Abstract

In this Letter, we propose a method for retrieving the phase of a wavefront from the diffraction patterns recorded when the object is sequentially illuminated by spatially modulated light. For wavefronts having a smooth phase, the retrieval is achieved by using a deterministic method. When the phase has discontinuities, an iterative process is used for the retrieval and enhancement of the spatial resolution. Both the deterministic and iterative phase reconstructions are demonstrated by experiments.

© 2014 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2013 (3)

T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, Phys. Rev. A 87, 053850 (2013).
[CrossRef]

C. Zuo, Q. Chen, Y. Yu, and A. Asundi, Opt. Express 21, 5346 (2013).
[CrossRef]

P. Gao, G. Pedrini, and W. Osten, Opt. Lett. 38, 5204 (2013).
[CrossRef]

2012 (3)

2010 (2)

J. Frank, S. Altmeyer, and G. Wernicke, J. Opt. Soc. Am. A 27, 2244 (2010).
[CrossRef]

F. Zhang and J. M. Rodenburg, Phys. Rev. B 82, 121104(R) (2010).
[CrossRef]

2009 (1)

2008 (3)

B. Kemper and G. von Bally, Appl. Opt. 47, A52 (2008).
[CrossRef]

P. Bao, F. Zhang, G. Pedrini, and W. Osten, Opt. Lett. 33, 309 (2008).
[CrossRef]

Y. J. Liu, B. Chen, E. R. Li, J. Y. Wang, A. Marcelli, S. W. Wilkins, H. Ming, Y. C. Tian, K. A. Nugent, P. P. Zhu, and Z. Y. Wu, Phys. Rev. A 78, 023817 (2008).
[CrossRef]

2007 (1)

F. Zhang, G. Pedrini, and W. Osten, Phys. Rev. A 75, 043805 (2007).
[CrossRef]

2006 (5)

G. Popescu, T. Ikeda, K. Goda, C. A. Best-Popescu, and M. Laposata, Phys. Rev. Lett. 97, 218101 (2006).
[CrossRef]

S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, Phys. Rev. Lett. 97, 168102 (2006).
[CrossRef]

A. V. Martin, F. R. Chen, W. K. Hsieh, J. J. Kai, S. D. Findlay, and L. J. Allen, Ultramicroscopy 106, 914 (2006).
[CrossRef]

C. Kohler, X. Schwab, and W. Osten, Appl. Opt. 45, 960 (2006).
[CrossRef]

P. Almoro, G. Pedrini, and W. Osten, Appl. Opt. 45, 8596 (2006).
[CrossRef]

2005 (2)

G. Pedrini, W. Osten, and Y. Zhang, Opt. Lett 30, 833 (2005).
[CrossRef]

J. B. Costa, Appl. Opt. 44, 60 (2005).
[CrossRef]

2004 (2)

J. M. Rodenburg and H. M. L. Faulkner, Appl. Phys. Lett. 85, 4795 (2004).
[CrossRef]

H. M. L. Faulkner and J. M. Rodenburg, Phys. Rev. Lett. 93, 023903 (2004).
[CrossRef]

2001 (2)

B. C. Platt and R. Shack, J. Refract. Surg. 17, S573 (2001).

L. J. Allen and M. P. Oxley, Opt. Commun. 199, 65 (2001).
[CrossRef]

1990 (1)

1988 (1)

R. T. Frankot and Z. Chellappa, IEEE Trans. Pattern Anal. Mach. Intell. 10, 439 (1988).
[CrossRef]

1985 (2)

1982 (1)

Agour, M.

Alexandrov, S. A.

S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, Phys. Rev. Lett. 97, 168102 (2006).
[CrossRef]

Allen, L. J.

A. V. Martin, F. R. Chen, W. K. Hsieh, J. J. Kai, S. D. Findlay, and L. J. Allen, Ultramicroscopy 106, 914 (2006).
[CrossRef]

L. J. Allen and M. P. Oxley, Opt. Commun. 199, 65 (2001).
[CrossRef]

Almoro, P.

Almoro, P. F.

Altmeyer, S.

Asundi, A.

Bao, P.

Batey, D. J.

T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, Phys. Rev. A 87, 053850 (2013).
[CrossRef]

Best-Popescu, C. A.

G. Popescu, T. Ikeda, K. Goda, C. A. Best-Popescu, and M. Laposata, Phys. Rev. Lett. 97, 218101 (2006).
[CrossRef]

Bon, P.

Bulirsch, R.

J. Stoer and R. Bulirsch, Introduction to Numerical Analysis (Springer, 1980).

Chellappa, Z.

R. T. Frankot and Z. Chellappa, IEEE Trans. Pattern Anal. Mach. Intell. 10, 439 (1988).
[CrossRef]

Chen, B.

Y. J. Liu, B. Chen, E. R. Li, J. Y. Wang, A. Marcelli, S. W. Wilkins, H. Ming, Y. C. Tian, K. A. Nugent, P. P. Zhu, and Z. Y. Wu, Phys. Rev. A 78, 023817 (2008).
[CrossRef]

Chen, F. R.

A. V. Martin, F. R. Chen, W. K. Hsieh, J. J. Kai, S. D. Findlay, and L. J. Allen, Ultramicroscopy 106, 914 (2006).
[CrossRef]

Chen, Q.

Costa, J. B.

Dong, S.

Edo, T. B.

T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, Phys. Rev. A 87, 053850 (2013).
[CrossRef]

Falldorf, C.

Faulkner, H. M. L.

J. M. Rodenburg and H. M. L. Faulkner, Appl. Phys. Lett. 85, 4795 (2004).
[CrossRef]

H. M. L. Faulkner and J. M. Rodenburg, Phys. Rev. Lett. 93, 023903 (2004).
[CrossRef]

Findlay, S. D.

A. V. Martin, F. R. Chen, W. K. Hsieh, J. J. Kai, S. D. Findlay, and L. J. Allen, Ultramicroscopy 106, 914 (2006).
[CrossRef]

Frank, J.

Frankot, R. T.

R. T. Frankot and Z. Chellappa, IEEE Trans. Pattern Anal. Mach. Intell. 10, 439 (1988).
[CrossRef]

Gao, P.

Goda, K.

G. Popescu, T. Ikeda, K. Goda, C. A. Best-Popescu, and M. Laposata, Phys. Rev. Lett. 97, 218101 (2006).
[CrossRef]

Gutzler, T.

S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, Phys. Rev. Lett. 97, 168102 (2006).
[CrossRef]

Haist, T.

Hanson, S. G.

Hillman, T. R.

S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, Phys. Rev. Lett. 97, 168102 (2006).
[CrossRef]

Hsieh, W. K.

A. V. Martin, F. R. Chen, W. K. Hsieh, J. J. Kai, S. D. Findlay, and L. J. Allen, Ultramicroscopy 106, 914 (2006).
[CrossRef]

Ikeda, T.

G. Popescu, T. Ikeda, K. Goda, C. A. Best-Popescu, and M. Laposata, Phys. Rev. Lett. 97, 218101 (2006).
[CrossRef]

Kai, J. J.

A. V. Martin, F. R. Chen, W. K. Hsieh, J. J. Kai, S. D. Findlay, and L. J. Allen, Ultramicroscopy 106, 914 (2006).
[CrossRef]

Kemper, B.

Kohler, C.

Laposata, M.

G. Popescu, T. Ikeda, K. Goda, C. A. Best-Popescu, and M. Laposata, Phys. Rev. Lett. 97, 218101 (2006).
[CrossRef]

Li, E. R.

Y. J. Liu, B. Chen, E. R. Li, J. Y. Wang, A. Marcelli, S. W. Wilkins, H. Ming, Y. C. Tian, K. A. Nugent, P. P. Zhu, and Z. Y. Wu, Phys. Rev. A 78, 023817 (2008).
[CrossRef]

Liu, Y. J.

Y. J. Liu, B. Chen, E. R. Li, J. Y. Wang, A. Marcelli, S. W. Wilkins, H. Ming, Y. C. Tian, K. A. Nugent, P. P. Zhu, and Z. Y. Wu, Phys. Rev. A 78, 023817 (2008).
[CrossRef]

Maiden, A. M.

T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, Phys. Rev. A 87, 053850 (2013).
[CrossRef]

Marcelli, A.

Y. J. Liu, B. Chen, E. R. Li, J. Y. Wang, A. Marcelli, S. W. Wilkins, H. Ming, Y. C. Tian, K. A. Nugent, P. P. Zhu, and Z. Y. Wu, Phys. Rev. A 78, 023817 (2008).
[CrossRef]

Martin, A. V.

A. V. Martin, F. R. Chen, W. K. Hsieh, J. J. Kai, S. D. Findlay, and L. J. Allen, Ultramicroscopy 106, 914 (2006).
[CrossRef]

Maucort, G.

Ming, H.

Y. J. Liu, B. Chen, E. R. Li, J. Y. Wang, A. Marcelli, S. W. Wilkins, H. Ming, Y. C. Tian, K. A. Nugent, P. P. Zhu, and Z. Y. Wu, Phys. Rev. A 78, 023817 (2008).
[CrossRef]

Monneret, S.

Nugent, K. A.

Y. J. Liu, B. Chen, E. R. Li, J. Y. Wang, A. Marcelli, S. W. Wilkins, H. Ming, Y. C. Tian, K. A. Nugent, P. P. Zhu, and Z. Y. Wu, Phys. Rev. A 78, 023817 (2008).
[CrossRef]

Osten, W.

Oxley, M. P.

L. J. Allen and M. P. Oxley, Opt. Commun. 199, 65 (2001).
[CrossRef]

Pedrini, G.

Pešic, Z. D.

T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, Phys. Rev. A 87, 053850 (2013).
[CrossRef]

Platt, B. C.

B. C. Platt and R. Shack, J. Refract. Surg. 17, S573 (2001).

Popescu, G.

G. Popescu, T. Ikeda, K. Goda, C. A. Best-Popescu, and M. Laposata, Phys. Rev. Lett. 97, 218101 (2006).
[CrossRef]

Rau, C.

T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, Phys. Rev. A 87, 053850 (2013).
[CrossRef]

Roddier, F.

Rodenburg, J. M.

T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, Phys. Rev. A 87, 053850 (2013).
[CrossRef]

F. Zhang and J. M. Rodenburg, Phys. Rev. B 82, 121104(R) (2010).
[CrossRef]

H. M. L. Faulkner and J. M. Rodenburg, Phys. Rev. Lett. 93, 023903 (2004).
[CrossRef]

J. M. Rodenburg and H. M. L. Faulkner, Appl. Phys. Lett. 85, 4795 (2004).
[CrossRef]

Sampson, D. D.

S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, Phys. Rev. Lett. 97, 168102 (2006).
[CrossRef]

Schwab, X.

Shack, R.

B. C. Platt and R. Shack, J. Refract. Surg. 17, S573 (2001).

Stoer, J.

J. Stoer and R. Bulirsch, Introduction to Numerical Analysis (Springer, 1980).

Teague, M. R.

Tian, Y. C.

Y. J. Liu, B. Chen, E. R. Li, J. Y. Wang, A. Marcelli, S. W. Wilkins, H. Ming, Y. C. Tian, K. A. Nugent, P. P. Zhu, and Z. Y. Wu, Phys. Rev. A 78, 023817 (2008).
[CrossRef]

von Bally, G.

Wagner, U.

T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, Phys. Rev. A 87, 053850 (2013).
[CrossRef]

Waigh, T. A.

T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, Phys. Rev. A 87, 053850 (2013).
[CrossRef]

Waller, L.

Wang, J. Y.

Y. J. Liu, B. Chen, E. R. Li, J. Y. Wang, A. Marcelli, S. W. Wilkins, H. Ming, Y. C. Tian, K. A. Nugent, P. P. Zhu, and Z. Y. Wu, Phys. Rev. A 78, 023817 (2008).
[CrossRef]

Wattellier, B.

Wernicke, G.

Wilkins, S. W.

Y. J. Liu, B. Chen, E. R. Li, J. Y. Wang, A. Marcelli, S. W. Wilkins, H. Ming, Y. C. Tian, K. A. Nugent, P. P. Zhu, and Z. Y. Wu, Phys. Rev. A 78, 023817 (2008).
[CrossRef]

Wu, Z. Y.

Y. J. Liu, B. Chen, E. R. Li, J. Y. Wang, A. Marcelli, S. W. Wilkins, H. Ming, Y. C. Tian, K. A. Nugent, P. P. Zhu, and Z. Y. Wu, Phys. Rev. A 78, 023817 (2008).
[CrossRef]

Yu, Y.

Zhang, F.

F. Zhang and J. M. Rodenburg, Phys. Rev. B 82, 121104(R) (2010).
[CrossRef]

P. Bao, F. Zhang, G. Pedrini, and W. Osten, Opt. Lett. 33, 309 (2008).
[CrossRef]

F. Zhang, G. Pedrini, and W. Osten, Phys. Rev. A 75, 043805 (2007).
[CrossRef]

Zhang, Y.

G. Pedrini, W. Osten, and Y. Zhang, Opt. Lett 30, 833 (2005).
[CrossRef]

Zhu, P. P.

Y. J. Liu, B. Chen, E. R. Li, J. Y. Wang, A. Marcelli, S. W. Wilkins, H. Ming, Y. C. Tian, K. A. Nugent, P. P. Zhu, and Z. Y. Wu, Phys. Rev. A 78, 023817 (2008).
[CrossRef]

Zuo, C.

Appl. Opt. (7)

Appl. Phys. Lett. (1)

J. M. Rodenburg and H. M. L. Faulkner, Appl. Phys. Lett. 85, 4795 (2004).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

R. T. Frankot and Z. Chellappa, IEEE Trans. Pattern Anal. Mach. Intell. 10, 439 (1988).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

J. Refract. Surg. (1)

B. C. Platt and R. Shack, J. Refract. Surg. 17, S573 (2001).

Opt. Commun. (1)

L. J. Allen and M. P. Oxley, Opt. Commun. 199, 65 (2001).
[CrossRef]

Opt. Express (2)

Opt. Lett (1)

G. Pedrini, W. Osten, and Y. Zhang, Opt. Lett 30, 833 (2005).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. A (3)

Y. J. Liu, B. Chen, E. R. Li, J. Y. Wang, A. Marcelli, S. W. Wilkins, H. Ming, Y. C. Tian, K. A. Nugent, P. P. Zhu, and Z. Y. Wu, Phys. Rev. A 78, 023817 (2008).
[CrossRef]

F. Zhang, G. Pedrini, and W. Osten, Phys. Rev. A 75, 043805 (2007).
[CrossRef]

T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, Phys. Rev. A 87, 053850 (2013).
[CrossRef]

Phys. Rev. B (1)

F. Zhang and J. M. Rodenburg, Phys. Rev. B 82, 121104(R) (2010).
[CrossRef]

Phys. Rev. Lett. (3)

H. M. L. Faulkner and J. M. Rodenburg, Phys. Rev. Lett. 93, 023903 (2004).
[CrossRef]

G. Popescu, T. Ikeda, K. Goda, C. A. Best-Popescu, and M. Laposata, Phys. Rev. Lett. 97, 218101 (2006).
[CrossRef]

S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, Phys. Rev. Lett. 97, 168102 (2006).
[CrossRef]

Proc. SPIE (1)

W. Osten, Proc. SPIE 0473, 52 (1985).
[CrossRef]

Ultramicroscopy (1)

A. V. Martin, F. R. Chen, W. K. Hsieh, J. J. Kai, S. D. Findlay, and L. J. Allen, Ultramicroscopy 106, 914 (2006).
[CrossRef]

Other (2)

J. Stoer and R. Bulirsch, Introduction to Numerical Analysis (Springer, 1980).

W. Osten and N. Reingand, eds., Optical Imaging and Metrology: Advanced Technologies (Wiley-VCH, 2012).

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Figures (4)

Fig. 1.
Fig. 1.

Setup of the phase retrieval by using spatially modulated illuminations. SLM, spatial light modulator; L1L4, achromatic lenses; IM, image plane; dz, the distance between imaging plane and CCD plane.

Fig. 2.
Fig. 2.

Reconstruction results of a slice of mouse kidney. (a) Intensity distributions of five illuminations, (b) intensity distributions of five generated diffraction patterns, (c) intensity image of the sample under plane wave illumination, (d) reconstructed phase derivative of the object wave in the x direction, (e) the reconstructed phase distributions obtained by the proposed method, and (f) those obtained by DHM. The arrow in (d) denotes the direction of the phase derivative.

Fig. 3.
Fig. 3.

Phase retrieval on the wing of a mosquito. (a) Diffraction patterns under five spatially modulated illuminations and (b) phase distribution of the wavefront transmitted by the wing of the mosquito.

Fig. 4.
Fig. 4.

Resolution enhancement. (a) Spectrum evolvement of the object wave during the imaging process. The red dashed rings denote the frequency limited by the NA of the imaging system. The four black dots denote the frequency components, which are beyond the theoretical resolution limit. (b) Phase images reconstructed by DHM with plane wave illumination and (c) the phase retrieval with spatially modulated illumination.

Equations (8)

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

·[ImIMφmIM]=2πλ·ImCCDImIMdz,
cm,1I0IM+cm,2xI0IM+cm,3yI0IM+cm,4I0IMxφ0IM+cm,5I0IMyφ0IM=dm.
cm,1=Imillum,IMΔφmillum,IM+Imillum,IM·φmillum,IM,cm,2=Imillum,IMxφmillum,IM,cm,3=Imillum,IMyφmillum,IM,cm,4=xImillum,IM,cm,5=yImillum,IM,dm=2πλ0·ImCCDI0CCDImillum,IMdz.
[c11c12c13c14c15c21c22c23c24c25c31c32c33c34c35c41c42c43c44c45c51c52c53c54c55][I0IMxI0IMyI0IMI0xφ0IMI0yφ0IM]=[d1d2d3d4d5].
I0IM=I0CCD+λdz2π[I0CCD·φ0IM+I0CCDΔφ0IM].
[a1,1a1,2a2,1a2,2][xφ0IMyφ0IM]=[e1e2].
am,1=cm,4I0CCD+cm,1λdz2πxI0CCD,am,2=cm,5I0CCD+cm,1λdz2πyI0CCD,em=dmc1,1I0CCDc1,2xI0CCDc1,3yI0CCD.
φ0IM(x,y)=FT1{FT{xφ0IM}ikx}+{FT{yφ0IM}iky}.

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