Abstract

The reduced scattering cross section per unit of volume Σs′ ≡ Σs(1 − g) is an important parameter to describe light propagation in media with scattering and absorption. Mie calculations of the asymmetry factory g for nonabsorbing spheres and Qsca, the ratio of the scattering cross section Σs and the particle cross section, show that Qsca(1 − g) = 3.28x0.37(m − 1)2.09 is true to within a few percent, when the Mie parameters for relative refractive index m and size x are in the ranges of 1 < m ≤ 1.1 and 5 < x < 50, respectively. A ratio of reduced scattering cross sections for radiation at two wavelengths is also independent of the size within the range mentioned, even for mixtures of different size spheres. The results seem promising for biomedical applications.

© 1992 Optical Society of America

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References

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  1. R. A. J. Groenhuis, H. A. Ferwerda, J. J. Ten Bosch, “Scattering and absorption of turbid materials determined from reflection measurements. 1: Theory,” Appl. Opt. 22, 2456–2462 (1983).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  3. H. W. Jentink, F. F. M. de Mul, R. Graaff, H. E. Suichies, J. G. Aarnoudse, J. Greve, “Laser Doppler flowmetry: measurements in a layered perfusion model and Monte Carlo simulations of measurements,” Appl. Opt. 30, 2592–2597 (1991).
    [CrossRef] [PubMed]
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  5. H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vols. 1 and 2.
  6. P. S. Mudgett, L. W. Richards, “Multiple scattering calculations for technology II,” J. Colloid Interface Sci. 39, 551–567 (1972).
    [CrossRef]
  7. L. Reynolds, C. Johnson, A. Ishimaru, “Diffuse reflectance from a finite blood medium: applications to the modeling of fiber optic catheters,” Appl. Opt. 15, 2059–2067 (1976).
    [CrossRef] [PubMed]
  8. R. Graaff, J. G. Aarnoudse, F. F. M. de Mul, H. W. Jentink, “Light propagation parameters for anisotropically scattering media, based on a rigorous solution of the transport equation,” Appl. Opt. 28, 2273–2279 (1989).
    [CrossRef] [PubMed]
  9. R. Graaff, J. G. Aarnoudse, F. F. M. de Mul, H. W. Jentink, “Improved expressions for anisotropic scattering in absorbing media,” in Scattering and Diffraction, H. A. Ferwerda, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1029, 103–110 (1989).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  24. J. M. Steinke, A. P. Shepherd, “Comparison of Mie theory and the light scattering of red blood cells,” Appl. Opt. 27, 4027–4033 (1988).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  27. S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of helium–neon laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

1991 (1)

1990 (1)

1989 (4)

1988 (1)

1987 (2)

S. T. Flock, B. C. Wilson, M. S. Patterson, “Total attenuation coefficient and scattering phase functions of tissues and phantom materials at 633 nm,” Med. Phys. 14, 835–841 (1987).
[CrossRef] [PubMed]

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of helium–neon laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

1984 (1)

W. A. de Rooij, C. C. A. H. van der Stap, “Expansion of Mie scattering matrices in generalized spherical functions,” Astron. Astrophys. 131, 237–248 (1984).

1983 (2)

1981 (1)

1980 (1)

1976 (1)

1972 (1)

P. S. Mudgett, L. W. Richards, “Multiple scattering calculations for technology II,” J. Colloid Interface Sci. 39, 551–567 (1972).
[CrossRef]

1965 (1)

1958 (1)

Aarnoudse, J. G.

Alter, C. A.

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of helium–neon laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

Ashkin, A.

Ashley, L. E.

Bohren, C. F.

C. F. Bohren, T. J. Nevitt, “Absorption by a sphere: a simple approximation,” Appl. Opt. 22, 774–775 (1983).
[CrossRef] [PubMed]

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Bolin, F. P.

Case, K. M.

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

Chu, C. M.

C. M. Chu, G. C. Clark, S. W. Churchill, Tables of Angular Distribution Coefficients for Light Scattering by Spheres (U. Michigan Press, Engineering Research Institute Publication, Ann Arbor, Mich., 1957).

Churchill, S. W.

C. M. Chu, G. C. Clark, S. W. Churchill, Tables of Angular Distribution Coefficients for Light Scattering by Spheres (U. Michigan Press, Engineering Research Institute Publication, Ann Arbor, Mich., 1957).

Clark, G. C.

C. M. Chu, G. C. Clark, S. W. Churchill, Tables of Angular Distribution Coefficients for Light Scattering by Spheres (U. Michigan Press, Engineering Research Institute Publication, Ann Arbor, Mich., 1957).

Cobb, C. M.

de Mul, F. F. M.

de Rooij, W. A.

W. A. de Rooij, C. C. A. H. van der Stap, “Expansion of Mie scattering matrices in generalized spherical functions,” Astron. Astrophys. 131, 237–248 (1984).

Dziedzic, J. M.

Ference, R. J.

Ferwerda, H. A.

Figdor, C. G.

Flock, S. T.

S. T. Flock, B. C. Wilson, M. S. Patterson, “Total attenuation coefficient and scattering phase functions of tissues and phantom materials at 633 nm,” Med. Phys. 14, 835–841 (1987).
[CrossRef] [PubMed]

Graaff, R.

Greve, J.

Groenhuis, R. A. J.

Gumprecht, R. O.

R. O. Gumprecht, C. M. Sliepcevich, Tables of Light Scattering Functions for Spherical Particles (U. Michigan Press, Ann Arbor, Mich., 1951).

Hermsen, R. G. A. M.

Hoekstra, A. G.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Irvine, W. M.

Ishimaru, A.

Jacques, S. L.

M. J. C. Van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[CrossRef] [PubMed]

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of helium–neon laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

Jentink, H. W.

Johnson, C.

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

Mudgett, P. S.

P. S. Mudgett, L. W. Richards, “Multiple scattering calculations for technology II,” J. Colloid Interface Sci. 39, 551–567 (1972).
[CrossRef]

Nevitt, T. J.

Patterson, M. S.

S. T. Flock, B. C. Wilson, M. S. Patterson, “Total attenuation coefficient and scattering phase functions of tissues and phantom materials at 633 nm,” Med. Phys. 14, 835–841 (1987).
[CrossRef] [PubMed]

Prahl, S. A.

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of helium–neon laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

Preuss, L. E.

Reynolds, L.

Richards, L. W.

P. S. Mudgett, L. W. Richards, “Multiple scattering calculations for technology II,” J. Colloid Interface Sci. 39, 551–567 (1972).
[CrossRef]

Shepherd, A. P.

Sliepcevich, C. M.

R. O. Gumprecht, C. M. Sliepcevich, Tables of Light Scattering Functions for Spherical Particles (U. Michigan Press, Ann Arbor, Mich., 1951).

Sloot, P. M. A.

Star, W. M.

M. J. C. Van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[CrossRef] [PubMed]

Steinke, J. M.

Sterenborg, H. J. C. M.

M. J. C. Van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[CrossRef] [PubMed]

Suichies, H. E.

Taylor, R. C.

Ten Bosch, J. J.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vols. 1 and 2.

van der Liet, H.

van der Stap, C. C. A. H.

W. A. de Rooij, C. C. A. H. van der Stap, “Expansion of Mie scattering matrices in generalized spherical functions,” Astron. Astrophys. 131, 237–248 (1984).

Van Gemert, M. J. C.

M. J. C. Van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[CrossRef] [PubMed]

Wickramasinghe, N. C.

N. C. Wickramasinghe, Light Scattering Functions for Small Particles (Hilger, London, 1973).

Wilson, B. C.

S. T. Flock, B. C. Wilson, M. S. Patterson, “Total attenuation coefficient and scattering phase functions of tissues and phantom materials at 633 nm,” Med. Phys. 14, 835–841 (1987).
[CrossRef] [PubMed]

Wiscombe, W. J.

Zweifel, P. F.

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

Appl. Opt. (11)

R. A. J. Groenhuis, H. A. Ferwerda, J. J. Ten Bosch, “Scattering and absorption of turbid materials determined from reflection measurements. 1: Theory,” Appl. Opt. 22, 2456–2462 (1983).
[CrossRef] [PubMed]

H. W. Jentink, F. F. M. de Mul, R. G. A. M. Hermsen, R. Graaff, J. Greve, “Monte Carlo simulations of laser Doppler blood flow measurements in tissue,” Appl. Opt. 29, 2371–2381 (1990).
[CrossRef] [PubMed]

H. W. Jentink, F. F. M. de Mul, R. Graaff, H. E. Suichies, J. G. Aarnoudse, J. Greve, “Laser Doppler flowmetry: measurements in a layered perfusion model and Monte Carlo simulations of measurements,” Appl. Opt. 30, 2592–2597 (1991).
[CrossRef] [PubMed]

L. Reynolds, C. Johnson, A. Ishimaru, “Diffuse reflectance from a finite blood medium: applications to the modeling of fiber optic catheters,” Appl. Opt. 15, 2059–2067 (1976).
[CrossRef] [PubMed]

R. Graaff, J. G. Aarnoudse, F. F. M. de Mul, H. W. Jentink, “Light propagation parameters for anisotropically scattering media, based on a rigorous solution of the transport equation,” Appl. Opt. 28, 2273–2279 (1989).
[CrossRef] [PubMed]

W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
[CrossRef] [PubMed]

P. M. A. Sloot, A. G. Hoekstra, H. van der Liet, C. G. Figdor, “Scattering matrix elements of biological particles measured in a flow-through system: theory and practice, Appl. Opt. 28, 1752–1762 (1989).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, “Observation of optical resonances of dielectric spheres by light scattering,” Appl. Opt. 20, 1803–1814 (1981).
[CrossRef] [PubMed]

C. F. Bohren, T. J. Nevitt, “Absorption by a sphere: a simple approximation,” Appl. Opt. 22, 774–775 (1983).
[CrossRef] [PubMed]

J. M. Steinke, A. P. Shepherd, “Comparison of Mie theory and the light scattering of red blood cells,” Appl. Opt. 27, 4027–4033 (1988).
[CrossRef] [PubMed]

F. P. Bolin, L. E. Preuss, R. C. Taylor, R. J. Ference, “Refractive index of some mammalian tissues using a fiber optic cladding method,” Appl. Opt. 28, 2297–2303 (1989).
[CrossRef] [PubMed]

Astron. Astrophys. (1)

W. A. de Rooij, C. C. A. H. van der Stap, “Expansion of Mie scattering matrices in generalized spherical functions,” Astron. Astrophys. 131, 237–248 (1984).

IEEE Trans. Biomed. Eng. (1)

M. J. C. Van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[CrossRef] [PubMed]

J. Colloid Interface Sci. (1)

P. S. Mudgett, L. W. Richards, “Multiple scattering calculations for technology II,” J. Colloid Interface Sci. 39, 551–567 (1972).
[CrossRef]

J. Opt. Soc. Am. (2)

Lasers Life Sci. (1)

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of helium–neon laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

Med. Phys. (1)

S. T. Flock, B. C. Wilson, M. S. Patterson, “Total attenuation coefficient and scattering phase functions of tissues and phantom materials at 633 nm,” Med. Phys. 14, 835–841 (1987).
[CrossRef] [PubMed]

Other (9)

C. M. Chu, G. C. Clark, S. W. Churchill, Tables of Angular Distribution Coefficients for Light Scattering by Spheres (U. Michigan Press, Engineering Research Institute Publication, Ann Arbor, Mich., 1957).

N. C. Wickramasinghe, Light Scattering Functions for Small Particles (Hilger, London, 1973).

R. O. Gumprecht, C. M. Sliepcevich, Tables of Light Scattering Functions for Spherical Particles (U. Michigan Press, Ann Arbor, Mich., 1951).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

R. Graaff, J. G. Aarnoudse, F. F. M. de Mul, H. W. Jentink, “Improved expressions for anisotropic scattering in absorbing media,” in Scattering and Diffraction, H. A. Ferwerda, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1029, 103–110 (1989).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

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

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vols. 1 and 2.

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Figures (5)

Fig. 1
Fig. 1

Qsca for Mie scattering of nonabsorbing spheres as a function of size parameter x and relative refractive index m.

Fig. 2
Fig. 2

Average cosine of the scatter angle g for Mie scattering of nonabsorbing spheres as a function of size parameter x and relative refractive Index m.

Fig. 3
Fig. 3

Qsca (1 − g) for Mie scattering of nonabsorbing spheres as a function of size parameter x and relative refractive index m. The dashed curves represent the predictions of Eq. (6).

Fig. 4
Fig. 4

Qsca (1 − g) for Rayleigh–Gans and Mie scattering (m = 1.05). Predictions according to Eq. (6) and Rayleigh scattering (assuming that g = 0) have been added.

Fig. 5
Fig. 5

Wavelength dependence of Qsca (1 − g) for mixtures of scatterers in the range of validity of Eq. (10) and various fractions of contributions by Rayleigh scatterers at 633 nm (0%, 10%, 20%, 30%, 40%, and 50%).

Tables (6)

Tables Icon

Table I Angular Intensity Functions for x = 35 and m = 1.2 for a Spherical Particle as Calculated by Several Mie Programs

Tables Icon

Table II Qsca from Mie Calculations with Eq. (1) as a Function of x and m

Tables Icon

Table III g from Mie Calculations with Eq. (2) as a Function of x and m

Tables Icon

Table IV Qsca and g from Mie Calculations Compared with the Results of Chu et al. and Wickramasinghe for Different Values of x with m = 1.2

Tables Icon

Table V Qsca(1 − g) from Mie Calculations with Eqs. (1) and (2) as Functions of x and m

Tables Icon

Table VI [Qsca(1 − g)]pred/[Qsca(1 − g)]Mle as a Function of x and m with [Qsca/(1 − g)]pred = 3.28x0.37(m − 1)2.09

Equations (14)

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Q sca = 2 x 2 n = 1 N ( 2 n + 1 ) ( | a n | 2 + | b n | 2 ) ,
g = 4 x 2 Q sca n = 1 N [ n ( n + 2 ) n + 1 Re ( a n a n + 1 * + b n b n + 1 * ) + 2 n + 1 n ( n + 1 ) Re ( a n b n * ) ] ,
Q sca = 1 x 2 0 π [ i 1 ( θ ) + i 2 ( θ ) ] sin ( θ ) d θ ,
g = 0 π [ i 1 ( θ ) + i 2 ( θ ) ] cos ( θ ) sin ( θ ) d θ 0 π [ i 1 ( θ ) + i 2 ( θ ) ] sin ( θ ) d θ .
Q pr Q abs + Q sca ( 1 g ) .
[ Q sca ( 1 g ) ] pred = 3 . 28 x 0 . 37 ( m 1 ) 2 . 09 ,
I ( θ ) = I 0 2 x 4 a 2 r 2 ( m 2 1 m 2 + 2 ) 2 × { 3 u 3 [ sin ( u ) u cos ( u ) ] } 2 [ 1 + cos 2 ( θ ) ] ,
u = 2 x sin ( θ / 2 ) .
i 1 ( θ ) + i 2 ( θ ) = 2 ( 2 π r ) 2 λ 2 I ( θ ) I 0 = x 6 | m 2 1 m 2 + 2 | 2 × { 3 u 3 [ sin ( u ) u cos ( u ) ] } 2 [ 1 + cos 2 ( θ ) ] .
σ s ( λ 1 ) σ s ( λ 2 ) = ( λ 2 λ 1 ) 0 . 37 ( m 1 1 m 2 1 ) 2 . 09 .
g = x 2 Q sca g x 2 Q sca = σ s g σ s ,
σ s ( mixture ) = σ s ( 1 g ) = σ s σ s g = σ s ( 1 g ) .
σ s , 2 ( λ 1 ) σ s , 2 ( λ 2 ) = ( λ 2 λ 1 ) 4 . 0 ( m 1 1 m 2 1 ) 2 . 0 .
σ s ( λ ) σ s ( λ 0 ) = σ s , 1 ( λ ) σ s , 1 ( λ 0 ) [ 1 f 2 ( λ 0 ) ] + σ s , 2 ( λ ) σ s , 2 ( λ 0 ) f 2 ( λ 0 ) ,

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