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At exactly $540\mathrm{THz}$, radiant intensity $1/683\mathrm{W}\mathrm{s}{\mathrm{r}}^{-1}$ corresponds to luminous intensity $1\mathrm{cd}$. This convention specifies a direction, rather than the source's total luminous flux. ^[ref]

In NIST's 2001 comparison, the Laser-Optimized Cryogenic Radiometer held its cavity near $5.4\mathrm{K}$ and controlled the heat sink separately. Opening the beam reduced heater power at fixed cavity temperature; an AC bridge read the temperature. Tests used stabilized, spatially filtered lasers at $488$, $514$ and $633\mathrm{nm}$, with horizontal input and a movable photodiode transfer detector in the cavity's optical plane. The paper contrasts this with the High Accuracy Cryogenic Radiometer's optical heating followed by two electrical heating cycles and interpolation.^[ref]

The two laboratories compared calibrations of the same transfer detectors, and checked their spatial response. Alignment differences and transported dust limited that comparison; its agreement did not prove universal equivalence. Its correction tables retain window transmission, cavity radiometric absorptance, alignment, electrical readout and optical-electrical heating non-equivalence. The heat sink was controlled, but §3 does not specify its numerical setpoint: the $5.0\mathrm{K}$ reference in the preceding example is stipulated.^[ref]

For a reproducible substitution measurement, record the beam's wavelength and entrance boundary, receiver and reference temperatures, heater terminal voltage and electric current, dark readings before and after illumination, and the settling rule. Check beam collection at the aperture, window transmission, radiometric absorptance, deposited lead heating, and sensitivity to beam position. Repeatability alone misses a stable correction error. Photometric calibration can subsequently use this radiant flux scale, but photometry still needs the appropriate spectral visual weighting; the cold receiver does not itself measure luminous flux.

An original declared model makes the distinction visible without substituting invented numbers for the photopic luminous efficiency function. Define a triangular weightThis $w$ is an illustrative function, not a fit or table of $V$. Three ideal spectral lines, each carrying radiant flux $1\mathrm{mW}$, at $450$, $555$ and $650\mathrm{nm}$ have weights $3/10$, $1$ and $11/30$. Their model weighted outputs are respectively $0.2049$, $0.6830$ and approximately $0.25043\mathrm{lm}$. These are model calculations, not standard luminous fluxes or measured data. For mixtures of these lines the model outputs add, just as the integral adds independent spectral contributions.

Illuminance is incident luminous flux per receiving area, measured in lux. It is the visually weighted counterpart of irradiance. Uniform illumination of an aperture of area $A$ gives collected luminous flux $A{E}_v$. An aperture measurement alone does not give a source's total emitted luminous flux.^[ref]

The lumen is the International System of Units unit of luminous flux: $1\mathrm{lm}=1\mathrm{cd}\mathrm{sr}$. A source with constant luminous intensity $1\mathrm{cd}$ within a cone of solid angle $1\mathrm{sr}$ supplies $1\mathrm{lm}$ into that cone. A source with the same luminous intensity in every direction supplies $4\pi \mathrm{lm}$ over a full sphere. ^[ref]

The luminous efficacy of radiation is luminous flux divided by total radiant flux, in $\mathrm{lm}{\mathrm{W}}^{-1}$, for the specified visual conditions. For nonzero radiant flux, it is a spectral weighted mean of spectral luminous efficacy. Specify the denominator: a lamp's luminous efficacy relative to electrical input power also includes conversion losses and is a different ratio. ^[ref]

Luminous flux is the visually weighted counterpart of radiant flux, measured in lumens. Its value depends on the optical spectrum and the specified visual response; it is distinct from radiant flux in watts.^[ref]

Luminous intensity is luminous flux per outgoing solid angle, measured in candelas. It is the visually weighted counterpart of radiant intensity. The same total luminous flux concentrated into a narrower cone can give a greater luminous intensity within that cone.^[ref]

A photometer measures a visually weighted radiation quantity. A filtered photodiode approximates the photopic luminous efficiency function with its spectral responsivity; calibration establishes how its electric current maps to illuminance or collected luminous flux. NIST's detector-based realization traces the spectral responsivity to a cryogenic radiometer and the aperture area to length measurements. ^[ref]

Photometry weights visible electromagnetic radiation by standardized visual response functions; radiometry measures its energy and power without visual weighting.^[ref]

The question is how much light a source supplies under a stated visual weighting. A beam containing $1\mathrm{mW}$ at the defining frequency $540\mathrm{THz}$ has luminous flux $0.683\mathrm{lm}$. Moving the same radiant flux to a region of smaller spectral luminous efficacy reduces its luminous flux. Changing the optical spectrum changes this conversion even when its total power stays fixed. This makes photometry useful for comparing lighting, while radiometry remains necessary for an energy budget. The lumen counts luminous flux; the candela additionally specifies its distribution over solid angle.

Two measurement routes implement the weighting: measure an optical spectrum and evaluate a weighted integral, or use a photometer whose spectral responsivity approximates the prescribed weighting. Both require a specified collection geometry. Neither is a complete prediction of an individual's perception of a scene.^[ref]

The photopic luminous efficiency function $V(\lambda )$ is dimensionless and describes standardized relative sensitivity for photopic vision. It is normalized to $V(555\mathrm{nm})=1$ and tabulated over $360$–$830\mathrm{nm}$. The wavelength coordinate is in standard air, rather than vacuum; changing coordinates requires converting the spectral radiant flux density as well. The official prescription uses linear interpolation between the specified 1 nm entries.^[ref]

Write ${\lambda }_a$ for the standard-air wavelength at $540\mathrm{THz}$. With spectral radiant flux ${\Phi }_{e,\lambda }$ in $\mathrm{W}\mathrm{n}{\mathrm{m}}^{-1}$ and $d\lambda$ in nanometres, the luminous flux isThe integral has units of watts, and $K_m$ converts them to lumens. For radiation at a single wavelength ${\lambda }_0$ with total radiant flux $P$, this reduces to ${\Phi }_v=K_{m}V({\lambda }_0)P$. For a sufficiently narrow finite band, this is an approximation requiring nearly constant $V$ across the band.^[ref]

Photopic vision means vision under light-adapted conditions. Standard photometry here uses the photopic luminous efficiency function; scotopic vision uses a different weighting for dark-adapted conditions. The response function and adaptation conditions must be stated when comparing luminous fluxes.^[ref]

Radiant flux is the power carried by electromagnetic radiation: radiation energy transferred per unit time, measured in watts, $\mathrm{W}=\mathrm{J}{\mathrm{s}}^{-1}$. Specify what is counted: all radiation emitted by a source, radiation within one beam, or radiation intercepted by a photodetector. Writing $Q(t)$ for the cumulative energy crossing the specified boundary givesThe first expression is an instantaneous derivative; the second is an average over the stated time interval. If $0.020\mathrm{J}$ arrives during $0.010\mathrm{s}$, the mean radiant flux is $2\mathrm{W}$. The photodetector's absorbed power equals this incident radiant flux only if it absorbs all of the incident radiation.

Keeping radiant flux fixed while widening a beam reduces its irradiance, the incident power per receiving area. Radiance additionally resolves the propagation direction and has units $\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}$; an optical camera can collect different radiant fluxes from sources with the same radiance but different apparent solid angles. Luminous flux, measured in lumens, weights the optical spectrum by the human eye's standardized response instead of counting energy equally at every wavelength. Equal radiant fluxes need not give equal luminous fluxes.^[ref]

Radiant power also differs from the optical power of an optical lens, which is reciprocal focal length in dioptres. Likewise, astronomical radiative flux is power per area. Stating the units prevents these shared words from hiding different physical quantities.

Spectral luminous efficacy converts radiant flux at one wavelength to luminous flux. The candela convention fixes $K_{\mathrm{cd}}$ exactly at $540\times 10^{12}\mathrm{Hz}$, not at an exactly specified wavelength of $555\mathrm{nm}$. ^[ref]

In vacuum, that frequency corresponds to $c/(540\times 10^{12}\mathrm{Hz})\simeq 555.1712\mathrm{nm}$, using the speed of light. In standard air it is approximately $555.017\mathrm{nm}$, close to the $555\mathrm{nm}$ maximum of the photopic luminous efficiency function. Thus $K_m$ is slightly larger than $683\mathrm{lm}{\mathrm{W}}^{-1}$; using $K_m\simeq 683\mathrm{lm}{\mathrm{W}}^{-1}$ in ordinary calculations is a rounded convention. The exact anchor is $K({\lambda }_a)=683\mathrm{lm}{\mathrm{W}}^{-1}$. ^[ref]

For collected spectral radiant flux $f(\lambda )$ and a linear photodiode with spectral responsivity $r(\lambda )$, the dark-subtracted electric current is $i=\int rfd\lambda$. A reference optical spectrum $f_A$ sets the conversion $C_A$ from electric current to collected luminous flux. Applying it to another optical spectrum $f_T$ givesAssume 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. ^[ref]