Abstract

A general equation is developed for the efficiency of an integrating sphere with a nonuniform coating. The only assumptions are that the interior is a perfect sphere and that all areas reflect perfectly diffusely. Three special cases of the general equation are examined for the basic applications of integrating spheres as mixing mechanisms in hemispherical reflectance measurements and in absolute reflectance techniques.

© 1967 Optical Society of America

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References

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  1. J. A. Jacques, H. F. Kuppenheim, J. Opt. Soc. Am. 45, 460 (1955).
    [CrossRef]
  2. B. J. Hisdal, J. Opt. Soc. Am. 55, 1122 (1965).
    [CrossRef]
  3. B. J. Hisdal, J. Opt. Soc. Am. 55, 1255 (1965).
    [CrossRef]
  4. A. H. Taylor, Sci. Papers Bur. Std. S16, 421 (1920).
  5. D. G. Goebel, J. Opt. Soc. Am. 56, 783 (1966).
    [CrossRef]

1966 (1)

1965 (2)

1955 (1)

1920 (1)

A. H. Taylor, Sci. Papers Bur. Std. S16, 421 (1920).

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Figures (3)

Fig. 1
Fig. 1

Geometric relations in a sphere used for calculating the irradiance on dA from da.

Fig. 2
Fig. 2

Geometric relations used for calculating the area of a spherical cap.

Fig. 3
Fig. 3

Graph of Eq. 4; sphere efficiency vs coating reflectance.

Tables (1)

Tables Icon

Table I f = 1 2 { 1 - [ 1 - ( d / D ) 2 ] ½ }

Equations (40)

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H = J o cos 2 θ / 4 R 2 cos 2 θ = J o / 4 R 2 .
f = a / 4 π R 2 .
f = ( ½ ) { 1 - [ 1 - ( r / R ) 2 ] 1 2 }
f r 2 / r 2 + D 2 , ( D = 2 R )
f r 2 / D 2 .
f = ½ [ 1 - ( 1 - 4 l ) ½ ] = ½ { 1 - [ 1 - ½ ( 4 l ) - ( 4 l ) 2 - ¹ / ₁₆ ( 4 l ) 3 ] } = ½ [ 2 l + 2 l 2 + 4 l 3 + ] = l + l 2 + 2 l 3 +
f - l = l = l 2 + 2 l 3 +
l l 2 l = r 2 / D 2 r 2 D 2
d 2 2 D 2 , where d = 2 r ,
d 2 D .
ρ ¯ w = o n ρ i f i + ρ w ( 1 - o n f i ) .
( 1 - o n f i ) ρ o P o
A s - o n a i .
o n ρ i f i ρ o P o
o n a i ,
ρ w ( 1 - o n f i ) ρ o P o
A s - o n a i .
ρ ¯ w = ρ w ( 1 - o n f i ) + o n f i ρ i
o n f i ρ ¯ w ρ o P o
o n a i ,
( 1 - o n f i ) ρ ¯ w ρ o P o
A s - o n ρ i .
o n ρ i f i ρ ¯ w ρ o P o
o n a i ,
ρ w ( 1 - o n f i ) ρ ¯ w ρ o P o
A s - o n a i .
[ ρ w ( 1 - o n f i ) + o n ρ i f i ] ρ ¯ w ρ o P o ,
P i = f i ρ o P o + f i ρ ¯ w ρ o P o + f i ρ ¯ w 2 ρ o P o + f i ρ ¯ w 3 ρ o P o + P i = f i ρ o P o ( 1 + ρ ¯ w + ρ ¯ w 2 + ρ ¯ w 3 + ) P i = f i ρ o P o / 1 - ρ ¯ w .
ρ ¯ w = ρ w ( 1 - o n f i ) + o n f i ρ i , F i = f i ρ o [ 1 - ρ w ( 1 - o n f i ) - o n f i ρ i ] - 1 .
F i = f i ρ w / 1 - ρ w ( 1 - 1 n f j ) .
F i ρ w f i / α w for ρ w 1.0 ,
Δ F i / F i Δ ρ w / ρ w + Δ f i / f i - Δ α w / α w ;
1 - ρ w = α w ,
Δ ρ w = - Δ α w ;
Δ F i / F i 1 / α w × Δ ρ w / ρ w + Δ f i / f i
ρ o ρ w , ρ 1 ρ w and ρ j = 0 for j 0 , 1 F i = f i ρ o [ 1 - ρ w ( 1 - o n f j ) - ρ o f o - ρ 1 f 1 ] - 1
ρ o = ρ w F i = f i ρ w [ 1 - ρ w ( 1 - 1 n f j ) - 1 n f j ρ j ] - 1
F = f m ρ w / 1 - ρ w ( 1 - f e - f m ) .
F = f m ρ w / 1 - ρ w ( 1 - f e - f m - f c ) ,
ρ w = Q - Q [ ( Q - Q ) ( 1 - f e - f m ) + Q f c ] - 1 = [ 1 - f e - f m + f c Q / ( Q - Q ) ] - 1 ,

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