Abstract

For several years, scientific, industrial, and biological fields have benefited from knowledge of phase information, which allows for the revealing of hidden features of various objects. An alternative to interferometry is single-beam phase retrieval techniques that are based on the transport of intensity equation, which describes the relation between the axial derivative of the intensity and the phase distribution for a given plane in the Fresnel region. The estimation of the axial intensity derivative is obtained from a series of intensity measurements, where the accuracy is subject to an optimum separation between the measurement planes depending on the number of planes, the level of noise, and the actual object phase distribution. In this Letter, a quantitative analysis of the error in estimated axial derivative is carried out and a model is reported that describes the interdependence between these parameters. The results of this work allow for estimation of the optimum separation between measurement planes with minimal error in the axial derivative.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. P. Allen and L. J. Oxley, Opt. Commun. 199, 65 (2001).
    [CrossRef]
  2. G. Pedrini, W. Osten, and Y. Zhang, Opt. Lett. 30, 833 (2005).
    [CrossRef]
  3. K. Falaggis, T. Kozacki, and M. Kujawinska, Opt. Lett. 38, 1660 (2013).
    [CrossRef]
  4. E. J. Candes, T. Strohmer, and V. Voroninski, Commun. Pure Appl. Math. 66, 1241 (2013).
    [CrossRef]
  5. M. R. Teague, J. Opt. Soc. Am. 73, 1434 (1983).
    [CrossRef]
  6. T. E. Gureyev, A. Roberts, and K. A. Nugent, J. Opt. Soc. Am. A 12, 1942 (1995).
    [CrossRef]
  7. J. C. Petruccelli, L. Tian, and G. Barbastathis, Opt. Express 21, 14430 (2013).
    [CrossRef]
  8. D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, J. Microsc. 214, 51 (2004).
    [CrossRef]
  9. T. Gureyev and S. Wilkins, Opt. Commun. 147, 229 (1998).
    [CrossRef]
  10. S. Bajt, A. Barty, K. Nugent, M. McCartney, M. Wall, and D. Paganin, Ultramicroscopy 83, 67 (2000).
    [CrossRef]
  11. M. Soto and E. Acosta, Appl. Opt. 46, 7978 (2007).
    [CrossRef]
  12. C. Zuo, Q. Chen, Y. Yu, and A. Asundi, Opt. Express 21, 5346 (2013).
    [CrossRef]
  13. L. Waller, L. Tian, and G. Barbastathis, Opt. Express 18, 12552 (2010).
    [CrossRef]
  14. L. Tian, J. C. Petruccelli, and G. Barbastathis, Opt. Lett. 37, 4131 (2012).
    [CrossRef]
  15. L. Tian, J. C. Petruccelli, Q. Miao, H. Kudrolli, V. Nagarkar, and G. Barbastathis, Opt. Lett. 38, 3418 (2013).
    [CrossRef]
  16. R. Bie, X.-H. Yuan, M. Zhao, and L. Zhang, Opt. Express 20, 8186 (2012).
    [CrossRef]
  17. S. Huang, F. Xi, C. Liu, and Z. Jiang, J. Mod. Opt. 59, 35 (2012).
    [CrossRef]
  18. C. Zuo, Q. Chen, W. Qu, and A. Asundi, Opt. Lett. 38, 3538 (2013).
    [CrossRef]

2013 (6)

2012 (3)

2010 (1)

2007 (1)

2005 (1)

2004 (1)

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, J. Microsc. 214, 51 (2004).
[CrossRef]

2001 (1)

M. P. Allen and L. J. Oxley, Opt. Commun. 199, 65 (2001).
[CrossRef]

2000 (1)

S. Bajt, A. Barty, K. Nugent, M. McCartney, M. Wall, and D. Paganin, Ultramicroscopy 83, 67 (2000).
[CrossRef]

1998 (1)

T. Gureyev and S. Wilkins, Opt. Commun. 147, 229 (1998).
[CrossRef]

1995 (1)

1983 (1)

Acosta, E.

Allen, M. P.

M. P. Allen and L. J. Oxley, Opt. Commun. 199, 65 (2001).
[CrossRef]

Asundi, A.

Bajt, S.

S. Bajt, A. Barty, K. Nugent, M. McCartney, M. Wall, and D. Paganin, Ultramicroscopy 83, 67 (2000).
[CrossRef]

Barbastathis, G.

Barty, A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, J. Microsc. 214, 51 (2004).
[CrossRef]

S. Bajt, A. Barty, K. Nugent, M. McCartney, M. Wall, and D. Paganin, Ultramicroscopy 83, 67 (2000).
[CrossRef]

Bie, R.

Candes, E. J.

E. J. Candes, T. Strohmer, and V. Voroninski, Commun. Pure Appl. Math. 66, 1241 (2013).
[CrossRef]

Chen, Q.

Falaggis, K.

Gureyev, T.

T. Gureyev and S. Wilkins, Opt. Commun. 147, 229 (1998).
[CrossRef]

Gureyev, T. E.

Huang, S.

S. Huang, F. Xi, C. Liu, and Z. Jiang, J. Mod. Opt. 59, 35 (2012).
[CrossRef]

Jiang, Z.

S. Huang, F. Xi, C. Liu, and Z. Jiang, J. Mod. Opt. 59, 35 (2012).
[CrossRef]

Kozacki, T.

Kudrolli, H.

Kujawinska, M.

Liu, C.

S. Huang, F. Xi, C. Liu, and Z. Jiang, J. Mod. Opt. 59, 35 (2012).
[CrossRef]

McCartney, M.

S. Bajt, A. Barty, K. Nugent, M. McCartney, M. Wall, and D. Paganin, Ultramicroscopy 83, 67 (2000).
[CrossRef]

McMahon, P. J.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, J. Microsc. 214, 51 (2004).
[CrossRef]

Miao, Q.

Nagarkar, V.

Nugent, K.

S. Bajt, A. Barty, K. Nugent, M. McCartney, M. Wall, and D. Paganin, Ultramicroscopy 83, 67 (2000).
[CrossRef]

Nugent, K. A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, J. Microsc. 214, 51 (2004).
[CrossRef]

T. E. Gureyev, A. Roberts, and K. A. Nugent, J. Opt. Soc. Am. A 12, 1942 (1995).
[CrossRef]

Osten, W.

Oxley, L. J.

M. P. Allen and L. J. Oxley, Opt. Commun. 199, 65 (2001).
[CrossRef]

Paganin, D.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, J. Microsc. 214, 51 (2004).
[CrossRef]

S. Bajt, A. Barty, K. Nugent, M. McCartney, M. Wall, and D. Paganin, Ultramicroscopy 83, 67 (2000).
[CrossRef]

Pedrini, G.

Petruccelli, J. C.

Qu, W.

Roberts, A.

Soto, M.

Strohmer, T.

E. J. Candes, T. Strohmer, and V. Voroninski, Commun. Pure Appl. Math. 66, 1241 (2013).
[CrossRef]

Teague, M. R.

Tian, L.

Voroninski, V.

E. J. Candes, T. Strohmer, and V. Voroninski, Commun. Pure Appl. Math. 66, 1241 (2013).
[CrossRef]

Wall, M.

S. Bajt, A. Barty, K. Nugent, M. McCartney, M. Wall, and D. Paganin, Ultramicroscopy 83, 67 (2000).
[CrossRef]

Waller, L.

Wilkins, S.

T. Gureyev and S. Wilkins, Opt. Commun. 147, 229 (1998).
[CrossRef]

Xi, F.

S. Huang, F. Xi, C. Liu, and Z. Jiang, J. Mod. Opt. 59, 35 (2012).
[CrossRef]

Yu, Y.

Yuan, X.-H.

Zhang, L.

Zhang, Y.

Zhao, M.

Zuo, C.

Appl. Opt. (1)

Commun. Pure Appl. Math. (1)

E. J. Candes, T. Strohmer, and V. Voroninski, Commun. Pure Appl. Math. 66, 1241 (2013).
[CrossRef]

J. Microsc. (1)

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, J. Microsc. 214, 51 (2004).
[CrossRef]

J. Mod. Opt. (1)

S. Huang, F. Xi, C. Liu, and Z. Jiang, J. Mod. Opt. 59, 35 (2012).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (2)

T. Gureyev and S. Wilkins, Opt. Commun. 147, 229 (1998).
[CrossRef]

M. P. Allen and L. J. Oxley, Opt. Commun. 199, 65 (2001).
[CrossRef]

Opt. Express (4)

Opt. Lett. (5)

Ultramicroscopy (1)

S. Bajt, A. Barty, K. Nugent, M. McCartney, M. Wall, and D. Paganin, Ultramicroscopy 83, 67 (2000).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1.
Fig. 1.

Dependence of ε on Δz for various levels of noise using the phase object shown on the top left corner for M=3. The optimum plane separations for Eqs. (3) (circles) and (4) (triangles) are plotted as well. The grating period is 26.7 μm.

Fig. 2.
Fig. 2.

Values of (a) Δzopt and (b) εmin obtained by simulation (circles) and using the calculated values of Eqs. (10) and (11) (solid line). The triangles are the minimum values given in [11]. Each circle corresponds to an average of five simulations. The grating period is 26.7 μm.

Fig. 3.
Fig. 3.

Dependence of ε on Δz for various frequencies for M=7 using a SNR=60dB. The optimum plane separations for each frequency are calculated with Eq. (10) (triangles). The dashed lines indicate the region of validity established by Eq. (8).

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

zI^=1Δzj=NNajI(x,y,z0+jΔz),
ε2=(zI^zI0)2=εN2+εD2,
ΔzoptSoto=4σ|z2I|z=ζ|max[2N+13N3(N+1)3]1/2,
Δzopt=(3σ/z3I|z=0RMS)1/3,
εS2=εN2+εI2.
ε2=σ2Δz2j=NNaj2+Δz4(3!)2(z3I|z0RMSj=NNajj3)2,
ε2=c1σ2Δz2+c2Δz4z3I|z0RMS2,
kσc1I02φRMSΔzI02φRMSkz3I|z0RMSc2.
σ10I032φRMS3N(N+1)(2N+1)k3z3IRMS(3N2+3N1).
Δzopt=((σz3I|z0RMS)2[1350N(N+1)(2N+1)(3N2+3N1)2])1/6.
(εmin)2=(920σ2z3I|z0RMSM27/3M21)2/31M2/3.
z3I|z0=3(2Re(zuz2u*))+2Re(u*z3u),

Metrics