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

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

http://dx.doi.org/10.1364/JOSAA.15.003086

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### Abstract

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

**History**

Original Manuscript: February 18, 1998

Revised Manuscript: September 4, 1998

Manuscript Accepted: August 17, 1998

Published: December 1, 1998

**Citation**

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)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-15-12-3086

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### References

- 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]
- J. W. T. Walsh, Photometry (Constable, London, 1958). Walsh gives a simple energy-balance derivation due to Ulbricht in 1920.
- 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.
- A. H. Taylor, “The measurement of diffuse reflection factors and a new absolute reflectometer,” J. Opt. Soc. Am. 4, 9–23 (1920). [CrossRef]
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- 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]
- R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer, 3rd ed. (Taylor & Francis, Washington, D.C., 1992).
- 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]
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- 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).
- 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]
- W. Budde, C. X. Dodd, “Absolute reflectance measurements in the D/0° geometry,” Farbe 19, 94–102 (1970).

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