Abstract

Vectorial ptychography has been recently introduced to reconstruct the Jones matrix of an anisotropic object by means of series of ptychographic measurements performed using a set of polarized illumination probes in conjugation with various analyzers. So far, the probes were assumed to be completely known (amplitude, wavefront, state of polarization), which is rarely the case in practice. Here we address the issue of the joint estimating of the set of polarized illumination probes together with the Jones matrix of an anisotropic object in vectorial ptychography. We propose an algorithm based on a conjugate gradient strategy. Experimental results are reported, showing an improvement on the object estimate, in addition to a precise reconstruction of the probes.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: A novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
    [Crossref] [PubMed]
  2. H. M. L. Faulkner and J. M. Rodenburg, “Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy,” Ultramicroscopy 103, 153–164 (2005).
    [Crossref] [PubMed]
  3. P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
    [Crossref] [PubMed]
  4. A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
    [Crossref] [PubMed]
  5. J. Marrison, L. Räty, P. Marriott, and P. O’Toole, “Ptychography – a label free, high-contrast imaging technique for live cells using quantitative phase information,” Sci. Rep. 3, 2369 (2013).
    [Crossref]
  6. L. Shemilt, E. Verbanis, J. Schwenke, A. K. Estandarte, G. Xiong, R. Harder, N. Parmar, M. Yusuf, F. Zhang, and I. K. Robinson, “Karyotyping human chromosomes by optical and x-ray ptychography methods,” Biophys. J. 108, 706–713 (2015).
    [Crossref] [PubMed]
  7. S. McDermott, P. Li, G. Williams, and A. Maiden, “Characterizing a spatial light modulator using ptychography,” Opt. Lett. 42, 371–374 (2017).
    [Crossref] [PubMed]
  8. P. Ferrand, M. Allain, and V. Chamard, “Ptychography in anisotropic media,” Opt. Lett. 40, 5144–5147 (2015).
    [Crossref] [PubMed]
  9. P. Ferrand, A. Baroni, M. Allain, and V. Chamard, “Quantitative imaging of anisotropic material properties with vectorial ptychography,” Opt. Lett. 43, 763–766 (2018).
    [Crossref] [PubMed]
  10. R. C. Jones, “A new calculus formalism for the treatment of optical systems. I. Description and discussion of the method,” J. Opt. Soc. Am. 31, 488–493 (1941).
    [Crossref]
  11. L. Sorber, M. Barel, and L. Lathauwer, “Unconstrained optimization of real functions in complex variables,” SIAM J. Optimiz. 22, 879–898 (2012).
    [Crossref]
  12. P. Godard, M. Allain, V. Chamard, and J. Rodenburg, “Noise models for low counting rate coherent diffraction imaging,” Opt. Express 20, 25914–25934 (2012).
    [Crossref] [PubMed]
  13. J. Nocedal and S. Wright, Numerical Optimization (Springer-Verlag, 2006), 2nd ed.
  14. J. W. Goodman, Introduction to Fourier Optics (W. H. Freeman & Co Ltd, 2017), 4th ed.
  15. M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express 16, 7264–7278 (2008).
    [Crossref] [PubMed]
  16. A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339, 1232009 (2013).
    [Crossref] [PubMed]
  17. E. Bricchi, J. D. Mills, P. G. Kazansky, B. G. Klappauf, and J. J. Baumberg, “Birefringent Fresnel zone plates in silica fabricated by femtosecond laser machining,” Opt. Lett. 27, 2200–2202 (2002).
    [Crossref]
  18. S. McDermott, P. Li, G. Williams, and A. Maiden, “Characterizing a spatial light modulator using ptychography,” Opt. Lett. 42, 371–374 (2017).
    [Crossref] [PubMed]
  19. D. Kogan, S. Sivankutty, V. Tsvirkun, G. Bouwmans, E. R. Andresen, H. Rigneault, and D. Oron, “Phase retrieval in multicore fiber bundles,” Opt. Lett. 42, 647–650 (2017).
    [Crossref] [PubMed]

2018 (1)

P. Ferrand, A. Baroni, M. Allain, and V. Chamard, “Quantitative imaging of anisotropic material properties with vectorial ptychography,” Opt. Lett. 43, 763–766 (2018).
[Crossref] [PubMed]

2017 (3)

S. McDermott, P. Li, G. Williams, and A. Maiden, “Characterizing a spatial light modulator using ptychography,” Opt. Lett. 42, 371–374 (2017).
[Crossref] [PubMed]

D. Kogan, S. Sivankutty, V. Tsvirkun, G. Bouwmans, E. R. Andresen, H. Rigneault, and D. Oron, “Phase retrieval in multicore fiber bundles,” Opt. Lett. 42, 647–650 (2017).
[Crossref] [PubMed]

S. McDermott, P. Li, G. Williams, and A. Maiden, “Characterizing a spatial light modulator using ptychography,” Opt. Lett. 42, 371–374 (2017).
[Crossref] [PubMed]

2015 (2)

P. Ferrand, M. Allain, and V. Chamard, “Ptychography in anisotropic media,” Opt. Lett. 40, 5144–5147 (2015).
[Crossref] [PubMed]

L. Shemilt, E. Verbanis, J. Schwenke, A. K. Estandarte, G. Xiong, R. Harder, N. Parmar, M. Yusuf, F. Zhang, and I. K. Robinson, “Karyotyping human chromosomes by optical and x-ray ptychography methods,” Biophys. J. 108, 706–713 (2015).
[Crossref] [PubMed]

2013 (2)

J. Marrison, L. Räty, P. Marriott, and P. O’Toole, “Ptychography – a label free, high-contrast imaging technique for live cells using quantitative phase information,” Sci. Rep. 3, 2369 (2013).
[Crossref]

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339, 1232009 (2013).
[Crossref] [PubMed]

2012 (2)

L. Sorber, M. Barel, and L. Lathauwer, “Unconstrained optimization of real functions in complex variables,” SIAM J. Optimiz. 22, 879–898 (2012).
[Crossref]

P. Godard, M. Allain, V. Chamard, and J. Rodenburg, “Noise models for low counting rate coherent diffraction imaging,” Opt. Express 20, 25914–25934 (2012).
[Crossref] [PubMed]

2009 (1)

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref] [PubMed]

2008 (2)

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref] [PubMed]

M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express 16, 7264–7278 (2008).
[Crossref] [PubMed]

2005 (1)

H. M. L. Faulkner and J. M. Rodenburg, “Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy,” Ultramicroscopy 103, 153–164 (2005).
[Crossref] [PubMed]

2004 (1)

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: A novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[Crossref] [PubMed]

2002 (1)

E. Bricchi, J. D. Mills, P. G. Kazansky, B. G. Klappauf, and J. J. Baumberg, “Birefringent Fresnel zone plates in silica fabricated by femtosecond laser machining,” Opt. Lett. 27, 2200–2202 (2002).
[Crossref]

1941 (1)

R. C. Jones, “A new calculus formalism for the treatment of optical systems. I. Description and discussion of the method,” J. Opt. Soc. Am. 31, 488–493 (1941).
[Crossref]

Allain, M.

P. Ferrand, A. Baroni, M. Allain, and V. Chamard, “Quantitative imaging of anisotropic material properties with vectorial ptychography,” Opt. Lett. 43, 763–766 (2018).
[Crossref] [PubMed]

P. Ferrand, M. Allain, and V. Chamard, “Ptychography in anisotropic media,” Opt. Lett. 40, 5144–5147 (2015).
[Crossref] [PubMed]

P. Godard, M. Allain, V. Chamard, and J. Rodenburg, “Noise models for low counting rate coherent diffraction imaging,” Opt. Express 20, 25914–25934 (2012).
[Crossref] [PubMed]

Andresen, E. R.

D. Kogan, S. Sivankutty, V. Tsvirkun, G. Bouwmans, E. R. Andresen, H. Rigneault, and D. Oron, “Phase retrieval in multicore fiber bundles,” Opt. Lett. 42, 647–650 (2017).
[Crossref] [PubMed]

Barel, M.

L. Sorber, M. Barel, and L. Lathauwer, “Unconstrained optimization of real functions in complex variables,” SIAM J. Optimiz. 22, 879–898 (2012).
[Crossref]

Baroni, A.

P. Ferrand, A. Baroni, M. Allain, and V. Chamard, “Quantitative imaging of anisotropic material properties with vectorial ptychography,” Opt. Lett. 43, 763–766 (2018).
[Crossref] [PubMed]

Baumberg, J. J.

E. Bricchi, J. D. Mills, P. G. Kazansky, B. G. Klappauf, and J. J. Baumberg, “Birefringent Fresnel zone plates in silica fabricated by femtosecond laser machining,” Opt. Lett. 27, 2200–2202 (2002).
[Crossref]

Boltasseva, A.

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339, 1232009 (2013).
[Crossref] [PubMed]

Bouwmans, G.

D. Kogan, S. Sivankutty, V. Tsvirkun, G. Bouwmans, E. R. Andresen, H. Rigneault, and D. Oron, “Phase retrieval in multicore fiber bundles,” Opt. Lett. 42, 647–650 (2017).
[Crossref] [PubMed]

Bricchi, E.

E. Bricchi, J. D. Mills, P. G. Kazansky, B. G. Klappauf, and J. J. Baumberg, “Birefringent Fresnel zone plates in silica fabricated by femtosecond laser machining,” Opt. Lett. 27, 2200–2202 (2002).
[Crossref]

Bunk, O.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref] [PubMed]

Chamard, V.

P. Ferrand, A. Baroni, M. Allain, and V. Chamard, “Quantitative imaging of anisotropic material properties with vectorial ptychography,” Opt. Lett. 43, 763–766 (2018).
[Crossref] [PubMed]

P. Ferrand, M. Allain, and V. Chamard, “Ptychography in anisotropic media,” Opt. Lett. 40, 5144–5147 (2015).
[Crossref] [PubMed]

P. Godard, M. Allain, V. Chamard, and J. Rodenburg, “Noise models for low counting rate coherent diffraction imaging,” Opt. Express 20, 25914–25934 (2012).
[Crossref] [PubMed]

David, C.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref] [PubMed]

Dierolf, M.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref] [PubMed]

Estandarte, A. K.

L. Shemilt, E. Verbanis, J. Schwenke, A. K. Estandarte, G. Xiong, R. Harder, N. Parmar, M. Yusuf, F. Zhang, and I. K. Robinson, “Karyotyping human chromosomes by optical and x-ray ptychography methods,” Biophys. J. 108, 706–713 (2015).
[Crossref] [PubMed]

Faulkner, H. M. L.

H. M. L. Faulkner and J. M. Rodenburg, “Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy,” Ultramicroscopy 103, 153–164 (2005).
[Crossref] [PubMed]

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: A novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[Crossref] [PubMed]

Ferrand, P.

P. Ferrand, A. Baroni, M. Allain, and V. Chamard, “Quantitative imaging of anisotropic material properties with vectorial ptychography,” Opt. Lett. 43, 763–766 (2018).
[Crossref] [PubMed]

P. Ferrand, M. Allain, and V. Chamard, “Ptychography in anisotropic media,” Opt. Lett. 40, 5144–5147 (2015).
[Crossref] [PubMed]

Fienup, J. R.

M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express 16, 7264–7278 (2008).
[Crossref] [PubMed]

Godard, P.

P. Godard, M. Allain, V. Chamard, and J. Rodenburg, “Noise models for low counting rate coherent diffraction imaging,” Opt. Express 20, 25914–25934 (2012).
[Crossref] [PubMed]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (W. H. Freeman & Co Ltd, 2017), 4th ed.

Guizar-Sicairos, M.

M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express 16, 7264–7278 (2008).
[Crossref] [PubMed]

Harder, R.

L. Shemilt, E. Verbanis, J. Schwenke, A. K. Estandarte, G. Xiong, R. Harder, N. Parmar, M. Yusuf, F. Zhang, and I. K. Robinson, “Karyotyping human chromosomes by optical and x-ray ptychography methods,” Biophys. J. 108, 706–713 (2015).
[Crossref] [PubMed]

Jones, R. C.

R. C. Jones, “A new calculus formalism for the treatment of optical systems. I. Description and discussion of the method,” J. Opt. Soc. Am. 31, 488–493 (1941).
[Crossref]

Kazansky, P. G.

E. Bricchi, J. D. Mills, P. G. Kazansky, B. G. Klappauf, and J. J. Baumberg, “Birefringent Fresnel zone plates in silica fabricated by femtosecond laser machining,” Opt. Lett. 27, 2200–2202 (2002).
[Crossref]

Kildishev, A. V.

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339, 1232009 (2013).
[Crossref] [PubMed]

Klappauf, B. G.

E. Bricchi, J. D. Mills, P. G. Kazansky, B. G. Klappauf, and J. J. Baumberg, “Birefringent Fresnel zone plates in silica fabricated by femtosecond laser machining,” Opt. Lett. 27, 2200–2202 (2002).
[Crossref]

Kogan, D.

D. Kogan, S. Sivankutty, V. Tsvirkun, G. Bouwmans, E. R. Andresen, H. Rigneault, and D. Oron, “Phase retrieval in multicore fiber bundles,” Opt. Lett. 42, 647–650 (2017).
[Crossref] [PubMed]

Lathauwer, L.

L. Sorber, M. Barel, and L. Lathauwer, “Unconstrained optimization of real functions in complex variables,” SIAM J. Optimiz. 22, 879–898 (2012).
[Crossref]

Li, P.

S. McDermott, P. Li, G. Williams, and A. Maiden, “Characterizing a spatial light modulator using ptychography,” Opt. Lett. 42, 371–374 (2017).
[Crossref] [PubMed]

S. McDermott, P. Li, G. Williams, and A. Maiden, “Characterizing a spatial light modulator using ptychography,” Opt. Lett. 42, 371–374 (2017).
[Crossref] [PubMed]

Maiden, A.

S. McDermott, P. Li, G. Williams, and A. Maiden, “Characterizing a spatial light modulator using ptychography,” Opt. Lett. 42, 371–374 (2017).
[Crossref] [PubMed]

S. McDermott, P. Li, G. Williams, and A. Maiden, “Characterizing a spatial light modulator using ptychography,” Opt. Lett. 42, 371–374 (2017).
[Crossref] [PubMed]

Maiden, A. M.

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref] [PubMed]

Marriott, P.

J. Marrison, L. Räty, P. Marriott, and P. O’Toole, “Ptychography – a label free, high-contrast imaging technique for live cells using quantitative phase information,” Sci. Rep. 3, 2369 (2013).
[Crossref]

Marrison, J.

J. Marrison, L. Räty, P. Marriott, and P. O’Toole, “Ptychography – a label free, high-contrast imaging technique for live cells using quantitative phase information,” Sci. Rep. 3, 2369 (2013).
[Crossref]

McDermott, S.

S. McDermott, P. Li, G. Williams, and A. Maiden, “Characterizing a spatial light modulator using ptychography,” Opt. Lett. 42, 371–374 (2017).
[Crossref] [PubMed]

S. McDermott, P. Li, G. Williams, and A. Maiden, “Characterizing a spatial light modulator using ptychography,” Opt. Lett. 42, 371–374 (2017).
[Crossref] [PubMed]

Menzel, A.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref] [PubMed]

Mills, J. D.

E. Bricchi, J. D. Mills, P. G. Kazansky, B. G. Klappauf, and J. J. Baumberg, “Birefringent Fresnel zone plates in silica fabricated by femtosecond laser machining,” Opt. Lett. 27, 2200–2202 (2002).
[Crossref]

Nocedal, J.

J. Nocedal and S. Wright, Numerical Optimization (Springer-Verlag, 2006), 2nd ed.

O’Toole, P.

J. Marrison, L. Räty, P. Marriott, and P. O’Toole, “Ptychography – a label free, high-contrast imaging technique for live cells using quantitative phase information,” Sci. Rep. 3, 2369 (2013).
[Crossref]

Oron, D.

D. Kogan, S. Sivankutty, V. Tsvirkun, G. Bouwmans, E. R. Andresen, H. Rigneault, and D. Oron, “Phase retrieval in multicore fiber bundles,” Opt. Lett. 42, 647–650 (2017).
[Crossref] [PubMed]

Parmar, N.

L. Shemilt, E. Verbanis, J. Schwenke, A. K. Estandarte, G. Xiong, R. Harder, N. Parmar, M. Yusuf, F. Zhang, and I. K. Robinson, “Karyotyping human chromosomes by optical and x-ray ptychography methods,” Biophys. J. 108, 706–713 (2015).
[Crossref] [PubMed]

Pfeiffer, F.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref] [PubMed]

Räty, L.

J. Marrison, L. Räty, P. Marriott, and P. O’Toole, “Ptychography – a label free, high-contrast imaging technique for live cells using quantitative phase information,” Sci. Rep. 3, 2369 (2013).
[Crossref]

Rigneault, H.

D. Kogan, S. Sivankutty, V. Tsvirkun, G. Bouwmans, E. R. Andresen, H. Rigneault, and D. Oron, “Phase retrieval in multicore fiber bundles,” Opt. Lett. 42, 647–650 (2017).
[Crossref] [PubMed]

Robinson, I. K.

L. Shemilt, E. Verbanis, J. Schwenke, A. K. Estandarte, G. Xiong, R. Harder, N. Parmar, M. Yusuf, F. Zhang, and I. K. Robinson, “Karyotyping human chromosomes by optical and x-ray ptychography methods,” Biophys. J. 108, 706–713 (2015).
[Crossref] [PubMed]

Rodenburg, J.

P. Godard, M. Allain, V. Chamard, and J. Rodenburg, “Noise models for low counting rate coherent diffraction imaging,” Opt. Express 20, 25914–25934 (2012).
[Crossref] [PubMed]

Rodenburg, J. M.

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref] [PubMed]

H. M. L. Faulkner and J. M. Rodenburg, “Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy,” Ultramicroscopy 103, 153–164 (2005).
[Crossref] [PubMed]

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: A novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[Crossref] [PubMed]

Schwenke, J.

L. Shemilt, E. Verbanis, J. Schwenke, A. K. Estandarte, G. Xiong, R. Harder, N. Parmar, M. Yusuf, F. Zhang, and I. K. Robinson, “Karyotyping human chromosomes by optical and x-ray ptychography methods,” Biophys. J. 108, 706–713 (2015).
[Crossref] [PubMed]

Shalaev, V. M.

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339, 1232009 (2013).
[Crossref] [PubMed]

Shemilt, L.

L. Shemilt, E. Verbanis, J. Schwenke, A. K. Estandarte, G. Xiong, R. Harder, N. Parmar, M. Yusuf, F. Zhang, and I. K. Robinson, “Karyotyping human chromosomes by optical and x-ray ptychography methods,” Biophys. J. 108, 706–713 (2015).
[Crossref] [PubMed]

Sivankutty, S.

D. Kogan, S. Sivankutty, V. Tsvirkun, G. Bouwmans, E. R. Andresen, H. Rigneault, and D. Oron, “Phase retrieval in multicore fiber bundles,” Opt. Lett. 42, 647–650 (2017).
[Crossref] [PubMed]

Sorber, L.

L. Sorber, M. Barel, and L. Lathauwer, “Unconstrained optimization of real functions in complex variables,” SIAM J. Optimiz. 22, 879–898 (2012).
[Crossref]

Thibault, P.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref] [PubMed]

Tsvirkun, V.

D. Kogan, S. Sivankutty, V. Tsvirkun, G. Bouwmans, E. R. Andresen, H. Rigneault, and D. Oron, “Phase retrieval in multicore fiber bundles,” Opt. Lett. 42, 647–650 (2017).
[Crossref] [PubMed]

Verbanis, E.

L. Shemilt, E. Verbanis, J. Schwenke, A. K. Estandarte, G. Xiong, R. Harder, N. Parmar, M. Yusuf, F. Zhang, and I. K. Robinson, “Karyotyping human chromosomes by optical and x-ray ptychography methods,” Biophys. J. 108, 706–713 (2015).
[Crossref] [PubMed]

Williams, G.

S. McDermott, P. Li, G. Williams, and A. Maiden, “Characterizing a spatial light modulator using ptychography,” Opt. Lett. 42, 371–374 (2017).
[Crossref] [PubMed]

S. McDermott, P. Li, G. Williams, and A. Maiden, “Characterizing a spatial light modulator using ptychography,” Opt. Lett. 42, 371–374 (2017).
[Crossref] [PubMed]

Wright, S.

J. Nocedal and S. Wright, Numerical Optimization (Springer-Verlag, 2006), 2nd ed.

Xiong, G.

L. Shemilt, E. Verbanis, J. Schwenke, A. K. Estandarte, G. Xiong, R. Harder, N. Parmar, M. Yusuf, F. Zhang, and I. K. Robinson, “Karyotyping human chromosomes by optical and x-ray ptychography methods,” Biophys. J. 108, 706–713 (2015).
[Crossref] [PubMed]

Yusuf, M.

L. Shemilt, E. Verbanis, J. Schwenke, A. K. Estandarte, G. Xiong, R. Harder, N. Parmar, M. Yusuf, F. Zhang, and I. K. Robinson, “Karyotyping human chromosomes by optical and x-ray ptychography methods,” Biophys. J. 108, 706–713 (2015).
[Crossref] [PubMed]

Zhang, F.

L. Shemilt, E. Verbanis, J. Schwenke, A. K. Estandarte, G. Xiong, R. Harder, N. Parmar, M. Yusuf, F. Zhang, and I. K. Robinson, “Karyotyping human chromosomes by optical and x-ray ptychography methods,” Biophys. J. 108, 706–713 (2015).
[Crossref] [PubMed]

Biophys. J. (1)

L. Shemilt, E. Verbanis, J. Schwenke, A. K. Estandarte, G. Xiong, R. Harder, N. Parmar, M. Yusuf, F. Zhang, and I. K. Robinson, “Karyotyping human chromosomes by optical and x-ray ptychography methods,” Biophys. J. 108, 706–713 (2015).
[Crossref] [PubMed]

J. Opt. Soc. Am. (1)

R. C. Jones, “A new calculus formalism for the treatment of optical systems. I. Description and discussion of the method,” J. Opt. Soc. Am. 31, 488–493 (1941).
[Crossref]

Opt. Express (2)

P. Godard, M. Allain, V. Chamard, and J. Rodenburg, “Noise models for low counting rate coherent diffraction imaging,” Opt. Express 20, 25914–25934 (2012).
[Crossref] [PubMed]

M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express 16, 7264–7278 (2008).
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E. Bricchi, J. D. Mills, P. G. Kazansky, B. G. Klappauf, and J. J. Baumberg, “Birefringent Fresnel zone plates in silica fabricated by femtosecond laser machining,” Opt. Lett. 27, 2200–2202 (2002).
[Crossref]

S. McDermott, P. Li, G. Williams, and A. Maiden, “Characterizing a spatial light modulator using ptychography,” Opt. Lett. 42, 371–374 (2017).
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S. McDermott, P. Li, G. Williams, and A. Maiden, “Characterizing a spatial light modulator using ptychography,” Opt. Lett. 42, 371–374 (2017).
[Crossref] [PubMed]

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P. Ferrand, A. Baroni, M. Allain, and V. Chamard, “Quantitative imaging of anisotropic material properties with vectorial ptychography,” Opt. Lett. 43, 763–766 (2018).
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H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: A novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
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[Crossref]

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A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
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Figures (6)

Fig. 1
Fig. 1 Initial guess of the three probes for k = 1 to 3, from left to right. From top to bottom, pk,x, pk,y, and SoP of pk. Inset shows the complex value color coding, with phase encoded as hue and modulus as brightness. Scale bar is 25 μm.
Fig. 2
Fig. 2 Reconstructed probes pk. Same formatting as Fig. 1. Scale bar is 25 μm.
Fig. 3
Fig. 3 Reconstructed Jones maps of the birefringent test target. Inset shows the complex value color coding, with phase encoded as hue and modulus as brightness. Scale bar is 25 μm.
Fig. 4
Fig. 4 Retrieved optical properties of the birefringent test target: Normalized power transmittance T, relative optical path length (OPL) variations, in nm, fast axis orientation θ, in degrees, and retardance R, in nm. Scale bar is 25 μm.
Fig. 5
Fig. 5 Retrieved optical properties of the test target, obtained with methods (b) and (c), to be compared with Fig. 4. Scale bar is 25 μm.
Fig. 6
Fig. 6 Evolution of the criterion L ( s ( n ) ) for different reconstruction strategies. For each method, labelled (a) to (c), colored patterns remind which algorithm is used and which variable is updated, either the object only or both object and probes.

Equations (21)

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ψ j k ( r ) = O ( r )   p k ( r r j ) ,
[ ψ j k , x ( r ) ψ j k , y ( r ) ] = [ ρ x x ( r ) ρ y x ( r ) ρ x y ( r ) ρ y y ( r ) ] [ p k , x ( r r j ) p k , y ( r r j ) ] ,
ψ j k ( r ) = P k ( r r j )   ρ ( r ) ,
P k ( r ) = [ p k , x ( r ) 0 p k , y ( r ) 0 0 p k , y ( r ) 0 p k , x ( r ) ]
ρ ( r ) = [ ρ x x ( r ) ρ y y ( r ) ρ y x ( r ) ρ x y ( r ) ] .
S = { O ( r ) ; p k ( r ) , k }
L ( S ) = j , k , l ( I ( calc ) j k l ( S ) I ( meas ) j k l ) 2 ,
I j k l (calc) ( q ) = | ξ j k l ( q ) | 2 + ϵ k l ( q ) ,
ξ j k l = h l T F ψ j k ,
s = [ ρ p ]
s ( n + 1 ) = s ( n ) α ( n ) d ( n ) D ( n ) ,
d ( n ) = ( n ) + β ( n ) d ( n 1 ) ,
( n ) = L ( s ( n ) ) s ,
( n ) = [ L ( s ( n ) ) ρ L ( s ( n ) ) p ] .
L ( s ( n ) ) ρ = 2 j , k P k ( n ) F l h l ξ j k l [ I ( meas ) j k l I ( calc ) j k l 1 ] ,
L ( s ( n ) ) p k = 2 j O j F l h l ξ j k l [ I ( meas ) j k l I ( calc ) j k l 1 ] ,
D O ( n ) = [ a x x j , k p k , ; x ( n ) ( r r j ) a x x j , k p k , y ( n ) ( r r j ) a y y j , k p k , x ( n ) ( r r j ) a y y j , k p k , y ( n ) ( r r j ) ] ,
D p ( n ) = diag [ j O j [ a x x a y x a y x a y y ] O j ] ,
ψ j k ( r ) = O ( r ) M M 1 p k ( r r j ) ,
O ( r ) = O ( r ) M ,
p k ( r ) = M 1 p k ( r )

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