Abstract

It is an effective scheme to the phase retrieval for axial intensity derivative computing. In this paper, we demonstrate a method for estimating the axial intensity derivative and improving the calculation accuracy in the transport of intensity equation (TIE) from multiple intensity measurements. The method takes both the higher-order intensity derivatives and the noise into account, and minimizes the impact of detecting noise. The simulation results demonstrate that the proposed method can effectively reduce the error of intensity derivative computing.

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References

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  1. M. Reed Teague, “Deterministic phase retrieval: a Green's function solution,” J. Opt. Soc. Am. 73(11), 1434–1441 (1983).
    [CrossRef]
  2. D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998).
    [CrossRef]
  3. 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]
  4. 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]
  5. T. E. Gureyev and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13(8), 1670–1682 (1996).
    [CrossRef]
  6. 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. A 12(9), 1932–1942 (1995).
    [CrossRef]
  7. M. Soto and E. Acosta, “Improved phase imaging from intensity measurements in multiple planes,” Appl. Opt. 46(33), 7978–7981 (2007).
    [CrossRef] [PubMed]
  8. L. Waller, L. Tian, and G. Barbastathis, “Transport of intensity imaging with higher order derivatives,” Opt. Express 18(12), 12552–12561 (2010).
    [CrossRef] [PubMed]
  9. B. D. Xue, S. L. Zheng, L. Y. Cui, X. Z. Bai, and F. G. Zhou, “Transport of intensity phase imaging from multiple intensities measured in unequally-spaced planes,” Opt. Express 19(21), 20244–20250 (2011).
    [CrossRef] [PubMed]
  10. S. L. Zheng, B. D. Xue, W. F. Xue, X. Z. Bai, and F. G. Zhou, “Transport of intensity phase imaging from multiple noisy intensities measured in unequally-spaced planes,” Opt. Express 20(2), 972–985 (2012).
    [CrossRef] [PubMed]

2012 (1)

2011 (1)

2010 (1)

2007 (1)

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]

1998 (1)

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

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)

1995 (1)

1983 (1)

Acosta, E.

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]

Bai, X. Z.

Barbastathis, G.

Cui, L. Y.

Gureyev, T. E.

Nugent, K. A.

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

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]

T. E. Gureyev and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13(8), 1670–1682 (1996).
[CrossRef]

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. A 12(9), 1932–1942 (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.

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

Reed Teague, M.

Roberts, A.

Soto, M.

Tian, L.

Waller, L.

Xue, B. D.

Xue, W. F.

Zheng, S. L.

Zhou, F. G.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

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

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 (3)

Phys. Rev. Lett. (1)

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

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Figures (2)

Fig. 1
Fig. 1

Axial intensity derivative error computed by noise suppression (described in this paper) and high order derivative suppression method (proposed by Bindang Xue) for different sampling number m: Each point is the average of 100 results. The blue lines show the results derived by noise suppression method for n = 4. (a)~(d) denotes the results for different sampling intervals Δ z .

Fig. 2
Fig. 2

Axial intensity derivative error computed by noise suppression method for different n, the highest order derivative considered, when sampling number m = 16

Equations (22)

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u ( r ) = [ I ( r ) ] 1 / 2 exp [ i ϕ ( r ) ] ,
k I z = ( I ϕ ) ,
I ( z i ) = I ( 0 ) + I ( 0 ) z z i + k = 2 n z i k k ! k I ( 0 ) z k + z i n + 1 ( n + 1 ) ! n + 1 I ( ξ i ) z n + 1 ,
i = 1 m a i I ( z i ) = I ( 0 ) i = 1 m a i + I ( 0 ) z i = 1 m a i z i + k = 2 n 1 k ! k I ( 0 ) z k i = 1 m a i z i k + 1 ( n + 1 ) ! i = 1 m a i z i n + 1 n + 1 I ( ξ i ) z n + 1 .
i = 1 m a i = 0 , i = 1 m a i z i = 1 , i = 1 m a i z i k = 0 k = 2 , 3... n ,
I ( 0 ) z = i = 1 m a i I ( z i ) 1 ( n + 1 ) ! i = 1 m a i z i n + 1 n + 1 I ( ξ i ) z n + 1 .
[ z I ] c = i = 1 m a i I ( z i ) .
[ z I ] c I ( 0 ) z = i = 1 m a i N ( z i ) + 1 ( n + 1 ) ! i = 1 m a i z i n + 1 n + 1 I ( ξ i ) z n + 1 ,
ε 2 = D ( N ) + D ( ξ ) ,
D ( N ) = σ 2 i = 1 m a i 2 ,
D ( ξ ) = [ 1 ( n + 1 ) ! i = 1 m a i z i n + 1 n + 1 I ( ξ i ) z n + 1 ] 2 ,
[ 1 1 1 1 z 1 z 2 z 3 z m z 1 2 z 2 2 z 3 2 z m 2 z 1 n z 2 n z 3 n z m n ] · [ a 1 a 2 a 3 a m ] = [ 0 1 0 0 ] .
[ A B ] · [ C D ] = E ,
D = B - 1 E B - 1 A C .
D ( N ) = σ 2 [ i = 1 n + 1 a i 2 + j = 1 m n 1 ( f j G j C ) 2 ] .
a k D ( N ) = σ 2 [ 2 a k j = 1 n + 1 2 ( f j G j C ) g j k ] = 0 k = 1 , 2 ( m n 1 ) ,
( G T G + I ) C = G T F .
[ G T G + I 0 A B ] · [ C D ] = [ G T F E ] .
u 00 ( x , y , z ) = c w ( z ) exp ( r 2 w 2 ( z ) ) exp { i [ k ( z + r 2 2 R ) arctan ( z f ) ] } ,
k = 2 π λ r 2 = x 2 + y 2 w ( z ) = w 0 1 + ( z f ) 2 R = R ( z ) = z [ 1 + ( f z ) 2 ] f = π w 0 2 λ w 0 = λ f π .
I ( z ) = c 2 w 2 ( z ) ,
w 0 = 0.005 m , λ = 1550 n m , f = π w 0 2 λ = 50.6708 m , c = 0.01.

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