Abstract

We present an image formation model for deterministic phase retrieval in propagation-based wavefront sensing, unifying analysis for classical wavefront sensors such as Shack-Hartmann (slopes tracking) and curvature sensors (based on Transport-of-Intensity Equation). We show how this model generalizes commonly seen formulas, including Transport-of-Intensity Equation, from small distances and beyond. Using this model, we analyze theoretically achievable lateral wavefront resolution in propagation-based deterministic wavefront sensing. Finally, via a prototype masked wavefront sensor, we show simultaneous bright field and phase imaging numerically recovered in real-time from a single-shot measurement.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (5)

C. Wang, Q. Fu, X. Dun, and W. Heidrich, “Quantitative phase and intensity microscopy using snapshot white light wavefront sensing,” Sci. Rep. 9(1), 13795 (2019).
[Crossref]

G. N. McKay, F. Mahmood, and N. J. Durr, “Large dynamic range autorefraction with a low-cost diffuser wavefront sensor,” Biomed. Opt. Express 10(4), 1718–1735 (2019).
[Crossref]

Y. Wu, M. K. Sharma, and A. Veeraraghavan, “Wish: wavefront imaging sensor with high resolution,” Light: Sci. Appl. 8(1), 44 (2019).
[Crossref]

D. M. Paganin and K. S. Morgan, “X-ray fokker–planck equation for paraxial imaging,” Sci. Rep. 9(1), 17537–18 (2019).
[Crossref]

K. S. Morgan and D. M. Paganin, “Applying the fokker–planck equation to grating-based x-ray phase and dark-field imaging,” Sci. Rep. 9(1), 17465–14 (2019).
[Crossref]

2018 (2)

F. Soldevila, V. Durán, P. Clemente, J. Lancis, and E. Tajahuerce, “Phase imaging by spatial wavefront sampling,” Optica 5(2), 164–174 (2018).
[Crossref]

C. Wang, Q. Fu, X. Dun, and W. Heidrich, “Megapixel adaptive optics: towards correcting large-scale distortions in computational cameras,” ACM Trans. Graph. 37(4), 1–12 (2018).
[Crossref]

2017 (5)

M. Bawart, S. Bernet, and M. Ritsch-Marte, “Programmable freeform optical elements,” Opt. Express 25(5), 4898–4906 (2017).
[Crossref]

C. Wang, X. Dun, Q. Fu, and W. Heidrich, “Ultra-high resolution coded wavefront sensor,” Opt. Express 25(12), 13736–13746 (2017).
[Crossref]

P. Berto, H. Rigneault, and M. Guillon, “Wavefront sensing with a thin diffuser,” Opt. Lett. 42(24), 5117–5120 (2017).
[Crossref]

Z. F. Phillips, M. Chen, and L. Waller, “Single-shot quantitative phase microscopy with color-multiplexed differential phase contrast (cdpc),” PLoS One 12(2), e0171228 (2017).
[Crossref]

C. Zuo, J. Sun, J. Li, J. Zhang, A. Asundi, and Q. Chen, “High-resolution transport-of-intensity quantitative phase microscopy with annular illumination,” Sci. Rep. 7(1), 7654 (2017).
[Crossref]

2016 (2)

J. Zhong, L. Tian, P. Varma, and L. Waller, “Nonlinear optimization algorithm for partially coherent phase retrieval and source recovery,” IEEE Trans. Comput. Imaging 2(3), 310–322 (2016).
[Crossref]

R. Horisaki, R. Egami, and J. Tanida, “Single-shot phase imaging with randomized light (spiral),” Opt. Express 24(4), 3765–3773 (2016).
[Crossref]

2015 (3)

2014 (3)

I. Zanette, T. Zhou, A. Burvall, U. Lundström, D. H. Larsson, M. Zdora, P. Thibault, F. Pfeiffer, and H. M. Hertz, “Speckle-based x-ray phase-contrast and dark-field imaging with a laboratory source,” Phys. Rev. Lett. 112(25), 253903 (2014).
[Crossref]

P. Gao, G. Pedrini, C. Zuo, and W. Osten, “Phase retrieval using spatially modulated illumination,” Opt. Lett. 39(12), 3615–3618 (2014).
[Crossref]

Y. Schwartzburg, R. Testuz, A. Tagliasacchi, and M. Pauly, “High-contrast computational caustic design,” ACM Trans. Graph. 33(4), 1–11 (2014).
[Crossref]

2013 (5)

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, and P. Thibault, “Near-field ptychography: phase retrieval for inline holography using a structured illumination,” Sci. Rep. 3(1), 1927 (2013).
[Crossref]

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution fourier ptychographic microscopy,” Nat. Photonics 7(9), 739–745 (2013).
[Crossref]

C. Zuo, Q. Chen, W. Qu, and A. Asundi, “High-speed transport-of-intensity phase microscopy with an electrically tunable lens,” Opt. Express 21(20), 24060–24075 (2013).
[Crossref]

K. S. Morgan, P. Modregger, S. C. Irvine, S. Rutishauser, V. A. Guzenko, M. Stampanoni, and C. David, “A sensitive x-ray phase contrast technique for rapid imaging using a single phase grid analyzer,” Opt. Lett. 38(22), 4605–4608 (2013).
[Crossref]

S. Bérujon, H. Wang, I. Pape, and K. Sawhney, “X-ray phase microscopy using the speckle tracking technique,” Appl. Phys. Lett. 102(15), 154105 (2013).
[Crossref]

2012 (2)

K. S. Morgan, D. M. Paganin, and K. K. Siu, “X-ray phase imaging with a paper analyzer,” Appl. Phys. Lett. 100(12), 124102 (2012).
[Crossref]

S. Bérujon, E. Ziegler, R. Cerbino, and L. Peverini, “Two-dimensional x-ray beam phase sensing,” Phys. Rev. Lett. 108(15), 158102 (2012).
[Crossref]

2011 (2)

2010 (4)

2009 (3)

2007 (1)

2005 (1)

2001 (1)

H. Richard and M. Raffel, “Principle and applications of the background oriented schlieren (bos) method,” Meas. Sci. Technol. 12(9), 1576–1585 (2001).
[Crossref]

2000 (1)

1997 (1)

T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133(1-6), 339–346 (1997).
[Crossref]

1996 (1)

R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43(2), 289–293 (1996).
[Crossref]

1995 (1)

1993 (1)

1990 (1)

1988 (1)

1983 (1)

M. R. Teague, “Deterministic phase retrieval: a green’s function solution,” J. Opt. Soc. Am. A 73(11), 1434–1441 (1983).
[Crossref]

1981 (1)

B. K. Horn and B. G. Schunck, “Determining optical flow,” Artif. Intelligence 17(1-3), 185–203 (1981).
[Crossref]

1977 (1)

J. P. Guigay, “Fourier-transform analysis of fresnel diffraction patterns and in-line holograms,” Optik 49, 121–125 (1977).

1971 (1)

R. V. Shack and B. Platt, “Production and use of a lenticular hartmann screen,” J. Opt. Soc. Am. A 61, 656–660 (1971).

Asundi, A.

C. Zuo, J. Sun, J. Li, J. Zhang, A. Asundi, and Q. Chen, “High-resolution transport-of-intensity quantitative phase microscopy with annular illumination,” Sci. Rep. 7(1), 7654 (2017).
[Crossref]

C. Zuo, Q. Chen, W. Qu, and A. Asundi, “High-speed transport-of-intensity phase microscopy with an electrically tunable lens,” Opt. Express 21(20), 24060–24075 (2013).
[Crossref]

Barbastathis, G.

Bawart, M.

Bennett, E. E.

Bernet, S.

Berto, P.

Berujon, S.

Bérujon, S.

S. Bérujon and E. Ziegler, “Near-field speckle-scanning-based x-ray imaging,” Phys. Rev. A 92(1), 013837 (2015).
[Crossref]

S. Bérujon, H. Wang, I. Pape, and K. Sawhney, “X-ray phase microscopy using the speckle tracking technique,” Appl. Phys. Lett. 102(15), 154105 (2013).
[Crossref]

S. Bérujon, E. Ziegler, R. Cerbino, and L. Peverini, “Two-dimensional x-ray beam phase sensing,” Phys. Rev. Lett. 108(15), 158102 (2012).
[Crossref]

Boistel, R.

Bon, P.

Brox, T.

T. Brox, A. Bruhn, N. Papenberg, and J. Weickert, “High accuracy optical flow estimation based on a theory for warping,” in European Conference on Computer Vision (Springer, 2004), pp. 25–36.

Bruhn, A.

T. Brox, A. Bruhn, N. Papenberg, and J. Weickert, “High accuracy optical flow estimation based on a theory for warping,” in European Conference on Computer Vision (Springer, 2004), pp. 25–36.

Burvall, A.

I. Zanette, T. Zhou, A. Burvall, U. Lundström, D. H. Larsson, M. Zdora, P. Thibault, F. Pfeiffer, and H. M. Hertz, “Speckle-based x-ray phase-contrast and dark-field imaging with a laboratory source,” Phys. Rev. Lett. 112(25), 253903 (2014).
[Crossref]

Cerbino, R.

S. Bérujon, E. Ziegler, R. Cerbino, and L. Peverini, “Two-dimensional x-ray beam phase sensing,” Phys. Rev. Lett. 108(15), 158102 (2012).
[Crossref]

Chanteloup, J.-C.

Chen, M.

Z. F. Phillips, M. Chen, and L. Waller, “Single-shot quantitative phase microscopy with color-multiplexed differential phase contrast (cdpc),” PLoS One 12(2), e0171228 (2017).
[Crossref]

Chen, Q.

C. Zuo, J. Sun, J. Li, J. Zhang, A. Asundi, and Q. Chen, “High-resolution transport-of-intensity quantitative phase microscopy with annular illumination,” Sci. Rep. 7(1), 7654 (2017).
[Crossref]

C. Zuo, Q. Chen, W. Qu, and A. Asundi, “High-speed transport-of-intensity phase microscopy with an electrically tunable lens,” Opt. Express 21(20), 24060–24075 (2013).
[Crossref]

Clemente, P.

Cloetens, P.

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, and P. Thibault, “Near-field ptychography: phase retrieval for inline holography using a structured illumination,” Sci. Rep. 3(1), 1927 (2013).
[Crossref]

J. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, “Mixed transfer function and transport of intensity approach for phase retrieval in the fresnel region,” Opt. Lett. 32(12), 1617–1619 (2007).
[Crossref]

Damberg, G.

David, C.

Dierolf, M.

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, and P. Thibault, “Near-field ptychography: phase retrieval for inline holography using a structured illumination,” Sci. Rep. 3(1), 1927 (2013).
[Crossref]

Dun, X.

C. Wang, Q. Fu, X. Dun, and W. Heidrich, “Quantitative phase and intensity microscopy using snapshot white light wavefront sensing,” Sci. Rep. 9(1), 13795 (2019).
[Crossref]

C. Wang, Q. Fu, X. Dun, and W. Heidrich, “Megapixel adaptive optics: towards correcting large-scale distortions in computational cameras,” ACM Trans. Graph. 37(4), 1–12 (2018).
[Crossref]

C. Wang, X. Dun, Q. Fu, and W. Heidrich, “Ultra-high resolution coded wavefront sensor,” Opt. Express 25(12), 13736–13746 (2017).
[Crossref]

C. Wang, Q. Fu, X. Dun, and W. Heidrich, “A model for classical wavefront sensors and snapshot incoherent wavefront sensing,” in Computational Optical Sensing and Imaging (Optical Society of America, 2019), pp. CM1A–4.

Durán, V.

Durr, N. J.

Egami, R.

Enders, B.

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, and P. Thibault, “Near-field ptychography: phase retrieval for inline holography using a structured illumination,” Sci. Rep. 3(1), 1927 (2013).
[Crossref]

Fu, Q.

C. Wang, Q. Fu, X. Dun, and W. Heidrich, “Quantitative phase and intensity microscopy using snapshot white light wavefront sensing,” Sci. Rep. 9(1), 13795 (2019).
[Crossref]

C. Wang, Q. Fu, X. Dun, and W. Heidrich, “Megapixel adaptive optics: towards correcting large-scale distortions in computational cameras,” ACM Trans. Graph. 37(4), 1–12 (2018).
[Crossref]

C. Wang, X. Dun, Q. Fu, and W. Heidrich, “Ultra-high resolution coded wavefront sensor,” Opt. Express 25(12), 13736–13746 (2017).
[Crossref]

C. Wang, Q. Fu, X. Dun, and W. Heidrich, “A model for classical wavefront sensors and snapshot incoherent wavefront sensing,” in Computational Optical Sensing and Imaging (Optical Society of America, 2019), pp. CM1A–4.

Gao, P.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier optics (Roberts and Company Publishers, 2005).

Guérineau, N.

Guigay, J. P.

Guillon, M.

Gureyev, T.

Gureyev, T. E.

T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133(1-6), 339–346 (1997).
[Crossref]

Guzenko, V. A.

Heidrich, W.

C. Wang, Q. Fu, X. Dun, and W. Heidrich, “Quantitative phase and intensity microscopy using snapshot white light wavefront sensing,” Sci. Rep. 9(1), 13795 (2019).
[Crossref]

C. Wang, Q. Fu, X. Dun, and W. Heidrich, “Megapixel adaptive optics: towards correcting large-scale distortions in computational cameras,” ACM Trans. Graph. 37(4), 1–12 (2018).
[Crossref]

C. Wang, X. Dun, Q. Fu, and W. Heidrich, “Ultra-high resolution coded wavefront sensor,” Opt. Express 25(12), 13736–13746 (2017).
[Crossref]

G. Damberg and W. Heidrich, “Efficient freeform lens optimization for computational caustic displays,” Opt. Express 23(8), 10224–10232 (2015).
[Crossref]

C. Wang, Q. Fu, X. Dun, and W. Heidrich, “A model for classical wavefront sensors and snapshot incoherent wavefront sensing,” in Computational Optical Sensing and Imaging (Optical Society of America, 2019), pp. CM1A–4.

Hertz, H. M.

I. Zanette, T. Zhou, A. Burvall, U. Lundström, D. H. Larsson, M. Zdora, P. Thibault, F. Pfeiffer, and H. M. Hertz, “Speckle-based x-ray phase-contrast and dark-field imaging with a laboratory source,” Phys. Rev. Lett. 112(25), 253903 (2014).
[Crossref]

Horisaki, R.

Horn, B. K.

B. K. Horn and B. G. Schunck, “Determining optical flow,” Artif. Intelligence 17(1-3), 185–203 (1981).
[Crossref]

Horstmeyer, R.

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution fourier ptychographic microscopy,” Nat. Photonics 7(9), 739–745 (2013).
[Crossref]

Irvine, S. C.

Kopace, R.

Kou, S. S.

Lancis, J.

Langer, M.

Larsson, D. H.

I. Zanette, T. Zhou, A. Burvall, U. Lundström, D. H. Larsson, M. Zdora, P. Thibault, F. Pfeiffer, and H. M. Hertz, “Speckle-based x-ray phase-contrast and dark-field imaging with a laboratory source,” Phys. Rev. Lett. 112(25), 253903 (2014).
[Crossref]

Li, J.

C. Zuo, J. Sun, J. Li, J. Zhang, A. Asundi, and Q. Chen, “High-resolution transport-of-intensity quantitative phase microscopy with annular illumination,” Sci. Rep. 7(1), 7654 (2017).
[Crossref]

Lundström, U.

I. Zanette, T. Zhou, A. Burvall, U. Lundström, D. H. Larsson, M. Zdora, P. Thibault, F. Pfeiffer, and H. M. Hertz, “Speckle-based x-ray phase-contrast and dark-field imaging with a laboratory source,” Phys. Rev. Lett. 112(25), 253903 (2014).
[Crossref]

Mahmood, F.

Matsushima, K.

Maucort, G.

McKay, G. N.

Mehta, S. B.

Modregger, P.

Monneret, S.

Morgan, K. S.

D. M. Paganin and K. S. Morgan, “X-ray fokker–planck equation for paraxial imaging,” Sci. Rep. 9(1), 17537–18 (2019).
[Crossref]

K. S. Morgan and D. M. Paganin, “Applying the fokker–planck equation to grating-based x-ray phase and dark-field imaging,” Sci. Rep. 9(1), 17465–14 (2019).
[Crossref]

K. S. Morgan, P. Modregger, S. C. Irvine, S. Rutishauser, V. A. Guzenko, M. Stampanoni, and C. David, “A sensitive x-ray phase contrast technique for rapid imaging using a single phase grid analyzer,” Opt. Lett. 38(22), 4605–4608 (2013).
[Crossref]

K. S. Morgan, D. M. Paganin, and K. K. Siu, “X-ray phase imaging with a paper analyzer,” Appl. Phys. Lett. 100(12), 124102 (2012).
[Crossref]

K. S. Morgan, D. M. Paganin, and K. K. Siu, “Quantitative single-exposure x-ray phase contrast imaging using a single attenuation grid,” Opt. Express 19(20), 19781–19789 (2011).
[Crossref]

Nugent, K.

Nugent, K. A.

T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133(1-6), 339–346 (1997).
[Crossref]

Osten, W.

Paganin, D. M.

K. S. Morgan and D. M. Paganin, “Applying the fokker–planck equation to grating-based x-ray phase and dark-field imaging,” Sci. Rep. 9(1), 17465–14 (2019).
[Crossref]

D. M. Paganin and K. S. Morgan, “X-ray fokker–planck equation for paraxial imaging,” Sci. Rep. 9(1), 17537–18 (2019).
[Crossref]

K. S. Morgan, D. M. Paganin, and K. K. Siu, “X-ray phase imaging with a paper analyzer,” Appl. Phys. Lett. 100(12), 124102 (2012).
[Crossref]

K. S. Morgan, D. M. Paganin, and K. K. Siu, “Quantitative single-exposure x-ray phase contrast imaging using a single attenuation grid,” Opt. Express 19(20), 19781–19789 (2011).
[Crossref]

Pai, V.

Pape, I.

S. Bérujon, H. Wang, I. Pape, and K. Sawhney, “X-ray phase microscopy using the speckle tracking technique,” Appl. Phys. Lett. 102(15), 154105 (2013).
[Crossref]

Papenberg, N.

T. Brox, A. Bruhn, N. Papenberg, and J. Weickert, “High accuracy optical flow estimation based on a theory for warping,” in European Conference on Computer Vision (Springer, 2004), pp. 25–36.

Pauly, M.

Y. Schwartzburg, R. Testuz, A. Tagliasacchi, and M. Pauly, “High-contrast computational caustic design,” ACM Trans. Graph. 33(4), 1–11 (2014).
[Crossref]

Pedrini, G.

Peverini, L.

S. Bérujon, E. Ziegler, R. Cerbino, and L. Peverini, “Two-dimensional x-ray beam phase sensing,” Phys. Rev. Lett. 108(15), 158102 (2012).
[Crossref]

Pfeiffer, F.

I. Zanette, T. Zhou, A. Burvall, U. Lundström, D. H. Larsson, M. Zdora, P. Thibault, F. Pfeiffer, and H. M. Hertz, “Speckle-based x-ray phase-contrast and dark-field imaging with a laboratory source,” Phys. Rev. Lett. 112(25), 253903 (2014).
[Crossref]

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, and P. Thibault, “Near-field ptychography: phase retrieval for inline holography using a structured illumination,” Sci. Rep. 3(1), 1927 (2013).
[Crossref]

Phillips, Z. F.

Z. F. Phillips, M. Chen, and L. Waller, “Single-shot quantitative phase microscopy with color-multiplexed differential phase contrast (cdpc),” PLoS One 12(2), e0171228 (2017).
[Crossref]

Platt, B.

R. V. Shack and B. Platt, “Production and use of a lenticular hartmann screen,” J. Opt. Soc. Am. A 61, 656–660 (1971).

Primot, J.

Qu, W.

Raffel, M.

H. Richard and M. Raffel, “Principle and applications of the background oriented schlieren (bos) method,” Meas. Sci. Technol. 12(9), 1576–1585 (2001).
[Crossref]

Ragazzoni, R.

R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43(2), 289–293 (1996).
[Crossref]

Richard, H.

H. Richard and M. Raffel, “Principle and applications of the background oriented schlieren (bos) method,” Meas. Sci. Technol. 12(9), 1576–1585 (2001).
[Crossref]

Rigneault, H.

Ritsch-Marte, M.

Roberts, A.

Roddier, C.

Roddier, F.

Rutishauser, S.

Sawhney, K.

S. Bérujon, H. Wang, I. Pape, and K. Sawhney, “X-ray phase microscopy using the speckle tracking technique,” Appl. Phys. Lett. 102(15), 154105 (2013).
[Crossref]

H. Wang, K. Sawhney, S. Berujon, E. Ziegler, S. Rutishauser, and C. David, “X-ray wavefront characterization using a rotating shearing interferometer technique,” Opt. Express 19(17), 16550–16559 (2011).
[Crossref]

Schunck, B. G.

B. K. Horn and B. G. Schunck, “Determining optical flow,” Artif. Intelligence 17(1-3), 185–203 (1981).
[Crossref]

Schwartzburg, Y.

Y. Schwartzburg, R. Testuz, A. Tagliasacchi, and M. Pauly, “High-contrast computational caustic design,” ACM Trans. Graph. 33(4), 1–11 (2014).
[Crossref]

Shack, R. V.

R. V. Shack and B. Platt, “Production and use of a lenticular hartmann screen,” J. Opt. Soc. Am. A 61, 656–660 (1971).

Sharma, M. K.

Y. Wu, M. K. Sharma, and A. Veeraraghavan, “Wish: wavefront imaging sensor with high resolution,” Light: Sci. Appl. 8(1), 44 (2019).
[Crossref]

Sheppard, C. J.

Shimobaba, T.

Siu, K. K.

K. S. Morgan, D. M. Paganin, and K. K. Siu, “X-ray phase imaging with a paper analyzer,” Appl. Phys. Lett. 100(12), 124102 (2012).
[Crossref]

K. S. Morgan, D. M. Paganin, and K. K. Siu, “Quantitative single-exposure x-ray phase contrast imaging using a single attenuation grid,” Opt. Express 19(20), 19781–19789 (2011).
[Crossref]

Soldevila, F.

Stampanoni, M.

Stein, A. F.

Stockmar, M.

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, and P. Thibault, “Near-field ptychography: phase retrieval for inline holography using a structured illumination,” Sci. Rep. 3(1), 1927 (2013).
[Crossref]

Sun, J.

C. Zuo, J. Sun, J. Li, J. Zhang, A. Asundi, and Q. Chen, “High-resolution transport-of-intensity quantitative phase microscopy with annular illumination,” Sci. Rep. 7(1), 7654 (2017).
[Crossref]

Tagliasacchi, A.

Y. Schwartzburg, R. Testuz, A. Tagliasacchi, and M. Pauly, “High-contrast computational caustic design,” ACM Trans. Graph. 33(4), 1–11 (2014).
[Crossref]

Tajahuerce, E.

Tanida, J.

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M. R. Teague, “Deterministic phase retrieval: a green’s function solution,” J. Opt. Soc. Am. A 73(11), 1434–1441 (1983).
[Crossref]

Testuz, R.

Y. Schwartzburg, R. Testuz, A. Tagliasacchi, and M. Pauly, “High-contrast computational caustic design,” ACM Trans. Graph. 33(4), 1–11 (2014).
[Crossref]

Thibault, P.

I. Zanette, T. Zhou, A. Burvall, U. Lundström, D. H. Larsson, M. Zdora, P. Thibault, F. Pfeiffer, and H. M. Hertz, “Speckle-based x-ray phase-contrast and dark-field imaging with a laboratory source,” Phys. Rev. Lett. 112(25), 253903 (2014).
[Crossref]

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, and P. Thibault, “Near-field ptychography: phase retrieval for inline holography using a structured illumination,” Sci. Rep. 3(1), 1927 (2013).
[Crossref]

Tian, L.

Varma, P.

J. Zhong, L. Tian, P. Varma, and L. Waller, “Nonlinear optimization algorithm for partially coherent phase retrieval and source recovery,” IEEE Trans. Comput. Imaging 2(3), 310–322 (2016).
[Crossref]

Veeraraghavan, A.

Y. Wu, M. K. Sharma, and A. Veeraraghavan, “Wish: wavefront imaging sensor with high resolution,” Light: Sci. Appl. 8(1), 44 (2019).
[Crossref]

Waller, L.

Wang, C.

C. Wang, Q. Fu, X. Dun, and W. Heidrich, “Quantitative phase and intensity microscopy using snapshot white light wavefront sensing,” Sci. Rep. 9(1), 13795 (2019).
[Crossref]

C. Wang, Q. Fu, X. Dun, and W. Heidrich, “Megapixel adaptive optics: towards correcting large-scale distortions in computational cameras,” ACM Trans. Graph. 37(4), 1–12 (2018).
[Crossref]

C. Wang, X. Dun, Q. Fu, and W. Heidrich, “Ultra-high resolution coded wavefront sensor,” Opt. Express 25(12), 13736–13746 (2017).
[Crossref]

C. Wang, Q. Fu, X. Dun, and W. Heidrich, “A model for classical wavefront sensors and snapshot incoherent wavefront sensing,” in Computational Optical Sensing and Imaging (Optical Society of America, 2019), pp. CM1A–4.

Wang, H.

S. Bérujon, H. Wang, I. Pape, and K. Sawhney, “X-ray phase microscopy using the speckle tracking technique,” Appl. Phys. Lett. 102(15), 154105 (2013).
[Crossref]

H. Wang, K. Sawhney, S. Berujon, E. Ziegler, S. Rutishauser, and C. David, “X-ray wavefront characterization using a rotating shearing interferometer technique,” Opt. Express 19(17), 16550–16559 (2011).
[Crossref]

Wattellier, B.

Weickert, J.

T. Brox, A. Bruhn, N. Papenberg, and J. Weickert, “High accuracy optical flow estimation based on a theory for warping,” in European Conference on Computer Vision (Springer, 2004), pp. 25–36.

Wen, H. H.

Wu, Y.

Y. Wu, M. K. Sharma, and A. Veeraraghavan, “Wish: wavefront imaging sensor with high resolution,” Light: Sci. Appl. 8(1), 44 (2019).
[Crossref]

Yang, C.

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution fourier ptychographic microscopy,” Nat. Photonics 7(9), 739–745 (2013).
[Crossref]

Zanette, I.

I. Zanette, T. Zhou, A. Burvall, U. Lundström, D. H. Larsson, M. Zdora, P. Thibault, F. Pfeiffer, and H. M. Hertz, “Speckle-based x-ray phase-contrast and dark-field imaging with a laboratory source,” Phys. Rev. Lett. 112(25), 253903 (2014).
[Crossref]

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, and P. Thibault, “Near-field ptychography: phase retrieval for inline holography using a structured illumination,” Sci. Rep. 3(1), 1927 (2013).
[Crossref]

Zdora, M.

I. Zanette, T. Zhou, A. Burvall, U. Lundström, D. H. Larsson, M. Zdora, P. Thibault, F. Pfeiffer, and H. M. Hertz, “Speckle-based x-ray phase-contrast and dark-field imaging with a laboratory source,” Phys. Rev. Lett. 112(25), 253903 (2014).
[Crossref]

Zhang, J.

C. Zuo, J. Sun, J. Li, J. Zhang, A. Asundi, and Q. Chen, “High-resolution transport-of-intensity quantitative phase microscopy with annular illumination,” Sci. Rep. 7(1), 7654 (2017).
[Crossref]

Zheng, G.

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution fourier ptychographic microscopy,” Nat. Photonics 7(9), 739–745 (2013).
[Crossref]

Zhong, J.

J. Zhong, L. Tian, P. Varma, and L. Waller, “Nonlinear optimization algorithm for partially coherent phase retrieval and source recovery,” IEEE Trans. Comput. Imaging 2(3), 310–322 (2016).
[Crossref]

Zhou, T.

I. Zanette, T. Zhou, A. Burvall, U. Lundström, D. H. Larsson, M. Zdora, P. Thibault, F. Pfeiffer, and H. M. Hertz, “Speckle-based x-ray phase-contrast and dark-field imaging with a laboratory source,” Phys. Rev. Lett. 112(25), 253903 (2014).
[Crossref]

Ziegler, E.

S. Bérujon and E. Ziegler, “Near-field speckle-scanning-based x-ray imaging,” Phys. Rev. A 92(1), 013837 (2015).
[Crossref]

S. Bérujon, E. Ziegler, R. Cerbino, and L. Peverini, “Two-dimensional x-ray beam phase sensing,” Phys. Rev. Lett. 108(15), 158102 (2012).
[Crossref]

H. Wang, K. Sawhney, S. Berujon, E. Ziegler, S. Rutishauser, and C. David, “X-ray wavefront characterization using a rotating shearing interferometer technique,” Opt. Express 19(17), 16550–16559 (2011).
[Crossref]

Zuo, C.

ACM Trans. Graph. (2)

C. Wang, Q. Fu, X. Dun, and W. Heidrich, “Megapixel adaptive optics: towards correcting large-scale distortions in computational cameras,” ACM Trans. Graph. 37(4), 1–12 (2018).
[Crossref]

Y. Schwartzburg, R. Testuz, A. Tagliasacchi, and M. Pauly, “High-contrast computational caustic design,” ACM Trans. Graph. 33(4), 1–11 (2014).
[Crossref]

Appl. Opt. (4)

Appl. Phys. Lett. (2)

K. S. Morgan, D. M. Paganin, and K. K. Siu, “X-ray phase imaging with a paper analyzer,” Appl. Phys. Lett. 100(12), 124102 (2012).
[Crossref]

S. Bérujon, H. Wang, I. Pape, and K. Sawhney, “X-ray phase microscopy using the speckle tracking technique,” Appl. Phys. Lett. 102(15), 154105 (2013).
[Crossref]

Artif. Intelligence (1)

B. K. Horn and B. G. Schunck, “Determining optical flow,” Artif. Intelligence 17(1-3), 185–203 (1981).
[Crossref]

Biomed. Opt. Express (1)

IEEE Trans. Comput. Imaging (1)

J. Zhong, L. Tian, P. Varma, and L. Waller, “Nonlinear optimization algorithm for partially coherent phase retrieval and source recovery,” IEEE Trans. Comput. Imaging 2(3), 310–322 (2016).
[Crossref]

J. Mod. Opt. (1)

R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43(2), 289–293 (1996).
[Crossref]

J. Opt. Soc. Am. A (4)

Light: Sci. Appl. (1)

Y. Wu, M. K. Sharma, and A. Veeraraghavan, “Wish: wavefront imaging sensor with high resolution,” Light: Sci. Appl. 8(1), 44 (2019).
[Crossref]

Meas. Sci. Technol. (1)

H. Richard and M. Raffel, “Principle and applications of the background oriented schlieren (bos) method,” Meas. Sci. Technol. 12(9), 1576–1585 (2001).
[Crossref]

Nat. Photonics (1)

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution fourier ptychographic microscopy,” Nat. Photonics 7(9), 739–745 (2013).
[Crossref]

Opt. Commun. (1)

T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133(1-6), 339–346 (1997).
[Crossref]

Opt. Express (12)

P. Bon, G. Maucort, B. Wattellier, and S. Monneret, “Quadriwave lateral shearing interferometry for quantitative phase microscopy of living cells,” Opt. Express 17(15), 13080–13094 (2009).
[Crossref]

K. Matsushima and T. Shimobaba, “Band-limited angular spectrum method for numerical simulation of free-space propagation in far and near fields,” Opt. Express 17(22), 19662–19673 (2009).
[Crossref]

L. Waller, L. Tian, and G. Barbastathis, “Transport of intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express 18(12), 12552–12561 (2010).
[Crossref]

L. Waller, S. S. Kou, C. J. Sheppard, and G. Barbastathis, “Phase from chromatic aberrations,” Opt. Express 18(22), 22817–22825 (2010).
[Crossref]

H. Wang, K. Sawhney, S. Berujon, E. Ziegler, S. Rutishauser, and C. David, “X-ray wavefront characterization using a rotating shearing interferometer technique,” Opt. Express 19(17), 16550–16559 (2011).
[Crossref]

K. S. Morgan, D. M. Paganin, and K. K. Siu, “Quantitative single-exposure x-ray phase contrast imaging using a single attenuation grid,” Opt. Express 19(20), 19781–19789 (2011).
[Crossref]

C. Zuo, Q. Chen, W. Qu, and A. Asundi, “High-speed transport-of-intensity phase microscopy with an electrically tunable lens,” Opt. Express 21(20), 24060–24075 (2013).
[Crossref]

G. Damberg and W. Heidrich, “Efficient freeform lens optimization for computational caustic displays,” Opt. Express 23(8), 10224–10232 (2015).
[Crossref]

L. Tian and L. Waller, “Quantitative differential phase contrast imaging in an led array microscope,” Opt. Express 23(9), 11394–11403 (2015).
[Crossref]

R. Horisaki, R. Egami, and J. Tanida, “Single-shot phase imaging with randomized light (spiral),” Opt. Express 24(4), 3765–3773 (2016).
[Crossref]

M. Bawart, S. Bernet, and M. Ritsch-Marte, “Programmable freeform optical elements,” Opt. Express 25(5), 4898–4906 (2017).
[Crossref]

C. Wang, X. Dun, Q. Fu, and W. Heidrich, “Ultra-high resolution coded wavefront sensor,” Opt. Express 25(12), 13736–13746 (2017).
[Crossref]

Opt. Lett. (7)

Optica (1)

Optik (1)

J. P. Guigay, “Fourier-transform analysis of fresnel diffraction patterns and in-line holograms,” Optik 49, 121–125 (1977).

Phys. Rev. A (1)

S. Bérujon and E. Ziegler, “Near-field speckle-scanning-based x-ray imaging,” Phys. Rev. A 92(1), 013837 (2015).
[Crossref]

Phys. Rev. Lett. (2)

I. Zanette, T. Zhou, A. Burvall, U. Lundström, D. H. Larsson, M. Zdora, P. Thibault, F. Pfeiffer, and H. M. Hertz, “Speckle-based x-ray phase-contrast and dark-field imaging with a laboratory source,” Phys. Rev. Lett. 112(25), 253903 (2014).
[Crossref]

S. Bérujon, E. Ziegler, R. Cerbino, and L. Peverini, “Two-dimensional x-ray beam phase sensing,” Phys. Rev. Lett. 108(15), 158102 (2012).
[Crossref]

PLoS One (1)

Z. F. Phillips, M. Chen, and L. Waller, “Single-shot quantitative phase microscopy with color-multiplexed differential phase contrast (cdpc),” PLoS One 12(2), e0171228 (2017).
[Crossref]

Sci. Rep. (5)

C. Zuo, J. Sun, J. Li, J. Zhang, A. Asundi, and Q. Chen, “High-resolution transport-of-intensity quantitative phase microscopy with annular illumination,” Sci. Rep. 7(1), 7654 (2017).
[Crossref]

C. Wang, Q. Fu, X. Dun, and W. Heidrich, “Quantitative phase and intensity microscopy using snapshot white light wavefront sensing,” Sci. Rep. 9(1), 13795 (2019).
[Crossref]

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, and P. Thibault, “Near-field ptychography: phase retrieval for inline holography using a structured illumination,” Sci. Rep. 3(1), 1927 (2013).
[Crossref]

D. M. Paganin and K. S. Morgan, “X-ray fokker–planck equation for paraxial imaging,” Sci. Rep. 9(1), 17537–18 (2019).
[Crossref]

K. S. Morgan and D. M. Paganin, “Applying the fokker–planck equation to grating-based x-ray phase and dark-field imaging,” Sci. Rep. 9(1), 17465–14 (2019).
[Crossref]

Other (4)

J. W. Goodman, Introduction to Fourier optics (Roberts and Company Publishers, 2005).

C. Wang, Q. Fu, X. Dun, and W. Heidrich, “A model for classical wavefront sensors and snapshot incoherent wavefront sensing,” in Computational Optical Sensing and Imaging (Optical Society of America, 2019), pp. CM1A–4.

T. Brox, A. Bruhn, N. Papenberg, and J. Weickert, “High accuracy optical flow estimation based on a theory for warping,” in European Conference on Computer Vision (Springer, 2004), pp. 25–36.

https://github.com/vccimaging/PhaseIntensityMicroscope (2019).

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Figures (7)

Fig. 1.
Fig. 1. General single-shot wavefront sensor model. The unknown wavefront $\phi ({\mathbf r})$ is numerically solved from cached $I_0({\mathbf r})$ and single-shot measurement $I({\mathbf r})$.
Fig. 2.
Fig. 2. Monochromatic free space light propagation as image formation model.
Fig. 3.
Fig. 3. Wavefront transfer function for different optics-sensor distance $z$ configurations. Area of valid regions (shaded) for Eq. (3) to hold decreases with an increase of $z$. This $z$-$|{\boldsymbol{\rho}}|$ resolution trade-off applies to all classical wavefront sensors based on Eq. (3), especially for slope tracking sensors whose coding optics are usually mm away.
Fig. 4.
Fig. 4. Errors of different formulas for a Gaussian wavefront propagating through $z$. As a mixed approach, our formula Eq. (3) outperforms alternative formulations in terms of $\ell _1$-fitting errors. With a gradual violation of ray optics and weak diffraction assumption (increasing $z$, and hence breaking of ${\nabla} ^2 d({\mathbf r}) \ll 1/z$), all formulas start to fail.
Fig. 5.
Fig. 5. Errors of different formulas for a real-captured cheek cell wavefront propagating through $z$. For small wavefront slopes, the advantage of our formula Eq. (3) is not obvious compared to TIE, but still maintain its advantage comparing to flow-tracking formula Eq. (7). However, once the assumption ${\nabla} ^2 d({\mathbf r}) \ll 1/z$ is violated, both TIE and our formula amplify the defocusing error, showing $I({\mathbf r})$ as a blurry image.
Fig. 6.
Fig. 6. The “optics” for our wavefront sensor (variant of [20]) in Fig. 1, and its capability to recover amplitude $A({\mathbf r})$ and OPD $d({\mathbf r}) = \frac {\lambda }{2\pi }\phi ({\mathbf r})$ from reference $I_0({\mathbf r})$ and measurement $I({\mathbf r})$. The condition number of our mask as defined in §3.2 is $\kappa = 12.2$, justifying a “good” enough coding optics for wavefront sensing. Notice the round biconcave bell shape of the human red blood cell has been successfully reconstructed.
Fig. 7.
Fig. 7. Experimental results using the proposed quantitative phase imaging pipeline. Images were taken under a $\times 100$ Mitutoyo plan apochromat objective, 0.70 NA. Inset close-up images indicate that the speckle patterns have been fully removed from the original raw data. Wavefronts are shown in terms of OPD.

Tables (2)

Tables Icon

Table 1. Optics and corresponding reference image of different wavefront sensors. δ p ( x ) is the Dirac comb function with period p (the lenslet or pinhole pitch). r = ( x , y ) .

Tables Icon

Table 2. Theoretical models used in classical wavefront sensing research.

Equations (36)

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

r = r + λ z 2 π ϕ ( r ) I ( r ) d r = | A ( r ) | 2 J ( r ) d r , (geometry relationship) (energy conservation)
I ( r + λ z 2 π ϕ ( r ) ) = ( 1 + λ z 2 π 2 ϕ ( r ) ) 1 | A ( r ) | 2 J ( r ) ( 1 λ z 2 π 2 ϕ ( r ) ) | A ( r ) | 2 J ( r ) ,
I ( r + λ z 2 π ϕ ) = | A ( r ) | 2 ( 1 λ z 2 π 2 ϕ ) I 0 ( r ) .
I ( r + z d ) = | A ( r ) | 2 ( 1 z 2 d ) I 0 ( r ) .
I 2 ϕ + I 1 2 ϕ = k z ( I 1 I 2 ) k I ¯ z ,
I ( r + λ z 2 π ϕ ) = I 0 ( r ) ,
I ( r + λ z 2 π ϕ ) = | A ( r ) | 2 I 0 ( r ) ,
I ( r ) = | A ( r ) | 2 I 0 ( r λ z 2 π ϕ ) ,
I ( r ) = | A ( r ) | 2 [ I ¯ 0 + D ( r ) Δ I 0 ( r λ z 2 π ϕ ) ] ,
d ( r ) 1 and 2 d ( r ) 1 z .
D ( ρ ) 1 2 π | ρ | min ( 1 , 1 / z 2 π | ρ | ) = { 1 2 π | ρ | if\quad | ρ | < 1 2 π z , 1 4 π 2 z | ρ | 2 otherwise .
I 0 ( r ) λ z 2 π I 0 ( r ) ϕ ( r ) + H.O.T. I ( r ) / | A ( r ) | 2 .
κ = max | I 0 ( r ) | min | I 0 ( r ) | .
a , ϕ = arg min a , ϕ I ( r + ϕ ) a I 0 ( r ) 2 + R 1 ( a ) + R 2 ( ϕ ) ,
a | A ( r ) | 2 ( 1 λ z 2 π 2 ϕ ) and ϕ λ z 2 π ϕ .
u z ( r ) = exp [ j k z ( 1 + 2 k 2 ) 1 / 2 ] u 0 ( r ) U z ( ρ ) = exp [ j k z ( 1 λ 2 | ρ | 2 ) 1 / 2 ] U 0 ( ρ ) , (spatial domain) (frequency domain)
u z ( r ) exp ( j z 2 k 2 ) Fresnel propagator u 0 ( r ) ( 1 + j z 2 k 2 ) 0 th a n d 1 st o r d e r u 0 ( r ) ,
max ( 1 k 2 , z 2 k ) | 2 u 0 ( r ) | = ( 18 ) z 2 k | 2 u 0 ( r ) | | u 0 ( r ) | ,
z λ π .
u 0 ( r ) = p 0 ( r ) optics f 0 ( r ) sample .
u z ( r ) exp [ j k z ( 1 + 2 k 2 ) 1 / 2 ] u 0 ( r ) = F exp ( j 2 π r ρ ) exp [ j k z ( 1 λ 2 | ρ | 2 ) 1 / 2 ] × P 0 ( ρ ) F 0 ( ρ ρ ) d ρ d ρ ( 21 ) exp ( j k z ) exp ( j 2 π r ρ ) exp [ j k z ( 1 λ 2 | ρ | 2 ) 1 / 2 ] × exp [ j 2 π ( r λ z ρ ) ρ ] exp [ j k z ( 1 λ 2 | ρ | 2 ) 1 / 2 ] F 0 ( ρ ) d ρ P 0 ( ρ ) d ρ = exp ( j k z ) exp ( j 2 π r ρ ) P z ( ρ ) × ( exp [ j 2 π ( r λ z ρ ) ρ ] F z ( ρ ) d ρ ) d ρ = F 1 exp ( j k z ) exp ( j 2 π r ρ ) P z ( ρ ) f z ( r λ z ρ ) d ρ ,
( 1 λ 2 | ρ | 2 ) 1 / 2 ( 1 λ 2 | ρ | 2 ) 1 / 2 + ( 1 λ 2 | ρ | 2 ) 1 / 2 λ 2 ρ ρ 1.
u z ( r ) = exp ( j 2 π r ρ ) P z ( ρ ) f z ( r λ z ρ ) d ρ .
f z ( r ) = ( 22 ) exp ( j 2 π r ρ ) A z ( ρ ) g z ( r λ z ρ ) d ρ ,
A z ( r ) ( 16 ) A ( r ) + j z 2 k 2 A ( r ) A ( r ) ,
g z ( r ) ( 16 ) exp [ j ϕ ( r ) ] ( 1 z 2 k 2 ϕ ( r ) j z 2 k ϕ ( r ) ϕ ( r ) ) .
z 2 k | ϕ ( r ) | 2 1 | ϕ ( r ) | ( 4 π λ z ) 1 / 2 ( 18 ) 2 π λ ,
z 2 k | 2 ϕ ( r ) | ( 17 ) 1 | 2 ϕ ( r ) | 4 π λ z 2 π λ z .
ϕ ( r λ z ρ ) ϕ ( r ) λ z ρ ϕ ( r ) , ϕ ( r λ z ρ ) ϕ ( r ) and 2 ϕ ( r λ z ρ ) 2 ϕ ( r ) .
f z ( r ) ( 28 ) ( 25 ) ( 1 z 2 k 2 ϕ ( r ) j z 2 k ϕ ( r ) ϕ ( r ) ) exp [ j ϕ ( r ) ] exp [ j 2 π ( r z k ϕ ( r ) ) ρ ] A z ( ρ ) d ρ ( 24 ) F 1 ( 1 z 2 k 2 ϕ ( r ) j z 2 k ϕ ( r ) ϕ ( r ) ) A ( r z k ϕ ( r ) ) exp [ j ϕ ( r ) ] .
f z ( r λ z ρ ) ( 28 ) ( 29 ) ( 1 z 2 k 2 ϕ ( r ) j z 2 k ϕ ( r ) ϕ ( r ) ) A ( r z k ϕ ( r ) ) exp [ j λ z ρ ϕ ( r ) ] ,
f z ( r λ z ρ ) ( 1 z 2 k 2 ϕ ( r ) j z 2 k ϕ ( r ) ϕ ( r ) ) A ( r ) exp [ j λ z ρ ϕ ( r ) ] .
u z ( r ) F 1 ( 30 ) A ( r z k ϕ ( r ) ) ( 1 z 2 k 2 ϕ ( r ) j z 2 k ϕ ( r ) ϕ ( r ) ) p z ( r λ z 2 π ϕ ( r ) ) .
I 0 ( r ) = | p z ( r ) | 2 .
I ( r ) | u z ( r ) | 2 = ( 33 ) ( 32 ) | A ( r z k ϕ ( r ) ) | 2 I 0 ( r λ z 2 π ϕ ( r ) ) [ ( 1 z 2 k 2 ϕ ( r ) ) 2 + ( z 2 k ϕ ( r ) ϕ ( r ) ) 2 ] | A ( r z k ϕ ( r ) ) | 2 I 0 ( r λ z 2 π ϕ ( r ) ) ( 1 z k 2 ϕ ( r ) ) .
I ( r + λ z 2 π ϕ ) = | A ( r ) | 2 ( 1 λ z 2 π 2 ϕ ) I 0 ( r ) .

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