Abstract

A theoretical analysis of the in-line method of x-ray phase-contrast imaging with a small (microfocus) source is presented. By applying the stationary-phase formula to the Fresnel integral, we derive a new variant of the transport of intensity equation (TIE) that explicitly takes image magnification into account and does not require the distance between the two observation planes to be infinitesimally small. The new derivation method provides a uniform technique for obtaining the TIE’s with incident waves of different types and also leads to new practically convenient validity conditions. A new boundary problem for the TIE, which is particularly suitable for the in-line imaging, is defined and solved analytically. Numerical examples of phase retrieval using simulated data in the in-line case with a point source are also given. The present approach should be useful in quantitative phase-contrast microscopy and tomography using microfocus sources. The question of partial coherence is briefly discussed but is essentially outside the scope of the present work.

© 1998 Optical Society of America

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References

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  1. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, A. W. Stevenson, “Phase contrast imaging using polychromatic hard x-rays,” Nature 384, 335–338 (1996).
    [CrossRef]
  2. T. Davis, D. Gao, T. Gureyev, A. Stevenson, S. Wilkins, “Phase contrast imaging of weakly absorbing materials using hard x-rays,” Nature 373, 595–598 (1995).
    [CrossRef]
  3. T. J. Davis, T. E. Gureyev, D. Gao, A. W. Stevenson, S. W. Wilkins, “X-ray image contrast from a simple phase object,” Phys. Rev. Lett. 74, 3173–3176 (1995).
    [CrossRef] [PubMed]
  4. K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
    [CrossRef] [PubMed]
  5. V. N. Ingal, E. A. Beliaevskaya, “X-ray plane-wave topography observation of the phase contrast from a noncrystalline object,” J. Phys. D: Appl. Phys. 28, 2314–2317 (1995).
    [CrossRef]
  6. A. Momose, T. Takeda, Y. Itai, “Phase contrast x-ray computed tomography for observing biological specimens and organic materials,” Rev. Sci. Instrum. 66, 1434–1436 (1995).
    [CrossRef]
  7. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–92 (1995).
    [CrossRef]
  8. P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, M. J. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D: Appl. Phys. 29, 133–46 (1996).
    [CrossRef]
  9. D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
    [CrossRef] [PubMed]
  10. A. Pogany, D. Gao, S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
    [CrossRef]
  11. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).
  12. M. V. Fedoryuk, “The stationary phase method and pseudodifferential operators,” Usp. Math. Nauk 26, 67–112 (1971).
  13. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
    [CrossRef]
  14. F. Roddier, “Wave-front sensing and the irradiance transport equation,” Appl. Opt. 29, 1402–1403 (1990).
    [CrossRef] [PubMed]
  15. J. M. Cowley, Diffraction Physics (North-Holland, Amsterdam, 1975), Sec. 3.4.2.
  16. T. E. Gureyev, A. Roberts, K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation and phase uniqueness,” J. Opt. Soc. Am. A 12, 1942–1946 (1995).
    [CrossRef]
  17. T. E. Gureyev, K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series so-lution for nonuniform illumination,” J. Opt. Soc. Am. A 13, 1670–1682 (1996).
    [CrossRef]
  18. C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35 (1954).
    [CrossRef]
  19. U. W. Arndt, B. T. M. Willis, Single Crystal Diffractometry (Cambridge University, Cambridge, 1966), p. 185.
  20. L. V. Kantorovich, V. I. Krylov, Approximate Methods of Higher Analysis (Wiley, New York, 1964).

1997 (1)

A. Pogany, D. Gao, S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[CrossRef]

1996 (4)

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

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

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, A. W. Stevenson, “Phase contrast imaging using polychromatic hard x-rays,” Nature 384, 335–338 (1996).
[CrossRef]

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, M. J. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D: Appl. Phys. 29, 133–46 (1996).
[CrossRef]

1995 (6)

T. Davis, D. Gao, T. Gureyev, A. Stevenson, S. Wilkins, “Phase contrast imaging of weakly absorbing materials using hard x-rays,” Nature 373, 595–598 (1995).
[CrossRef]

T. J. Davis, T. E. Gureyev, D. Gao, A. W. Stevenson, S. W. Wilkins, “X-ray image contrast from a simple phase object,” Phys. Rev. Lett. 74, 3173–3176 (1995).
[CrossRef] [PubMed]

V. N. Ingal, E. A. Beliaevskaya, “X-ray plane-wave topography observation of the phase contrast from a noncrystalline object,” J. Phys. D: Appl. Phys. 28, 2314–2317 (1995).
[CrossRef]

A. Momose, T. Takeda, Y. Itai, “Phase contrast x-ray computed tomography for observing biological specimens and organic materials,” Rev. Sci. Instrum. 66, 1434–1436 (1995).
[CrossRef]

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–92 (1995).
[CrossRef]

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

1990 (1)

1983 (1)

1971 (1)

M. V. Fedoryuk, “The stationary phase method and pseudodifferential operators,” Usp. Math. Nauk 26, 67–112 (1971).

1954 (1)

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35 (1954).
[CrossRef]

1948 (1)

D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
[CrossRef] [PubMed]

Arndt, U. W.

U. W. Arndt, B. T. M. Willis, Single Crystal Diffractometry (Cambridge University, Cambridge, 1966), p. 185.

Barnea, Z.

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

Barrett, R.

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, M. J. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D: Appl. Phys. 29, 133–46 (1996).
[CrossRef]

Baruchel, J.

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, M. J. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D: Appl. Phys. 29, 133–46 (1996).
[CrossRef]

Beliaevskaya, E. A.

V. N. Ingal, E. A. Beliaevskaya, “X-ray plane-wave topography observation of the phase contrast from a noncrystalline object,” J. Phys. D: Appl. Phys. 28, 2314–2317 (1995).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Bouwkamp, C. J.

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35 (1954).
[CrossRef]

Cloetens, P.

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, M. J. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D: Appl. Phys. 29, 133–46 (1996).
[CrossRef]

Cookson, D. F.

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

Cowley, J. M.

J. M. Cowley, Diffraction Physics (North-Holland, Amsterdam, 1975), Sec. 3.4.2.

Davis, T.

T. Davis, D. Gao, T. Gureyev, A. Stevenson, S. Wilkins, “Phase contrast imaging of weakly absorbing materials using hard x-rays,” Nature 373, 595–598 (1995).
[CrossRef]

Davis, T. J.

T. J. Davis, T. E. Gureyev, D. Gao, A. W. Stevenson, S. W. Wilkins, “X-ray image contrast from a simple phase object,” Phys. Rev. Lett. 74, 3173–3176 (1995).
[CrossRef] [PubMed]

Fedoryuk, M. V.

M. V. Fedoryuk, “The stationary phase method and pseudodifferential operators,” Usp. Math. Nauk 26, 67–112 (1971).

Gabor, D.

D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
[CrossRef] [PubMed]

Gao, D.

A. Pogany, D. Gao, S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[CrossRef]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, A. W. Stevenson, “Phase contrast imaging using polychromatic hard x-rays,” Nature 384, 335–338 (1996).
[CrossRef]

T. Davis, D. Gao, T. Gureyev, A. Stevenson, S. Wilkins, “Phase contrast imaging of weakly absorbing materials using hard x-rays,” Nature 373, 595–598 (1995).
[CrossRef]

T. J. Davis, T. E. Gureyev, D. Gao, A. W. Stevenson, S. W. Wilkins, “X-ray image contrast from a simple phase object,” Phys. Rev. Lett. 74, 3173–3176 (1995).
[CrossRef] [PubMed]

Guigay, J.-P.

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, M. J. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D: Appl. Phys. 29, 133–46 (1996).
[CrossRef]

Gureyev, T.

T. Davis, D. Gao, T. Gureyev, A. Stevenson, S. Wilkins, “Phase contrast imaging of weakly absorbing materials using hard x-rays,” Nature 373, 595–598 (1995).
[CrossRef]

Gureyev, T. E.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, A. W. Stevenson, “Phase contrast imaging using polychromatic hard x-rays,” Nature 384, 335–338 (1996).
[CrossRef]

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

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

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

T. J. Davis, T. E. Gureyev, D. Gao, A. W. Stevenson, S. W. Wilkins, “X-ray image contrast from a simple phase object,” Phys. Rev. Lett. 74, 3173–3176 (1995).
[CrossRef] [PubMed]

Ingal, V. N.

V. N. Ingal, E. A. Beliaevskaya, “X-ray plane-wave topography observation of the phase contrast from a noncrystalline object,” J. Phys. D: Appl. Phys. 28, 2314–2317 (1995).
[CrossRef]

Itai, Y.

A. Momose, T. Takeda, Y. Itai, “Phase contrast x-ray computed tomography for observing biological specimens and organic materials,” Rev. Sci. Instrum. 66, 1434–1436 (1995).
[CrossRef]

Kantorovich, L. V.

L. V. Kantorovich, V. I. Krylov, Approximate Methods of Higher Analysis (Wiley, New York, 1964).

Kohn, V.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–92 (1995).
[CrossRef]

Krylov, V. I.

L. V. Kantorovich, V. I. Krylov, Approximate Methods of Higher Analysis (Wiley, New York, 1964).

Kuznetsov, S.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–92 (1995).
[CrossRef]

Momose, A.

A. Momose, T. Takeda, Y. Itai, “Phase contrast x-ray computed tomography for observing biological specimens and organic materials,” Rev. Sci. Instrum. 66, 1434–1436 (1995).
[CrossRef]

Nugent, K. A.

Paganin, D.

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

Pogany, A.

A. Pogany, D. Gao, S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[CrossRef]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, A. W. Stevenson, “Phase contrast imaging using polychromatic hard x-rays,” Nature 384, 335–338 (1996).
[CrossRef]

Roberts, A.

Roddier, F.

Schelokov, I.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–92 (1995).
[CrossRef]

Schlenker, M. J.

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, M. J. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D: Appl. Phys. 29, 133–46 (1996).
[CrossRef]

Snigirev, A.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–92 (1995).
[CrossRef]

Snigireva, I.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–92 (1995).
[CrossRef]

Stevenson, A.

T. Davis, D. Gao, T. Gureyev, A. Stevenson, S. Wilkins, “Phase contrast imaging of weakly absorbing materials using hard x-rays,” Nature 373, 595–598 (1995).
[CrossRef]

Stevenson, A. W.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, A. W. Stevenson, “Phase contrast imaging using polychromatic hard x-rays,” Nature 384, 335–338 (1996).
[CrossRef]

T. J. Davis, T. E. Gureyev, D. Gao, A. W. Stevenson, S. W. Wilkins, “X-ray image contrast from a simple phase object,” Phys. Rev. Lett. 74, 3173–3176 (1995).
[CrossRef] [PubMed]

Takeda, T.

A. Momose, T. Takeda, Y. Itai, “Phase contrast x-ray computed tomography for observing biological specimens and organic materials,” Rev. Sci. Instrum. 66, 1434–1436 (1995).
[CrossRef]

Teague, M. R.

Wilkins, S.

T. Davis, D. Gao, T. Gureyev, A. Stevenson, S. Wilkins, “Phase contrast imaging of weakly absorbing materials using hard x-rays,” Nature 373, 595–598 (1995).
[CrossRef]

Wilkins, S. W.

A. Pogany, D. Gao, S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[CrossRef]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, A. W. Stevenson, “Phase contrast imaging using polychromatic hard x-rays,” Nature 384, 335–338 (1996).
[CrossRef]

T. J. Davis, T. E. Gureyev, D. Gao, A. W. Stevenson, S. W. Wilkins, “X-ray image contrast from a simple phase object,” Phys. Rev. Lett. 74, 3173–3176 (1995).
[CrossRef] [PubMed]

Willis, B. T. M.

U. W. Arndt, B. T. M. Willis, Single Crystal Diffractometry (Cambridge University, Cambridge, 1966), p. 185.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

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

J. Phys. D: Appl. Phys. (2)

V. N. Ingal, E. A. Beliaevskaya, “X-ray plane-wave topography observation of the phase contrast from a noncrystalline object,” J. Phys. D: Appl. Phys. 28, 2314–2317 (1995).
[CrossRef]

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, M. J. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D: Appl. Phys. 29, 133–46 (1996).
[CrossRef]

Nature (3)

D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
[CrossRef] [PubMed]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, A. W. Stevenson, “Phase contrast imaging using polychromatic hard x-rays,” Nature 384, 335–338 (1996).
[CrossRef]

T. Davis, D. Gao, T. Gureyev, A. Stevenson, S. Wilkins, “Phase contrast imaging of weakly absorbing materials using hard x-rays,” Nature 373, 595–598 (1995).
[CrossRef]

Phys. Rev. Lett. (2)

T. J. Davis, T. E. Gureyev, D. Gao, A. W. Stevenson, S. W. Wilkins, “X-ray image contrast from a simple phase object,” Phys. Rev. Lett. 74, 3173–3176 (1995).
[CrossRef] [PubMed]

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

Rep. Prog. Phys. (1)

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35 (1954).
[CrossRef]

Rev. Sci. Instrum. (3)

A. Momose, T. Takeda, Y. Itai, “Phase contrast x-ray computed tomography for observing biological specimens and organic materials,” Rev. Sci. Instrum. 66, 1434–1436 (1995).
[CrossRef]

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–92 (1995).
[CrossRef]

A. Pogany, D. Gao, S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[CrossRef]

Usp. Math. Nauk (1)

M. V. Fedoryuk, “The stationary phase method and pseudodifferential operators,” Usp. Math. Nauk 26, 67–112 (1971).

Other (4)

J. M. Cowley, Diffraction Physics (North-Holland, Amsterdam, 1975), Sec. 3.4.2.

U. W. Arndt, B. T. M. Willis, Single Crystal Diffractometry (Cambridge University, Cambridge, 1966), p. 185.

L. V. Kantorovich, V. I. Krylov, Approximate Methods of Higher Analysis (Wiley, New York, 1964).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

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Figures (8)

Fig. 1
Fig. 1

Principal scheme of the method.

Fig. 2
Fig. 2

Original phase on the object plane.

Fig. 3
Fig. 3

Scaled intensity on the image plane R2=20 cm.

Fig. 4
Fig. 4

Laplacian of the original phase shown in Fig. 2.

Fig. 5
Fig. 5

Reconstructed phase (uniform intensity case).

Fig. 6
Fig. 6

Scaled intensity at R2=20 cm (nonuniform incident intensity).

Fig. 7
Fig. 7

Reconstructed phase (nonuniform intensity case).

Fig. 8
Fig. 8

Phase reconstructed from data with 10% noise.

Equations (24)

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

u0(x, y)=exp[ik(x2+y2+R12)1/2][1+(x2+y2)/R12]1/2×a(x, y)exp[ikψ(x, y)]
(2+k2)u=0.
u(x, y, z)-ik2π exp(ikr)rzru0(x, y)dxdy,
ψ(x, y)0anda(x, y)1,
when(x2+y2)d2,anddR1;
|xmynψ(x, y)|(R)1-(n+m),m2+n2>0,
|xmyna(x, y)|C(R)-(n+m),m2+n2>0,
u(x, y, R2)=-ik2πexp(ikR)R2× exp[ikS(x, y, x, y)]×a(x, y)dxdy,
S(x, y, x, y)=(x-x)2+(y-y)22RR2+x2+y22RR1+ψ(x, y)R.
u(x, y, R2)= exp{i[kR+kSs+(π/2)sgn Ss]}RR2(det Ss)1/2×a(xs, ys)+O(1/k),
xs=M-1x-Rψx(xs, ys)ys=M-1y-Rψy(xs, ys),M=(R1+R2)/R1,
S=(R)-21+RψxxRψxyRψxy1+Rψyy.
det S(R)-4(1+R2ψ),R|2ψ|1.
u(x, y, R2)a(xs, ys)M(1+R2ψ)1/2exp[i(kR+kSs)],
I(x, y, R2)M-2I(xs, ys)[1-R2ψ(xs, ys)].
I(xs, ys)
=I(M-1x-M-1R2ψx, M-1y-M-1R2ψy)=I(M-1x, M-1y)-R(I·ψ)(M-1x, M-1y).
M2I(Mx, My, R2)I(x, y)[1-R2ψ(x, y)-R log I(x, y)·ψ(x, y)].
ψ(x, y)=λ2re2π-0ρ(x, y, z)dz,
δ=σR2/(R1+R2).
K2πλR/σ2.
-·[I(x, y)ψ(x, y)]=F(x, y),
ψ(x, y)|Γ=0.
ψmn=Dπ2 F˜mn(m2+n2),m2+n2>0,

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