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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 15, Iss. 12 — Dec. 1, 1998
  • pp: 3086–3096

Comparison of four analytic methods for the calculation of irradiance in integrating spheres

John F. Clare  »View Author Affiliations

JOSA A, Vol. 15, Issue 12, pp. 3086-3096 (1998)

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The relative merits of four methods—energy balance, summation of reflections, inversion of the irradiance-transfer matrix, and solution of the integral equation—are compared by using each to determine irradiance in a multizone true sphere and in a sphere with a flat port; in the process several new solutions are presented. Although limited in applicability, the energy-balance method is by far the most direct. For the flat-port configuration the relationships among various published expressions are established; furthermore, the curved-surface interreflection irradiance is shown to be nonuniform when the initial irradiance is restricted to a part of the curved surface.

© 1998 Optical Society of America

OCIS Codes
(120.3150) Instrumentation, measurement, and metrology : Integrating spheres
(120.3930) Instrumentation, measurement, and metrology : Metrological instrumentation
(120.5700) Instrumentation, measurement, and metrology : Reflection

Original Manuscript: February 18, 1998
Revised Manuscript: September 4, 1998
Manuscript Accepted: August 17, 1998
Published: December 1, 1998

John F. Clare, "Comparison of four analytic methods for the calculation of irradiance in integrating spheres," J. Opt. Soc. Am. A 15, 3086-3096 (1998)

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  1. E. Karrer, “Use of the Ulbricht sphere in measuring reflection and transmission factors,” Sci. Pap. Bur. Stand. No. 415 (1921). Karrer derives the (1-ρ)-1 form by energy balance. [CrossRef]
  2. J. W. T. Walsh, Photometry (Constable, London, 1958). Walsh gives a simple energy-balance derivation due to Ulbricht in 1920.
  3. E. E. N. Mascart, “Sur la mesure de l’éclairement,” Lum. Elect.28, 180–187 (1888). Mascart uses summation of reflections to obtain the (1-ρ)-1 expression for the total flux in a room with diffusely reflecting surfaces.Karrer1 cites A. Palaz in Photométrie Industrielle (Carré, Paris, 1892) to the effect that Mascart was the first to derive this expression.
  4. A. H. Taylor, “The measurement of diffuse reflection factors and a new absolute reflectometer,” J. Opt. Soc. Am. 4, 9–23 (1920). [CrossRef]
  5. A. H. Taylor, “Errors in reflectometry,” J. Opt. Soc. Am. 25, 51–56 (1935). [CrossRef]
  6. P. Moon, The Scientific Basis of Illuminating Engineering (McGraw-Hill, New York, 1936), p. 331.
  7. J. A. Jacquez, H. F. Kuppenheim, “Theory of the integrating sphere,” J. Opt. Soc. Am. 45, 460–470 (1955). [CrossRef]
  8. B. J. Hisdal, “Reflectance of perfect diffuse and specular samples in the integrating sphere,” J. Opt. Soc. Am. 55, 1122–1128 (1965). [CrossRef]
  9. D. G. Goebel, “Generalized integrating-sphere theory,” Appl. Opt. 6, 125–128 (1967). [CrossRef] [PubMed]
  10. H. L. Tardy, “Matrix method for integrating sphere calculations,” J. Opt. Soc. Am. A 8, 1411–1418 (1991). [CrossRef]
  11. H. L. Tardy, “Flat-sample and limited-field effects in integrating sphere measurements,” J. Opt. Soc. Am. A 5, 241–245 (1988). Note the omission of a prime in his Eq. (2.7) [cf. Eq. (22) of the present paper]. [CrossRef]
  12. R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer, 3rd ed. (Taylor & Francis, Washington, D.C., 1992).
  13. J. G. Symons, E. A. Christie, M. K. Peck, “Integrating sphere for solar transmittance measurement of planar and nonplanar samples,” Appl. Opt. 21, 2827–2832 (1982). [CrossRef] [PubMed]
  14. L. M. Hanssen, “Effects of restricting the detector field of view when using integrating spheres,” Appl. Opt. 28, 2097–2103 (1989). [CrossRef] [PubMed]
  15. This is known in illumination engineering as McAllister’s equilux theorem; see Ref. 6, p. 301. It can also be derived by using the contour integration theorem to replace the flat F with the cap K. The latter theorem states that the flux from a surface of uniform radiance L bounded by a contour C′ is the same as that from any other surface of uniform radiance L that is also bounded by C′ (Ref. 6, p. 312).
  16. M. W. Finkel, “Integrating sphere theory,” Opt. Commun. 2, 25–28 (1970). Finkel omits all but the first two groups of terms in Eq. (55) of the present paper. [CrossRef]
  17. W. Budde, C. X. Dodd, “Absolute reflectance measurements in the D/0° geometry,” Farbe 19, 94–102 (1970).

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