Abstract

We present a computational method for field-varying aberration recovery in optical systems by imaging a weak (index-matched) diffuser. Using multiple images acquired under plane wave illumination at distinct angles, the aberrations of the imaging system can be uniquely determined up to a sign. Our method is based on a statistical model for image formation that relates the spectrum of the speckled intensity image to the local aberrations at different locations in the field-of-view. The diffuser is treated as a wide-sense stationary scattering object, eliminating the need for precise knowledge of its surface shape. We validate our method both numerically and experimentally, showing that this relatively simple algorithmic calibration method can be reliably used to recover system aberrations quantitatively.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photonics 7, 739 (2013).
    [Crossref]
  2. X. Ou, G. Zheng, and C. Yang, “Embedded pupil function recovery for Fourier ptychographic microscopy,” Opt. Express 22, 4960–4972 (2014).
    [Crossref] [PubMed]
  3. J. Chung, J. Kim, X. Ou, R. Horstmeyer, and C. Yang, “Wide field-of-view fluorescence image deconvolution with aberration-estimation from Fourier ptychography,” Biomed. Opt. Express 7, 352–368 (2016).
    [Crossref] [PubMed]
  4. A. Thust, M. Overwijk, W. Coene, and M. Lentzen, “Numerical correction of lens aberrations in phase-retrieval HRTEM,” Ultramicroscopy 64, 249–264 (1996).
    [Crossref]
  5. P. Hariharan, B. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
    [Crossref] [PubMed]
  6. M. P. Rimmer and J. C. Wyant, “Evaluation of large aberrations using a lateral-shear interferometer having variable shear,” Appl. Opt. 14, 142–150 (1975).
    [Crossref] [PubMed]
  7. B. M. Hanser, M. G. Gustafsson, D. Agard, and J. W. Sedat, “Phase-retrieved pupil functions in wide-field fluorescence microscopy,” J. Microsc. 216, 32–48 (2004).
    [Crossref] [PubMed]
  8. J. R. Fienup, J. C. Marron, T. J. Schulz, and J. H. Seldin, “Hubble space telescope characterized by using phase-retrieval algorithms,” Appl. Opt. 32, 1747–1767 (1993).
    [Crossref] [PubMed]
  9. R. K. Tyson, Principles of Adaptive Optics (CRC, 2015).
  10. H. W. Babcock, “Adaptive optics revisited,” Science 249, 253–257 (1990).
    [Crossref] [PubMed]
  11. N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7, 141 (2010).
    [Crossref]
  12. A. Roorda, F. Romero-Borja, W. J. Donnelly, H. Queener, T. J. Hebert, and M. C. Campbell, “Adaptive optics scanning laser ophthalmoscopy,” Opt. Express 10, 405–412 (2002).
    [Crossref] [PubMed]
  13. F. Zemlin, K. Weiss, P. Schiske, W. Kunath, and K.-H. Herrmann, “Coma-free alignment of high resolution electron microscopes with the aid of optical diffractograms,” Ultramicroscopy 3, 49–60 (1978).
    [Crossref]
  14. E. Voelkl, “Using diffractograms to evaluate optical systems with coherent illumination,” Opt. Lett. 28, 2318–2320 (2003).
    [Crossref] [PubMed]
  15. J. C. Spence, Experimental High-Resolution Electron Microscopy(Oxford University, 1988).
  16. R. Erni, Aberration-Corrected Imaging in Transmission Electron Microscopy: An Introduction(World Scientific, 2010).
    [Crossref]
  17. R. G. Paxman, T. J. Schulz, and J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” JOSA A 9, 1072–1085 (1992).
    [Crossref]
  18. D. Hamilton, C. Sheppard, and T. Wilson, “Improved imaging of phase gradients in scanning optical microscopy,” J. Microsc. 135, 275–286 (1984).
    [Crossref]
  19. N. Streibl, “Three-dimensional imaging by a microscope,” JOSA A 2, 121–127 (1985).
    [Crossref]
  20. H. Rose, “Nonstandard imaging methods in electron microscopy,” Ultramicroscopy 2, 251–267 (1976).
    [Crossref]
  21. G. Gunjala, A. Shanker, V. Jaedicke, N. Antipa, and L. Waller, “Optical transfer function characterization using a weak diffuser,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XXIII, (SPIE, 2016), p. 971315.
  22. A. Shanker, A. Wojdyla, G. Gunjala, J. Dong, M. Benk, A. Neureuther, K. Goldberg, and L. Waller, “Off-axis aberration estimation in an EUV microscope using natural speckle,” in Imaging Systems and Applications, (Optical Society of America, 2016), pp. ITh1F–2.
  23. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” JOSA 66, 207–211 (1976).
    [Crossref]
  24. J. Goodman, Introduction to Fourier Optics(McGraw-hill, 2008).
  25. J. W. Goodman, Statistical Optics(John Wiley & Sons, 2015).
  26. E. J. Kirkland, Advanced Computing in Electron Microscopy(Springer Science & Business Media, 2010).
    [Crossref]
  27. R. Eckert, Z. F. Phillips, and L. Waller, “efficient illumination angle self-calibration in fourier ptychography,” Appl. Opt. 57, 5434–5442 (2018).
    [Crossref]
  28. G. Aubert and J.-F. Aujol, “A variational approach to removing multiplicative noise,” SIAM J. on Appl. Math. 68, 925–946 (2008).
    [Crossref]
  29. R. H. Shumway and D. S. Stoffer, Time Series Analysis and its Applications(Springer, 2011).
    [Crossref]
  30. S. Wright and J. Nocedal, Numerical Optimization(Springer Science, 1999).
  31. M. Siddiqui, “Statistical inference for Rayleigh distributions,” J. Res. Natl. Bureau Standards, Sec. D 68, 1007 (1964).
  32. K. Lange, Applied Probability(Springer Science & Business Media2010).
    [Crossref]
  33. F. H. Clarke, “Generalized gradients and applications,” Transactions Am. Math. Soc. 205, 247–262 (1975).
    [Crossref]
  34. J.-B. Hiriart-Urruty and C. Lemaréchal, Fundamentals of Convex Analysis(Springer-Verlag, 2001).
    [Crossref]

2018 (1)

2016 (1)

2014 (1)

2013 (1)

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photonics 7, 739 (2013).
[Crossref]

2010 (1)

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7, 141 (2010).
[Crossref]

2008 (1)

G. Aubert and J.-F. Aujol, “A variational approach to removing multiplicative noise,” SIAM J. on Appl. Math. 68, 925–946 (2008).
[Crossref]

2004 (1)

B. M. Hanser, M. G. Gustafsson, D. Agard, and J. W. Sedat, “Phase-retrieved pupil functions in wide-field fluorescence microscopy,” J. Microsc. 216, 32–48 (2004).
[Crossref] [PubMed]

2003 (1)

2002 (1)

1996 (1)

A. Thust, M. Overwijk, W. Coene, and M. Lentzen, “Numerical correction of lens aberrations in phase-retrieval HRTEM,” Ultramicroscopy 64, 249–264 (1996).
[Crossref]

1993 (1)

1992 (1)

R. G. Paxman, T. J. Schulz, and J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” JOSA A 9, 1072–1085 (1992).
[Crossref]

1990 (1)

H. W. Babcock, “Adaptive optics revisited,” Science 249, 253–257 (1990).
[Crossref] [PubMed]

1987 (1)

1985 (1)

N. Streibl, “Three-dimensional imaging by a microscope,” JOSA A 2, 121–127 (1985).
[Crossref]

1984 (1)

D. Hamilton, C. Sheppard, and T. Wilson, “Improved imaging of phase gradients in scanning optical microscopy,” J. Microsc. 135, 275–286 (1984).
[Crossref]

1978 (1)

F. Zemlin, K. Weiss, P. Schiske, W. Kunath, and K.-H. Herrmann, “Coma-free alignment of high resolution electron microscopes with the aid of optical diffractograms,” Ultramicroscopy 3, 49–60 (1978).
[Crossref]

1976 (2)

H. Rose, “Nonstandard imaging methods in electron microscopy,” Ultramicroscopy 2, 251–267 (1976).
[Crossref]

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” JOSA 66, 207–211 (1976).
[Crossref]

1975 (2)

1964 (1)

M. Siddiqui, “Statistical inference for Rayleigh distributions,” J. Res. Natl. Bureau Standards, Sec. D 68, 1007 (1964).

Agard, D.

B. M. Hanser, M. G. Gustafsson, D. Agard, and J. W. Sedat, “Phase-retrieved pupil functions in wide-field fluorescence microscopy,” J. Microsc. 216, 32–48 (2004).
[Crossref] [PubMed]

Antipa, N.

G. Gunjala, A. Shanker, V. Jaedicke, N. Antipa, and L. Waller, “Optical transfer function characterization using a weak diffuser,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XXIII, (SPIE, 2016), p. 971315.

Aubert, G.

G. Aubert and J.-F. Aujol, “A variational approach to removing multiplicative noise,” SIAM J. on Appl. Math. 68, 925–946 (2008).
[Crossref]

Aujol, J.-F.

G. Aubert and J.-F. Aujol, “A variational approach to removing multiplicative noise,” SIAM J. on Appl. Math. 68, 925–946 (2008).
[Crossref]

Babcock, H. W.

H. W. Babcock, “Adaptive optics revisited,” Science 249, 253–257 (1990).
[Crossref] [PubMed]

Benk, M.

A. Shanker, A. Wojdyla, G. Gunjala, J. Dong, M. Benk, A. Neureuther, K. Goldberg, and L. Waller, “Off-axis aberration estimation in an EUV microscope using natural speckle,” in Imaging Systems and Applications, (Optical Society of America, 2016), pp. ITh1F–2.

Betzig, E.

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7, 141 (2010).
[Crossref]

Campbell, M. C.

Chung, J.

Clarke, F. H.

F. H. Clarke, “Generalized gradients and applications,” Transactions Am. Math. Soc. 205, 247–262 (1975).
[Crossref]

Coene, W.

A. Thust, M. Overwijk, W. Coene, and M. Lentzen, “Numerical correction of lens aberrations in phase-retrieval HRTEM,” Ultramicroscopy 64, 249–264 (1996).
[Crossref]

Dong, J.

A. Shanker, A. Wojdyla, G. Gunjala, J. Dong, M. Benk, A. Neureuther, K. Goldberg, and L. Waller, “Off-axis aberration estimation in an EUV microscope using natural speckle,” in Imaging Systems and Applications, (Optical Society of America, 2016), pp. ITh1F–2.

Donnelly, W. J.

Eckert, R.

Eiju, T.

Erni, R.

R. Erni, Aberration-Corrected Imaging in Transmission Electron Microscopy: An Introduction(World Scientific, 2010).
[Crossref]

Fienup, J. R.

J. R. Fienup, J. C. Marron, T. J. Schulz, and J. H. Seldin, “Hubble space telescope characterized by using phase-retrieval algorithms,” Appl. Opt. 32, 1747–1767 (1993).
[Crossref] [PubMed]

R. G. Paxman, T. J. Schulz, and J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” JOSA A 9, 1072–1085 (1992).
[Crossref]

Goldberg, K.

A. Shanker, A. Wojdyla, G. Gunjala, J. Dong, M. Benk, A. Neureuther, K. Goldberg, and L. Waller, “Off-axis aberration estimation in an EUV microscope using natural speckle,” in Imaging Systems and Applications, (Optical Society of America, 2016), pp. ITh1F–2.

Goodman, J.

J. Goodman, Introduction to Fourier Optics(McGraw-hill, 2008).

Goodman, J. W.

J. W. Goodman, Statistical Optics(John Wiley & Sons, 2015).

Gunjala, G.

A. Shanker, A. Wojdyla, G. Gunjala, J. Dong, M. Benk, A. Neureuther, K. Goldberg, and L. Waller, “Off-axis aberration estimation in an EUV microscope using natural speckle,” in Imaging Systems and Applications, (Optical Society of America, 2016), pp. ITh1F–2.

G. Gunjala, A. Shanker, V. Jaedicke, N. Antipa, and L. Waller, “Optical transfer function characterization using a weak diffuser,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XXIII, (SPIE, 2016), p. 971315.

Gustafsson, M. G.

B. M. Hanser, M. G. Gustafsson, D. Agard, and J. W. Sedat, “Phase-retrieved pupil functions in wide-field fluorescence microscopy,” J. Microsc. 216, 32–48 (2004).
[Crossref] [PubMed]

Hamilton, D.

D. Hamilton, C. Sheppard, and T. Wilson, “Improved imaging of phase gradients in scanning optical microscopy,” J. Microsc. 135, 275–286 (1984).
[Crossref]

Hanser, B. M.

B. M. Hanser, M. G. Gustafsson, D. Agard, and J. W. Sedat, “Phase-retrieved pupil functions in wide-field fluorescence microscopy,” J. Microsc. 216, 32–48 (2004).
[Crossref] [PubMed]

Hariharan, P.

Hebert, T. J.

Herrmann, K.-H.

F. Zemlin, K. Weiss, P. Schiske, W. Kunath, and K.-H. Herrmann, “Coma-free alignment of high resolution electron microscopes with the aid of optical diffractograms,” Ultramicroscopy 3, 49–60 (1978).
[Crossref]

Hiriart-Urruty, J.-B.

J.-B. Hiriart-Urruty and C. Lemaréchal, Fundamentals of Convex Analysis(Springer-Verlag, 2001).
[Crossref]

Horstmeyer, R.

Jaedicke, V.

G. Gunjala, A. Shanker, V. Jaedicke, N. Antipa, and L. Waller, “Optical transfer function characterization using a weak diffuser,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XXIII, (SPIE, 2016), p. 971315.

Ji, N.

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7, 141 (2010).
[Crossref]

Kim, J.

Kirkland, E. J.

E. J. Kirkland, Advanced Computing in Electron Microscopy(Springer Science & Business Media, 2010).
[Crossref]

Kunath, W.

F. Zemlin, K. Weiss, P. Schiske, W. Kunath, and K.-H. Herrmann, “Coma-free alignment of high resolution electron microscopes with the aid of optical diffractograms,” Ultramicroscopy 3, 49–60 (1978).
[Crossref]

Lange, K.

K. Lange, Applied Probability(Springer Science & Business Media2010).
[Crossref]

Lemaréchal, C.

J.-B. Hiriart-Urruty and C. Lemaréchal, Fundamentals of Convex Analysis(Springer-Verlag, 2001).
[Crossref]

Lentzen, M.

A. Thust, M. Overwijk, W. Coene, and M. Lentzen, “Numerical correction of lens aberrations in phase-retrieval HRTEM,” Ultramicroscopy 64, 249–264 (1996).
[Crossref]

Marron, J. C.

Milkie, D. E.

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7, 141 (2010).
[Crossref]

Neureuther, A.

A. Shanker, A. Wojdyla, G. Gunjala, J. Dong, M. Benk, A. Neureuther, K. Goldberg, and L. Waller, “Off-axis aberration estimation in an EUV microscope using natural speckle,” in Imaging Systems and Applications, (Optical Society of America, 2016), pp. ITh1F–2.

Nocedal, J.

S. Wright and J. Nocedal, Numerical Optimization(Springer Science, 1999).

Noll, R. J.

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” JOSA 66, 207–211 (1976).
[Crossref]

Oreb, B.

Ou, X.

Overwijk, M.

A. Thust, M. Overwijk, W. Coene, and M. Lentzen, “Numerical correction of lens aberrations in phase-retrieval HRTEM,” Ultramicroscopy 64, 249–264 (1996).
[Crossref]

Paxman, R. G.

R. G. Paxman, T. J. Schulz, and J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” JOSA A 9, 1072–1085 (1992).
[Crossref]

Phillips, Z. F.

Queener, H.

Rimmer, M. P.

Romero-Borja, F.

Roorda, A.

Rose, H.

H. Rose, “Nonstandard imaging methods in electron microscopy,” Ultramicroscopy 2, 251–267 (1976).
[Crossref]

Schiske, P.

F. Zemlin, K. Weiss, P. Schiske, W. Kunath, and K.-H. Herrmann, “Coma-free alignment of high resolution electron microscopes with the aid of optical diffractograms,” Ultramicroscopy 3, 49–60 (1978).
[Crossref]

Schulz, T. J.

J. R. Fienup, J. C. Marron, T. J. Schulz, and J. H. Seldin, “Hubble space telescope characterized by using phase-retrieval algorithms,” Appl. Opt. 32, 1747–1767 (1993).
[Crossref] [PubMed]

R. G. Paxman, T. J. Schulz, and J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” JOSA A 9, 1072–1085 (1992).
[Crossref]

Sedat, J. W.

B. M. Hanser, M. G. Gustafsson, D. Agard, and J. W. Sedat, “Phase-retrieved pupil functions in wide-field fluorescence microscopy,” J. Microsc. 216, 32–48 (2004).
[Crossref] [PubMed]

Seldin, J. H.

Shanker, A.

A. Shanker, A. Wojdyla, G. Gunjala, J. Dong, M. Benk, A. Neureuther, K. Goldberg, and L. Waller, “Off-axis aberration estimation in an EUV microscope using natural speckle,” in Imaging Systems and Applications, (Optical Society of America, 2016), pp. ITh1F–2.

G. Gunjala, A. Shanker, V. Jaedicke, N. Antipa, and L. Waller, “Optical transfer function characterization using a weak diffuser,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XXIII, (SPIE, 2016), p. 971315.

Sheppard, C.

D. Hamilton, C. Sheppard, and T. Wilson, “Improved imaging of phase gradients in scanning optical microscopy,” J. Microsc. 135, 275–286 (1984).
[Crossref]

Shumway, R. H.

R. H. Shumway and D. S. Stoffer, Time Series Analysis and its Applications(Springer, 2011).
[Crossref]

Siddiqui, M.

M. Siddiqui, “Statistical inference for Rayleigh distributions,” J. Res. Natl. Bureau Standards, Sec. D 68, 1007 (1964).

Spence, J. C.

J. C. Spence, Experimental High-Resolution Electron Microscopy(Oxford University, 1988).

Stoffer, D. S.

R. H. Shumway and D. S. Stoffer, Time Series Analysis and its Applications(Springer, 2011).
[Crossref]

Streibl, N.

N. Streibl, “Three-dimensional imaging by a microscope,” JOSA A 2, 121–127 (1985).
[Crossref]

Thust, A.

A. Thust, M. Overwijk, W. Coene, and M. Lentzen, “Numerical correction of lens aberrations in phase-retrieval HRTEM,” Ultramicroscopy 64, 249–264 (1996).
[Crossref]

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics (CRC, 2015).

Voelkl, E.

Waller, L.

R. Eckert, Z. F. Phillips, and L. Waller, “efficient illumination angle self-calibration in fourier ptychography,” Appl. Opt. 57, 5434–5442 (2018).
[Crossref]

A. Shanker, A. Wojdyla, G. Gunjala, J. Dong, M. Benk, A. Neureuther, K. Goldberg, and L. Waller, “Off-axis aberration estimation in an EUV microscope using natural speckle,” in Imaging Systems and Applications, (Optical Society of America, 2016), pp. ITh1F–2.

G. Gunjala, A. Shanker, V. Jaedicke, N. Antipa, and L. Waller, “Optical transfer function characterization using a weak diffuser,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XXIII, (SPIE, 2016), p. 971315.

Weiss, K.

F. Zemlin, K. Weiss, P. Schiske, W. Kunath, and K.-H. Herrmann, “Coma-free alignment of high resolution electron microscopes with the aid of optical diffractograms,” Ultramicroscopy 3, 49–60 (1978).
[Crossref]

Wilson, T.

D. Hamilton, C. Sheppard, and T. Wilson, “Improved imaging of phase gradients in scanning optical microscopy,” J. Microsc. 135, 275–286 (1984).
[Crossref]

Wojdyla, A.

A. Shanker, A. Wojdyla, G. Gunjala, J. Dong, M. Benk, A. Neureuther, K. Goldberg, and L. Waller, “Off-axis aberration estimation in an EUV microscope using natural speckle,” in Imaging Systems and Applications, (Optical Society of America, 2016), pp. ITh1F–2.

Wright, S.

S. Wright and J. Nocedal, Numerical Optimization(Springer Science, 1999).

Wyant, J. C.

Yang, C.

Zemlin, F.

F. Zemlin, K. Weiss, P. Schiske, W. Kunath, and K.-H. Herrmann, “Coma-free alignment of high resolution electron microscopes with the aid of optical diffractograms,” Ultramicroscopy 3, 49–60 (1978).
[Crossref]

Zheng, G.

X. Ou, G. Zheng, and C. Yang, “Embedded pupil function recovery for Fourier ptychographic microscopy,” Opt. Express 22, 4960–4972 (2014).
[Crossref] [PubMed]

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photonics 7, 739 (2013).
[Crossref]

Appl. Opt. (4)

Biomed. Opt. Express (1)

J. Microsc. (2)

D. Hamilton, C. Sheppard, and T. Wilson, “Improved imaging of phase gradients in scanning optical microscopy,” J. Microsc. 135, 275–286 (1984).
[Crossref]

B. M. Hanser, M. G. Gustafsson, D. Agard, and J. W. Sedat, “Phase-retrieved pupil functions in wide-field fluorescence microscopy,” J. Microsc. 216, 32–48 (2004).
[Crossref] [PubMed]

J. Res. Natl. Bureau Standards, Sec. D (1)

M. Siddiqui, “Statistical inference for Rayleigh distributions,” J. Res. Natl. Bureau Standards, Sec. D 68, 1007 (1964).

JOSA (1)

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” JOSA 66, 207–211 (1976).
[Crossref]

JOSA A (2)

R. G. Paxman, T. J. Schulz, and J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” JOSA A 9, 1072–1085 (1992).
[Crossref]

N. Streibl, “Three-dimensional imaging by a microscope,” JOSA A 2, 121–127 (1985).
[Crossref]

Nat. Methods (1)

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7, 141 (2010).
[Crossref]

Nat. Photonics (1)

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photonics 7, 739 (2013).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Science (1)

H. W. Babcock, “Adaptive optics revisited,” Science 249, 253–257 (1990).
[Crossref] [PubMed]

SIAM J. on Appl. Math. (1)

G. Aubert and J.-F. Aujol, “A variational approach to removing multiplicative noise,” SIAM J. on Appl. Math. 68, 925–946 (2008).
[Crossref]

Transactions Am. Math. Soc. (1)

F. H. Clarke, “Generalized gradients and applications,” Transactions Am. Math. Soc. 205, 247–262 (1975).
[Crossref]

Ultramicroscopy (3)

F. Zemlin, K. Weiss, P. Schiske, W. Kunath, and K.-H. Herrmann, “Coma-free alignment of high resolution electron microscopes with the aid of optical diffractograms,” Ultramicroscopy 3, 49–60 (1978).
[Crossref]

A. Thust, M. Overwijk, W. Coene, and M. Lentzen, “Numerical correction of lens aberrations in phase-retrieval HRTEM,” Ultramicroscopy 64, 249–264 (1996).
[Crossref]

H. Rose, “Nonstandard imaging methods in electron microscopy,” Ultramicroscopy 2, 251–267 (1976).
[Crossref]

Other (12)

G. Gunjala, A. Shanker, V. Jaedicke, N. Antipa, and L. Waller, “Optical transfer function characterization using a weak diffuser,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XXIII, (SPIE, 2016), p. 971315.

A. Shanker, A. Wojdyla, G. Gunjala, J. Dong, M. Benk, A. Neureuther, K. Goldberg, and L. Waller, “Off-axis aberration estimation in an EUV microscope using natural speckle,” in Imaging Systems and Applications, (Optical Society of America, 2016), pp. ITh1F–2.

R. K. Tyson, Principles of Adaptive Optics (CRC, 2015).

J. C. Spence, Experimental High-Resolution Electron Microscopy(Oxford University, 1988).

R. Erni, Aberration-Corrected Imaging in Transmission Electron Microscopy: An Introduction(World Scientific, 2010).
[Crossref]

J. Goodman, Introduction to Fourier Optics(McGraw-hill, 2008).

J. W. Goodman, Statistical Optics(John Wiley & Sons, 2015).

E. J. Kirkland, Advanced Computing in Electron Microscopy(Springer Science & Business Media, 2010).
[Crossref]

J.-B. Hiriart-Urruty and C. Lemaréchal, Fundamentals of Convex Analysis(Springer-Verlag, 2001).
[Crossref]

R. H. Shumway and D. S. Stoffer, Time Series Analysis and its Applications(Springer, 2011).
[Crossref]

S. Wright and J. Nocedal, Numerical Optimization(Springer Science, 1999).

K. Lange, Applied Probability(Springer Science & Business Media2010).
[Crossref]

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Figures (10)

Fig. 1
Fig. 1 Overview of our wavefront error function (WEF) recovery procedure. (a) A weak diffuser is placed in the image plane of the system to be characterized. (b) A calibration speckle image is captured and its spectrum is used to estimate the statistical parameters of the diffuser. (c) Speckle images are measured for N distinct illumination angles and their spectra are shown. (d) Measurements are processed and input to (e) a nonlinear least-squares problem in order to recover the Zernike coefficients and (f) reconstruct the WEF.
Fig. 2
Fig. 2 (a) Measured phase, φ( x ), of a section of a 10° holographic diffuser (prior to index matching). (b) Spectrum of full intensity measurement (DC-suppressed and zeroed outside normalized cutoff frequency) displays rings due to defocus. (c) Radial average of spectrum.
Fig. 3
Fig. 3 WEF reconstruction from simulated data. (a) Example intensity measurement; (b) WEF with 18 random Zernike coefficients (Noll indices 4-21); (c) spectra of 4 intensity images with illumination angles described by azimuthal angle θ and deflection angle φ; (d) processed spectra, Mj( u ), as in Eq. (13); (e) spectra produced by the forward model using the true WEF coefficients; (f) spectra produced by the forward model using the recovered WEF coefficients; (g) recovered WEF coefficients demonstrate accurate recovery (3.11% relative error) of 5th order WEF using four input images and reasonable initialization values.
Fig. 4
Fig. 4 (a) Fraction of distinct WEFs successfully recovered as aberrations increase. Success is defined as convergence to within 5% relative error of ground truth for at least one of 100 initializations generated by 0-mean normally-distributed noise. Random initializations for small aberration magnitudes are likely to converge, with the probability decreasing as the aberration magnitude increases. (b) Fraction of distinct WEFs successfully recovered for the case of large aberrations (all 6π radians RMS) with increasing initialization distance. 100 initializations are generated by adding 0-mean normally-distributed noise of increasing magnitude to the true solution. This demonstrates that initializations with as much as 3π radians RMS error converge with high probability, suggesting that modestly accurate initializations will enable fitting to large aberrations.
Fig. 5
Fig. 5 Experimental setup. (a) Melles-Griot HeNe laser (632 nm), (b) 4f system with mirror tilt used to control illumination angle at the object (diffuser) plane; (c) 10° holographic diffuser (Edmund Optics, #54-493) index-matched with oil; (d) Deformable Mirror (Iris AO PTT111, 7 mm pupil diameter, gold-coated) in Fourier plane to introduce controlled aberrations; (e) ThorLabs DCC1240C CMOS camera (1280 × 1024 pixels, 5.3 µm pixels).
Fig. 6
Fig. 6 Experimental fit to aberrations Z5, Z6, Z7 and Z9 using (a) 200 centered initializations (actual coefficients plus zero-mean white noise) and (b) 10,000 zero-mean random initializations. Bars and points of a single color represent recovered and actual coefficients, respectively, for a particular aberration magnitude. Recovered values of Z5 and Z6 demonstrate the most accurate recovery, while Z7 and Z9 (polynomials of higher degree) incur larger relative error, especially when initializations have zero mean. The relative error increases when fringe contrast in the measurements is low.
Fig. 7
Fig. 7 Experimental fit to (a) oblique astigmatism (Z5), (b) vertical astigmatism (Z6), (c) vertical coma (Z7), and (d) vertical trefoil (Z9). (Left column) Input polynomial coefficient written to DM plotted vs. recovered coefficient in π radians RMS. (Right) Examples of measured and reconstructed intensity spatial spectra for 3 illumination angles and aberrations of varying magnitude (π radians RMS). Rows (a) and (b) show excellent quantitative agreement between expected and recovered coefficients. Rows (c) and (d) are somewhat worse, likely due to lower contrast in the interference fringes at the edges of the pupil.
Fig. 8
Fig. 8 (a) Simulated calibration procedure, in which the window function of the scatterer spectrum is estimated from a low-order (defocus only) model. (b) The Rayleigh noise parameter is estimated from the residuals upon dividing the measured spectrum by the damped defocus kernel in regions where division is stable.
Fig. 9
Fig. 9 (a) Trace[(ATA−1)] evaluated for sample plans consisting of uniformly-spaced angles at radii varying from 0.05–0.5 of the pupil. (b) Uniformly-spaced sample plans with optimized radii (blue) and gradient-based optimization from these initializations.
Fig. 10
Fig. 10 (a) Initializing at the true coefficients to evaluate how coefficient error scales with the number of images. (b) Initializing randomly to evaluate how the probability of convergence scales with the number of images.

Tables (1)

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Table 1 2-Reconstruction Error a for Second- and Third-Order Aberrations

Equations (26)

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I ( x ) = | E i ( x ) | 2 = | P ( x ) * E o ( x ) | 2 ,
I ^ ( u ) = E ^ i ( u ) * E i * ^ ( u ) = P ^ ( u ) E o ^ ( u ) * P * ^ ( u ) E o * ^ ( u ) ,
P ^ ( u ) = exp [ i W ( u ) ] Circ ( u ) ,
E o ( x ) = exp [ i 2 π ( u 0 ) ] exp [ i φ ( x ) ] .
E o ^ ( u ) = δ ( u u 0 ) + i φ ^ ( u u 0 ) .
I ^ ( u ) = δ ( u ) [ | P ^ ( u 0 ) | 2 + γ ] + i φ ^ ( u ) [ P * ^ ( u 0 ) P ^ ( u + u 0 ) P ^ ( u 0 ) P * ^ ( u + u 0 ) ] .
I ^ ( u ) = i φ ^ ( u ) [ P * ^ ( u 0 ) P ^ ( u + u 0 ) P ^ ( u 0 ) P * ^ ( u + u 0 ) ] ,
S { f ( u + u 0 ) } = f ( u + u 0 ) + f ( u + u 0 ) 2 , A { f ( u + u 0 ) } = f ( u + u 0 ) f ( u + u 0 ) 2 .
I ^ ( u ) = 2 i exp [ i A { u + u 0 } ] φ ^ ( u ) sin ( S { W ( u + u 0 ) } ) .
| I ^ ( u ) | = 2 | φ ^ ( u ) | | sin ( S { W ( u + u 0 ) } ) | , u U .
| φ ^ ( u ) | = | φ d ^ ( u ) | η ( u ) ,
M j ( u ) | I , j ^ ( u ) | 2 | φ d ^ ( u ) | = η ( u ) | sin ( S { W ( u + u j ) } ) | , u U j .
m j | I , j ^ | 2 | φ d ^ | = η | sin ( Ψ diag ( s ) j c ) | = η | sin ( A j c ) | .
η | sin ( A j c ) | ( E [ η ] + v ) | sin ( A j c ) | , v ~ N ( 0 , 4 π 2 σ 2 ) .
η | sin ( A j c ) | ( E [ η ] | sin ( A j c ) | + v | sin ( A j c ) | ,
c = arg min c j = 1 K 1 [ U j ] ( m j σ ( π 2 ) 2 | sin ( A j c ) | ) 2 ,
I ^ ( u ) = [ δ ( u u 0 ) P ^ ( u ) * δ ( u u 0 ) P ^ * ( u ) ] + j [ δ ( u u 0 ) P ^ * ( u ) * φ ^ ( u u 0 ) P ^ ( u ) ] j [ δ ( u u 0 ) P ^ ( u ) * φ ^ * ( u u 0 ) P ^ * ( u ) ] + [ φ ^ ( u u 0 ) P ^ ( u ) * φ ^ * ( u u 0 ) P ^ * ( u ) ] .
I ^ ( u ) = δ ( u ) P ^ ( u 0 ) P ^ * ( u 0 ) + γ ] + i P ^ * ( u 0 ) [ φ ^ ( u ) P ^ ( u + u 0 ) ] i P ^ ( u 0 ) [ φ ^ * ( u ) P ^ * ( u + u 0 ) ] .
I ^ ( u ) = δ ( u ) [ | P ^ ( u 0 ) | 2 + γ ] + i φ ^ ( u ) [ P ^ * ( u 0 ) P ^ ( u + u 0 ) P ^ ( u 0 ) P ^ * ( u + u 0 ) ] .
| I ^ ( u ) | 2 | φ d ^ ( u ) | | sin ( S { W ( u ) } ) | = η ( u ) ~ Rayleigh ( σ )
σ ^ = ( j = 1 K η ( u j ) 2 2 K ) 1 / 2 4 K K ! ( K 1 ) ! K ( 2 K ) ! π ( j = 1 K η ( u j ) 2 2 K ) 1 / 2 e 1 K K 1 ( K 1 K ) K ,
e j = M j ( u ) σ ( π 2 ) 1 / 2 | sin ( A j c ) | .
c f = 2 σ ( π 2 ) 1 / 2 j = 1 K e j diag ( cos ( A j c ) sgn ( sin ( A j c ) ) ) A j .
c 2 f = 2 σ ( π 2 ) 1 / 2 j = 1 K A j diag ( cos 2 ( A j c ) + e j sin ( 2 A j c ) sgn ( sin ( A j c ) ) ) A j .
A = ( A 1 A 2 A K ) .
E [ δ c 2 2 ] = Tr [ A ( A ) ] ,

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