Abstract

The equilibrium angular distribution of radiation at the surface of a closed, partially specular integrating sphere is described by a Fredholm-type integral equation. The solutions to this equation are shown to be polarized and concentrated off normal. As a consequence, the diffuse character of the radiance in actual integrating spheres is questioned. It is proposed that a more nearly diffuse radiation pattern can be achieved through a simple modification to the construction of integrating spheres.

© 1994 Optical Society of America

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References

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  1. F. B. Hildebrand, Methods of Applied Mathematics (Prentice-Hall, Englewood Cliffs, N.J., 1965), Chap. 3.
  2. H. Buckley, “On the radiation from the inside of a circular cylinder,” Phil. Mag. 4, 753–757 (1927).
  3. P. Moon, “On interreflections,” J. Opt. Soc. Am. 30, 195–205 (1940).
    [CrossRef]
  4. J. A. Jacquez, H. F. Kuppenheim, “Theory of the integrating sphere,” J. Opt. Soc. Am. 45, 460–470 (1955).
    [CrossRef]
  5. B. J. Hisdal, “Reflectance of perfect diffuse and specular samples in the integrating sphere,” J. Opt. Soc. Am. 55, 1122–1128 (1965).
    [CrossRef]
  6. B. J. Hisdal, “Reflectance of nonperfect surfaces in the integrating sphere,” J. Opt. Soc. Am. 55, 1255–1260 (1965).
    [CrossRef]
  7. N. Halyo, D. B. Taylor, “Explicit solutions of the spectral radiance in integrating spheres with application to the earth radiation budget experiment ground calibration,” J. Opt. Soc. Am. A 5, 520–534 (1988).
    [CrossRef]
  8. H. L. Tardy, “Matrix method for integrating-sphere calculations,” J. Opt. Soc. Am. A 8, 1411–1418 (1991).
    [CrossRef]
  9. D. K. Edwards, J. T. Gier, K. E. Nelson, R. D. Roddick, “Integrating sphere for imperfectly diffuse samples,” Appl. Opt. 51, 1279–1288 (1961).
  10. A. Roos, C. G. Ribbing, “Interpretation of integrating sphere signal output for non-Lambertian samples,” Appl. Opt. 27, 3833–3837 (1988).
    [CrossRef] [PubMed]
  11. F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Mogr. 160 (1977).
  12. G. I. Pokrowski, “Theory of diffuse reflection,” Z. Phys. 30, 66–72 (1924).
    [CrossRef]
  13. G. I. Pokrowski, “Diffuse reflection of light,” Z. Phys. 35, 34–37 (1925).
  14. G. I. Pokrowski, “Diffuse light reflection,” Z. Phys. 36, 472–476 (1926).
    [CrossRef]
  15. W. W. Barkas, “Analysis of light scattered from a surface of low gloss into its specular and diffuse components,” Proc. Phys. Soc. London 51, 274–292 (1939).
    [CrossRef]
  16. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).
  17. W. S. Bickel, W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized, scattered light,” Am. J. Phys. 53, (5) 468–478 (1985).
    [CrossRef]
  18. G. Kortüm, Reflectance Spectroscopy (Springer-Verlag, New York, 1969).
    [CrossRef]
  19. H. Wright, “Diffused reflection of light,” Ann. Physik 1, 17–41 (1900).
    [CrossRef]

1991 (1)

1988 (2)

1985 (1)

W. S. Bickel, W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized, scattered light,” Am. J. Phys. 53, (5) 468–478 (1985).
[CrossRef]

1977 (1)

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Mogr. 160 (1977).

1965 (2)

1961 (1)

D. K. Edwards, J. T. Gier, K. E. Nelson, R. D. Roddick, “Integrating sphere for imperfectly diffuse samples,” Appl. Opt. 51, 1279–1288 (1961).

1955 (1)

1940 (1)

1939 (1)

W. W. Barkas, “Analysis of light scattered from a surface of low gloss into its specular and diffuse components,” Proc. Phys. Soc. London 51, 274–292 (1939).
[CrossRef]

1927 (1)

H. Buckley, “On the radiation from the inside of a circular cylinder,” Phil. Mag. 4, 753–757 (1927).

1926 (1)

G. I. Pokrowski, “Diffuse light reflection,” Z. Phys. 36, 472–476 (1926).
[CrossRef]

1925 (1)

G. I. Pokrowski, “Diffuse reflection of light,” Z. Phys. 35, 34–37 (1925).

1924 (1)

G. I. Pokrowski, “Theory of diffuse reflection,” Z. Phys. 30, 66–72 (1924).
[CrossRef]

1900 (1)

H. Wright, “Diffused reflection of light,” Ann. Physik 1, 17–41 (1900).
[CrossRef]

Bailey, W. M.

W. S. Bickel, W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized, scattered light,” Am. J. Phys. 53, (5) 468–478 (1985).
[CrossRef]

Barkas, W. W.

W. W. Barkas, “Analysis of light scattered from a surface of low gloss into its specular and diffuse components,” Proc. Phys. Soc. London 51, 274–292 (1939).
[CrossRef]

Bickel, W. S.

W. S. Bickel, W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized, scattered light,” Am. J. Phys. 53, (5) 468–478 (1985).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Buckley, H.

H. Buckley, “On the radiation from the inside of a circular cylinder,” Phil. Mag. 4, 753–757 (1927).

Edwards, D. K.

D. K. Edwards, J. T. Gier, K. E. Nelson, R. D. Roddick, “Integrating sphere for imperfectly diffuse samples,” Appl. Opt. 51, 1279–1288 (1961).

Gier, J. T.

D. K. Edwards, J. T. Gier, K. E. Nelson, R. D. Roddick, “Integrating sphere for imperfectly diffuse samples,” Appl. Opt. 51, 1279–1288 (1961).

Ginsberg, I. W.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Mogr. 160 (1977).

Halyo, N.

Hildebrand, F. B.

F. B. Hildebrand, Methods of Applied Mathematics (Prentice-Hall, Englewood Cliffs, N.J., 1965), Chap. 3.

Hisdal, B. J.

Hsia, J. J.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Mogr. 160 (1977).

Jacquez, J. A.

Kortüm, G.

G. Kortüm, Reflectance Spectroscopy (Springer-Verlag, New York, 1969).
[CrossRef]

Kuppenheim, H. F.

Limperis, T.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Mogr. 160 (1977).

Moon, P.

Nelson, K. E.

D. K. Edwards, J. T. Gier, K. E. Nelson, R. D. Roddick, “Integrating sphere for imperfectly diffuse samples,” Appl. Opt. 51, 1279–1288 (1961).

Nicodemus, F. E.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Mogr. 160 (1977).

Pokrowski, G. I.

G. I. Pokrowski, “Diffuse light reflection,” Z. Phys. 36, 472–476 (1926).
[CrossRef]

G. I. Pokrowski, “Diffuse reflection of light,” Z. Phys. 35, 34–37 (1925).

G. I. Pokrowski, “Theory of diffuse reflection,” Z. Phys. 30, 66–72 (1924).
[CrossRef]

Ribbing, C. G.

Richmond, J. C.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Mogr. 160 (1977).

Roddick, R. D.

D. K. Edwards, J. T. Gier, K. E. Nelson, R. D. Roddick, “Integrating sphere for imperfectly diffuse samples,” Appl. Opt. 51, 1279–1288 (1961).

Roos, A.

Tardy, H. L.

Taylor, D. B.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Wright, H.

H. Wright, “Diffused reflection of light,” Ann. Physik 1, 17–41 (1900).
[CrossRef]

Am. J. Phys. (1)

W. S. Bickel, W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized, scattered light,” Am. J. Phys. 53, (5) 468–478 (1985).
[CrossRef]

Ann. Physik (1)

H. Wright, “Diffused reflection of light,” Ann. Physik 1, 17–41 (1900).
[CrossRef]

Appl. Opt. (2)

A. Roos, C. G. Ribbing, “Interpretation of integrating sphere signal output for non-Lambertian samples,” Appl. Opt. 27, 3833–3837 (1988).
[CrossRef] [PubMed]

D. K. Edwards, J. T. Gier, K. E. Nelson, R. D. Roddick, “Integrating sphere for imperfectly diffuse samples,” Appl. Opt. 51, 1279–1288 (1961).

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (2)

Natl. Bur. Stand. (U.S.) Mogr. (1)

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Mogr. 160 (1977).

Phil. Mag. (1)

H. Buckley, “On the radiation from the inside of a circular cylinder,” Phil. Mag. 4, 753–757 (1927).

Proc. Phys. Soc. London (1)

W. W. Barkas, “Analysis of light scattered from a surface of low gloss into its specular and diffuse components,” Proc. Phys. Soc. London 51, 274–292 (1939).
[CrossRef]

Z. Phys. (3)

G. I. Pokrowski, “Theory of diffuse reflection,” Z. Phys. 30, 66–72 (1924).
[CrossRef]

G. I. Pokrowski, “Diffuse reflection of light,” Z. Phys. 35, 34–37 (1925).

G. I. Pokrowski, “Diffuse light reflection,” Z. Phys. 36, 472–476 (1926).
[CrossRef]

Other (3)

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

F. B. Hildebrand, Methods of Applied Mathematics (Prentice-Hall, Englewood Cliffs, N.J., 1965), Chap. 3.

G. Kortüm, Reflectance Spectroscopy (Springer-Verlag, New York, 1969).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Geometrical demonstration that, in a perfect sphere, the angle to the normal of light on emission from the sphere is equal to the angle to the normal at reception.

Fig. 2
Fig. 2

Polar plots of the relative radiance at the surface of an integrating sphere with an index of refraction of 1.65 and s values of 0.2 (solid curve), 0.5 (even-dashed curve), and 0.8 (graduated-dashed curve). As s increases, the radiation distribution grows progressively more intense at large incidence angles.

Fig. 3
Fig. 3

Polar plots of the percent degree of polarization at the surface of an integrating sphere with an index of refraction of 1.65 and s values of 0.2 (solid curve), 0.5 (even-dashed curve), and 0.8 (graduated-dashed curve). Larger s values produce a more polarized distribution.

Fig. 4
Fig. 4

Polar plots of the relative radiance at the surface of an integrating sphere with an index of refraction of 2.85 and s values of 0.2, 0.5, and 0.8. As s increases, the radiation distribution grows progressively more intense at large incidence angles. Curves as in Fig. 2.

Fig. 5
Fig. 5

Polar plots of the percent degree of polarization at the surface of an integrating sphere with an index of refraction of 2.85 and s values of 0.2, 0.5, and 0.8. Larger s values produce a more polarized distribution. Curves as in Fig. 3.

Fig. 6
Fig. 6

Polar plots of the relative radiance at the surface of an integrating sphere with an index of refraction of 10 and s values of 0.2, 0.5, and 0.8. As s increases, the radiation distribution grows progressively more intense at large incidence angles. Curves as in Fig. 2.

Fig. 7
Fig. 7

Polar plots of the percent degree of polarization at the surface of an integrating sphere with an index of refraction of 10 and s values of 0.2, 0.5, and 0.8. Larger s values produce a more polarized distribution. Curves as in Fig. 3.

Equations (17)

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d L ( θ r , ϕ r ) = f r ( θ r , ϕ r , θ i , ϕ i ) d E .
L ( θ r ) = Ω i f r ( θ r , ϕ r ; θ i , ϕ i ) L ( θ i ) cos θ i d Ω i .
0 2 π 0 π / 2 f r ( θ r , ϕ r ; θ i , ϕ i ) cos θ r sin θ r d θ r d ϕ r = 1 .
f r ( θ r , ϕ r ; θ i , ϕ i ) = d ( θ i ) + R ( θ i ) δ ( θ r θ i ) × δ ( ϕ i ϕ i π ) / cos θ i sin θ i ,
d ( θ i ) = [ 1 R ( θ i ) ] / π .
L ( θ r ) = 2 0 π / 2 [ 1 R ( θ i ) ] L ( θ i ) cos θ i sin θ i d θ i + R ( θ r ) L ( θ r ) .
L ( θ ) = C / [ 1 R ( θ ) ] ,
f r , = 0.5 d ( θ i ) + R ( θ i ) δ ( θ r θ i ) × δ ( ϕ r ϕ i π ) / sin θ i cos θ i ,
f r , = 0.5 d ( θ i ) ,
f r , = 0.5 d ( θ i ) + R ( θ i ) δ ( θ r θ i ) × δ ( ϕ i ϕ r π ) / sin θ i cos θ i ,
f r , = 0.5 d ( θ i ) .
d ( θ i ) = [ 1 R ( θ i ) ] / π ,
d ( θ i ) = [ 1 R ( θ i ) ] / π .
L ( θ r ) = 0 π / 2 { [ 1 R ( θ i ) ] L ( θ i ) + [ 1 R ( θ i ) ] L ( θ i ) } × cos θ i sin θ i d θ i + R ( θ r ) L ( θ r ) ,
L ( θ r ) = 0 π / 2 { [ 1 R ( θ i ) ] L ( θ i ) + [ 1 R ( θ i ) ] L ( θ i ) } × cos θ i sin θ i d θ i + R ( θ r ) L ( θ r ) .
L ( θ r ) = C / [ 1 R ( θ r ) ] ,
L ( θ r ) = C / [ 1 R ( θ r ) ] .

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