Abstract

The effect of large, flat diffuse samples on integrating sphere measurements is calculated exactly, based on an extension of the integral equation formalism of Jacquez and Kuppenheim [ J. Opt. Soc. Am. 45, 460 ( 1955)]. The theory is further extended to include the effect of a limited detector field of view, and examples for different sphere configurations are provided. A variation of Taylor’s method for absolute reflectance measurements is presented that has the advantage of an improved signal-to-noise ratio.

© 1988 Optical Society of America

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References

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  1. J. A. Jacquez, H. F. Kuppenheim, “Theory of the integrating sphere,” J. Opt. Soc. Am. 45, 460–470 (1955).
    [CrossRef]
  2. H. W. Wyld, Mathematical Methods for Physics (Benjamin, Reading, Mass., 1976).
  3. D. G. Goebel, “Generalized integrating-sphere theory,” Appl. Opt. 6, 125–128 (1967).
    [CrossRef] [PubMed]
  4. K. Gindele, M. Köhl, M. Mast, “Methods and problems of photometric integration for spectral reflectance measurements,” presented at the Fifth International Solar Forum, Berlin (preprint, 1984).
  5. A. H. Taylor, “The measurement of diffuse reflection factors and a new absolute reflectometer,” J. Opt. Soc. Am. 4, 9–23 (1920).
    [CrossRef]
  6. A. Reule, “Absolute reflectance measurements using integrating spheres—goniometric corrections in the evaluation of results of different methods,” Optik 49, 499–504 (1978).
  7. R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1972).
  8. P. W. Kruse, “The photon detection process,” in Optical and Infrared Detectors, 2nd ed., R. J. Keyes, ed. (Springer-Verlag, Berlin, 1980), pp. 5–69.
    [CrossRef]
  9. H. L. Tardy, “Flux concentrators in integrating sphere experiments: potential for increased detector signal,” Appl. Opt. 24, 3914–3916 (1985).
    [CrossRef] [PubMed]

1985

1978

A. Reule, “Absolute reflectance measurements using integrating spheres—goniometric corrections in the evaluation of results of different methods,” Optik 49, 499–504 (1978).

1967

1955

1920

Gindele, K.

K. Gindele, M. Köhl, M. Mast, “Methods and problems of photometric integration for spectral reflectance measurements,” presented at the Fifth International Solar Forum, Berlin (preprint, 1984).

Goebel, D. G.

Howell, J. R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1972).

Jacquez, J. A.

Köhl, M.

K. Gindele, M. Köhl, M. Mast, “Methods and problems of photometric integration for spectral reflectance measurements,” presented at the Fifth International Solar Forum, Berlin (preprint, 1984).

Kruse, P. W.

P. W. Kruse, “The photon detection process,” in Optical and Infrared Detectors, 2nd ed., R. J. Keyes, ed. (Springer-Verlag, Berlin, 1980), pp. 5–69.
[CrossRef]

Kuppenheim, H. F.

Mast, M.

K. Gindele, M. Köhl, M. Mast, “Methods and problems of photometric integration for spectral reflectance measurements,” presented at the Fifth International Solar Forum, Berlin (preprint, 1984).

Reule, A.

A. Reule, “Absolute reflectance measurements using integrating spheres—goniometric corrections in the evaluation of results of different methods,” Optik 49, 499–504 (1978).

Siegel, R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1972).

Tardy, H. L.

Taylor, A. H.

Wyld, H. W.

H. W. Wyld, Mathematical Methods for Physics (Benjamin, Reading, Mass., 1976).

Appl. Opt.

J. Opt. Soc. Am.

Optik

A. Reule, “Absolute reflectance measurements using integrating spheres—goniometric corrections in the evaluation of results of different methods,” Optik 49, 499–504 (1978).

Other

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1972).

P. W. Kruse, “The photon detection process,” in Optical and Infrared Detectors, 2nd ed., R. J. Keyes, ed. (Springer-Verlag, Berlin, 1980), pp. 5–69.
[CrossRef]

H. W. Wyld, Mathematical Methods for Physics (Benjamin, Reading, Mass., 1976).

K. Gindele, M. Köhl, M. Mast, “Methods and problems of photometric integration for spectral reflectance measurements,” presented at the Fifth International Solar Forum, Berlin (preprint, 1984).

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Figures (3)

Fig. 1
Fig. 1

Geometric relationships discussed in the text.

Fig. 2
Fig. 2

Baffle and FOV configuration for the standard Taylor method. For the geometry shown, α ≈ 11°.

Fig. 3
Fig. 3

Baffle and FOV configuration for the Taylor method modified to produce a higher S/N ratio, as described in the text. Note the much larger FOV and the smaller baffle in this figure compared with that in Fig. 2. With α ≈ 45° as shown, the modified geometry yields a 14-fold increase in signal and a factor-of-3.7 increase in the S/N ratio for a background fluctuation-limited detector.

Equations (28)

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H ( Ω ) = s [ H 0 ( Ω ) + H ( Ω ) ] ρ ( Ω ) G ( Ω , Ω ) d A ,
G ( Ω , Ω ) = ( χ n ̂ ) ( χ n ̂ ) / ( π χ 4 ) ,
H ( Ω ) = f ( Ω ) + s R ( Ω , Ω ) f ( Ω ) d A ,
f ( Ω ) = s H 0 ( Ω ) ρ ( Ω ) G ( Ω , Ω ) d A .
R ( Ω , Ω ) = m = 0 ( 1 ) m D m ( Ω , Ω ) / m ! / m = 0 ( 1 ) m D m / m ! ,
D 0 = 1 , D 0 ( Ω , Ω ) = ρ ( Ω ) G ( Ω , Ω ) ,
D m = s D m 1 ( Ω , Ω ) d A ,
D m ( Ω , Ω ) = D m ρ ( Ω ) G ( Ω , Ω ) m × s D m 1 ( Ω , Ω ) ρ ( Ω ) G ( Ω , Ω ) d A .
H ( Ω on detector ) = f υ ( Ω ) + υ R ( Ω , Ω ) f ( Ω ) d A ,
f υ ( Ω ) = υ H 0 ( Ω ) ρ ( Ω ) G ( Ω , Ω ) d A .
G ( Ω , Ω ) = { 1 / A , Ω , Ω not on A f 0 , Ω , Ω both on A f A s / ( A f A ) , Ω on A f , Ω not on A f g ( Ω , Ω ) , Ω not on A f , Ω on A f
g ( Ω , Ω ) = 4 ( cos θ 0 cos θ ) { 1 cos θ 0 [ cos θ + sin θ tan θ cos ( ϕ ϕ ) ] } A { 1 2 cos θ 0 [ cos θ + sin θ tan θ cos ( ϕ ϕ ) ] + cos 2 θ 0 sec 2 θ } 2 .
g ( Ω ) = 4 ( cos θ 0 cos θ ) ( 1 cos θ 0 cos θ ) A ( 1 2 cos θ 0 cos θ + cos 2 θ 0 ) 2 .
ρ ( Ω ) = { 0 , Ω on A p ρ , Ω on A w , ρ s , Ω on A f
Φ f = Φ 0 ρ s A f [ 1 ρ A w / A ρ ρ s A s w d A f d A g ( Ω , Ω ) / ( A f A ) ] × [ ( 1 ρ A w / A ) d d A f d A g ( Ω , Ω ) + ρ A d w d A f d A g ( Ω , Ω ) / A ] .
y d A x d A g ( Ω , Ω ) = A x F x , y ,
Φ f = Φ 0 ρ s [ ( 1 ρ A w / A ) A f F f , d + ρ ( A d / A ) A f F f , w ] A f ( 1 ρ A w / A ρ ρ s A s F f , w / A ) .
F x , s = F x , f .
F f , x = A x A A s A f .
Φ f = Φ 0 ρ s ( A d / A ) ( A s / A f ) { 1 ρ A w / A [ 1 + ρ s A s 2 / ( A f A ) ] }
Φ r = Φ 0 ρ A d / A [ 1 + ρ s A s 2 / ( A f A ) ] { 1 ρ A w / A [ 1 + ρ s A s 2 / ( A f A ) ] }
Φ f / Φ r = ρ s A s / A f ρ [ 1 + ρ s A s 2 / ( A f A ) ]
Φ f = Φ 0 ρ s A d A s / A f { x + ρ / A ( A υ x A w ) [ 1 + ρ s A s 2 / ( A f A ) ] } A { 1 ρ A w / A [ 1 + ρ s A s 2 / ( A f A ) ] }
Φ r = Φ 0 ρ A d [ 1 + ρ s A s 2 / ( A f A ) ] [ x + ρ / A ( A υ x A w ) ] A { 1 ρ A w / A [ 1 + ρ s A s 2 / ( A f A ) ] }
Φ f / Φ r = ρ s A s / A f { x + ρ / A ( A υ x A w ) [ 1 + ρ s A s 2 / ( A f A ) ] } ρ [ 1 + ρ s A s 2 / ( A f A ) ] [ x + ρ / A ( A υ x A w ) ]
Φ f / Φ r = ρ s ρ A s A f
Φ f / Φ r = ρ s ( A s / A f ) ( A υ / A ) x + ρ / A ( A υ x A w )
Φ f / Φ r = ρ s A w A A s A f

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