Abstract

We demonstrate a method for improving the accuracy of phase retrieval based on the Transport of Intensity equation by using intensity measurements at multiple planes to estimate and remove the artifacts due to higher order axial derivatives. We suggest two similar methods of higher order correction, and demonstrate their ability for accurate phase retrieval well beyond the ‘linear’ range of defocus that TIE imaging traditionally requires. Computation is fast and efficient, and sensitivity to noise is reduced by using many images.

© 2010 OSA

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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. A 73(11), 1434–1441 (1983).
    [CrossRef]
  2. N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49(1), 6–10 (1984).
    [CrossRef]
  3. D. Paganin and K. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998).
    [CrossRef]
  4. E. Barone-Nugent, A. Barty, and K. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206(3), 194–203 (2002).
    [CrossRef] [PubMed]
  5. D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
    [CrossRef] [PubMed]
  6. T. Gureyev and K. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133(1-6), 339–346 (1997).
    [CrossRef]
  7. L. Allen and M. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1-4), 65–75 (2001).
    [CrossRef]
  8. G. Strang, Computational Science and Engineering, (Wellesley-Cambridge Press, 2007).
  9. M. Soto, E. Acosta, and S. Ríos, “Performance analysis of curvature sensors: optimum positioning of the measurement planes,” Opt. Express 11(20), 2577–2588 (2003).
    [CrossRef] [PubMed]
  10. T. Gureyev, A. Roberts, and K. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12(9), 1942–1946 (1995).
    [CrossRef]
  11. M. Soto and E. Acosta, “Improved phase imaging from intensity measurements in multiple planes,” Appl. Opt. 46(33), 7978–7981 (2007).
    [CrossRef] [PubMed]
  12. T. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun. 220(1-3), 49–58 (2003).
    [CrossRef]
  13. T. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2 Partially coherent illumination,” Opt. Commun. 259(2), 569–580 (2006).
    [CrossRef]
  14. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21(15), 2758–2769 (1982).
    [CrossRef] [PubMed]
  15. R. Paxman, T. Schulz, and J. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A 9(7), 1072–1085 (1992).
    [CrossRef]
  16. J. Goodman, Introduction to Fourier Optics, (McGraw-Hill, 1996.)
  17. S. S. Kou, L. Waller, G. Barbastathis, and C. J. Sheppard “Transport-of-intensity approach to differential interference contrast (TI-DIC) microscopy for quantitative phase imaging,” Opt. Lett. 35(3), 447-449 (2010).

2010 (1)

S. S. Kou, L. Waller, G. Barbastathis, and C. J. Sheppard “Transport-of-intensity approach to differential interference contrast (TI-DIC) microscopy for quantitative phase imaging,” Opt. Lett. 35(3), 447-449 (2010).

2007 (1)

2006 (1)

T. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2 Partially coherent illumination,” Opt. Commun. 259(2), 569–580 (2006).
[CrossRef]

2004 (1)

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[CrossRef] [PubMed]

2003 (2)

2002 (1)

E. Barone-Nugent, A. Barty, and K. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206(3), 194–203 (2002).
[CrossRef] [PubMed]

2001 (1)

L. Allen and M. 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. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998).
[CrossRef]

1997 (1)

T. Gureyev and K. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133(1-6), 339–346 (1997).
[CrossRef]

1995 (1)

1992 (1)

1984 (1)

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49(1), 6–10 (1984).
[CrossRef]

1983 (1)

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

1982 (1)

Acosta, E.

Allen, L.

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

Barbastathis, G..

S. S. Kou, L. Waller, G. Barbastathis, and C. J. Sheppard “Transport-of-intensity approach to differential interference contrast (TI-DIC) microscopy for quantitative phase imaging,” Opt. Lett. 35(3), 447-449 (2010).

Barone-Nugent, E.

E. Barone-Nugent, A. Barty, and K. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206(3), 194–203 (2002).
[CrossRef] [PubMed]

Barty, A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[CrossRef] [PubMed]

E. Barone-Nugent, A. Barty, and K. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206(3), 194–203 (2002).
[CrossRef] [PubMed]

Fienup, J.

Fienup, J. R.

Gureyev, T.

T. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2 Partially coherent illumination,” Opt. Commun. 259(2), 569–580 (2006).
[CrossRef]

T. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun. 220(1-3), 49–58 (2003).
[CrossRef]

T. Gureyev and K. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133(1-6), 339–346 (1997).
[CrossRef]

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

Kou, S. S.

S. S. Kou, L. Waller, G. Barbastathis, and C. J. Sheppard “Transport-of-intensity approach to differential interference contrast (TI-DIC) microscopy for quantitative phase imaging,” Opt. Lett. 35(3), 447-449 (2010).

McMahon, P. J.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[CrossRef] [PubMed]

Nesterets, Y.

T. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2 Partially coherent illumination,” Opt. Commun. 259(2), 569–580 (2006).
[CrossRef]

Nugent, K.

E. Barone-Nugent, A. Barty, and K. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206(3), 194–203 (2002).
[CrossRef] [PubMed]

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

T. Gureyev and K. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133(1-6), 339–346 (1997).
[CrossRef]

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

Nugent, K. A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[CrossRef] [PubMed]

Oxley, M.

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

Paganin, D.

T. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2 Partially coherent illumination,” Opt. Commun. 259(2), 569–580 (2006).
[CrossRef]

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[CrossRef] [PubMed]

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

Paxman, R.

Pogany, A.

T. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2 Partially coherent illumination,” Opt. Commun. 259(2), 569–580 (2006).
[CrossRef]

Ríos, S.

Roberts, A.

Schulz, T.

Sheppard, C. J.

S. S. Kou, L. Waller, G. Barbastathis, and C. J. Sheppard “Transport-of-intensity approach to differential interference contrast (TI-DIC) microscopy for quantitative phase imaging,” Opt. Lett. 35(3), 447-449 (2010).

Soto, M.

Streibl, N.

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49(1), 6–10 (1984).
[CrossRef]

Teague, M. R.

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

Waller, L..

S. S. Kou, L. Waller, G. Barbastathis, and C. J. Sheppard “Transport-of-intensity approach to differential interference contrast (TI-DIC) microscopy for quantitative phase imaging,” Opt. Lett. 35(3), 447-449 (2010).

Wilkins, S.

T. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2 Partially coherent illumination,” Opt. Commun. 259(2), 569–580 (2006).
[CrossRef]

Appl. Opt. (2)

J. Microsc. (2)

E. Barone-Nugent, A. Barty, and K. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206(3), 194–203 (2002).
[CrossRef] [PubMed]

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (3)

Opt. Commun. (5)

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49(1), 6–10 (1984).
[CrossRef]

T. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun. 220(1-3), 49–58 (2003).
[CrossRef]

T. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2 Partially coherent illumination,” Opt. Commun. 259(2), 569–580 (2006).
[CrossRef]

T. Gureyev and K. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133(1-6), 339–346 (1997).
[CrossRef]

L. Allen and M. 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)

S. S. Kou, L. Waller, G. Barbastathis, and C. J. Sheppard “Transport-of-intensity approach to differential interference contrast (TI-DIC) microscopy for quantitative phase imaging,” Opt. Lett. 35(3), 447-449 (2010).

Phys. Rev. Lett. (1)

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

Other (2)

J. Goodman, Introduction to Fourier Optics, (McGraw-Hill, 1996.)

G. Strang, Computational Science and Engineering, (Wellesley-Cambridge Press, 2007).

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Figures (8)

Fig. 1
Fig. 1

Error mechanisms in TIE imaging. (a) Noise corrupted phase reconstruction at small Δz, (b) actual phase, (c) Root mean squared (RMS) error in phase reconstruction for increasing Δz. 1% Poisson noise was added to each intensity image. Error bars denote standard deviation over 100 trials and FFT and Multigrid refer to the method of Poisson solver. (d) Nonlinearity corrupted phase reconstruction at large Δz, and (e) phase reconstruction at optimal Δz.

Fig. 2
Fig. 2

Higher order derivative components of the axial intensity for a propagated test image with. 0th order is related to the amplitude of the wave-field and 1st order specifies phase.

Fig. 3
Fig. 3

Error simulation for increasing Δz with a) no noise, b) noise (standard deviation σ = 0.001) and no averaging, c) noise with averaging. Error bars denote σ. Note that with no noise, or with averaging, the higher order solutions are always better than lower order ones.

Fig. 4
Fig. 4

(a) Simulated intensity focal stack with Δz = 10µm, size 90µm and pure phase object (max phase 0.36 radians at focus). (b) Axial intensity profile for a few randomly selected pixels. (c) Single pixel intensity profile with corresponding fits to 1st, 7th and 13th orders, having respectively, 0.0657,0.0296, and 0.0042 RMS fit error.

Fig. 5
Fig. 5

(Left) Simulation of phase recovery improvement by fitting to higher order polynomials. (Top) Recovered phase (radians). (Bottom) Error maps for corresponding phase retrieved (radians). (Right) RMS error and RMS curve fit error for a similar data set with noisy images, as fitting order is increased.

Fig. 6
Fig. 6

Phase object reconstructions from simulated noisy intensity images (radians). (a) Actual phase, (b) technique 2, 20th order TIE, (c) technique 1, 6th order TIE, (d) traditional TIE, (e) method given in [11], (f) Phase diversity (500 iterations).

Fig. 7
Fig. 7

Experimental reconstruction of a test phase object height from multiple intensity images, using different reconstruction algorithms. (a) Intensity image stack. (b) Traditional TIE reconstruction, (c) method given in [11], (d) phase diversity (500 iterations), and (e) 20th order TIE using technique 2 (nm).

Fig. 8
Fig. 8

Experimental test phase object in a brightfield microscope. (a) Amplitude reconstruction and phase height reconstructions (nm) with TIE of (b) 1st order, (c) 3rd order and (d) 5th order.

Tables (1)

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Table 1 Finding image weights for a given desired order of accuracy.

Equations (7)

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I(x,y) z = λ 2π ·( I(x,y) φ(x,y) ),
2π λI I(x,y) z = 2 φ(x,y),
I(x,y) z I(x,y,Δz)I(x,y,0) Δz ,
I(Δz)=I(0)+ Δz 1! I z + (Δz) 2 2! 2 I z 2 + (Δz) 3 3! 3 I z 3 + (Δz) 4 4! 4 I z 4 +...
I z = I(Δz)I(0) Δz [ Δz 2! 2 I z 2 + (Δz) 2 3! 3 I z 3 + (Δz) 3 4! 4 I z 4 +... ].
I z a m I m + a (m+1) I (m+1) +... a j I j +...+ a n1 I n1 + a n I n Δz ,
[ (m) 0 (m+1) 0 (n1) 0 n 0 (m) 1 (m+1) 1 (n1) 1 n 1 (m) 2 (m+1) 2 (n1) 2 n 2 (m) m+n (m+1) m+n n m+n ]( a m a m+1 a m+2 a n )=( 0 1 0 0 ),

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