Abstract

A reconstruction theory for intensity diffraction tomography (I-DT) has been proposed that permits reconstruction of a weakly scattering object without explicit knowledge of phase information. We investigate the I-DT reconstruction problem assuming an incident (paraxial) spherical wave and scanning geometries that employ fixed source-to-object distances. Novel reconstruction methods are derived by identifying and exploiting tomographic symmetries and the rotational invariance of the problem. An underlying theme is that symmetries in tomographic imaging systems can facilitate solutions for phase-retrieval problems. A preliminary numerical investigation of the developed reconstruction methods is presented.

© 2005 Optical Society of America

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  1. E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
    [CrossRef]
  2. E. Wolf, “Principles and development of diffraction tomography,” in Trends in Optics, A. Consortini, ed. (Academic, 1996).
    [CrossRef]
  3. A. J. Devaney, “Diffraction tomography,” in Inverse Methods in Electromagnetic Imaging, Part 2, W. M. Boerner, ed., NATO ASI Series (Reidel, 1983), pp. 1107–1135.
  4. A. J. Devaney, “Geophysical diffraction tomography,” IEEE Trans. Geosci. Remote Sens. 22, 3–13 (1984).
    [CrossRef]
  5. T. Wedberg, J. Stamnes, “Quantitative imaging by optical diffraction tomography,” Opt. Rev. 2, 28–31 (1995).
    [CrossRef]
  6. G. Gbur, E. Wolf, “Diffraction tomography without phase information,” Opt. Lett. 27, 1890–1892 (2002).
    [CrossRef]
  7. G. Gbur, E. Wolf, “Hybrid diffraction tomography without phase information,” J. Opt. Soc. Am. A 19, 2194–2202 (2002).
    [CrossRef]
  8. V. Lauer, “New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope,” J. Microsc. (Oxford) 205, 165–176 (2001).
    [CrossRef]
  9. T. C. Wedberg, J. J. Stamnes, “Recent results in optical diffraction microtomography,” Meas. Sci. Technol. 7, 414–418 (1996).
    [CrossRef]
  10. A. S. T. Beetz, C. Jacobsen, “Soft x-ray diffraction tomography: simulations and first experimental results,” J. Phys. IV 104, 31–34 (2003).
  11. K. A. Nugent, T. E. Gureyev, D. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
    [CrossRef] [PubMed]
  12. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
    [CrossRef]
  13. G. Gbur, M. A. Anastasio, Y. Huang, D. Shi, “Spherical-wave intensity diffraction tomography,” J. Opt. Soc. Am. A 22, 230–238 (2005).
    [CrossRef]
  14. A. J. Devaney, “Generalized projection-slice theorem for fan-beam diffraction tomography,” Ultrason. Imaging 7, 264–275 (1985).
    [CrossRef] [PubMed]
  15. M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, 1999).
    [CrossRef]
  16. M. A. Anastasio, X. Pan, “An improved reconstruction algorithm for 3D diffraction tomography using spherical-wave sources,” IEEE Trans. Biomed. Eng. 50, 517–521 (2003).
    [CrossRef] [PubMed]
  17. Z. Lu, “Multidimensional structure diffraction tomography for varying object orientation through generalised scattered waves,” Inverse Probl. 1, 339–356 (1985).
    [CrossRef]
  18. M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, 1998).
    [CrossRef]
  19. M. Slaney, A. C. Kak, L. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. 32, 860–874 (1984).
    [CrossRef]
  20. B. Chen, J. Stamnes, “Validity of diffraction tomography based on the first-Born and first-Rytov approximations,” Appl. Opt. 37, 2996–3006 (1998).
    [CrossRef]
  21. M. A. Anastasio, X. Pan, “Full- and minimal-scan reconstruction algorithms for fan-beam diffraction tomography,” Appl. Opt. 40, 3334–3345 (2001).
    [CrossRef]
  22. S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Paganin, A. Pogany, A. Stevenson, S. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11, 2289–2302 (2003).
    [CrossRef] [PubMed]
  23. A. Barty, K. A. Nugent, A. Roberts, D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175, 329–336 (2000).
    [CrossRef]
  24. T. Gureyev, T. Davis, A. Pogany, S. Mayo, S. Wilkins, “Optical phase retrieval by use of first Born- and Rytov-type approximations,” Appl. Opt. 43, 2418–2430 (2004).
    [CrossRef] [PubMed]

2005 (1)

2004 (1)

2003 (3)

M. A. Anastasio, X. Pan, “An improved reconstruction algorithm for 3D diffraction tomography using spherical-wave sources,” IEEE Trans. Biomed. Eng. 50, 517–521 (2003).
[CrossRef] [PubMed]

S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Paganin, A. Pogany, A. Stevenson, S. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11, 2289–2302 (2003).
[CrossRef] [PubMed]

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

2002 (2)

2001 (2)

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

M. A. Anastasio, X. Pan, “Full- and minimal-scan reconstruction algorithms for fan-beam diffraction tomography,” Appl. Opt. 40, 3334–3345 (2001).
[CrossRef]

2000 (1)

A. Barty, K. A. Nugent, A. Roberts, D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175, 329–336 (2000).
[CrossRef]

1998 (1)

1996 (2)

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

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

1995 (1)

T. Wedberg, J. Stamnes, “Quantitative imaging by optical diffraction tomography,” Opt. Rev. 2, 28–31 (1995).
[CrossRef]

1985 (2)

Z. Lu, “Multidimensional structure diffraction tomography for varying object orientation through generalised scattered waves,” Inverse Probl. 1, 339–356 (1985).
[CrossRef]

A. J. Devaney, “Generalized projection-slice theorem for fan-beam diffraction tomography,” Ultrason. Imaging 7, 264–275 (1985).
[CrossRef] [PubMed]

1984 (2)

M. Slaney, A. C. Kak, L. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. 32, 860–874 (1984).
[CrossRef]

A. J. Devaney, “Geophysical diffraction tomography,” IEEE Trans. Geosci. Remote Sens. 22, 3–13 (1984).
[CrossRef]

1983 (1)

1969 (1)

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

Anastasio, M. A.

Barnea, Z.

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

Barty, A.

A. Barty, K. A. Nugent, A. Roberts, D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175, 329–336 (2000).
[CrossRef]

Beetz, A. S. T.

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

Bertero, M.

M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, 1998).
[CrossRef]

Boccacci, P.

M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, 1998).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, 1999).
[CrossRef]

Chen, B.

Cookson, D.

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

Davis, T.

Devaney, A. J.

A. J. Devaney, “Generalized projection-slice theorem for fan-beam diffraction tomography,” Ultrason. Imaging 7, 264–275 (1985).
[CrossRef] [PubMed]

A. J. Devaney, “Geophysical diffraction tomography,” IEEE Trans. Geosci. Remote Sens. 22, 3–13 (1984).
[CrossRef]

A. J. Devaney, “Diffraction tomography,” in Inverse Methods in Electromagnetic Imaging, Part 2, W. M. Boerner, ed., NATO ASI Series (Reidel, 1983), pp. 1107–1135.

Gbur, G.

Gureyev, T.

Gureyev, T. E.

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

Huang, Y.

Jacobsen, C.

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

Kak, A. C.

M. Slaney, A. C. Kak, L. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. 32, 860–874 (1984).
[CrossRef]

Larsen, L.

M. Slaney, A. C. Kak, L. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. 32, 860–874 (1984).
[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. (Oxford) 205, 165–176 (2001).
[CrossRef]

Lu, Z.

Z. Lu, “Multidimensional structure diffraction tomography for varying object orientation through generalised scattered waves,” Inverse Probl. 1, 339–356 (1985).
[CrossRef]

Mayo, S.

Miller, P.

Nugent, K. A.

A. Barty, K. A. Nugent, A. Roberts, D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175, 329–336 (2000).
[CrossRef]

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

Paganin, D.

S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Paganin, A. Pogany, A. Stevenson, S. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11, 2289–2302 (2003).
[CrossRef] [PubMed]

A. Barty, K. A. Nugent, A. Roberts, D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175, 329–336 (2000).
[CrossRef]

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

Pan, X.

M. A. Anastasio, X. Pan, “An improved reconstruction algorithm for 3D diffraction tomography using spherical-wave sources,” IEEE Trans. Biomed. Eng. 50, 517–521 (2003).
[CrossRef] [PubMed]

M. A. Anastasio, X. Pan, “Full- and minimal-scan reconstruction algorithms for fan-beam diffraction tomography,” Appl. Opt. 40, 3334–3345 (2001).
[CrossRef]

Pogany, A.

Roberts, A.

A. Barty, K. A. Nugent, A. Roberts, D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175, 329–336 (2000).
[CrossRef]

Shi, D.

Slaney, M.

M. Slaney, A. C. Kak, L. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. 32, 860–874 (1984).
[CrossRef]

Stamnes, J.

B. Chen, J. Stamnes, “Validity of diffraction tomography based on the first-Born and first-Rytov approximations,” Appl. Opt. 37, 2996–3006 (1998).
[CrossRef]

T. Wedberg, J. Stamnes, “Quantitative imaging by optical diffraction tomography,” Opt. Rev. 2, 28–31 (1995).
[CrossRef]

Stamnes, J. J.

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

Stevenson, A.

Teague, M. R.

Wedberg, T.

T. Wedberg, J. Stamnes, “Quantitative imaging by optical diffraction tomography,” Opt. Rev. 2, 28–31 (1995).
[CrossRef]

Wedberg, T. C.

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

Wilkins, S.

Wolf, E.

G. Gbur, E. Wolf, “Diffraction tomography without phase information,” Opt. Lett. 27, 1890–1892 (2002).
[CrossRef]

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

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

E. Wolf, “Principles and development of diffraction tomography,” in Trends in Optics, A. Consortini, ed. (Academic, 1996).
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, 1999).
[CrossRef]

Appl. Opt. (3)

IEEE Trans. Biomed. Eng. (1)

M. A. Anastasio, X. Pan, “An improved reconstruction algorithm for 3D diffraction tomography using spherical-wave sources,” IEEE Trans. Biomed. Eng. 50, 517–521 (2003).
[CrossRef] [PubMed]

IEEE Trans. Geosci. Remote Sens. (1)

A. J. Devaney, “Geophysical diffraction tomography,” IEEE Trans. Geosci. Remote Sens. 22, 3–13 (1984).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

M. Slaney, A. C. Kak, L. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. 32, 860–874 (1984).
[CrossRef]

Inverse Probl. (1)

Z. Lu, “Multidimensional structure diffraction tomography for varying object orientation through generalised scattered waves,” Inverse Probl. 1, 339–356 (1985).
[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. (Oxford) 205, 165–176 (2001).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Phys. IV (1)

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

Meas. Sci. Technol. (1)

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

Opt. Commun. (2)

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

A. Barty, K. A. Nugent, A. Roberts, D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175, 329–336 (2000).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Opt. Rev. (1)

T. Wedberg, J. Stamnes, “Quantitative imaging by optical diffraction tomography,” Opt. Rev. 2, 28–31 (1995).
[CrossRef]

Phys. Rev. Lett. (1)

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

Ultrason. Imaging (1)

A. J. Devaney, “Generalized projection-slice theorem for fan-beam diffraction tomography,” Ultrason. Imaging 7, 264–275 (1985).
[CrossRef] [PubMed]

Other (4)

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, 1999).
[CrossRef]

M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, 1998).
[CrossRef]

E. Wolf, “Principles and development of diffraction tomography,” in Trends in Optics, A. Consortini, ed. (Academic, 1996).
[CrossRef]

A. J. Devaney, “Diffraction tomography,” in Inverse Methods in Electromagnetic Imaging, Part 2, W. M. Boerner, ed., NATO ASI Series (Reidel, 1983), pp. 1107–1135.

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Figures (9)

Fig. 1
Fig. 1

Rotated coordinate system used to describe the tomographic measurement geometry.

Fig. 2
Fig. 2

Measurement geometry of spherical-wave I-DT.

Fig. 3
Fig. 3

Intersection of the ellipsoidal surfaces S 1 and S 2 with a plane of constant u in the 3D Fourier space ( u = 0.3 in this example). The diagram was generated using k = 1 , and S 1 and S 2 correspond to values α = 0.7 and α = 0.5 , respectively.

Fig. 4
Fig. 4

(a) Real component of the phantom object. (b) Imaginary component of the phantom object. (c) Profile through the central row in (a). (d) Profile through the central row in (b).

Fig. 5
Fig. 5

Shown is the 2D scanning geometry employed in the numerical simulations. Intensity data were acquired on three detector planes at each view angle.

Fig. 6
Fig. 6

(a) Real and (b) imaginary components of the refractive-index distribution n ( r ) reconstructed from noiseless data by use of method 1. (c) and (d) The corresponding images reconstructed by use of method 2.

Fig. 7
Fig. 7

(a) Profiles through the central rows of Figs. 6a, 6c are represented by the solid and dashed curves, respectively. (b) Profiles through the central rows of Figs. 6b, 6d are represented by the solid and dashed curves, respectively. In both graphs, the dashed curves are completely obscured by the solid curves.

Fig. 8
Fig. 8

(a) Real and (b) imaginary components of the refractive-index distribution n ( r ) reconstructed from noisy data by use of method 1. (c) and (d) The corresponding images reconstructed by use of method 2.

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 (49)

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U i ( r ) = exp [ j k r r 0 ] r r 0 ,
f ( r ) = k 2 4 π [ n 2 ( r ) 1 ] .
U ( r ) = U i ( r ) exp [ ψ z 0 ( r ) ] ,
ψ ̂ z 0 ( u , v r , ϕ ; z r ) = 1 ( 2 π ) 2 R 2 d x d y r ψ z 0 ( x , y r , ϕ ; z r ) exp [ j ( u x + v r y r ) ]
F ̂ ( K ) = 1 ( 2 π ) 3 R 3 d r f ( r ) exp [ j K r ]
Q z 0 ( x , y r , ϕ ; z r ) = ψ z 0 ( x , y r , ϕ ; z r ) r d r 0 ,
α [ z 0 z r + z 0 ] 1 2
w α [ k 2 ( u α ) 2 ( v r α ) 2 ] 1 2 ,
F ̂ [ u α 2 , v r α 2 , ϕ ] F ̂ ( u α 2 s 1 + v r α 2 s 2 , r + ( w α k ) s 0 , r )
D z 0 ( x , y r , ϕ ; z r ) = 1 r d r 0 ln [ I z 0 ( x , y r , ϕ ; z r ) ] ,
D ̂ z 0 ( u , v r , ϕ ; z r ) = ( 2 π ) 2 j z 0 w α α 2 { F ̂ [ u α 2 , v r α 2 , ϕ ] exp [ j ( w α k ) z r ] F ̂ * [ u α 2 , v r α 2 , ϕ ] exp [ j ( w α k ) z r ] } ,
α 1 [ z 0 d + z 0 ] 1 2 , α 2 [ z 0 ( d + Δ ) + z 0 ] 1 2
D ̂ z 0 ( u , v r , ϕ ; d ) = ( 2 π ) 2 j z 0 w α 1 α 1 2 { F ̂ [ u α 1 2 , v r α 1 2 , ϕ ] exp [ j ( w α 1 k ) d ] F ̂ * [ u α 1 2 , v r α 1 2 , ϕ ] exp [ j ( w α 1 k ) d ] } ,
D ̂ z 0 ( u , v r , ϕ ; d + Δ ) = ( 2 π ) 2 j z 0 w α 2 α 2 2 { F ̂ [ u α 2 2 , v r α 2 2 , ϕ ] exp [ j ( w α 2 k ) ( d + Δ ) ] F ̂ * [ u α 2 2 , v r α 2 2 , ϕ ] exp [ j ( w α 2 k ) ( d + Δ ) ] } .
F ̂ [ u α 1 2 , v r α 1 2 , ϕ ] = D ̂ z 0 ( u , v r , ϕ ; d ) D ̂ z 0 ( u , v r , ϕ ; d + Δ ) exp [ j ( w α 1 k ) Δ ] z 0 z 0 j ( 2 π ) 2 exp [ j ( w α 1 k ) d ] ( 1 exp [ j 2 ( w α 1 k ) Δ ] ) ( z 0 w α 1 α 1 2 ) .
R [ ( v r α 1 2 ) 2 + ( w α 1 k ) 2 ] 1 2 .
w α 2 [ k 2 ( u α 2 ) 2 ( v r α 2 ) 2 ] 1 2
u = u ( α 2 α 1 ) 2 .
R = [ ( v r α 2 2 ) 2 + ( w α 2 k ) 2 ] 1 2 .
F ̂ [ u α 1 2 , v r α 1 2 , ϕ + ϕ ] = F ̂ [ u α 2 2 , v r α 2 2 , ϕ ] ,
v r = sgn ( v r ) [ b + [ b 2 4 a c ] 1 2 2 a ] 1 2 ,
a = ( 1 α 2 4 1 α 2 2 ) 2 ,
b = 4 k 2 α 2 4 2 ( R 2 + ( u α 2 ) 2 ) ( 1 α 2 4 1 α 2 2 ) ,
c = ( u α 2 ) 4 + 2 R 2 ( u α 2 ) 2 + R 4 4 k 2 R 2
ϕ = arctan ( v r α 2 2 w α 2 , r k ) + arctan ( v r α 1 2 w α 1 k ) .
F ̂ [ u α 1 2 , v r α 1 2 , ϕ ϕ ] = F ̂ [ u α 2 2 , v r α 2 2 , ϕ ] ,
ϕ = arctan ( v r α 2 2 w α 2 k ) arctan ( v r α 1 2 w α 1 k ) .
D ̂ ( u , v r , ϕ ; z r ) D ̂ z 0 ( u , v r , ϕ ; z r ) z 0 w α α 2 ( 2 π ) 2 j exp [ j ( w α k ) z r ] .
D ̂ ( u , v r , ϕ ; d ) = F ̂ [ u α 1 2 , v r α 1 2 , ϕ ] F ̂ * [ u α 1 2 , v r α 1 2 , ϕ ] exp [ 2 j ( w α 1 k ) d ] ,
D ̂ ( u , v r , ϕ ; d + Δ ) = F ̂ [ u α 2 2 , v r α 2 2 , ϕ ] F ̂ * [ u α 2 2 , v r α 2 2 , ϕ ] exp [ 2 j ( w α 2 k ) ( d + Δ ) ] ,
D ̂ ( u , v r , ϕ ; d + Δ ) = F ̂ [ u α 2 2 , v r α 2 2 , ϕ ] F ̂ * [ u α 2 2 , v r α 2 2 , ϕ ] exp [ 2 j ( w α 2 k ) ( d + Δ ) ] ,
D ̂ ( u , v r , ϕ ; d + Δ ) = F ̂ [ u α 1 2 , v r α 1 2 , ϕ + ϕ ( v r ) ] F ̂ * [ u α 1 2 , v r α 1 2 , ϕ ϕ ( v r ) ] exp [ 2 j ( w α 2 k ) ( d + Δ ) ] ,
D ̂ ( u , v r , ϕ ; z r ) = n = D ̂ n ( u , v r ; z r ) exp [ j n ϕ ] ,
D ̂ n ( u , u r ; z r ) = 1 2 π 0 2 π d ϕ D ̂ ( u , v r , ϕ ; z r ) exp [ j n ϕ ] .
D ̂ n ( u , v r ; d ) = F ̂ n [ u α 1 2 , v r α 1 2 ] ( F ̂ n [ u α 1 2 , v r α 1 2 ] ) * exp [ 2 j ( w α 1 k ) d ] ,
F ̂ n [ u α 1 2 , v r α 1 2 ] = 1 2 π 0 2 π d ϕ F ̂ [ u α 1 2 , v r α 1 2 , ϕ ] exp [ j n ϕ ] .
D ̂ n ( u , v r ; d + Δ ) = exp [ j n ϕ ( v r ) ] F ̂ n [ u α 1 2 , v r α 1 2 ] exp [ j n ϕ ( v r ) ] ( F ̂ n [ u α 1 2 , v r α 1 2 ] ) * exp [ 2 j ( w α 2 , r k ) ( d + Δ ) ] .
F ̂ n [ u α 1 2 , v r α 1 2 ] = D ̂ n ( u , v r ; d ) D ̂ n ( u , v r ; d + Δ ) exp [ j n ϕ ( v r ) + 2 j ( w α 2 , r k ) ( d + Δ ) 2 j ( w α 1 k ) d ] 1 exp [ 2 j ( n ϕ ( v r ) + ( w α 2 , r k ) ( d + Δ ) ( w α 1 k ) d ) ]
F ̂ [ u α 1 2 , v r α 1 2 , ϕ ] = n = F ̂ n [ u α 1 2 , v r α 1 2 ] exp [ j n ϕ ] ,
D ̂ ( u , v r , ϕ ; d + Δ ) = F ̂ [ u α 2 2 , v r α 2 2 , ϕ ] F ̂ * [ u α 2 2 , v r α 2 2 , ϕ ] exp [ j ( w α 2 , r k ) ( d + Δ ) ] .
D ̂ ( u , v r , ϕ ; d + Δ ) = F ̂ [ u α 1 2 , v r α 1 2 , ϕ ϕ ( v r ) ] F ̂ * [ u α 1 2 , v r α 1 2 , ϕ + ϕ ( v r ) ] exp [ 2 j ( w α 2 , r k ) ( d + Δ ) ] ,
D ̂ n ( u , v r ; d + Δ ) = exp [ j n ϕ ( v r ) ] F ̂ n [ u α 1 2 , v r α 1 2 ] exp [ + j n ϕ ( v r ) ] ( F ̂ n [ u α 1 2 , v r α 1 2 ] ) * exp [ 2 j ( w α 2 , r k ) ( d + Δ ) ] .
F ̂ n [ u α 1 2 , v r α 1 2 ] = D ̂ n ( u , v r ; d ) D ̂ n ( u , v r ; d + Δ ) exp [ j n ϕ ( v r ) + 2 j ( w α 2 , r k ) ( d + Δ ) 2 j ( w α 1 k ) d ] 1 exp [ 2 j ( n ϕ ( v r ) + ( w α 2 , r k ) ( d + Δ ) ( w α 1 k ) d ) ]
F n ( i ) [ v r α 1 2 ] N n ( i ) [ v r ] D n ( i ) [ v r ] ,
F n [ v r α 1 2 ] = { F n ( 1 ) [ v r α 1 2 ] : D n ( 1 ) [ v r ] D n ( 3 ) [ v r ] F n ( 3 ) [ v r α 1 2 ] : otherwise . } .
R 2 = ( v r α 2 2 ) 2 + ( w α 2 k ) 2 = ( v r α 2 2 ) + ( [ k 2 ( u α 2 ) ( v r α 2 ) 2 ] 1 2 k ) 2 ,
a ( v r ) 4 + b ( v r ) 2 + c = 0 ,
v r ( 1 ) = v r ( 2 ) = [ b + [ b 2 4 a c ] 1 2 2 a ] 1 2 ,
v r ( 3 ) = v r ( 4 ) = [ b [ b 2 4 a c ] 1 2 2 a ] 1 2 .

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