Abstract

A multispectral intensity diffraction tomography (I-DT) reconstruction theory for quasi-nondispersive scattering objects is developed and investigated. By “quasi-nondispersive” we refer to an object that is characterized by a refractive index distribution that is approximately nondispersive over a predefined finite temporal frequency interval in which the tomographic measurements are acquired. The scanning requirements and measurement data are shown to be different than in conventional I-DT. Unlike conventional I-DT that requires intensity measurements on a pair of detector planes for each probing wave field, this new method uses measurements on a single detector plane at two frequencies. Computer simulation studies are conducted to demonstrate the method.

© 2006 Optical Society of America

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  1. E. Wolf, "Principles and development of diffraction tomography," in Trends in Optics, A.Consortini, ed. (Academic, 1996), pp. 83-110.
    [CrossRef]
  2. A. J. Devaney, "Reconstructive tomography with diffracting wavefields," Inverse Probl. 2, 161-183 (1986).
    [CrossRef]
  3. T. C. Wedberg and J. J. Stamnes, "Recent results in optical diffraction microtomography," Meas. Sci. Technol. 7, 414-418 (1996).
    [CrossRef]
  4. T. Wedberg and J. Stamnes, "Quantitative imaging by optical diffraction tomography," Opt. Rev. 2, 28-31 (1995).
    [CrossRef]
  5. V. Lauer, "New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope," J. Microsc. 205, 165-176 (2001).
    [CrossRef]
  6. A. S. T. Beetz and C. Jacobsen, "Soft x-ray diffraction tomography: simulations and first experimental results," J. Phys. (Paris) 104, 31-34 (2003).
  7. M. A. Anastasio and D. Shi, "On the relationship between intensity diffraction tomography and phase-contrast tomography," in Developments in X-ray Tomography IV, U.Bonse, ed., Proc. SPIE 5535, 361-368 (2004).
  8. P. Grassin, B. Duchene, and W. Tabbara, "Diffraction tomography: some applications and extension to 3-d ultrasound imaging," in Mathematical Methods in Tomography (Springer-Verlag, 1991), pp. 98-105.
    [CrossRef]
  9. T. Mast, "Wideband quantitative ultrasonic imaging by time-domain diffraction tomography," J. Acoust. Soc. Am. 106, 3061-3071 (1999).
    [CrossRef]
  10. D. Shi, M. A. Anastasio, Y. Huang, and G. Gbur, "Half-scan and single-plane intensity diffraction tomography for phase objects," Phys. Med. Biol. 49, 2733-2752 (2004).
    [CrossRef] [PubMed]
  11. G. Gbur, M. A. Anastasio, Y. Huang, and D. Shi, "Spherical-wave intensity diffraction tomography," J. Opt. Soc. Am. A 22, 230-238 (2005).
    [CrossRef]
  12. G. Gbur and E. Wolf, "Diffraction tomography without phase information," Opt. Lett. 27, 1890-1892 (2002).
    [CrossRef]
  13. G. Gbur and E. Wolf, "Hybrid diffraction tomography without phase information," J. Opt. Soc. Am. A 19, 2194-2202 (2002).
    [CrossRef]
  14. M. R. Teague, "Deterministic phase retrieval: a Green's function solution," J. Opt. Soc. Am. 73, 1434-1441 (1983).
    [CrossRef]
  15. A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, "Quantitative phase tomography," Opt. Commun. 175, 329-336 (2000).
    [CrossRef]
  16. Y. Waseda, Novel Application of Anomalous (Resonance) X-ray Scattering for Structural Characterization of Disordered Materials (Springer-Verlag, 1984).
    [CrossRef] [PubMed]
  17. G. Gbur and E. Wolf, "Determination of density correlation functions from scattering of polychromatic light," Opt. Commun. 168, 39-45 (1999).
    [CrossRef]
  18. P. S. Carney and J. C. Schotland, "Inverse scattering for near-field microscopy," Appl. Phys. Lett. 77, 2798-2800 (2000).
    [CrossRef]
  19. M. A. Anastasio and X. Pan, "Computationally efficient and statistically robust image reconstruction in 3D diffraction tomography," J. Opt. Soc. Am. A 17, 391-400 (2000).
    [CrossRef]
  20. X. Pan, "A unified reconstruction theory for diffraction tomography with considerations of noise control," J. Opt. Soc. Am. A 15, 2312-2326 (1998).
    [CrossRef]
  21. A. J. Devaney, "A filtered backpropagation algorithm for diffraction tomography," Ultrason. Imaging 4, 336-350 (1982).
    [CrossRef] [PubMed]
  22. M. A. Anastasio, D. Shi, Y. Huang, and G. Gbur, "Image reconstruction in spherical wave intensity diffraction tomography," J. Opt. Soc. Am. A 22, 2651-2661 (2005).
    [CrossRef]
  23. B. Chen and J. Stamnes, "Validity of diffraction tomography based on the first Born and first Rytov approximations," Appl. Opt. 37, 2996-3006 (1998).
    [CrossRef]
  24. S. Pan and A. Kak, "A computational study of reconstruction algorithms for diffraction tomography: interpolation versus filtered backpropagation," IEEE Trans. Acoust. Speech Signal Process. 31, 1262-1275 (1983).
    [CrossRef]
  25. M. A. Anastasio and X. Pan, "Investigation of the noise properties of a new class of reconstruction methods in diffraction tomography," Int. J. Imaging Syst. Technol. 10, 437-446 (1999).
    [CrossRef]
  26. E. Clarkson, "Projections onto the range of the exponential Radon transform and reconstruction algorithms," Inverse Probl. 15, 563-571 (1999).
    [CrossRef]

2005 (2)

2004 (1)

D. Shi, M. A. Anastasio, Y. Huang, and G. Gbur, "Half-scan and single-plane intensity diffraction tomography for phase objects," Phys. Med. Biol. 49, 2733-2752 (2004).
[CrossRef] [PubMed]

2003 (1)

A. S. T. Beetz and C. Jacobsen, "Soft x-ray diffraction tomography: simulations and first experimental results," J. Phys. (Paris) 104, 31-34 (2003).

2002 (2)

2001 (1)

V. Lauer, "New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope," J. Microsc. 205, 165-176 (2001).
[CrossRef]

2000 (3)

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, "Quantitative phase tomography," Opt. Commun. 175, 329-336 (2000).
[CrossRef]

P. S. Carney and J. C. Schotland, "Inverse scattering for near-field microscopy," Appl. Phys. Lett. 77, 2798-2800 (2000).
[CrossRef]

M. A. Anastasio and X. Pan, "Computationally efficient and statistically robust image reconstruction in 3D diffraction tomography," J. Opt. Soc. Am. A 17, 391-400 (2000).
[CrossRef]

1999 (4)

M. A. Anastasio and X. Pan, "Investigation of the noise properties of a new class of reconstruction methods in diffraction tomography," Int. J. Imaging Syst. Technol. 10, 437-446 (1999).
[CrossRef]

E. Clarkson, "Projections onto the range of the exponential Radon transform and reconstruction algorithms," Inverse Probl. 15, 563-571 (1999).
[CrossRef]

G. Gbur and E. Wolf, "Determination of density correlation functions from scattering of polychromatic light," Opt. Commun. 168, 39-45 (1999).
[CrossRef]

T. Mast, "Wideband quantitative ultrasonic imaging by time-domain diffraction tomography," J. Acoust. Soc. Am. 106, 3061-3071 (1999).
[CrossRef]

1998 (2)

1996 (1)

T. C. Wedberg and J. J. Stamnes, "Recent results in optical diffraction microtomography," Meas. Sci. Technol. 7, 414-418 (1996).
[CrossRef]

1995 (1)

T. Wedberg and J. Stamnes, "Quantitative imaging by optical diffraction tomography," Opt. Rev. 2, 28-31 (1995).
[CrossRef]

1986 (1)

A. J. Devaney, "Reconstructive tomography with diffracting wavefields," Inverse Probl. 2, 161-183 (1986).
[CrossRef]

1983 (2)

M. R. Teague, "Deterministic phase retrieval: a Green's function solution," J. Opt. Soc. Am. 73, 1434-1441 (1983).
[CrossRef]

S. Pan and A. Kak, "A computational study of reconstruction algorithms for diffraction tomography: interpolation versus filtered backpropagation," IEEE Trans. Acoust. Speech Signal Process. 31, 1262-1275 (1983).
[CrossRef]

1982 (1)

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

Anastasio, M. A.

M. A. Anastasio, D. Shi, Y. Huang, and G. Gbur, "Image reconstruction in spherical wave intensity diffraction tomography," J. Opt. Soc. Am. A 22, 2651-2661 (2005).
[CrossRef]

G. Gbur, M. A. Anastasio, Y. Huang, and D. Shi, "Spherical-wave intensity diffraction tomography," J. Opt. Soc. Am. A 22, 230-238 (2005).
[CrossRef]

D. Shi, M. A. Anastasio, Y. Huang, and G. Gbur, "Half-scan and single-plane intensity diffraction tomography for phase objects," Phys. Med. Biol. 49, 2733-2752 (2004).
[CrossRef] [PubMed]

M. A. Anastasio and X. Pan, "Computationally efficient and statistically robust image reconstruction in 3D diffraction tomography," J. Opt. Soc. Am. A 17, 391-400 (2000).
[CrossRef]

M. A. Anastasio and X. Pan, "Investigation of the noise properties of a new class of reconstruction methods in diffraction tomography," Int. J. Imaging Syst. Technol. 10, 437-446 (1999).
[CrossRef]

M. A. Anastasio and D. Shi, "On the relationship between intensity diffraction tomography and phase-contrast tomography," in Developments in X-ray Tomography IV, U.Bonse, ed., Proc. SPIE 5535, 361-368 (2004).

Barty, A.

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, "Quantitative phase tomography," Opt. Commun. 175, 329-336 (2000).
[CrossRef]

Beetz, A. S. T.

A. S. T. Beetz and C. Jacobsen, "Soft x-ray diffraction tomography: simulations and first experimental results," J. Phys. (Paris) 104, 31-34 (2003).

Carney, P. S.

P. S. Carney and J. C. Schotland, "Inverse scattering for near-field microscopy," Appl. Phys. Lett. 77, 2798-2800 (2000).
[CrossRef]

Chen, B.

Clarkson, E.

E. Clarkson, "Projections onto the range of the exponential Radon transform and reconstruction algorithms," Inverse Probl. 15, 563-571 (1999).
[CrossRef]

Devaney, A. J.

A. J. Devaney, "Reconstructive tomography with diffracting wavefields," Inverse Probl. 2, 161-183 (1986).
[CrossRef]

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

Duchene, B.

P. Grassin, B. Duchene, and W. Tabbara, "Diffraction tomography: some applications and extension to 3-d ultrasound imaging," in Mathematical Methods in Tomography (Springer-Verlag, 1991), pp. 98-105.
[CrossRef]

Gbur, G.

Grassin, P.

P. Grassin, B. Duchene, and W. Tabbara, "Diffraction tomography: some applications and extension to 3-d ultrasound imaging," in Mathematical Methods in Tomography (Springer-Verlag, 1991), pp. 98-105.
[CrossRef]

Huang, Y.

Jacobsen, C.

A. S. T. Beetz and C. Jacobsen, "Soft x-ray diffraction tomography: simulations and first experimental results," J. Phys. (Paris) 104, 31-34 (2003).

Kak, A.

S. Pan and A. Kak, "A computational study of reconstruction algorithms for diffraction tomography: interpolation versus filtered backpropagation," IEEE Trans. Acoust. Speech Signal Process. 31, 1262-1275 (1983).
[CrossRef]

Lauer, V.

V. Lauer, "New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope," J. Microsc. 205, 165-176 (2001).
[CrossRef]

Mast, T.

T. Mast, "Wideband quantitative ultrasonic imaging by time-domain diffraction tomography," J. Acoust. Soc. Am. 106, 3061-3071 (1999).
[CrossRef]

Nugent, K. A.

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, "Quantitative phase tomography," Opt. Commun. 175, 329-336 (2000).
[CrossRef]

Paganin, D.

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, "Quantitative phase tomography," Opt. Commun. 175, 329-336 (2000).
[CrossRef]

Pan, S.

S. Pan and A. Kak, "A computational study of reconstruction algorithms for diffraction tomography: interpolation versus filtered backpropagation," IEEE Trans. Acoust. Speech Signal Process. 31, 1262-1275 (1983).
[CrossRef]

Pan, X.

Roberts, A.

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, "Quantitative phase tomography," Opt. Commun. 175, 329-336 (2000).
[CrossRef]

Schotland, J. C.

P. S. Carney and J. C. Schotland, "Inverse scattering for near-field microscopy," Appl. Phys. Lett. 77, 2798-2800 (2000).
[CrossRef]

Shi, D.

M. A. Anastasio, D. Shi, Y. Huang, and G. Gbur, "Image reconstruction in spherical wave intensity diffraction tomography," J. Opt. Soc. Am. A 22, 2651-2661 (2005).
[CrossRef]

G. Gbur, M. A. Anastasio, Y. Huang, and D. Shi, "Spherical-wave intensity diffraction tomography," J. Opt. Soc. Am. A 22, 230-238 (2005).
[CrossRef]

D. Shi, M. A. Anastasio, Y. Huang, and G. Gbur, "Half-scan and single-plane intensity diffraction tomography for phase objects," Phys. Med. Biol. 49, 2733-2752 (2004).
[CrossRef] [PubMed]

M. A. Anastasio and D. Shi, "On the relationship between intensity diffraction tomography and phase-contrast tomography," in Developments in X-ray Tomography IV, U.Bonse, ed., Proc. SPIE 5535, 361-368 (2004).

Stamnes, J.

Stamnes, J. J.

T. C. Wedberg and J. J. Stamnes, "Recent results in optical diffraction microtomography," Meas. Sci. Technol. 7, 414-418 (1996).
[CrossRef]

Tabbara, W.

P. Grassin, B. Duchene, and W. Tabbara, "Diffraction tomography: some applications and extension to 3-d ultrasound imaging," in Mathematical Methods in Tomography (Springer-Verlag, 1991), pp. 98-105.
[CrossRef]

Teague, M. R.

Waseda, Y.

Y. Waseda, Novel Application of Anomalous (Resonance) X-ray Scattering for Structural Characterization of Disordered Materials (Springer-Verlag, 1984).
[CrossRef] [PubMed]

Wedberg, T.

T. Wedberg and J. Stamnes, "Quantitative imaging by optical diffraction tomography," Opt. Rev. 2, 28-31 (1995).
[CrossRef]

Wedberg, T. C.

T. C. Wedberg and J. J. Stamnes, "Recent results in optical diffraction microtomography," Meas. Sci. Technol. 7, 414-418 (1996).
[CrossRef]

Wolf, E.

G. Gbur and E. Wolf, "Diffraction tomography without phase information," Opt. Lett. 27, 1890-1892 (2002).
[CrossRef]

G. Gbur and E. Wolf, "Hybrid diffraction tomography without phase information," J. Opt. Soc. Am. A 19, 2194-2202 (2002).
[CrossRef]

G. Gbur and E. Wolf, "Determination of density correlation functions from scattering of polychromatic light," Opt. Commun. 168, 39-45 (1999).
[CrossRef]

E. Wolf, "Principles and development of diffraction tomography," in Trends in Optics, A.Consortini, ed. (Academic, 1996), pp. 83-110.
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

P. S. Carney and J. C. Schotland, "Inverse scattering for near-field microscopy," Appl. Phys. Lett. 77, 2798-2800 (2000).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process. (1)

S. Pan and A. Kak, "A computational study of reconstruction algorithms for diffraction tomography: interpolation versus filtered backpropagation," IEEE Trans. Acoust. Speech Signal Process. 31, 1262-1275 (1983).
[CrossRef]

Int. J. Imaging Syst. Technol. (1)

M. A. Anastasio and X. Pan, "Investigation of the noise properties of a new class of reconstruction methods in diffraction tomography," Int. J. Imaging Syst. Technol. 10, 437-446 (1999).
[CrossRef]

Inverse Probl. (2)

E. Clarkson, "Projections onto the range of the exponential Radon transform and reconstruction algorithms," Inverse Probl. 15, 563-571 (1999).
[CrossRef]

A. J. Devaney, "Reconstructive tomography with diffracting wavefields," Inverse Probl. 2, 161-183 (1986).
[CrossRef]

J. Acoust. Soc. Am. (1)

T. Mast, "Wideband quantitative ultrasonic imaging by time-domain diffraction tomography," J. Acoust. Soc. Am. 106, 3061-3071 (1999).
[CrossRef]

J. Microsc. (1)

V. Lauer, "New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope," J. Microsc. 205, 165-176 (2001).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

J. Phys. (Paris) (1)

A. S. T. Beetz and C. Jacobsen, "Soft x-ray diffraction tomography: simulations and first experimental results," J. Phys. (Paris) 104, 31-34 (2003).

Meas. Sci. Technol. (1)

T. C. Wedberg and J. J. Stamnes, "Recent results in optical diffraction microtomography," Meas. Sci. Technol. 7, 414-418 (1996).
[CrossRef]

Opt. Commun. (2)

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, "Quantitative phase tomography," Opt. Commun. 175, 329-336 (2000).
[CrossRef]

G. Gbur and E. Wolf, "Determination of density correlation functions from scattering of polychromatic light," Opt. Commun. 168, 39-45 (1999).
[CrossRef]

Opt. Lett. (1)

Opt. Rev. (1)

T. Wedberg and J. Stamnes, "Quantitative imaging by optical diffraction tomography," Opt. Rev. 2, 28-31 (1995).
[CrossRef]

Phys. Med. Biol. (1)

D. Shi, M. A. Anastasio, Y. Huang, and G. Gbur, "Half-scan and single-plane intensity diffraction tomography for phase objects," Phys. Med. Biol. 49, 2733-2752 (2004).
[CrossRef] [PubMed]

Ultrason. Imaging (1)

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

Other (4)

M. A. Anastasio and D. Shi, "On the relationship between intensity diffraction tomography and phase-contrast tomography," in Developments in X-ray Tomography IV, U.Bonse, ed., Proc. SPIE 5535, 361-368 (2004).

P. Grassin, B. Duchene, and W. Tabbara, "Diffraction tomography: some applications and extension to 3-d ultrasound imaging," in Mathematical Methods in Tomography (Springer-Verlag, 1991), pp. 98-105.
[CrossRef]

E. Wolf, "Principles and development of diffraction tomography," in Trends in Optics, A.Consortini, ed. (Academic, 1996), pp. 83-110.
[CrossRef]

Y. Waseda, Novel Application of Anomalous (Resonance) X-ray Scattering for Structural Characterization of Disordered Materials (Springer-Verlag, 1984).
[CrossRef] [PubMed]

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Figures (9)

Fig. 1
Fig. 1

Rotated coordinate system used to describe the tomographic measurement geometry.

Fig. 2
Fig. 2

In the measurement geometry, a plane wave with wavenumber k 1 irradiates the object, and the intensity of the forward-scattered wave field is measured on the plane z r = d . The measurement is repeated using an incident plane wave with wavenumber k 2 . Tomographic scanning is achieved by simultaneously rotating the source and detector plane about the x axis.

Fig. 3
Fig. 3

Intersection of the hemispherical Ewald surfaces of radius k 1 and k 2 with a plane of constant u in the 3D Fourier space. The usefulness of this figure for interpreting Fourier space symmetries is explained in Subsection 3B.

Fig. 4
Fig. 4

Two-dimensional scanning geometry utilized in the computer simulation studies.

Fig. 5
Fig. 5

Mathematical phantom object that represents (a) Re { n ( r ) } 1 and (b) and Im { n ( r ) } . (c) Profile through the central row in (a). (d) Profile through the central row in (b).

Fig. 6
Fig. 6

Images of (a) Re { n ( r ) } 1 and (b) Im { n ( r ) } reconstructed from the noiseless simulation data by use of method 1. The corresponding images reconstructed by use of method 2 are shown in (c) and (d).

Fig. 7
Fig. 7

Dashed curves in (a) and (b) display profiles through the central rows of Figs. 6a, 6b, respectively. In each case, the profile through the true phantom is represented by a solid line.

Fig. 8
Fig. 8

Images of (a) Re { n ( r ) } 1 and (b) Im { n ( r ) } reconstructed from the noisy simulation data by use of method 1. The corresponding images reconstructed by use of method 2 are shown in (c) and (d).

Fig. 9
Fig. 9

(a) Profiles through the central rows of Figs. 8a, 8c are represented by the solid and dashed curves, respectively. (b) Profiles through the central rows of Figs. 8b, 8d are represented by the solid and dashed curves, respectively.

Equations (46)

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

U ( r ; ω ) U i ( r ; ω ) exp [ ψ ( r ; ω ) ] ,
ψ ( r ; ω ) = k 2 4 π U i ( r ; ω ) V d 3 r f ω ( r ) U i ( r ; ω ) exp ( i k r r ) r r ,
f ω ( r ) n 2 ( r ; ω ) 1 .
I ( x , y r ; d , ϕ , ω ) = U ( x , y r ; d , ϕ , ω ) 2 = exp [ ψ ( x , y r ; d , ϕ , ω ) + ψ * ( x , y r ; d , ϕ , ω ) ] ,
D ( x , y r ; d , ϕ , ω ) = log [ I ( x , y r ; d , ϕ , ω ) ] = ψ ( x , y r ; d , ϕ , ω ) + ψ * ( x , y r ; d , ϕ , ω ) .
D ̂ ( u , v r ; d , ϕ , ω ) = 1 ( 2 π ) 2 d x d y r D ( x , y r ; d , ϕ , ω ) exp [ i ( u x + v r y r ) ] ,
F ̂ ω ( K ) = 1 ( 2 π ) 3 V d 3 r f ω ( r ) exp [ i K r ]
D ̂ ( u , v r ; d , ϕ , ω ) = i π k 2 ν { F ̂ ω [ u , v r ; ϕ , ω ] exp [ i ( ν k ) d ] ( F ̂ ω [ u , v r ; ϕ , ω ] ) * exp [ i ( ν k ) d ] } ,
F ̂ ω [ u , v r ; ϕ , ω ] F ̂ ω ( u s 1 + v r s 2 , r ( ϕ ) + ( ν k ) s 0 , r ( ϕ ) ) ,
ν k 2 u 2 v r 2 ,
f ( r ) f ω 1 ( r ) = f ω 2 ( r ) ,
F ̂ [ u , v r ; ϕ , ω 1 ] = F ̂ ω 1 [ u , v r ; ϕ , ω 1 ] ,
F ̂ [ u , v r ; ϕ , ω 2 ] = F ̂ ω 2 [ u , v r ; ϕ , ω 2 ] ,
ν j [ k j 2 u 2 v r 2 ] 1 2 ,
D ̂ ( u , v r ; d , ϕ , ω 1 ) = i π k 1 2 ν 1 { F ̂ [ u , v r ; ϕ , ω 1 ] exp [ i ( ν 1 k 1 ) d ] ( F ̂ [ u , v r ; ϕ , ω 1 ] ) * exp [ i ( ν 1 k 1 ) d ] } ,
D ̂ ( u , v r ; d , ϕ , ω 2 ) = i π k 2 2 ν 2 { F ̂ [ u , v r ; ϕ , ω 2 ] exp [ i ( ν 2 k 2 ) d ] ( F ̂ [ u , v r ; ϕ , ω 2 ] ) * exp [ i ( ν 2 k 2 ) d ] }
F ̂ [ u , v r ; ϕ + ϕ , ω 2 ] = F ̂ [ u , v r ; ϕ , ω 1 ] ,
v r = [ R 2 ( R 2 + u 2 ) 4 k 1 2 ] 1 2 sgn ( v r ) ,
R [ v r 2 + ( ν 2 k 2 ) 2 ] 1 2 .
ϕ = arctan ( v r ν 1 k 1 ) + arctan ( v r ν 2 k 2 ) ,
ν 1 = [ k 1 2 u 2 ( v r ) 2 ] 1 2 .
F ̂ [ u , v r ; ϕ ϕ , ω 2 ] = F ̂ [ u , v r ; ϕ , ω 1 ] ,
ϕ = arctan ( v r ν 1 k 1 ) arctan ( v r ν 2 k 2 ) .
D ̂ ( u , v r ; d , ϕ , ω 1 ) = i π k 1 2 ν 1 { F ̂ [ u , v r ; ϕ , ω 1 ] exp [ i ( ν 1 k 1 ) d ] ( F ̂ [ u , v r ; ϕ , ω 1 ] ) * exp [ i ( ν 1 k 1 ) d ] } ,
D ̂ ( u , v r ; d , ϕ , ω 1 ) = i π k 1 2 ν 1 { F ̂ [ u , v r ; ϕ + ϕ ( v r ) , ω 2 ] exp [ i ( ν 1 k 1 ) d ] ( F ̂ [ u , v r ; ϕ ϕ ( v r ) , ω 2 ] ) * exp [ i ( ν 1 k 1 ) d ] } ,
D ̂ ( u , v r ; d , ϕ , ω j ) = n = D ̂ n ( u , v r ; d , ω j ) exp [ i n ϕ ] ,
F ̂ [ u , v r ; ϕ , ω 2 ] = n = F ̂ n [ u , v r ; ω 2 ] exp [ i n ϕ ] ,
D ̂ n ( u , v r ; d , ω j ) = 1 2 π 0 2 π d ϕ D ̂ ( u , v r ; d , ϕ , ω j ) exp [ i n ϕ ] ,
F ̂ n [ u , v r ; ω 2 ] = 1 2 π 0 2 π d ϕ F ̂ [ u , v r ; ϕ , ω 2 ] exp [ i n ϕ ] .
D ̂ n ( u , v r ; d , ω 1 ) = i π k 1 2 ν 1 { F ̂ n [ u , v r ; ω 2 ] exp [ i n ϕ ( v r ) + i ( ν 1 k 1 ) d ] ( F ̂ n [ u , v r ; ω 2 ] ) * exp [ i n ϕ ( v r ) i ( ν 1 k 1 ) d ] } .
D ̂ n ( u , v r ; d , ω 2 ) = i π k 2 2 ν 2 { F ̂ n [ u , v r ; ω 2 ] exp [ i ( ν 2 k 2 ) d ] ( F ̂ n [ u , v r ; ω 2 ] ) * exp [ i ( ν 2 k 2 ) d ] } .
F ̂ n [ u , v r ; ω 2 ] = i ν 2 π k 2 2 exp [ i ( ν 2 k 2 ) d ] D ̂ n ( u , v r ; d , ω 2 ) ( k 2 k 1 ) 2 ( ν 1 ν 2 ) D ̂ n ( u , v r ; d , ω 1 ) exp [ i n ϕ ( v r ) + i ( ν 1 k 1 ν 2 + k 2 ) d ] 1 exp [ 2 i n ϕ ( v r ) + 2 i ( ν 1 k 1 ν 2 + k 2 ) d ] .
D ̂ ( u , v r ; d , ϕ , ω 1 ) = i π k 1 2 ν 1 { F ̂ [ u , v r ; ϕ , ω 1 ] exp [ i ( ν 1 k 1 ) d ] ( F ̂ [ u , v r ; ϕ , ω 1 ] ) * exp [ i ( ν 1 k 1 ) d ] } .
D ̂ ( u , v r ; d , ϕ , ω 1 ) = i π k 1 2 ν 1 { F ̂ [ u , v r ; ϕ ϕ ( v r ) , ω 2 ] exp [ i ( ν 1 k 1 ) d ] ( F ̂ [ u , v r ; ϕ + ϕ ( v r ) , ω 2 ] ) * exp [ i ( ν 1 k 1 ) d ] } .
D ̂ n ( u , v r ; d , ω 1 ) = i π k 1 2 ν 1 { F ̂ n [ u , v r ; ω 2 ] exp [ i n ϕ ( v r ) + i ( ν 1 k 1 ) d ] ( F ̂ n [ u , v r ; ω 2 ] ) * exp [ i n ϕ ( v r ) i ( ν 1 k 1 ) d ] } .
F ̂ n [ u , v r ; ω 2 ] = i ν 2 π k 2 2 exp [ i ( ν 2 k 2 ) d ] D ̂ n ( u , v r ; d , ω 2 ) ( k 2 k 1 ) 2 ( ν 1 ν 2 ) D ̂ n ( u , v r ; d , ω 1 ) exp [ i n ϕ ( v r ) + i ( ν 1 k 1 ν 2 + k 2 ) d ] 1 exp [ 2 i n ϕ ( v r ) + 2 i ( ν 1 k 1 ν 2 + k 2 ) d ] .
K A ( ϕ ) = u s 1 v r s 2 , r ( ϕ ) + ( ν 2 k 2 ) s 0 , r ( ϕ ) ,
K C ( ϕ ) = u s 1 + v r s 2 , r ( ϕ ) + ( ν 2 k 2 ) s 0 , r ( ϕ ) .
R = K C ( K C s 1 ) s 1 = v r s 2 , r ( ϕ ) + ( ν 2 k 2 ) s 0 , r ( ϕ ) = [ v r 2 + ( ν 2 k 2 ) 2 ] 1 2 ,
K B ( ϕ ) = u s 1 v r s 2 , r ( ϕ ) + ( ν 1 k 1 ) s 0 , r ( ϕ ) ,
K D ( ϕ ) = u s 1 + v r s 2 , r ( ϕ ) + ( ν 1 k 1 ) s 0 , r ( ϕ ) ,
R 2 = v r 2 + ( ν 1 k 1 ) 2 = v r 2 + ( k 1 2 u 2 v r 2 k 1 ) 2 .
( R 2 + u 2 ) 2 + 4 k 1 2 ( v r 2 R 2 ) = 0 ,
v r = [ R 2 ( R 2 + u 2 ) 2 4 k 1 2 ] 1 2 .
K A ( ϕ + ϕ ) = K B ( ϕ ) , K C ( ϕ + ϕ ) = K D ( ϕ ) ,
K A ( ϕ ϕ ) = K D ( ϕ ) , K C ( ϕ ϕ ) = K B ( ϕ ) ,

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