Abstract

Integrating sphere theory is developed for restricted field of view (FOV) detectors using a simple series solution technique. The sphere throughput, sample reflectance, and sphere wall reflectance are calculated. The effects of the sample’s scattering characteristics on sphere measurements are determined. It is shown that although the generalized equations incorporating detector FOV dependence reduce to the hemispherical FOV equations in some cases, in general integrating sphere behavior is altered through restriction of the detector FOV.

© 1989 Optical Society of America

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References

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  1. R. Ulbricht, “Photometer for Mean Spherical Candle-Power,” Electrotech. Zeit. 21, 595–597 (1900).
  2. D. K. Edwards, J. T. Gier, K. E. Nelson, R. D. Roddick, “Integrating Sphere for Imperfectly Diffuse Samples,” J. Opt. Soc. Am. 51, 1279–1288 (1961).
    [CrossRef]
  3. J. A. Jacquez, H. F. Kuppenheim, “Theory of the Integrating Sphere,” J. Opt. Soc. Am. 45, 460–470 (1955).
    [CrossRef]
  4. R. T. Neher, D. K. Edwards, “Far Infrared Reflectometer for Imperfectly Diffuse Specimens,” Appl. Opt. 4, 775–780 (1965).
    [CrossRef]
  5. M. W. Finkel, “Integrating Sphere Theory,” Opt. Commun. 2, 25–28 (1970).
    [CrossRef]
  6. A. H. Taylor, “The Measurement of Diffuse Reflection Factors and a New Absolute Reflectometer,” J. Opt. Soc. Am. 4, 9–23 (1920).
    [CrossRef]
  7. A. H. Taylor, “Errors in Reflectometry,” J. Opt. Soc. Am. 25, 51–56 (1935).
    [CrossRef]
  8. F. J. J. Clarke, J. A. Compton, “Correction Methods for Integrating-Sphere Measurement for Hemispherical Reflectance,” Color Res. Appl. 11, 253–262 (1986).
    [CrossRef]
  9. G. J. Kneissl, J. C. Richmond, “A Laser-Source Integrating Sphere Reflectometer,” Natl. Bur. Stand. US Tech. Note 439 (Feb.1968).
  10. K. A. Snail, K. F. Carr, “Optical Design of an Integrating Sphere-Fourier Transform Spectrophotometer (FTS) Emissometer,” Proc. Soc. Photo-Opt. Instrum. Eng. 643, 75–83 (1985).
  11. L. M. Hanssen, K. A. Snail, “Infrared Diffuse Reflectometer for Spectral, Angular and Temperature Resolved Measurements,” Proc. Soc. Photo-Opt. Instrum. Eng. 807, 148–159 (1987).
  12. K. A. Snail, K. F. Carr, L. M. Hanssen, “Integrating Sphere Measurements Using Restricted Field of View Detectors,” submitted to Applied Optics.
  13. B. J. Hisdal, “Reflectance of Perfect Diffuse Specular Samples in the Integrating Sphere,” J. Opt. Soc. Am. 55, 1122–1128 (1965).
    [CrossRef]
  14. The equation within the text between Eqs. (4) and (5) of Ref. 5 assumes that the flat sample and/or reference conform to the sphere (i.e., are curved to match the sphere wall). This is in contrast to the more accurate approximation of Eq. (3.1) of Ref. 3.
  15. H. L. Tardy, “Flat-Sample and Limited-Field Effects in Integrating Sphere Measurements,” J. Opt. Soc. Am. A 5, 241–245 (1988).
    [CrossRef]
  16. K. A. Snail, L. M. Hanssen, “Integrating Sphere Designs with Isotropic Throughput,” Appl. Opt. 28, 1793 (1959).
    [CrossRef]
  17. W. Budde, “Calibration of Reflectance Standards,” J. Res. Natl. Bur. Stand Sect. A 80, 585 (1976).
    [CrossRef]
  18. Hemispherical total reflectance refers to the hemispherically integrated radiation reflected off a sample which is ratioed to the diffusely incident radiation from the same hemisphere. In general, this is not equal to the directional/hemispherical reflectance, which is the ratio of the hemispherically collected radiation off a sample to radiation incident from a specific direction (often normal). The directional/hemispherical reflectance is the quantity referred to by the ρ terms in this paper. See F. E. Nicodemus, “Reflectance Nomenclature and Directional Reflectance and Emissivity,” Appl. Opt. 9, 1474–1475 (1970); J. J. Hsia, J. C. Richmond, “Bidirectional Reflectometry. Part 1,” J. Res. Natl. Bur. Stand. Sect. A 80, 189–205 (1976); R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1981), pp. 64+.
    [CrossRef] [PubMed]
  19. F. A. Benford, “An Absolute Method for Determining Coefficients of Diffuse Reflection,” Gen. Elect. Rev. 23, 72–76 (1920).
  20. D. G. Goebel, “Generalized Integrating Sphere Theory,” Appl. Opt. 6, 125–128 (1967). Note that his equations as shown are missing many parentheses and a / in the last equation.
    [CrossRef] [PubMed]

1988 (1)

1987 (1)

L. M. Hanssen, K. A. Snail, “Infrared Diffuse Reflectometer for Spectral, Angular and Temperature Resolved Measurements,” Proc. Soc. Photo-Opt. Instrum. Eng. 807, 148–159 (1987).

1986 (1)

F. J. J. Clarke, J. A. Compton, “Correction Methods for Integrating-Sphere Measurement for Hemispherical Reflectance,” Color Res. Appl. 11, 253–262 (1986).
[CrossRef]

1985 (1)

K. A. Snail, K. F. Carr, “Optical Design of an Integrating Sphere-Fourier Transform Spectrophotometer (FTS) Emissometer,” Proc. Soc. Photo-Opt. Instrum. Eng. 643, 75–83 (1985).

1976 (1)

W. Budde, “Calibration of Reflectance Standards,” J. Res. Natl. Bur. Stand Sect. A 80, 585 (1976).
[CrossRef]

1970 (2)

1968 (1)

G. J. Kneissl, J. C. Richmond, “A Laser-Source Integrating Sphere Reflectometer,” Natl. Bur. Stand. US Tech. Note 439 (Feb.1968).

1967 (1)

1965 (2)

1961 (1)

1959 (1)

1955 (1)

1935 (1)

1920 (2)

F. A. Benford, “An Absolute Method for Determining Coefficients of Diffuse Reflection,” Gen. Elect. Rev. 23, 72–76 (1920).

A. H. Taylor, “The Measurement of Diffuse Reflection Factors and a New Absolute Reflectometer,” J. Opt. Soc. Am. 4, 9–23 (1920).
[CrossRef]

1900 (1)

R. Ulbricht, “Photometer for Mean Spherical Candle-Power,” Electrotech. Zeit. 21, 595–597 (1900).

Benford, F. A.

F. A. Benford, “An Absolute Method for Determining Coefficients of Diffuse Reflection,” Gen. Elect. Rev. 23, 72–76 (1920).

Budde, W.

W. Budde, “Calibration of Reflectance Standards,” J. Res. Natl. Bur. Stand Sect. A 80, 585 (1976).
[CrossRef]

Carr, K. F.

K. A. Snail, K. F. Carr, “Optical Design of an Integrating Sphere-Fourier Transform Spectrophotometer (FTS) Emissometer,” Proc. Soc. Photo-Opt. Instrum. Eng. 643, 75–83 (1985).

K. A. Snail, K. F. Carr, L. M. Hanssen, “Integrating Sphere Measurements Using Restricted Field of View Detectors,” submitted to Applied Optics.

Clarke, F. J. J.

F. J. J. Clarke, J. A. Compton, “Correction Methods for Integrating-Sphere Measurement for Hemispherical Reflectance,” Color Res. Appl. 11, 253–262 (1986).
[CrossRef]

Compton, J. A.

F. J. J. Clarke, J. A. Compton, “Correction Methods for Integrating-Sphere Measurement for Hemispherical Reflectance,” Color Res. Appl. 11, 253–262 (1986).
[CrossRef]

Edwards, D. K.

Finkel, M. W.

M. W. Finkel, “Integrating Sphere Theory,” Opt. Commun. 2, 25–28 (1970).
[CrossRef]

Gier, J. T.

Goebel, D. G.

Hanssen, L. M.

L. M. Hanssen, K. A. Snail, “Infrared Diffuse Reflectometer for Spectral, Angular and Temperature Resolved Measurements,” Proc. Soc. Photo-Opt. Instrum. Eng. 807, 148–159 (1987).

K. A. Snail, L. M. Hanssen, “Integrating Sphere Designs with Isotropic Throughput,” Appl. Opt. 28, 1793 (1959).
[CrossRef]

K. A. Snail, K. F. Carr, L. M. Hanssen, “Integrating Sphere Measurements Using Restricted Field of View Detectors,” submitted to Applied Optics.

Hisdal, B. J.

Jacquez, J. A.

Kneissl, G. J.

G. J. Kneissl, J. C. Richmond, “A Laser-Source Integrating Sphere Reflectometer,” Natl. Bur. Stand. US Tech. Note 439 (Feb.1968).

Kuppenheim, H. F.

Neher, R. T.

Nelson, K. E.

Nicodemus, F. E.

Richmond, J. C.

G. J. Kneissl, J. C. Richmond, “A Laser-Source Integrating Sphere Reflectometer,” Natl. Bur. Stand. US Tech. Note 439 (Feb.1968).

Roddick, R. D.

Snail, K. A.

L. M. Hanssen, K. A. Snail, “Infrared Diffuse Reflectometer for Spectral, Angular and Temperature Resolved Measurements,” Proc. Soc. Photo-Opt. Instrum. Eng. 807, 148–159 (1987).

K. A. Snail, K. F. Carr, “Optical Design of an Integrating Sphere-Fourier Transform Spectrophotometer (FTS) Emissometer,” Proc. Soc. Photo-Opt. Instrum. Eng. 643, 75–83 (1985).

K. A. Snail, L. M. Hanssen, “Integrating Sphere Designs with Isotropic Throughput,” Appl. Opt. 28, 1793 (1959).
[CrossRef]

K. A. Snail, K. F. Carr, L. M. Hanssen, “Integrating Sphere Measurements Using Restricted Field of View Detectors,” submitted to Applied Optics.

Tardy, H. L.

Taylor, A. H.

Ulbricht, R.

R. Ulbricht, “Photometer for Mean Spherical Candle-Power,” Electrotech. Zeit. 21, 595–597 (1900).

Appl. Opt. (4)

Color Res. Appl. (1)

F. J. J. Clarke, J. A. Compton, “Correction Methods for Integrating-Sphere Measurement for Hemispherical Reflectance,” Color Res. Appl. 11, 253–262 (1986).
[CrossRef]

Electrotech. Zeit. (1)

R. Ulbricht, “Photometer for Mean Spherical Candle-Power,” Electrotech. Zeit. 21, 595–597 (1900).

Gen. Elect. Rev. (1)

F. A. Benford, “An Absolute Method for Determining Coefficients of Diffuse Reflection,” Gen. Elect. Rev. 23, 72–76 (1920).

J. Opt. Soc. Am. (5)

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

J. Res. Natl. Bur. Stand Sect. A (1)

W. Budde, “Calibration of Reflectance Standards,” J. Res. Natl. Bur. Stand Sect. A 80, 585 (1976).
[CrossRef]

Natl. Bur. Stand. US Tech. Note 439 (1)

G. J. Kneissl, J. C. Richmond, “A Laser-Source Integrating Sphere Reflectometer,” Natl. Bur. Stand. US Tech. Note 439 (Feb.1968).

Opt. Commun. (1)

M. W. Finkel, “Integrating Sphere Theory,” Opt. Commun. 2, 25–28 (1970).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (2)

K. A. Snail, K. F. Carr, “Optical Design of an Integrating Sphere-Fourier Transform Spectrophotometer (FTS) Emissometer,” Proc. Soc. Photo-Opt. Instrum. Eng. 643, 75–83 (1985).

L. M. Hanssen, K. A. Snail, “Infrared Diffuse Reflectometer for Spectral, Angular and Temperature Resolved Measurements,” Proc. Soc. Photo-Opt. Instrum. Eng. 807, 148–159 (1987).

Other (2)

K. A. Snail, K. F. Carr, L. M. Hanssen, “Integrating Sphere Measurements Using Restricted Field of View Detectors,” submitted to Applied Optics.

The equation within the text between Eqs. (4) and (5) of Ref. 5 assumes that the flat sample and/or reference conform to the sphere (i.e., are curved to match the sphere wall). This is in contrast to the more accurate approximation of Eq. (3.1) of Ref. 3.

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Figures (2)

Fig. 1
Fig. 1

Integrating sphere geometry. This cut view of an example geometry shows a typical configuration of entrance port, sample, and detector. Not shown are specular, reference, secondary detector, and other ports, some of which would be out of the plane of the page. The detector field of view is defined by the solid angle Ө and extends out of the plane of the page.

Fig. 2
Fig. 2

Reflectance measurement error limits for restricted FOV detection. The curves labeled R are calculated for sphere wall reflectance, ρw = 0.99 (vis) and ρw = 0.95 (IR) according to Eqs. (32) and (33) in the text. The curves labeled r are ρw and 1/ρw and appear as a single line at a measured ratio near 1 (which corresponds to 0 error).

Tables (1)

Tables Icon

Table I Nomenclature

Equations (55)

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Φ 1 = ρ s Φ 0 f d η s 0
ρ ¯ w = ρ w ( 1 i f i ) + i ρ i f i
Φ 2 = ρ s Φ 0 f d ρ ¯ υ f υ
Φ 2 = ( ρ ¯ w ρ s Φ 0 f d ) ( f υ ρ ¯ υ / ρ ¯ w )
ρ ¯ υ = ρ w [ 1 ( i f i δ i υ ) / f υ ] + ( i f i ρ i δ i υ ) / f υ ,
Φ 3 = ( ρ ¯ w 2 ρ s Φ 0 f d ) ( f υ ρ ¯ υ / ρ ¯ w )
Φ d s = Φ 1 + Φ 2 + Φ 3 +
Φ d s = f d ρ s Φ 0 [ η s 0 + f υ ρ ¯ υ / ρ ¯ w ( ρ ¯ w + ρ ¯ w 2 + ρ ¯ w 3 + ) ] .
τ D = Φ d s / ( ρ s Φ 0 ) = f d [ η s 0 + f υ ρ ¯ υ / ( 1 ρ ¯ w ) ] .
Φ d s = f d ρ s Φ 0 / ( 1 ρ ¯ w ) ,
Φ d s = f d ρ s Φ 0 f υ ρ ¯ υ / ( 1 ρ ¯ w )
Φ d s = f d ρ s Φ 0 [ 1 ρ ¯ w + f υ ρ ¯ υ ] / ( 1 ρ ¯ w ) .
Φ 1 = ( ρ s Φ 0 ) η s 0 η s d
η s i = ρ s i / ρ s .
ρ s i = Ω i F r ( θ 0 , ϕ 0 ; θ 1 , ϕ 1 ) cos ( θ 1 ) d Ω 1 ,
ρ ¯ s b = ρ w ( 1 i η s i ) + i ρ i f i η s i ,
Φ 2 = ρ ¯ s b υ η s υ ρ s f d Φ 0
ρ ¯ s b υ = ρ w [ 1 ( i f i η s i ) / η s υ ] + ( i ρ i f i η s i ) / η s υ .
τ N = f d [ η s υ ρ ¯ s b υ + η s 0 η s d / f d + ρ ¯ s b f υ ρ ¯ υ / ( 1 ρ ¯ w ) ] ,
τ S = f d ρ w [ η s υ + f υ ρ ¯ υ / ( 1 ρ ¯ w ) ] .
τ S = f d ρ w / ( 1 ρ ¯ w ) .
Φ d s = τ D ρ s D Φ 0 + τ N ρ s N Φ 0 ,
Φ d s = f d Φ 0 [ ρ s N ( ρ ¯ s b υ η s υ + η s 0 η s d f d ) + ρ s D η s 0 + ( ρ s N ρ ¯ s b + ρ s D ) f υ ρ ¯ υ 1 ρ ¯ w ] .
R = Φ d s Φ d r = ρ s N ( ρ ¯ s b υ η s υ + η s 0 η s d / f d ) + ρ s D η s 0 + ( ρ s N ρ ¯ s b + ρ s D ) f υ ρ ¯ υ / ( 1 ρ ¯ w ) ρ r N ( ρ ¯ r b υ η r υ + η r 0 η r d / f d ) + ρ r D η r 0 + ( ρ r N ρ ¯ r b + ρ r D ) f υ ρ ¯ υ / ( 1 ρ ¯ w ) .
R = ρ s [ η s 0 + f υ ρ ¯ υ / ( 1 ρ ¯ w ) ] ρ r [ η r 0 + f υ ρ ¯ υ / ( 1 ρ ¯ w ) ] .
R = ρ s / ρ r .
R = ρ s ρ w ρ r ,
R = ρ s .
R = ρ s ρ r ρ w .
R max = ρ s N ρ ¯ s b + ( ρ s N ρ ¯ s b + ρ s D ) f υ ρ ¯ υ / ( 1 ρ ¯ w ) ( ρ r N ρ ¯ r b + ρ r D ) f υ ρ ¯ υ / ( 1 ρ ¯ w ) ,
R min = ( ρ s N ρ ¯ s b + ρ s D ) f υ ρ ¯ υ / ( 1 ρ ¯ w ) ρ r N ρ ¯ r b + ( ρ r N ρ ¯ r b + ρ r D ) f υ ρ ¯ υ / ( 1 ρ ¯ w ) .
R max = ρ s ρ r ( 1 + 1 ρ ¯ w f υ ρ ¯ υ ) ,
R min = ρ s ρ r ( 1 + 1 ρ ¯ w f υ ρ ¯ υ ) 1 ,
τ i n = f d ρ ¯ s b υ [ 1 + f υ ρ ¯ υ / ( 1 ρ ¯ w ) ] ,
τ out = f d [ η s 0 η s d / f d + ρ ¯ s b o f υ ρ ¯ υ / ( 1 ρ ¯ w ) ] ,
R = ρ s υ ρ ¯ s b υ + ρ s o η s 0 η s d f d + ( ρ s υ ρ ¯ s b υ + ρ s o ρ ¯ s b o ) f υ ρ ¯ υ 1 ρ ¯ w ρ r υ ρ ¯ r b υ + ρ r o η r 0 η r d f d + ( ρ r υ ρ ¯ r b υ + ρ r o ρ ¯ r b o ) f υ ρ ¯ υ 1 ρ ¯ w .
R = ρ s [ η s υ + f υ ρ ¯ υ / ( 1 ρ ¯ w ) ] ρ r [ η r υ + f υ ρ ¯ υ / ( 1 ρ ¯ w ) ] ,
R = Φ C Φ R = [ η 0 + f υ ρ ¯ υ C / ( 1 ρ ¯ w C ) ] [ η 0 + f υ ρ ¯ υ R / ( 1 ρ ¯ w R ) ] .
ρ w = b ± ( b 2 4 a c ) 2 a ,
a = C 2 C 3 R C 1 C 4 C 1 C 2 C 5 , b = R C 4 C 3 + ( C 1 + C 2 ) C 5 + ( C 8 C 6 C 5 ) C 2 + ( C 3 C 1 C 5 ) C 7 R ( C 1 C 9 + C 4 C 6 ) , c = ( 1 C 7 ) ( C 5 C 6 C 5 C 8 ) + R ( 1 C 6 ) C 9 , C 1 = 1 C ; i f i , C 2 = 1 R ; i f i , C 3 = f υ [ 1 ( C ; i f i δ i υ ) / f υ ] , C 4 = f υ [ 1 ( R ; i f i δ i υ ) / f υ ] , C 5 = η 0 ( 1 R ) , C 6 = C ; i f i ρ i , C 7 = R ; i f i ρ i , C 8 = C ; i f i ρ i δ i υ , C 9 = R ; i f i ρ i δ i υ .
ρ ¯ w C , R = ρ w ( 1 C , R ; i f i ) and ρ ¯ υ C , R = ρ w [ 1 ( C , R ; i f i δ i υ ) / f υ ] ,
b = R C 4 C 3 + ( C 1 + C 2 ) C 5 , c = C 5 . }
ρ w = ( C 1 + C 1 C 2 R C 4 / C 3 1 ) 1 ,
ρ w = ( 1 C ; i f i + f cap R C 4 / C 3 1 ) 1 ,
ρ w = ( 1 C ; i f i + f cap R 1 ) 1 ,
ρ w = ( 1 C ; i f i + f cap R 1 ) 1 ,
b = [ R C 4 C 3 + ( C 1 + C 2 ) C 5 ] ( 1 C 6 ) , c = C 5 ( 1 C 6 ) . }
ρ w = ( 1 C 6 ) ( C 1 + C 1 C 2 R C 4 / C 3 1 ) 1 ,
ρ w = ( 1 C ; i f i ρ i ) ( 1 C ; i f i + f cap R C 4 / C 3 1 ) 1 .
ρ w = ( 1 C ; i f i ρ i ) ( 1 C ; i f i + f cap R 1 ) 1 .
a = C 1 C 3 ( 1 R ) , b = C 3 ( R 1 + C 7 R C 6 ) , c = 0 , }
ρ w = ( 1 R C 6 C 7 R 1 ) 1 C 1 ,
ρ w = ( 1 C ; i f i ρ i f cap ρ cap R 1 ) ( 1 C ; i f i ) 1 .
a = C 1 ( 1 R ) ( C 3 C 1 ) , b = ( C 3 C 1 ( R 1 + C 7 R C 6 ) + C 1 ( 1 R ) , c = R 1 + C 7 R C 6 ,
ρ w = [ 1 C ; i f i ρ i f cap ρ cap R 1 ] ( 1 C ; i f i ) 1 .

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