementation and measurement capability,” Opt. Eng.35(10), 2962–2969 (1996).
[CrossRef]

C. R. Mercer and K. Creath, “Liquid-crystal point-diffraction interferometer for wave-front measurements,” Appl. Opt.35(10), 1633–1642 (1996).
[CrossRef] [PubMed]

1988

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two‐dimensional phase unwrapping,” Radio Sci.23(4), 713–720 (1988).
[CrossRef]

1987

1975

R. N. Smartt and W. H. Steel, “Theory and Application of point-diffraction interferometers,” Jpn. J. Appl. Phys.14, 351–356 (1975).

1933

W. P. Linnik, “A simple interferometer for the investigation of optical systems,” C. R. Acad. Sci. URSS5, 208–210 (1933).

Barrett, H. H.

Brock, N. J.

J. E. Millerd, N. J. Brock, J. B. Hayes, and J. C. Wyant, “Instantaneous phase-shift point-diffraction interferometer,” Proc. SPIE5531, 264–272 (2004).
[CrossRef]

Creath, K.

Dainty, C.

De Nicola, S.

Doyle, K. B.

V. Genberg, G. Michels, and K. B. Doyle, “Orthogonality of Zernike polynomials,” Proc. SPIE4771, 276–286 (2002).
[CrossRef]

Ferraro, P.

Gappinger, R. O.

Genberg, V.

V. Genberg, G. Michels, and K. B. Doyle, “Orthogonality of Zernike polynomials,” Proc. SPIE4771, 276–286 (2002).
[CrossRef]

Goldstein, R. M.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two‐dimensional phase unwrapping,” Radio Sci.23(4), 713–720 (1988).
[CrossRef]

Goncharov, A. V.

Greivenkamp, J. E.

Grilli, S.

Hayes, J. B.

J. E. Millerd, N. J. Brock, J. B. Hayes, and J. C. Wyant, “Instantaneous phase-shift point-diffraction interferometer,” Proc. SPIE5531, 264–272 (2004).
[CrossRef]

Lara, D.

Linnik, W. P.

W. P. Linnik, “A simple interferometer for the investigation of optical systems,” C. R. Acad. Sci. URSS5, 208–210 (1933).

Lowman, A. E.

J. E. Greivenkamp, A. E. Lowman, and R. J. Palum, “Sub‐Nyquist interferometry: implementation and measurement capability,” Opt. Eng.35(10), 2962–2969 (1996).
[CrossRef]

Mercer, C. R.

Michels, G.

V. Genberg, G. Michels, and K. B. Doyle, “Orthogonality of Zernike polynomials,” Proc. SPIE4771, 276–286 (2002).
[CrossRef]

Millerd, J. E.

J. E. Millerd, N. J. Brock, J. B. Hayes, and J. C. Wyant, “Instantaneous phase-shift point-diffraction interferometer,” Proc. SPIE5531, 264–272 (2004).
[CrossRef]

Neal, R. M.

Palum, R. J.

J. E. Greivenkamp, A. E. Lowman, and R. J. Palum, “Sub‐Nyquist interferometry: implementation and measurement capability,” Opt. Eng.35(10), 2962–2969 (1996).
[CrossRef]

Paturzo, M.

Pignatiello, F.

Ruijters, D.

D. Ruijters, B. M. ter Harr Romeny, and P. Suetens, "Efficient GPU-accelerated elastic image registration," in Proceedings Sixth IASTED international conference on biomedical engineering (BioMed), pp. 419-424 (2008).

Sakamoto, J. A.

Smartt, R. N.

R. N. Smartt and W. H. Steel, “Theory and Application of point-diffraction interferometers,” Jpn. J. Appl. Phys.14, 351–356 (1975).

Steel, W. H.

R. N. Smartt and W. H. Steel, “Theory and Application of point-diffraction interferometers,” Jpn. J. Appl. Phys.14, 351–356 (1975).

Suetens, P.

D. Ruijters, B. M. ter Harr Romeny, and P. Suetens, "Efficient GPU-accelerated elastic image registration," in Proceedings Sixth IASTED international conference on biomedical engineering (BioMed), pp. 419-424 (2008).

ter Harr Romeny, B. M.

D. Ruijters, B. M. ter Harr Romeny, and P. Suetens, "Efficient GPU-accelerated elastic image registration," in Proceedings Sixth IASTED international conference on biomedical engineering (BioMed), pp. 419-424 (2008).

Werner, C. L.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two‐dimensional phase unwrapping,” Radio Sci.23(4), 713–720 (1988).
[CrossRef]

Wyant, J. C.

R. M. Neal and J. C. Wyant, “Polarization phase-shifting point-diffraction interferometer,” Appl. Opt.45(15), 3463–3476 (2006).
[CrossRef] [PubMed]

J. E. Millerd, N. J. Brock, J. B. Hayes, and J. C. Wyant, “Instantaneous phase-shift point-diffraction interferometer,” Proc. SPIE5531, 264–272 (2004).
[CrossRef]

Zebker, H. A.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two‐dimensional phase unwrapping,” Radio Sci.23(4), 713–720 (1988).
[CrossRef]

Appl. Opt.

C. R. Acad. Sci. URSS

W. P. Linnik, “A simple interferometer for the investigation of optical systems,” C. R. Acad. Sci. URSS5, 208–210 (1933).

J. Opt. Soc. Am. A

Jpn. J. Appl. Phys.

R. N. Smartt and W. H. Steel, “Theory and Application of point-diffraction interferometers,” Jpn. J. Appl. Phys.14, 351–356 (1975).

Opt. Eng.

J. E. Greivenkamp, A. E. Lowman, and R. J. Palum, “Sub‐Nyquist interferometry: implementation and measurement capability,” Opt. Eng.35(10), 2962–2969 (1996).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. SPIE

J. E. Millerd, N. J. Brock, J. B. Hayes, and J. C. Wyant, “Instantaneous phase-shift point-diffraction interferometer,” Proc. SPIE5531, 264–272 (2004).
[CrossRef]

V. Genberg, G. Michels, and K. B. Doyle, “Orthogonality of Zernike polynomials,” Proc. SPIE4771, 276–286 (2002).
[CrossRef]

Radio Sci.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two‐dimensional phase unwrapping,” Radio Sci.23(4), 713–720 (1988).
[CrossRef]

Other

D. Ruijters, B. M. ter Harr Romeny, and P. Suetens, "Efficient GPU-accelerated elastic image registration," in Proceedings Sixth IASTED international conference on biomedical engineering (BioMed), pp. 419-424 (2008).

G. E. Sommargren, "Phase shifting diffraction interferometry for measuring extreme ultraviolet optics," No. UCRL-JC--123549, CONF-9604150--1, Lawrence Livermore National Lab., CA (1996).

D. Malacara, Optical Shop Testing (Wiley, 1978).

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).

P. Su, J. Burge, R. A. Sprowl, and J. Sasian,"Maximum likelihood estimation as a general method of combining subaperture data for interferometric testing," Proc. SPIE 6342, 1X-1X-6 (2006).

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