Abstract

A method based on the transport of intensity equation (TIE) for phase retrieval is presented, which can retrieve the optical phase from intensity measurements in multiple unequally-spaced planes in the near-field region. In this method, the intensity derivative in the TIE is represented by a linear combination of intensity measurements, and the coefficient of the combination can be expressed by explicitly analytical form related to the defocused distances. The proposed formula is a generalization of the TIE with high order intensity derivatives. The numerical experiments demonstrate that the proposed method can improve the accuracy of phase retrieval with higher-order intensity derivatives and is more convenient for practical application.

©2011 Optical Society of America

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References

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  1. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73(11), 1434–1441 (1983).
    [Crossref]
  2. T. E. Gureyev, A. Roberts, and K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12(9), 1942–1946 (1995).
    [Crossref]
  3. A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23(11), 817–819 (1998).
    [Crossref] [PubMed]
  4. K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. (Tokyo) 54(3), 191–197 (2005).
    [Crossref] [PubMed]
  5. B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408(6809), 158–159 (2000).
    [Crossref] [PubMed]
  6. K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
    [Crossref] [PubMed]
  7. 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]
  8. D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998).
    [Crossref]
  9. L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1-4), 65–75 (2001).
    [Crossref]
  10. W. Xiao, M. Heng, and Z. Dazun, “Phase retrieval based on intensity transport equation,” Acta Opt. Sin. 27, 2117–2122 (2007).
  11. T. E. Gureyev, A. Roberts, and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials,” J. Opt. Soc. Am. 12(9), 1932–1941 (1995).
    [Crossref]
  12. S. V. Pinhasi, R. Alimi, L. Perelmutter, and S. Eliezer, “Topography retrieval using different solutions of the transport intensity equation,” J. Opt. Soc. Am. A 27(10), 2285–2292 (2010).
    [Crossref] [PubMed]
  13. J. Frank, G. Wernicke, J. Matrisch, S. Wette, J. Beneke, and S. Altmeyer, “Quantitative determination of the optical properties of phase objects by using a real-time phase retrieval technique,” Proc. SPIE 8082, 80820N, 80820N-9 (2011).
    [Crossref]
  14. M. Soto and E. Acosta, “Improved phase imaging from intensity measurements in multiple planes,” Appl. Opt. 46(33), 7978–7981 (2007).
    [Crossref] [PubMed]
  15. 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] [PubMed]
  16. W. X. Cong and G. Wang, “Higher-order phase shift reconstruction approach,” Med. Phys. 37(10), 5238–5242 (2010).
    [Crossref] [PubMed]
  17. A. E. Knuth, The Art of Computer Programming: Volume 1: Fundamental Algorithms (Addison-Wesley Professional, 1997), pp. 38.
  18. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996), pp. 55–61.

2011 (1)

J. Frank, G. Wernicke, J. Matrisch, S. Wette, J. Beneke, and S. Altmeyer, “Quantitative determination of the optical properties of phase objects by using a real-time phase retrieval technique,” Proc. SPIE 8082, 80820N, 80820N-9 (2011).
[Crossref]

2010 (3)

2007 (2)

M. Soto and E. Acosta, “Improved phase imaging from intensity measurements in multiple planes,” Appl. Opt. 46(33), 7978–7981 (2007).
[Crossref] [PubMed]

W. Xiao, M. Heng, and Z. Dazun, “Phase retrieval based on intensity transport equation,” Acta Opt. Sin. 27, 2117–2122 (2007).

2005 (1)

K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. (Tokyo) 54(3), 191–197 (2005).
[Crossref] [PubMed]

2001 (1)

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1-4), 65–75 (2001).
[Crossref]

2000 (1)

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408(6809), 158–159 (2000).
[Crossref] [PubMed]

1998 (2)

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998).
[Crossref]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23(11), 817–819 (1998).
[Crossref] [PubMed]

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)

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[Crossref] [PubMed]

1995 (2)

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials,” J. Opt. Soc. Am. 12(9), 1932–1941 (1995).
[Crossref]

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12(9), 1942–1946 (1995).
[Crossref]

1983 (1)

Acosta, E.

Alimi, R.

Allen, L. J.

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1-4), 65–75 (2001).
[Crossref]

Allman, B.

K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. (Tokyo) 54(3), 191–197 (2005).
[Crossref] [PubMed]

Allman, B. E.

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408(6809), 158–159 (2000).
[Crossref] [PubMed]

Altmeyer, S.

J. Frank, G. Wernicke, J. Matrisch, S. Wette, J. Beneke, and S. Altmeyer, “Quantitative determination of the optical properties of phase objects by using a real-time phase retrieval technique,” Proc. SPIE 8082, 80820N, 80820N-9 (2011).
[Crossref]

Arif, M.

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408(6809), 158–159 (2000).
[Crossref] [PubMed]

Barbastathis, G.

Barnea, Z.

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[Crossref] [PubMed]

Barty, A.

Beneke, J.

J. Frank, G. Wernicke, J. Matrisch, S. Wette, J. Beneke, and S. Altmeyer, “Quantitative determination of the optical properties of phase objects by using a real-time phase retrieval technique,” Proc. SPIE 8082, 80820N, 80820N-9 (2011).
[Crossref]

Cong, W. X.

W. X. Cong and G. Wang, “Higher-order phase shift reconstruction approach,” Med. Phys. 37(10), 5238–5242 (2010).
[Crossref] [PubMed]

Cookson, D. J.

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[Crossref] [PubMed]

Dazun, Z.

W. Xiao, M. Heng, and Z. Dazun, “Phase retrieval based on intensity transport equation,” Acta Opt. Sin. 27, 2117–2122 (2007).

Eliezer, S.

Frank, J.

J. Frank, G. Wernicke, J. Matrisch, S. Wette, J. Beneke, and S. Altmeyer, “Quantitative determination of the optical properties of phase objects by using a real-time phase retrieval technique,” Proc. SPIE 8082, 80820N, 80820N-9 (2011).
[Crossref]

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]

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[Crossref] [PubMed]

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials,” J. Opt. Soc. Am. 12(9), 1932–1941 (1995).
[Crossref]

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12(9), 1942–1946 (1995).
[Crossref]

Heng, M.

W. Xiao, M. Heng, and Z. Dazun, “Phase retrieval based on intensity transport equation,” Acta Opt. Sin. 27, 2117–2122 (2007).

Ishizuka, K.

K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. (Tokyo) 54(3), 191–197 (2005).
[Crossref] [PubMed]

Jacobson, D. L.

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408(6809), 158–159 (2000).
[Crossref] [PubMed]

Matrisch, J.

J. Frank, G. Wernicke, J. Matrisch, S. Wette, J. Beneke, and S. Altmeyer, “Quantitative determination of the optical properties of phase objects by using a real-time phase retrieval technique,” Proc. SPIE 8082, 80820N, 80820N-9 (2011).
[Crossref]

McMahon, P. J.

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408(6809), 158–159 (2000).
[Crossref] [PubMed]

Nugent, K. A.

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408(6809), 158–159 (2000).
[Crossref] [PubMed]

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998).
[Crossref]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23(11), 817–819 (1998).
[Crossref] [PubMed]

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]

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[Crossref] [PubMed]

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials,” J. Opt. Soc. Am. 12(9), 1932–1941 (1995).
[Crossref]

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12(9), 1942–1946 (1995).
[Crossref]

Oxley, M. P.

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1-4), 65–75 (2001).
[Crossref]

Paganin, D.

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408(6809), 158–159 (2000).
[Crossref] [PubMed]

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998).
[Crossref]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23(11), 817–819 (1998).
[Crossref] [PubMed]

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[Crossref] [PubMed]

Perelmutter, L.

Pinhasi, S. V.

Roberts, A.

Soto, M.

Teague, M. R.

Tian, L.

Waller, L.

Wang, G.

W. X. Cong and G. Wang, “Higher-order phase shift reconstruction approach,” Med. Phys. 37(10), 5238–5242 (2010).
[Crossref] [PubMed]

Werner, S. A.

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408(6809), 158–159 (2000).
[Crossref] [PubMed]

Wernicke, G.

J. Frank, G. Wernicke, J. Matrisch, S. Wette, J. Beneke, and S. Altmeyer, “Quantitative determination of the optical properties of phase objects by using a real-time phase retrieval technique,” Proc. SPIE 8082, 80820N, 80820N-9 (2011).
[Crossref]

Wette, S.

J. Frank, G. Wernicke, J. Matrisch, S. Wette, J. Beneke, and S. Altmeyer, “Quantitative determination of the optical properties of phase objects by using a real-time phase retrieval technique,” Proc. SPIE 8082, 80820N, 80820N-9 (2011).
[Crossref]

Xiao, W.

W. Xiao, M. Heng, and Z. Dazun, “Phase retrieval based on intensity transport equation,” Acta Opt. Sin. 27, 2117–2122 (2007).

Acta Opt. Sin. (1)

W. Xiao, M. Heng, and Z. Dazun, “Phase retrieval based on intensity transport equation,” Acta Opt. Sin. 27, 2117–2122 (2007).

Appl. Opt. (1)

J. Electron Microsc. (Tokyo) (1)

K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. (Tokyo) 54(3), 191–197 (2005).
[Crossref] [PubMed]

J. Opt. Soc. Am. (2)

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials,” J. Opt. Soc. Am. 12(9), 1932–1941 (1995).
[Crossref]

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

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

Med. Phys. (1)

W. X. Cong and G. Wang, “Higher-order phase shift reconstruction approach,” Med. Phys. 37(10), 5238–5242 (2010).
[Crossref] [PubMed]

Nature (1)

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408(6809), 158–159 (2000).
[Crossref] [PubMed]

Opt. Commun. (2)

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]

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1-4), 65–75 (2001).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. Lett. (2)

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998).
[Crossref]

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[Crossref] [PubMed]

Proc. SPIE (1)

J. Frank, G. Wernicke, J. Matrisch, S. Wette, J. Beneke, and S. Altmeyer, “Quantitative determination of the optical properties of phase objects by using a real-time phase retrieval technique,” Proc. SPIE 8082, 80820N, 80820N-9 (2011).
[Crossref]

Other (2)

A. E. Knuth, The Art of Computer Programming: Volume 1: Fundamental Algorithms (Addison-Wesley Professional, 1997), pp. 38.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996), pp. 55–61.

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Figures (4)

Fig. 1
Fig. 1 Simulated phase in the focal plane.

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Fig. 2
Fig. 2 An example of the simulated intensity focal stack. The defocused distances for the intensities are respectively (from left to right): 0,4.02μm,8.03μm,12.00μm,16.00μm,20.03μm,24.07μm .

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Fig. 3
Fig. 3 Recovered phases (radians) and phase errors (radians) for case 2: The first row are listed the recovered phases for M=1,2,6 . The second row are listed the corresponding phase errors relative to the true phase.

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Fig. 4
Fig. 4 RMSEs as a function of the number of intensity measurements with a logarithmic scale. (a) case 1 and 2; (b) case 2 and case 3.

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Tables (1)

Tables Icon

Table 1 Comparison of the weight coefficients for the given order of accuracy.

View Table

Equations (11)

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u z (r)= I z 1/2 (r)exp(i ϕ z (r)),
∇·[ I z (r)∇ ϕ z (r) ]=−k ∂ I z (r) / ∂z ,
I z (r)− I 0 (r)=z∂ I 0 (r)+ ∑ n=2 ∞ z n ∂ n I 0 (r) / n!
[ z 1 z 1 2 ⋯ z 1 M z 2 z 2 2 ⋯ z 2 M ⋮ ⋮ ⋯ ⋮ z M z M 2 ⋯ z M M ][ ∂ I 0 (r) ∂ 2 I 0 (r) / 2! ⋮ ∂ M I 0 (r) / M! ]=[ I z 1 (r)− I 0 (r) I z 2 (r)− I 0 (r) ⋮ I z M (r)− I 0 (r) ].
B= [ b ] M , b ij = ∑ 1≤ k 1 <...< k M−i ≤M k 1 ,..., k M−i ≠j (−1) i−1 z k 1 ... z k M−i / [ z j ∏ 1≤k≤M k≠j ( z k − z j ) ] .
[ ∂ I 0 (r) ∂ 2 I 0 (r) / 2! ⋮ ∂ M I 0 (r) / M! ]=B[ I z 1 (r)− I 0 (r) I z 2 (r)− I 0 (r) ⋮ I z M (r)− I 0 (r) ].
∂ I 0 (r)= ∑ m=1 M c m [ I z m (r)− I 0 (r) ] .
c m = b 1j = ∑ 1≤ k 1 <...< k M−1 ≤M k 1 ,..., k M−1 ≠m z k 1 ... z k M−1 / [ z m ∏ 1≤k≤M k≠m ( z k − z m ) ] ,m=j.
∇·[ I 0 (r)∇ ϕ 0 (r) ]=−k ∑ m=1 M c m [ I z m (r)− I 0 (r) ] .
RMSE= ∑ x,y ( θ 0 ′ (x,y)− θ 0 (x,y)−a) 2 /(X×Y) a= ∑ x,y ( θ 0 ′ (x,y)− θ 0 (x,y)) /(X×Y),
I z m = u z m (r) u z m * (r), u z m (r)= F −1 { F{ u 0 (r) } H z m (g) }, H z m (g)=exp( ik z m (1− λ 2 g 2 ) 1/2 ).

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