Abstract

It has been noted that certain spectrophotometric curves, especially those of the sharp cut-off type of colored glasses, have essentially the same shape—merely displaced in wave-length for different thicknesses. If this were strictly true, it must be that the curve between log log transmittance and wave-length is a straight line, since only a straight line can be moved vertically (by change of glass thickness) and match the same curve moved horizontally (by change of wave-length). If it is assumed that the log log transmittance versus wave-length curve is a straight line the curve between transmittance and wave-length has an S shape which is not a bad fit for actual spectrophotometric curves of certain glasses. If the fit were perfect, the wave-length displacement of the curve caused by change in thickness would be proportional to the change of the logarithm of thickness, the constant in the equation can be determined by the slope of the curve. With a series of infinitely sharp cut-off curves in which there is complete transmission for all wave-lengths greater than a certain wave-length, and complete absorption for all shorter than this, the values of Y, x, y, and z (Transmission and x, y, and z I.C.I. Coordinates), may be calculated. A wave-length of cut can be found which will produce the same chromaticity, x and y coordinates as any of the sharp cut glasses but the transmission will differ, being the maximum possible for the given chromaticity. The change in chromaticity expressed as change in y coordinate can be determined relative to change in wave-length of cut-off. By combining this latter relation with the change of wave-length of cut-off caused by change of thickness of the given glass as determined from the shape of the log log curve it is possible to calculate the required change in thickness to produce any desired change in the y coordinate within the capabilities of the glass.

© 1945 Optical Society of America

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References

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  1. A. C. Hardy, Handbook of Colorimetry (The Technology Press, Massachusetts Institute of Technology, Cambridge Massachusetts, 1936).
  2. H. P. Gage, “Colorimetric Efficiency of Red Glass,” J. Opt. Soc. Am. 21, 433 (1931).
  3. Report of the Committee on Colorimetry for 1920–21, J. Opt. Soc. Am.6, 527 (1922).
    [Crossref]
  4. D. L. MacAdam, “The Theory of the Maximum Visual Efficiency of Colored Materials,” J. Opt. Soc. Am. 25, 249 (1935); and J. Opt. Soc. Am. 25, 361 (1935).
    [Crossref]
  5. Deane B. Judd, “The 1931 Standard Observer and Coordinate System for Colorimetry,” J. Opt. Soc. Am. 23, 359–374 (1933). See Table III, p. 365.
    [Crossref]
  6. Committee on Colorimetry, “Quantitative Data and Methods for Colorimetry,” J. Opt. Soc. Am. 34, 633–688 (1944). See Table VII, p. 643.

1944 (1)

1935 (1)

1933 (1)

1931 (1)

H. P. Gage, “Colorimetric Efficiency of Red Glass,” J. Opt. Soc. Am. 21, 433 (1931).

Gage, H. P.

H. P. Gage, “Colorimetric Efficiency of Red Glass,” J. Opt. Soc. Am. 21, 433 (1931).

Hardy, A. C.

A. C. Hardy, Handbook of Colorimetry (The Technology Press, Massachusetts Institute of Technology, Cambridge Massachusetts, 1936).

Judd, Deane B.

MacAdam, D. L.

J. Opt. Soc. Am. (4)

Other (2)

Report of the Committee on Colorimetry for 1920–21, J. Opt. Soc. Am.6, 527 (1922).
[Crossref]

A. C. Hardy, Handbook of Colorimetry (The Technology Press, Massachusetts Institute of Technology, Cambridge Massachusetts, 1936).

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Figures (5)

Fig. 1
Fig. 1

(a) Sharp cut-off type glasses, reproduction of curve sheet made on Hardy G.E. spectrophotometer from nearly colorless Noviol A no. 3389 to dark Pyrometer red no. 2403. The two glasses with lowest transmission at long wave-length end are of the same melt but different thickness. All other glasses are of separate melts. (b) Four pieces of the same melt of an orange glass, thickness roughly .5 mm, 1 mm, 2 mm, and 5.14 mm. The .5 mm piece is too thin to have complete and abrupt absorption at the short wave-length end and has a gradual toe. The others show the small but definite shift in cut-off with increasing thickness. Also, that the curves rise to a nearly flat maximum with but small absorption for the longer wave-lengths.

Fig. 2
Fig. 2

A mathematical curve illustrating the relationships in Table I. On the horizontal axis is plotted the values of log log T′ and along the vertical axis is plotted the corresponding values of T′ 0 to 1.00. Also, a straight line representing .8 log log T′+.08 which it is noticed nearly coincides with the S shape T′ curve between the limits of T′=.15 and .64 but the T′ curve departs from the straight line at the toe below .16 and on the long shoulder above .64. The differences are to be noted in the last two columns of Table I. Curve a is condensed in horizontal scale and has a close resemblance to the actual spectrophotometric curves in Fig. 1; curve b is on a more extended horizontal scale.

Fig. 3
Fig. 3

Measured spectral transmission curve and T′=T/Tmax curve. The derived or T′ curve has the same relative transmission distribution as the measured transmission curve but for each wave-length the values are increased sufficiently so the maximum point is 100 percent or unity. The chromaticity of light transmitted with either, i.e., the x, y, and z coordinates, will be the same and also the same as a theoretical transmitter in which all wave-lengths shorter than the cut-off wave-length λc are removed, and all wave-lengths greater than λc are completely transmitted. The luminous transmissions, i.e., the Y values, will differ.

Fig. 4
Fig. 4

The log log T′ is plotted against wave-length λ for two pieces of the same glass of differing thickness, assuming these curves to be straight lines. For a single glass the difference in height of the points is Δ log β. For two thicknesses of glass having the same transmission for two different wave-lengths, λ1 and λ2 differing by Δλ the vertical separation is Δ log t.

Fig. 5
Fig. 5

Relationships shown in Table II. For theoretical sharp cut-off filters having zero transmission at the wave-length indicated and complete transmissions for all greater wave-lengths the trichromatic coefficients x, y, and z and the transmission Y are as shown. This is calculated for illuminant A, 2842°K. The dashed curve Δy is the difference in the y value caused by a change in cut-off position by 0.01 micron or 10 mμ. The slope n, i.e., Δy/Δλ when λ is measured in milimicrons is 1/10 of this value.

Tables (1)

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Table II Colorimetric properties of theoretical sharp cut filters with illuminant A, at 2842°K.

Equations (25)

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10 x = a ;             10 β t = T .
X = log 10 a ;             β t = log 10 T .
d 10 x = 10 x log e 10 d x ; d 10 β t = 10 β t log e 10 d β t ; = 10 x d x / .4343 ; d 10 β t = 10 β t d β t / .4343.
d T / d β t = T / .4343.
γ = log 10 β t ,
10 v = β t ,
d β t / d γ = β t / .4343.
d T d β t × d β t d γ = d T d γ ,
T .4343 × β t .4343 = d T d γ = T β t .4343 2 = T β t .18861 .
d T = T β t d γ / .4343 2 ,
d β t = β t d γ / .4343
d U V = U d v + V d u . d 2 T = d ( T β t d γ ) / .4343 2 ,
= ( d T β t d γ + T d β t d γ ) / .4343 2 .
d 2 T = 1 .4343 2 × ( T β t d γ × β t d γ .4343 2 + T β t d γ d γ .4343 ) ,
d 2 T / d γ 2 = T .4343 3 ( β t .4343 + 1 ) .
d T / d γ = T β t / .4343 2 = ( T × - .4343 ) / .4343 2 = T / - .4343 = - .848.
k = ( Δ log β t ) / Δ λ ,
k = ( log β 2 t 1 - log β 1 t 1 ) / ( λ 2 - λ 1 ) ;
k = ( log β 2 + log t 1 - log β 1 - log t 1 ) / ( λ 2 - λ 1 ) .
k = ( log β 2 - log β 1 ) / ( λ 2 - λ 1 ) ,             k = Δ β / Δ λ .
k = log β 2 t 1 - log β 1 t 1 λ 2 - λ 1 = log β 1 t 2 - log β 1 t 1 λ 2 - λ 1 ,
k = log β 1 + log t 2 - log β 1 - log t 1 λ 2 - λ 1 ;
k = log t 2 - log t 1 λ 2 - λ 1 or             k = Δ log t Δ λ .
Δ λ / Δ log t = 1 / k . n = Δ y / Δ λ from Table II . Δ y / Δ λ × Δ λ / Δ log t = Δ y / Δ log t ,
Δ y / Δ log t = n / k             or             Δ y = n k Δ log t .