Abstract

Presented here are expressions for the PN approximation for light propagation in scattering media in the frequency domain. To elucidate parametric dependencies, the derivation uses normalization of the resulting expressions to either the total interaction coefficient or the reduced total interaction coefficient. For the latter case, a set of reduced phase function coefficients are introduced. Expression of the PN approximation as a conventional eigenvalue problem facilitates computation of the eigenvalues or attenuation coefficients. This approach is used to determine the attenuation coefficients in the asymptotic regime over the full values of the scattering albedo and reduced scattering albedo (0 to 1) and all positive values of the asymmetry factor (0 to 1). Frequency-domain measurements yield a sensitivity to turbid media optical properties for reduced scattering albedos as small as 0.2. PN calculations are used to assess the magnitude of errors associated with the P1 and P3 approximations over a range of scattering albedo, phase function, and modulation frequency.

© 2005 Optical Society of America

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  5. L. Marti-Lopez, J. Bouza-Dominguez, J. C. Hebden, S. R. Arridge, R. A. Martinez-Celorio, “Validity conditions for the radiative transfer equation,” J. Opt. Soc. Am. A 20, 2046–2056 (2003).
    [CrossRef]
  6. L.-H. Wang, S. L. Jacques, L.-Q. Zheng, “MCML—Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
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    [CrossRef]
  9. A. E. Cerussi, B. J. Tromberg, “Photon migration spectroscopy frequency-domain techniques,” in Biomedical Photonics Handbook, T. Vo-Dinh, ed. (CRC Press, Boca Raton, Fla., 2003), pp. 22-21–22-17.
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  21. D. A. Boas, “Diffuse photon probes of structural and dynamical properties of turbid media: theory and biomedical applications,” Ph.D. dissertation (University of Pennsylvania, Philadelphia, Pa., 1996).
  22. J.-M. Kaltenbach, M. Kaschke, “Frequency- and time-domain modelling of light transport in random media,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Müller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds. (SPIE, Bellingham, Wash., 1993), Vol. IS11, pp. 65–86.
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  29. J. B. Fishkin, S. Fantini, M. J. van de Ven, E. Gratton, “Gigahertz photon density waves in a turbid medium: theory and experiments,” Phys. Rev. E 53, 2307–2319 (1996).
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    [CrossRef]
  34. This axis transformation allows both negative and positive values to be plotted on a single logarithmic plot. The dual-signed logarithmic axis is created by use of the transformation y= sinh−1(x/2)/ln(10), where the value x= 1 gives the first decade value above or below zero. This mapping corresponds to the inverse of x= 10y− 10−y, while the mapping of a conventional logarithmic axis is the inverse of x=10y. The dual-signed logarithmic axis may be considered a generalization of the conventional logarithmic axis, the latter being the limiting case for large y. Note that this axis transformation has 10 tick marks in the first decade on either side of 0, whereas the remaining (higher) decades have 9 ticks per decade.
  35. M. Abramowitz, I. A. Stegun, “Equation 8.1.2,” in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, Washington, D.C., 1972), p. 332.
  36. B. Davison, J. B. Sykes, Neutron Transport Theory (Oxford U. Press, London, 1957), pp. 122–126.
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2004 (1)

2003 (2)

L. Marti-Lopez, J. Bouza-Dominguez, J. C. Hebden, S. R. Arridge, R. A. Martinez-Celorio, “Validity conditions for the radiative transfer equation,” J. Opt. Soc. Am. A 20, 2046–2056 (2003).
[CrossRef]

Z. Sun, S. Torrance, F. K. McNeil-Watson, E. M. Sevick-Muraca, “Application of frequency domain photon migration to particle size analysis and monitoring of pharmaceutical powders,” Anal. Chem. 75, 1720–1725 (2003).
[CrossRef] [PubMed]

2001 (1)

2000 (1)

1999 (5)

1998 (3)

1996 (1)

J. B. Fishkin, S. Fantini, M. J. van de Ven, E. Gratton, “Gigahertz photon density waves in a turbid medium: theory and experiments,” Phys. Rev. E 53, 2307–2319 (1996).
[CrossRef]

1995 (1)

L.-H. Wang, S. L. Jacques, L.-Q. Zheng, “MCML—Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

1994 (1)

1993 (1)

1989 (1)

W. M. Star, “Comparing the P3-approximation with diffusion theory and with Monte Carlo calculations of light propagation in a slab geometry,” SPIE Institute Series IS 5, 146–154 (1989).

1965 (1)

W. M. Irvine, “Multiple scattering by large particles,” Astrophys. J. 142, 1563–1575 (1965).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, “Equation 4.6.35,” in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, Washington, D.C., 1972), p. 88.

M. Abramowitz, I. A. Stegun, “Equation 8.1.2,” in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, Washington, D.C., 1972), p. 332.

M. Abramowitz, I. A. Stegun, “Equations 10.2.15, 10.2.17, 10.2.20, and 10.2.21,” in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, Washington, D.C., 1972), p. 444.

Amaldi, E.

E. Amaldi, “The production and slowing down of neutrons,” in Encyclopedia of Physics, S. Flugge, ed. (Springer-Verlag, Berlin, 1959), Vol. 38/2, pp. 504–580.

Aronson, R.

Arridge, S. R.

Barbieri, B.

Bevilacqua, F.

Boas, D. A.

D. A. Boas, H. Liu, M. A. O’Leary, B. Chance, A. G. Yodh, “Photon migration within the P3 approximation,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, ed., Proc. SPIE2389, 240–247 (1995).
[CrossRef]

D. A. Boas, “Diffuse photon probes of structural and dynamical properties of turbid media: theory and biomedical applications,” Ph.D. dissertation (University of Pennsylvania, Philadelphia, Pa., 1996).

Bouza-Dominguez, J.

Case, K. M.

K. M. Case, P. F. Zweifel, “Numerical methods,” in Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Chap. 8, pp. 194–229.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), pp. 174–180.

Cerussi, A. E.

A. E. Cerussi, B. J. Tromberg, “Photon migration spectroscopy frequency-domain techniques,” in Biomedical Photonics Handbook, T. Vo-Dinh, ed. (CRC Press, Boca Raton, Fla., 2003), pp. 22-21–22-17.

Chance, B.

B. Chance, M. Cope, E. Gratton, N. Ramanujam, B. Tromberg, “Phase measurement of light absorption and scatter in human tissue,” Rev. Sci. Instrum. 69, 3457–3481 (1998).
[CrossRef]

D. A. Boas, H. Liu, M. A. O’Leary, B. Chance, A. G. Yodh, “Photon migration within the P3 approximation,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, ed., Proc. SPIE2389, 240–247 (1995).
[CrossRef]

Cope, M.

B. Chance, M. Cope, E. Gratton, N. Ramanujam, B. Tromberg, “Phase measurement of light absorption and scatter in human tissue,” Rev. Sci. Instrum. 69, 3457–3481 (1998).
[CrossRef]

Corngold, N.

Davison, B.

B. Davison, J. B. Sykes, Neutron Transport Theory (Oxford U. Press, London, 1957), pp. 116–173.

B. Davison, J. B. Sykes, Neutron Transport Theory (Oxford U. Press, London, 1957), pp. 122–126.

Depeursinge, C.

Fantini, S.

Faris, G. W.

Fishkin, J. B.

Foster, T. H.

Franceschini, M. A.

Gerken, M.

Godfrey, D.

Gratton, E.

B. Chance, M. Cope, E. Gratton, N. Ramanujam, B. Tromberg, “Phase measurement of light absorption and scatter in human tissue,” Rev. Sci. Instrum. 69, 3457–3481 (1998).
[CrossRef]

J. B. Fishkin, S. Fantini, M. J. van de Ven, E. Gratton, “Gigahertz photon density waves in a turbid medium: theory and experiments,” Phys. Rev. E 53, 2307–2319 (1996).
[CrossRef]

S. Fantini, M. A. Franceschini, J. B. Fishkin, B. Barbieri, E. Gratton, “Quantitative determination of the absorption spectra of chromophores in strongly scattering media: a light-emitting-diode based technique,” Appl. Opt. 33, 5204–5213 (1994).
[CrossRef] [PubMed]

Gross, J. D.

Haskell, R. C.

Hebden, J. C.

Hull, E. L.

Irvine, W. M.

W. M. Irvine, “Multiple scattering by large particles,” Astrophys. J. 142, 1563–1575 (1965).
[CrossRef]

Ishimaru, A.

A. Ishimaru, “Transport equation for a partially polarized electromagnetic wave,” in Wave Propagation and Scattering in Random Media (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 164–165.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), pp. 88, 90, 99, 101.

Jacques, S. L.

L.-H. Wang, S. L. Jacques, L.-Q. Zheng, “MCML—Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Kaltenbach, J.-M.

J.-M. Kaltenbach, M. Kaschke, “Frequency- and time-domain modelling of light transport in random media,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Müller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds. (SPIE, Bellingham, Wash., 1993), Vol. IS11, pp. 65–86.

Kaschke, M.

J.-M. Kaltenbach, M. Kaschke, “Frequency- and time-domain modelling of light transport in random media,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Müller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds. (SPIE, Bellingham, Wash., 1993), Vol. IS11, pp. 65–86.

Kim, A. D.

Lionheart, W. R. B.

Liu, H.

D. A. Boas, H. Liu, M. A. O’Leary, B. Chance, A. G. Yodh, “Photon migration within the P3 approximation,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, ed., Proc. SPIE2389, 240–247 (1995).
[CrossRef]

Marquet, P.

Marti-Lopez, L.

Martinez-Celorio, R. A.

McNeil-Watson, F. K.

Z. Sun, S. Torrance, F. K. McNeil-Watson, E. M. Sevick-Muraca, “Application of frequency domain photon migration to particle size analysis and monitoring of pharmaceutical powders,” Anal. Chem. 75, 1720–1725 (2003).
[CrossRef] [PubMed]

Mobley, C.

C. Mobley, “Optical properties of water,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 1, pp. 43.43–43.56.

Murrer, L. H. P.

O’Leary, M. A.

D. A. Boas, H. Liu, M. A. O’Leary, B. Chance, A. G. Yodh, “Photon migration within the P3 approximation,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, ed., Proc. SPIE2389, 240–247 (1995).
[CrossRef]

Piguet, D.

Ramanujam, N.

B. Chance, M. Cope, E. Gratton, N. Ramanujam, B. Tromberg, “Phase measurement of light absorption and scatter in human tissue,” Rev. Sci. Instrum. 69, 3457–3481 (1998).
[CrossRef]

Rinzema, K.

Sevick-Muraca, E. M.

Z. Sun, S. Torrance, F. K. McNeil-Watson, E. M. Sevick-Muraca, “Application of frequency domain photon migration to particle size analysis and monitoring of pharmaceutical powders,” Anal. Chem. 75, 1720–1725 (2003).
[CrossRef] [PubMed]

Star, W. M.

K. Rinzema, L. H. P. Murrer, W. M. Star, “Direct experimental verification of light transport theory in an optical phantom,” J. Opt. Soc. Am. A 15, 2078–2088 (1998).
[CrossRef]

W. M. Star, “Comparing the P3-approximation with diffusion theory and with Monte Carlo calculations of light propagation in a slab geometry,” SPIE Institute Series IS 5, 146–154 (1989).

Stegun, I. A.

M. Abramowitz, I. A. Stegun, “Equation 4.6.35,” in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, Washington, D.C., 1972), p. 88.

M. Abramowitz, I. A. Stegun, “Equation 8.1.2,” in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, Washington, D.C., 1972), p. 332.

M. Abramowitz, I. A. Stegun, “Equations 10.2.15, 10.2.17, 10.2.20, and 10.2.21,” in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, Washington, D.C., 1972), p. 444.

Sun, Z.

Z. Sun, S. Torrance, F. K. McNeil-Watson, E. M. Sevick-Muraca, “Application of frequency domain photon migration to particle size analysis and monitoring of pharmaceutical powders,” Anal. Chem. 75, 1720–1725 (2003).
[CrossRef] [PubMed]

Svaasand, L. O.

Sykes, J. B.

B. Davison, J. B. Sykes, Neutron Transport Theory (Oxford U. Press, London, 1957), pp. 122–126.

B. Davison, J. B. Sykes, Neutron Transport Theory (Oxford U. Press, London, 1957), pp. 116–173.

Torrance, S.

Z. Sun, S. Torrance, F. K. McNeil-Watson, E. M. Sevick-Muraca, “Application of frequency domain photon migration to particle size analysis and monitoring of pharmaceutical powders,” Anal. Chem. 75, 1720–1725 (2003).
[CrossRef] [PubMed]

Tromberg, B.

B. Chance, M. Cope, E. Gratton, N. Ramanujam, B. Tromberg, “Phase measurement of light absorption and scatter in human tissue,” Rev. Sci. Instrum. 69, 3457–3481 (1998).
[CrossRef]

Tromberg, B. J.

Tsay, T.-T.

van de Hulst, H. C.

H. C. van de Hulst, “Phase Functions,” in Multiple Light Scattering: Tables, Formulas, and Applications, Volume 2 (Academic, New York, 1980), Chap. 10, pp. 303–330.

van de Ven, M. J.

J. B. Fishkin, S. Fantini, M. J. van de Ven, E. Gratton, “Gigahertz photon density waves in a turbid medium: theory and experiments,” Phys. Rev. E 53, 2307–2319 (1996).
[CrossRef]

Wang, L.-H.

L.-H. Wang, S. L. Jacques, L.-Q. Zheng, “MCML—Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Weinberg, A. M.

A. M. Weinberg, E. P. Wigner, “Transport theory and the diffusion of monoenergetic neutrons,” in The Physical Theory of Neutron Chain Reactors (The University of Chicago, Chicago, 1958), Chap. IX, pp. 219–278.

Wigner, E. P.

A. M. Weinberg, E. P. Wigner, “Transport theory and the diffusion of monoenergetic neutrons,” in The Physical Theory of Neutron Chain Reactors (The University of Chicago, Chicago, 1958), Chap. IX, pp. 219–278.

Yodh, A. G.

D. A. Boas, H. Liu, M. A. O’Leary, B. Chance, A. G. Yodh, “Photon migration within the P3 approximation,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, ed., Proc. SPIE2389, 240–247 (1995).
[CrossRef]

Zheng, L.-Q.

L.-H. Wang, S. L. Jacques, L.-Q. Zheng, “MCML—Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Zweifel, P. F.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), pp. 174–180.

K. M. Case, P. F. Zweifel, “Numerical methods,” in Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Chap. 8, pp. 194–229.

Anal. Chem. (1)

Z. Sun, S. Torrance, F. K. McNeil-Watson, E. M. Sevick-Muraca, “Application of frequency domain photon migration to particle size analysis and monitoring of pharmaceutical powders,” Anal. Chem. 75, 1720–1725 (2003).
[CrossRef] [PubMed]

Appl. Opt. (3)

Astrophys. J. (1)

W. M. Irvine, “Multiple scattering by large particles,” Astrophys. J. 142, 1563–1575 (1965).
[CrossRef]

Comput. Methods Programs Biomed. (1)

L.-H. Wang, S. L. Jacques, L.-Q. Zheng, “MCML—Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (6)

Opt. Lett. (4)

Phys. Rev. E (1)

J. B. Fishkin, S. Fantini, M. J. van de Ven, E. Gratton, “Gigahertz photon density waves in a turbid medium: theory and experiments,” Phys. Rev. E 53, 2307–2319 (1996).
[CrossRef]

Rev. Sci. Instrum. (1)

B. Chance, M. Cope, E. Gratton, N. Ramanujam, B. Tromberg, “Phase measurement of light absorption and scatter in human tissue,” Rev. Sci. Instrum. 69, 3457–3481 (1998).
[CrossRef]

SPIE Institute Series IS (1)

W. M. Star, “Comparing the P3-approximation with diffusion theory and with Monte Carlo calculations of light propagation in a slab geometry,” SPIE Institute Series IS 5, 146–154 (1989).

Other (19)

D. A. Boas, H. Liu, M. A. O’Leary, B. Chance, A. G. Yodh, “Photon migration within the P3 approximation,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, ed., Proc. SPIE2389, 240–247 (1995).
[CrossRef]

D. A. Boas, “Diffuse photon probes of structural and dynamical properties of turbid media: theory and biomedical applications,” Ph.D. dissertation (University of Pennsylvania, Philadelphia, Pa., 1996).

J.-M. Kaltenbach, M. Kaschke, “Frequency- and time-domain modelling of light transport in random media,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Müller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds. (SPIE, Bellingham, Wash., 1993), Vol. IS11, pp. 65–86.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), pp. 88, 90, 99, 101.

M. Abramowitz, I. A. Stegun, “Equations 10.2.15, 10.2.17, 10.2.20, and 10.2.21,” in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, Washington, D.C., 1972), p. 444.

This axis transformation allows both negative and positive values to be plotted on a single logarithmic plot. The dual-signed logarithmic axis is created by use of the transformation y= sinh−1(x/2)/ln(10), where the value x= 1 gives the first decade value above or below zero. This mapping corresponds to the inverse of x= 10y− 10−y, while the mapping of a conventional logarithmic axis is the inverse of x=10y. The dual-signed logarithmic axis may be considered a generalization of the conventional logarithmic axis, the latter being the limiting case for large y. Note that this axis transformation has 10 tick marks in the first decade on either side of 0, whereas the remaining (higher) decades have 9 ticks per decade.

M. Abramowitz, I. A. Stegun, “Equation 8.1.2,” in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, Washington, D.C., 1972), p. 332.

B. Davison, J. B. Sykes, Neutron Transport Theory (Oxford U. Press, London, 1957), pp. 122–126.

M. Abramowitz, I. A. Stegun, “Equation 4.6.35,” in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, Washington, D.C., 1972), p. 88.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), pp. 174–180.

A. E. Cerussi, B. J. Tromberg, “Photon migration spectroscopy frequency-domain techniques,” in Biomedical Photonics Handbook, T. Vo-Dinh, ed. (CRC Press, Boca Raton, Fla., 2003), pp. 22-21–22-17.

B. Davison, J. B. Sykes, Neutron Transport Theory (Oxford U. Press, London, 1957), pp. 116–173.

A. M. Weinberg, E. P. Wigner, “Transport theory and the diffusion of monoenergetic neutrons,” in The Physical Theory of Neutron Chain Reactors (The University of Chicago, Chicago, 1958), Chap. IX, pp. 219–278.

K. M. Case, P. F. Zweifel, “Numerical methods,” in Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Chap. 8, pp. 194–229.

E. Amaldi, “The production and slowing down of neutrons,” in Encyclopedia of Physics, S. Flugge, ed. (Springer-Verlag, Berlin, 1959), Vol. 38/2, pp. 504–580.

A. Ishimaru, “Transport equation for a partially polarized electromagnetic wave,” in Wave Propagation and Scattering in Random Media (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 164–165.

V. V. Tuchin, ed., Handbook of Optical Biomedical Diagnostics (SPIE, Bellingham, Wash., 2002).

C. Mobley, “Optical properties of water,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 1, pp. 43.43–43.56.

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Figures (8)

Fig. 1
Fig. 1

(a) Vectors and (b) angles for the PN approximation in spherical geometry.

Fig. 2
Fig. 2

Second-order phase function coefficient f2 as a function of g or f1 for four phase functions and Mie scattering (in color on-line).

Fig. 3
Fig. 3

(a) Real and (b) imaginary attenuation coefficients normalized to the total interaction coefficient μt as functions of the scattering albedo a and the normalized angular frequency ω ̂ (in color on-line).

Fig. 4
Fig. 4

(a) Real and (b) imaginary attenuation coefficients normalized to the reduced total interaction coefficient μt′ as functions of the reduced scattering albedo a′ and the normalized angular frequency ω ̂ (in color on-line).

Fig. 5
Fig. 5

(a) Ratio of attenuation coefficients as functions of the reduced scattering albedo a′ and the normalized angular frequency ω ̂ . (b) Data shown on an expanded scale in the region of a′ = 0 for ω ̂ = 0.1 (in color on-line).

Fig. 6
Fig. 6

Fractional errors for (a) the real and (b) the imaginary attenuation coefficients in the P1 approximation as functions of the reduced scattering albedo a′ and the normalized angular frequency ω ̂ (in color on-line).

Fig. 7
Fig. 7

Fractional error for ratio of attenuation coefficients in P1 approximation as functions of the reduced scattering albedo a′ and the normalized angular frequency ω ̂ (in color on-line).

Fig. 8
Fig. 8

Lowest eight orders of the Legendre polynomials.

Tables (4)

Tables Icon

Table 1 Expressions for fn and fn′ for the Henyey–Greenstein and Power Function Phase Functions

Tables Icon

Table 2 Analytical Solutions of κ in Special Cases of Radiative Transport

Tables Icon

Table 3 Values of g Corresponding to the Values of f2′ Used in Fig. 4

Tables Icon

Table 4 Approximate Eigenvalue Ratios in Low- and High-Frequency Regimes

Equations (67)

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1 ν L ( r , ŝ , t ) t + ŝ L ( r , ŝ , t ) + μ t L ( r , ŝ , t ) = μ s 4 π L ( r , ŝ , t ) f ( ŝ ŝ ) d Ω + S ( r , ŝ , t ) ,
1 ν L ( r , η , t ) t + η L ( r , η , t ) r + 1 η 2 r L ( r , η , t ) η = μ t L ( r , η , t ) = μ s 4 π L ( r , η , t ) f ( ŝ ŝ ) d Ω + S ( r , η , t ) .
Y lm ( θ , ϕ ) = [ 2 l + 1 ( l m ) ! 4 π ( l + m ) ! ] 1 / 2 P l m ( cos θ ) exp ( i m ϕ )
L ( r , η , t ) = l = 0 N 2 l + 1 4 π Ψ l ( r ) P 1 ( η ) exp ( i ω t ) ,
S ( r , η , t ) = l = 0 N 2 l + 1 4 π s l ( r ) P 1 ( η ) exp ( i ω t ) ,
f ( ŝ ŝ ) = l = 0 N 2 l + 1 4 π f l P l ( ŝ ŝ ) ,
f l = 4 π P l ( ŝ ŝ ) f ( ŝ ŝ ) d Ω .
n + 1 2 n + 1 ( d d r + n + 2 r ) Ψ n + 1 ( r ) + n 2 n + 1 ( d d r n 1 r ) × Ψ n 1 ( r ) + ( μ t μ s f n + i ω ν ) Ψ n ( r ) = s n ( r ) .
Ψ n ( r ) = j A j G n ( κ j ) Q n ( κ j r ) ,
κ j [ ( n + 1 ) G n + 1 ( κ j ) + n G n 1 ( κ j ) ] + ( 2 n + 1 ) × ( μ t μ s f n + i ω ν ) G n ( κ j ) = 0 .
[ χ 0 κ 0 0 . . 0 0 0 κ χ 1 2 κ 0 . . 0 0 0 0 2 κ χ 2 3 κ . . 0 0 0 0 0 3 κ χ 3 . . 0 0 0 . . . . . . . . . . . . . . . . . . 0 0 0 0 . . χ N 2 ( N 1 ) κ 0 0 0 0 0 . . ( N 1 ) κ χ N 1 N κ 0 0 0 0 . . 0 N κ χ N ] = 0 ,
χ n ( 2 n + 1 ) [ μ a + ( 1 f n ) μ s + i ω ν ] = ( 2 n + 1 ) [ μ a + f n μ s + i ω ν ] ,
f n 1 f n 1 g .
χ 0 = μ a + i ω ν , χ 1 = 3 ( μ t + i ω ν ) ,
M = | 0 1 / χ 0 0 0 . . 0 0 0 1 / χ 1 0 2 / χ 1 0 . . 0 0 0 0 2 / χ 2 0 3 / χ 2 . . 0 0 0 0 0 3 / χ 3 0 . . 0 0 0 . . . . . . . . . . . . . . . . . . 0 0 0 0 . . 0 ( N 1 ) / ( χ N 2 ) 0 0 0 0 0 . . ( N 1 ) / ( χ N 1 ) 0 N / χ N 1 0 0 0 0 . . 0 N / χ N 0 | .
G 0 ( κ ) = 1 , G 1 ( κ ) = χ 0 κ , G 2 ( κ ) = χ 0 χ 1 2 κ 2 1 2 , G 3 ( κ ) = 1 6 κ ( χ 0 χ 1 χ 2 κ 2 + 4 χ 0 + χ 2 ) .
L ( r , η , t ) = exp ( i ω t ) j > 0 A j n = 0 N 2 n + 1 4 π G n ( κ j ) × ( 1 ) n + 1 [ ( 1 / 2 ) π / ( κ j r ) ] 1 / 2 K n + 1 / 2 ( κ j r ) P n ( η ) .
( π 2 x ) 1 / 2 K 1 / 2 ( x ) = π 2 exp ( x ) x , ( π 2 x ) 1 / 2 K 3 / 2 ( x ) = π 2 exp ( x ) x ( 1 + x 1 ) , ( π 2 x ) 1 / 2 K 5 / 2 ( x ) = π 2 exp ( x ) x ( 1 + 3 x 1 + 3 x 2 ) , ( π 2 x ) 1 / 2 K 7 / 2 ( x ) = π 2 exp ( x ) x ( 1 + 6 x 1 + 15 x 2 + 15 x 3 ) .
L [ r , cos ( θ ) , t ] = A 8 exp ( κ r + i ω t ) κ r [ 1 + 3 κ ( i ω ν + μ a ) × ( 1 + 1 κ r ) cos ( θ ) ] ,
κ = [ 3 ( μ a + i ω ν ) ( μ a + μ s + i ω ν ) ] 1 / 2 .
κ r κ i = ( 3 / 2 ) 1 / 2 { [ μ a 2 + ( ω ν ) 2 ] 1 / 2 [ μ t 2 + ( ω ν ) 2 ] 1 / 2 ± [ μ a μ t ( ω ν ) 2 ] } 1 / 2 ,
κ r κ i = ( 3 2 μ t μ a ) 1 / 2 { [ 1 + ( ω ν μ a ) 2 ] 1 / 2 ± 1 } 1 / 2 ,
κ r ( 3 μ t μ a ) 1 / 2 , κ i ( 3 μ t μ a ) 1 / 2 ω 2 ν μ a ,
κ r κ i = ( 3 2 ω ν μ t ) 1 / 2 { [ 1 + ( ω ν μ t ) 2 ] 1 / 2 ω ν μ t } 1 / 2 ( 3 2 ω ν μ t ) 1 / 2 ,
κ j 2 = 1 18 { 9 χ 0 χ 1 + 4 χ 0 χ 3 + χ 2 χ 3 ± [ ( 9 χ 0 χ 1 + 4 χ 0 χ 3 + χ 2 χ 3 ) 2 36 * χ 0 χ 1 χ 2 χ 3 ] 1 / 2 } ,
χ n μ t = ( 2 n + 1 ) ( 1 a f n + i ω ̂ ) ,
χ n μ t = ( 2 n + 1 ) [ 1 a ( f n 1 ) + i ω ̂ ] ,
ω ̂ = ω ν μ t , ω ̂ = ω ν μ t .
P n ( η ) = P n ( 1 ) + ( η 1 ) 1 ! d P n d η | η = 1 + ( η 1 ) 2 2 ! d 2 P n d η 2 | η = 1 + ( η 1 ) 3 3 ! d 3 P n d η 3 | η = 1 + .
f n = d P n d η | η = 1 1 2 d 2 P n d η 2 | η = 1 1 1 f ( η ) ( 1 η ) 2 d η 1 1 f ( η ) ( 1 η ) d η + 1 3 ! d 3 P n d η 3 | η = 1 1 1 f ( η ) ( 1 η ) 3 d η 1 1 f ( η ) ( 1 η ) d η + .
f n | max = d P n d η | η = 1 = n ( n + 1 ) 2 .
f n | HGmax = n .
1 = c κ tanh 1 κ = c 2 κ ln 1 + κ 1 κ ,
c = μ s i ω / ν + μ t = a ( 1 + i ω ̂ ) ,
κ = κ i ω / ν + μ t = κ μ t 1 ( 1 + i ω ̂ ) .
f ( cos φ ) = α f HG ( cos φ ) + ( 1 α ) f PF ( cos φ ) ,
f ( cos φ ) = α f HG ( cos φ ) + ( 1 α ) f R ( cos φ ) ,
f ( cos φ ) = ( 3 / 16 π ) + ( 1 + cos 2 φ ) .
1 g 2 4 π ( 1 + g 2 2 g cos φ ) 3 / 2
m = 0 n 1 g m
1 4 π N + 1 2 N ( 1 + cos φ ) N ; N = 2 g 1 g
m = 1 n N + 1 m N + 1 + m
N + 2 2 ( 1 f n )
μ a + i ω ν
μ a + i ω ν
( ω 2 ν μ a ) 1 / 2
( ω 2 ν μ a ) 1 / 2
ω 2 ν μ a
ω ν μ a
ω 2 ν μ a
ω ν μ a
x P l ( x ) = 1 2 l + 1 [ ( l + 1 ) P l + 1 ( x ) + l P l 1 ( x ) ] ,
( x 2 1 ) d d x P l ( x ) = l [ x P l ( x ) P l 1 ( x ) ] , = l ( l + 1 ) 2 l + 1 [ P l + 1 ( x ) P l 1 ( x ) ] .
1 1 P l ( x ) P l ( x ) d x = 2 2 l + 1 δ l l ,
Y l m * ( ŝ ) Y l m ( ŝ ) d Ω = δ l l δ m m .
P l ( r ̂ 1 · r ̂ 2 ) = 4 π 2 l + 1 m = l l Y l m * ( r ̂ 1 ) Y l m ( r ̂ 2 ) .
μ s 4 π L ( r , r ̂ · ŝ , t ) f ( ŝ · ŝ ) d Ω = μ s exp ( i ω t ) l , l = 0 N Ψ l ( r ) f l 2 l + 1 4 π 2 l + 1 4 π × 4 π P l ( r ̂ · ŝ ) P l ( ŝ · ŝ ) d Ω = μ s exp ( i ω t ) l , l = 0 N Ψ l ( r ) f l 4 π m = l l m = l l × Y l m * ( r ̂ ) Y l m ( ŝ ) Y l m * ( ŝ ) Y l m ( ŝ ) d Ω = μ s exp ( i ω t ) l = 0 N Ψ l ( r ) f l 2 l + 1 4 π P l ( r ̂ · ŝ ) ,
P l ( x ) = F ( l , l + 1 , 1 , 1 x 2 ) = k = 0 l ( 1 ) k l ! ( l k ) ! ( l + k ) ! l ! 1 k ! 1 k ! ( 1 x 2 ) k ,
d m P l d x m | x = 1 = ( l + m ) ! ( l m ) ! m ! 2 m .
d d x Q n ( x ) + n + 1 x Q n ( x ) = Q n 1 ( x ) , d d x Q n ( x ) n x Q n ( x ) = Q n + 1 ( x )
( π 2 x ) 1 / 2 K n + 1 / 2 ( x ) = π 2 exp ( x ) x k = 0 n ( n + k ) ! ( n k ) ! k ! ( 2 x ) k .
D n = χ n D n 1 ( n κ ) 2 D n 2
R n = D n / D n 1 μ t + i ω / ν
R n 1 = ( n κ ) 2 ( 2 n + 1 ) R n ,
R 0 = 1 c .
1 c = R 0 = κ 2 3 R 1 = κ 2 3 4 κ 2 5 9 κ 2 7 16 κ 2 9 .
tanh 1 z = z 1 z 2 3 4 z 2 5 9 z 2 7 16 z 2 9

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