Abstract

We propose a novel and effective method to quantitatively recover a complex-valued object from diffraction intensity maps recorded in the fractional Fourier domain. A wavefront modulation is introduced in the wave path, and several diffraction intensity maps are recorded through variable function orders in the fractional Fourier transform. A new phase retrieval algorithm is then proposed, and advantages of the proposed algorithm are also discussed. A proof-of-principle study is presented to show the feasibility and effectiveness of the proposed method.

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References

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  1. J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300(5624), 1419–1421 (2003).
    [CrossRef] [PubMed]
  2. M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442(7098), 63–66 (2006).
    [CrossRef] [PubMed]
  3. D. Gabor, “A new microscopic principle,” Nature 161(4098), 777–778 (1948).
    [CrossRef] [PubMed]
  4. C. J. Mann, L. Yu, C. Lo, and M. K. Kim, “High-resolution quantitative phase-contrast microscopy by digital holography,” Opt. Express 13, 8693–8698 (2005), http://www.opticsinfobase.org/oe/abstract.cfm ? URI = oe-13–22–8693.
  5. W. Chen, C. Quan, and C. J. Tay, “Extended depth of focus in a particle field measurement using a single-shot digital hologram,” Appl. Phys. Lett. 95(20), 201103 (2009).
    [CrossRef]
  6. W. Chen, C. Quan, and C. J. Tay, “Measurement of curvature and twist of a deformed object using digital holography,” Appl. Opt. 47(15), 2874–2881 (2008).
    [CrossRef] [PubMed]
  7. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22(16), 1268–1270 (1997).
    [CrossRef] [PubMed]
  8. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–246 (1972).
  9. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21(15), 2758–2769 (1982).
    [CrossRef] [PubMed]
  10. J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400(6742), 342–344 (1999).
    [CrossRef]
  11. 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, “Phase retrieval in X-ray imaging based on using structured illumination,” Phys. Rev. A 78(2), 023817 (2008).
    [CrossRef]
  12. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  13. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10(10), 2181–2186 (1993).
    [CrossRef]
  14. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).
  15. M. G. Ertosun, H. Atlı, H. M. Ozaktas, and B. Barshan, “Complex signal recovery from two fractional Fourier transform intensities: order and noise dependence,” Opt. Commun. 244(1-6), 61–70 (2005).
    [CrossRef]
  16. M. G. Ertosun, H. Atlı, H. M. Ozaktas, and B. Barshan, “Complex signal recovery from multiple fractional Fourier-transform intensities,” Appl. Opt. 44(23), 4902–4908 (2005).
    [CrossRef] [PubMed]
  17. J. Hua, L. Liu, and G. Li, “Extended fractional Fourier transforms,” J. Opt. Soc. Am. A 14(12), 3316–3322 (1997).
    [CrossRef]

2009 (1)

W. Chen, C. Quan, and C. J. Tay, “Extended depth of focus in a particle field measurement using a single-shot digital hologram,” Appl. Phys. Lett. 95(20), 201103 (2009).
[CrossRef]

2008 (2)

W. Chen, C. Quan, and C. J. Tay, “Measurement of curvature and twist of a deformed object using digital holography,” Appl. Opt. 47(15), 2874–2881 (2008).
[CrossRef] [PubMed]

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, “Phase retrieval in X-ray imaging based on using structured illumination,” Phys. Rev. A 78(2), 023817 (2008).
[CrossRef]

2006 (1)

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442(7098), 63–66 (2006).
[CrossRef] [PubMed]

2005 (2)

M. G. Ertosun, H. Atlı, H. M. Ozaktas, and B. Barshan, “Complex signal recovery from two fractional Fourier transform intensities: order and noise dependence,” Opt. Commun. 244(1-6), 61–70 (2005).
[CrossRef]

M. G. Ertosun, H. Atlı, H. M. Ozaktas, and B. Barshan, “Complex signal recovery from multiple fractional Fourier-transform intensities,” Appl. Opt. 44(23), 4902–4908 (2005).
[CrossRef] [PubMed]

2003 (1)

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300(5624), 1419–1421 (2003).
[CrossRef] [PubMed]

1999 (1)

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400(6742), 342–344 (1999).
[CrossRef]

1997 (2)

1993 (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 (Stuttg.) 35, 237–246 (1972).

1948 (1)

D. Gabor, “A new microscopic principle,” Nature 161(4098), 777–778 (1948).
[CrossRef] [PubMed]

Atli, H.

M. G. Ertosun, H. Atlı, H. M. Ozaktas, and B. Barshan, “Complex signal recovery from two fractional Fourier transform intensities: order and noise dependence,” Opt. Commun. 244(1-6), 61–70 (2005).
[CrossRef]

M. G. Ertosun, H. Atlı, H. M. Ozaktas, and B. Barshan, “Complex signal recovery from multiple fractional Fourier-transform intensities,” Appl. Opt. 44(23), 4902–4908 (2005).
[CrossRef] [PubMed]

Barshan, B.

M. G. Ertosun, H. Atlı, H. M. Ozaktas, and B. Barshan, “Complex signal recovery from two fractional Fourier transform intensities: order and noise dependence,” Opt. Commun. 244(1-6), 61–70 (2005).
[CrossRef]

M. G. Ertosun, H. Atlı, H. M. Ozaktas, and B. Barshan, “Complex signal recovery from multiple fractional Fourier-transform intensities,” Appl. Opt. 44(23), 4902–4908 (2005).
[CrossRef] [PubMed]

Charalambous, P.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400(6742), 342–344 (1999).
[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, “Phase retrieval in X-ray imaging based on using structured illumination,” Phys. Rev. A 78(2), 023817 (2008).
[CrossRef]

Chen, W.

W. Chen, C. Quan, and C. J. Tay, “Extended depth of focus in a particle field measurement using a single-shot digital hologram,” Appl. Phys. Lett. 95(20), 201103 (2009).
[CrossRef]

W. Chen, C. Quan, and C. J. Tay, “Measurement of curvature and twist of a deformed object using digital holography,” Appl. Opt. 47(15), 2874–2881 (2008).
[CrossRef] [PubMed]

Ertosun, M. G.

M. G. Ertosun, H. Atlı, H. M. Ozaktas, and B. Barshan, “Complex signal recovery from multiple fractional Fourier-transform intensities,” Appl. Opt. 44(23), 4902–4908 (2005).
[CrossRef] [PubMed]

M. G. Ertosun, H. Atlı, H. M. Ozaktas, and B. Barshan, “Complex signal recovery from two fractional Fourier transform intensities: order and noise dependence,” Opt. Commun. 244(1-6), 61–70 (2005).
[CrossRef]

Fienup, J. R.

Gabor, D.

D. Gabor, “A new microscopic principle,” Nature 161(4098), 777–778 (1948).
[CrossRef] [PubMed]

Gao, M.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300(5624), 1419–1421 (2003).
[CrossRef] [PubMed]

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 (Stuttg.) 35, 237–246 (1972).

Harder, R.

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442(7098), 63–66 (2006).
[CrossRef] [PubMed]

Hua, J.

J. Tay, C.

Kirz, J.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400(6742), 342–344 (1999).
[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, “Phase retrieval in X-ray imaging based on using structured illumination,” Phys. Rev. A 78(2), 023817 (2008).
[CrossRef]

Li, G.

Liu, L.

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, “Phase retrieval in X-ray imaging based on using structured illumination,” Phys. Rev. A 78(2), 023817 (2008).
[CrossRef]

Lohmann, A. W.

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, “Phase retrieval in X-ray imaging based on using structured illumination,” Phys. Rev. A 78(2), 023817 (2008).
[CrossRef]

Miao, J.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400(6742), 342–344 (1999).
[CrossRef]

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, “Phase retrieval in X-ray imaging based on using structured illumination,” Phys. Rev. A 78(2), 023817 (2008).
[CrossRef]

Nagahara, L. A.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300(5624), 1419–1421 (2003).
[CrossRef] [PubMed]

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, “Phase retrieval in X-ray imaging based on using structured illumination,” Phys. Rev. A 78(2), 023817 (2008).
[CrossRef]

Ozaktas, H. M.

M. G. Ertosun, H. Atlı, H. M. Ozaktas, and B. Barshan, “Complex signal recovery from two fractional Fourier transform intensities: order and noise dependence,” Opt. Commun. 244(1-6), 61–70 (2005).
[CrossRef]

M. G. Ertosun, H. Atlı, H. M. Ozaktas, and B. Barshan, “Complex signal recovery from multiple fractional Fourier-transform intensities,” Appl. Opt. 44(23), 4902–4908 (2005).
[CrossRef] [PubMed]

Pfeifer, M. A.

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442(7098), 63–66 (2006).
[CrossRef] [PubMed]

Quan, C.

W. Chen, C. Quan, and C. J. Tay, “Extended depth of focus in a particle field measurement using a single-shot digital hologram,” Appl. Phys. Lett. 95(20), 201103 (2009).
[CrossRef]

W. Chen, C. Quan, and C. J. Tay, “Measurement of curvature and twist of a deformed object using digital holography,” Appl. Opt. 47(15), 2874–2881 (2008).
[CrossRef] [PubMed]

Robinson, I. K.

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442(7098), 63–66 (2006).
[CrossRef] [PubMed]

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 (Stuttg.) 35, 237–246 (1972).

Sayre, D.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400(6742), 342–344 (1999).
[CrossRef]

Tay, C. J.

W. Chen, C. Quan, and C. J. Tay, “Extended depth of focus in a particle field measurement using a single-shot digital hologram,” Appl. Phys. Lett. 95(20), 201103 (2009).
[CrossRef]

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, “Phase retrieval in X-ray imaging based on using structured illumination,” Phys. Rev. A 78(2), 023817 (2008).
[CrossRef]

Vartanyants, I.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300(5624), 1419–1421 (2003).
[CrossRef] [PubMed]

Vartanyants, I. A.

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442(7098), 63–66 (2006).
[CrossRef] [PubMed]

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, “Phase retrieval in X-ray imaging based on using structured illumination,” Phys. Rev. A 78(2), 023817 (2008).
[CrossRef]

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, “Phase retrieval in X-ray imaging based on using structured illumination,” Phys. Rev. A 78(2), 023817 (2008).
[CrossRef]

Williams, G. J.

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442(7098), 63–66 (2006).
[CrossRef] [PubMed]

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, “Phase retrieval in X-ray imaging based on using structured illumination,” Phys. Rev. A 78(2), 023817 (2008).
[CrossRef]

Yamaguchi, I.

Zhang, R.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300(5624), 1419–1421 (2003).
[CrossRef] [PubMed]

Zhang, T.

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, “Phase retrieval in X-ray imaging based on using structured illumination,” Phys. Rev. A 78(2), 023817 (2008).
[CrossRef]

Zuo, J. M.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300(5624), 1419–1421 (2003).
[CrossRef] [PubMed]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

W. Chen, C. Quan, and C. J. Tay, “Extended depth of focus in a particle field measurement using a single-shot digital hologram,” Appl. Phys. Lett. 95(20), 201103 (2009).
[CrossRef]

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

Nature (3)

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442(7098), 63–66 (2006).
[CrossRef] [PubMed]

D. Gabor, “A new microscopic principle,” Nature 161(4098), 777–778 (1948).
[CrossRef] [PubMed]

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400(6742), 342–344 (1999).
[CrossRef]

Opt. Commun. (1)

M. G. Ertosun, H. Atlı, H. M. Ozaktas, and B. Barshan, “Complex signal recovery from two fractional Fourier transform intensities: order and noise dependence,” Opt. Commun. 244(1-6), 61–70 (2005).
[CrossRef]

Opt. Lett. (1)

Optik (Stuttg.) (1)

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

Phys. Rev. A (1)

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, “Phase retrieval in X-ray imaging based on using structured illumination,” Phys. Rev. A 78(2), 023817 (2008).
[CrossRef]

Science (1)

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300(5624), 1419–1421 (2003).
[CrossRef] [PubMed]

Other (3)

C. J. Mann, L. Yu, C. Lo, and M. K. Kim, “High-resolution quantitative phase-contrast microscopy by digital holography,” Opt. Express 13, 8693–8698 (2005), http://www.opticsinfobase.org/oe/abstract.cfm ? URI = oe-13–22–8693.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

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Figures (5)

Fig. 1
Fig. 1

A schematic experimental setup.

Fig. 2
Fig. 2

A flow chart of the proposed quantitative phase retrieval algorithm.

Fig. 3
Fig. 3

(a) Original object amplitude; (b) original object phase; (c) a typical one of recorded diffraction intensity maps; (d) recovered object amplitude; (e) recovered object phase; and (f) a comparison along the dashed line between Figs. 3(b) and 3(e).

Fig. 4
Fig. 4

Relationships between the iteration number and SSE using (a) 3 and (b) 4 recordings.

Fig. 5
Fig. 5

Relationships between the iteration number and SSE using (a) 3 and (b) 4 recordings with noise.

Equations (9)

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F ( x , y ) = j λ + + C ( ξ , η ) exp ( j     k ρ ) ρ         d ξ     d η ,
F ( x , y ) = F T 1 [ C ( ξ , η )     ¯     T ( f ξ , f η ) ] ,
T ( f ξ , f η ) = exp {     ( j 2 π     z / λ )             [ 1 ( λ     f ξ ) 2 ( λ     f η ) 2 ] 1 / 2 }                         if         ( f ξ 2 + f η 2 ) 1 / 2 < ( 1 / λ )     .
T ( g , h ) = exp ( j     k     z     { 1 [ λ     ( g G / 2 ) / ( G Δ ξ ) ] 2 [ λ     ( h H / 2 ) / ( H Δ η ) ] 2 } 1 / 2 ) ,
FRFT a [ F ( x ) M ( x ) ] = + F ( x ) M ( x )       P a ( μ a , x )     d x ,
I ( μ , ν ) = | F R F T a , b [ F ( x , y ) M ( x , y ) ] | 2 ,
      [     | F n 3 ( x , y ) ¯ | | F n ( x , y ) | ] 2 σ           ( i f       n = 1 ) ;
      [     | F n 3 ( x , y ) ¯ | | F n 1 3 ( x , y ) ¯ | ] 2 σ         ( if       n > 1 ) .
S S E = { [ | C ( ξ , η ) | | C ' ( ξ , η ) | ]     2 } / { [ | C ( ξ , η ) | ] 2 } ,

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