Luminous Efficacy
1. Definition
Luminous efficacy is the ratio of luminous flux $\Phi_\mathrm{V}$ to power, expressed in lumens per watt (lm·W⁻¹).
$$K = \frac{\Phi_V}{\Phi_e} = \frac{\int_{380,\mathrm{nm}}^{780,\mathrm{nm}} K(\lambda),\Phi_{e,\lambda},\mathrm{d}\lambda}{\int_{0}^{\infty} \Phi_{e,\lambda},\mathrm{d}\lambda}$$
where $\Phi_{\mathrm{e},\lambda}$ is the spectral radiant flux and $V(\lambda)$ is the luminous efficiency function for photopic vision.
- Radiant flux $\Phi_{\mathrm{e}}$ — The total electromagnetic power emitted by the source, measured in watts ($\text{W}$), integrated over all wavelengths.
- Luminous flux $\Phi_{\mathrm{V}}$ — The part of the radiant flux that is perceivable by the human eye. It is calculated by weighting the radiant power according to the spectral sensitivity of the eye, measured in lumens ($\text{lm}$).
Depending on which “power” is used in the denominator, two distinct meanings are used:
- Luminous efficacy of radiation — The ratio of luminous flux to the total emitted radiant flux. This value depends only on the spectral distribution of the light source.
- Luminous efficacy of a source (or overall efficacy) — The ratio of luminous flux to the total input electrical power. This value accounts for both the light production and the internal conversion losses. For example, a white LED delivers far more lumens per watt than an incandescent lamp, because most of the energy in an incandescent lamp is emitted as infrared radiation, which is invisible to the human eye.
2. Maximum Luminous Efficacy
The maximum luminous efficacy $K_\mathrm{m}$ is the maximum ratio of luminous flux to radiant flux attainable by optical radiation, expressed in lumens per watt (lm·W⁻¹). It is reached only by monochromatic radiation at the wavelength of peak visual sensitivity.
The figure shows the luminous efficiency function $V(\lambda)$ for photopic vision, which gives the relative spectral sensitivity of the human eye, normalized to unity at its maximum. The horizontal axis is wavelength in nanometres (nm); the vertical axis is relative sensitivity.
$V(\lambda)$ peaks at 555 nm (yellow-green), which is therefore the wavelength at which $K_\mathrm{m}$ applies. Sensitivity falls off towards both ends of the spectrum, so wavelengths away from the peak — for example violet near 400 nm or red near 700 nm — contribute little luminous flux per watt. Since a white-light source must radiate across the whole visible range, including these low-sensitivity regions, its luminous efficacy is necessarily well below $K_\mathrm{m}$.
Under CIE standards, different visual conditions are assigned different symbols and values:
| Visual condition | Symbol | Value (lm·W⁻¹) |
|---|---|---|
| Photopic vision | $K_\mathrm{m}$ | 683 |
| Scotopic vision | $K'_\mathrm{m}$ | 1700 |
| Mesopic vision | $K_{\mathrm{m,mes}}$ | varies with adaptation level, between 683 and 1700 |
| Photopic vision, 10° field | $K_{\mathrm{m,10}}$ | ≈ 683 |
| Photopic vision, 2° field | $K_{\mathrm{m,2}}$ | ≈ 683 |
Note: where no photometric condition is specified, values refer to photopic vision with the CIE 1931 (2°) standard observer. Under scotopic vision the sensitivity function $V'(\lambda)$ peaks at 507 nm rather than 555 nm.
The value $K_\mathrm{m}=683\ \mathrm{lm{\cdot}W^{-1}}$ is not an experimental measurement but a defined constant of the SI. The candela is defined by fixing the luminous efficacy of monochromatic radiation of frequency 540 THz to be exactly 683 lm·W⁻¹. In terms of the photopic function,
$$K_\mathrm{m}=\frac{683}{V(\lambda_\mathrm{cd})}\ \mathrm{cd{\cdot}sr{\cdot}W^{-1}}$$
where $\lambda_\mathrm{cd}$ is the reference wavelength corresponding to 540 THz, approximately 555 nm, at which $V(\lambda)$ is normalized to 1. Hence $V(\lambda_\mathrm{cd})\approx1$ and $K_\mathrm{m}=683\ \mathrm{lm{\cdot}W^{-1}}$, or equivalently 683 cd·sr·W⁻¹ since 1 lm = 1 cd·sr.
Luminous efficacy and luminous efficiency
These two terms are distinct and should not be interchanged:
- Luminous efficacy is an absolute quantity, in lm·W⁻¹, giving the luminous flux produced per unit power.
- Luminous efficiency is a dimensionless ratio, usually expressed as a percentage, of a source's luminous efficacy to the theoretical maximum $K_\mathrm{m}$.
Two caveats apply when comparing values. First, 683 lm·W⁻¹ is the ceiling for monochromatic 555 nm radiation (the efficacy of radiation); white light is lower, typically 250–370 lm·W⁻¹ depending on colour temperature and colour rendering, and lower still for the efficacy of a source, which also includes losses in converting electrical input into radiation. Second, $K_\mathrm{m}$ applies to photopic vision; under other adaptation states the reference maximum changes.
See Also
Luminous Flux
