# Which luminous efficacy function does BIPM reference?

The definition of candela given by BIPM is as follows:

The candela, symbol cd, is the SI unit of luminous intensity in a given direction. It is defined by taking the fixed numerical value of the luminous efficacy of monochromatic radiation of frequency $$540\times10^{12}\,\mathrm{Hz}$$, $$K_{\mathrm{cd}}$$, to be $$683$$ when expressed in the unit $$\mathrm{lm\,W^{–1}}$$, which is equal to $$\mathrm{cd\,sr\,W^{–1}}$$, or $$\mathrm{cd\,sr\,kg^{–1}\,m^{–2}\,s^3}$$, where the kilogram, metre and second are defined in terms of $$h$$, $$c$$ and $$\Delta \nu_{\mathrm{Cs}}$$.

So, from this definition we can only directly characterize luminous intensity of 540 THz light. To find luminous intensity of light with any other spectrum we need the normalized luminous efficacy function (to be scaled by $$K_{\mathrm{cd}}$$). But this one is not so definite: there're at least three versions of it, shown below in black: CIE 1931 (solid), Judd-Vos modified (dashed), and Sharpe, Stockman, Jagla & Jägle 2005 (dotted).

So which of these, if any, does BIPM reference with its definition of candela?

• Those look like different realizations of the same standard rather than different functions to me. (but to be sure this probably needs a professional metrologist.) – Emilio Pisanty Sep 29 at 8:54

The BIPM booklet on photometry [MEP 2019. p 6] states: "The above definition of the candela applies to photopic, scotopic and mesopic vision."

In essence, this means that for any spectral luminosity function whatever, the value for 540 THz is by definition 683 lm/W. In other words, the graphs of these functions, when using SI units, all go through the same point (540 THz, 683 lm/W). The SI system does not depend on any specific luminosity function; it can be applied to all of them.

For current realizations of the SI units lm, cd, lx with various luminosity functions, see [REAL 2019]; for an introduction into concepts of photometry, see [MEP 2019].

References:

    [REAL 2019] "Mise en pratique for the definition of the candela
and associated derived units for photometric and
Part of Appendix 2 to the SI Brochure 9, 2019.
<https://www.bipm.org/utils/en/pdf/si-mep/SI-App2-candela>

[MEP 2019] "BUREAU INTERNATIONAL DES POIDS ET MESURES.
PRINCIPLES GOVERNING PHOTOMETRY".
2nd Edition. 2019.
<https://www.bipm.org/utils/common/pdf/rapportBIPM/RapportB IPM-2019-05.pdf>

• Actually the rules appear to be more specific, as given in clause 4.5 of your link. Moreover, they even provide tables of the luminosity functions, which is actually what I was looking for. So it's not as bad as "10 lm means nothing until you tell your luminosity function". It would be nice if you edited your answer to include these rules from 4.5, or at least mention them. – Ruslan Nov 8 at 17:27
• You are right, 10 lm means something for 540 THz; but what it means for other frequencies is not defined by the SI system. Nor should it be, in my opinion. There is a similar case for the sievert: in this case, the SI system does not even give a reference for any radiation with quality factor 1 Sv/Gy; the choice of quality factors just is not a matter of the unit definition. Thanks for your proposal! I added photometric references as in the SI Brochure -- I hope that is what you meant. – Michael Deckers Nov 8 at 21:01
• I actually meant that $\S4.5$ of MEP 2019 defines that, for instance, "If the luminous efficiency function used is not specified, it is assumed to be the $V(\lambda)$ function.". And $V(\lambda)$ is the photopic luminosity function defined in Table 1 of the same reference. So 10 lm means, unless specified differently, the value one would obtain from a spectral power distribution with the spectral weight function given in Table 1 of MEP 2019 ­— not only the value for 540 THz monochromatic light. – Ruslan Nov 8 at 21:32

The answer by Michael Deckers provides some useful references, but is not completely correct.

2. The conditions when to use each of these functions in section 4.5. In particular, there's the following rule for the 2° FOV photopic luminous efficiency function $$V(\lambda)$$:
... the $$V(\lambda)$$ function applies at all luminance levels for foveal view, i.e. for all on-axis visual tasks (where objects seen by the eye are in a narrow field of view in central vision).
If the luminous efficiency function used is not specified, it is assumed to be the $$V(\lambda)$$ function.