Abstract

When the light propagates in media where absorption is not negligible and∕or scattering is weak, a contribution to the energy density coming from ballistic photons cannot be neglected. A point source effectively spreads out over a scattering volume and its spatial distribution is described by the source function. We consider a boundary value problem of light propagation in half-space for such a source on the basis of the telegraph equation. A solution is found by convolution of Green's function with the source function. The final result shows a significant difference in the behavior of the radiant energy density between the solution obtained for a distributed source and the diffusion approximation. Our results agree well with the Monte Carlo simulations over a broad range of scattering and∕or absorption conditions. The obtained results are of practical importance in luminescence optical tomography because an erroneous shape of the energy density function may lead to an incorrect estimate of the light source depth after image reconstruction. The range of applications of the diffusion approximation is also discussed.

© 2006 Optical Society of America

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2005

W. Cai, M. Xu, and R. R. Alfano, "Analytical form of the particle distribution based on the cumulant solution of the elastic Boltzmann transport equation," Phys. Rev. E. 71, 041202 (2005).
[CrossRef]

2004

2003

2002

M. Xu, W. Cai, M. Lax, and R. R. Alfano, "Photon migration in turbid media using a cumulant approximation to the radiative transfer," Phys. Rev. E 65, 066609 (2002).
[CrossRef]

X. Wang, G. Yao, and L.-H. Wang, "Monte Carlo model and single-scattering approximation of polarized light propagation in turbid media containing glucose," Appl. Opt. 41, 792-801 (2002).
[CrossRef] [PubMed]

A. D. Klose, U. Netz, J. Beuthan, and A. H. Heilscher, "Optical tomography using time-independent equation of radiative transfer--Part 1: forward model," J. Quant. Spectrosc. Radiat. Transfer 72, 691-713 (2002).
[CrossRef]

A. D. Klose and A. H. Heilscher, "Optical tomography using time-independent equation of radiative transfer--Part 2: inverse model," J. Quant. Spectrosc. Radiat. Transfer 72, 715-732 (2002).
[CrossRef]

2001

1999

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

1997

1995

L. H. Wang, S. L. Jacques, and L. Q. Zheng, "MCML-Monte Carlo modeling of light transport in multilayered tissues," Comput. Methods Programs Biomed. 47, 131-145 (1995).
[CrossRef] [PubMed]

1994

D. A. Boas, M. A. O'Leary, B. Chance, and A. G. Yodh, "Scattering of diffuse photon density waves by spherical inhomogeneties within turbid media: Analytical solution and applications," Proc. Natl. Acad. Sci. USA 91, 4887-4891 (1994).
[CrossRef] [PubMed]

1992

S. R. Arridge, M. Cope, and D. T. Delpy, "The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis," Phys. Med. Biol. 37, 1531-1560 (1992).
[CrossRef] [PubMed]

Abdoulaev, G. S.

Alfano, R. R.

W. Cai, M. Xu, and R. R. Alfano, "Analytical form of the particle distribution based on the cumulant solution of the elastic Boltzmann transport equation," Phys. Rev. E. 71, 041202 (2005).
[CrossRef]

M. Xu, W. Cai, and R. R. Alfano, "Multiple passage of the light through an absorption inhomogeneity in optical imaging of turbid media," Opt. Lett. 29, 1757-1759 (2004).
[CrossRef] [PubMed]

M. Xu, W. Cai, M. Lax, and R. R. Alfano, "Photon migration in turbid media using a cumulant approximation to the radiative transfer," Phys. Rev. E 65, 066609 (2002).
[CrossRef]

M. Xu, W. Cai, M. Lax, and R. R. Alfano, "Photon-transport forward model for imaging in turbid media," Opt. Lett. 26, 1066-1068 (2001).
[CrossRef]

Arridge, S. R.

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

S. R. Arridge, M. Cope, and D. T. Delpy, "The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis," Phys. Med. Biol. 37, 1531-1560 (1992).
[CrossRef] [PubMed]

Bal, G.

Beuthan, J.

A. D. Klose, U. Netz, J. Beuthan, and A. H. Heilscher, "Optical tomography using time-independent equation of radiative transfer--Part 1: forward model," J. Quant. Spectrosc. Radiat. Transfer 72, 691-713 (2002).
[CrossRef]

Boas, D. A.

D. A. Boas, M. A. O'Leary, B. Chance, and A. G. Yodh, "Scattering of diffuse photon density waves by spherical inhomogeneties within turbid media: Analytical solution and applications," Proc. Natl. Acad. Sci. USA 91, 4887-4891 (1994).
[CrossRef] [PubMed]

Cai, W.

W. Cai, M. Xu, and R. R. Alfano, "Analytical form of the particle distribution based on the cumulant solution of the elastic Boltzmann transport equation," Phys. Rev. E. 71, 041202 (2005).
[CrossRef]

M. Xu, W. Cai, and R. R. Alfano, "Multiple passage of the light through an absorption inhomogeneity in optical imaging of turbid media," Opt. Lett. 29, 1757-1759 (2004).
[CrossRef] [PubMed]

M. Xu, W. Cai, M. Lax, and R. R. Alfano, "Photon migration in turbid media using a cumulant approximation to the radiative transfer," Phys. Rev. E 65, 066609 (2002).
[CrossRef]

M. Xu, W. Cai, M. Lax, and R. R. Alfano, "Photon-transport forward model for imaging in turbid media," Opt. Lett. 26, 1066-1068 (2001).
[CrossRef]

Carminati, R.

Chance, B.

D. A. Boas, M. A. O'Leary, B. Chance, and A. G. Yodh, "Scattering of diffuse photon density waves by spherical inhomogeneties within turbid media: Analytical solution and applications," Proc. Natl. Acad. Sci. USA 91, 4887-4891 (1994).
[CrossRef] [PubMed]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

Cope, M.

S. R. Arridge, M. Cope, and D. T. Delpy, "The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis," Phys. Med. Biol. 37, 1531-1560 (1992).
[CrossRef] [PubMed]

Delpy, D. T.

S. R. Arridge, M. Cope, and D. T. Delpy, "The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis," Phys. Med. Biol. 37, 1531-1560 (1992).
[CrossRef] [PubMed]

Durian, D. J.

Elaloufi, R.

Fedoryuk, M. V.

M. V. Fedoryuk, Asimptotika: Integraly, Summy i Ryady (Nauka, 1987), in Russian.

Greffet, J.-J.

Heilscher, A. H.

A. D. Klose and A. H. Heilscher, "Optical tomography using time-independent equation of radiative transfer--Part 2: inverse model," J. Quant. Spectrosc. Radiat. Transfer 72, 715-732 (2002).
[CrossRef]

A. D. Klose, U. Netz, J. Beuthan, and A. H. Heilscher, "Optical tomography using time-independent equation of radiative transfer--Part 1: forward model," J. Quant. Spectrosc. Radiat. Transfer 72, 691-713 (2002).
[CrossRef]

Hielscher, A. H.

Jacques, S. L.

L. H. Wang, S. L. Jacques, and L. Q. Zheng, "MCML-Monte Carlo modeling of light transport in multilayered tissues," Comput. Methods Programs Biomed. 47, 131-145 (1995).
[CrossRef] [PubMed]

Keller, J. B.

Kim, A. D.

Klose, A. D.

A. D. Klose, U. Netz, J. Beuthan, and A. H. Heilscher, "Optical tomography using time-independent equation of radiative transfer--Part 1: forward model," J. Quant. Spectrosc. Radiat. Transfer 72, 691-713 (2002).
[CrossRef]

A. D. Klose and A. H. Heilscher, "Optical tomography using time-independent equation of radiative transfer--Part 2: inverse model," J. Quant. Spectrosc. Radiat. Transfer 72, 715-732 (2002).
[CrossRef]

Lax, M.

M. Xu, W. Cai, M. Lax, and R. R. Alfano, "Photon migration in turbid media using a cumulant approximation to the radiative transfer," Phys. Rev. E 65, 066609 (2002).
[CrossRef]

M. Xu, W. Cai, M. Lax, and R. R. Alfano, "Photon-transport forward model for imaging in turbid media," Opt. Lett. 26, 1066-1068 (2001).
[CrossRef]

Lebedev, N. N.

N. N. Lebedev, Special Functions and Their Applications (Dover, 1972).

Moscoso, M.

Netz, U.

A. D. Klose, U. Netz, J. Beuthan, and A. H. Heilscher, "Optical tomography using time-independent equation of radiative transfer--Part 1: forward model," J. Quant. Spectrosc. Radiat. Transfer 72, 691-713 (2002).
[CrossRef]

O'Leary, M. A.

D. A. Boas, M. A. O'Leary, B. Chance, and A. G. Yodh, "Scattering of diffuse photon density waves by spherical inhomogeneties within turbid media: Analytical solution and applications," Proc. Natl. Acad. Sci. USA 91, 4887-4891 (1994).
[CrossRef] [PubMed]

Olver, F. W.

F. W. Olver, Asymptotics and Special Functions (Academic, 1974).

Ren, K.

Rundick, J.

Sobolev, V. V.

V. V. Sobolev, A Treatise on Radiative Transfer (Van Nostrand, 1963).

Soloviev, V.

Soloviev, V. Y.

Sun, C.-W.

X. Wang, L. V. Wang, C.-W. Sun, and C.-C. Yang, "Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments," J. Biomed. Opt. 8, 608-617 (2003).
[CrossRef] [PubMed]

Vinogradov, S.

Vinogradov, S. A.

Wang, L. H.

L. H. Wang, S. L. Jacques, and L. Q. Zheng, "MCML-Monte Carlo modeling of light transport in multilayered tissues," Comput. Methods Programs Biomed. 47, 131-145 (1995).
[CrossRef] [PubMed]

Wang, L. V.

X. Wang, L. V. Wang, C.-W. Sun, and C.-C. Yang, "Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments," J. Biomed. Opt. 8, 608-617 (2003).
[CrossRef] [PubMed]

Wang, L.-H.

Wang, X.

X. Wang, L. V. Wang, C.-W. Sun, and C.-C. Yang, "Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments," J. Biomed. Opt. 8, 608-617 (2003).
[CrossRef] [PubMed]

X. Wang, G. Yao, and L.-H. Wang, "Monte Carlo model and single-scattering approximation of polarized light propagation in turbid media containing glucose," Appl. Opt. 41, 792-801 (2002).
[CrossRef] [PubMed]

Wilson, D.

Wilson, D. F.

Xu, M.

W. Cai, M. Xu, and R. R. Alfano, "Analytical form of the particle distribution based on the cumulant solution of the elastic Boltzmann transport equation," Phys. Rev. E. 71, 041202 (2005).
[CrossRef]

M. Xu, W. Cai, and R. R. Alfano, "Multiple passage of the light through an absorption inhomogeneity in optical imaging of turbid media," Opt. Lett. 29, 1757-1759 (2004).
[CrossRef] [PubMed]

M. Xu, W. Cai, M. Lax, and R. R. Alfano, "Photon migration in turbid media using a cumulant approximation to the radiative transfer," Phys. Rev. E 65, 066609 (2002).
[CrossRef]

M. Xu, W. Cai, M. Lax, and R. R. Alfano, "Photon-transport forward model for imaging in turbid media," Opt. Lett. 26, 1066-1068 (2001).
[CrossRef]

Yang, C.-C.

X. Wang, L. V. Wang, C.-W. Sun, and C.-C. Yang, "Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments," J. Biomed. Opt. 8, 608-617 (2003).
[CrossRef] [PubMed]

Yao, G.

Yodh, A. G.

D. A. Boas, M. A. O'Leary, B. Chance, and A. G. Yodh, "Scattering of diffuse photon density waves by spherical inhomogeneties within turbid media: Analytical solution and applications," Proc. Natl. Acad. Sci. USA 91, 4887-4891 (1994).
[CrossRef] [PubMed]

Zheng, L. Q.

L. H. Wang, S. L. Jacques, and L. Q. Zheng, "MCML-Monte Carlo modeling of light transport in multilayered tissues," Comput. Methods Programs Biomed. 47, 131-145 (1995).
[CrossRef] [PubMed]

Appl. Opt.

Comput. Methods Programs Biomed.

L. H. Wang, S. L. Jacques, and L. Q. Zheng, "MCML-Monte Carlo modeling of light transport in multilayered tissues," Comput. Methods Programs Biomed. 47, 131-145 (1995).
[CrossRef] [PubMed]

Inverse Probl.

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

J. Biomed. Opt.

X. Wang, L. V. Wang, C.-W. Sun, and C.-C. Yang, "Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments," J. Biomed. Opt. 8, 608-617 (2003).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

J. Quant. Spectrosc. Radiat. Transfer

A. D. Klose, U. Netz, J. Beuthan, and A. H. Heilscher, "Optical tomography using time-independent equation of radiative transfer--Part 1: forward model," J. Quant. Spectrosc. Radiat. Transfer 72, 691-713 (2002).
[CrossRef]

A. D. Klose and A. H. Heilscher, "Optical tomography using time-independent equation of radiative transfer--Part 2: inverse model," J. Quant. Spectrosc. Radiat. Transfer 72, 715-732 (2002).
[CrossRef]

Opt. Lett.

Phys. Med. Biol.

S. R. Arridge, M. Cope, and D. T. Delpy, "The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis," Phys. Med. Biol. 37, 1531-1560 (1992).
[CrossRef] [PubMed]

Phys. Rev. E

M. Xu, W. Cai, M. Lax, and R. R. Alfano, "Photon migration in turbid media using a cumulant approximation to the radiative transfer," Phys. Rev. E 65, 066609 (2002).
[CrossRef]

Phys. Rev. E.

W. Cai, M. Xu, and R. R. Alfano, "Analytical form of the particle distribution based on the cumulant solution of the elastic Boltzmann transport equation," Phys. Rev. E. 71, 041202 (2005).
[CrossRef]

Proc. Natl. Acad. Sci. USA

D. A. Boas, M. A. O'Leary, B. Chance, and A. G. Yodh, "Scattering of diffuse photon density waves by spherical inhomogeneties within turbid media: Analytical solution and applications," Proc. Natl. Acad. Sci. USA 91, 4887-4891 (1994).
[CrossRef] [PubMed]

Other

V. V. Sobolev, A Treatise on Radiative Transfer (Van Nostrand, 1963).

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

N. N. Lebedev, Special Functions and Their Applications (Dover, 1972).

M. V. Fedoryuk, Asimptotika: Integraly, Summy i Ryady (Nauka, 1987), in Russian.

F. W. Olver, Asymptotics and Special Functions (Academic, 1974).

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Figures (5)

Fig. 1
Fig. 1

Normalized photon number density versus depth z at the lateral distance R = 5 mm (top), versus R at the surface of the semispace (middle), and at the depth z = 10 mm (bottom). The point source is located at (0, 0, 5). The transport coefficient is set to α = 0.5 mm−1. The three curves are computed by the diffusion approximation (DE), the asymptotic solution presented in this work (U), and the Monte Carlo method (MC), respectively. The albedo of single scattering event λ is set to 0.95.

Fig. 2
Fig. 2

Same as in Fig. 1 but for the albedo λ = 0.9.

Fig. 3
Fig. 3

Same as in Fig. 1 but for the albedo λ = 0.85.

Fig. 4
Fig. 4

Same as in Fig. 1 but for the albedo λ = 0.8.

Fig. 5
Fig. 5

Dependence of |γ| and |ϰ| on ω for the transport coefficient α = 0.5 and several values of the albedo λ = {0.999, 0.99, 0.95, 0.9, 0.85, 0.8}.

Equations (95)

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

( s , I ) + 1 c I t = α I + α λ 4 π 0 2 π d φ × 1 1 I ( n , φ ) d η + S ,
S = α λ P 0 exp ( α | r r 0 | ) 4 π | r r 0 | 2 δ ( t t 0 | r r 0 | / c ) ,
U = 1 c 0 2 π d φ 1 1 I ( η , φ ) d η .
Δ U ϰ 2 U = q ,
ϰ 2 = 3 γ ( γ α λ ) ,
γ = α ( 1 + i ω α c ) ,
q = Q 0 4 π | r r 0 | 2 exp ( γ | r r 0 | i ω t 0 ) ,
Q 0 = 3 c N 0 h ν α λ γ .
U z | z = 0 = γ U | z = 0 ,
U | z = 0 .
Δ G ϰ 2 G = δ ( r r 0 )
G = G ( 0 ) + G ( 1 ) G ( 2 ) ,
G ( 0 ) = exp ( ϰ | r r 0 | ) 4 π | r r 0 | ,
G ( 1 ) = exp ( ϰ | r r 1 | ) 4 π | r r 1 | ,
G ( 2 ) = γ 2 π 0 exp [ ( z + z 0 ) μ 2 + ϰ 2 ] μ 2 + ϰ 2 [ μ 2 + ϰ 2 + γ ] J 0 ( μ R ) μ d μ ,
| r r 0 | = [ R 2 + ( z z 0 ) 2 ] 1 / 2 ,
| r r 1 | = [ R 2 + ( z + z 0 ) 2 ] 1 / 2 ,
R = [ r 2 + r 0     2 2 r r 0 cos ( φ φ 0 ) ] 1 / 2 ,
G ( 2 ) γ 2 π exp ( ϰ | r r 1 | ) ϰ ( z + z 0 ) + γ | r r 1 | .
U = U ( 0 ) + U ( 1 ) U ( 2 ) ,
K λ ( x ) = 1 2 ( x 2 ) λ 0 exp ( t x 2 4 t ) t λ 1 d t .
q ( | r r 0 | ) = Q 0 γ 2 64 π 2 exp ( i ω t 0 ) 0 0 d ν d τ ( ν τ ) 3 / 2 × exp [ ( ν + τ ) γ 2 16 ν + τ ν τ | r r 0 | 2 ] .
exp { ϰ [ R 2 + ( z ± z 0 ) 2 ] 1 / 2 } [ R 2 + ( z ± z 0 ) 2 ] 1 / 2 = 0 E ( μ , | z ± z 0 | ) × J 0 ( μ R ) μ d μ ,
E ( μ , z ) = exp ( z ϰ 2 + μ 2 ) ϰ 2 + μ 2 ,
U ( j ) = Q 0 γ 2 ( 16 ) 2 π 3 n = exp ( i n φ i ω t 0 ) × 0 C j ϰ 2 + μ 2 J n ( μ r ) μ d μ × 0 0 exp [ ( ν + τ ) γ 2 r 0     2 16 ν + τ ν τ ] d ν d τ ( ν τ ) 3 / 2 × 0 exp ( γ 2 r 2 16 ν + τ ν τ ) J n ( μ r ) r d r × 0 2 π exp [ γ 2 8 ν + τ ν τ r 0 r cos ( φ 0 φ ) i n φ ] d φ × 0 exp [ γ 2 16 ν + τ ν τ ( z z 0 ) 2 | z z | ϰ 2 + μ 2 ] d z ,
C 0 = C 1 = 1 , C 2 = 2 γ ( μ 2 + ϰ 2 + γ ) 1 .
J n ( x ) = 1 2 π 0 2 π exp ( i x sin φ i n φ ) d φ ,
0 exp ( p 2 r 2 ) J n ( λ r ) J n ( μ r ) r d r = 1 2 p 2 exp ( λ 2 + μ 2 4 p 2 ) I n ( λ μ 2 p 2 ) ,
U ( j ) = exp ( i ω t 0 ) Q 0 16 π 2 0 C j μ d μ J 0 ( μ R ) × 0 d z E ( μ , | z z | ) S ( μ , z z 0 ) ,
S ( μ , x ) = 0 0 d ν d τ ( ν + τ ) ν τ × exp [ ( ν + τ ) ( μ γ ) 2 4 ν τ ν + τ γ 2 x 2 16 ν + τ ν τ ] .
S ( μ , x ) = 0 d t t exp [ t μ 2 x 2 4 t ] × 0 d s ( 1 + s ) s exp [ 1 + s 4 t γ 2 x 2 ] .
Φ ( a x ) = 2 π 0 a x exp ( s 2 ) d s .
S ( μ , x ) = 2 π K 0 ( μ | x | ) π 0 d t t × exp [ t x 2 μ 2 4 t ] Φ ( γ | x | 2 t ) ,
K λ ( x ) = Γ ( λ + 1 2 ) π ( 2 x ) λ 0 cos ( x t ) d t ( 1 + t 2 ) λ + ( 1 / 2 ) ,
S ( μ , x ) = 4 0 cos ( x ν ) μ 2 + ν 2 arctan ( 1 γ μ 2 + ν 2 ) d ν .
U = exp ( i ω t 0 ) Q 0 2 π 2 0 F ( μ , z z 0 ) J 0 ( μ R ) μ d μ ,
U ( 1 ) = exp ( i ω t 0 ) Q 0 2 π 2 z 0 × 0 E ( μ , z ) F ( μ , z 0 ) J 0 ( μ R ) μ d μ ,
U ( 2 ) = exp ( i ω t 0 ) Q 0 4 π 2 ( z + z 0 ) × 0 C 2 E ( μ , z ) F ( μ , z 0 ) J 0 ( μ R ) μ d μ ,
F ( μ , x ) = 0 cos ( x ν ) arctan ( 1 γ μ 2 + ν 2 ) μ 2 + ν 2 ( μ 2 + ϰ 2 + ν 2 )  d ν .
F ( ν , x ) = F 1 ( ν , x ) F 2 ( ν , x ) ,
F 1 ( μ , x ) = 1 4 i ln ( 1 + i γ μ 2 + ν 2 ) μ 2 + ν 2 ( μ 2 + ϰ 2 + ν 2 ) ×  exp ( i ν | x | ) d ν ,
F 2 ( μ , x ) = 1 4 i ln ( 1 i γ μ 2 + ν 2 ) μ 2 + ν 2 ( μ 2 + ϰ 2 + ν 2 ) ×  exp ( i ν | x | ) d ν .
F 1 ( μ , x ) = 1 4 i ln ( 1 + i μ γ cosh ζ ) μ 2 cosh 2 ζ + ϰ 2 ×  exp ( i μ | x | sinh ζ ) d ζ ,
F 2 ( μ , x ) = 1 4 i ln ( 1 i μ γ cosh ζ ) μ 2 cosh 2 ζ + ϰ 2 ×  exp ( i μ | x | sinh ζ ) d ζ .
F 1 ( μ , x ) = π 4 ϰ ln ( γ + ϰ γ ϰ ) E ( μ , | x | ) + 1 4 i + i π + i π ln ( 1 + i μ γ cosh ζ ) μ 2 cosh 2 ζ + ϰ 2 ×  exp ( i μ | x | sinh ζ ) d ζ + 1 4 i    Γ ln ( 1 + i μ γ     cosh ζ ) μ 2 cosh 2 ζ + ϰ 2 ×  exp ( i μ | x | sinh ζ ) d ζ ,
F ( μ , x ) = π 4 ϰ ln ( γ + ϰ γ ϰ ) E ( μ , | x | ) + π 2 γ exp ( | x | μ 2 + ν 2 ) d ν ( ϰ 2 ν 2 ) μ 2 + ν 2 .
U = exp ( i ω t 0 ) Q 0 [ 1 2 ϰ ln ( γ + ϰ γ ϰ ) G ( 0 ) + D ( 0 ) ] ,
D ( 0 ) = 1 4 π | r r 0 | γ exp ( ν | r r 0 | ) ϰ 2 ν 2  d ν ,
D ( 0 ) = exp ( ϰ | r r 0 | ) 8 π ϰ | r r 0 |  Ei [ ( γ ϰ ) | r r 0 | ] exp ( ϰ | r r 0 | ) 8 π ϰ | r r 0 |  Ei [ ( γ + ϰ ) | r r 0 | ] .
U Q 0 exp ( γ | r r 0 | i ω t 0 ) 8 π ϰ ( γ + ϰ ) | r r 0 | 2 ,   when   | γ | | ϰ | ,
U exp ( - i ω t 0 ) Q 0 2 ϰ ln [ ( γ + ϰ ) | r r 0 | ] G ( 0 ) ,   when   | γ | | ϰ | ,
U exp ( i ω t 0 ) Q 0 G ( 0 ) ,   when   | γ | | ϰ | .
J 0 ( μ R ) = 1 2 [ H 0 ( 1 ) ( μ R ) + H 0 ( 2 ) ( μ R ) ] ,
H 0 ( 1 ) ( μ R ) = H 0 ( 2 ) ( μ R ) ,
U ( j ) = exp ( i ω t 0 ) Q 0 ϰ 3 / 2 16 i π 2 I j ,
I j = exp ( i π 4 ) 2 π R Γ ξ Γ ζ A j exp ( Ψ ) sinh ξ  d ξ d ζ ,
A 1 = i W sinh ξ sinh ζ ,
A 2 = γ W cosh ξ i sinh ξ sinh ζ ϰ cosh ξ + γ ,
W = s = 1 , 2 ( 1 ) s 1 ln [ γ + ( 1 ) s 1 i ϰ sinh ξ cosh ζ ] ,
Ψ = Ψ ( 1 ) + Ψ ( 2 ) ,
Ψ ( 1 ) = ϰ z cosh ξ + i ϰ z 0 sinh ξ sinh ζ + i ϰ R sinh ξ ,
Ψ ( 2 ) = ln ( cosh 2 ξ + sinh 2 ξ sinh 2 ζ ) 2 ln ϰ ,
( Ψ / ξ ) | ξ 0 = 0 , ( Ψ / ζ ) | ξ 0 = 0.
ϰ z sinh ξ 0 i ϰ z 0 cosh ξ 0 sinh ζ + i ϰ R cosh ξ 0 = 0 .
i ϰ z 0 = 2 sinh ξ 0 sinh ζ cosh 2 ξ 0 + sinh 2 ξ 0 sinh 2 ζ ,
z 1 ( ζ ) = i z 0 cosh ξ 0 sinh ξ 0 sinh ζ ,
Ψ / ξ | ξ 0 = ϰ | r r | sinh ( ξ 0 i θ 1 ) = 0 ,
| r r | = [ R 2 + ( z + z 1 ) 2 ] 1 / 2 ,
cos θ 1 = ( z + z 1 ) / | r r | , sin θ 1 = R / | r r | .
ξ 0 = i θ 1 .
Ψ ( 1 ) Ψ 0 ( 1 ) + 1 2 Ψ 0 ( 1 ) ( ξ ξ 0 ) 2 ,
Ψ 0 ( 1 ) = ϰ | r r | cos ( θ 2 θ 1 ) ,
| r r | = [ R 2 + ( z + z 0 2 / z 1 ) 2 ] 1 / 2 ,
cos θ 2 = z + z 0 2 / z 1 | r r | ,
sin θ 2 = R | r r | .
I j 2 ϰ 2 Γ ζ z 1     2 ( z 1     2 z 0     2 ) A j ( ξ 0 ) exp ( Ψ 0 ( 1 ) ) d ζ ( Ψ 0 ( 1 ) | r r | ) 1 / 2 cos 2 θ 1 .
cos ψ = Re ϰ / | ϰ | , sin ψ = Im ϰ / | ϰ | .
ζ 1 = sinh 1 [ ( z + z 0 ) / R ] , ζ 2 = i π ζ 1 ,
U ( j ) exp ( i ω t 0 ) Q 0 [ 1 2 ϰ ln ( γ + ϰ γ ϰ ) G ( j ) + D ( j ) ] ,
D ( 1 ) = 1 4 π ϰ 0 τ 0 Λ d τ 1 τ 2 exp ( ϰ σ | r r | ) | r r | ,
D ( 2 ) = γ 4 π ϰ 0 τ 0 Λ d τ 1 τ exp ( ϰ σ | r r | ) ϰ ( z + z 0 τ ) + γ | r r | ,
σ = ( z + z 0 / τ ) ( z + z 0 τ ) + R 2 | r r | 2 ,
Λ = | r r | σ [ ( z + z 0 τ ) 2 + R 2 τ 2 ] 1 / 2 ,
| r r | = [ ( z + z 0 τ ) 2 + R 2 ] 1 / 2 .
τ 0 = z R sinh ζ 0 z 0 ,
cosh ζ 0 = γ ϰ sin θ 1 .
ϰ γ = τ 0 f ( τ 0 ) ,
f ( τ 0 ) = [ R 2 τ 0     2 + ( z + z 0 τ 0 ) 2 R 2 + ( z + z 0 τ 0 ) 2 ] 1 / 2
τ 0 = z R 2 + z 2 ϰ γ + z z 0 R 2 ( R 2 + z 2 ) 2 ( ϰ γ ) 2 + O ( ϰ 3 γ 3 ) .
τ 0 ( n + 1 ) = x γ    f ( τ 0 ( n ) ) ,
D ( 0 ) 1 4 π 1 2 ϰ ln ( γ + ϰ γ ϰ ) exp ( γ | r r 0 | ) | r r 0 | ,
D ( 1 ) 1 4 π Λ 0 2 ϰ ln ( 1 + τ 0 1 τ 0 ) exp ( ϰ σ 0 | r r 2 | ) | r r 2 | ,
D ( 2 ) γ 2 π Λ 0 2 ϰ ln ( 1 τ 0 ) exp ( ϰ σ 0 | r r 2 | ) ϰ ( z + z 0 τ 0 ) + γ | r r 2 | ,
Λ 0 = ϰ γ 1 τ 0 σ 0 .
t t 0 3 | r r 0 | / c ,

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