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Generalized Integrating-Sphere Theory David G. Goebel A general equation is developed for the efficiency of an integrating sphere with a nonuniform coating. The only assumptions are that the interior is a perfect sphere and that all areas reflect perfectly diffusely. Three special cases of the general equation are examined for the basic applications of integrating spheres as mixing mechanisms in hemispherical reflectance measurements and in absolute reflectance techniques. Introduction A sphere that is not uniform in the reflectance of its coating can be represented by a sphere with a number of areas of varying reflectance. The number and size of these areas can be chosen to approximate any nonuni- form coating. If J is the intensity (flux-per-unit solid angle) of the reflected flux from an element da, the resulting ir- radiance H on any element dA of the sphere wall is H- J cosa/r2 (see Fig. 1), where J is the intensity in the given direction. Since r = 2R cosO, and = a for a sphere, H = J cos9/4R2 cos20, where R is the radius of the sphere. If the reflecting element da has a per- fectly diffuse surface, J = Jo cosO, where Jo is the in- tensity in the perpendicular direction. Then the ir- radiance on any element of the sphere wall is H = Jo cos'0/4R1 cos'o = J 0 /4R2. Thus, the irradiance is constant, and the total flux reflected by the perfectly diffusing element da is simply P = 4rRIH = 7rJo. Also, the fraction f of the total reflected flux that is incident on an area a of the sphere wall is simply equal to a divided by 4 7rR, thus, f = a/47rR'. (1) If a is a circular cap in the sphere wall with radius r, a equals 27rRh (see Fig. 2), where h = R -(R2 _ 2) r'2. Therefore, for a circular cap f = 2 7rR [R - (r2 - R12)/21/ 47rR1 or f = (/2){1 - [- (rR)1l} (2) (see Table I). Two other convenient forms forf which are equivalent to Eq. (2) for a circular cap are f = /,(1 - cos2O) andf= sin'2. The author is with the Colorimetry and Spectrophotometry Section, National Bureau of Standards, Washington, D.C. 20234. Received 11 July 1966. Two useful approximate forms when r<<R are f r/r' + DI, (D = 2R) and f r2/92. The last approximate form is the familiar relation obtained if the area of a flat disk xr 2 is used for a instead of the spherical area. The error introduced by this approximation can be estimated by expanding Eq. (2) in a power series in 1, where 1 = r2/D2: f = '/2[l (1 -41)'/] = /211 [I -1/:(41) '/(41)2 - /(41)3 .. = 1/2[21 + 212 + 413± . = + 12 + 213 + . . . If we approximatef by I to within a decimal error e, f - 1 = j = 12 + 213 + or e z 1 = r/D2 r2 eD' or dI - 2eD2, where d = 2r, then d V -/2eD. Use of the approximate form for f results in an error greater than 1%, if the diameter of a is more than 20% of the sphere diameter. For an error of less than 0.1% the maximum value for d is 0.06 D. For a 20-cm sphere the use of r2/D' for f yields a 1% error for d = 4.0 cm and a 0.1% error for d = 1.2 cm. January 1967 / Vol. 6, No. 1 / APPLIED OPTICS 125
