Abstract

An analytical solution to the attenuation of flux within an integrating sphere due to spherical integrating source coating, exit port escape, and atmospheric absorption is derived employing a geometric probability distribution of completed sphere transits. This is used to determine the mean number of completed sphere transits and its variance. Equations that provide the attenuation ratios for the three extinction mechanisms are derived using the energy balance and summation of reflection methods. The mean length of a transit of the sphere and its variance are presented and used to derive expressions for the mean and variance of photon path lengths in the sphere.

© 2008 Optical Society of America

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References

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2001 (1)

1998 (1)

1996 (1)

1995 (1)

1967 (1)

Appl. Opt. (4)

J. Opt. Soc. Am. A (1)

Other (6)

J. S. Rustagi, "Random variables and discrete probability distributions," in Introduction to Statistical Methods (Rowman and AllanHeld, 1984), Vol. 1, pp. 67-114.

E. R. Hansen, A Table of Series and Products (Prentice Hall, 1975).

A. Berk, L. S. Bernstein, and D. C. Robertson, MODTRAN: A Moderate Resolution Model for LOWTRAN 7, Rep. GL-TR-89-0122 (Geophysical Directorate Phillips Laboratory, 1989).

D. R. Myers and A. A. Andreas, "Sensitivity of spectroradiometric calibrations in the near infrared to variations in atmospheric water vapor," presented at the American Solar Energy Society Annual Conference, Portland, Ore., 11-14 July 2004.

Labsphere Inc., "A guide to integrating sphere theory and applications," http://www.labsphere.com/tecdocs.aspx.

J. W. T. Walsh, Photometry, 3rd ed. (Dover, 1958).

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Figures (4)

Fig. 1
Fig. 1

Fractional reflectance for a typical SIS, as computed using theprocedure derived by Goebel [3]. At the curve peak, 19% of the source flux remains unabsorbed by the SIS coating and is subject to further attenuation by atmospheric effects during SIS transit.

Fig. 2
Fig. 2

Integrating sphere of diameter D S and exit port radius R X P , with arbitrary ray length l.

Fig. 3
Fig. 3

(a) Section through the integrating sphere showing that the normal to a reflecting element δA at P is a radius of the sphere and that any reflection from P at an angle θ to the normal follows a chord to next impact on the sphere wall with a path length of l = D S cos θ . (b) Unit hemisphere of all emission directions from P showing, as a shaded zone, the element 2 π sin θ d θ of solid angle at a diffuse reflectance angle of θ.

Fig. 4
Fig. 4

Sample distribution P ( n ) , geometric PDF, probability of photon exit on ( n + 1 ) th reflection for f = 0.015 .

Tables (2)

Tables Icon

Table 1 Fractional Loss per Transit a

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Table 2 Sample Set of Computed Values a

Equations (74)

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n = 2 [ 1 ( 1 4 R X P 2 D S 2 ) 1 / 2 ] .
  l ^ = l = 0 π / 2 D S cos θ I cos θ 2 π sin θ d θ 0 π / 2 I cos θ 2 π sin θ d θ = 2 D S 3 .
  l 2 = 0 π / 2 ( D S cos θ ) 2 I cos θ 2 π sin θ d θ 0 π / 2 I cos θ 2 π sin θ d θ = D S 2 2 ,
σ l 2 = l 2 l 2 = D S 2 18 .
  P ( n ) = p q n 1 = p ( 1 p ) n 1 ,   n = 1 , 2 , 3 ,   …   .
n = 1 P ( n ) = p n = 1 ( 1 p ) n 1 = 1 ,
r = n 1.
P ( r ) = p ( 1 p ) r ,   n = 0 , 1 , 2 ,   …   .
r 2 = ( n 1 ) 2 = n 2 2 n + 1 ,
σ r 2 = σ n 2 .
n = n = 1 n P ( n ) = p 1 p n = 1 n ( 1 p ) n = 1 p ,
n = 1 n x n = x ( 1 x ) 2 ,   | x | < 1 ,
  n 2 = n = 1 n 2 P ( n ) = p 1 p n = 1 n 2 ( 1 p ) n = 2 p p 2 ,
n = 1 n 2 x n = x ( 1 + x ) ( 1 x ) 3 ,   | x | < 1 .
σ n 2 = n 2 n 2 = 1 p p 2 .
σ n = 1 p p n 1 2 .
q = r = ( 1 / f ) 1 = ( 1 / p ) 1 .
f = [ 1 ( 1 4 R X P 2 D S 2 ) 1 / 2 ] 2 .
p a = l ^ α λ ,
p b = ( 1 l ^ α λ ) f .
p c = ( 1 l ^ α λ ) ( 1 ρ λ ) ,
p c = ( 1 l ^ α λ ) ( 1 f ) ( 1 ρ λ ) .
P ( n ) = g ,   n = 1 ,
= ( 1 g ) p q n 2 n 2 ,
n = 1 P ( n ) = g + n = 2 ( 1 g ) p q n 2 = 1 .
  g = a + b + c = 1 ρ λ ( 1 l ^ α λ ) ,   n = 1 ,
p = a + b + c = 1 ( 1 l ^ α λ ) ρ λ ( 1 f ) ,   n 2 .
A atmos = a + ( 1 a c ) a a + b + c = l ^ α λ [ 1 ρ λ f ( 1 l ^ α λ ) ] 1 ( 1 l ^ α λ ) ρ λ ( 1 f ) .
A refl = c + ( 1 a c ) c a + b + c = ( 1 ρ λ ) ( 1 l ^ α λ ) 1 ( 1 l ^ α λ ) ρ λ ( 1 f ) ,
A port = 0 + ( 1 a c ) b a + b + c = ρ λ f ( 1 l ^ α λ ) 2 1 ( 1 l ^ α λ ) ρ λ ( 1 f ) .
γ 1 = γ 0 ( 1 l ^ α λ ) ρ λ
γ n = ρ λ ( 1 l ^ α λ ) ( 1 f ) γ n 1 ,
γ ˜ = n = 1 γ n = ρ λ ( 1 l ^ α λ ) 1 ρ λ ( 1 f ) ( 1 l ^ α λ ) γ 0 ,
A atmos = l ^ α λ ( γ 0 + γ ˜ ) = l ^ α λ [ 1 ρ λ f ( 1 l ^ α λ ) ] 1 ( 1 l ^ α λ ) ρ λ ( 1 f ) ,
A refl = 1 ρ λ ρ λ γ ˜ = ( 1 ρ λ ) ( 1 l ^ α λ ) 1 ( 1 l ^ α λ ) ρ λ ( 1 f ) ,
A port = f ( 1 l ^ α λ ) γ ˜ = ρ λ f ( 1 l ^ α λ ) 2 1 ( 1 l ^ α λ ) ρ λ ( 1 f ) .
n = g + n = 2 n ( 1 g ) p q n 2 = 1 + ( p g ) p ,
n 2 = g + n = 2 n 2 ( 1 g ) p q n 2 = 1 + ( 1 g ) ( 2 + p ) p 2 ,
σ n 2 = n 2 n 2 = 1 p g 2 + p g p 2 .
d = l ^ n ,
d = l ^ 1 + p g p = 2 D 3 p ( 1 + p g ) .
σ d 2 = n 2 σ l 2 + l ^ 2 σ n 2 .
σ d 2 = n σ l 2 + l ^ 2 σ n 2 .
σ d 2 = D 2 18 p 2 [ ( p g ) ( 8 g + p ) 7 p + 8 ] .
d = 1 α λ ,
σ d 2 = 1 α λ 2 .
σ d 2 = n σ l 2 + l ^ 2 σ n 2 = 1 p σ l 2 + l ^ 2 1 p p 2 = 1 α λ 2 ( 1 l ^ α λ ) + 1 l ^ α λ σ l 2 .
n a = 1 f .
n b = 1 ( 1 ρ λ ) ( 1 f ) .
n c = 1 l ^ α λ ρ λ ( 1 f ) .
A atmos = l ^ α λ [ 1 ρ λ f ( 1 l ^ α λ ) ] 1 ( 1 l ^ α λ ) ρ λ ( 1 f ) .
A refl = ( 1 ρ λ ) ( 1 l ^ α λ ) 1 ( 1 l ^ α λ ) ρ λ ( 1 f ) .
A port = ρ λ f ( 1 l ^ α λ ) 2 1 ( 1 l ^ α λ ) ρ λ ( 1 f ) .
z = | / d ϕ ( ω ) d ω / i | ω = 0 ,
z 2 = | d 2 ϕ ( ω ) d ω 2 | ω = 0 .
ϕ ( ω ) = FT { P n ( z ) } ,
= f z = ( 1 f ) z e i w x δ ( x x z ) d x ,
= f z = 0 ( 1 f ) z e i w z ,
= f z = 0 [ ( 1 f ) e i w ] z ,
z = 0 [ ( 1 f ) e i w ] z 1 / [ 1 ( 1 f ) e i w ] for [ ( 1 f ) e i w ] < 1
ϕ ( ω ) = f / [ 1 ( 1 f ) e i w ] .
ϕ ( ω ) = f i ( e i w f e i w ) ( 1 + f e i w e i ϖ ) 2 .
z = | / d ϕ ( ω ) d ω / i | ω = 0 ,
= 1 f f = 1 f 1 .
n = z + 1 ,
= z + 1 ,
= 1 f .
z 2 = 2 ( 1 f ) 2 f 2 + ( 1 f ) f ,
= ( n 1 ) 2 ,
= n 2 + f 2 f .
n 2 = 2 ( 1 f ) 2 f 2 + ( 1 f ) f f 2 f ,
= 2 ( 1 f ) 2 f 2 + ( 3 2 f ) f ,
σ n 2 = n 2 n 2 ,
= 1 f f 2 .

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