Abstract

We consider the problem of phase retrieval for classical and quantum wave fields that obey a wide class of nonlinear wave equations. This problem is tackled by means of a suitable generalization of existing methods based on the linear transport-of-intensity equation. The extension of these ideas to systems of coupled nonlinear wave equations is also given.

© 2002 Optical Society of America

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References

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  1. D. Paganin and K. A. Nugent, in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, San Diego, Calif., 2001), Vol. 118, pp. 85–127.
    [CrossRef]
  2. I. Bloch, T. W. Hänsch, and T. Esslinger, Phys. Rev. Lett. 82, 3008 (1999).
    [CrossRef]
  3. S. Mancini and P. Tombesi, Europhys. Lett. 40, 351 (1997).
    [CrossRef]
  4. S. Mancini, M. Fortunato, P. Tombesi, and G. M. D’Ariano, J. Opt. Soc. Am. A 17, 2529 (2000).
    [CrossRef]
  5. D. Paganin and K. A. Nugent, Phys. Rev. Lett. 80, 2586 (1998).
    [CrossRef]
  6. K. A. Nugent and D. Paganin, Phys. Rev. A 61, 063614 (2000).
    [CrossRef]
  7. M. R. Teague, J. Opt. Soc. Am. 73, 1434 (1983).
    [CrossRef]
  8. T. E. Gureyev, A. Roberts, and K. A. Nugent, J. Opt. Soc. Am. A 12, 1942 (1995).
    [CrossRef]
  9. Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998).
    [CrossRef]
  10. G. Fibich, B. Ilan, and S. Tsynkov, “Computation of nonlinear backscattering using a high-order numerical method,” document NASA/CR-2001–211036, ICASE rep. 2001–21 (Langely Research Center, Hampton, Va., 2001).
  11. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer-Verlag, Berlin, 1977), Chap. 8.
    [CrossRef]
  12. T. E. Gureyev, A. Roberts, and K. A. Nugent, J. Opt. Soc. Am. A 12, 1932 (1995).
    [CrossRef]
  13. T. E. Gureyev and K. A. Nugent, J. Opt. Soc. Am. A 13, 1670 (1996).
    [CrossRef]
  14. T. E. Gureyev and K. A. Nugent, Opt. Commun. 133, 339 (1997).
    [CrossRef]
  15. T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, J. Phys. D 32, 563 (1999).
    [CrossRef]
  16. L. J. Allen and M. P. Oxley, Opt. Commun. 199, 65 (2001).
    [CrossRef]
  17. E. Cornell, J. R. Ensher, and C. E. Wieman, in Proceedings of the International School of Physics “Enrico Fermi,” M. Inguscio, S. Stringari, and C. E. Wieman, eds. (IOS Press, Amsterdam, 1999), pp. 15–66.
  18. See, e.g., the stationary-state version of Eq. 8 in E. V. Goldstein, K. Plättner, and P. Meystre, J. Res. Natl. Inst. Stand. Technol. (US) 101, 583 (1996), with backpropagating components neglected.
    [CrossRef]
  19. M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, M. J. Holland, J. E. Williams, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett. 83, 3358 (1999).
    [CrossRef]
  20. L. J. Allen, H. M. L. Faulkner, M. P. Oxley, and D. Paganin, Ultramicroscopy 88, 85 (2001).
    [CrossRef] [PubMed]
  21. L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, and D. Paganin, Phys. Rev. E 63, 037602 (2001).
    [CrossRef]
  22. L. J. Allen, M. P. Oxley, and D. Paganin, Phys. Rev. Lett. 87, 1239–02 (2001).
    [CrossRef]

2001

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

L. J. Allen, H. M. L. Faulkner, M. P. Oxley, and D. Paganin, Ultramicroscopy 88, 85 (2001).
[CrossRef] [PubMed]

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, and D. Paganin, Phys. Rev. E 63, 037602 (2001).
[CrossRef]

L. J. Allen, M. P. Oxley, and D. Paganin, Phys. Rev. Lett. 87, 1239–02 (2001).
[CrossRef]

2000

1999

I. Bloch, T. W. Hänsch, and T. Esslinger, Phys. Rev. Lett. 82, 3008 (1999).
[CrossRef]

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, J. Phys. D 32, 563 (1999).
[CrossRef]

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, M. J. Holland, J. E. Williams, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett. 83, 3358 (1999).
[CrossRef]

1998

Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998).
[CrossRef]

D. Paganin and K. A. Nugent, Phys. Rev. Lett. 80, 2586 (1998).
[CrossRef]

1997

T. E. Gureyev and K. A. Nugent, Opt. Commun. 133, 339 (1997).
[CrossRef]

S. Mancini and P. Tombesi, Europhys. Lett. 40, 351 (1997).
[CrossRef]

1996

See, e.g., the stationary-state version of Eq. 8 in E. V. Goldstein, K. Plättner, and P. Meystre, J. Res. Natl. Inst. Stand. Technol. (US) 101, 583 (1996), with backpropagating components neglected.
[CrossRef]

T. E. Gureyev and K. A. Nugent, J. Opt. Soc. Am. A 13, 1670 (1996).
[CrossRef]

1995

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

T. E. Gureyev, A. Roberts, and K. A. Nugent, J. Opt. Soc. Am. A 12, 1932 (1995).
[CrossRef]

1983

Allen, L. J.

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

L. J. Allen, H. M. L. Faulkner, M. P. Oxley, and D. Paganin, Ultramicroscopy 88, 85 (2001).
[CrossRef] [PubMed]

L. J. Allen, M. P. Oxley, and D. Paganin, Phys. Rev. Lett. 87, 1239–02 (2001).
[CrossRef]

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, and D. Paganin, Phys. Rev. E 63, 037602 (2001).
[CrossRef]

Anderson, B. P.

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, M. J. Holland, J. E. Williams, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett. 83, 3358 (1999).
[CrossRef]

Bloch, I.

I. Bloch, T. W. Hänsch, and T. Esslinger, Phys. Rev. Lett. 82, 3008 (1999).
[CrossRef]

Cornell, E.

E. Cornell, J. R. Ensher, and C. E. Wieman, in Proceedings of the International School of Physics “Enrico Fermi,” M. Inguscio, S. Stringari, and C. E. Wieman, eds. (IOS Press, Amsterdam, 1999), pp. 15–66.

Cornell, E. A.

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, M. J. Holland, J. E. Williams, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett. 83, 3358 (1999).
[CrossRef]

D'Ariano, G. M.

Ensher, J. R.

E. Cornell, J. R. Ensher, and C. E. Wieman, in Proceedings of the International School of Physics “Enrico Fermi,” M. Inguscio, S. Stringari, and C. E. Wieman, eds. (IOS Press, Amsterdam, 1999), pp. 15–66.

Esslinger, T.

I. Bloch, T. W. Hänsch, and T. Esslinger, Phys. Rev. Lett. 82, 3008 (1999).
[CrossRef]

Faulkner, H. M. L.

L. J. Allen, H. M. L. Faulkner, M. P. Oxley, and D. Paganin, Ultramicroscopy 88, 85 (2001).
[CrossRef] [PubMed]

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, and D. Paganin, Phys. Rev. E 63, 037602 (2001).
[CrossRef]

Fibich, G.

G. Fibich, B. Ilan, and S. Tsynkov, “Computation of nonlinear backscattering using a high-order numerical method,” document NASA/CR-2001–211036, ICASE rep. 2001–21 (Langely Research Center, Hampton, Va., 2001).

Fortunato, M.

Gilbarg, D.

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer-Verlag, Berlin, 1977), Chap. 8.
[CrossRef]

Goldstein, E. V.

See, e.g., the stationary-state version of Eq. 8 in E. V. Goldstein, K. Plättner, and P. Meystre, J. Res. Natl. Inst. Stand. Technol. (US) 101, 583 (1996), with backpropagating components neglected.
[CrossRef]

Gureyev, T. E.

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, J. Phys. D 32, 563 (1999).
[CrossRef]

T. E. Gureyev and K. A. Nugent, Opt. Commun. 133, 339 (1997).
[CrossRef]

T. E. Gureyev and K. A. Nugent, J. Opt. Soc. Am. A 13, 1670 (1996).
[CrossRef]

T. E. Gureyev, A. Roberts, and K. A. Nugent, J. Opt. Soc. Am. A 12, 1932 (1995).
[CrossRef]

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

Haljan, P. C.

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, M. J. Holland, J. E. Williams, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett. 83, 3358 (1999).
[CrossRef]

Hall, D. S.

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, M. J. Holland, J. E. Williams, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett. 83, 3358 (1999).
[CrossRef]

Hänsch, T. W.

I. Bloch, T. W. Hänsch, and T. Esslinger, Phys. Rev. Lett. 82, 3008 (1999).
[CrossRef]

Holland, M. J.

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, M. J. Holland, J. E. Williams, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett. 83, 3358 (1999).
[CrossRef]

Ilan, B.

G. Fibich, B. Ilan, and S. Tsynkov, “Computation of nonlinear backscattering using a high-order numerical method,” document NASA/CR-2001–211036, ICASE rep. 2001–21 (Langely Research Center, Hampton, Va., 2001).

Kivshar, Y. S.

Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998).
[CrossRef]

Luther-Davies, B.

Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998).
[CrossRef]

Mancini, S.

Matthews, M. R.

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, M. J. Holland, J. E. Williams, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett. 83, 3358 (1999).
[CrossRef]

Meystre, P.

See, e.g., the stationary-state version of Eq. 8 in E. V. Goldstein, K. Plättner, and P. Meystre, J. Res. Natl. Inst. Stand. Technol. (US) 101, 583 (1996), with backpropagating components neglected.
[CrossRef]

Nugent, K. A.

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, and D. Paganin, Phys. Rev. E 63, 037602 (2001).
[CrossRef]

K. A. Nugent and D. Paganin, Phys. Rev. A 61, 063614 (2000).
[CrossRef]

D. Paganin and K. A. Nugent, Phys. Rev. Lett. 80, 2586 (1998).
[CrossRef]

T. E. Gureyev and K. A. Nugent, Opt. Commun. 133, 339 (1997).
[CrossRef]

T. E. Gureyev and K. A. Nugent, J. Opt. Soc. Am. A 13, 1670 (1996).
[CrossRef]

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

T. E. Gureyev, A. Roberts, and K. A. Nugent, J. Opt. Soc. Am. A 12, 1932 (1995).
[CrossRef]

D. Paganin and K. A. Nugent, in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, San Diego, Calif., 2001), Vol. 118, pp. 85–127.
[CrossRef]

Oxley, M. P.

L. J. Allen, H. M. L. Faulkner, M. P. Oxley, and D. Paganin, Ultramicroscopy 88, 85 (2001).
[CrossRef] [PubMed]

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

L. J. Allen, M. P. Oxley, and D. Paganin, Phys. Rev. Lett. 87, 1239–02 (2001).
[CrossRef]

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, and D. Paganin, Phys. Rev. E 63, 037602 (2001).
[CrossRef]

Paganin, D.

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, and D. Paganin, Phys. Rev. E 63, 037602 (2001).
[CrossRef]

L. J. Allen, M. P. Oxley, and D. Paganin, Phys. Rev. Lett. 87, 1239–02 (2001).
[CrossRef]

L. J. Allen, H. M. L. Faulkner, M. P. Oxley, and D. Paganin, Ultramicroscopy 88, 85 (2001).
[CrossRef] [PubMed]

K. A. Nugent and D. Paganin, Phys. Rev. A 61, 063614 (2000).
[CrossRef]

D. Paganin and K. A. Nugent, Phys. Rev. Lett. 80, 2586 (1998).
[CrossRef]

D. Paganin and K. A. Nugent, in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, San Diego, Calif., 2001), Vol. 118, pp. 85–127.
[CrossRef]

Plättner, K.

See, e.g., the stationary-state version of Eq. 8 in E. V. Goldstein, K. Plättner, and P. Meystre, J. Res. Natl. Inst. Stand. Technol. (US) 101, 583 (1996), with backpropagating components neglected.
[CrossRef]

Raven, C.

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, J. Phys. D 32, 563 (1999).
[CrossRef]

Roberts, A.

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

T. E. Gureyev, A. Roberts, and K. A. Nugent, J. Opt. Soc. Am. A 12, 1932 (1995).
[CrossRef]

Snigirev, A.

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, J. Phys. D 32, 563 (1999).
[CrossRef]

Snigireva, I.

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, J. Phys. D 32, 563 (1999).
[CrossRef]

Teague, M. R.

Tombesi, P.

Trudinger, N. S.

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer-Verlag, Berlin, 1977), Chap. 8.
[CrossRef]

Tsynkov, S.

G. Fibich, B. Ilan, and S. Tsynkov, “Computation of nonlinear backscattering using a high-order numerical method,” document NASA/CR-2001–211036, ICASE rep. 2001–21 (Langely Research Center, Hampton, Va., 2001).

Wieman, C. E.

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, M. J. Holland, J. E. Williams, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett. 83, 3358 (1999).
[CrossRef]

E. Cornell, J. R. Ensher, and C. E. Wieman, in Proceedings of the International School of Physics “Enrico Fermi,” M. Inguscio, S. Stringari, and C. E. Wieman, eds. (IOS Press, Amsterdam, 1999), pp. 15–66.

Wilkins, S. W.

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, J. Phys. D 32, 563 (1999).
[CrossRef]

Williams, J. E.

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, M. J. Holland, J. E. Williams, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett. 83, 3358 (1999).
[CrossRef]

Europhys. Lett.

S. Mancini and P. Tombesi, Europhys. Lett. 40, 351 (1997).
[CrossRef]

J. Opt. Soc. Am. A

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

T. E. Gureyev, A. Roberts, and K. A. Nugent, J. Opt. Soc. Am. A 12, 1932 (1995).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. D

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, J. Phys. D 32, 563 (1999).
[CrossRef]

J. Res. Natl. Inst. Stand. Technol. (US)

See, e.g., the stationary-state version of Eq. 8 in E. V. Goldstein, K. Plättner, and P. Meystre, J. Res. Natl. Inst. Stand. Technol. (US) 101, 583 (1996), with backpropagating components neglected.
[CrossRef]

Opt. Commun.

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

T. E. Gureyev and K. A. Nugent, Opt. Commun. 133, 339 (1997).
[CrossRef]

Phys. Rep.

Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998).
[CrossRef]

Phys. Rev. A

K. A. Nugent and D. Paganin, Phys. Rev. A 61, 063614 (2000).
[CrossRef]

Phys. Rev. E

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, and D. Paganin, Phys. Rev. E 63, 037602 (2001).
[CrossRef]

Phys. Rev. Lett.

L. J. Allen, M. P. Oxley, and D. Paganin, Phys. Rev. Lett. 87, 1239–02 (2001).
[CrossRef]

D. Paganin and K. A. Nugent, Phys. Rev. Lett. 80, 2586 (1998).
[CrossRef]

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, M. J. Holland, J. E. Williams, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett. 83, 3358 (1999).
[CrossRef]

I. Bloch, T. W. Hänsch, and T. Esslinger, Phys. Rev. Lett. 82, 3008 (1999).
[CrossRef]

Ultramicroscopy

L. J. Allen, H. M. L. Faulkner, M. P. Oxley, and D. Paganin, Ultramicroscopy 88, 85 (2001).
[CrossRef] [PubMed]

Other

E. Cornell, J. R. Ensher, and C. E. Wieman, in Proceedings of the International School of Physics “Enrico Fermi,” M. Inguscio, S. Stringari, and C. E. Wieman, eds. (IOS Press, Amsterdam, 1999), pp. 15–66.

D. Paganin and K. A. Nugent, in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, San Diego, Calif., 2001), Vol. 118, pp. 85–127.
[CrossRef]

G. Fibich, B. Ilan, and S. Tsynkov, “Computation of nonlinear backscattering using a high-order numerical method,” document NASA/CR-2001–211036, ICASE rep. 2001–21 (Langely Research Center, Hampton, Va., 2001).

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer-Verlag, Berlin, 1977), Chap. 8.
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

Typical imaging model for quantitative phase retrieval by use of the nonlinear transport-of-intensity equation.

Equations (10)

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

IrΦr=0,
2ikz+2I expiΦ=0,
-IΦ=kIz,
iαz+β+γ2+fψ+igψψ=0,
ψ=I expiΦ,
-IΦ=1γα2Iz+IImβ+g.
Ωdxdyγα2Iz+IImβ+g+ΩIΦnds=0.
IΦ1-Φ2=0.
iαmz+βm+γm2+j=1Mfjmψj+igjmψjψm=0,  m=1,,M.
-ImΦm=1γmαm2Imz+Im Imβm+Imj=1MgjmIj,  m=1,,M.

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