Abstract

Problems stemming from quantitative phase imaging from intensity measurements play a key role in many fields of physics. Techniques based on the transport of intensity equation require an estimate of the axial derivative of the intensity to invert the problem. Derivation formulas in two adjacent planes are commonly used to experimentally compute the derivative of the irradiance. Here we propose a formula that improves the estimate of the derivative by using a higher number of planes and taking the noisy nature of the measurements into account. We also establish an upper and lower limit for the estimate error and provide the distance between planes that optimizes the estimate of the derivative.

© 2007 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  3. V. V. Volkov and Y. Zhu, "Phase imaging and nanoscale currents in phase objects imaged with fast electrons," Phys. Rev. Lett. 91, 043904 (2003).
    [CrossRef] [PubMed]
  4. K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, "Quantitative phase imaging using hard x rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
    [CrossRef] [PubMed]
  5. B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Warner, "Phase radiography with neutrons," Nature 408, 158-159 (2000).
    [CrossRef] [PubMed]
  6. K. F. Riley, M. P. Hobson, and S. J. Bence, Mathematical Methods for Physics and Engineering (Cambridge U. Press, 1997).
  7. D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, "Quantitative phase-amplitude microscopy. III: The effect of noise," J. Micros. 241, 51-61 (2004).
    [CrossRef]
  8. M. Soto, E. Acosta, and S. Ríos, "Performance analysis of curvature sensors: optimum positioning of the measurement planes," Opt. Express 11, 2577-2588 (2003).
    [CrossRef] [PubMed]
  9. J. H. Mathews and K. D. Fink, Numerical Methods Using Matlab, 4th ed. (Prentice Hall, 2004).
  10. D. Van Dick and W. Coene, "A new procedure for wave function restoration in high resolution electron microscopy," Optik (Stuttgart) 77, 125-128 (1987).
  11. M. Soto, "Curvature sensors: performance optimization," Ph.D. dissertation (Universidad de Santiago de Compostela, 2007).
  12. M. Bellegia, M. A. Shofield, V. V. Volkov, and Y. Zhu, "On the transport of intensity equation for phase retrieval," Ultramicroscopy 102, 37-39 (2004).
    [CrossRef]

2004 (2)

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, "Quantitative phase-amplitude microscopy. III: The effect of noise," J. Micros. 241, 51-61 (2004).
[CrossRef]

M. Bellegia, M. A. Shofield, V. V. Volkov, and Y. Zhu, "On the transport of intensity equation for phase retrieval," Ultramicroscopy 102, 37-39 (2004).
[CrossRef]

2003 (2)

M. Soto, E. Acosta, and S. Ríos, "Performance analysis of curvature sensors: optimum positioning of the measurement planes," Opt. Express 11, 2577-2588 (2003).
[CrossRef] [PubMed]

V. V. Volkov and Y. Zhu, "Phase imaging and nanoscale currents in phase objects imaged with fast electrons," Phys. Rev. Lett. 91, 043904 (2003).
[CrossRef] [PubMed]

2000 (1)

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Warner, "Phase radiography with neutrons," Nature 408, 158-159 (2000).
[CrossRef] [PubMed]

1996 (1)

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, "Quantitative phase imaging using hard x rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

1993 (1)

1987 (1)

D. Van Dick and W. Coene, "A new procedure for wave function restoration in high resolution electron microscopy," Optik (Stuttgart) 77, 125-128 (1987).

1983 (1)

J. Micros. (1)

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, "Quantitative phase-amplitude microscopy. III: The effect of noise," J. Micros. 241, 51-61 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Nature (1)

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Warner, "Phase radiography with neutrons," Nature 408, 158-159 (2000).
[CrossRef] [PubMed]

Opt. Express (1)

Optik (1)

D. Van Dick and W. Coene, "A new procedure for wave function restoration in high resolution electron microscopy," Optik (Stuttgart) 77, 125-128 (1987).

Phys. Rev. Lett. (2)

V. V. Volkov and Y. Zhu, "Phase imaging and nanoscale currents in phase objects imaged with fast electrons," Phys. Rev. Lett. 91, 043904 (2003).
[CrossRef] [PubMed]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, "Quantitative phase imaging using hard x rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

Ultramicroscopy (1)

M. Bellegia, M. A. Shofield, V. V. Volkov, and Y. Zhu, "On the transport of intensity equation for phase retrieval," Ultramicroscopy 102, 37-39 (2004).
[CrossRef]

Other (3)

M. Soto, "Curvature sensors: performance optimization," Ph.D. dissertation (Universidad de Santiago de Compostela, 2007).

J. H. Mathews and K. D. Fink, Numerical Methods Using Matlab, 4th ed. (Prentice Hall, 2004).

K. F. Riley, M. P. Hobson, and S. J. Bence, Mathematical Methods for Physics and Engineering (Cambridge U. Press, 1997).

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Equations (30)

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I ( x , y , z = 0 ) z = I ( x , y , Δ z ) I ( x , y , Δ z ) 2 Δ z ,
I ( x , y , z = 0 ) z = I ( x , y , Δ z ) I ( x , y , 0 ) Δ z ,
I ( x , y , z = 0 ) z = I ( x , y , Δ z ) I ( x , y , 0 ) Δ z ,
I ( x , y , z j ) = I ( x , y , z j ) + n j ,
with   z j = ± j Δ z , j = 1 , 2 ,   …   ,   N ,
I ^ 0 = 1 Δ z j = N N a j I j   for   a   central   difference   formula ,
I ^ 0 = 1 Δ z j = 1 N a j ( I j I 0 )   for   a   forward   difference   formula ,
I ^ 0 = 1 Δ z j = 1 N a j ( I j I 0 )   for   a   backward   difference   formula ,
lim Δ z 0 I ^ 0 = I ( x , y , z = 0 ) z ,
ε 2 = [ I ^ 0 I ( x , y , z = 0 ) z ] 2   be   minimum ,
I ^ 0 = 1 Δ z j = N N a j n j + 1 Δ z j = N N a j I 0 + j = N N a j j I 0 + Δ z 2 ! j = N N a j j 2 2 I ( x , y , z = c j ( x , y ) ) z 2 .
j = N N a j = 0 , j = N N a j j = 1 .
ε 2 ( x , y ) = σ 2 Δ z 2 j = N N a j 2 + Δ z 2 4 [ j = N N a j j 2 2 I ( x , y , z = c j ( x , y ) ) z 2 ] 2 = minimum ,
j = N N a j 2 = minimum .
a j = 3 j N ( N + 1 ) ( 2 N + 1 ) .
| 2 I ( x , y , z ) z 2 | U ,
ε u p 2 ( Δ z ) = 3 N ( N + 1 ) ( 2 N + 1 ) σ 2 Δ z 2 + 9 N 2 ( N + 1 ) 2 16 ( 2 N + 1 ) 2 Δ z 2 U 2 ,
ε l o 2 ( Δ z ) = 3 N ( N + 1 ) ( 2 N + 1 ) σ 2 Δ z 2 ,
Δ z o p t 2 = 4 σ U [ 2 N + 1 3 N 3 ( N + 1 ) 3 ] 1 / 2 ,
ε o p t ( u p ) 2 = 3 2 σ U [ 3 N ( N + 1 ) ( 2 N + 1 ) 3 ] 1 / 2 ,
ε o p t ( l o ) 2 = 1 2 ε o p t ( u p ) 2 .
a j = 6 j N ( N + 1 ) ( 2 N + 1 ) .
ε u p 2 ( Δ z ) = 12 N ( N + 1 ) ( 2 N + 1 ) σ 2 Δ z 2 + 9 N 2 ( N + 1 ) 2 16 ( 2 N + 1 ) 2 Δ z 2 U 2 ,
ε l o 2 ( Δ z ) = 12 N ( N + 1 ) ( 2 N + 1 ) σ 2 Δ z 2 .
Δ z o p t 2 = 8 σ U [ 2 N + 1 3 N 3 ( N + 1 ) 3 ] 1 / 2 ,
ε o p t ( u p ) 2 = 3 σ U [ 3 N ( N + 1 ) ( 2 N + 1 ) 3 ] 1 / 2 ,
ε o p t ( l o ) 2 = 1 2 ε o p t ( u p ) 2 .
φ ( x ) = A π   sin ( 2 π L x ) ,
T ( k ) = exp { i π λ z k 2 2 ( π θ 0 z ) 2 k 2 } ,
I ( x , z ) = 1 + 2 A π   sin ( 2 π x L )   sin ( π λ z L 2 ) × exp { 2 ( π θ 0 z ) 2 L 2 } .

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