Abstract

Average transmittance of polarized diffuse light across a dielectric surface is calculated in both two and three dimensions. The incident light in both cases is confined to an angular range measured from the surface normal. Limiting values in three dimensions correspond to known results for two cases, (i) normal incidence, and (ii) diffuse light incident from a 180° cone. The two-dimensional formulation is solvable in terms of elliptic functions and incomplete elliptic integrals of the first, second, and third kinds. Results are displayed graphically for values of transmittances in refraction in the range <i>m</i> = 1.0 to <i>m</i> = 2.6.

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  1. W. A. Allen, H. W. Gausman, A. J. Richardson, and J. R. Thomas, J. Opt. Soc. Am. 59, 1376 (1969).
  2. J. W. T. Walsh, The Reflection Factor of a Polished Glass Surface for Diffused Light, Department of Scientific and Industrial Research (Brit.), Illumination Research, Technical paper (printed under the authority of Her Majesty's Stationery Office, Harrison and Sons, Ltd., London, 1926), Vol. 2.
  3. J. C. Richmond, J. Res. Natl. Bur. Stand. (U.S.) C 67, 222 (1963).
  4. F. Stern, Appl. Opt. 3, 111 (1964).
  5. J. W. Ryde and B. S. Cooper, Proc. R. Soc. A 131, 463 (1931).
  6. S. Q. Duntley, J. Opt. Soc. Am. 32, 61 (1942).
  7. D. B. Judd, J. Res. Natl. Bur. Stand. (U.S.) 29, 329 (1942).
  8. Gustav Kortürm, Reflectance Spectroscopy, translated from the German by James E. Lohr (Springer, New York, 1969).
  9. W. A. Allen and A. J. Richardson, J. Opt. Soc. Am. 62, 713 (1972).
  10. R. Willstätter and A. Stoll, Untersuchunger über die Assimilation der Kohlensäure (Springer, Berlin, 1913).
  11. P. S. Mudgett and L. W. Richards, Appl. Opt. 10, 1485 (1971).
  12. Paul F. Byrd and Morris D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists, 1st ed. (Springer, New York, 1954), pp. 191–192.
  13. Handbook of Mathematical Functions, 1st ed., edited by M. Abramowitz and I. A. Stegun (Dover, New York, 1965), Ch. 17, p. 589.
  14. R. G. Selfridge and J. E. Maxfield, Table of the Incomplete Elliptic Integral of the Third Kind (Dover, New York, 1958).
  15. R. Burirsch, Numer. Math. 7, 78 (1965).
  16. The author is indebted to Frank Stern for this approach. (Personal communication dated 3 February 1972.)
  17. J. T. Woolley, Plant Physiol. 47, 656 (1971).

Allen, W. A.

W. A. Allen, H. W. Gausman, A. J. Richardson, and J. R. Thomas, J. Opt. Soc. Am. 59, 1376 (1969).

W. A. Allen and A. J. Richardson, J. Opt. Soc. Am. 62, 713 (1972).

Burirsch, R.

R. Burirsch, Numer. Math. 7, 78 (1965).

Byrd, Paul F.

Paul F. Byrd and Morris D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists, 1st ed. (Springer, New York, 1954), pp. 191–192.

Cooper, B. S.

J. W. Ryde and B. S. Cooper, Proc. R. Soc. A 131, 463 (1931).

Duntley, S. Q.

S. Q. Duntley, J. Opt. Soc. Am. 32, 61 (1942).

Friedman, Morris D.

Paul F. Byrd and Morris D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists, 1st ed. (Springer, New York, 1954), pp. 191–192.

Gausman, H. W.

W. A. Allen, H. W. Gausman, A. J. Richardson, and J. R. Thomas, J. Opt. Soc. Am. 59, 1376 (1969).

Judd, D. B.

D. B. Judd, J. Res. Natl. Bur. Stand. (U.S.) 29, 329 (1942).

Kortürm, Gustav

Gustav Kortürm, Reflectance Spectroscopy, translated from the German by James E. Lohr (Springer, New York, 1969).

Maxfield, J. E.

R. G. Selfridge and J. E. Maxfield, Table of the Incomplete Elliptic Integral of the Third Kind (Dover, New York, 1958).

Mudgett, P. S.

P. S. Mudgett and L. W. Richards, Appl. Opt. 10, 1485 (1971).

Richards, L. W.

P. S. Mudgett and L. W. Richards, Appl. Opt. 10, 1485 (1971).

Richardson, A. J.

W. A. Allen and A. J. Richardson, J. Opt. Soc. Am. 62, 713 (1972).

W. A. Allen, H. W. Gausman, A. J. Richardson, and J. R. Thomas, J. Opt. Soc. Am. 59, 1376 (1969).

Richmond, J. C.

J. C. Richmond, J. Res. Natl. Bur. Stand. (U.S.) C 67, 222 (1963).

Ryde, J. W.

J. W. Ryde and B. S. Cooper, Proc. R. Soc. A 131, 463 (1931).

Selfridge, R. G.

R. G. Selfridge and J. E. Maxfield, Table of the Incomplete Elliptic Integral of the Third Kind (Dover, New York, 1958).

Stern, F.

F. Stern, Appl. Opt. 3, 111 (1964).

Stoll, A.

R. Willstätter and A. Stoll, Untersuchunger über die Assimilation der Kohlensäure (Springer, Berlin, 1913).

Thomas, J. R.

W. A. Allen, H. W. Gausman, A. J. Richardson, and J. R. Thomas, J. Opt. Soc. Am. 59, 1376 (1969).

Walsh, J. W. T.

J. W. T. Walsh, The Reflection Factor of a Polished Glass Surface for Diffused Light, Department of Scientific and Industrial Research (Brit.), Illumination Research, Technical paper (printed under the authority of Her Majesty's Stationery Office, Harrison and Sons, Ltd., London, 1926), Vol. 2.

Willstätter, R.

R. Willstätter and A. Stoll, Untersuchunger über die Assimilation der Kohlensäure (Springer, Berlin, 1913).

Woolley, J. T.

J. T. Woolley, Plant Physiol. 47, 656 (1971).

Other

W. A. Allen, H. W. Gausman, A. J. Richardson, and J. R. Thomas, J. Opt. Soc. Am. 59, 1376 (1969).

J. W. T. Walsh, The Reflection Factor of a Polished Glass Surface for Diffused Light, Department of Scientific and Industrial Research (Brit.), Illumination Research, Technical paper (printed under the authority of Her Majesty's Stationery Office, Harrison and Sons, Ltd., London, 1926), Vol. 2.

J. C. Richmond, J. Res. Natl. Bur. Stand. (U.S.) C 67, 222 (1963).

F. Stern, Appl. Opt. 3, 111 (1964).

J. W. Ryde and B. S. Cooper, Proc. R. Soc. A 131, 463 (1931).

S. Q. Duntley, J. Opt. Soc. Am. 32, 61 (1942).

D. B. Judd, J. Res. Natl. Bur. Stand. (U.S.) 29, 329 (1942).

Gustav Kortürm, Reflectance Spectroscopy, translated from the German by James E. Lohr (Springer, New York, 1969).

W. A. Allen and A. J. Richardson, J. Opt. Soc. Am. 62, 713 (1972).

R. Willstätter and A. Stoll, Untersuchunger über die Assimilation der Kohlensäure (Springer, Berlin, 1913).

P. S. Mudgett and L. W. Richards, Appl. Opt. 10, 1485 (1971).

Paul F. Byrd and Morris D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists, 1st ed. (Springer, New York, 1954), pp. 191–192.

Handbook of Mathematical Functions, 1st ed., edited by M. Abramowitz and I. A. Stegun (Dover, New York, 1965), Ch. 17, p. 589.

R. G. Selfridge and J. E. Maxfield, Table of the Incomplete Elliptic Integral of the Third Kind (Dover, New York, 1958).

R. Burirsch, Numer. Math. 7, 78 (1965).

The author is indebted to Frank Stern for this approach. (Personal communication dated 3 February 1972.)

J. T. Woolley, Plant Physiol. 47, 656 (1971).

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