Abstract

We continue our study of the inverse scattering problem for diffuse light. In particular, we derive inversion formulas for this problem that are based on the functional singular-value decomposition of the linearized forward-scattering operator in the slab, cylindrical, and spherical geometries. Computer simulations are used to illustrate our results in model systems.

© 2003 Optical Society of America

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References

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  1. B. Chance, R. R. Alfano, M. Tamura, E. M. Sevick-Muraca, Optical Tomography and Spectroscopy of Tissue, Proc. SPIE 4250, (2001).
  2. Biomedical Optics, Vol. 71 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002).
  3. V. A. Markel, J. C. Schotland, “The inverse problem in optical diffusion tomography. I. Fourier–Laplace inversion formulas,” J. Opt. Soc. Am. A 18, 1336–1347 (2001).
    [CrossRef]
  4. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
    [CrossRef]
  5. V. A. Markel, J. C. Schotland, “The inverse problem in optical diffusion tomography. II. Inversion with boundary conditions,” J. Opt. Soc. Am. A 19, 558–566 (2002).
    [CrossRef]
  6. J. C. Schotland, V. A. Markel, “Inverse scattering with diffusing waves,” J. Opt. Soc. Am. A 18, 2767–2777 (2001).
    [CrossRef]
  7. V. A. Markel, J. C. Schotland, “Inverse scattering for the diffusion equation with general boundary conditions,” Phys. Rev. E 64, R035601 (2001).
    [CrossRef]
  8. R. Aronson, “Boundary conditions for diffuse light,” J. Opt. Soc. Am. A 12, 2532–2539 (1995).
    [CrossRef]
  9. F. Natterer, The Mathematics of Computerized Tomography (Wiley, New York, 1986).

2002 (1)

2001 (4)

B. Chance, R. R. Alfano, M. Tamura, E. M. Sevick-Muraca, Optical Tomography and Spectroscopy of Tissue, Proc. SPIE 4250, (2001).

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

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

J. C. Schotland, V. A. Markel, “Inverse scattering with diffusing waves,” J. Opt. Soc. Am. A 18, 2767–2777 (2001).
[CrossRef]

1999 (1)

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

1995 (1)

Alfano, R. R.

B. Chance, R. R. Alfano, M. Tamura, E. M. Sevick-Muraca, Optical Tomography and Spectroscopy of Tissue, Proc. SPIE 4250, (2001).

Aronson, R.

Arridge, S. R.

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

Chance, B.

B. Chance, R. R. Alfano, M. Tamura, E. M. Sevick-Muraca, Optical Tomography and Spectroscopy of Tissue, Proc. SPIE 4250, (2001).

Markel, V. A.

Natterer, F.

F. Natterer, The Mathematics of Computerized Tomography (Wiley, New York, 1986).

Schotland, J. C.

Sevick-Muraca, E. M.

B. Chance, R. R. Alfano, M. Tamura, E. M. Sevick-Muraca, Optical Tomography and Spectroscopy of Tissue, Proc. SPIE 4250, (2001).

Tamura, M.

B. Chance, R. R. Alfano, M. Tamura, E. M. Sevick-Muraca, Optical Tomography and Spectroscopy of Tissue, Proc. SPIE 4250, (2001).

Inverse Probl. (1)

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

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

Optical Tomography and Spectroscopy of Tissue (1)

B. Chance, R. R. Alfano, M. Tamura, E. M. Sevick-Muraca, Optical Tomography and Spectroscopy of Tissue, Proc. SPIE 4250, (2001).

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]

Other (2)

F. Natterer, The Mathematics of Computerized Tomography (Wiley, New York, 1986).

Biomedical Optics, Vol. 71 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002).

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Figures (7)

Fig. 1
Fig. 1

Set σ ( l ) of allowable values of l 1 and l 2 for l = 6 . Note that σ ( l ) is infinite, and the plot must be continued to infinitely large numbers of l 1 and l 2 lying on or between the lines l 2 = l 1 ± l and l 2 = l - l 1 as shown.

Fig. 2
Fig. 2

Depth resolution for free boundary conditions. Left column: forward data simulated for the infinite plane absorber (109); right column: forward data generated for the point absorber (116); top row: sources and detectors on the same plane; middle row: sources and detectors on different planes; bottom row: simultaneous SVD solution for sources and detectors on plane z = 0 , sources and detectors on plane z = L , and sources on plane z = 0 and detectors on plane z = L . Solid curves correspond to z A = 0.2 L , long-dashed curves correspond to z A = 0.5 L , and short-dashed curves correspond to z A = 0.8 L .

Fig. 3
Fig. 3

Depth resolution for absorbing boundary conditions; parameters are the same as those in Fig. 2.

Fig. 4
Fig. 4

Depth resolution for reflecting boundary conditions; parameters are the same as those in Fig. 2.

Fig. 5
Fig. 5

Reconstruction of the absorption coefficient δα with the assumption δ D = 0 according to Eq. (78) in the cylindrical geometry. A point absorber is placed at ρ = 0.5 R , φ = π , z = 0 . The slices shown are perpendicular to the cylinder axis and have different z coordinates as indicated in the figure (the z = 0 slice is drawn directly through the absorber). The same linear gray scale is employed in all the plots.

Fig. 6
Fig. 6

Reconstruction of the absorption coefficient δα with the assumption δ D = 0 according to Eq. (78) in the cylindrical geometry. The forward data are generated for a point absorber located at φ = 0 , z = 0 , and ρ = 0.25 R , 0.5 R , 0.75 R as indicated. Each reconstructed function is normalized to its maximum, and a linear gray scale is employed.

Fig. 7
Fig. 7

Simultaneous reconstruction of δα and δ D . In images a and d, the forward data are generated for a point absorber located at ρ = 0.5 R , φ = π , z = 0 and a point diffusing inhomogeneity located at ρ = 0.5 R , φ = 0 , z = 0 . In images b and e only the absorbing inhomogeneity, and in c and f only the diffusion inhomogeneity, were used to generate the forward data. Images a–c (and d–f) use the same linear gray scale.

Equations (161)

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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 ,
ϕ ( r 1 ,   r 2 ) = β G 0 ( r 1 ,   r ) V ( r ) G 0 ( r ,   r 2 ) d 3 r .
V ( r ) δ α ( r ) - δ D ( r ) ,
β = 1 for free boundaries ( 1 + l * / l ) 2 for boundary conditions of the type ( 2 ) ,
( 2 - k 2 ) G 0 ( r ,   r ) = - 1 D 0   δ ( r - r ) ,
k 2 = α 0 - i ω D 0
A ( x ,   y ) = n σ n g n ( x ) f n * ( y ) ,
A * Af n = σ n 2 f n ,
AA * g n = σ n 2 g n .
Af n = σ n g n ,
A * g n = σ n f n .
A + ( x ,   y ) = n 1 σ n   f n ( x ) g n * ( y ) .
0 L K n ( z ) f ( z ) d z = F n , n = 1 , ,   N .
K n ( z ) = l σ l ( g l ) n f l * ( z ) .
M mn = 0 L K m ( z ) K n * ( z ) d z .
f l ( z ) = 1 σ l n K n * ( z ) ( g l ) n .
K n + ( z ) = l 1 σ l   ( g l * ) n f l ( z ) .
M mn - 1 = l 1 σ l 2   ( g l ) m ( g l * ) n ,
K n + ( z ) = m K m * ( z ) M mn - 1 .
f ( z ) = m , n K m * ( z ) M mn - 1 F n .
M mn - 1 = l Θ ( σ l - )   1 σ l 2   ( g l ) m ( g l * ) n ,
f ( z ) = L - 1 n = 1 N exp ( - ip n z ) F n .
0 L p = 1 P K n ( p ) ( z ) f ( p ) ( z ) d z = F n , n = 1 , ,   N .
f ( p ) ( z ) = m , n K m ( p ) * ( z ) M mn - 1 F n ,
M mn = 0 L p = 1 P K m ( p ) ( z ) K n ( p ) * ( z ) d z .
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 ) = 1 2 Q ( q ) D 0 exp [ - Q ( q ) | z - z | ] + Q ( q ) l - 1 Q ( q ) l + 1 exp [ - Q ( q ) | z + z | ] ,
Q ( q ) q 2 + k 2 .
g ( q ;   z ,   0 ) = g ( q ;   0 ,   z ) = g ˜ ( q ;   z ) ,
g ˜ ( q ;   z ) = exp [ - Q ( q ) | z | ] 2 D 0 Q ( q ) ( free boundaries ) ,
g ˜ ( q ;   z ) = l D 0 exp [ - Q ( q ) | z | ] Q ( q ) l + 1 ( boundary conditions ) .
ϕ ( ρ 1 ,   ρ 2 ) = β d 2 q 1 d 2 q 2 ( 2 π ) 4 z > 0 d 3 r g ˜ ( q 1 ;   z ) × exp [ i q 1 ( ρ - ρ 1 ) ] V ( r ) g ˜ ( q 2 ;   z ) × exp [ i q 2 ( ρ 2 - ρ ) ] .
ϕ ( q 1 ,   q 2 ) = d 2 ρ 1 d 2 ρ 2 ϕ ( ρ 1 ,   ρ 2 ) × exp [ i ( q 1   ρ 1 + q 2   ρ 2 ) ]
ϕ ( q 1 ,   q 2 ) = β d 3 r g ˜ ( q 1 ;   z ) exp ( i q 1   ρ ) × V ( r ) g ˜ ( q 2 ;   z ) exp ( i q 2   ρ ) .
ϕ ( q 1 ,   q 2 ) = d 3 r [ κ A ( q 1 ,   q 2 ;   z ) δ α ( r ) + κ D ( q 1 ,   q 2 ;   z ) δ D ( r ) ] exp [ i ( q 1 + q 2 ) ρ ] ,
κ A ( q 1 ,   q 2 ;   z ) = β g ˜ ( q 1 ;   z ) g ˜ ( q 2 ;   z ) ,
κ D ( q 1 ,   q 2 ;   z ) = β g ˜ ( q 1 ;   z ) z g ˜ ( q 2 ;   z ) z - q 1   q 2 g ˜ ( q 1 ;   z ) g ˜ ( q 2 ;   z ) .
F n ( q ) = 0 L d z [ K n ( A ) ( q ;   z ) a ( q ;   z ) + K n ( D ) ( q ;   z ) b ( q ;   z ) ] ,
a ( q ;   z ) = d 2 ρ δ α ( r ) exp ( i q     ρ ) ,
b ( q ;   z ) = d 2 ρ δ D ( r ) exp ( i q     ρ ) ,
K n ( A ) ( q ;   z ) = κ A ( q / 2 + p n ,   q / 2 - p n ;   z ) ,
K n ( D ) ( q ;   z ) = κ D ( q / 2 + p n ,   q / 2 - p n ;   z ) ,
F n ( q ) = ϕ ( q / 2 + p n ,   q / 2 - p n ) .
a ( q ;   z ) = m , n K m ( A ) * ( q ;   z ) M mn - 1 ( q ) F n ( q ) ,
b ( q ;   z ) = m , n K m ( D ) * ( q ;   z ) M mn - 1 ( q ) F n ( q ) ,
M mn ( q ) = 0 L [ K m ( A ) ( q ;   z ) K n ( A ) * ( q ;   z ) + K m ( D ) ( q ;   z ) K n ( D ) * ( q ;   z ) ] d z .
δ α ( r ) = d 2 q ( 2 π ) 2 exp ( - i q     ρ ) × ml K m ( A ) * ( q ;   z ) M ml - 1 ( q ) F l ( q ) ,  
δ D ( r ) = d 2 q ( 2 π ) 2 exp ( - i q     ρ ) × ml K m ( D ) * ( q ;   z ) M ml - 1 ( q ) F l ( q ) .  
δ α ( r ) = d 2 q ( 2 π ) 2 exp ( - i q     ρ ) × m , n κ A * q 2 + p m ,   q 2 - p m ;   z × M mn - 1 ( q ) ϕ q 2 + p n ,   q 2 - p n ,
δ D ( r ) = d 2 q ( 2 π ) 2 exp ( - i q     ρ ) × m , n κ D * q 2 + p m ,   q 2 - p m ;   z × M mn - 1 ( q ) ϕ q 2 + p n ,   q 2 - p n ,
M mn ( q ) = 0 L κ A q 2 + p m ,   q 2 - p m ;   z × κ A * q 2 + p n ,   q 2 - p n ;   z + κ D q 2 + p m ,   q 2 - p m ;   z × κ D * q 2 + p n ,   q 2 - p n ;   z d z .
g ( q ;   z ,   z ) = l D 0 [ 1 + ( Ql ) 2 ] cosh [ Q ( L - | z - z | ) ] - [ 1 - ( Ql ) 2 ] cosh [ Q ( L - | z + z | ) ] + 2 Ql   sinh [ Q ( L - | z - z | ) ] 2 D 0 Q [ sinh ( QL ) + 2 Ql   cosh ( QL ) + ( Ql ) 2 sinh ( QL ) ]
g ( q ;   z ,   z ) | z = 0 , L = g ( q ;   z ,   z ) | z = 0 , L = g ˜ ( q ;   z ,   z ) ,
g ˜ ( q ;   z ,   z ) = exp [ - Q ( q ) | z - z | ] 2 D 0 Q ( q ) ( free boundaries ) ,
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 ] ( boundary conditions ) .
κ A ( q 1 ,   q 2 ;   z ) = β g ˜ ( q 1 ;   z 1 ,   z ) g ˜ ( q 2 ;   z ,   z 2 ) ,
κ D ( 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 ) .
M mn - 1 ( q ) = l 1 σ q l 2   [ c l ( q ) ] m [ c l * ( q ) ] n ,
n M mn ( q ) [ c l ( q ) ] n = σ q l 2 [ c l ( q ) ] m .
δ α ( r ) δ D ( r ) = μ 1 σ μ   f μ ( r ) g μ ,   ϕ .
f q l ( r ) = exp ( - i q     ρ ) ( 2 π ) 2 σ q l n [ c l ( q ) ] n × κ A * ( q / 2 + p n ,   q / 2 - p n ;   z ) κ D * ( q / 2 + p n ,   q / 2 - p n ;   z ) ,
g q l ( q 1 ,   q 2 ) = n [ c l ( q ) ] n δ ( q 1 - q / 2 - p n ) × δ ( q 2 - q / 2 + p n ) .
G 0 ( r ,   r ) = 1 2 π m = = - d q 2 π exp [ im ( φ - φ ) + iq ( z - z ) ] g ( m ,   q ;   ρ ,   ρ ) ,
g ( m ,   q ;   ρ ,   ρ )
= 1 D 0   K m ( Q ρ > ) I m ( Q ρ < ) ( free boundaries ) ,
g ( m ,   q ;   ρ ,   ρ )
= 1 D 0 K m ( Q ρ > ) I m ( Q ρ < ) - K m ( QR ) + QlK m ( QR ) I m ( QR ) + QlI m ( QR ) × I m ( Q ρ ) I m ( Q ρ )   ( boundary conditions ) ,
g ( m ,   q ;   ρ ,   R ) = g ( m ,   q ;   R ,   ρ ) = g ˜ ( m ,   q ;   ρ ) ,
g ˜ ( m ,   q ;   ρ ) = 1 D 0   K m ( QR ) I m ( Q ρ ) ( free boundaries ) ,
g ˜ ( m ,   q ;   ρ ) = l D 0 R I m ( Q ρ ) I m ( QR ) + QlI m ( QR )
( boundary conditions ) .
ϕ ( z 1 ,   φ 1 ;   z 2 ,   φ 2 )
= β ( 2 π ) 4 m 1 , m 2 = - × d q 1 d q 2 ρ < R ρ d ρ d φ d z g ˜ ( m 1 ,   q 1 ;   ρ ) × exp [ im 1 ( φ - φ 1 ) + iq 1 ( z - z 1 ) ] × V g ˜ ( m 2 ,   q 2 ;   ρ ) exp [ im 2 ( φ 2 - φ ) + iq 2 ( z 2 - z ) ] .
ϕ ( m 1 ,   q 1 ;   m 2 ,   q 2 )
= - d z 1 d z 2 0 2 π d φ 1 d φ 2 ϕ ( φ 1 ,   z 1 ;   φ 2 ,   z 2 ) × exp { i [ q 1 z 1 + q 2 z 2 + m 1 φ 1 + m 2 φ 2 ] } .
ϕ ( m 1 ,   q 1 ;   m 2 ,   q 2 )
= β ρ d ρ d φ d z   exp [ i ( m 1 φ + q 1 z ) ] g ˜ ( m 1 ,   q 1 ;   ρ ) V
× exp [ i ( m 2 φ + q 2 z ) ] g ˜ ( m 2 ,   q 2 ;   ρ ) ,
ϕ ( m 1 ,   q 1 ;   m 2 ,   q 2 )
= 0 R ρ d ρ 0 2 π d φ - d z × [ κ A ( m 1 ,   q 1 ,   m 2 ,   q 2 ;   ρ ) δ α ( ρ ,   φ ,   z ) + κ D ( m 1 ,   q 1 ,   m 2 ,   q 2 ;   ρ ) δ D ( ρ ,   φ ,   z ) ] × exp [ i ( m 1 + m 2 ) φ + i ( q 1 + q 2 ) z ] ,
κ A ( m 1 ,   q 1 ,   m 2 ,   q 2 ;   ρ )
= β g ˜ ( m 1 ,   q 1 ;   ρ ) g ˜ ( m 2 ,   q 2 ;   ρ ) ,
κ D ( m 1 ,   q 1 ,   m 2 ,   q 2 ;   ρ )
= β g ˜ ( m 1 ,   q 1 ;   ρ ) ρ g ˜ ( m 2 ,   q 2 ;   ρ ) ρ - q 1 q 2 + m 1 m 2 ρ 2 g ˜ ( m 1 ,   q 1 ;   ρ ) g ˜ ( m 2 ,   q 2 ;   ρ ) .
 
ϕ ( m - n ,   q - p l ;   n ,   p k )
= 0 R [ κ A ( m - n ,   q - p l ,   n ,   p l ;   ρ ) a ( ρ ,   m ,   q )
+ κ D ( m - n ,   q - p l ,   n ,   p l ;   ρ ) b ( ρ ,   m ,   q ) ] ρ d ρ ,
 
a ( ρ ,   m ,   q ) = 0 2 π d φ - d z   δ α ( ρ ,   φ ,   z ) exp [ i ( m φ + qz ) ] ,
b ( ρ ,   m ,   q ) = 0 2 π d φ - d z   δ D ( ρ ,   φ ,   z ) × exp [ i ( m φ + qz ) ] .
δ α ( ρ ,   φ ,   z )
= 1 ( 2 π ) 2 ρ m = - exp ( - im φ ) - d q   exp ( - iqz ) × nl , n l κ A * ( m - n ,   q - p l ,   n ,   p l ;   ρ ) M nl , n l - 1 ( m ,   q ) × ϕ ( m - n ,   q - p l ;   n ,   p l ) ,
δ D ( ρ ,   φ ,   z )
= 1 ( 2 π ) 2 ρ m = - exp ( - im φ ) - d q   exp ( - iqz ) × nl , n l κ D * ( m - n ,   q - p l ,   n ,   p l ;   ρ ) M nl , n l - 1 ( m ,   q ) × ϕ ( m - n ,   q - p l ;   n ,   p l ) ,
M nl , n l ( m ,   q )
= 0 R [ κ A ( m - n ,   q - p l ,   n ,   p l ;   ρ ) × κ A * ( m - n ,   q - p l ,   n ,   p l ;   ρ ) + κ D ( m - n ,   q - p l ,   n ,   p l ;   ρ ) × κ D * ( m - n ,   q - p l ,   n ,   p l ;   ρ ) ] d ρ .
G 0 ( r ,   r ) = l = 0 m = - 1 l g ( l ;   r ,   r ) Y lm ( r ˆ ) Y lm * ( r ^ ) ,
g ( l ;   r ,   r ) = 2 k π D 0   i l ( kr < ) k l ( kr > )
( free boundaries ) ,  
g ( l ;   r ,   r ) = 2 k π D 0 i l ( kr < ) k l ( kr > ) - k l ( kR ) + klk l ( kR ) i l ( kR ) + kli l ( kR )   i l ( kr ) i l ( kr )
( boundary conditions ) ,
g ( l ;   r ,   R ) = g ( l ;   R ,   r ) = g ˜ ( l ;   r ) ,
g ˜ ( l ;   r ) = 2 k 2 π D 0   k l ( kR ) i l ( kr ) ( free boundaries ) ,
g ˜ ( l ;   r ) = l D 0 R 2 i l ( kr ) i l ( kR ) + kli l ( kR )
( boundary conditions ) .
ϕ ( r ^ 1 ,   r ^ 2 ) = β l 1 , l 2 m 1 , m 2 d 3 r g ˜ ( l 1 ;   r ) Y l 1 m 1 ( r ^ 1 ) × Y l 1 m 1 * ( r ˆ ) V g ˜ ( l 2 ;   r ) Y l 2 m 2 ( r ˆ ) Y l 2 m 2 * ( r ^ 2 ) .
ϕ ( l 1 ,   m 1 ;   l 2 ,   m 2 )
= ϕ ( r ^ 1 ,   r ^ 2 ) Y l 1 m 1 * ( r ^ 1 ) Y l 2 m 2 ( r ^ 2 ) d 2 r ^ 1 d 2 r ^ 2 .
ϕ ( l 1 ,   m 1 ;   l 2 ,   m 2 )
= β d 3 r g ˜ ( l 1 ;   r ) Y l 1 m 1 * ( r ˆ ) V g ˜ ( l 2 ;   r ) Y l 2 m 2 ( r ˆ ) .
d 3 r g ˜ ( l 1 ;   r ) Y l 1 m 1 * ( r ˆ ) ( -     δ D ) g ˜ ( l 2 ;   r ) Y l 2 m 2 ( r ˆ ) = d 3 r δ D g ˜ ( l 1 ;   r ) r g ˜ ( l 2 ;   r ) r - g ˜ ( l 1 ;   r ) g ˜ ( l 2 ;   r ) r 2 × L 2 - l 1 ( l 1 + 1 ) - l 2 ( l 2 + 1 ) 2 Y l 1 m 1 * Y l 2 m 2 .
Y l 1 m 1 * Y l 2 m 2 = lm C lm l 1 m 1 l 2 m 2 Y lm * ,
ϕ ( l 1 ,   m 1 ;   l 2 ,   m 2 )
= β lm C lm l 1 m 1 l 2 m 2 d 3 r g ˜ ( l 1 ;   r ) g ˜ ( l 2 ;   r ) δ α ( r ) + δ D ( r ) g ˜ ( l 1 ;   r ) r g ˜ ( l 2 ;   r ) r - g ˜ ( l 1 ;   r ) g ˜ ( l 2 ;   r ) r 2   × l ( l + 1 ) - l 1 ( l 1 + 1 ) - l 2 ( l 2 + 1 ) 2 Y lm * ( r ˆ ) .
C lm l 1 m 1 l 2 m 2 = ( - 1 ) m + m 2 ( 2 l 1 + 1 ) ( 2 l + 1 ) ( 2 l 2 + 1 ) 4 π 1 / 2 × l 1 l l 2 0 0 0 l 1 l l 2 - m 1 m m 2 .
l 1 l l 2 - m 1 m m 2 = l 1 l 2 l m 1 - m 2 - m ,
m 1 = - l 1 l 1 m 2 = - l 2 l 2 l 1 l 2 l m 1 m 2 - m l 1 l 2 l m 1 m 2 - m
= δ ll δ mm 2 l + 1 ;
ψ ( l ,   m ;   l 1 ,   l 2 )
= 4 π ( 2 l + 1 ) ( 2 l 1 + 1 ) ( 2 l 2 + 1 ) 1 / 2 × l 1 l l 2 0 0 0 - 1 m 1 = - l 1 l 1 m 2 = - l 2 l 2 ( - 1 ) m 2 - m × l 1 l 2 l m 1 m 2 - m ϕ ( l 1 ,   m 1 ;   l 2 ,   - m 2 ) .
ψ ( l ,   m ;   l 1 ,   l 2 ) = 0 R r 2 d r d Ω [ κ A ( l 1 ,   l 2 ;   r ) δ α ( r ) + κ D ( l ,   l 1 ,   l 2 ;   r ) δ D ( r ) ] Y lm * ( r ˆ ) ,
κ A ( l 1 ,   l 2 ;   r ) = β g ˜ ( l 1 ;   r ) g ˜ ( l 2 ;   r ) ,
κ D ( l ,   l 1 ,   l 2 ;   r )
= β g ˜ ( l 1 ;   r ) r g ˜ ( l 2 ;   r ) r - g ˜ ( l 1 ;   r ) g ˜ ( l 2 ;   r ) r 2 × l ( l + 1 ) - l 1 ( l 1 + 1 ) - l 2 ( l 2 + 1 ) 2 .
a ( l ,   m ;   r ) = δ α ( r ) Y lm * ( r ˆ ) d Ω ,
b ( l ,   m ;   r ) = δ D ( r ) Y lm * ( r ˆ ) d Ω ,
ψ ( l ,   m ;   l 1 ,   l 2 ) = 0 R r 2 d r [ κ A ( l 1 ,   l 2 ;   r ) a ( l ,   m ;   r ) + κ D ( l ,   l 1 ,   l 2 ;   r ) b ( l ,   m ;   r ) ] ,
a ( l ,   m ;   r ) = 1 r 2 l 1 , l 2 , l 1 , l 2 κ A * ( l 1 ,   l 2 ;   r )
×   M l 1 l 2 , l 1 l 2 - 1 ( l ) ψ ( l ,   m ;   l 1 ,   l 2 ) ,
b ( l ,   m ;   r ) = 1 r 2 l 1 , l 2 , l 1 , l 2 κ D * ( l 1 ,   l 2 ;   r ) ×   M l 1 l 2 , l 1 l 2 - 1 ( l ) ψ ( l ,   m ;   l 1 ,   l 2 ) ,
M l 1 l 2 , l 1 l 2 - 1 ( l ) = 0 r [ κ A ( l 1 ,   l 2 ;   r ) κ A * ( l 1 ,   l 2 ;   r ) + κ D ( l ,   l 1 ,   l 2 ;   r ) κ D * ( l ,   l 1 ,   l 2 ;   r ) ] d r .
δ α ( r ) = lm a ( l ,   m ;   r ) Y lm ( r ˆ ) = lm Y lm ( r ˆ ) l 1 , l 2 , l 1 , l 2 σ ( l ) κ A * ( l 1 ,   l 2 ;   r ) r 2 × M l 1 l 2 , l 1 l 2 - 1 ( l ) ψ ( l ,   m ;   l 1 ,   l 2 ) ,
δ D ( r ) = lm b ( l ,   m ;   r ) Y lm ( r ˆ ) = lm Y lm ( r ˆ ) l 1 , l 2 , l 1 , l 2 σ ( l ) κ D * ( l ,   l 1 ,   l 2 ;   r ) r 2 × M l 1 l 2 , l 1 l 2 - 1 ( l ) ψ ( l ,   m ;   l 1 ,   l 2 ) .
l 1 l l 2 0 0 0
ψ ( l ,   m ;   l 1 ,   l 2 )
= 4 π ( 2 l + 1 ) ( 2 l 1 + 1 ) ( 2 l 2 + 1 ) 1 / 2 l 1 l l 2 0 0 0 - 1 × m 1 = max ( - l 1 ,   m - l 2 ) min ( l 1 ,   m + l 2 ) ( - 1 ) m 1 l 1 l 2 l m 1 m - m 1 - m
×   ϕ ( l 1 ,   m 1 ;   l 2 ,   m 1 - m ) ,
l 1 l 2 l m 1 m - m 1 - m
δ α ( r ) = a A δ ( z - z A ) , δ D ( r ) = a D δ ( z - z D ) ,
ϕ ( q 1 ,   q 2 ) = ( 2 π ) 2 δ ( q 1 + q 2 ) [ a A κ A ( q 1 ,   q 2 ;   z A ) + a D κ D ( q 1 ,   q 2 ;   z D ) ] .
δ α δ D = m , n κ A * ( p m ,   - p m ;   z ) κ D * ( p m ,   - p m ;   z ) M mn - 1 ( 0 ) [ a A κ A ( p n ,   - p n ;   z A ) + a D κ D ( p n ,   - p n ;   z D ) ] .
δ α = a A mn κ A * ( p m ,   - p m ;   z ) M mn - 1 κ A ( p n ,   - p n ;   z A ) ,
M mn = 0 L κ A ( p m ,   - p m ;   z ) κ A * ( p n ,   - p n ;   z ) d z .
κ ( p m ,   - p m ;   z ) = n A mn φ n ( z ) ,
δ α = a A m , n φ n * ( z ) φ m ( z A ) [ A * ( AA * ) - 1 A ] mn .
δ α = a A δ ( z - z A ) δ ( p ) .
d 3 r g ˜ ( l 1 ;   r ) Y l 1 m 1 * ( r ˆ ) ( - δ D ) g ˜ ( l 2 ;   r ) Y l 2 m 2 ( r ˆ ) ,
F G = ( r F ) ( r G ) + ( r × F ) ( r × G ) r 2
r × F = i g ˜ ( l 1 ;   r ) L Y l 1 m 1 * ,
r × G = i g ˜ ( l 2 ;   r ) L Y l 2 m 2 ;
r F = r   g ˜ ( l 1 ;   r ) r   Y l 1 m 1 * ,
r G = r   g ˜ ( l 2 ;   r ) r   Y l 2 m 2 ,
F G = g ˜ ( l 1 ;   r ) r g ˜ ( l 2 ;   r ) r   Y l 1 m 1 * Y l 2 m 2 - g ˜ ( l 1 ;   r ) g ˜ ( l 2 ;   r ) r 2 L Y l 1 m 1 * L Y l 2 m 2 .
L 2 Y l 1 m 1 * Y l 2 m 2 = 2 L Y l 1 m 1 * L Y l 2 m 2 + Y l 1 m 1 * L 2 Y l 2 m 2 + Y l 2 m 2 L 2 Y l 1 m 1 * ,
L Y l 2 m 2 L Y l 1 m 1 *
= 1 2 [ L 2 - l 1 ( l 1 + 1 ) - l 2 ( l 2 + 1 ) ] Y l 1 m 1 * Y l 2 m 2 .
d 3 r g ˜ ( l 1 ;   r ) Y l 1 m 1 * ( r ) ( -     δ D ) g ˜ ( l 2 ;   r ) Y l 2 m 2 ( r )
= d 3 r δ D g ˜ ( l 1 ;   r ) r g ˜ ( l 2 ;   r ) r - g ˜ ( l 1 ;   r ) g ˜ ( l 2 ;   r ) r 2 × L 2 - l 1 ( l 1 + 1 ) - l 2 ( l 2 + 1 ) 2 Y l 1 m 1 * Y l 2 m 2 .

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