Abstract

The three-dimensional radiative transfer equation is solved for modeling the light propagation in anisotropically scattering semi-infinite media such as biological tissue, considering the effect of internal reflection at the interfaces. The two-dimensional Fourier transform and the modified spherical harmonics method are applied to derive the general solution to the associated homogeneous problem in terms of analytical functions. The obtained solution is used for solving boundary-value problems, which are important for applications in the biomedical optics field. The derived equations are successfully verified by comparisons with Monte Carlo simulations.

© 2012 Optical Society of America

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References

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  1. K. M. Case and P. F. Zweifel, Linear Transport Theory(Addison-Wesley, 1967).
  2. E. D’Eon and G. Irving, “A quantized-diffusion model for rendering translucent materials,” ACM Trans. Graph. 30, 56:1–56:12 (2011).
    [CrossRef]
  3. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).
  4. A. Kienle and R. Hibst, “Light guiding in biological tissue due to scattering,” Phys. Rev. Lett. 97, 018104 (2006).
    [CrossRef]
  5. B. Wilson and G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
    [CrossRef]
  6. F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation Through Biological Tissue (SPIE, 2010).
  7. P. S. Mohan, T. Tarvainen, M. Schweiger, A. Pulkkinen, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comput. Phys. 230, 7364–7383 (2011).
    [CrossRef]
  8. A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285–1302 (1998).
    [CrossRef]
  9. B. D. Ganapol, “Radiative transfer with internal reflection via the converge discrete ordinates method,” J. Quant. Spectrosc. Radiat. Transfer 112, 693–713 (2011).
    [CrossRef]
  10. M. M. R. Williams, “The searchlight problem in radiative transfer with internal reflection,” J. Phys. A Math. Theor. 40, 6407–6425 (2007).
    [CrossRef]
  11. V. A. Markel, “Modified spherical harmonics method for solving the radiative transport equation,” Waves Random Complex Media 14, L13–L19 (2004).
  12. A. Liemert and A. Kienle, “Radiative transfer in two-dimensional infinitely extended scattering media,” J. Phys. A Math. Theor. 44, 505206 (2011).
    [CrossRef]
  13. G. Panasyuk, J. C. Schotland, and V. A. Markel, “Radiative transport equation in rotated reference frames,” J. Phys. A Math. Gen. 39, 115–137 (2006).
    [CrossRef]
  14. M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “The Green’s function for the radiative transport equation in the slab geometry,” J. Phys. A Math. Theor. 43, 065402 (2010).
    [CrossRef]
  15. A. Liemert and A. Kienle, “Analytical approach for solving the radiative transfer equation in two-dimensional layered media,” J. Quant. Spectrosc. Radiat. Transfer 113, 559–564 (2012).
    [CrossRef]
  16. A. Liemert and A. Kienle, “Green’s functions for the two-dimensional radiative transfer equation in bounded media,” J. Phys. A Math. Theor. 45, 175201 (2012).
    [CrossRef]
  17. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
    [CrossRef]
  18. M. A. Blanco, M. Flórez, and M. Bermejo, “Evaluation of the rotation matrices in the basis of real spherical harmonics,” J. Mol. Struct. 419, 19–27 (1997).
  19. N. Baddour, “Operational and convolution properties of two-dimensional Fourier transforms in polar coordinates,” J. Opt. Soc. Am. A 26, 1767–1777 (2009).
    [CrossRef]
  20. M. M. R. Williams, “The Milne problem with Fresnel reflection,” J. Phys. A Math. Gen. 38, 3841–3856 (2005).
    [CrossRef]

2012 (2)

A. Liemert and A. Kienle, “Analytical approach for solving the radiative transfer equation in two-dimensional layered media,” J. Quant. Spectrosc. Radiat. Transfer 113, 559–564 (2012).
[CrossRef]

A. Liemert and A. Kienle, “Green’s functions for the two-dimensional radiative transfer equation in bounded media,” J. Phys. A Math. Theor. 45, 175201 (2012).
[CrossRef]

2011 (4)

E. D’Eon and G. Irving, “A quantized-diffusion model for rendering translucent materials,” ACM Trans. Graph. 30, 56:1–56:12 (2011).
[CrossRef]

P. S. Mohan, T. Tarvainen, M. Schweiger, A. Pulkkinen, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comput. Phys. 230, 7364–7383 (2011).
[CrossRef]

B. D. Ganapol, “Radiative transfer with internal reflection via the converge discrete ordinates method,” J. Quant. Spectrosc. Radiat. Transfer 112, 693–713 (2011).
[CrossRef]

A. Liemert and A. Kienle, “Radiative transfer in two-dimensional infinitely extended scattering media,” J. Phys. A Math. Theor. 44, 505206 (2011).
[CrossRef]

2010 (1)

M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “The Green’s function for the radiative transport equation in the slab geometry,” J. Phys. A Math. Theor. 43, 065402 (2010).
[CrossRef]

2009 (1)

2007 (1)

M. M. R. Williams, “The searchlight problem in radiative transfer with internal reflection,” J. Phys. A Math. Theor. 40, 6407–6425 (2007).
[CrossRef]

2006 (2)

G. Panasyuk, J. C. Schotland, and V. A. Markel, “Radiative transport equation in rotated reference frames,” J. Phys. A Math. Gen. 39, 115–137 (2006).
[CrossRef]

A. Kienle and R. Hibst, “Light guiding in biological tissue due to scattering,” Phys. Rev. Lett. 97, 018104 (2006).
[CrossRef]

2005 (1)

M. M. R. Williams, “The Milne problem with Fresnel reflection,” J. Phys. A Math. Gen. 38, 3841–3856 (2005).
[CrossRef]

2004 (1)

V. A. Markel, “Modified spherical harmonics method for solving the radiative transport equation,” Waves Random Complex Media 14, L13–L19 (2004).

1999 (1)

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

1998 (1)

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285–1302 (1998).
[CrossRef]

1997 (1)

M. A. Blanco, M. Flórez, and M. Bermejo, “Evaluation of the rotation matrices in the basis of real spherical harmonics,” J. Mol. Struct. 419, 19–27 (1997).

1983 (1)

B. Wilson and G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef]

Adam, G.

B. Wilson and G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef]

Alcouffe, R. E.

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285–1302 (1998).
[CrossRef]

Arridge, S. R.

P. S. Mohan, T. Tarvainen, M. Schweiger, A. Pulkkinen, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comput. Phys. 230, 7364–7383 (2011).
[CrossRef]

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

Baddour, N.

Barbour, R. L.

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285–1302 (1998).
[CrossRef]

Bermejo, M.

M. A. Blanco, M. Flórez, and M. Bermejo, “Evaluation of the rotation matrices in the basis of real spherical harmonics,” J. Mol. Struct. 419, 19–27 (1997).

Blanco, M. A.

M. A. Blanco, M. Flórez, and M. Bermejo, “Evaluation of the rotation matrices in the basis of real spherical harmonics,” J. Mol. Struct. 419, 19–27 (1997).

Case, K. M.

K. M. Case and P. F. Zweifel, Linear Transport Theory(Addison-Wesley, 1967).

D’Eon, E.

E. D’Eon and G. Irving, “A quantized-diffusion model for rendering translucent materials,” ACM Trans. Graph. 30, 56:1–56:12 (2011).
[CrossRef]

Del Bianco, S.

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation Through Biological Tissue (SPIE, 2010).

Flórez, M.

M. A. Blanco, M. Flórez, and M. Bermejo, “Evaluation of the rotation matrices in the basis of real spherical harmonics,” J. Mol. Struct. 419, 19–27 (1997).

Ganapol, B. D.

B. D. Ganapol, “Radiative transfer with internal reflection via the converge discrete ordinates method,” J. Quant. Spectrosc. Radiat. Transfer 112, 693–713 (2011).
[CrossRef]

Hibst, R.

A. Kienle and R. Hibst, “Light guiding in biological tissue due to scattering,” Phys. Rev. Lett. 97, 018104 (2006).
[CrossRef]

Hielscher, A. H.

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285–1302 (1998).
[CrossRef]

Irving, G.

E. D’Eon and G. Irving, “A quantized-diffusion model for rendering translucent materials,” ACM Trans. Graph. 30, 56:1–56:12 (2011).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

Ismaelli, A.

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation Through Biological Tissue (SPIE, 2010).

Kienle, A.

A. Liemert and A. Kienle, “Analytical approach for solving the radiative transfer equation in two-dimensional layered media,” J. Quant. Spectrosc. Radiat. Transfer 113, 559–564 (2012).
[CrossRef]

A. Liemert and A. Kienle, “Green’s functions for the two-dimensional radiative transfer equation in bounded media,” J. Phys. A Math. Theor. 45, 175201 (2012).
[CrossRef]

A. Liemert and A. Kienle, “Radiative transfer in two-dimensional infinitely extended scattering media,” J. Phys. A Math. Theor. 44, 505206 (2011).
[CrossRef]

A. Kienle and R. Hibst, “Light guiding in biological tissue due to scattering,” Phys. Rev. Lett. 97, 018104 (2006).
[CrossRef]

Liemert, A.

A. Liemert and A. Kienle, “Green’s functions for the two-dimensional radiative transfer equation in bounded media,” J. Phys. A Math. Theor. 45, 175201 (2012).
[CrossRef]

A. Liemert and A. Kienle, “Analytical approach for solving the radiative transfer equation in two-dimensional layered media,” J. Quant. Spectrosc. Radiat. Transfer 113, 559–564 (2012).
[CrossRef]

A. Liemert and A. Kienle, “Radiative transfer in two-dimensional infinitely extended scattering media,” J. Phys. A Math. Theor. 44, 505206 (2011).
[CrossRef]

Machida, M.

M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “The Green’s function for the radiative transport equation in the slab geometry,” J. Phys. A Math. Theor. 43, 065402 (2010).
[CrossRef]

Markel, V. A.

M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “The Green’s function for the radiative transport equation in the slab geometry,” J. Phys. A Math. Theor. 43, 065402 (2010).
[CrossRef]

G. Panasyuk, J. C. Schotland, and V. A. Markel, “Radiative transport equation in rotated reference frames,” J. Phys. A Math. Gen. 39, 115–137 (2006).
[CrossRef]

V. A. Markel, “Modified spherical harmonics method for solving the radiative transport equation,” Waves Random Complex Media 14, L13–L19 (2004).

Martelli, F.

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation Through Biological Tissue (SPIE, 2010).

Mohan, P. S.

P. S. Mohan, T. Tarvainen, M. Schweiger, A. Pulkkinen, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comput. Phys. 230, 7364–7383 (2011).
[CrossRef]

Panasyuk, G.

G. Panasyuk, J. C. Schotland, and V. A. Markel, “Radiative transport equation in rotated reference frames,” J. Phys. A Math. Gen. 39, 115–137 (2006).
[CrossRef]

Panasyuk, G. Y.

M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “The Green’s function for the radiative transport equation in the slab geometry,” J. Phys. A Math. Theor. 43, 065402 (2010).
[CrossRef]

Pulkkinen, A.

P. S. Mohan, T. Tarvainen, M. Schweiger, A. Pulkkinen, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comput. Phys. 230, 7364–7383 (2011).
[CrossRef]

Schotland, J. C.

M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “The Green’s function for the radiative transport equation in the slab geometry,” J. Phys. A Math. Theor. 43, 065402 (2010).
[CrossRef]

G. Panasyuk, J. C. Schotland, and V. A. Markel, “Radiative transport equation in rotated reference frames,” J. Phys. A Math. Gen. 39, 115–137 (2006).
[CrossRef]

Schweiger, M.

P. S. Mohan, T. Tarvainen, M. Schweiger, A. Pulkkinen, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comput. Phys. 230, 7364–7383 (2011).
[CrossRef]

Tarvainen, T.

P. S. Mohan, T. Tarvainen, M. Schweiger, A. Pulkkinen, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comput. Phys. 230, 7364–7383 (2011).
[CrossRef]

Williams, M. M. R.

M. M. R. Williams, “The searchlight problem in radiative transfer with internal reflection,” J. Phys. A Math. Theor. 40, 6407–6425 (2007).
[CrossRef]

M. M. R. Williams, “The Milne problem with Fresnel reflection,” J. Phys. A Math. Gen. 38, 3841–3856 (2005).
[CrossRef]

Wilson, B.

B. Wilson and G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef]

Zaccanti, G.

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation Through Biological Tissue (SPIE, 2010).

Zweifel, P. F.

K. M. Case and P. F. Zweifel, Linear Transport Theory(Addison-Wesley, 1967).

ACM Trans. Graph. (1)

E. D’Eon and G. Irving, “A quantized-diffusion model for rendering translucent materials,” ACM Trans. Graph. 30, 56:1–56:12 (2011).
[CrossRef]

Inverse Probl. (1)

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

J. Comput. Phys. (1)

P. S. Mohan, T. Tarvainen, M. Schweiger, A. Pulkkinen, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comput. Phys. 230, 7364–7383 (2011).
[CrossRef]

J. Mol. Struct. (1)

M. A. Blanco, M. Flórez, and M. Bermejo, “Evaluation of the rotation matrices in the basis of real spherical harmonics,” J. Mol. Struct. 419, 19–27 (1997).

J. Opt. Soc. Am. A (1)

J. Phys. A Math. Gen. (2)

M. M. R. Williams, “The Milne problem with Fresnel reflection,” J. Phys. A Math. Gen. 38, 3841–3856 (2005).
[CrossRef]

G. Panasyuk, J. C. Schotland, and V. A. Markel, “Radiative transport equation in rotated reference frames,” J. Phys. A Math. Gen. 39, 115–137 (2006).
[CrossRef]

J. Phys. A Math. Theor. (4)

M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “The Green’s function for the radiative transport equation in the slab geometry,” J. Phys. A Math. Theor. 43, 065402 (2010).
[CrossRef]

A. Liemert and A. Kienle, “Green’s functions for the two-dimensional radiative transfer equation in bounded media,” J. Phys. A Math. Theor. 45, 175201 (2012).
[CrossRef]

M. M. R. Williams, “The searchlight problem in radiative transfer with internal reflection,” J. Phys. A Math. Theor. 40, 6407–6425 (2007).
[CrossRef]

A. Liemert and A. Kienle, “Radiative transfer in two-dimensional infinitely extended scattering media,” J. Phys. A Math. Theor. 44, 505206 (2011).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (2)

B. D. Ganapol, “Radiative transfer with internal reflection via the converge discrete ordinates method,” J. Quant. Spectrosc. Radiat. Transfer 112, 693–713 (2011).
[CrossRef]

A. Liemert and A. Kienle, “Analytical approach for solving the radiative transfer equation in two-dimensional layered media,” J. Quant. Spectrosc. Radiat. Transfer 113, 559–564 (2012).
[CrossRef]

Med. Phys. (1)

B. Wilson and G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef]

Phys. Med. Biol. (1)

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285–1302 (1998).
[CrossRef]

Phys. Rev. Lett. (1)

A. Kienle and R. Hibst, “Light guiding in biological tissue due to scattering,” Phys. Rev. Lett. 97, 018104 (2006).
[CrossRef]

Waves Random Complex Media (1)

V. A. Markel, “Modified spherical harmonics method for solving the radiative transport equation,” Waves Random Complex Media 14, L13–L19 (2004).

Other (3)

K. M. Case and P. F. Zweifel, Linear Transport Theory(Addison-Wesley, 1967).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation Through Biological Tissue (SPIE, 2010).

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Figures (4)

Fig. 1.
Fig. 1.

Spatially resolved reflectance caused by an isotropic point source located at z0=l* inside the semi-infinite medium. The approximation order is N=21.

Fig. 2.
Fig. 2.

Relative differences between the analytical approach and the Monte Carlo simulation. The inset shows the relative differences for a higher spatial resolution within the Monte Carlo simulation and the radial distances 0ρ3mm.

Fig. 3.
Fig. 3.

Fluence inside the semi-infinite medium with nin=1.0 caused by a Gaussian incident beam with radius η=1mm. The approximation order is N=13.

Fig. 4.
Fig. 4.

Angle-resolved Green’s function for the RTE in the semi-infinite geometry caused by a perpendicular incident δ beam. The approximation order is N=13.

Equations (58)

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s^·ψ(r,s^)+μtψ(r,s^)=μsf(s^·s^)ψ(r,s^)d2s,
ψ(r,s^)=1(2π)2ψ(q,z,s^)exp(iq·ρ)d2q,
[cosθz+iqsinθcos(ϕϕq)+μt]ψ(q,z,s^)=μsf(s^·s^)ψ(q,z,s^)d2s.
ψ(q,z,s^)=eξzψ(q,s^),
[μt+k·s^]ψ(q,s^)=μsf(s^·s^)ψ(q,s^)d2s,
k=(iq1iq2ξ)=k(sinθkcosϕqsinθksinϕqcosθk),
ψ(q,s^)=l=0M=ll(1)lψlMYlM(s^;k^),
(k^·s^)Ylm(s^;k^)=l2m24l21Yl1,m(s^;k^)+(l+1)2m24(l+1)21Yl+1,m(s^;k^).
f(s^·s^)=l=0M=llflYlM(s^;k^)YlM*(s^;k^),
fl=2π11f(ζ)Pl(ζ)dζ,
kl2M24l21ψl1,M+k(l+1)2M24(l+1)21ψl+1,MσlψlM=0,
BM=(0βlM000βlM0βl+1,M00βl+1,M00000βNM000βNM0),
βlM=l2M2(4l21)σl1σl.
BN1=(0βN,N1βN,N10).
ψ(q,s^)=l=MN(1)ll|νσl[YlM(s^;k^)+(1)M(1δM0)Yl,M(s^;k^)].
ξ=ξ(q)=q2+1λ2.
YlM(s^;k^)=m=lldmMl(θk)Ylm(θ,ϕϕq),
θk=θk(qλ)=arccos(ξk)=πiarsinh(qλ).
dmMl[θk(qλ)]=iMm(l+m)!(lm)!(l+M)!(lM)!k[1+(qλ)21]lk+Mm2[1+1+(qλ)2]k+mM22lk!(lmk)!(l+Mk)!(mM+k)!,
ψ(q,z,s^)=λi>0Ci(q)eq2+1/λi2zl=MNm=llψlm(qλi)Ylm(θ,ϕϕq),
ψlm(qλi)=(1)ll|νiσl[dmMl[θk(qλi)]+(1)M(1δM0)dm,Ml[θk(qλi)]],
ψ(r,s^)=l=0N2l+14πm=0l(2δm0)ψlm(ρ,z)dm0l(θ)cos(mχ),
ψlm(ρ,z)=im2π0Jm(qρ)i=1ilCi(q)ψlm(qλi)eq2+1/λi2zqdq,
il=(N+1)(l+1)2l+12l+12,
ψ(p)(r,s^)=l=0Nm=llψlm(p)(ρ,z)Ylm(θ,χ),
ψlm(p)(ρ,z)=14ππσ0σl(lm)!(l+m)!Plm[zz0R]i=1N+12l|νiνi|0λi3kl[Rλi]
ψ(ρ,z=0,s^)=R(s^·n^)ψ(ρ,z=0,s^),s^·n^<0,
ψ(p)(q,z,s^)=ψ(p)(r,s^)exp(iq·ρ)d2ρ
ψ(p)(q,z,s^)=l=0Nm=llψlm(p)(q,z)Ylm(θ,ϕϕq),
ψlm(p)(q,z)=(1)l4πσ0σl[sgn(zz0)]l+mi=1N+12l|νiνi|0λi2eq2+1/λi2|zz0|q2+1/λi2dm0l[θk(qλi)]
ψ(q,z=0,s^)=R(s^·n^)ψ(q,z=0,s^),s^·n^<0.
λi>0Ci(q)l=l¯Nψlm(h)(qλi)Rllm=l=mNψlm(p)(q,z=0)Rllm,
Rllm=12(2l+1)(2l+1)(lm)!(lm)!(l+m)!(l+m)!01[1(1)l+mR(μ)]×Plm(μ)Plm(μ)dμ.
R(μ)=12(μnμ0μ+nμ0)2+12(μ0nμμ0+nμ)2,μ>μc,
R(ρ)=s^·n^>0[1R(s^·n^)](s^·n^)ψ(ρ,z=0,s^)d2s.
R(ρ)=(s^·n^)ψ(ρ,z=0,s^)d2s,
R(ρ)=4π3[ψ10(h)(ρ,0)+ψ10(p)(ρ,0)],
ψ10(h)(ρ,0)=12π0J0(qρ)i=1NCi(q)ψ10(qλi)qdq.
Φ(r)=ψ(r,s^)d2s=4π[ψ00(h)(ρ,z)+ψ00(p)(ρ,z)],
ψ00(h)(ρ,z)=12π0J0(qρ)i=1N+12Ci(q)ψ00(qλi)eq2+1/λi2zqdq.
ψ(ρ,z=0,s^)=ψinc(ρ,s^)+R(s^·n^)ψ(ρ,z=0,s^),s^·n^<0,
ψ(q,z=0,s^)=ψinc(q,s^)+R(s^·n^)ψ(q,z=0,s^),s^·n^<0.
λi>0Ci(q)l=l¯Nψlm(qλi)Rllm=S(q)Ylm*(s^0)exp(imϕq).
λi>0Ci(q)l=l¯Nψlm(qλi)Rllm=S(q)2l+14πδm0,
R(ρ)=(s^·n^)ψ(ρ,z=0,s^)d2s(s^0·n^)S(ρ).
R(ρ)=S(ρ)4π3ψ10(h)(ρ,0).
Φ(r)=4πψ00(h)(ρ,z).
ψinc(ρ,s^)=2πη2exp(2ρ2η2)δ(s^z^)
ψinc(ρ,s^)=δ(ρ)δ(s^z^)
d001[θk(qλ)]=1+(qλ)2,d101[θk(qλ)]=iqλ2,d111[θk(qλ)]=11+(qλ)22,d1,11[θk(qλ)]=1+1+(qλ)22.
dmMl=l(2l1)(l2m2)(l2M2)[(d001mMl(l1))dmMl1[(l1)2m2][(l1)2M2](l1)(2l1)dmMl2].
dl1,l1l=(1l+ld001)dl1,l1l1,
dlll=d111dl1,l1l1.
dl,M1l=l+MlM+1iqλ1+(qλ)21dl,Ml,
dl1,M1l=ld001M+1ld001Ml+MlM+1iqλ1+(qλ)21dl1,Ml,
dMml=(1)m+MdmMl,
dM,ml=dm,Ml,
dm0l(θ)=(lm)!(l+m)!Plm(cosθ).

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