Abstract

The phase-retrieval problem for a physical system with strong support constraints is investigated. Propagation of an optical field in a system with no variation along one transverse axis results in cylindrical wave fronts. Scalar propagation in such systems is a purely two-dimensional process. We show that, given the optical intensity in a plane, the phase of the wave field can be calculated directly if the system has this special symmetry. The procedure relies on a simple geometric relation between the system pupil function (or angular spectrum) and the system optical transfer function in the Debye theory of scalar wave focusing. The inherent autocorrelation operation can be undone, and the phase directly retrieved, with a simple coordinate transformation.

© 1999 Optical Society of America

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References

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  1. M. G. Raymer, M. Beck, D. F. McAlister, “Complex wavefield reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
    [CrossRef] [PubMed]
  2. D. F. McAlister, M. Beck, L. Clarke, A. Mayer, M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20, 1181–1183 (1995).
    [CrossRef] [PubMed]
  3. E. Collett, E. Wolf, “Is complete spatial coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27–29 (1978).
    [CrossRef] [PubMed]
  4. F. Gori, M. Santarsiero, G. Guattari, “Coherence and the spatial distribution of intensity,” J. Opt. Soc. Am. A 10, 673–679 (1993).
    [CrossRef]
  5. 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]
  6. M. R. Teague, “Irradiance moments: their propagation and use for unique retrieval of phases,” J. Opt. Soc. Am. A 72, 1199–1209 (1982).
    [CrossRef]
  7. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
    [CrossRef]
  8. A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1962).
    [CrossRef]
  9. I. M. Bruck, L. G. Sodin, “On the ambiguity of the image restoration problem,” Opt. Commun. 30, 304–308 (1979).
    [CrossRef]
  10. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [CrossRef] [PubMed]
  11. M. A. Fiddy, B. J. Brames, J. C. Dainty, “Enforcing irreducibility for phase retrieval in two dimensions,” Opt. Lett. 8, 96–98 (1983).
    [CrossRef] [PubMed]
  12. R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension: underlying theory,” Optik 61, 247–262 (1982).
  13. R. W. Gerschberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  14. J. R. Fienup, “Phase retrieval using boundary conditions,” J. Opt. Soc. Am. A 3, 284–288 (1986).
    [CrossRef]
  15. M. Kaveh, M. Soumekh, “Computer-assisted diffraction tomography,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, Orlando, Fla., 1987), pp. 369–413.
  16. M. H. Maleki, A. J. Devaney, A. Schatzberg, “Tomographic reconstruction from optical scattered intensities,” J. Opt. Soc. Am. A 9, 1356–1363 (1992).
    [CrossRef]
  17. A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336–350 (1982).
    [CrossRef] [PubMed]
  18. H. P. Baltes, “Introduction,” in Inverse Source Problems, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), pp. 1–10.
  19. H. A. Ferweda, “The phase reconstruction problem for wave amplitudes and coherence functions,” in Inverse Source Problems, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), pp. 13–38.
  20. E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
    [CrossRef]
  21. M. H. Maleki, A. J. Devaney, “Phase retrieval in inverse scattering,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE1767, 398–408 (1992).
    [CrossRef]
  22. M. H. Maleki, A. J. Devaney, “Holographic techniques for inverse scattering and tomographic imaging,” in Practical Holography VIII, S. A. Benton, ed., Proc. SPIE2176, 184–194 (1994).
    [CrossRef]
  23. N. Streibl, “Three-dimensional imaging by a microscope,” J. Opt. Soc. Am. A 2, 121–127 (1985).
    [CrossRef]
  24. P. J. Shaw, D. A. Agard, Y. Hiraoka, J. W. Sedat, “Tilted view reconstruction in optical microscopy,” Biophys. J. 55, 101–110 (1989).
    [CrossRef] [PubMed]
  25. C. J. Cogswell, K. G. Larkin, H. U. Klemm, “Fluorescence microtomography: multiangle image acquisition and 3D digital reconstruction,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. S. Kino, eds., Proc. SPIE2655, 109–115 (1996).
    [CrossRef]
  26. T. C. Wedberg, J. Stamnes, “Experimental examination of the quantitative imaging properties of optical diffraction tomography,” J. Opt. Soc. Am. A 12, 493–500 (1995).
    [CrossRef]
  27. In practice we have access only to discretely sampled measurements over a finite region of space. It is a straightforward procedure to include these limitations and the computation artifacts that ensue. The main effect, which is due to the finite area, is a blurring in the angular spectrum domain.
  28. C. J. R. Sheppard, K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik 107, 79–87 (1997).
  29. C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am. 54, 240–244 (1964).
    [CrossRef]
  30. C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging,” Optik 72, 131–133 (1986).
  31. B. R. Frieden, “Optical transfer of the three-dimensional object,” J. Opt. Soc. Am. 57, 56–66 (1967).
    [CrossRef]
  32. L. Mertz, Transformations in Optics (Wiley, New York, 1965).
  33. C. J. R. Sheppard, T. J. Connolly, M. Gu, “Scattering by a one-dimensional rough surface, and surface profile reconstruction confocal imaging,” Phys. Rev. Lett. 70, 1409–1412 (1993).
    [CrossRef] [PubMed]

1997 (1)

C. J. R. Sheppard, K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik 107, 79–87 (1997).

1995 (3)

1994 (1)

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wavefield reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

1993 (2)

C. J. R. Sheppard, T. J. Connolly, M. Gu, “Scattering by a one-dimensional rough surface, and surface profile reconstruction confocal imaging,” Phys. Rev. Lett. 70, 1409–1412 (1993).
[CrossRef] [PubMed]

F. Gori, M. Santarsiero, G. Guattari, “Coherence and the spatial distribution of intensity,” J. Opt. Soc. Am. A 10, 673–679 (1993).
[CrossRef]

1992 (1)

1989 (1)

P. J. Shaw, D. A. Agard, Y. Hiraoka, J. W. Sedat, “Tilted view reconstruction in optical microscopy,” Biophys. J. 55, 101–110 (1989).
[CrossRef] [PubMed]

1986 (2)

J. R. Fienup, “Phase retrieval using boundary conditions,” J. Opt. Soc. Am. A 3, 284–288 (1986).
[CrossRef]

C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging,” Optik 72, 131–133 (1986).

1985 (1)

1983 (2)

1982 (3)

M. R. Teague, “Irradiance moments: their propagation and use for unique retrieval of phases,” J. Opt. Soc. Am. A 72, 1199–1209 (1982).
[CrossRef]

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336–350 (1982).
[CrossRef] [PubMed]

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension: underlying theory,” Optik 61, 247–262 (1982).

1979 (1)

I. M. Bruck, L. G. Sodin, “On the ambiguity of the image restoration problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

1978 (2)

1972 (1)

R. W. Gerschberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1969 (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

1967 (1)

1964 (1)

1962 (1)

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1962).
[CrossRef]

Agard, D. A.

P. J. Shaw, D. A. Agard, Y. Hiraoka, J. W. Sedat, “Tilted view reconstruction in optical microscopy,” Biophys. J. 55, 101–110 (1989).
[CrossRef] [PubMed]

Baltes, H. P.

H. P. Baltes, “Introduction,” in Inverse Source Problems, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), pp. 1–10.

Bates, R. H. T.

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension: underlying theory,” Optik 61, 247–262 (1982).

Beck, M.

Brames, B. J.

Bruck, I. M.

I. M. Bruck, L. G. Sodin, “On the ambiguity of the image restoration problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

Clarke, L.

Cogswell, C. J.

C. J. Cogswell, K. G. Larkin, H. U. Klemm, “Fluorescence microtomography: multiangle image acquisition and 3D digital reconstruction,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. S. Kino, eds., Proc. SPIE2655, 109–115 (1996).
[CrossRef]

Collett, E.

Connolly, T. J.

C. J. R. Sheppard, T. J. Connolly, M. Gu, “Scattering by a one-dimensional rough surface, and surface profile reconstruction confocal imaging,” Phys. Rev. Lett. 70, 1409–1412 (1993).
[CrossRef] [PubMed]

Dainty, J. C.

Devaney, A. J.

M. H. Maleki, A. J. Devaney, A. Schatzberg, “Tomographic reconstruction from optical scattered intensities,” J. Opt. Soc. Am. A 9, 1356–1363 (1992).
[CrossRef]

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336–350 (1982).
[CrossRef] [PubMed]

M. H. Maleki, A. J. Devaney, “Phase retrieval in inverse scattering,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE1767, 398–408 (1992).
[CrossRef]

M. H. Maleki, A. J. Devaney, “Holographic techniques for inverse scattering and tomographic imaging,” in Practical Holography VIII, S. A. Benton, ed., Proc. SPIE2176, 184–194 (1994).
[CrossRef]

Ferweda, H. A.

H. A. Ferweda, “The phase reconstruction problem for wave amplitudes and coherence functions,” in Inverse Source Problems, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), pp. 13–38.

Fiddy, M. A.

Fienup, J. R.

Frieden, B. R.

Gerschberg, R. W.

R. W. Gerschberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Gori, F.

Gu, M.

C. J. R. Sheppard, T. J. Connolly, M. Gu, “Scattering by a one-dimensional rough surface, and surface profile reconstruction confocal imaging,” Phys. Rev. Lett. 70, 1409–1412 (1993).
[CrossRef] [PubMed]

Guattari, G.

Gureyev, T. E.

Hiraoka, Y.

P. J. Shaw, D. A. Agard, Y. Hiraoka, J. W. Sedat, “Tilted view reconstruction in optical microscopy,” Biophys. J. 55, 101–110 (1989).
[CrossRef] [PubMed]

Kaveh, M.

M. Kaveh, M. Soumekh, “Computer-assisted diffraction tomography,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, Orlando, Fla., 1987), pp. 369–413.

Klemm, H. U.

C. J. Cogswell, K. G. Larkin, H. U. Klemm, “Fluorescence microtomography: multiangle image acquisition and 3D digital reconstruction,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. S. Kino, eds., Proc. SPIE2655, 109–115 (1996).
[CrossRef]

Larkin, K. G.

C. J. R. Sheppard, K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik 107, 79–87 (1997).

C. J. Cogswell, K. G. Larkin, H. U. Klemm, “Fluorescence microtomography: multiangle image acquisition and 3D digital reconstruction,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. S. Kino, eds., Proc. SPIE2655, 109–115 (1996).
[CrossRef]

Maleki, M. H.

M. H. Maleki, A. J. Devaney, A. Schatzberg, “Tomographic reconstruction from optical scattered intensities,” J. Opt. Soc. Am. A 9, 1356–1363 (1992).
[CrossRef]

M. H. Maleki, A. J. Devaney, “Phase retrieval in inverse scattering,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE1767, 398–408 (1992).
[CrossRef]

M. H. Maleki, A. J. Devaney, “Holographic techniques for inverse scattering and tomographic imaging,” in Practical Holography VIII, S. A. Benton, ed., Proc. SPIE2176, 184–194 (1994).
[CrossRef]

Mayer, A.

McAlister, D. F.

McCutchen, C. W.

Mertz, L.

L. Mertz, Transformations in Optics (Wiley, New York, 1965).

Nugent, K. A.

Raymer, M. G.

Roberts, A.

Santarsiero, M.

Saxton, W. O.

R. W. Gerschberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Schatzberg, A.

Sedat, J. W.

P. J. Shaw, D. A. Agard, Y. Hiraoka, J. W. Sedat, “Tilted view reconstruction in optical microscopy,” Biophys. J. 55, 101–110 (1989).
[CrossRef] [PubMed]

Shaw, P. J.

P. J. Shaw, D. A. Agard, Y. Hiraoka, J. W. Sedat, “Tilted view reconstruction in optical microscopy,” Biophys. J. 55, 101–110 (1989).
[CrossRef] [PubMed]

Sheppard, C. J. R.

C. J. R. Sheppard, K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik 107, 79–87 (1997).

C. J. R. Sheppard, T. J. Connolly, M. Gu, “Scattering by a one-dimensional rough surface, and surface profile reconstruction confocal imaging,” Phys. Rev. Lett. 70, 1409–1412 (1993).
[CrossRef] [PubMed]

C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging,” Optik 72, 131–133 (1986).

Sodin, L. G.

I. M. Bruck, L. G. Sodin, “On the ambiguity of the image restoration problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

Soumekh, M.

M. Kaveh, M. Soumekh, “Computer-assisted diffraction tomography,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, Orlando, Fla., 1987), pp. 369–413.

Stamnes, J.

Streibl, N.

Teague, M. R.

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

M. R. Teague, “Irradiance moments: their propagation and use for unique retrieval of phases,” J. Opt. Soc. Am. A 72, 1199–1209 (1982).
[CrossRef]

Walther, A.

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1962).
[CrossRef]

Wedberg, T. C.

Wolf, E.

E. Collett, E. Wolf, “Is complete spatial coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27–29 (1978).
[CrossRef] [PubMed]

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

Biophys. J. (1)

P. J. Shaw, D. A. Agard, Y. Hiraoka, J. W. Sedat, “Tilted view reconstruction in optical microscopy,” Biophys. J. 55, 101–110 (1989).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (7)

Opt. Acta (1)

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1962).
[CrossRef]

Opt. Commun. (2)

I. M. Bruck, L. G. Sodin, “On the ambiguity of the image restoration problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

Opt. Lett. (4)

Optik (4)

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension: underlying theory,” Optik 61, 247–262 (1982).

R. W. Gerschberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

C. J. R. Sheppard, K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik 107, 79–87 (1997).

C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging,” Optik 72, 131–133 (1986).

Phys. Rev. Lett. (2)

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wavefield reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

C. J. R. Sheppard, T. J. Connolly, M. Gu, “Scattering by a one-dimensional rough surface, and surface profile reconstruction confocal imaging,” Phys. Rev. Lett. 70, 1409–1412 (1993).
[CrossRef] [PubMed]

Ultrason. Imaging (1)

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336–350 (1982).
[CrossRef] [PubMed]

Other (8)

H. P. Baltes, “Introduction,” in Inverse Source Problems, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), pp. 1–10.

H. A. Ferweda, “The phase reconstruction problem for wave amplitudes and coherence functions,” in Inverse Source Problems, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), pp. 13–38.

M. H. Maleki, A. J. Devaney, “Phase retrieval in inverse scattering,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE1767, 398–408 (1992).
[CrossRef]

M. H. Maleki, A. J. Devaney, “Holographic techniques for inverse scattering and tomographic imaging,” in Practical Holography VIII, S. A. Benton, ed., Proc. SPIE2176, 184–194 (1994).
[CrossRef]

M. Kaveh, M. Soumekh, “Computer-assisted diffraction tomography,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, Orlando, Fla., 1987), pp. 369–413.

L. Mertz, Transformations in Optics (Wiley, New York, 1965).

C. J. Cogswell, K. G. Larkin, H. U. Klemm, “Fluorescence microtomography: multiangle image acquisition and 3D digital reconstruction,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. S. Kino, eds., Proc. SPIE2655, 109–115 (1996).
[CrossRef]

In practice we have access only to discretely sampled measurements over a finite region of space. It is a straightforward procedure to include these limitations and the computation artifacts that ensue. The main effect, which is due to the finite area, is a blurring in the angular spectrum domain.

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Figures (12)

Fig. 1
Fig. 1

Regions of support for (a) the pupil function F(m, s) and (b) its autocorrelation.

Fig. 2
Fig. 2

Spatial-frequency coordinate system and angular coordinate system.

Fig. 3
Fig. 3

Overlap region in the autocorrelation of F(m, s). Note the rhombic shape.

Fig. 4
Fig. 4

Effect of coordinate transformation for a typical grid pattern.

Fig. 5
Fig. 5

Gray-scale plot of a typical intensity pattern. The quartic root of the intensity is displayed to enhance low-level features of interest. This particular pattern corresponds to the focal region of a phase-aberrated cylindrical wave.

Fig. 6
Fig. 6

Gray-scale plot of the modulus of the Fourier-transformed intensity, |G(m, s)|. The gray scale is nonlinear to enhance low-level features of interest. The predicted correlation petal-shaped outline is clearly shown. Note that G(m, s) is, in general, complex.

Fig. 7
Fig. 7

Gray-scale plot of the initial angular spectrum F(m, s) as a magnitude (left-hand-side) and a phase (right-hand-side) component. The gray scales are linear in this case. Note that the circular arc subtends π rad and has been given a narrow Gaussian profile to reduce line-aliasing artifacts. The imposed phase modulation has a sinusoidal form with five periods over the arc length. The linear gray scale is shown at the far left.

Fig. 8
Fig. 8

Gray-scale plot of F˜(θ1)F˜*(θ2) from the direct phase-retrieval algorithm of Eq. (12). The magnitude is on the left; the phase, on the right. Note that the magnitude is near constant (light gray) over the upper right-hand quadrant corresponding to the range -π/2<θ1,2<π/2 in this implementation. The phase component clearly shows the five-period grid variation that we expect in this example. The patterns have Hermitian symmetry and could be adequately defined over a region half this area.

Fig. 9
Fig. 9

Phase profile recovered by use of Eq. (14). The predicted five-period sinusoid is clear. The 64 samples cover the angular coordinate range -π/2<θ1<π/2.

Fig. 10
Fig. 10

Gray-scale plot of an intensity pattern with a uniform random noise. The random noise range is ±2% of the peak intensity. The quartic root of the intensity is displayed to enhance low-level features of interest.

Fig. 11
Fig. 11

Gray-scale plot of F˜(θ1)F˜*(θ2) from the direct phase-retrieval algorithm applied to the noisy intensity map. The magnitude (left-hand-side) and phase (right-hand-side) areas shown correspond to the top right-hand quadrants in Fig. 8.

Fig. 12
Fig. 12

Phase profile recovered from the noisy intensity map. Again the predicted five-period sinusoid is clear. The 64 samples cover the angular coordinate range -π/2<θ1<π/2.

Equations (18)

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

I(x, y, z)|E(x, y, z)|2=|f(x, y, z)|2=I(x, z)=|f(x, z)|2.
2f(x, y, z)+k2f(x, y, z)=0.
F(m, n, s)=f(x, y, z)×exp[-2πi(mx+ny+sz)]dxdydz.
F(m, n, s)[k2-(2π)2(m2+n2+s2)]=0.
F(m, n, s)[1-λ2(m2+n2+s2)]δ(s)
=0=F(m, s)[1-λ2(m2+s2)].
mλ=sin θ1-sin θ2
sλ=cos θ2-cos θ1.
G(m, s)=g(x, z)exp[-2πi(mx+sz)]dxdz=F(m, s)F*(m-m, s-s)dmds.
G˜(θ1, θ2)=F˜(θ1)F˜*(θ2)|sin(θ1-θ2)|.
G(m, s)=G˜(θ1, θ2)=g(x, z)exp{-(2πi/λ)[(sin θ1-sin θ2)x+(cos θ2-cos θ1)z]}dxdz.
F˜(θ1)F˜*(θ2)=|sin(θ1-θ2)|G˜(θ1, θ2).
Gρ,σ=G[ρ/(NΔ),σ/(NΔ)]=Δ2N q=0N-1p=0N-1g(pΔ, qΔ)exp[2πi(pρ+qσ)/N],
F˜αF˜β*=G˜(α, β)=Δ2N sin2π(α-β)N×p=-(N/2)+1(N/2)-1p=-(N/2)+1(N/2)-1gp,q×exp2πiLN pcos2παN-cos2πβN+qsin2πβN-sin2παN.
θ1=2παN,
θ2=2πβN,
1λ=LNΔ,
-ΘΘG˜(θ1, θ2)|sin(θ1-θ2)|dθ1=F˜(θ1)-ΘΘF˜(θ2)dθ2.

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