= Spectral mismatch correction
{title2=$F=\Phi_v/\Phi_{v,\mathrm{indicated}}$}
For collected <spectral radiant flux> $f(\lambda)$ and a linear <photodiode> with <spectral responsivity> $r(\lambda)$, the dark-subtracted <electric current> is $i=\int r f\,d\lambda$. A reference <optical spectrum> $f_A$ sets the conversion $C_A$ from <electric current> to collected <luminous flux>. Applying it to another <optical spectrum> $f_T$ gives
$$
C_A=\frac{K_m\int V f_A\,d\lambda}{\int r f_A\,d\lambda},\qquad
\Phi_{v,\mathrm{indicated}}=C_A\int r f_T\,d\lambda,
\qquad
F=\frac{\int V f_T\,d\lambda}{\int r f_T\,d\lambda}
\frac{\int r f_A\,d\lambda}{\int V f_A\,d\lambda}.
$$
Assume nonzero calibration and test signals. Then $F\Phi_{v,\mathrm{indicated}}$ is the corrected <luminous flux>. Multiplying either <optical spectrum> by an overall positive scale leaves $F$ unchanged. If $r$ is proportional to $V$, $F=1$ for every <optical spectrum>. Uniformly overfilled aperture measurements use <wavelength>-resolved <irradiance> instead, multiplying the <electric current> expression by aperture <area>; the same ratio cancellation gives the <illuminance> correction. This is the algebra behind source-dependent calibration, rather than a new measured correction curve. <Realization of the candela>{ref}
Back to article page
