Abstract

Coherent Fourier scatterometry is an optical metrology technique that utilizes the measured intensity of the scattered optical field to reconstruct certain parameters of test structures written on a wafer with nano-scale accuracy. The intensity of the scattered field is recorded with a camera and this information is used to retrieve the grating parameters. To improve sensitivity in the parameter reconstruction, the phase of the scattered field can also be acquired. Interferometry can be used for this purpose, but with the cost of cumbersomeness. In this paper, we show that iterative phase retrieval methods can be applied to retrieve the scattered complex fields from only intensity measurement data. We show that the accuracy of the retrieved complex fields using phase retrieval is comparable to that measured directly using interferometry.

© 2017 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Phase retrieval applied to coherent Fourier scatterometry using the extended ptychographic iterative engine

P. Dwivedi, J. E. H. Cardoso Sakamoto, and S. F. Pereira
OSA Continuum 2(5) 1590-1599 (2019)

Determination of the full scattering matrix using coherent Fourier scatterometry

Nitish Kumar, Luca Cisotto, Sarathi Roy, Gopika K. P. Ramanandan, Silvania F. Pereira, and H. Paul Urbach
Appl. Opt. 55(16) 4408-4413 (2016)

Phase retrieval with complexity guidance

Mansi Butola, Sunaina Rajora, and Kedar Khare
J. Opt. Soc. Am. A 36(2) 202-211 (2019)

References

  • View by:
  • |
  • |
  • |

  1. O. El Gawhary, N. Kumar, S. F. Pereira, W. M. J. Coene, and H. P. Urbach, “Performance analysis of coherent optical scatterometry,” Appl. Phys. B 105(4), 775–781 (2011).
    [Crossref]
  2. N. Kumar, O. el Gawhary, S. Roy, S. F. Pereira, and H. P. Urbach, “Phase retrieval between overlapping orders in coherent Fourier scatterometry using scanning,” J. Eur. Opt. Soc. Rapid Publ. 8, 13048 (2013).
    [Crossref]
  3. S. Roy, N. Kumar, S. F. Pereira, and H. P. Urbach, “Interferometric coherent Fourier scatterometry: a method for obtaining high sensitivity in the optical inverse-grating problem,” J. Opt. 15(7), 075707 (2013).
    [Crossref]
  4. C. Falldorf, M. Agour, C. Von Kopylow, and R. B. Bergmann, “Phase retrieval for optical inspection of technical components,” J. Opt. 14(16), 065701 (2012).
    [Crossref]
  5. M. Agour, P. Almoro, and C. Falldorf, “Investigation of smooth wave fronts using SLM-based phase retireval and a phase diffuser,” J. Eur. Opt. Soc. Rapid Publ. 7, 12046 (2012).
    [Crossref]
  6. M. Agour, C. Falldorf, and R. B. Bergmann, “Investigation of composite materials using SLM-based phase retireval,” Opt. Lett. 38(13), 2203–2205 (2013).
    [Crossref] [PubMed]
  7. A. Alpers, G. T. Herman, H. F. Poulsen, and S. Schmidt, “Phase retrieval for superposed signals from multiple objects,” J. Opt. Soc. Am. A 27(9), 1927–1937 (2010).
    [Crossref]
  8. J. Gulden, O. M. Yefanov, A. P. Mancuso, R. Dronyak, A. Singer, V. Bernátová, A. Burkhardt, O. Polozhentsev, A. Soldatov, M. Sprung, and I. A. Vartanyants, “Three dimensional structure of a single colloidal crystal strain studied by coherent x-ray diffraction,” Opt. Express 20(4), 4039–4049 (2012).
    [Crossref] [PubMed]
  9. J. C. Petruccelli, L. Tian, and G. Barbastathis, “The transport of intensity equation for optical path length recovery using partially coherent illumination,” Opt. Express 21(12), 14430–14441 (2013).
    [Crossref] [PubMed]
  10. H. H. Bauschke and J. M. Borwein, “On the convergence of von Neumann’s alternating projection algorithm for two sets,” Set-Valued Anal. 1(2), 185–212 (1993).
    [Crossref]
  11. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21(15), 2758–2769 (1982).
    [Crossref] [PubMed]
  12. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35(2), 237–246 (1972).
  13. S. Marchesini, “Invited article: A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78(1), 011301 (2007).
    [Crossref] [PubMed]
  14. D. E. Adams, L. S. Martin, M. D. Seaberg, D. F. Gardner, H. C. Kapteyn, and M. M. Murnane, “A generalization for optimized phase retrieval algorithms,” Opt. Express 20(22), 24778–24790 (2012).
    [Crossref] [PubMed]
  15. V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A 20(1), 40–55 (2003).
    [Crossref]
  16. N. Kumar, L. Cisotto, S. Roy, G. K. P. Ramanandan, S. F. Pereira, and H. P. Urbach, “Determination of the full scattering matrix using coherent Fourier scatterometry,” Appl. Opt. 55(16), 4408–4413 (2016).
    [Crossref] [PubMed]
  17. D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl. 21(1), 37–50 (2004).
    [Crossref]
  18. A. V. Martin, F. Wang, N. D. Loh, T. Ekeberg, F. R. N. C. Maia, M. Hantke, G. van der Schot, C. Y. Hampton, R. G. Sierra, A. Aquila, S. Bajt, M. Barthelmess, C. Bostedt, J. D. Bozek, N. Coppola, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, M. Frank, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, P. Holl, S. Kassemeyer, N. Kimmel, M. Liang, L. Lomb, S. Marchesini, K. Nass, E. Pedersoli, C. Reich, D. Rolles, B. Rudek, A. Rudenko, J. Schulz, R. L. Shoeman, H. Soltau, D. Starodub, J. Steinbrener, F. Stellato, L. Strüder, J. Ullrich, G. Weidenspointner, T. A. White, C. B. Wunderer, A. Barty, I. Schlichting, M. J. Bogan, and H. N. Chapman, “Noise-robust coherent diffractive imaging with a single diffraction pattern,” Opt. Express 20(15), 16650–16661 (2012).
    [Crossref]
  19. J. A. Rodriguez, R. Xu, C. C. Chen, Y. Zou, and J. Miao, “Oversampling smoothness: an effective algorithm for phase retrieval of noisy diffraction intensities,” J. Appl. Crystallogr. 46(2), 312–318 (2013).
    [Crossref] [PubMed]
  20. J. R. Fienup and C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3(11), 1897–1907 (1986).
    [Crossref]
  21. A. P. Konijnenberg, W. M. J. Coene, S. F. Pereira, and H. P. Urbach, “Combining ptychographical algorithms with the Hybrid-Output (HIO) algorithm,” Ultramicoscopy 171, 43–54 (2016).
    [Crossref]

2016 (2)

N. Kumar, L. Cisotto, S. Roy, G. K. P. Ramanandan, S. F. Pereira, and H. P. Urbach, “Determination of the full scattering matrix using coherent Fourier scatterometry,” Appl. Opt. 55(16), 4408–4413 (2016).
[Crossref] [PubMed]

A. P. Konijnenberg, W. M. J. Coene, S. F. Pereira, and H. P. Urbach, “Combining ptychographical algorithms with the Hybrid-Output (HIO) algorithm,” Ultramicoscopy 171, 43–54 (2016).
[Crossref]

2013 (5)

J. A. Rodriguez, R. Xu, C. C. Chen, Y. Zou, and J. Miao, “Oversampling smoothness: an effective algorithm for phase retrieval of noisy diffraction intensities,” J. Appl. Crystallogr. 46(2), 312–318 (2013).
[Crossref] [PubMed]

J. C. Petruccelli, L. Tian, and G. Barbastathis, “The transport of intensity equation for optical path length recovery using partially coherent illumination,” Opt. Express 21(12), 14430–14441 (2013).
[Crossref] [PubMed]

N. Kumar, O. el Gawhary, S. Roy, S. F. Pereira, and H. P. Urbach, “Phase retrieval between overlapping orders in coherent Fourier scatterometry using scanning,” J. Eur. Opt. Soc. Rapid Publ. 8, 13048 (2013).
[Crossref]

S. Roy, N. Kumar, S. F. Pereira, and H. P. Urbach, “Interferometric coherent Fourier scatterometry: a method for obtaining high sensitivity in the optical inverse-grating problem,” J. Opt. 15(7), 075707 (2013).
[Crossref]

M. Agour, C. Falldorf, and R. B. Bergmann, “Investigation of composite materials using SLM-based phase retireval,” Opt. Lett. 38(13), 2203–2205 (2013).
[Crossref] [PubMed]

2012 (5)

J. Gulden, O. M. Yefanov, A. P. Mancuso, R. Dronyak, A. Singer, V. Bernátová, A. Burkhardt, O. Polozhentsev, A. Soldatov, M. Sprung, and I. A. Vartanyants, “Three dimensional structure of a single colloidal crystal strain studied by coherent x-ray diffraction,” Opt. Express 20(4), 4039–4049 (2012).
[Crossref] [PubMed]

C. Falldorf, M. Agour, C. Von Kopylow, and R. B. Bergmann, “Phase retrieval for optical inspection of technical components,” J. Opt. 14(16), 065701 (2012).
[Crossref]

M. Agour, P. Almoro, and C. Falldorf, “Investigation of smooth wave fronts using SLM-based phase retireval and a phase diffuser,” J. Eur. Opt. Soc. Rapid Publ. 7, 12046 (2012).
[Crossref]

D. E. Adams, L. S. Martin, M. D. Seaberg, D. F. Gardner, H. C. Kapteyn, and M. M. Murnane, “A generalization for optimized phase retrieval algorithms,” Opt. Express 20(22), 24778–24790 (2012).
[Crossref] [PubMed]

A. V. Martin, F. Wang, N. D. Loh, T. Ekeberg, F. R. N. C. Maia, M. Hantke, G. van der Schot, C. Y. Hampton, R. G. Sierra, A. Aquila, S. Bajt, M. Barthelmess, C. Bostedt, J. D. Bozek, N. Coppola, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, M. Frank, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, P. Holl, S. Kassemeyer, N. Kimmel, M. Liang, L. Lomb, S. Marchesini, K. Nass, E. Pedersoli, C. Reich, D. Rolles, B. Rudek, A. Rudenko, J. Schulz, R. L. Shoeman, H. Soltau, D. Starodub, J. Steinbrener, F. Stellato, L. Strüder, J. Ullrich, G. Weidenspointner, T. A. White, C. B. Wunderer, A. Barty, I. Schlichting, M. J. Bogan, and H. N. Chapman, “Noise-robust coherent diffractive imaging with a single diffraction pattern,” Opt. Express 20(15), 16650–16661 (2012).
[Crossref]

2011 (1)

O. El Gawhary, N. Kumar, S. F. Pereira, W. M. J. Coene, and H. P. Urbach, “Performance analysis of coherent optical scatterometry,” Appl. Phys. B 105(4), 775–781 (2011).
[Crossref]

2010 (1)

2007 (1)

S. Marchesini, “Invited article: A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78(1), 011301 (2007).
[Crossref] [PubMed]

2004 (1)

D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl. 21(1), 37–50 (2004).
[Crossref]

2003 (1)

1993 (1)

H. H. Bauschke and J. M. Borwein, “On the convergence of von Neumann’s alternating projection algorithm for two sets,” Set-Valued Anal. 1(2), 185–212 (1993).
[Crossref]

1986 (1)

1982 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35(2), 237–246 (1972).

Adams, D. E.

Agour, M.

M. Agour, C. Falldorf, and R. B. Bergmann, “Investigation of composite materials using SLM-based phase retireval,” Opt. Lett. 38(13), 2203–2205 (2013).
[Crossref] [PubMed]

M. Agour, P. Almoro, and C. Falldorf, “Investigation of smooth wave fronts using SLM-based phase retireval and a phase diffuser,” J. Eur. Opt. Soc. Rapid Publ. 7, 12046 (2012).
[Crossref]

C. Falldorf, M. Agour, C. Von Kopylow, and R. B. Bergmann, “Phase retrieval for optical inspection of technical components,” J. Opt. 14(16), 065701 (2012).
[Crossref]

Almoro, P.

M. Agour, P. Almoro, and C. Falldorf, “Investigation of smooth wave fronts using SLM-based phase retireval and a phase diffuser,” J. Eur. Opt. Soc. Rapid Publ. 7, 12046 (2012).
[Crossref]

Alpers, A.

Aquila, A.

Bajt, S.

Barbastathis, G.

Barthelmess, M.

Barty, A.

Bauschke, H. H.

H. H. Bauschke and J. M. Borwein, “On the convergence of von Neumann’s alternating projection algorithm for two sets,” Set-Valued Anal. 1(2), 185–212 (1993).
[Crossref]

Bergmann, R. B.

M. Agour, C. Falldorf, and R. B. Bergmann, “Investigation of composite materials using SLM-based phase retireval,” Opt. Lett. 38(13), 2203–2205 (2013).
[Crossref] [PubMed]

C. Falldorf, M. Agour, C. Von Kopylow, and R. B. Bergmann, “Phase retrieval for optical inspection of technical components,” J. Opt. 14(16), 065701 (2012).
[Crossref]

Bernátová, V.

Bogan, M. J.

Borwein, J. M.

H. H. Bauschke and J. M. Borwein, “On the convergence of von Neumann’s alternating projection algorithm for two sets,” Set-Valued Anal. 1(2), 185–212 (1993).
[Crossref]

Bostedt, C.

Bozek, J. D.

Burkhardt, A.

Chapman, H. N.

Chen, C. C.

J. A. Rodriguez, R. Xu, C. C. Chen, Y. Zou, and J. Miao, “Oversampling smoothness: an effective algorithm for phase retrieval of noisy diffraction intensities,” J. Appl. Crystallogr. 46(2), 312–318 (2013).
[Crossref] [PubMed]

Cisotto, L.

Coene, W. M. J.

A. P. Konijnenberg, W. M. J. Coene, S. F. Pereira, and H. P. Urbach, “Combining ptychographical algorithms with the Hybrid-Output (HIO) algorithm,” Ultramicoscopy 171, 43–54 (2016).
[Crossref]

O. El Gawhary, N. Kumar, S. F. Pereira, W. M. J. Coene, and H. P. Urbach, “Performance analysis of coherent optical scatterometry,” Appl. Phys. B 105(4), 775–781 (2011).
[Crossref]

Coppola, N.

Dronyak, R.

Ekeberg, T.

el Gawhary, O.

N. Kumar, O. el Gawhary, S. Roy, S. F. Pereira, and H. P. Urbach, “Phase retrieval between overlapping orders in coherent Fourier scatterometry using scanning,” J. Eur. Opt. Soc. Rapid Publ. 8, 13048 (2013).
[Crossref]

O. El Gawhary, N. Kumar, S. F. Pereira, W. M. J. Coene, and H. P. Urbach, “Performance analysis of coherent optical scatterometry,” Appl. Phys. B 105(4), 775–781 (2011).
[Crossref]

Elser, V.

Epp, S. W.

Erk, B.

Falldorf, C.

M. Agour, C. Falldorf, and R. B. Bergmann, “Investigation of composite materials using SLM-based phase retireval,” Opt. Lett. 38(13), 2203–2205 (2013).
[Crossref] [PubMed]

M. Agour, P. Almoro, and C. Falldorf, “Investigation of smooth wave fronts using SLM-based phase retireval and a phase diffuser,” J. Eur. Opt. Soc. Rapid Publ. 7, 12046 (2012).
[Crossref]

C. Falldorf, M. Agour, C. Von Kopylow, and R. B. Bergmann, “Phase retrieval for optical inspection of technical components,” J. Opt. 14(16), 065701 (2012).
[Crossref]

Fienup, J. R.

Fleckenstein, H.

Foucar, L.

Frank, M.

Gardner, D. F.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35(2), 237–246 (1972).

Graafsma, H.

Gulden, J.

Gumprecht, L.

Hampton, C. Y.

Hantke, M.

Hartmann, A.

Hartmann, R.

Hauser, G.

Herman, G. T.

Hirsemann, H.

Holl, P.

Kapteyn, H. C.

Kassemeyer, S.

Kimmel, N.

Konijnenberg, A. P.

A. P. Konijnenberg, W. M. J. Coene, S. F. Pereira, and H. P. Urbach, “Combining ptychographical algorithms with the Hybrid-Output (HIO) algorithm,” Ultramicoscopy 171, 43–54 (2016).
[Crossref]

Kumar, N.

N. Kumar, L. Cisotto, S. Roy, G. K. P. Ramanandan, S. F. Pereira, and H. P. Urbach, “Determination of the full scattering matrix using coherent Fourier scatterometry,” Appl. Opt. 55(16), 4408–4413 (2016).
[Crossref] [PubMed]

N. Kumar, O. el Gawhary, S. Roy, S. F. Pereira, and H. P. Urbach, “Phase retrieval between overlapping orders in coherent Fourier scatterometry using scanning,” J. Eur. Opt. Soc. Rapid Publ. 8, 13048 (2013).
[Crossref]

S. Roy, N. Kumar, S. F. Pereira, and H. P. Urbach, “Interferometric coherent Fourier scatterometry: a method for obtaining high sensitivity in the optical inverse-grating problem,” J. Opt. 15(7), 075707 (2013).
[Crossref]

O. El Gawhary, N. Kumar, S. F. Pereira, W. M. J. Coene, and H. P. Urbach, “Performance analysis of coherent optical scatterometry,” Appl. Phys. B 105(4), 775–781 (2011).
[Crossref]

Liang, M.

Loh, N. D.

Lomb, L.

Luke, D. R.

D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl. 21(1), 37–50 (2004).
[Crossref]

Maia, F. R. N. C.

Mancuso, A. P.

Marchesini, S.

Martin, A. V.

Martin, L. S.

Miao, J.

J. A. Rodriguez, R. Xu, C. C. Chen, Y. Zou, and J. Miao, “Oversampling smoothness: an effective algorithm for phase retrieval of noisy diffraction intensities,” J. Appl. Crystallogr. 46(2), 312–318 (2013).
[Crossref] [PubMed]

Murnane, M. M.

Nass, K.

Pedersoli, E.

Pereira, S. F.

N. Kumar, L. Cisotto, S. Roy, G. K. P. Ramanandan, S. F. Pereira, and H. P. Urbach, “Determination of the full scattering matrix using coherent Fourier scatterometry,” Appl. Opt. 55(16), 4408–4413 (2016).
[Crossref] [PubMed]

A. P. Konijnenberg, W. M. J. Coene, S. F. Pereira, and H. P. Urbach, “Combining ptychographical algorithms with the Hybrid-Output (HIO) algorithm,” Ultramicoscopy 171, 43–54 (2016).
[Crossref]

N. Kumar, O. el Gawhary, S. Roy, S. F. Pereira, and H. P. Urbach, “Phase retrieval between overlapping orders in coherent Fourier scatterometry using scanning,” J. Eur. Opt. Soc. Rapid Publ. 8, 13048 (2013).
[Crossref]

S. Roy, N. Kumar, S. F. Pereira, and H. P. Urbach, “Interferometric coherent Fourier scatterometry: a method for obtaining high sensitivity in the optical inverse-grating problem,” J. Opt. 15(7), 075707 (2013).
[Crossref]

O. El Gawhary, N. Kumar, S. F. Pereira, W. M. J. Coene, and H. P. Urbach, “Performance analysis of coherent optical scatterometry,” Appl. Phys. B 105(4), 775–781 (2011).
[Crossref]

Petruccelli, J. C.

Polozhentsev, O.

Poulsen, H. F.

Ramanandan, G. K. P.

Reich, C.

Rodriguez, J. A.

J. A. Rodriguez, R. Xu, C. C. Chen, Y. Zou, and J. Miao, “Oversampling smoothness: an effective algorithm for phase retrieval of noisy diffraction intensities,” J. Appl. Crystallogr. 46(2), 312–318 (2013).
[Crossref] [PubMed]

Rolles, D.

Roy, S.

N. Kumar, L. Cisotto, S. Roy, G. K. P. Ramanandan, S. F. Pereira, and H. P. Urbach, “Determination of the full scattering matrix using coherent Fourier scatterometry,” Appl. Opt. 55(16), 4408–4413 (2016).
[Crossref] [PubMed]

N. Kumar, O. el Gawhary, S. Roy, S. F. Pereira, and H. P. Urbach, “Phase retrieval between overlapping orders in coherent Fourier scatterometry using scanning,” J. Eur. Opt. Soc. Rapid Publ. 8, 13048 (2013).
[Crossref]

S. Roy, N. Kumar, S. F. Pereira, and H. P. Urbach, “Interferometric coherent Fourier scatterometry: a method for obtaining high sensitivity in the optical inverse-grating problem,” J. Opt. 15(7), 075707 (2013).
[Crossref]

Rudek, B.

Rudenko, A.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35(2), 237–246 (1972).

Schlichting, I.

Schmidt, S.

Schulz, J.

Seaberg, M. D.

Shoeman, R. L.

Sierra, R. G.

Singer, A.

Soldatov, A.

Soltau, H.

Sprung, M.

Starodub, D.

Steinbrener, J.

Stellato, F.

Strüder, L.

Tian, L.

Ullrich, J.

Urbach, H. P.

N. Kumar, L. Cisotto, S. Roy, G. K. P. Ramanandan, S. F. Pereira, and H. P. Urbach, “Determination of the full scattering matrix using coherent Fourier scatterometry,” Appl. Opt. 55(16), 4408–4413 (2016).
[Crossref] [PubMed]

A. P. Konijnenberg, W. M. J. Coene, S. F. Pereira, and H. P. Urbach, “Combining ptychographical algorithms with the Hybrid-Output (HIO) algorithm,” Ultramicoscopy 171, 43–54 (2016).
[Crossref]

S. Roy, N. Kumar, S. F. Pereira, and H. P. Urbach, “Interferometric coherent Fourier scatterometry: a method for obtaining high sensitivity in the optical inverse-grating problem,” J. Opt. 15(7), 075707 (2013).
[Crossref]

N. Kumar, O. el Gawhary, S. Roy, S. F. Pereira, and H. P. Urbach, “Phase retrieval between overlapping orders in coherent Fourier scatterometry using scanning,” J. Eur. Opt. Soc. Rapid Publ. 8, 13048 (2013).
[Crossref]

O. El Gawhary, N. Kumar, S. F. Pereira, W. M. J. Coene, and H. P. Urbach, “Performance analysis of coherent optical scatterometry,” Appl. Phys. B 105(4), 775–781 (2011).
[Crossref]

van der Schot, G.

Vartanyants, I. A.

Von Kopylow, C.

C. Falldorf, M. Agour, C. Von Kopylow, and R. B. Bergmann, “Phase retrieval for optical inspection of technical components,” J. Opt. 14(16), 065701 (2012).
[Crossref]

Wackerman, C. C.

Wang, F.

Weidenspointner, G.

White, T. A.

Wunderer, C. B.

Xu, R.

J. A. Rodriguez, R. Xu, C. C. Chen, Y. Zou, and J. Miao, “Oversampling smoothness: an effective algorithm for phase retrieval of noisy diffraction intensities,” J. Appl. Crystallogr. 46(2), 312–318 (2013).
[Crossref] [PubMed]

Yefanov, O. M.

Zou, Y.

J. A. Rodriguez, R. Xu, C. C. Chen, Y. Zou, and J. Miao, “Oversampling smoothness: an effective algorithm for phase retrieval of noisy diffraction intensities,” J. Appl. Crystallogr. 46(2), 312–318 (2013).
[Crossref] [PubMed]

Appl. Opt. (2)

Appl. Phys. B (1)

O. El Gawhary, N. Kumar, S. F. Pereira, W. M. J. Coene, and H. P. Urbach, “Performance analysis of coherent optical scatterometry,” Appl. Phys. B 105(4), 775–781 (2011).
[Crossref]

Inverse Probl. (1)

D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl. 21(1), 37–50 (2004).
[Crossref]

J. Appl. Crystallogr. (1)

J. A. Rodriguez, R. Xu, C. C. Chen, Y. Zou, and J. Miao, “Oversampling smoothness: an effective algorithm for phase retrieval of noisy diffraction intensities,” J. Appl. Crystallogr. 46(2), 312–318 (2013).
[Crossref] [PubMed]

J. Eur. Opt. Soc. Rapid Publ. (2)

N. Kumar, O. el Gawhary, S. Roy, S. F. Pereira, and H. P. Urbach, “Phase retrieval between overlapping orders in coherent Fourier scatterometry using scanning,” J. Eur. Opt. Soc. Rapid Publ. 8, 13048 (2013).
[Crossref]

M. Agour, P. Almoro, and C. Falldorf, “Investigation of smooth wave fronts using SLM-based phase retireval and a phase diffuser,” J. Eur. Opt. Soc. Rapid Publ. 7, 12046 (2012).
[Crossref]

J. Opt. (2)

S. Roy, N. Kumar, S. F. Pereira, and H. P. Urbach, “Interferometric coherent Fourier scatterometry: a method for obtaining high sensitivity in the optical inverse-grating problem,” J. Opt. 15(7), 075707 (2013).
[Crossref]

C. Falldorf, M. Agour, C. Von Kopylow, and R. B. Bergmann, “Phase retrieval for optical inspection of technical components,” J. Opt. 14(16), 065701 (2012).
[Crossref]

J. Opt. Soc. Am. A (3)

Opt. Express (4)

J. Gulden, O. M. Yefanov, A. P. Mancuso, R. Dronyak, A. Singer, V. Bernátová, A. Burkhardt, O. Polozhentsev, A. Soldatov, M. Sprung, and I. A. Vartanyants, “Three dimensional structure of a single colloidal crystal strain studied by coherent x-ray diffraction,” Opt. Express 20(4), 4039–4049 (2012).
[Crossref] [PubMed]

D. E. Adams, L. S. Martin, M. D. Seaberg, D. F. Gardner, H. C. Kapteyn, and M. M. Murnane, “A generalization for optimized phase retrieval algorithms,” Opt. Express 20(22), 24778–24790 (2012).
[Crossref] [PubMed]

A. V. Martin, F. Wang, N. D. Loh, T. Ekeberg, F. R. N. C. Maia, M. Hantke, G. van der Schot, C. Y. Hampton, R. G. Sierra, A. Aquila, S. Bajt, M. Barthelmess, C. Bostedt, J. D. Bozek, N. Coppola, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, M. Frank, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, P. Holl, S. Kassemeyer, N. Kimmel, M. Liang, L. Lomb, S. Marchesini, K. Nass, E. Pedersoli, C. Reich, D. Rolles, B. Rudek, A. Rudenko, J. Schulz, R. L. Shoeman, H. Soltau, D. Starodub, J. Steinbrener, F. Stellato, L. Strüder, J. Ullrich, G. Weidenspointner, T. A. White, C. B. Wunderer, A. Barty, I. Schlichting, M. J. Bogan, and H. N. Chapman, “Noise-robust coherent diffractive imaging with a single diffraction pattern,” Opt. Express 20(15), 16650–16661 (2012).
[Crossref]

J. C. Petruccelli, L. Tian, and G. Barbastathis, “The transport of intensity equation for optical path length recovery using partially coherent illumination,” Opt. Express 21(12), 14430–14441 (2013).
[Crossref] [PubMed]

Opt. Lett. (1)

Optik (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35(2), 237–246 (1972).

Rev. Sci. Instrum. (1)

S. Marchesini, “Invited article: A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78(1), 011301 (2007).
[Crossref] [PubMed]

Set-Valued Anal. (1)

H. H. Bauschke and J. M. Borwein, “On the convergence of von Neumann’s alternating projection algorithm for two sets,” Set-Valued Anal. 1(2), 185–212 (1993).
[Crossref]

Ultramicoscopy (1)

A. P. Konijnenberg, W. M. J. Coene, S. F. Pereira, and H. P. Urbach, “Combining ptychographical algorithms with the Hybrid-Output (HIO) algorithm,” Ultramicoscopy 171, 43–54 (2016).
[Crossref]

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 (8)

Fig. 1
Fig. 1 Schematic overview of the setup used both in simulation and experiment.
Fig. 2
Fig. 2 Schematic overview of the structure of the grating
Fig. 3
Fig. 3 The original and retrieved modulus of image in the TMTM case with different algorithms after 600 iterations. The original image was obtained from simulation (described in [16]). Poisson noise and white Gaussian noise with signal to noise ratio of 100 was applied to the initial intensity pattern. β used for HIO and RAAR were 0.7 and 0.96, respectively.
Fig. 4
Fig. 4 The original and retrieved phase of the same images as in Fig. 3.
Fig. 5
Fig. 5 The change of error of the retrieved image with iterations. This figure shows the error decline within the retrieval procedure before finally obtaining the images as shown in Figs. 3 and 4.
Fig. 6
Fig. 6 (a) intensity profile of the TMTM field exposed for 76000μs. The red part in the center denotes strong over-exposure; (b) cross sections of images taken at different exposure times. The cross section is taken following the red line as shown in (a); (c) the relative intensity ratio of the cross section denoted as the green line in (a) to the sample region as in (b) for a given exposure time. (d) the reconstructed image.
Fig. 7
Fig. 7 Retrieved modulus and phase of (a) plane wave, (b) TMTM and (c) TMTE images from the combination of HIO and RAAR algorithms. For retrieval of plane wave, HIO with β of 0.98 was performed for 250 iterations, followed by RAAR, with β of 0.96, for the next 250 iterations. The same parameters were applied for the retrieval of image in the TMTM case. To reconstruct the image in the TMTE case however, HIO was applied twice with different β: firstly β = 0.9 was used for the first 150 iterations, followed by β = 0.96 for the next 100 iterations, and lastly RAAR was applied with β = 0.96 for the remaining 250 iterations.
Fig. 8
Fig. 8 Comparison of (a) modulus and (b) phase of images obtained from phase retrieval, interferometric measurement and simulation (the latter two cases are reproduced from [16]).

Equations (9)

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

f k + 1 ( x , y ) = { f k ( x , y ) ( x , y ) γ f k ( x , y ) β f k ( x , y ) ( x , y ) γ ,
f k + 1 ( x , y ) = { f k ( x , y ) ( x , y ) γ β f k ( x , y ) + ( 1 2 β ) f k ( x , y ) ( x , y ) γ .
f k = f k _ error + σ noise ,
f k + 1 ( x , y ) = { f k ( x , y ) ( x , y ) γ f k ( x , y ) β f k ( x , y ) ( x , y ) γ & | f k ( x , y ) | > 3 σ noise 0 ( x , y ) γ & | f k ( x , y ) | 3 σ noise .
f k ( x , y ) = { f k ( x , y ) ( x , y ) γ f k ( x , y ) β f k ( x , y ) ( x , y ) γ ,
f k + 1 ( x , y ) = { f k + 1 ( x , y ) ( x , y ) γ 1 { F k ( u , v ) W ( u , v ) } ( x , y ) γ ,
W ( u , v ) = e 0.5 u 2 + v 2 ( w α o ) 2 ,
Error = Σ j = 0 m Σ i = 0 n | f i j c f i j | 2 Σ j = 0 m Σ i = 0 n | f i j | 2 ,
c * = Σ j = 0 m Σ i = 0 n f i j * f i j Σ j = 0 m Σ i = 0 n | f i j | 2 .

Metrics