Abstract

The repartition of the single reflections in an integrating sphere is discussed using a model sphere where wall and screen are divided into twenty-six zones. The results lead to a simplified model with three zones and three partial luminous fluxes. In this simplified model sphere, the net hold is defined as that part of the volume of the sphere that is hidden by the screen when one looks into the sphere through the sphere window. The optimal place and the optimal size of the screen are determined with assumption of a constant net hold. Then the screen is mainly given by the ratio between the diameter of the net hold and the observation window. Further influences will only slightly affect place and size of the screen.

© 1971 Optical Society of America

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References

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  1. R. Ulbricht, ETZ 21, 595 (1900).
  2. G. Krenzke, “Die Optimierung der Messanordnung in runden und eckigen Hohlräumen zur Lichtstrombestimmung ausgedehnter Lichtquellen,” Dissertation, Dresden University (1966).
  3. H. Korte, M. Schmidt, Lichttechnik 6, 88 (1954).
  4. K. Fränz, Licht 4, 17 (1934).
  5. J. A. Jaquez, H. F. Kuppenheim, J. Opt. Soc. Am. 45, 460(1955).
    [CrossRef]
  6. P. F. O’Brien, J. Opt. Soc. Am. 45, 419 (1955).
    [CrossRef]
  7. P. F. O’Brien, J. Opt. Soc. Am. 46, 343 (1956).
    [CrossRef]
  8. R. Ulbricht, ETZ 26, 512 (1905).
  9. R. Ulbricht, Das Kugelphotometer (Verlag R. Oldenbourg, Munchen und Berlin, 1920).
  10. R. Ulbricht, ETZ 28, 777 (1907).
  11. E. B. Rosa, A. H. Taylor, Sci. Papers Nat. Bur. Stand. U.S. 18, 281 (1922).
  12. E. B. Rosa, A. H. Taylor, Trans. Illum. Eng. Soc. 11, 453 (1916).
  13. F. Rotter, Lichttechnik 21, 44A (1969).

1969 (1)

F. Rotter, Lichttechnik 21, 44A (1969).

1956 (1)

1955 (2)

1954 (1)

H. Korte, M. Schmidt, Lichttechnik 6, 88 (1954).

1934 (1)

K. Fränz, Licht 4, 17 (1934).

1922 (1)

E. B. Rosa, A. H. Taylor, Sci. Papers Nat. Bur. Stand. U.S. 18, 281 (1922).

1916 (1)

E. B. Rosa, A. H. Taylor, Trans. Illum. Eng. Soc. 11, 453 (1916).

1907 (1)

R. Ulbricht, ETZ 28, 777 (1907).

1905 (1)

R. Ulbricht, ETZ 26, 512 (1905).

1900 (1)

R. Ulbricht, ETZ 21, 595 (1900).

Fränz, K.

K. Fränz, Licht 4, 17 (1934).

Jaquez, J. A.

Korte, H.

H. Korte, M. Schmidt, Lichttechnik 6, 88 (1954).

Krenzke, G.

G. Krenzke, “Die Optimierung der Messanordnung in runden und eckigen Hohlräumen zur Lichtstrombestimmung ausgedehnter Lichtquellen,” Dissertation, Dresden University (1966).

Kuppenheim, H. F.

O’Brien, P. F.

Rosa, E. B.

E. B. Rosa, A. H. Taylor, Sci. Papers Nat. Bur. Stand. U.S. 18, 281 (1922).

E. B. Rosa, A. H. Taylor, Trans. Illum. Eng. Soc. 11, 453 (1916).

Rotter, F.

F. Rotter, Lichttechnik 21, 44A (1969).

Schmidt, M.

H. Korte, M. Schmidt, Lichttechnik 6, 88 (1954).

Taylor, A. H.

E. B. Rosa, A. H. Taylor, Sci. Papers Nat. Bur. Stand. U.S. 18, 281 (1922).

E. B. Rosa, A. H. Taylor, Trans. Illum. Eng. Soc. 11, 453 (1916).

Ulbricht, R.

R. Ulbricht, ETZ 28, 777 (1907).

R. Ulbricht, ETZ 26, 512 (1905).

R. Ulbricht, ETZ 21, 595 (1900).

R. Ulbricht, Das Kugelphotometer (Verlag R. Oldenbourg, Munchen und Berlin, 1920).

ETZ (3)

R. Ulbricht, ETZ 21, 595 (1900).

R. Ulbricht, ETZ 26, 512 (1905).

R. Ulbricht, ETZ 28, 777 (1907).

J. Opt. Soc. Am. (3)

Licht (1)

K. Fränz, Licht 4, 17 (1934).

Lichttechnik (2)

H. Korte, M. Schmidt, Lichttechnik 6, 88 (1954).

F. Rotter, Lichttechnik 21, 44A (1969).

Sci. Papers Nat. Bur. Stand. U.S. (1)

E. B. Rosa, A. H. Taylor, Sci. Papers Nat. Bur. Stand. U.S. 18, 281 (1922).

Trans. Illum. Eng. Soc. (1)

E. B. Rosa, A. H. Taylor, Trans. Illum. Eng. Soc. 11, 453 (1916).

Other (2)

G. Krenzke, “Die Optimierung der Messanordnung in runden und eckigen Hohlräumen zur Lichtstrombestimmung ausgedehnter Lichtquellen,” Dissertation, Dresden University (1966).

R. Ulbricht, Das Kugelphotometer (Verlag R. Oldenbourg, Munchen und Berlin, 1920).

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Figures (10)

Fig. 1
Fig. 1

Integrating sphere. L, lamp under test; F, observation window; S, internal screen; n, diameter of the net hold (the net hold is that part of the volume of the sphere which is hidden by the internal screen when one looks into the sphere through the observation window); ϕs, luminous flux which is received by the screen; ωs, solid angle of ϕs; ϕw, luminous flux which is received by that part of the wall which is not visible from the center of the window; ωw, solid angle of ϕw.

Fig. 2
Fig. 2

Sphere with zones. The sphere wall is divided into twenty zones numbered from 1 to 20. Zone 1 contains the observation window. The lamp in the center of the sphere emits the twenty partial luminous fluxes ϕ1, ϕ2,… ϕ20 into the corresponding zones of the wall. The internal screen S is also divided into zones. To show these zones, quarters of the screen are attached to the cross section of the sphere. The window side F of the screen contains zones c, b, and a, the lamp side L zones A, B, and C. The hatched areas are the cross sections of the partial luminous fluxes ϕ1, ϕ3, ϕ7, ϕ14, and ϕ20. The flux ϕ1 cannot reach the wall, for it is received by the screen.

Figs. 3–7
Figs. 3–7

There is at the bottom of each diagram a simplified cross section of the sphere in Fig. 2. The hatched areas indicate the partial luminous fluxes ϕ1, ϕ3, ϕ7, ϕ14 and ϕ20, respectively. Lines R(1), R(2), R(3) and R(4) show the values of the corresponding R factors. These R factors can be understood as relative illuminances within the different zones of wall and screen, produced by the first four reflections of the above-mentioned fluxes. The zones in question are given on the abscissas. The ordinate scales of the R factors are displaced as indicated.

Figs. 8–10
Figs. 8–10

There is at the bottom of each diagram a simplified cross section of the sphere in Fig. 2, but without the quarters of the screen. The diagrams show only relative illuminances on the window. Thus, the abscissas give the zone where the illuminating partial luminous fluxes are directed. The ordinate scales of the presented lines are displaced as indicated. Lines R(1) to R(4) in Fig. 8 show the relative illuminances on the window, produced by the first four reflections of the partial luminous fluxes ϕ1 to ϕ20. The lines in Figs. 9 and 10 show the total relative illuminances on the window. These values result from the single reflections, if the reflection factor ρ of the sphere wall is assumed to be 0.8, 0.85, 0.9, and 0.95. The lines refer to the sphere with zones (Fig. 9), or to the simplified sphere (Fig. 10). The hatched areas show the losses of illuminance on the window due to the presence of the screen. The corresponding hatched areas in Figs. 9 and 10 have different shapes but nearly equal sizes.

Fig. 11
Fig. 11

The parameter W gives the loss of illuminance on the window due to the presence of the screen in terms of the illuminance which would occur without any screen. The values of W, calculated once for the sphere with twenty zones (a on figure), once for the simplified sphere (b on figure), agree very closely. The value of W depends on the reflection factor ρ of the wall of the sphere.

Fig. 12
Fig. 12

Optimum positions of the circular screen. The diagram contains a quarter of the cross section of the sphere. The window F is at the left. The big black dots mark the minima of V, defined in Eq. (25), over the corresponding position of the screen. The value of V depends on p, q, and e/r, defined in Eqs. (16)(18). There are always three dots on a line of equal values of p. These lines refer to p = 1, 2, 4, 8, and 16. The dots on each line refer from the top to the bottom to q = ∞, q = 4, and q = 2. The cross sections of the screen, which result for q = 2, are shown by vertical lines rising from the abscissa. The graph demonstrates that the optimum screen position is mainly given by the value of p, while the value of q is of less influence.

Fig. 13
Fig. 13

Shape factor g of the net hold. The cross section of the net hold is described by its area N and by its shape factor g. A circular cross section has g = 1. The graph shows cross sections of equal area but of different values of g.

Fig. 14
Fig. 14

Optimum positions of the oblong screen. The minimum values of V are given in function of p and g with assumption of q = ∞. The dots on the lines of equal values of p refer to g = 3, g = 2, and g = 1. For further explanation, see Fig. 12. Consider that the optimum screen position is mainly given by the value of p, while the value of g is of less influence.

Fig. 15
Fig. 15

Optimum screen positions for big lamps (ρ = 0.85). The lamp is assumed to be a blackbody which fills the net hold completely. The graph corresponds widely to Figs. 12 and 14, but gives values of W·q2 instead of V. All values refer to the reflection factor ρ = 0.85 of the wall of the sphere. The dots on the lines of equal values of p correspond to q = 2, 4, 8, and ∞, respectively. The optimum screen position is again mainly given by the value of p. The value of q is of less influence.

Fig. 16
Fig. 16

Optimum screen positions for big lamps (q = 4). All values refer to the value of q = 4. For further explanation, see Fig. 15. The dots on the lines of equal values of p correspond to ρ = 0.8, 0.85, 0.9, and 0.95, respectively. The optimum screen position is again mainly given by the value of p. The value of the reflection factor ρ of the wall of the sphere is of less influence.

Equations (41)

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E i ( k ) = ρ k · ( ϕ i / A ) .
E i , j ( k ) = E i ( k ) · R i , j ( k ) .
lim k = R i , j ( k ) = A / ( A + 2 S ) .
E i , j = E i , j ( 0 ) + E i , j ( 1 ) + E i , j ( 2 ) + .
E i , 1 = E i , 1 ( 1 ) + E i , 1 ( 2 ) + E i , 1 ( 3 ) + .
E i , 1 = ( ϕ i / A ) [ ρ R i , 1 ( 1 ) + ρ 2 R i , 1 ( 2 ) + ρ 3 R i , 1 ( 3 ) + ] .
E i , 1 = ( ϕ i / A ) ( ρ + ρ 2 + ρ 3 + ) = [ ρ / ( 1 - ρ ) ] · ( ϕ i / A ) .
R i ( ρ ) = E i , 1 / E i , 1
R i ( ρ ) = ( 1 - ρ ) [ R i , 1 ( 1 ) + ρ R i , 1 ( 2 ) + ρ 2 R i , 1 ( 3 ) + ] .
E o ( k ) = ϕ o · ( ρ k / A ) · R o ( k ) , E s ( k ) = ϕ s · ( ρ k / A ) · R s ( k ) , E w ( k ) = ϕ w · ( ρ k / A ) · R w ( k ) .
R o ( k ) = R s ( k ) = R w ( k ) = A / ( A + 2 S )
E o ( k ) = ϕ o ρ k ( A + 2 S ) - 1 if k 1 , E s ( k ) = ϕ s ρ ( k ) ( A + 2 S ) - 1 if k 2 , E w ( k ) = ϕ w ρ k ( A + 2 S ) - 1 if k 2.
E o ( 0 ) = E s ( 0 ) = E w ( 0 ) = E s ( 1 ) = E w ( 1 ) = 0.
E o = ρ 1 - ρ · ϕ o A · R o with R o = A A + 2 S , E s = ρ 1 - ρ · ϕ s A · R s with R s = ρ · A A + 2 S , E ω = ρ 1 - ρ · ϕ w A · R w with R w = ρ · A A + 2 S .
E = E o + E s + E w = ρ 1 - ρ · ϕ o + ρ ϕ s + ρ ϕ w A + 2 S .
E = ρ α · ϕ A + 2 s ( 1 - W )             with             W = α ϕ s + ϕ w ϕ .
p = ( N / F ) 1 2 = n / f ,
q = ( G / N ) 1 2 = d / n .
z = e / ( r - e )             and             e / r = z / ( z + 1 ) ,
g = c · ( 4 π N ) - 1 2 ,
S = F [ ( z 2 + 2 g p z + p 2 ) / ( 1 + z ) 2 ] = ( G / p 2 q 2 ) · [ ( z 2 + 2 g p z + p 2 ) / ( 1 + z ) 2 ] .
S = f [ ( z + p ) / ( 1 + z ) ] = ( d / p q ) · [ ( z + p ) / ( 1 + z ) ] .
W = α ϕ s + ϕ w ϕ .
W = α [ ( ω s I s + ω w I w ) / 4 π sr · I ] .
I s = I w = I
W = α [ ( ω s + ω w ) / 4 π sr ] · ( I / I ) .
W = α V q 2 · I I             with             V = q 2 ω s + ω w 4 π s r .
V = { ( p + z ) 2 / [ p 2 + ( p + z ) 2 q - 2 ] } + [ ( p + z ) 2 / ( 2 p z ) 2 ] · { 2 / [ 1 + C + ( 1 + C ) 1 2 ] }
z 3 = ( p / 4 ) ( 1 + C 2 z 2 ) 2 · ( 1 + C 2 ) - 2 3 .
V = [ ( p + z ) 2 / p 2 ] [ 1 + ( 1 / 4 z 2 ) ] .
z = ( p / 4 ) 1 3 .
V = [ 1 + 2 g ( z / p ) + ( z 2 / p 2 ) ] [ 1 + ( 1 / 4 z 2 ) ] ,
z 3 = p ( g z + p ) / 4 ( z + g p ) .
ω s = ( π / 4 ) · ( s 2 / e 2 ) = π [ ( p + z ) / p q z ] 2
ω w = π [ s 2 / ( r - e ) 2 ] = 4 π [ ( p + z ) / p q ] 2 .
ω L s = N / e 2 = ( π / q 2 ) · [ 1 + ( 1 / z ) ] 2
ω L w = N / r 2 = π / q 2 .
ϕ loss = α ϕ s + α ϕ w + ( ρ / π ) ϕ s ω L s + ( ρ / π ) ϕ w ω L w .
W = ϕ loss / ϕ = α ( [ ϕ s + ϕ w ) / ϕ ] + ( ρ / π ) · [ ( ϕ s ω L s + ϕ w ω L w ) / ϕ ] .
W = ( α / q 2 ) · V · ( I / I ) + ( ρ / q 4 ) · V · ( I / I ) ,
V = [ ( p + z ) / p ] 2 { 1 + ( 1 / 4 z 2 ) [ 1 + ( 1 / z ) ] 2 } .

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