Abstract

Experimental difficulties encountered in the determination of spectral sensitivity functions (Sλ) of photoelectric detectors make it desirable to have independent checks on a measured Sλ. A multifilter method is described which allows not only a check on a given Sλ, but also makes it possible to determine Sλ directly and independently of any previous measurements. The filters employed in the method have to satisfy the condition that their spectral transmittance functions form a set of linearly independent functions over the spectral range considered. The practical importance of the method and ways of checking its precision are discussed, using a numerical example which involves 14 linearly independent filters forming a 14×14 matrix of spectral transmittances for the spectral range 390 to 670 mμ.

© 1960 Optical Society of America

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References

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  1. H. Wright, J. Opt. Soc. Am. 49, 980 (1959).
    [Crossref]
  2. H. S. Moran, J. Opt. Soc. Am. 45, 12 (1955).
    [Crossref]
  3. Here and in the following it is always assumed that the responses of the detector are directly proportional to the level of illumination on the photosensitive surface.
  4. Functions of high spectral selectivity show high curvatures, that is, large second derivatives with respect to λ, at one or more wavelengths. Their nonzero and/or near-zero ordinates are confined to one or more rather limited spectral ranges.
  5. J. v. Neumann and H. H. Goldstine, Bull. Am. Math. Soc. 53, 1021 (1947).
    [Crossref]
  6. Since the actual spectral transmittances obtained are valid only for the particular set used in this study, publication of these data has been omitted. Any duplicate set of filters would have to be newly calibrated, since manufacturer’s specifications are not sufficient to assume a definite spectral transmittance curve. However, the linear independence of the filters of duplicate sets would not be likely to be affected.
  7. G. Wyszecki, Visual Problems of Colour, NPL Symposium (Her Majesty’s Stationary Office, London, 1958), No. 8, Vol. 1, p. 363.

1959 (1)

1955 (1)

1947 (1)

J. v. Neumann and H. H. Goldstine, Bull. Am. Math. Soc. 53, 1021 (1947).
[Crossref]

Goldstine, H. H.

J. v. Neumann and H. H. Goldstine, Bull. Am. Math. Soc. 53, 1021 (1947).
[Crossref]

Moran, H. S.

Neumann, J. v.

J. v. Neumann and H. H. Goldstine, Bull. Am. Math. Soc. 53, 1021 (1947).
[Crossref]

Wright, H.

Wyszecki, G.

G. Wyszecki, Visual Problems of Colour, NPL Symposium (Her Majesty’s Stationary Office, London, 1958), No. 8, Vol. 1, p. 363.

Bull. Am. Math. Soc. (1)

J. v. Neumann and H. H. Goldstine, Bull. Am. Math. Soc. 53, 1021 (1947).
[Crossref]

J. Opt. Soc. Am. (2)

Other (4)

Here and in the following it is always assumed that the responses of the detector are directly proportional to the level of illumination on the photosensitive surface.

Functions of high spectral selectivity show high curvatures, that is, large second derivatives with respect to λ, at one or more wavelengths. Their nonzero and/or near-zero ordinates are confined to one or more rather limited spectral ranges.

Since the actual spectral transmittances obtained are valid only for the particular set used in this study, publication of these data has been omitted. Any duplicate set of filters would have to be newly calibrated, since manufacturer’s specifications are not sufficient to assume a definite spectral transmittance curve. However, the linear independence of the filters of duplicate sets would not be likely to be affected.

G. Wyszecki, Visual Problems of Colour, NPL Symposium (Her Majesty’s Stationary Office, London, 1958), No. 8, Vol. 1, p. 363.

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram of a source(S)-monochromator-thermocouple (T) arrangement to measure the relative spectral sensitivity function of photoelectric detector (P).

Fig. 2
Fig. 2

Schematic diagram of the optical arrangement for determining the relative spectral sensitivity function by the multifilter method. S=source, Fi=ith filter, P=detector to be calibrated.

Fig. 3
Fig. 3

Spectral transmittances tiλ of 14 linearly independent filters each one consisting of a one or two component glass combination as specified in Table I.

Fig. 4
Fig. 4

Spectral transmittances t of a special cutoff filter to limit the spectral range to 390 to 670 mμ.

Fig. 5
Fig. 5

Testing the condition of the T matrix. Two examples taken from Table II. Responses R6 and R7 were varied by +0.5% of their respective values and new solutions Pj(6) and Pj(7) computed, respectively. These solutions are shown together with the original Pj.

Fig. 6
Fig. 6

Result of a determination of a Pλ function involving the spectral sensitivity of a Gillod-Boutry photocell. Pj is the actually computed values, Pλ(c) is the smoothed solution function plotted through Pj and assumed to be the true Pλ function.

Tables (4)

Tables Icon

Table I Specifications of 14 linearly independent glass filters and cutoff filter.

Tables Icon

Table II Test of condition of matrix T. A change of Ri by +0.5% causes a change of Pj to Pj(i). The percent deviation is given by Dj(i)=100(Pj(i)Pj)/Pj.

Tables Icon

Table III Ratios ri and ri′ of measured responses Ri(m) and computed responses Ri(c) and Ri(c)′ based on smoothed Pλ(c) and Pλ(c)′ functions, respectively.

Tables Icon

Table IV Typical ratios αi of responses calculated from 1-mμ wavelength intervals and responses calculated from 20 mμ wavelength intervals.

Equations (15)

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R P = λ 1 λ 2 P λ t λ ( P ) H λ d λ
R T = λ 1 λ 2 T λ t λ ( T ) H λ d λ             ( λ 1 λ λ 2 )
R i ( m ) = k λ t i λ H λ S λ d λ .
k = 1 / λ t p λ H λ S λ d λ ,
R i = k λ t i λ H λ S λ Δ λ             ( k = 1 ) .
R i = j = 1 n t i j H j S j .
R i = j = 1 n t i j t 0 j H j S j
R i = j = 1 n t i j P j             with             P j t 0 j H j S j .
r = Tp ,
p = T - 1 r ,
t i j = 1 40 [ t i , λ j - 10 + 2 λ = λ j - 9 λ = λ j + 9 t i λ + t λ j + 10 ] ,
r i = R i ( m ) / R i ( c ) = R i ( m ) / k * λ = 390 670 t i λ P λ ( c ) Δ λ
R i ( m ) = k λ = 390 670 t i λ P λ d λ
R i = k j = 1 14 t i j P j .
λ = 390 670 t i λ P λ d λ = 1 2 [ t i , 390 P 390 + 2 λ = 391 669 t i λ P λ + t i , 670 P 670 ] , 1 / k = 1 2 [ P 390 + 2 λ = 391 669 P λ + P 670 ] .