Abstract

The inversion of multiple-scattered light measurements to extract the optical constant (complex refractive index) is computationally intensive. A significant portion of this time is due to the effort required for computing the single particle characteristics (absorption and scattering cross sections, anisotropy factor, and the phase function). We investigate approximations for computing these characteristics so as to significantly speed up the calculations without introducing large inaccuracies. Two suspensions of spherical particles viz., polystyrene and poly(methyl methacrylate) were used for this investigation. It was found that using the exact Mie theory to compute the absorption and scattering cross sections and the anisotropy factor with the phase function computed using the Henyey–Greenstein approximation yielded the best results. Analysis suggests that errors in the phase functions and thus in the estimated optical constants depend mainly on how closely the approximations match the Mie phase function at small scattering angles.

© 2007 Optical Society of America

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References

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  1. M. A. Velazco-Roa and S. N. Thennadil, "Estimation of complex refractive index of polydisperse particulate systems from multiple-scattered ultraviolet-visible-near-infrared measurements," Appl. Opt. 46, 3730-3735 (2007).
    [CrossRef] [PubMed]
  2. A. Ishimaru, Wave Propagation and Scattering in Random Media, IEEE/OUP Series on Electromagnetic Theory (IEEE, 1997), pp. xxv, 574.
  3. S. A. Prahl, "The adding-doubling method," in Optical Thermal Response of Laser Irradiated Tissue, A. J. Welch and M. J. C. van Gemert, eds. (Plenum, 1995), pp. 101-129.
  4. C. Bohren and D. Huffman, "Rayleigh-Gans theory," in Absorption and Scattering by Small Particles (Wiley-VCH, 2004), pp. 158-165.
  5. I. G. Henyey and J. L. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J. 85, 70-83 (1941).
    [CrossRef]
  6. D. Toublanc, "Henyey-Greenstein and Mie phase functions in Monte Carlo radiative transfer computations," Appl. Opt. 35, 3270-3274 (1996).
    [CrossRef] [PubMed]
  7. W. M. Cornette and J. G. Shanks, "Physically reasonable analytic expression for the single-scattering phase function," Appl. Opt. 31, 3152-3160 (1992).
    [CrossRef] [PubMed]
  8. M. Kerker, "Rayleigh-Debye scattering," in The Scattering of Light and Other Electromagnetic Radiation, M. L. Ernest, ed. (Academic, 1970), pp. 414-486.
  9. W. J. Wiscombe, "The delta-M method: rapid yet accurate radiative flux calculations for strongly asymmetric phase functions," J. Atmos. Sci. 34, 1408-1422 (1977).
    [CrossRef]
  10. J. H. Joseph and W. J. Wiscombe, "The delta-Eddington approximation for radiative flux transfer," J. Atmos. Sci. 33, 2452-2459 (1976).
    [CrossRef]
  11. K. N. Liou, "Approximations for radiative transfer," in An Introduction to Atmospheric Sciences (Academic, 2002), pp. 310-313.
  12. D. Segelstein, "The complex refractive index of water," M. S. thesis (University of Missouri, 1981).
  13. N. G. Sultanova, I. D. Nikolov, and C. D. Ivanov, "Measuring the refractometric characteristics of optical plastics," Opt. Quantum Electron. 35, 21-24 (2003).
    [CrossRef]

2007 (1)

2004 (1)

C. Bohren and D. Huffman, "Rayleigh-Gans theory," in Absorption and Scattering by Small Particles (Wiley-VCH, 2004), pp. 158-165.

2003 (1)

N. G. Sultanova, I. D. Nikolov, and C. D. Ivanov, "Measuring the refractometric characteristics of optical plastics," Opt. Quantum Electron. 35, 21-24 (2003).
[CrossRef]

2002 (1)

K. N. Liou, "Approximations for radiative transfer," in An Introduction to Atmospheric Sciences (Academic, 2002), pp. 310-313.

1997 (1)

A. Ishimaru, Wave Propagation and Scattering in Random Media, IEEE/OUP Series on Electromagnetic Theory (IEEE, 1997), pp. xxv, 574.

1996 (1)

1995 (1)

S. A. Prahl, "The adding-doubling method," in Optical Thermal Response of Laser Irradiated Tissue, A. J. Welch and M. J. C. van Gemert, eds. (Plenum, 1995), pp. 101-129.

1992 (1)

1981 (1)

D. Segelstein, "The complex refractive index of water," M. S. thesis (University of Missouri, 1981).

1977 (1)

W. J. Wiscombe, "The delta-M method: rapid yet accurate radiative flux calculations for strongly asymmetric phase functions," J. Atmos. Sci. 34, 1408-1422 (1977).
[CrossRef]

1976 (1)

J. H. Joseph and W. J. Wiscombe, "The delta-Eddington approximation for radiative flux transfer," J. Atmos. Sci. 33, 2452-2459 (1976).
[CrossRef]

1970 (1)

M. Kerker, "Rayleigh-Debye scattering," in The Scattering of Light and Other Electromagnetic Radiation, M. L. Ernest, ed. (Academic, 1970), pp. 414-486.

1941 (1)

I. G. Henyey and J. L. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J. 85, 70-83 (1941).
[CrossRef]

Bohren, C.

C. Bohren and D. Huffman, "Rayleigh-Gans theory," in Absorption and Scattering by Small Particles (Wiley-VCH, 2004), pp. 158-165.

Cornette, W. M.

Greenstein, J. L.

I. G. Henyey and J. L. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J. 85, 70-83 (1941).
[CrossRef]

Henyey, I. G.

I. G. Henyey and J. L. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J. 85, 70-83 (1941).
[CrossRef]

Huffman, D.

C. Bohren and D. Huffman, "Rayleigh-Gans theory," in Absorption and Scattering by Small Particles (Wiley-VCH, 2004), pp. 158-165.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media, IEEE/OUP Series on Electromagnetic Theory (IEEE, 1997), pp. xxv, 574.

Ivanov, C. D.

N. G. Sultanova, I. D. Nikolov, and C. D. Ivanov, "Measuring the refractometric characteristics of optical plastics," Opt. Quantum Electron. 35, 21-24 (2003).
[CrossRef]

Joseph, J. H.

J. H. Joseph and W. J. Wiscombe, "The delta-Eddington approximation for radiative flux transfer," J. Atmos. Sci. 33, 2452-2459 (1976).
[CrossRef]

Kerker, M.

M. Kerker, "Rayleigh-Debye scattering," in The Scattering of Light and Other Electromagnetic Radiation, M. L. Ernest, ed. (Academic, 1970), pp. 414-486.

Liou, K. N.

K. N. Liou, "Approximations for radiative transfer," in An Introduction to Atmospheric Sciences (Academic, 2002), pp. 310-313.

Nikolov, I. D.

N. G. Sultanova, I. D. Nikolov, and C. D. Ivanov, "Measuring the refractometric characteristics of optical plastics," Opt. Quantum Electron. 35, 21-24 (2003).
[CrossRef]

Prahl, S. A.

S. A. Prahl, "The adding-doubling method," in Optical Thermal Response of Laser Irradiated Tissue, A. J. Welch and M. J. C. van Gemert, eds. (Plenum, 1995), pp. 101-129.

Segelstein, D.

D. Segelstein, "The complex refractive index of water," M. S. thesis (University of Missouri, 1981).

Shanks, J. G.

Sultanova, N. G.

N. G. Sultanova, I. D. Nikolov, and C. D. Ivanov, "Measuring the refractometric characteristics of optical plastics," Opt. Quantum Electron. 35, 21-24 (2003).
[CrossRef]

Thennadil, S. N.

Toublanc, D.

Velazco-Roa, M. A.

Wiscombe, W. J.

W. J. Wiscombe, "The delta-M method: rapid yet accurate radiative flux calculations for strongly asymmetric phase functions," J. Atmos. Sci. 34, 1408-1422 (1977).
[CrossRef]

J. H. Joseph and W. J. Wiscombe, "The delta-Eddington approximation for radiative flux transfer," J. Atmos. Sci. 33, 2452-2459 (1976).
[CrossRef]

Appl. Opt. (3)

Astrophys. J. (1)

I. G. Henyey and J. L. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J. 85, 70-83 (1941).
[CrossRef]

J. Atmos. Sci. (2)

W. J. Wiscombe, "The delta-M method: rapid yet accurate radiative flux calculations for strongly asymmetric phase functions," J. Atmos. Sci. 34, 1408-1422 (1977).
[CrossRef]

J. H. Joseph and W. J. Wiscombe, "The delta-Eddington approximation for radiative flux transfer," J. Atmos. Sci. 33, 2452-2459 (1976).
[CrossRef]

Opt. Quantum Electron. (1)

N. G. Sultanova, I. D. Nikolov, and C. D. Ivanov, "Measuring the refractometric characteristics of optical plastics," Opt. Quantum Electron. 35, 21-24 (2003).
[CrossRef]

Other (6)

M. Kerker, "Rayleigh-Debye scattering," in The Scattering of Light and Other Electromagnetic Radiation, M. L. Ernest, ed. (Academic, 1970), pp. 414-486.

K. N. Liou, "Approximations for radiative transfer," in An Introduction to Atmospheric Sciences (Academic, 2002), pp. 310-313.

D. Segelstein, "The complex refractive index of water," M. S. thesis (University of Missouri, 1981).

A. Ishimaru, Wave Propagation and Scattering in Random Media, IEEE/OUP Series on Electromagnetic Theory (IEEE, 1997), pp. xxv, 574.

S. A. Prahl, "The adding-doubling method," in Optical Thermal Response of Laser Irradiated Tissue, A. J. Welch and M. J. C. van Gemert, eds. (Plenum, 1995), pp. 101-129.

C. Bohren and D. Huffman, "Rayleigh-Gans theory," in Absorption and Scattering by Small Particles (Wiley-VCH, 2004), pp. 158-165.

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Figures (8)

Fig. 1
Fig. 1

Complex refractive index, m ( λ ) = n ( λ ) + i k ( λ ) of PMMA microspheres (a) real part, n ( λ ) , and (b) imaginary part, k ( λ ) .

Fig. 2
Fig. 2

Comparison between the complex refractive index estimated using the RGD approximation and the Mie theory for polystyrene. (a) Real part, n, (b) percent relative error, Δ n = | n Mie n RGD | / n Mie 100 , (c) imaginary part, k and (d) percent relative error, Δ k = | k Mie k RGD | / k Mie 100 .

Fig. 3
Fig. 3

Complex refractive index of PMMA computed using Mie theory and the RGD approximation (a) real part n ( λ ) , and (b) imaginary part k ( λ ) .

Fig. 4
Fig. 4

Comparison between the optical constants estimated using the Mie phase function, the H-G and the H-Gm approximation for polystyrene. (a) n ( λ ) , (b) percent relative error Δ n = | n Mie n HG ( or H-Gm ) | / n Mie 100 , (c) k ( λ ) , and (d) percent relative error, Δ k = | k Mie k HG ( or H-Gm ) | / k Mie 100 .

Fig. 5
Fig. 5

Comparison between the optical constants estimated using the Mie phase function, the H-G and the H-Gm approximation for PMMA. (a) n ( λ ) and (b) k ( λ ) .

Fig. 6
Fig. 6

(a) Comparison between normalized scattering cross section computed by Mie theory and the RGD approximation. (b) Percent relative error in σ s . (c) Comparison between the normalized absorption cross section computed by Mie theory and the RGD approximation. (d) Percent relative error in σ a .

Fig. 7
Fig. 7

Comparison between phase functions computed using Mie, RGD, H-G, H-Gm, for particles of radius, a = 0.225 μm . (a) x = 4.1432 , g = 0.86 , and λ = 450 nm . (b) x = 3.1731 , g = 0.8 , and λ = 589 nm . (c) x = 2.1238 , g = 0.69 , and λ = 880 nm . (d) x = 1.5574 , g = 0.44 , and λ = 1200 n m .

Fig. 8
Fig. 8

Expansion coefficients, χ n * , computed using the δ - M method to compute the redistribution function of the Mie, H-G, and H-Gm phase function.

Tables (3)

Tables Icon

Table 1 Performance Comparison for the Estimation of the Optical Constants Using Mie Theory, RGD Approximation, Mie Theory Combined with H-G, and H-Gm Approximation for the Phase Function

Tables Icon

Table 2 Data Used to Compute the Different Phase Functions Shown in Fig. 7

Tables Icon

Table 3 Comparison between Total Diffuse Reflectance, R , and Transmittance, T Computed by the ADD Method Using Mie, H-G, and H-Gm Phase Functions

Equations (18)

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d I ( r , s ^ ) d s = μ t I ( r , s ^ ) + μ s 4 π 4 π p ( s , s ^ ) I ( r , s ^ ) d ω ,
μ a ( λ ) = μ a , p ( λ ) + μ a , w ( λ ) , μ a , p ( λ ) = ρ σ a ( λ ) ,
μ a , w ( λ ) = 4 π k w ( λ ) ( 1 c ) λ , μ s ( λ ) = ρ σ s ( λ ) ,
= abs ( R meas R c a l ) + abs ( T meas T c a l ) .
  σ a ( λ ) = 4 π n ( λ ) k ( λ ) V p λ ,
σ sca = 2 π ( n 1 ) 2 { 5 2 + 2 x 2 sin 4 x 4 x 7 16 x 2 ( 1 cos 4 x ) + ( 1 2 x 2 2 ) ( γ + log 4 x C i [ 4 x ] ) } ,
p ( s , s ) = p ( cos θ ) = F ( θ , a , n , λ ) σ sca .
g = cos θ = 1 4 π 2 x 4 ( n 1 ) 2 ( 1 cos 2 θ ) P ( θ ) cos θ d θ σ scat .
h ( cos θ i , cos θ j ) = 1 2 π 0 2 π p ( cos θ i cos θ j + 1 cos θ i 2 1 cos θ j 2 cos ϕ ) d ϕ .
P * ( cos θ ) = 2 f δ ( 1 cos θ ) + ( 1 f ) m = 0 2 M 1 ( 2 m + 1 ) × χ m * P n ( cos θ ) ,
p ( cos θ ) = n = 0 ( 2 n + 1 ) χ n P n ( cos θ ) ,
χ n = 1 2 0 π p ( cos θ ) P n sin θ d θ .
χ m * = χ m f 1 f , m = 0 ,   …   ,   2 M 1 ,
h * ( cos θ i , cos θ j ) = 2 f δ ( cos θ i cos θ j ) + ( 1 f ) m = 0 2 M 1 ( 2 m + 1 ) χ m * × P m ( cos θ i ) P m ( cos θ j ) .
p HG ( cos θ ) = 1 g 2 [ 1 + g 2 2 g cos θ ] 3 / 2 .
f = χ m = g M ,
χ m * = g n g M 1 g M .
p H-Gm ( cos θ ) = 3 2 1 g 2 2 + g 2 1 + cos 2 θ [ 1 + g 2 2 g cos θ ] 3 / 2 .

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