## Abstract

Experimental difficulties encountered in the determination of spectral sensitivity functions (*S*_{λ}) of photoelectric detectors make it desirable to have independent checks on a measured *S*_{λ}. A multifilter method is described which allows not only a check on a given *S*_{λ}, but also makes it possible to determine *S*_{λ} 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 m*μ*.

© 1960 Optical Society of America

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### Equations (15)

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(1)
$${R}_{P}={\int}_{{\mathrm{\lambda}}_{1}}^{{\mathrm{\lambda}}_{2}}{P}_{\mathrm{\lambda}}{{t}_{\mathrm{\lambda}}}^{(P)}{H}_{\mathrm{\lambda}}d\mathrm{\lambda}$$
(2)
$${R}_{T}={\int}_{{\mathrm{\lambda}}_{1}}^{{\mathrm{\lambda}}_{2}}{T}_{\mathrm{\lambda}}{{t}_{\mathrm{\lambda}}}^{(T)}{H}_{\mathrm{\lambda}}d\mathrm{\lambda}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}({\mathrm{\lambda}}_{1}\le \mathrm{\lambda}\le {\mathrm{\lambda}}_{2})$$
(3)
$${{R}_{i}}^{(m)}=k{\int}_{\mathrm{\lambda}}{t}_{i\mathrm{\lambda}}{H}_{\mathrm{\lambda}}{S}_{\mathrm{\lambda}}d\mathrm{\lambda}.$$
(4)
$$k=1/{\int}_{\mathrm{\lambda}}{t}_{p\mathrm{\lambda}}{H}_{\mathrm{\lambda}}{S}_{\mathrm{\lambda}}d\mathrm{\lambda},$$
(5)
$${R}_{i}=k{\sum}_{\mathrm{\lambda}}{t}_{i\mathrm{\lambda}}{H}_{\mathrm{\lambda}}{S}_{\mathrm{\lambda}}\mathrm{\Delta}\mathrm{\lambda}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}(k=1).$$
(6)
$${R}_{i}=\sum _{j=1}^{n}{t}_{ij}{H}_{j}{S}_{j}.$$
(7)
$${R}_{i}=\sum _{j=1}^{n}{t}_{ij}{t}_{0j}{H}_{j}{S}_{j}$$
(8)
$${R}_{i}=\sum _{j=1}^{n}{t}_{ij}{P}_{j}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\text{with}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}{P}_{j}\equiv {t}_{0j}{H}_{j}{S}_{j}.$$
(9)
$$\mathbf{r}=\mathbf{Tp},$$
(10)
$$\mathbf{p}={\mathbf{T}}^{-1}\mathbf{r},$$
(11)
$${t}_{ij}=\frac{1}{40}\left[{t}_{i,\mathrm{\lambda}j-10}+2\sum _{\mathrm{\lambda}=\mathrm{\lambda}j-9}^{\mathrm{\lambda}=\mathrm{\lambda}j+9}{t}_{i\mathrm{\lambda}}+{t}_{\mathrm{\lambda}j+10}\right],$$
(12)
$${r}_{i}={{R}_{i}}^{(m)}/{{R}_{i}}^{(c)}={{R}_{i}}^{(m)}/k*\sum _{\mathrm{\lambda}=390}^{670}{t}_{i\mathrm{\lambda}}{{P}_{\mathrm{\lambda}}}^{(c)}\mathrm{\Delta}\mathrm{\lambda}$$
(13)
$${{R}_{i}}^{(m)}=k\underset{\mathrm{\lambda}=390}{\overset{670}{\mathit{\int}}}{t}_{i\mathrm{\lambda}}{P}_{\mathrm{\lambda}}d\mathrm{\lambda}$$
(14)
$${R}_{i}={k}^{\prime}\sum _{j=1}^{14}{t}_{ij}{P}_{j}.$$
(15)
$$\begin{array}{ll}\hfill \underset{\mathrm{\lambda}=390}{\overset{670}{\mathit{\int}}}{t}_{i\mathrm{\lambda}}{P}_{\mathrm{\lambda}}d\mathrm{\lambda}& =\frac{1}{2}\left[{t}_{i,390}{P}_{390}+2\sum _{\mathrm{\lambda}=391}^{669}{t}_{i\mathrm{\lambda}}{P}_{\mathrm{\lambda}}+{t}_{i,\hspace{0.17em}670}{P}_{670}\right],\\ \hfill 1/k& =\frac{1}{2}\left[{P}_{390}+2\sum _{\mathrm{\lambda}=391}^{669}{P}_{\mathrm{\lambda}}+{P}_{670}\right].\end{array}$$