Abstract

A method is presented for the estimation of optical constants in the ultraviolet-visible-near-infrared (UV-Vis-NIR) region of nonspherical particles in a suspension at concentrations where multiple scattering is significant. The optical constants are obtained by an inversion technique using the adding-doubling method to solve the radiative transfer equation in combination with the single scattering theories for modelling scattering by nonspherical particles. Two methods for describing scattering by single scattering are considered: the T-matrix method and the approximate but computationally simpler Rayleigh–Gans–Debye (RGD) approximation. The method is then applied to obtain the optical constants of Bacillus subtilis spores in the wavelength region 4001200nm. It is found that the optical constants obtained using the RGD approximation matches those obtained using the T-matrix method to within experimental error.

© 2008 Optical Society of America

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References

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  1. R. Xu, Particle Characterization: Light Scattering Techniques (Kluwer Academic, 2000).
  2. X. Ma, Q. L. Lu, S. R. Brock, K. M. Jacobs, P. Yang, and X. H. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. 48, 4165-4172 (2003).
    [CrossRef]
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2007

2005

2003

X. Ma, Q. L. Lu, S. R. Brock, K. M. Jacobs, P. Yang, and X. H. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. 48, 4165-4172 (2003).
[CrossRef]

1997

1993

1978

1968

Alfano, R. R.

Alimova, A.

Arakawa, E. T.

Barber, P.

Brock, S. R.

X. Ma, Q. L. Lu, S. R. Brock, K. M. Jacobs, P. Yang, and X. H. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. 48, 4165-4172 (2003).
[CrossRef]

Gottlieb, P.

Hovenier, J. W.

M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, “Overview of scattering by nonspherical particles,” in Light Scattering by Nonspherical Particles. Theory, Measurements and Applications, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, and W. J. Wiscombe, eds. (Academic, 2000), pp. 40.

Hu, X. H.

X. Ma, Q. L. Lu, S. R. Brock, K. M. Jacobs, P. Yang, and X. H. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. 48, 4165-4172 (2003).
[CrossRef]

Jacobs, K. M.

X. Ma, Q. L. Lu, S. R. Brock, K. M. Jacobs, P. Yang, and X. H. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. 48, 4165-4172 (2003).
[CrossRef]

Katz, A.

Kerker, M.

M. Kerker, The Scattering of Light and other Electromagnetic Radiation (Academic, 1970), pp. 414-486.

Khare, B. N.

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).

Lu, Q. L.

X. Ma, Q. L. Lu, S. R. Brock, K. M. Jacobs, P. Yang, and X. H. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. 48, 4165-4172 (2003).
[CrossRef]

Ma, X.

X. Ma, Q. L. Lu, S. R. Brock, K. M. Jacobs, P. Yang, and X. H. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. 48, 4165-4172 (2003).
[CrossRef]

Mackowski, D. W.

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix codes for computing electromagnetic scattering by nonspherical particles and aggregated particles,” www.giss.nasa.gov/~crmin/t_matrix.html.

Milham, M. E.

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix codes for computing electromagnetic scattering by nonspherical particles and aggregated particles,” www.giss.nasa.gov/~crmin/t_matrix.html.

M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, “Overview of scattering by nonspherical particles,” in Light Scattering by Nonspherical Particles. Theory, Measurements and Applications, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, and W. J. Wiscombe, eds. (Academic, 2000), pp. 40.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).

Prahl, S. A.

Querry, M. R.

Rudolph, E.

Steiner, J. C.

Thennadil, S. N.

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix codes for computing electromagnetic scattering by nonspherical particles and aggregated particles,” www.giss.nasa.gov/~crmin/t_matrix.html.

M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, “Overview of scattering by nonspherical particles,” in Light Scattering by Nonspherical Particles. Theory, Measurements and Applications, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, and W. J. Wiscombe, eds. (Academic, 2000), pp. 40.

Tuminello, P. S.

van Germert, M. J. C.

Velazco-Roa, M. A.

Wang, D. S.

Welch, A. J.

Wrobel, J. M.

Wyatt, P.

Xu, M.

Xu, R.

R. Xu, Particle Characterization: Light Scattering Techniques (Kluwer Academic, 2000).

Yang, P.

X. Ma, Q. L. Lu, S. R. Brock, K. M. Jacobs, P. Yang, and X. H. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. 48, 4165-4172 (2003).
[CrossRef]

Appl. Opt.

Opt. Lett.

Phys. Med. Biol.

X. Ma, Q. L. Lu, S. R. Brock, K. M. Jacobs, P. Yang, and X. H. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. 48, 4165-4172 (2003).
[CrossRef]

Other

R. Xu, Particle Characterization: Light Scattering Techniques (Kluwer Academic, 2000).

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix codes for computing electromagnetic scattering by nonspherical particles and aggregated particles,” www.giss.nasa.gov/~crmin/t_matrix.html.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).

M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, “Overview of scattering by nonspherical particles,” in Light Scattering by Nonspherical Particles. Theory, Measurements and Applications, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, and W. J. Wiscombe, eds. (Academic, 2000), pp. 40.

M. Kerker, The Scattering of Light and other Electromagnetic Radiation (Academic, 1970), pp. 414-486.

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Figures (6)

Fig. 1
Fig. 1

SEM of Bacillus subtilis spores in the suspension used in this study.

Fig. 2
Fig. 2

(a) Total diffuse reflectance and (b) total diffuse transmittance for three different concentrations of Bacillus subtilis spores in water suspensions.

Fig. 3
Fig. 3

Optical constants (a) n and (b) k as a function of the wavelength of Bacillus subtilis spores in water suspensions estimated using the T-matrix method and RGD approximation along with those reported by Tuminello et al. [12].

Fig. 4
Fig. 4

(a) Absorption cross section computed using the T-matrix method and the RGD approximation for prolate ellipsoids in water. (b) Percent error in using the RGD approximation instead of the T-matrix method.

Fig. 5
Fig. 5

(a) Scattering cross section computed using the T-matrix method and the RGD approximation for prolate ellipsoids. (b) Percent error in using the RGD approximation instead of the T-matrix method.

Fig. 6
Fig. 6

(a) Anisotropy factor computed using the T-matrix method and the RGD approximation for prolate ellipsoids. (b) Percent error in using the RGD approximation instead of the T-matrix method.

Equations (20)

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= abs ( R tdm R tds ) + abs ( T tdm T tds ) .
E inc ( r ) = n = 1 m = n n [ a m n RgM m n ( κ r ) + b m n RgN m n ( κ r ) ] ,
E sca ( r ) = n = 1 m = n n [ p m n M m n ( κ r ) + q m n N m n ( κ r ) ] , | r ^ | > r 0 .
[ p q ] = [ T 11 T 12 T 21 T 22 ] [ a b ] .
σ ext = 2 π κ 2 Re n = 1 n max m = n n [ T m n m n 11 + T m n m n 12 ] ,
σ sca = 2 π κ 2 n = 1 n max n = 1 n max m = max ( n , n ) max ( n , n ) i , j = 1 , 2 | T m n m n i j | 2 ,
σ abs = σ ext σ sca .
g = 1 2 1 + 1 a 1 ( θ ) cos θ d θ .
| 1 | 1 and x | 1 | 1 ,
F u n = κ 4 8 π 2 V p 2 ( 1 + cos 2 θ ) ( m r 1 ) 2 P ( θ ) ,
V p = 4 3 π a c b ,
P ( θ ) = { 3 u 3 ( sin u u cos u ) } 2 ,
u = 2 k a sin θ 2 ( cos 2 β + b 2 a 2 sin 2 β ) 1 / 2 ,
cos β = cos α sin θ 2 + sin α cos θ 2 cos φ ,
P ( θ ) randomly oriented = 0 π / 2 P ( θ ) sin β d β .
σ sca = 0 2 π 0 π F ( θ , a , , λ ) sin θ d θ d ϕ ,
σ abs ( λ ) = 4 π n ( λ ) k ( λ ) V p λ .
p ( cos θ ) = 1 4 π 2 a 4 V p 2 x 4 ( 1 ) 2 ( 1 cos 2 θ ) P ( θ ) σ sca .
g = cos θ = 0 2 π 0 π p ( cos θ ) cos θ sin θ d θ d ϕ .
k ( λ ) = λ ln ( T water / T ) 4 π f v d ,

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