Abstract

Angular conditions of incidence are described as hemispherical, conical, or directional; the same adjectives are used to describe the angular conditions of collection. This classification of angular conditions leads to nine kinds of reflectance; symbols for them are proposed in which 2π, g, and θ0, ϕ0 refer to hemispherical, conical, and directional incidence, 2π, g′, and θr,ϕr refer to the corresponding kinds of collection. Use of the perfectly reflecting mirror and of the perfectly reflecting diffuser as reference standards in reflectometry is discussed. Three of the nine reflectance ratios, specimen to perfect diffuser, in which the collection is directional have already been named radiance [luminance] factor. It is proposed to differentiate them by angular condition of incidence. It is also proposed to name the other six ratios: reflectance factor qualified by the same adjectives identifying the type of incidence and collection as are used for reflectance. The interrelationships of these 18 concepts are shown both by formulas for computing one from another and by diagrams indicating the process (integration, summation, averaging, equality, reflectance of perfect difluser, and reciprocity) by which values of one concept may be computed from those of another.

© 1967 Optical Society of America

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References

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  1. Committee on Colorimetry, Optical Society of America, The Science of Color (New York: Thomas Y. Crowell Company, 1953), p. 379. Currently available from Optical Society of America, 1155 Sixteenth St., N.W., Washington D. C. 20036.
  2. H. J. McNicholas, J. Res. Natl. Bur. Std. (U.S.) 1, 29 (1928).
  3. C. S. McCamy, Phot. Sci. Eng. 10, 314 (Nov.–Dec.1966).
  4. W. M. Brandenberg and J. T. Neu, J. Opt. Soc. Am. 56, 97 (1966).
    [Crossref]

1966 (2)

C. S. McCamy, Phot. Sci. Eng. 10, 314 (Nov.–Dec.1966).

W. M. Brandenberg and J. T. Neu, J. Opt. Soc. Am. 56, 97 (1966).
[Crossref]

1928 (1)

H. J. McNicholas, J. Res. Natl. Bur. Std. (U.S.) 1, 29 (1928).

Brandenberg, W. M.

McCamy, C. S.

C. S. McCamy, Phot. Sci. Eng. 10, 314 (Nov.–Dec.1966).

McNicholas, H. J.

H. J. McNicholas, J. Res. Natl. Bur. Std. (U.S.) 1, 29 (1928).

Neu, J. T.

J. Opt. Soc. Am. (1)

J. Res. Natl. Bur. Std. (U.S.) (1)

H. J. McNicholas, J. Res. Natl. Bur. Std. (U.S.) 1, 29 (1928).

Phot. Sci. Eng. (1)

C. S. McCamy, Phot. Sci. Eng. 10, 314 (Nov.–Dec.1966).

Other (1)

Committee on Colorimetry, Optical Society of America, The Science of Color (New York: Thomas Y. Crowell Company, 1953), p. 379. Currently available from Optical Society of America, 1155 Sixteenth St., N.W., Washington D. C. 20036.

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Figures (3)

Fig. 1
Fig. 1

The upper diagrams indicate hemispherical (2π), conical (g), and directional (θ0) incidence, respectively, and the lower diagrams indicate hemispherical (2π), conical (g′), and directional (θr) collection. The indications are for a particular pair of azimuths: ϕ0 and ϕ0+π, for incidence; and ϕr and ϕr+π, for collection. The arrows in the upper diagrams all point toward the center of the specimen represented by a short horizontal line; those in the lower diagrams all point away from that center.

Fig. 2
Fig. 2

The symbols refer to the nine kinds of reflectance ρ, the six kinds of reflectance factor (R with finite solid-angular subtense of collection), and the three kinds of radiance factor (β with elemental solid-angular subtense of collection). Each arrow starts from the symbol for the concept whose average value corresponds to another of the concepts, and the arrow head indicates the symbol for that concept. In the labeling of the arrows, A stands for average, and the appropriate weighting factor (either cosθ0 or cosθr) is also shown. All six kinds of reflectance factors can be expressed as averages of directional radiance factor; but only two of the eight other kinds of reflectance can be expressed as averages of bi-directional reflectance.

Fig. 3
Fig. 3

Symbols shown are the same as those in Fig. 2. The arrows, like those in Fig. 2, pass from the symbols for concepts whose values may be processed so as to compute a value for the concept whose symbol is indicated by the arrow head. The arrows are labeled to indicate the process of computation: integral sign for integration, summation sign for summation, equal sign for numerical equality, PD for multiplication or division by the reflectance of the perfect diffuser, and R for numerical equivalents that, because of the Helmholtz reciprocity relation, hold when the angular conditions of incidence and collection are interchanged. There exist three reciprocal relations that cannot be shown by this diagrammatic method:

Tables (3)

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Table I Values of the nine kinds of reflectance for the perfect mirror and for the perfect diffuser.

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Table II Interrelationships among the nine kinds of reflectance.

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Table III Interrelationships among the six kinds of reflectance factor, the three kinds of radiance factor, and the nine kinds of reflectance.

Equations (2)

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ρ ( θ 0 , ϕ 0 : 2 π ) = β ( 2 π : θ r , ϕ r )
ρ ( θ 0 = m , ϕ 0 = n , θ r = u , ϕ r = v ) = ρ ( u , v : m , n ) β ( θ 0 = m , ϕ 0 = n , θ r = u , ϕ r = v ) = β ( u , v : m , n ) R ( g = a : g = b ) = R ( b : a ) .