Abstract

We continue our study of the inverse scattering problem for diffuse light. In contrast to our earlier work, in which we considered the linear inverse problem, we now consider the nonlinear problem. We obtain a solution to this problem in the form of a functional series expansion. The first term in this expansion is the pseudoinverse of the linearized forward-scattering operator and leads to the linear inversion formulas that we have reported previously. The higher-order terms represent nonlinear corrections to this result. We illustrate our results with computer simulations in model systems.

© 2003 Optical Society of America

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  1. V. A. Markel, J. C. Schotland, “Inverse problem in optical diffusion tomography. I. Fourier–Laplace inversion formulas,” J. Opt. Soc. Am. A 18, 1336–1347 (2001).
    [CrossRef]
  2. V. A. Markel, J. C. Schotland, “Inverse problem in optical diffusion tomography. II. Inversion with boundary conditions,” J. Opt. Soc. Am. A 19, 558–566 (2002).
    [CrossRef]
  3. V. A. Markel, J. C. Schotland, “Inverse problem in optical diffusion tomography. III. Inversion formulas and singular-value decomposition,” J. Opt. Soc. Am. A 20, 890–902 (2002).
    [CrossRef]
  4. R. Aronson, “Boundary conditions for diffuse light,” J. Opt. Soc. Am. A 12, 2532–2539 (1995).
    [CrossRef]
  5. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
    [CrossRef]
  6. H. E. Moses, “Calculation of the scattering potential from reflection coefficients,” Phys. Rev. 102, 550–567 (1956).
    [CrossRef]
  7. R. T. Prosser, “Formal solutions of the inverse scattering problem,” J. Math. Phys. 10, 1819–1822 (1969).
    [CrossRef]
  8. R. Snieder, “A perturbative analysis of non-linear inversion,” Geophys. J. Int. 101, 545–556 (1990).
    [CrossRef]
  9. G. A. Tsihrintzis, A. J. Devaney, “A Volterra series approach to nonlinear travel time tomography,” IEEE Trans. Geosci. Remote Sens. 38, 1733–1742 (2000).
    [CrossRef]
  10. V. A. Markel, J. C. Schotland, “Inverse scattering for the diffusion equation with general boundary conditions,” Phys. Rev. E 64, R035601 (2001).
    [CrossRef]
  11. P. S. Carney, J. C. Schotland, “Inverse scattering for near-field microscopy,” Appl. Phys. Lett. 77, 2798–2800 (2000).
    [CrossRef]
  12. P. S. Carney, V. A. Markel, J. C. Schotland, “Near-field tomography without phase retrieval,” Phys. Rev. Lett. 86, 5874–5877 (2001).
    [CrossRef] [PubMed]
  13. P. S. Carney, J. C. Schotland, “Three-dimensional total internal reflection microscopy,” Opt. Lett. 26, 1072–1074 (2001).
    [CrossRef]
  14. P. S. Carney, J. C. Schotland, “Determination of three-dimensional structure in photon scanning tunneling microscopy,” J. Opt. A 4, S140–S144 (2002).
    [CrossRef]
  15. V. A. Markel, J. C. Schotland, “Effects of sampling and limited data in optical tomography,” Appl. Phys. Lett. 81, 1180–1182 (2002).
    [CrossRef]
  16. D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffusive photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 19, 4887–4891 (1994).
    [CrossRef]

2002 (4)

P. S. Carney, J. C. Schotland, “Determination of three-dimensional structure in photon scanning tunneling microscopy,” J. Opt. A 4, S140–S144 (2002).
[CrossRef]

V. A. Markel, J. C. Schotland, “Effects of sampling and limited data in optical tomography,” Appl. Phys. Lett. 81, 1180–1182 (2002).
[CrossRef]

V. A. Markel, J. C. Schotland, “Inverse problem in optical diffusion tomography. II. Inversion with boundary conditions,” J. Opt. Soc. Am. A 19, 558–566 (2002).
[CrossRef]

V. A. Markel, J. C. Schotland, “Inverse problem in optical diffusion tomography. III. Inversion formulas and singular-value decomposition,” J. Opt. Soc. Am. A 20, 890–902 (2002).
[CrossRef]

2001 (4)

V. A. Markel, J. C. Schotland, “Inverse problem in optical diffusion tomography. I. Fourier–Laplace inversion formulas,” J. Opt. Soc. Am. A 18, 1336–1347 (2001).
[CrossRef]

P. S. Carney, J. C. Schotland, “Three-dimensional total internal reflection microscopy,” Opt. Lett. 26, 1072–1074 (2001).
[CrossRef]

V. A. Markel, J. C. Schotland, “Inverse scattering for the diffusion equation with general boundary conditions,” Phys. Rev. E 64, R035601 (2001).
[CrossRef]

P. S. Carney, V. A. Markel, J. C. Schotland, “Near-field tomography without phase retrieval,” Phys. Rev. Lett. 86, 5874–5877 (2001).
[CrossRef] [PubMed]

2000 (2)

G. A. Tsihrintzis, A. J. Devaney, “A Volterra series approach to nonlinear travel time tomography,” IEEE Trans. Geosci. Remote Sens. 38, 1733–1742 (2000).
[CrossRef]

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

1999 (1)

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[CrossRef]

1995 (1)

1994 (1)

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffusive photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 19, 4887–4891 (1994).
[CrossRef]

1990 (1)

R. Snieder, “A perturbative analysis of non-linear inversion,” Geophys. J. Int. 101, 545–556 (1990).
[CrossRef]

1969 (1)

R. T. Prosser, “Formal solutions of the inverse scattering problem,” J. Math. Phys. 10, 1819–1822 (1969).
[CrossRef]

1956 (1)

H. E. Moses, “Calculation of the scattering potential from reflection coefficients,” Phys. Rev. 102, 550–567 (1956).
[CrossRef]

Aronson, R.

Arridge, S. R.

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[CrossRef]

Boas, D. A.

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffusive photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 19, 4887–4891 (1994).
[CrossRef]

Carney, P. S.

P. S. Carney, J. C. Schotland, “Determination of three-dimensional structure in photon scanning tunneling microscopy,” J. Opt. A 4, S140–S144 (2002).
[CrossRef]

P. S. Carney, V. A. Markel, J. C. Schotland, “Near-field tomography without phase retrieval,” Phys. Rev. Lett. 86, 5874–5877 (2001).
[CrossRef] [PubMed]

P. S. Carney, J. C. Schotland, “Three-dimensional total internal reflection microscopy,” Opt. Lett. 26, 1072–1074 (2001).
[CrossRef]

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

Chance, B.

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffusive photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 19, 4887–4891 (1994).
[CrossRef]

Devaney, A. J.

G. A. Tsihrintzis, A. J. Devaney, “A Volterra series approach to nonlinear travel time tomography,” IEEE Trans. Geosci. Remote Sens. 38, 1733–1742 (2000).
[CrossRef]

Markel, V. A.

V. A. Markel, J. C. Schotland, “Effects of sampling and limited data in optical tomography,” Appl. Phys. Lett. 81, 1180–1182 (2002).
[CrossRef]

V. A. Markel, J. C. Schotland, “Inverse problem in optical diffusion tomography. II. Inversion with boundary conditions,” J. Opt. Soc. Am. A 19, 558–566 (2002).
[CrossRef]

V. A. Markel, J. C. Schotland, “Inverse problem in optical diffusion tomography. III. Inversion formulas and singular-value decomposition,” J. Opt. Soc. Am. A 20, 890–902 (2002).
[CrossRef]

V. A. Markel, J. C. Schotland, “Inverse problem in optical diffusion tomography. I. Fourier–Laplace inversion formulas,” J. Opt. Soc. Am. A 18, 1336–1347 (2001).
[CrossRef]

P. S. Carney, V. A. Markel, J. C. Schotland, “Near-field tomography without phase retrieval,” Phys. Rev. Lett. 86, 5874–5877 (2001).
[CrossRef] [PubMed]

V. A. Markel, J. C. Schotland, “Inverse scattering for the diffusion equation with general boundary conditions,” Phys. Rev. E 64, R035601 (2001).
[CrossRef]

Moses, H. E.

H. E. Moses, “Calculation of the scattering potential from reflection coefficients,” Phys. Rev. 102, 550–567 (1956).
[CrossRef]

O’Leary, M. A.

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffusive photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 19, 4887–4891 (1994).
[CrossRef]

Prosser, R. T.

R. T. Prosser, “Formal solutions of the inverse scattering problem,” J. Math. Phys. 10, 1819–1822 (1969).
[CrossRef]

Schotland, J. C.

V. A. Markel, J. C. Schotland, “Effects of sampling and limited data in optical tomography,” Appl. Phys. Lett. 81, 1180–1182 (2002).
[CrossRef]

P. S. Carney, J. C. Schotland, “Determination of three-dimensional structure in photon scanning tunneling microscopy,” J. Opt. A 4, S140–S144 (2002).
[CrossRef]

V. A. Markel, J. C. Schotland, “Inverse problem in optical diffusion tomography. II. Inversion with boundary conditions,” J. Opt. Soc. Am. A 19, 558–566 (2002).
[CrossRef]

V. A. Markel, J. C. Schotland, “Inverse problem in optical diffusion tomography. III. Inversion formulas and singular-value decomposition,” J. Opt. Soc. Am. A 20, 890–902 (2002).
[CrossRef]

P. S. Carney, J. C. Schotland, “Three-dimensional total internal reflection microscopy,” Opt. Lett. 26, 1072–1074 (2001).
[CrossRef]

V. A. Markel, J. C. Schotland, “Inverse problem in optical diffusion tomography. I. Fourier–Laplace inversion formulas,” J. Opt. Soc. Am. A 18, 1336–1347 (2001).
[CrossRef]

P. S. Carney, V. A. Markel, J. C. Schotland, “Near-field tomography without phase retrieval,” Phys. Rev. Lett. 86, 5874–5877 (2001).
[CrossRef] [PubMed]

V. A. Markel, J. C. Schotland, “Inverse scattering for the diffusion equation with general boundary conditions,” Phys. Rev. E 64, R035601 (2001).
[CrossRef]

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

Snieder, R.

R. Snieder, “A perturbative analysis of non-linear inversion,” Geophys. J. Int. 101, 545–556 (1990).
[CrossRef]

Tsihrintzis, G. A.

G. A. Tsihrintzis, A. J. Devaney, “A Volterra series approach to nonlinear travel time tomography,” IEEE Trans. Geosci. Remote Sens. 38, 1733–1742 (2000).
[CrossRef]

Yodh, A. G.

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffusive photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 19, 4887–4891 (1994).
[CrossRef]

Appl. Phys. Lett. (2)

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

V. A. Markel, J. C. Schotland, “Effects of sampling and limited data in optical tomography,” Appl. Phys. Lett. 81, 1180–1182 (2002).
[CrossRef]

Geophys. J. Int. (1)

R. Snieder, “A perturbative analysis of non-linear inversion,” Geophys. J. Int. 101, 545–556 (1990).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

G. A. Tsihrintzis, A. J. Devaney, “A Volterra series approach to nonlinear travel time tomography,” IEEE Trans. Geosci. Remote Sens. 38, 1733–1742 (2000).
[CrossRef]

Inverse Probl. (1)

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[CrossRef]

J. Math. Phys. (1)

R. T. Prosser, “Formal solutions of the inverse scattering problem,” J. Math. Phys. 10, 1819–1822 (1969).
[CrossRef]

J. Opt. A (1)

P. S. Carney, J. C. Schotland, “Determination of three-dimensional structure in photon scanning tunneling microscopy,” J. Opt. A 4, S140–S144 (2002).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Lett. (1)

Phys. Rev. (1)

H. E. Moses, “Calculation of the scattering potential from reflection coefficients,” Phys. Rev. 102, 550–567 (1956).
[CrossRef]

Phys. Rev. E (1)

V. A. Markel, J. C. Schotland, “Inverse scattering for the diffusion equation with general boundary conditions,” Phys. Rev. E 64, R035601 (2001).
[CrossRef]

Phys. Rev. Lett. (1)

P. S. Carney, V. A. Markel, J. C. Schotland, “Near-field tomography without phase retrieval,” Phys. Rev. Lett. 86, 5874–5877 (2001).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. USA (1)

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffusive photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 19, 4887–4891 (1994).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Linearized inversion with δ α ( r ) = δ α ( 1 ) ( r ) . Tomographic slices are drawn at different depths z for an absorbing sphere with different values of k 2 . The solid black circles indicate the physical boundary of the absorbing sphere. The reconstructed images are normalized by the “true” value of δα, D 0 ( k 2 2 - k 1 2 ) . A linear gray scale is employed; white corresponds to 1 and black to 0.  

Fig. 2
Fig. 2

First nonlinear correction δ α ( r ) = δ α ( 1 ) ( r ) + δ α ( 2 ) ( r ) . The same parameters as those in Fig. 1 were employed.

Fig. 3
Fig. 3

Reconstructed function δα(r) calculated along the line z = constant (as indicated in the legend), y = 0 , x [ - L ,   L ] . Long-dashed curves: δ α = δ α ( 1 ) (linearized inversion), solid curves: δ α = δ α ( 1 ) + δ α ( 2 ) (first nonlinear correction), short-dashed curves: true profile of δα.

Equations (114)

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u ( r ,   t ) t =     [ D ( r ) u ( r ,   t ) ] - α ( r ) u ( r ,   t ) + S ( r ,   t ) ,
u + l n ˆ     u = 0 ,
G ( r 1 ,   r 2 ) = G 0 ( r 1 ,   r 2 ) - d 3 rG 0 ( r 1 ,   r ) V ( r ) G ( r ,   r 2 ) ,
V ( r ) δ α ( r ) - δ D ( r ) ,
( 2 - k 2 ) G 0 ( r ,   r ) = - 1 D 0   δ ( r - r ) ,
k 2 = α 0 - i ω D 0 .
ϕ ( r 1 ,   r 2 ) = β G 0 ( r 1 ,   r ) V ( r ) G ( r ,   r 2 ) d 3 r .
β = 1 for free boundaries ( 1 + l * / l ) 2 for boundary conditions of the type ( 2 )
ϕ ( r 1 ,   r 2 ) = β d 3 rG 0 ( r 1 ,   r ) V ( r ) G 0 ( r ,   r 2 ) + β d 3 r d 3 r G 0 ( r 1 ,   r ) V ( r ) G 0 ( r ,   r ) × V ( r ) G 0 ( r ,   r 2 ) +   .
η n + 1 = η n + M n + ( ϕ - K [ η n ] ) , n = 1 ,   2 , ,
M n = δ K δ η η = η n .
ϕ ( r 1 ,   r 2 ) = d 3 rK 1 i ( r 1 ,   r 2 ;   r ) η i ( r ) + d 3 r d 3 r × K 2 ij ( r 1 ,   r 2 ;   r ,   r ) η i ( r ) η j ( r ) + ,
η ( r ) = η 1 ( r ) η 2 ( r ) = δ α ( r ) δ D ( r ) ,
K 1 1 ( r 1 ,   r 2 ;   r ) = β G 0 ( r 1 ,   r ) G 0 ( r ,   r 2 ) ,
K 1 2 ( r 1 ,   r 2 ;   r ) = β r G 0 ( r 1 ,   r )     r G 0 ( r ,   r 2 ) ,
K 2 11 ( r 1 ,   r 2 ;   r ,   r ) = - β G 0 ( r 1 ,   r ) G 0 ( r , r ) G 0 ( r ,   r 2 ) ,
K 2 12 ( r 1 ,   r 2 ;   r ,   r )
= - β G 0 ( r 1 , r ) r G 0 ( r ,   r )   r G 0 ( r ,   r 2 ) ,
K 2 21 ( r 1 ,   r 2 ;   r ,   r )
= - β r G 0 ( r 1 ,   r )   r G 0 ( r ,   r ) G 0 ( r ,   r 2 ) ,
K 2 22 ( r 1 ,   r 2 ;   r ,   r )
= - β r G 0 ( r 1 ,   r )     r [ r G 0 ( r ,   r )   r G 0 ( r ,   r 2 ) ] .
K n i 1     i n ( r 1 ,   r 2 ;   R 1 , ,   R n )
= ( - 1 ) n α 1 , , α n i 1 - 1 G 0 ( r 1 ,   R 1 ) R 1 α 1 i 1 - 1 i 1 + i 2 - 2 G 0 ( R 1 ,   R 2 ) R 1 α 1 i 1 - 1 R 2 α 2 i 2 - 1 × × i n - 1 + i n - 2 G 0 ( R n - 1 ,   R n ) R n - 1 , α n - 1 i n - 1 - 1 R n α n i n - 1 i n - 1 G 0 ( R n ,   r 2 ) R n α n i n - 1 ,
 
ϕ = K 1 η + K 2 η η +   .
η = K 1 + ϕ - K 1 + K 2 η η +   .
η = K 1 + ϕ - K 1 + K 2 K 1 + K 1 + ϕ ϕ + ,
K n = - p = 1 n - 1 K p i 1 + + i p = n K i 1 K i p K 1 K 1 ,
η = K 1 + ( ϕ - K 2 η η + )
= K 1 + { ϕ - ( K [ η ] - K 1 η ) }
= η + K 1 + ( ϕ - K [ η ] )
T [ η ] ,
η n + 1 = η n + K 1 + ( ϕ - K [ η n ] ) , n = 1 , 2 ,   .
G 0 ( r ,   r ) = d 2 q ( 2 π ) 2   g ( q ;   z ,   z ) exp [ i q     ( ρ - ρ ) ] ,
g ( q ;   z ,   z ) = exp [ - Q ( q ) | z - z | ] 2 Q ( q ) D 0 ,
g ( q ;   z ,   z ) = l D 0 sinh [ Q ( q ) ( L - | z - z | ) ] + Q ( q ) l   cosh [ Q ( q ) ( L - | z - z | ) ] sinh [ Q ( q ) L ] + 2 Q ( q ) l   cosh [ Q ( q ) L ] + [ Q ( q ) l ] 2 sinh [ Q ( q ) L ] ,
Q ( q ) ( q 2 + k 2 ) 1 / 2
η = η ( 1 ) + η ( 2 ) + ,
η ( 1 ) = K 1 + ϕ ,
η ( 2 ) = K 1 + K 2 η ( 1 ) η ( 1 ) ,
η ( 1 ) ( r ) = d 2 q 1 d 2 q 2 K 1 + ( r ;   q 1 ,   q 2 ) ϕ ( q 1 ,   q 2 ) ,
η ( 2 ) ( r ) = d 2 q 1 d 2 q 2 d 3 r d 3 r K 1 + ( r ;   q 1 ,   q 2 ) × K 2 ( q 1 ,   q 2 ;   r ,   r ) η ( 1 ) ( r ) η ( 1 ) ( r ) .
ϕ ( q 1 ,   q 2 ) = d 2 ρ 1 d 2 ρ 2 exp [ i ( q 1   ρ 1 + q 2   ρ 2 ) ] × ϕ ( ρ 1 ,   z 1 ,   ρ 2 ,   z 2 ) ,
K ( q 1 ,   q 2 ;   ) = d 2 ρ 1 d 2 ρ 2 exp [ i ( q 1   ρ 1 + q 2   ρ 2 ) ] × K ( ρ 1 ,   z 1 ,   ρ 2 ,   z 2 ;   ) .
K 1 + ( r ;   q 1 ,   q 2 ) = 1 σ   f σ ( r ) g σ * ( q 1 ,   q 2 ) d σ ,
K 1 * K 1 f σ = σ 2 f σ ,
K 1 K 1 * g σ = σ 2 g σ .
K 1 f σ = σ g σ ,
K 1 * g σ = σ f σ .
K 1 ( q 1 ,   q 2 ;   r ) = κ ( q 1 ,   q 2 ;   z ) exp [ i ( q 1 + q 2 )     ρ ] ,
κ 1 ( q 1 ,   q 2 ;   z ) = β g ( q 1 ;   z 1 ,   z ) g ( q 2 ;   z ,   z 2 ) ,
κ 1 ( q 1 ,   q 2 ;   z ) = β g ( q 1 ;   z 1 ,   z ) z g ( q 2 ;   z ,   z 2 ) z - q 1   q 2 g ( q 1 ;   z 1 ,   z ) g ( q 2 ;   z ,   z 2 ) .
K 1 K 1 * ( q 1 ,   q 2 ;   q 1 ,   q 2 )
= d 2 Q δ ( Q - q 1 - q 2 ) δ ( Q - q 1 - q 2 ) × M 1 2   ( q 1 - q 2 ) , 1 2   ( q 1 - q 2 ) ;   Q ,
M ( P ,   P ;   Q ) = ( 2 π ) 2 0 L d z κ ( Q / 2 + P ,   Q / 2 - P ;   z ) × κ * ( Q / 2 + P ,   Q / 2 - P ;   z ) .
g QQ ( q 1 ,   q 2 ) = d 2 PC Q ( P ;   Q ) δ ( q 1 - Q / 2 - P ) × δ ( q 2 - Q / 2 + P ) ,
d 2 P M ( P ,   P ;   Q ) C Q ( P ;   Q ) = σ QQ 2 C Q ( P ;   Q ) ,
f QQ ( r ) = 1 σ QQ d 2 P   exp ( - i Q     ρ ) × κ * ( Q / 2 + P ,   Q / 2 - P ;   z ) C Q ( P ;   Q ) .
K 1 + ( r ;   q 1 ,   q 2 ) = d 2 Q d 2 Q 1 σ QQ   f QQ ( r ) g QQ * ( q 1 ,   q 2 ) .
M - 1 ( P ,   P ;   Q ) = d 2 Q 1 σ QQ 2   C Q ( P ;   Q ) C Q * ( P ;   Q )
K 1 + ( r ;   q 1 ,   q 2 )
= d 2 Q d 2 P d 2 P exp ( - i Q     ρ ) M - 1 ( P ,   P ;   Q ) × κ * ( Q / 2 + P ,   Q / 2 - P ;   z ) δ ( q 1 - Q / 2 - P )
× δ ( q 2 - Q / 2 + P ) .
η ( 1 ) ( r ) = d 2 Q d 2 P d 2 P exp ( - i Q     ρ ) M - 1 ( P ,   P ;   Q ) × κ * ( Q / 2 + P ,   Q / 2 - P ;   z ) × ϕ ( Q / 2 + P ,   Q / 2 - P ) .
δ α ( 1 ) ( r ) = d 2 Q d 2 P d 2 P exp ( - i Q     ρ ) M - 1 ( P ,   P ;   Q ) × κ 1 * ( Q / 2 + P ,   Q / 2 - P ;   z ) × ϕ ( Q / 2 + P ,   Q / 2 - P ) ,
M ( P ,   P ;   Q ) = ( 2 π ) 2 0 L d z κ 1 ( Q / 2 + P ,   Q / 2 - P ;   z ) × κ 1 * ( Q / 2 + P ,   Q / 2 - P ;   z ) .
R ( σ ) = 1 σ   θ ( σ - σ c ) ,
δ α ( 1 ) ( r ) = d 2 Q d 2 P d 2 P exp ( - i Q     ρ ) M - 1 ( P ,   P ;   Q ) × κ 1 * ( Q / 2 + P ,   Q / 2 - P ;   z ) × ϕ ( 1 ) ( Q / 2 + P ,   Q / 2 - P ) ,
ϕ ( 1 ) ( q 1 ,   q 2 ) = d 3 r d 3 r K 2 11 ( q 1 ,   q 2 ;   r ,   r ) × δ α ( 1 ) ( r ) δ α ( 1 ) ( r ) .
K 2 11 ( q 1 ,   q 2 ;   r ,   r )
= G 0 ( r ,   r ) ( 2 π D 0 ) 2 Q ( q 1 ) Q ( q 2 ) exp [ i ( q 1   ρ + q 2   ρ ) ]
× exp [ - Q ( q 1 ) | z - z 1 | - Q ( q 2 ) | z - z 2 | ] .
ϕ = K 1 η + K 2 η η + K 3 η η η +   .
η = K 1 ϕ + K 2 ϕ ϕ + K 3 ϕ ϕ ϕ + ,
K 1 K 1 = I ,
K 2 K 1 K 1 + K 1 K 2 = 0 ,
K 3 K 1 K 1 K 1 + K 2 K 1 K 2
+ K 2 K 2 K 1 + K 1 K 3 = 0 ,
p = 1 n - 1 K p i 1 + + i p = n K i 1 K i p
+ K n K 1 K 1 = 0 ,
K 1 = K 1 + ,
K 2 = - K 1 K 2 K 1 K 1 ,
K 3 = - ( K 2 K 1 K 2 + K 2 K 2 K 1 + K 1 K 3 ) K 1 K 1 K 1 ,
K n = - p = 1 n - 1 K p i 1 + + i p = n K i 1 K i p K 1 K 1 .
G ( r ,   r ) = l = 0 m = - l l g l ( r ,   r ) Y lm * ( r ^ ) Y lm ( r ˆ ) ,
g l ( r ,   r ) = 2 k 1 π D 0   [ i l ( k 1 r < ) k l ( k 1 r > ) - F l k l ( k 1 r ) k l ( k 1 r ) ] ,
F l = k 2 i l ( k 1 R ) i l ( k 2 R ) - k 1 i l ( k 2 R ) i l ( k 1 R ) k 2 i l ( k 2 R ) k l ( k 1 R ) - k 1 i l ( k 2 R ) k l ( k 1 R ) ,
k 1 2 = α 0 - i ω D 0 , k 2 2 = α 0 + a - i ω D 0 .
G 0 ( r ,   r ) = 2 k 1 π D 0 l = 0 m = - 1 l i l ( k 1 r < ) k l ( k 1 r > ) × Y lm * ( r ^ ) Y lm ( r ˆ ) ,
ϕ ( r 1 ,   r 2 ) = 2 k 1 π D 0 l = 0 m = - l l F l k l ( k 1 r 1 ) k l ( k 1 r 2 ) × Y lm * ( r ^ 2 ) Y lm ( r ^ 1 ) .
ϕ ( q 1 ,   q 2 ) = d 2 ρ 1 d 2 ρ 2 exp [ i ( q 1   ρ 1 + q 2   ρ 2 ) ] × ϕ ( ρ 1 ,   z 1 ,   ρ 2 ,   z 2 ) .
ϕ ( q 1 ,   q 2 )
= π 2 2 D 0 k 1 Q ( q 1 ) Q ( q 2 ) × exp [ i ( q 2 - q 1 ) ρ c - Q ( q 1 ) | z 0 - z 1 | - Q ( q 2 ) | z 2 - z 0 | ] × l = 0 ( 2 l + 1 ) F l P l q 1   q 2 + γ ( z 1 ,   z 2 ) Q ( q 1 ) Q ( q 2 ) k 1 2 ,
γ ( z 1 ,   z 2 ) = 1 if z 1 = z 2 - 1 if z 1 z 2 .
k l ( k 1 r 1 ) Y lm ( r ^ 1 ) = d 3 p 1 ( 2 π ) 3   I lm 1 ( p 1 ) exp ( - i p 1   r 1 ) ,
k l ( k 1 r 2 ) Y lm * ( r ^ 2 ) = d 3 p 2 ( 2 π ) 3   I lm 2 ( p 2 ) exp ( - i p 2   r 2 ) ,
I lm 1 ( p 1 ) = k l ( k 1 r ) Y lm ( r ˆ ) exp ( i p 1   r ) d 3 r ,
I lm 2 ( p 2 ) = k l ( k 1 r ) Y lm * ( r ˆ ) exp ( i p 2   r ) d 3 r .
exp ( i p     r ) = 4 π l = 0 m = - l l i l j l ( pr ) Y lm * ( p ˆ ) Y lm ( r ˆ ) ,
I lm 1 ( p 1 ) = 4 π i l Y lm ( p ^ 1 ) 0 r 2 k l ( k 1 r ) j l ( p 1 r ) d r ,
I lm 2 ( p 2 ) = 4 π i l Y lm * ( p ^ 2 ) 0 r 2 k l ( k 1 r ) j l ( p 2 r ) d r .
0 x 2 k l ( ax ) j l ( bx ) d x = π 2 a ( b / a ) l a 2 + b 2 .
ϕ ( r 1 ,   r 2 )
= 1 ( 2 π ) 3 D 0 k 1 lm ( - 1 ) l F l × d 3 p 1 d 3 p 2 ( p 1 p 2 / k 1 2 ) l ( p 1 2 + k 1 2 ) ( p 2 2 + k 1 2 ) Y lm ( p ^ 1 ) Y lm * ( p ^ 2 ) × exp { - i [ p 1     ( r 1 - r 0 ) + p 2     ( r 2 - r 0 ) ] } ,
ϕ ( q 1 ,   q 2 ) = 2 π   exp [ i ( q 1 + q 2 )     ρ 0 ] D 0 k 1 × lm ( - 1 ) l F l J lm 1 ( q 1 ) J lm 2 ( q 2 ) ,
J lm 1 ( q 1 ) = - d t   ( q 1 2 + t 2 / k 1 ) l q 1 2 + t 2 + k 1 2   Y lm - q 1 + t e ^ z q 1 2 + t 2 × exp [ it ( z 0 - z 1 ) ] ,
J lm 2 ( q 2 ) = - d t   ( q 2 2 + t 2 / k 1 ) l q 2 2 + t 2 + k 1 2   Y lm * q 2 + t e ^ z q 2 2 + t 2 × exp [ it ( z 0 - z 2 ) ] .
J lm 1 ( q 1 ) = π i l Q ( q 1 )   Y lm - q 1 + i   sgn ( z 0 - z 1 ) Q ( q 1 ) e ^ z ik 1 × exp [ - Q ( q 1 ) | z 0 - z 1 | ] ,
J lm 2 ( q 2 ) = π i l Q ( q 1 )   Y lm * q 2 + i   sgn ( z 0 - z 2 ) Q ( q 2 ) e ^ z ik 1 × exp [ - Q ( q 2 ) | z 0 - z 2 | ] .
m = - l l Y lm - q 1 + i   sgn ( z 0 - z 1 ) Q ( q 1 ) e ^ z ik 1
×   Y lm * q 2 + i   sgn ( z 0 - z 2 ) Q ( q 2 ) e ^ z ik 1
= 2 l + 1 4 π   P l ( cos   θ ) ,
cos   θ = q 1   q 2 + sgn ( z 0 - z 1 ) sgn ( z 0 - z 2 ) Q ( q 1 ) Q ( q 2 ) k 1 2 .

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