Abstract

We present a model for the diffuse reflectance when a continuous beam is incident normally on a half space composed of a uniform scattering and absorbing medium. This model is the result of an asymptotic analysis of the radiative transport equation for strong scattering, weak absorption, and a narrow beam width. Through comparison with the diffuse reflectance computed using the numerical solution of the radiative transport equation, we show that this diffuse reflectance model gives results that are accurate for small source–-detector separation distances.

© 2012 Optical Society of America

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References

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  1. L. V. Wang and H. Wu, Biomedical Optics: Principles and Imaging (Wiley, 2007).
  2. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, 1997).
  3. R. A. J. Groenhuis, H. A. Ferwerda, and J. J. Tenbosch, “Scattering and absorption of turbid materials determined from reflection measurements 1. Theory,” Appl. Opt. 22, 2456–2462 (1983).
    [CrossRef]
  4. T. J. Farrell, M. S. Patterson, and B. C. Wilson, “A diffusion theory model of spatially-resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
    [CrossRef]
  5. A. Kienle, L. Lilge, M. S. Patterson, R. Hibst, R. Steiner, and B. C. Wilson, “Spatially resolved absolute diffuse reflectance measurements for noninvasive determination of the optical scattering and absorption coefficients of biological tissue,” Appl. Opt. 35, 2304–2314 (1996).
    [CrossRef]
  6. A. Kienle and M. S. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equations for reflectance from a semi-infinite turbid medium,” J. Opt. Soc. Am. A 14, 246–254 (1997).
    [CrossRef]
  7. A. Kienle and M. S. Patterson, “Determination of the optical properties of semi-infinite turbid media from frequency-domain reflectance close to the source,” Phys. Med. Biol. 42, 1801–1819 (1997).
    [CrossRef]
  8. V. Venugopalan, J. S. You, and B. J. Tromberg, “Radiative transport in the diffusion approximation: an extension for highly absorbing media and small source-detector separations,” Phys. Rev. E 58, 2395–2407 (1998).
    [CrossRef]
  9. J. S. You, C. K. Hayakawa, and V. Venugopalan, “Frequency domain photon migration in the δ-P1 approximation: analysis of ballistic, transport and diffuse regimes,” Phys. Rev. E 72, 021903 (2005).
    [CrossRef]
  10. I. Seo, C. K. Hayakawa, and V. Venugopalan, “Radiative transport in the delta-P1 approximation for semi-infinite turbid media,” Med. Phys. 35, 681–693 (2008).
    [CrossRef]
  11. Y. L. Kim, Y. Liu, R. K. Wali, H. K. Roy, M. J. Goldberg, A. K. Kromin, K. Chen, and V. Backman, “Simultaneous measurement of angular and spectral properties of light scattering for characterization of tissue microarchitecture and its alteration in early precancer,” IEEE J. Sel. Top. Quantum Electron. 9, 243–256 (2003).
    [CrossRef]
  12. W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
    [CrossRef]
  13. A. Myakov, L. Nieman, L. Wicky, U. Utzinger, R. Richards-Kortum, and K. Sokolov, “Fiber optic probe for polarized reflectance spectroscopy in vivo: design and performance,” J. Biomed. Opt. 7, 388–397 (2002).
    [CrossRef]
  14. A. D. Kim, “Correcting the diffusion approximation at the boundary,” J. Opt. Soc. Am. A 28, 1007–1015 (2011).
    [CrossRef]
  15. A. D. Kim and M. Moscoso, “Beam propagation in sharply peaked forward scattering media,” J. Opt. Soc. Am. A 21, 797–803 (2004).
    [CrossRef]
  16. A. E. Siegman, “Quasi fast Hankel transform,” Opt. Lett. 1, 13–15 (1977).
    [CrossRef]
  17. A. D. Kim and M. Moscoso, “Radiative transfer computations for optical beams,” J. Comput. Phys. 185, 50–60 (2003).
    [CrossRef]
  18. P. González-Rodríguez and A. D. Kim, “Light propagation in tissues with forward-peaked and large-angle scattering,” Appl. Opt. 47, 2599–2609 (2008).
    [CrossRef]
  19. M. Martinelli, A. Gardner, D. Cuccia, C. Hayakawa, J. Spanier, and V. Venugopalan, “Analysis of single Monte Carlo methods for prediction of reflectance from turbid media,” Opt. Express 19, 19627–19642 (2011).
    [CrossRef]
  20. S. A. Prahl, “The diffusion approximation in three dimensions,” Chap. 7 in Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch and M. J. C. van Gemert, eds. (Plenum, 1995), pp. 207–231.
  21. A. D. Kim and J. B. Keller, “Light propagation in biological tissue,” J. Opt. Soc. Am. A 20, 92–98 (2003).
    [CrossRef]

2011 (2)

2008 (2)

P. González-Rodríguez and A. D. Kim, “Light propagation in tissues with forward-peaked and large-angle scattering,” Appl. Opt. 47, 2599–2609 (2008).
[CrossRef]

I. Seo, C. K. Hayakawa, and V. Venugopalan, “Radiative transport in the delta-P1 approximation for semi-infinite turbid media,” Med. Phys. 35, 681–693 (2008).
[CrossRef]

2005 (1)

J. S. You, C. K. Hayakawa, and V. Venugopalan, “Frequency domain photon migration in the δ-P1 approximation: analysis of ballistic, transport and diffuse regimes,” Phys. Rev. E 72, 021903 (2005).
[CrossRef]

2004 (1)

2003 (3)

A. D. Kim and J. B. Keller, “Light propagation in biological tissue,” J. Opt. Soc. Am. A 20, 92–98 (2003).
[CrossRef]

Y. L. Kim, Y. Liu, R. K. Wali, H. K. Roy, M. J. Goldberg, A. K. Kromin, K. Chen, and V. Backman, “Simultaneous measurement of angular and spectral properties of light scattering for characterization of tissue microarchitecture and its alteration in early precancer,” IEEE J. Sel. Top. Quantum Electron. 9, 243–256 (2003).
[CrossRef]

A. D. Kim and M. Moscoso, “Radiative transfer computations for optical beams,” J. Comput. Phys. 185, 50–60 (2003).
[CrossRef]

2002 (1)

A. Myakov, L. Nieman, L. Wicky, U. Utzinger, R. Richards-Kortum, and K. Sokolov, “Fiber optic probe for polarized reflectance spectroscopy in vivo: design and performance,” J. Biomed. Opt. 7, 388–397 (2002).
[CrossRef]

1998 (1)

V. Venugopalan, J. S. You, and B. J. Tromberg, “Radiative transport in the diffusion approximation: an extension for highly absorbing media and small source-detector separations,” Phys. Rev. E 58, 2395–2407 (1998).
[CrossRef]

1997 (2)

A. Kienle and M. S. Patterson, “Determination of the optical properties of semi-infinite turbid media from frequency-domain reflectance close to the source,” Phys. Med. Biol. 42, 1801–1819 (1997).
[CrossRef]

A. Kienle and M. S. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equations for reflectance from a semi-infinite turbid medium,” J. Opt. Soc. Am. A 14, 246–254 (1997).
[CrossRef]

1996 (1)

1992 (1)

T. J. Farrell, M. S. Patterson, and B. C. Wilson, “A diffusion theory model of spatially-resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef]

1990 (1)

W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

1983 (1)

1977 (1)

Backman, V.

Y. L. Kim, Y. Liu, R. K. Wali, H. K. Roy, M. J. Goldberg, A. K. Kromin, K. Chen, and V. Backman, “Simultaneous measurement of angular and spectral properties of light scattering for characterization of tissue microarchitecture and its alteration in early precancer,” IEEE J. Sel. Top. Quantum Electron. 9, 243–256 (2003).
[CrossRef]

Chen, K.

Y. L. Kim, Y. Liu, R. K. Wali, H. K. Roy, M. J. Goldberg, A. K. Kromin, K. Chen, and V. Backman, “Simultaneous measurement of angular and spectral properties of light scattering for characterization of tissue microarchitecture and its alteration in early precancer,” IEEE J. Sel. Top. Quantum Electron. 9, 243–256 (2003).
[CrossRef]

Cheong, W. F.

W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Cuccia, D.

Farrell, T. J.

T. J. Farrell, M. S. Patterson, and B. C. Wilson, “A diffusion theory model of spatially-resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef]

Ferwerda, H. A.

Gardner, A.

Goldberg, M. J.

Y. L. Kim, Y. Liu, R. K. Wali, H. K. Roy, M. J. Goldberg, A. K. Kromin, K. Chen, and V. Backman, “Simultaneous measurement of angular and spectral properties of light scattering for characterization of tissue microarchitecture and its alteration in early precancer,” IEEE J. Sel. Top. Quantum Electron. 9, 243–256 (2003).
[CrossRef]

González-Rodríguez, P.

Groenhuis, R. A. J.

Hayakawa, C.

Hayakawa, C. K.

I. Seo, C. K. Hayakawa, and V. Venugopalan, “Radiative transport in the delta-P1 approximation for semi-infinite turbid media,” Med. Phys. 35, 681–693 (2008).
[CrossRef]

J. S. You, C. K. Hayakawa, and V. Venugopalan, “Frequency domain photon migration in the δ-P1 approximation: analysis of ballistic, transport and diffuse regimes,” Phys. Rev. E 72, 021903 (2005).
[CrossRef]

Hibst, R.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, 1997).

Keller, J. B.

Kienle, A.

Kim, A. D.

Kim, Y. L.

Y. L. Kim, Y. Liu, R. K. Wali, H. K. Roy, M. J. Goldberg, A. K. Kromin, K. Chen, and V. Backman, “Simultaneous measurement of angular and spectral properties of light scattering for characterization of tissue microarchitecture and its alteration in early precancer,” IEEE J. Sel. Top. Quantum Electron. 9, 243–256 (2003).
[CrossRef]

Kromin, A. K.

Y. L. Kim, Y. Liu, R. K. Wali, H. K. Roy, M. J. Goldberg, A. K. Kromin, K. Chen, and V. Backman, “Simultaneous measurement of angular and spectral properties of light scattering for characterization of tissue microarchitecture and its alteration in early precancer,” IEEE J. Sel. Top. Quantum Electron. 9, 243–256 (2003).
[CrossRef]

Lilge, L.

Liu, Y.

Y. L. Kim, Y. Liu, R. K. Wali, H. K. Roy, M. J. Goldberg, A. K. Kromin, K. Chen, and V. Backman, “Simultaneous measurement of angular and spectral properties of light scattering for characterization of tissue microarchitecture and its alteration in early precancer,” IEEE J. Sel. Top. Quantum Electron. 9, 243–256 (2003).
[CrossRef]

Martinelli, M.

Moscoso, M.

A. D. Kim and M. Moscoso, “Beam propagation in sharply peaked forward scattering media,” J. Opt. Soc. Am. A 21, 797–803 (2004).
[CrossRef]

A. D. Kim and M. Moscoso, “Radiative transfer computations for optical beams,” J. Comput. Phys. 185, 50–60 (2003).
[CrossRef]

Myakov, A.

A. Myakov, L. Nieman, L. Wicky, U. Utzinger, R. Richards-Kortum, and K. Sokolov, “Fiber optic probe for polarized reflectance spectroscopy in vivo: design and performance,” J. Biomed. Opt. 7, 388–397 (2002).
[CrossRef]

Nieman, L.

A. Myakov, L. Nieman, L. Wicky, U. Utzinger, R. Richards-Kortum, and K. Sokolov, “Fiber optic probe for polarized reflectance spectroscopy in vivo: design and performance,” J. Biomed. Opt. 7, 388–397 (2002).
[CrossRef]

Patterson, M. S.

A. Kienle and M. S. Patterson, “Determination of the optical properties of semi-infinite turbid media from frequency-domain reflectance close to the source,” Phys. Med. Biol. 42, 1801–1819 (1997).
[CrossRef]

A. Kienle and M. S. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equations for reflectance from a semi-infinite turbid medium,” J. Opt. Soc. Am. A 14, 246–254 (1997).
[CrossRef]

A. Kienle, L. Lilge, M. S. Patterson, R. Hibst, R. Steiner, and B. C. Wilson, “Spatially resolved absolute diffuse reflectance measurements for noninvasive determination of the optical scattering and absorption coefficients of biological tissue,” Appl. Opt. 35, 2304–2314 (1996).
[CrossRef]

T. J. Farrell, M. S. Patterson, and B. C. Wilson, “A diffusion theory model of spatially-resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef]

Prahl, S. A.

W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

S. A. Prahl, “The diffusion approximation in three dimensions,” Chap. 7 in Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch and M. J. C. van Gemert, eds. (Plenum, 1995), pp. 207–231.

Richards-Kortum, R.

A. Myakov, L. Nieman, L. Wicky, U. Utzinger, R. Richards-Kortum, and K. Sokolov, “Fiber optic probe for polarized reflectance spectroscopy in vivo: design and performance,” J. Biomed. Opt. 7, 388–397 (2002).
[CrossRef]

Roy, H. K.

Y. L. Kim, Y. Liu, R. K. Wali, H. K. Roy, M. J. Goldberg, A. K. Kromin, K. Chen, and V. Backman, “Simultaneous measurement of angular and spectral properties of light scattering for characterization of tissue microarchitecture and its alteration in early precancer,” IEEE J. Sel. Top. Quantum Electron. 9, 243–256 (2003).
[CrossRef]

Seo, I.

I. Seo, C. K. Hayakawa, and V. Venugopalan, “Radiative transport in the delta-P1 approximation for semi-infinite turbid media,” Med. Phys. 35, 681–693 (2008).
[CrossRef]

Siegman, A. E.

Sokolov, K.

A. Myakov, L. Nieman, L. Wicky, U. Utzinger, R. Richards-Kortum, and K. Sokolov, “Fiber optic probe for polarized reflectance spectroscopy in vivo: design and performance,” J. Biomed. Opt. 7, 388–397 (2002).
[CrossRef]

Spanier, J.

Steiner, R.

Tenbosch, J. J.

Tromberg, B. J.

V. Venugopalan, J. S. You, and B. J. Tromberg, “Radiative transport in the diffusion approximation: an extension for highly absorbing media and small source-detector separations,” Phys. Rev. E 58, 2395–2407 (1998).
[CrossRef]

Utzinger, U.

A. Myakov, L. Nieman, L. Wicky, U. Utzinger, R. Richards-Kortum, and K. Sokolov, “Fiber optic probe for polarized reflectance spectroscopy in vivo: design and performance,” J. Biomed. Opt. 7, 388–397 (2002).
[CrossRef]

Venugopalan, V.

M. Martinelli, A. Gardner, D. Cuccia, C. Hayakawa, J. Spanier, and V. Venugopalan, “Analysis of single Monte Carlo methods for prediction of reflectance from turbid media,” Opt. Express 19, 19627–19642 (2011).
[CrossRef]

I. Seo, C. K. Hayakawa, and V. Venugopalan, “Radiative transport in the delta-P1 approximation for semi-infinite turbid media,” Med. Phys. 35, 681–693 (2008).
[CrossRef]

J. S. You, C. K. Hayakawa, and V. Venugopalan, “Frequency domain photon migration in the δ-P1 approximation: analysis of ballistic, transport and diffuse regimes,” Phys. Rev. E 72, 021903 (2005).
[CrossRef]

V. Venugopalan, J. S. You, and B. J. Tromberg, “Radiative transport in the diffusion approximation: an extension for highly absorbing media and small source-detector separations,” Phys. Rev. E 58, 2395–2407 (1998).
[CrossRef]

Wali, R. K.

Y. L. Kim, Y. Liu, R. K. Wali, H. K. Roy, M. J. Goldberg, A. K. Kromin, K. Chen, and V. Backman, “Simultaneous measurement of angular and spectral properties of light scattering for characterization of tissue microarchitecture and its alteration in early precancer,” IEEE J. Sel. Top. Quantum Electron. 9, 243–256 (2003).
[CrossRef]

Wang, L. V.

L. V. Wang and H. Wu, Biomedical Optics: Principles and Imaging (Wiley, 2007).

Welch, A. J.

W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Wicky, L.

A. Myakov, L. Nieman, L. Wicky, U. Utzinger, R. Richards-Kortum, and K. Sokolov, “Fiber optic probe for polarized reflectance spectroscopy in vivo: design and performance,” J. Biomed. Opt. 7, 388–397 (2002).
[CrossRef]

Wilson, B. C.

A. Kienle, L. Lilge, M. S. Patterson, R. Hibst, R. Steiner, and B. C. Wilson, “Spatially resolved absolute diffuse reflectance measurements for noninvasive determination of the optical scattering and absorption coefficients of biological tissue,” Appl. Opt. 35, 2304–2314 (1996).
[CrossRef]

T. J. Farrell, M. S. Patterson, and B. C. Wilson, “A diffusion theory model of spatially-resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef]

Wu, H.

L. V. Wang and H. Wu, Biomedical Optics: Principles and Imaging (Wiley, 2007).

You, J. S.

J. S. You, C. K. Hayakawa, and V. Venugopalan, “Frequency domain photon migration in the δ-P1 approximation: analysis of ballistic, transport and diffuse regimes,” Phys. Rev. E 72, 021903 (2005).
[CrossRef]

V. Venugopalan, J. S. You, and B. J. Tromberg, “Radiative transport in the diffusion approximation: an extension for highly absorbing media and small source-detector separations,” Phys. Rev. E 58, 2395–2407 (1998).
[CrossRef]

Appl. Opt. (3)

IEEE J. Quantum Electron. (1)

W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

Y. L. Kim, Y. Liu, R. K. Wali, H. K. Roy, M. J. Goldberg, A. K. Kromin, K. Chen, and V. Backman, “Simultaneous measurement of angular and spectral properties of light scattering for characterization of tissue microarchitecture and its alteration in early precancer,” IEEE J. Sel. Top. Quantum Electron. 9, 243–256 (2003).
[CrossRef]

J. Biomed. Opt. (1)

A. Myakov, L. Nieman, L. Wicky, U. Utzinger, R. Richards-Kortum, and K. Sokolov, “Fiber optic probe for polarized reflectance spectroscopy in vivo: design and performance,” J. Biomed. Opt. 7, 388–397 (2002).
[CrossRef]

J. Comput. Phys. (1)

A. D. Kim and M. Moscoso, “Radiative transfer computations for optical beams,” J. Comput. Phys. 185, 50–60 (2003).
[CrossRef]

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

Med. Phys. (2)

T. J. Farrell, M. S. Patterson, and B. C. Wilson, “A diffusion theory model of spatially-resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef]

I. Seo, C. K. Hayakawa, and V. Venugopalan, “Radiative transport in the delta-P1 approximation for semi-infinite turbid media,” Med. Phys. 35, 681–693 (2008).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Med. Biol. (1)

A. Kienle and M. S. Patterson, “Determination of the optical properties of semi-infinite turbid media from frequency-domain reflectance close to the source,” Phys. Med. Biol. 42, 1801–1819 (1997).
[CrossRef]

Phys. Rev. E (2)

V. Venugopalan, J. S. You, and B. J. Tromberg, “Radiative transport in the diffusion approximation: an extension for highly absorbing media and small source-detector separations,” Phys. Rev. E 58, 2395–2407 (1998).
[CrossRef]

J. S. You, C. K. Hayakawa, and V. Venugopalan, “Frequency domain photon migration in the δ-P1 approximation: analysis of ballistic, transport and diffuse regimes,” Phys. Rev. E 72, 021903 (2005).
[CrossRef]

Other (3)

L. V. Wang and H. Wu, Biomedical Optics: Principles and Imaging (Wiley, 2007).

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, 1997).

S. A. Prahl, “The diffusion approximation in three dimensions,” Chap. 7 in Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch and M. J. C. van Gemert, eds. (Plenum, 1995), pp. 207–231.

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Figures (4)

Fig. 1.
Fig. 1.

Comparison of the diffuse reflectance (normalized by the incident flux F0) computed using Monte Carlo simulations (circle symbols) and the numerical method to solve the radiative transport equation described in [15] (solid curve). Here, ρ is the source–-detector separation distance normalized by the beam width w. The optical properties for these results are μs=100mm1, μa=0.01mm1, g=0.8, and nrel=1.4. The beam width for these simulations is w=0.4247mm.

Fig. 2.
Fig. 2.

The upper plot shows a comparison of the diffuse reflectance (normalized by the incident flux F0) computed using the numerical solution of the radiative transport equation (circle symbols), the asymptotic model (solid curve), and the diffusion approximation (dashed curve). The lower plot shows the absolute errors of the diffuse reflectance computed using the asymptotic model and the diffusion approximation made with respect to the diffuse reflectance computed using the numerical solution of the radiative transport equation. Here, ρ is the source–-detector separation distance normalized by the beam width w. The optical properties are μs=500mm1, μa=0.01mm1, g=0.8, and nrel=1.4. The beam width is w=0.5mm. Therefore, α=2×105, β=4×103.

Fig. 3.
Fig. 3.

The same as Fig. 2, except that μs=100mm1 so that α=104 and β=2×102.

Fig. 4.
Fig. 4.

The same as Fig. 2, except that μs=50mm1 and μa=0.1mm1 so that α=2×103 and β=4×102.

Equations (50)

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

R=RaϕRbnϕ+Rcf.
μz˜I˜+1μ2(cosφx˜I˜+sinφy˜I˜)+μaI˜+μsLI˜=0.
LI˜=I˜ππ11p(μ,μ,φφ)I˜(μ,φ,x˜,y˜,z˜)dμdφ.
I˜(μ,φ,x˜,y˜,0)r(μ)I˜(μ,φ,x˜,y˜,0)=δ(μ1)2πf(x˜w,y˜w)on0<μ1.
βμzI+β1μ2(cosφxI+sinφyI)+αI+LI=0inz>0
I(μ,φ,x,y,0)r(μ)I(μ,φ,x,y,0)=δ(μ1)2πf(x,y)on0<μ1.
R(x,y)=2π1μNAt(μ)I¯(μ,x,y,0)μdμ.
I¯(μ,x,y,z)=12πππI(μ,φ,x,y,z)dφ.
I=Φ+Ψ.
Φϕ0+βϕ1+O(α)+O(β2).
Lϕ0=0,
Lϕ1=μzϕ01μ2(cosφxϕ0+sinφyϕ0).
Lϕ2=μzϕ11μ2(cosφxϕ1+sinφyϕ1).
Lϕ2=μzϕ11μ2(cosφxϕ1+sinφyϕ1)αβ2ϕ0.
Φ=ϕ0βs^·(3κϕ0)+O(α)+O(β2)
·(κϕ0)=0inz>0
·(κϕ0)αβ2ϕ0=0inz>0
μζψ+β1μ2(cosφxψ+sinφyψ)+αψ+Lψ=0.
ψψ0+βψ1+O(α)+O(β2).
μζψ0+Lψ0=0
μζψ1+Lψ1=1μ2(cosφxψ0+sinφyψ0),
ψ0(μ,x,y,0)r(μ)ψ0(μ,x,y,0)=δ(μ1)2πf(x,y)[1r(μ)]ϕ0(x,y,0)on0<μ1
ψ1(μ,x,y,0)r(μ)ψ1(μ,x,y,0)=3κ[1+r(μ)]μzϕ0(x,y,0)3κ[1r(μ)]1μ2[cosφxϕ0(x,y,0)sinφyϕ0(x,y,0)]on0<μ1.
I¯=ϕ3βκμzϕ+ψ+O(α)+O(β2).
·(κϕ)=0inz>0,
·(κϕ)αβ2ϕ=0inz>0,
aϕ(x,y,0)bzϕ(x,y,0)=cf(x,y).
a=P[1r(μ)],
b=3βκP[μ+μr(μ)],
c=12πP[δ(μ1)].
μζψ+ψ11h(μ,μ)ψ(μ,x,y,ζ)dμ=0inζ>0
ψ(μ,x,y,0)r(μ)ψ(μ,x,y,0)=δ(μ1)2πf(x,y)[1r(μ)]ϕ(x,y,0)+3βκμ[1+r(μ)]zϕ(x,y,0)on0<μ1.
h(μ,μ)=12πππp(μ,μ,φφ)d(φφ).
ψ0asζ,
F(ξ,η)=1(2π)2f(x,y)eiξxiηydxdy.
ϕ(x,y,z)=cF(ξ,η)a+bkzekzz+iξx+iηydξdη
zϕ(x,y,z)=kzcF(ξ,η)a+bkzekzz+iξx+iηydξdη.
μζH+H11h(μ,μ)H(μ,ζ;μ0,ζ0)dμ=δ(μμ0)δ(ζζ0)inζ>0
H(μ,0;μ0,ζ0)r(μ)H(μ,0;μ0,ζ0)=0on0<μ1.
A(μ,ζ)=01H(μ,ζ;μ,0)[1r(μ)]μdμ,
B(μ,ζ)=01H(μ,ζ;μ,0)[μ+μr(μ)]μdμ,
C(μ,ζ)=12πH(μ,ζ;1,0).
ψ(μ,x,y,ζ)=C(μ,ζ)f(x,y)A(μ,ζ)ϕ(x,y,0)+3βκB(μ,ζ)zϕ(x,y,0).
I¯(μ,x,y,0)=[1A(μ,0)]ϕ(x,y,0)3βκ[μB(μ,0)]zϕ(x,y,0)+C(μ,0)f(x,y)+O(α)+O(β2).
R(x,y)=Raϕ(x,y,0)Rbzϕ(x,y,0)+Rcf(x,y)+O(α)+O(β2).
Ra=2π1μNAt(μ)[1A(μ,0)]μdμ,
Rb=6πβκ1μNAt(μ)[μB(μ,0)]μdμ,
Rc=2π1μNAt(μ)C(μ,0)μdμ.
kz2={ξ2+η2ifαβ2,ακβ2+ξ2+η2ifα=O(β2)orβ2αβ.
f(x,y)=F02πexp[12(x2+y2)].

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