Abstract

This paper, second of two parts, presents a parametric study of the forwardscattering corrections for experimentally measured optical extinction coefficients in polydisperse particulate media, since some forward scattered light invariably enters, along with the direct beam, into the finite aperture of the detector. Forwardscattering corrections are computed by two methods: (1) using the exact Mie theory, and (2) the approximate Rayleigh diffraction formula for spherical particles. A parametric study of the dependence of the corrections on mode radii, real and imaginary parts of the complex refractive index, and half-angle of the detector’s view cone has been carried out for three different size distribution functions of the modified Gamma type. In addition, a study has been carried out to investigate the range of these parameters in which the approximate formulation is valid. The agreement is especially good for small-view cone angles and large particles, which improves significantly for slightly absorbing aerosol particles. Also discussed is the dependence of these corrections on the experimental design of the transmissometer systems.

© 1978 Optical Society of America

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References

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  1. A. Deepak, M. A. Box, “Forwardscattering Corrections for Optical Extinction Measurements in Aerosol Media. Part 1: Monodispersions” Appl. Opt. 17, 2900 (1978).
    [CrossRef] [PubMed]
  2. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).
  3. A. Deepak, O. H. Vaughan, Appl. Opt. 17, 374 (1977).
    [CrossRef]
  4. A. E. S. Green, A. Deepak, B. J. Lipofsky, Appl. Opt. 10, 1263 (1971).
    [CrossRef] [PubMed]
  5. A. Deepak, G. P. Box, “Analytical Modeling of Aerosol Size Distributions” (Paper IFAORS-106-77, in preparation).
  6. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

1978 (1)

1977 (1)

1971 (1)

Box, G. P.

A. Deepak, G. P. Box, “Analytical Modeling of Aerosol Size Distributions” (Paper IFAORS-106-77, in preparation).

Box, M. A.

Deepak, A.

Deirmendjian, D.

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).

Green, A. E. S.

Lipofsky, B. J.

van de Hulst, H. C.

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

Vaughan, O. H.

Appl. Opt. (3)

Other (3)

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).

A. Deepak, G. P. Box, “Analytical Modeling of Aerosol Size Distributions” (Paper IFAORS-106-77, in preparation).

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

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Figures (21)

Fig. 1
Fig. 1

Plots of the three Deirmendjian size-distribution models, Hazes M, H, and C, for a mode radius of 1 μm.

Fig. 2
Fig. 2

Plots of the correction factor R ¯ vs mode radius for three values of θ and five refractive indices: Haze M.

Fig. 3
Fig. 3

Plots of the correction factor R ¯ vs mode radius for three values of θ and five refractive indices: Haze H.

Fig. 4
Fig. 4

Plots of the correction factor R ¯ vs mode radius for three values of θ and five refractive indices: Haze C.

Fig. 5
Fig. 5

Plots of percentage error (100Ē) vs mode radius for three values of θ and five refractive indices: Haze M.

Fig. 6
Fig. 6

Plots of percentage error (100Ē) vs mode radius for three values of θ and five refractive indices: Haze H.

Fig. 7
Fig. 7

Plots of percentage error (100Ē) vs mode radius for three values of θ and five refractive indices: Haze C.

Fig. 8
Fig. 8

Plots of percentage error (100Ē) vs half-angle θ for six mode radii (m = 1.55): Haze M.

Fig. 9
Fig. 9

Plots of percentage error (100Ē) vs half-angle θ for six mode radii (m = 1.55): Haze H.

Fig. 10
Fig. 10

Plots of percentage error (100Ē) vs half-angle θ for six mode radii (m = 1.55): Haze C.

Fig. 11
Fig. 11

Plots of correction factor R ¯ vs θ for m = 1.55, λ = 0.55 μm, and five mode radii: Haze M. (Symbols without crosses represent Mie results; symbols with crosses, approximate results.)

Fig. 12
Fig. 12

Plots of correction factor R ¯ vs θ for m = 1.55, λ = 0.55 μm, and five mode radii: Haze H. (Symbols without crosses represent Mie results; symbols with crosses, approximate results.)

Fig. 13
Fig. 13

Plots of correction factor R ¯ vs θ for m = 1.55, λ = 0.55 μm, and five mode radii: Haze C. (Symbols without crosses represent Mie results; symbols with crosses, approximate results.)

Fig. 14
Fig. 14

Plots of correction factor R ¯ vs θ for m = 1.55–i(0.05), λ = 0.55 μm, and five mode radii: Haze M. (Symbols without crosses represent Mie results; symbols with crosses, approximate results.)

Fig. 15
Fig. 15

Plots of correction factor R ¯ vs θ for m = 1.55–i(0.05), λ = 0.55 μm, and five mode radii: Haze H. (Symbols without crosses represent Mie results; symbols with cross, approximate results.)

Fig. 16
Fig. 16

Plots of correction factor R ¯ vs θ for m = 1.55–i(0.05), λ = 0.55 μm, and five mode radii: Haze C. (Symbols without crosses represent Mie results; symbols with cross, approximate results.)

Fig. 17
Fig. 17

Error contours in the rmθ plane for m = 1.55, λ = 0.55 μm, for models Haze M, Haze H, and Haze C.

Fig. 18
Fig. 18

Plots of the path-averaged correction factor R ¯ ˆ vs y, where y = krma/f and krmR2/L for the lens–pinhole detector and open detector systems, respectively, for models Haze M, Haze H, and Haze C. λ = 0.55 μm.

Fig. 19
Fig. 19

Error contours in the νθ plane for m = 1.33, λ = 0.55 μm, for the power law size distribution.

Fig. 20
Fig. 20

Error contours in the νθ plane for m = 1.55–i(0.05), λ = 0.55 μm, for the power law size distribution.

Fig. 21
Fig. 21

Plots of correction factor R ¯ vs θ for m = 1.55–i(0.05), λ = 0.55 μm, and five values of the power ν for a power law size distribution. (Symbols without crosses represent Mie results; symbols with crosses, approximate results.)

Equations (23)

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I ( λ ) = I 0 ( λ ) exp [ τ ( λ ) ] ,
τ ( λ ) = β ext ( λ ) L ,
β ext ( λ ) = β scat ( λ ) + β abs ( λ ) .
β j ( λ ) = r 1 r 2 π r 2 Q j ( x , m ) n ( r ) d r , j = ext , scat , and abs .
r 1 r 2 n ( r ) d r ,
β ext = β scat + β abs ( apparent ) .
Q ext Q ext = Q scat Q scat = 1 x 2 0 θ ( i 1 + i 2 ) sin θ d θ = I ( x , m , θ ) x 2 ,
β ext β ext = β scat β scat = π r 1 r 2 r 2 n ( r ) ( Q ext Q ext ) d r = π r 1 r 1 r 2 n ( r ) Q E d r = π k 2 r 1 r 2 n ( r ) I ( x , m , θ ) d r ,
R ¯ = β ext / β ext = 1 E ¯ ,
E ¯ = π β ext k 2 r 1 r 2 n ( r ) I ( x , m , θ ) d r .
R ¯ = π β ext 0 r 2 n ( r ) Q ( x , m ) 1 2 [ 1 + J 0 2 ( x θ ) + J 1 2 ( x θ ) ] d r ,
I ( λ ) = I 0 ( λ ) exp [ τ ( λ ) ] ,
τ ( λ ) = 0 L d l β ext = 0 L d l β ext R ¯ [ n ( r ) , θ ] = 0 L d l r 1 r 2 d r π r 2 Q ( x , m ) n ( r ) R ( x , θ ) .
τ ( λ ) = π L r 1 r 2 d r r 2 Q ( x , m ) n ( r ) R ˆ ( x , L ) ,
R ˆ ( x , L ) = 1 L 0 L d l R [ x , θ ( l ) ] .
( a ) Haze M : n ( r ) = r exp [ ( b r ) 1 / 2 ] , r m = 4 / b ,
( b ) Haze H : n ( r ) = r 2 exp ( b r ) , r m = 2 / b ,
( c ) Haze C : n ( r ) = r 8 exp [ ( b r ) 3 ] , r m = ( 8 / 3 ) 1 / 3 b 1 .
n ( r ) = r ν , 10 2 μ m r 15 μ m
R ¯ ˆ = r 1 r 2 d r r 2 n ( r / r m ) Q ( k r ) R ˆ ( k r θ ) / r 1 r 2 d r r 2 n ( r / r m ) Q ( k r ) .
R ¯ ˆ = R ¯ ˆ ( k r m θ ) = z 1 z 2 d z z 2 n ( z ) R ˆ ( k r m θ z ) / z 1 z 2 d z z 2 n ( z ) d z ,
z = r / r m , z 1 = r 1 / r m , z 2 = r 2 / r m .
r m / λ = r m / λ .

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