Abstract

A simple matrix technique is presented for modeling integrating-sphere performance. The method is applicable to any sphere configuration, including those with flat areas, specular samples, and baffles, and is especially effective when used in computer simulations of sphere irradiance. The formalism can accommodate the angular sensitivity of any detector or the bidirectional-reflectance distribution function of any sample. Examples of simple analytical solutions are presented, and computer simulation is demonstrated with calculations of the irradiance inhomogeneities caused by underfilling a flat sample. In particular, the simulation shows that, when the input beam does not completely fill a flat sample, the sample is surrounded by a band of reduced irradiance. Outside this dark band, the irradiance is increased slightly. The width of the dark band, but not its depth, increases as the beam size decreases relative to the sample size. The depth depends on sample size and reflectance. Outside the dark-band region, the irradiance shifts due to sample underfilling are much smaller than the easily avoidable, first-order errors caused by neglecting the flat-sample effects.

© 1991 Optical Society of America

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References

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  1. B. J. Hisdal, “Reflectance of perfect diffuse and specular samples in the integrating sphere,” J. Opt. Soc. Am. 55, 1122–1128 (1965).Hisdal’s Eq. (2.2) is an equation for the radiance and so differs slightly from Eq. (12) in the present paper.
    [CrossRef]
  2. B. J. Hisdal, “Reflectance of nonperfect surfaces in the integrating sphere,” J. Opt. Soc. Am. 55, 1255–1260 (1965).
    [CrossRef]
  3. D. G. Goebel, “Generalized integrating-sphere theory,” Appl. Opt. 6, 125–128 (1967).
    [CrossRef] [PubMed]
  4. G. J. Kneissl, J. C. Richmond, “A laser-source integrating sphere reflectometer,” Natl. Bur. Stand. (U.S.) Tech. Note439 (U.S. Government Printing Office, Washington, D.C., 1968).
  5. M.W. Finkel, “Integrating sphere theory,” Opt. Commun. 2, 25–28 (1970).Note: the results of this paper are incorrect. The flat-to-flat configuration factor is correctly evaluated as zero, but the flat-to-sphere configuration factors lack the prefactor as/asf.
    [CrossRef]
  6. W. Budde, C. X. Dodd, “Absolute reflectance measurements in the D/0° geometry,” Farbe 19, 94–102 (1970).
  7. L. M. Hanssen, “Effects of restricting detector field of view when using integrating spheres,” Appl. Opt. 28, 2097–2103 (1989).
    [CrossRef] [PubMed]
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    [CrossRef]
  12. M. N. Özisik, Radiative Transfer and Interactions with Conduction and Convection (Wiley, New York, 1973).
  13. R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1972).
  14. M. W. Wildin, “Error analysis of integrating sphere with center-mounted sample,” in Jet Propulsion Laboratory Space Programs Summary 37–49. Vol. III. Supporting Research and Advanced Development for the Period December 1, 1967 to January 30, 1968, Tech. Rep. NASA-CR-94676 (NASAPasadena, Calif., 1968), pp. 153–159.
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  16. W H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

1989 (1)

1988 (1)

1982 (1)

1981 (1)

1970 (2)

M.W. Finkel, “Integrating sphere theory,” Opt. Commun. 2, 25–28 (1970).Note: the results of this paper are incorrect. The flat-to-flat configuration factor is correctly evaluated as zero, but the flat-to-sphere configuration factors lack the prefactor as/asf.
[CrossRef]

W. Budde, C. X. Dodd, “Absolute reflectance measurements in the D/0° geometry,” Farbe 19, 94–102 (1970).

1967 (1)

1965 (2)

1955 (1)

1920 (1)

Budde, W.

W. Budde, C. X. Dodd, “Absolute reflectance measurements in the D/0° geometry,” Farbe 19, 94–102 (1970).

Christie, E. A.

Dodd, C. X.

W. Budde, C. X. Dodd, “Absolute reflectance measurements in the D/0° geometry,” Farbe 19, 94–102 (1970).

Finkel, M.W.

M.W. Finkel, “Integrating sphere theory,” Opt. Commun. 2, 25–28 (1970).Note: the results of this paper are incorrect. The flat-to-flat configuration factor is correctly evaluated as zero, but the flat-to-sphere configuration factors lack the prefactor as/asf.
[CrossRef]

Flannery, B. P.

W H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

Goebel, D. G.

Hanssen, L. M.

Hisdal, B. J.

Howell, J. R.

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

Hsia, J. J.

Jacquez, J. A.

Kneissl, G. J.

G. J. Kneissl, J. C. Richmond, “A laser-source integrating sphere reflectometer,” Natl. Bur. Stand. (U.S.) Tech. Note439 (U.S. Government Printing Office, Washington, D.C., 1968).

Kuppenheim, H. F.

Özisik, M. N.

M. N. Özisik, Radiative Transfer and Interactions with Conduction and Convection (Wiley, New York, 1973).

Peck, M. K.

Press, W H.

W H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

Richmond, J. C.

G. J. Kneissl, J. C. Richmond, “A laser-source integrating sphere reflectometer,” Natl. Bur. Stand. (U.S.) Tech. Note439 (U.S. Government Printing Office, Washington, D.C., 1968).

Siegel, R.

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

Symons, J. G.

Tardy, H. L.

Taylor, A. H.

Teukolsky, S. A.

W H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

Vetterling, W. T.

W H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

Weidner, V. R.

Wildin, M. W.

M. W. Wildin, “Error analysis of integrating sphere with center-mounted sample,” in Jet Propulsion Laboratory Space Programs Summary 37–49. Vol. III. Supporting Research and Advanced Development for the Period December 1, 1967 to January 30, 1968, Tech. Rep. NASA-CR-94676 (NASAPasadena, Calif., 1968), pp. 153–159.

Appl. Opt. (3)

Farbe (1)

W. Budde, C. X. Dodd, “Absolute reflectance measurements in the D/0° geometry,” Farbe 19, 94–102 (1970).

J. Opt. Soc. Am. (5)

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

Opt. Commun. (1)

M.W. Finkel, “Integrating sphere theory,” Opt. Commun. 2, 25–28 (1970).Note: the results of this paper are incorrect. The flat-to-flat configuration factor is correctly evaluated as zero, but the flat-to-sphere configuration factors lack the prefactor as/asf.
[CrossRef]

Other (5)

G. J. Kneissl, J. C. Richmond, “A laser-source integrating sphere reflectometer,” Natl. Bur. Stand. (U.S.) Tech. Note439 (U.S. Government Printing Office, Washington, D.C., 1968).

M. N. Özisik, Radiative Transfer and Interactions with Conduction and Convection (Wiley, New York, 1973).

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

M. W. Wildin, “Error analysis of integrating sphere with center-mounted sample,” in Jet Propulsion Laboratory Space Programs Summary 37–49. Vol. III. Supporting Research and Advanced Development for the Period December 1, 1967 to January 30, 1968, Tech. Rep. NASA-CR-94676 (NASAPasadena, Calif., 1968), pp. 153–159.

W H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

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Figures (7)

Fig. 1
Fig. 1

Integrating-sphere geometry and definitions. For clarity, the entrance and reference ports are not depicted.

Fig. 2
Fig. 2

Axisymmetric geometry for computer simulation. θs, θe, and θ1 are the boundary latitudes of the sample, entrance port, and illuminated area, respectively. Latitudes are defined in the usual geographic sense, with −90° at the center of the entrance port and 90° at the center of the sample. The dashed lines indicate a typical zone, with center latitude at θj.

Fig. 3
Fig. 3

Distributed irradiance (HHo) versus latitude for an axisymmetric sphere with unit input flux (Ho = 1/aSf), ρs = 0.5, ρw = 0.96, ae = 0.01, and 0.0125 ≤ as ≤ 0.1. The sample area is fully illuminated. H was calculated from expressions (B1)(B7) with 200 zones. Note that the irradiance is expressed in units of Φo/A, where Φo is the input flux and A is total sphere area.

Fig. 4
Fig. 4

Relative irradiance error versus latitude for the case shown in Fig. 3 (Ho = 1/asf, ρs = 0.5, ρw = 0.96, ae = 0.01, and 0.0125 ≤ as ≤ 0.1). H(Theory) is the result of an analytical model [Eqs. (A1)(A5)] with three zones: entrance, wall, and sample.

Fig. 5
Fig. 5

Relative irradiance error versus latitude for an underfilled sample. The sphere parameters are the same as Figs. 3 and 4 (Ho = 1/asf, ρs = 0.5, ρw = 0.96, ae = 0.01, and 0.0125 ≤ as ≤ 0.1), except that the input irradiance covers only 90% of the sample plane area. The analytical model consists of four zones: entrance, wall, unfilled sample area, and illuminated area.

Fig. 6
Fig. 6

Relative irradiance error versus latitude, showing the effect of sample reflectance. ρw = 0.96, ae = 0.01, as = 0.1, and the input beam covers 50% of the sample plane area. Ho = 2/asf (unit input flux).

Fig. 7
Fig. 7

Relative irradiance error versus latitude, showing the effect of input beam size. ρs = 0.5, ρw = 0.96, ae = 0.01, as = 0.04, and Ho = 1/aif.

Equations (47)

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a e = A e / A .
a w = 1 i a i ,
a FOV = i x i a i ,
a s = { 1 cos [ sin 1 ( c / b ) ] } / 2 .
H ( Ω ) = H o ( Ω ) + s H ( Ω ) ρ ( Ω ) G ( Ω , Ω ) d A ,
G ( Ω , Ω ) = ( χ n ) ( χ n ) π χ 4 ,
H ( Ω ) = H o ( Ω ) + j H j ρ j j G ( Ω , Ω ) d A j .
H i A i = H o , i A i + j H j ρ j i j G ( Ω , Ω ) d A j d A i .
i j G ( Ω , Ω ) d A j d A i = A j F j i = A i F i j .
H i = H o , i + j ρ j F i j H j .
H = H o + MH .
M i j = ρ j F i j .
H = ( I M ) 1 H o ,
Φ d = i H i ρ i x i A i F i d .
Φ d = H V ,
V i = ρ i x i A i F i d .
M = [ ρ s a s ρ w a w ρ s a s ρ w a w ] .
( H s , H w ) = [ H o / ( 1 ρ ) ] ( 1 ρ w a w , ρ s a s ) ,
ρ = ρ w a w + ρ s a s
1 ρ = | I M | .
Φ d = [ Φ o ρ s α d / ( 1 ρ ) ] [ x s + ρ w a w ( x w x s ) ] .
ρ s = R ρ r / [ 1 + ρ r a ( 1 R ) / ( 1 a ) ] .
M = [ 0 ( a s / a s f ) ( a w ρ w + a r ρ r ) a s ρ s ( a w ρ w + a r ρ r ) ] .
a s f = a s ( 1 a s ) .
a s = sin 2 ( α s / 2 ) .
ρ = ( ρ r a r + ρ w a w ) [ 1 + ρ s tan 2 ( α s / 2 ) ] ,
( H s , H w ) = [ H o / ( 1 ρ ) ] ( 1 ρ w a w ρ r a r , ρ s a s ) .
H T = [ H o / ( 1 ρ ) ] { ρ r a r sec 2 ( α s / 2 ) , 1 ρ w a w [ 1 + ρ s tan 2 ( α s / 2 ) ] , ρ r a r [ 1 + ρ s tan 2 ( α s / 2 ) ] } .
M = [ 0 ρ r sec 2 ( α s / 2 ) sin 2 ( α r / 2 ) ρ w a w sec 2 ( α s / 2 ) ρ s sec 2 ( α r / 2 ) sin 2 ( α s / 2 ) 0 ρ w a w sec 2 ( α r / 2 ) ρ s sin 2 ( α s / 2 ) ρ r sin 2 ( α r / 2 ) ρ w a w ] ,
ρ = ρ w a w [ 1 + ρ s tan 2 ( α s / 2 ) + ρ r tan 2 ( α r / 2 ) ] + ρ s ρ r tan 2 ( α s / 2 ) tan 2 ( α r / 2 ) ( 1 + ρ w a w ) .
H = H o / ( 1 ρ ) [ 1 ρ w a w [ 1 + ρ r tan 2 ( α r / 2 ) ] ρ s sin 2 ( α s / 2 ( sec 2 ( α r / 2 ) ρ s sin 2 ( α s / 2 ) [ 1 + ρ r tan 2 ( α r / 2 ) ] ] .
H ( Ω i ) H o ( Ω i ) + j H ( Ω j ) ρ ( Ω j ) G ( Ω i , Ω j ) λ j ,
( ρ i λ i ) 1 / 2 H i = ( ρ i λ i ) 1 / 2 H o , i + j ( ρ i ρ j λ i λ j ) 1 / 2 G i j ( ρ j λ i ) 1 / 2 H j ,
H = H o + M H ,
H i = ( ρ i λ i ) 1 / 2 H i ,
M i j = M i j = ( ρ i ρ j λ i λ j ) 1 / 2 G i j .
λ i = a i = { ( sin θ i sin θ i 1 ) / 2 , θ s θ i 1 90 ° sin 2 θ s ( cot 2 θ i 1 cost 2 θ i ) / 4 , 90 ° θ i θ s .
G i j = 0 2 π G ( Ω , Ω ) b 2 d ϕ = ( sin θ s sin θ j ) [ 2 ( 1 sin θ s sin θ j ) u υ 2 ] ( u 2 υ 2 ) 3 / 2 ,
u = 1 2 sin θ s sin θ j + sin 2 θ s csc 2 θ i ,
υ 2 = 4 sin 2 θ s cos 2 θ j cot 2 θ i ,
h = sin θ s sin θ j ( axial distance between zones ) ,
r i = sin θ s cot θ i ( midpoint radius of ring i ) ,
r j = cos θ j ( midpoint radius of ring j ) ,
c = cos θ s ( radius of sample ) .
u = h 2 + r i 2 + r j 2 ,
υ 2 = 4 r i 2 r j 2 ,
G i j = h [ ( h 2 + r j 2 + c 2 ) u υ 2 ] ( u 2 υ 2 ) 3 / 2 .

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