Abstract

Two nonlinear curve-fitting computer programs have been developed for designing subtractive glass filter combinations for use in photoelectric photometers. One program adjusts the filter thicknesses to minimize the sum of squares of errors at each wavelength and is thus useful in designing a general purpose instrument. The second program minimizes the errors in the photometric measurements of preselected light sources. Filters for correcting various photoelectric detectors to the CIE luminous efficiency function are specified. Several spectral response functions have been utilized including S-4, S-10, S-11, S-20 types, and those of typical silicon diodes, and selenium cells. Practical photometer designs with predicted measurement errors of less than 0.1% are presented.

© 1969 Optical Society of America

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References

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  1. W. E. R. Davies, G. Wyszecki, J. Opt. Soc. Amer. 52, 679 (1962).
    [CrossRef]
  2. If R = N+ 1 and the filters are appropriately chosen so the spectral transmittances are linearly independent it may be possible to select filter thicknesses which will provide a photometer with zero or near zero photometric error but this is not,a sufficient condition. If R > N+ 1 a least squares fit will usually be possible with a solution in which there are residual photometric errors PE(K). It is possible that the E(K,λ) may be linearly dependent making the number of independent spectral distributions less than R. If this happens the perfect fit may be made with N< R-1 filters.
  3. G. Wyszecki, W. S. Stiles, Color Science: concepts and methods, quantitative data and formulas (John Wiley & Sons, Inc., New York, 1967).
  4. H. J. Keegan, J. C. Schleter, D. B. Judd, J. Res. Nat. Bur. Std. 66A, 203 (1962).
    [CrossRef]
  5. W. Budde, CIE Compte Rendus 1967, CIE Publication 14A, 77 (1968).

1968

W. Budde, CIE Compte Rendus 1967, CIE Publication 14A, 77 (1968).

1962

W. E. R. Davies, G. Wyszecki, J. Opt. Soc. Amer. 52, 679 (1962).
[CrossRef]

H. J. Keegan, J. C. Schleter, D. B. Judd, J. Res. Nat. Bur. Std. 66A, 203 (1962).
[CrossRef]

Budde, W.

W. Budde, CIE Compte Rendus 1967, CIE Publication 14A, 77 (1968).

Davies, W. E. R.

W. E. R. Davies, G. Wyszecki, J. Opt. Soc. Amer. 52, 679 (1962).
[CrossRef]

Judd, D. B.

H. J. Keegan, J. C. Schleter, D. B. Judd, J. Res. Nat. Bur. Std. 66A, 203 (1962).
[CrossRef]

Keegan, H. J.

H. J. Keegan, J. C. Schleter, D. B. Judd, J. Res. Nat. Bur. Std. 66A, 203 (1962).
[CrossRef]

Schleter, J. C.

H. J. Keegan, J. C. Schleter, D. B. Judd, J. Res. Nat. Bur. Std. 66A, 203 (1962).
[CrossRef]

Stiles, W. S.

G. Wyszecki, W. S. Stiles, Color Science: concepts and methods, quantitative data and formulas (John Wiley & Sons, Inc., New York, 1967).

Wyszecki, G.

W. E. R. Davies, G. Wyszecki, J. Opt. Soc. Amer. 52, 679 (1962).
[CrossRef]

G. Wyszecki, W. S. Stiles, Color Science: concepts and methods, quantitative data and formulas (John Wiley & Sons, Inc., New York, 1967).

CIE Compte Rendus 1967

W. Budde, CIE Compte Rendus 1967, CIE Publication 14A, 77 (1968).

J. Opt. Soc. Amer.

W. E. R. Davies, G. Wyszecki, J. Opt. Soc. Amer. 52, 679 (1962).
[CrossRef]

J. Res. Nat. Bur. Std.

H. J. Keegan, J. C. Schleter, D. B. Judd, J. Res. Nat. Bur. Std. 66A, 203 (1962).
[CrossRef]

Other

If R = N+ 1 and the filters are appropriately chosen so the spectral transmittances are linearly independent it may be possible to select filter thicknesses which will provide a photometer with zero or near zero photometric error but this is not,a sufficient condition. If R > N+ 1 a least squares fit will usually be possible with a solution in which there are residual photometric errors PE(K). It is possible that the E(K,λ) may be linearly dependent making the number of independent spectral distributions less than R. If this happens the perfect fit may be made with N< R-1 filters.

G. Wyszecki, W. S. Stiles, Color Science: concepts and methods, quantitative data and formulas (John Wiley & Sons, Inc., New York, 1967).

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Figures (7)

Fig. 1
Fig. 1

Relative spectral responses of photomultipliers: No. 1 and No. 2, RCA 6217; and No. 3, RCA 1P21.

Fig. 2
Fig. 2

Relative spectral responses of detectors: No. 4, EMI 9558 PMT; No. 5, barrier layer cell; and No. 8, Gillod-Boutry photocell.

Fig. 3
Fig. 3

Relative spectral responses of detectors: No. 6, Siemens silicon diode with Schott WG 1 filter 2 mm thick; No. 7, EMI 9524 PMT; No. 9, VB59 Rank-Cintell photocell.

Fig. 4
Fig. 4

Spectral transmittances of filters: No. 1, Corning 3307, 2.76 mm; No. 2, Corning 3962, 2.51 mm; No. 3, Corning 4784, 5.03 mm; and No. 4, Corning 9788, 5.05 mm.

Fig. 5
Fig. 5

Spectral transmittances of filters: No. 5, Chance OGr 3, 2.21 mm; No. 6, Chance OY 13, 1.88 mm; No. 7, Chance OY 21, 2.10 mm; and No. 8, Schott BG 38, 2.03 mm.

Fig. 6
Fig. 6

Spectral transmittances of filters: No. 9, Schott FG 9, 1.98 mm; No. 10, Schott GG 4, 2.02 mm; No. 11, Schott GG 7, 1.94 mm; and No. 12, Schott GG 17, 2.09 mm.

Fig. 7
Fig. 7

Relative spectral irradiances from three fluorescent lamps: Source 5, F.L. 3100 K; Source 6, F.L. 4400 K; and Source 7, F.L. 7000 K.

Tables (4)

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Table I Approximation to V(λ) Function Using Detector No. 9 and Four Filters by Minimizing Spectral Errors

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Table II Photometric Errors S(K) for Seven Sources by Two Minimization Methods

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Table III Spectral Response of Detector No. 9 Modified by Four Filters to Give Minimum Photometric Error

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Table IV Parameters Corresponding to Filter Combinations for Nine Detectors Giving Best Least-Squares Fit, Best Photometric Fit, and Maximum Photometric Error

Equations (6)

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F ( λ ) = 0.92 α S ( λ ) [ u ( 1 , λ ) ] P ( 1 ) [ u ( 2 , λ ) ] P ( 2 ) . . . [ u ( n , λ ) ] P ( n ) . . . [ u ( N , λ ) ] P ( N )
F ( λ ) = 0.92 α S ( λ ) n = 1 N [ u ( n , λ ) ] P ( n ) ,
Q = λ [ V ( λ ) F ( λ ) ] 2 d λ
S ( K ) = 100 λ E ( K , λ ) F ( λ ) d λ λ E ( 1 , λ ) V ( λ ) d λ λ E ( K , λ ) V ( λ ) d λ λ E ( 1 , λ ) F ( λ ) d λ × 100 ( percent ) ,
P E ( K ) = λ E ( K , λ ) V ( λ ) d λ λ E ( K , λ ) F ( λ ) d λ λ E ( K , λ ) V ( λ ) d λ × 100 ( percent ) ,
PES = K = 1 R [ P E ( K ) ] 2 .

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