Physical size
Formula symbol ${\ displaystyle K, K (\ lambda)}$
Size and
unit system
unit dimension
SI lm · W −1 M −1 · L −2 · T 3 · J

The photometric radiation equivalent ( English luminous efficacy of radiation ) of a wavelength mixture of electromagnetic radiation is the quotient of the luminous flux of the radiation and its radiant power . Its SI unit is lumen by Watt (lm / W). ${\ displaystyle K}$ ${\ displaystyle \ Phi _ {\ mathrm {v}}}$ ${\ displaystyle \ Phi _ {\ mathrm {e}}}$

The greater this number, the greater the luminous flux that can be used by the eye for a given radiation output from a light source.

The spectral photometric radiation equivalent is the quotient of the luminous flux and the radiation power of monochromatic radiation of the wavelength . It directly indicates the sensitivity of the eye to radiation of the relevant wavelength, i.e. the strength of the light stimulus exerted by the radiation on the eye for a given radiation power. With its help it is possible to calculate the associated photometric quantity ( luminous flux , illuminance , etc.) from a given radiometric quantity whose wavelength distribution is known (e.g. radiant power , irradiance , etc.) . ${\ displaystyle K (\ lambda)}$${\ displaystyle \ lambda}$

The photometric radiation equivalent measured in lm / W should not be confused with the light output of a technical light source, also measured in lumens per watt (lm / W) . The photometric radiation equivalent describes how many lumens emitted per watt of the emitted electromagnetic radiation power of the light source. The luminous efficacy describes how many lumens are emitted for each watt of the (mostly electrical) power consumed by the light source, i.e. it includes technical conversion losses. The English term luminous efficacy can mean both, so luminous efficacy of radiation is the more precise term. On the other hand, the synonym “spectral light output” is also used for the spectral photometric radiation equivalent.

## Light sensitivity dependent on wavelength

The eye can only perceive part of the electromagnetic spectrum. It is most sensitive to yellow-green, perceives blue and red with lower intensity even with the same radiation power, and is insensitive to shorter wavelengths than violet and longer wavelengths than deep red.

From the broad wavelength spectrum of electromagnetic radiation , the wavelength range from around 380 to 780  nanometers (nm) is “visible”, that is, radiation from this range triggers a sensation of brightness in the eye and is perceived as light . However, the eye is not equally sensitive to all visible wavelengths. At wavelengths at the edge of the visible range, a higher radiation intensity is necessary in order to produce the same sensation of brightness as in its center.

The eye is most sensitive at a wavelength of 555 nm, corresponding to a yellow-green spectral color. At around 510 nm (green) on one side and around 610 nm (orange-red) on the other side of the maximum, the eye only achieves half the sensitivity. At 665 nm, the color of typical red light-emitting diodes , the sensitivity is only 4.5% of that at 555 nm. At around 380 nm (violet) or 780 nm (deep red), the sensitivity is almost zero.

If the eye is offered a mixture of electromagnetic radiation of different wavelengths, the impression of brightness produced depends on the sensitivity of the eye to the wavelengths contained in the mixture. Wavelengths close to 555 nm contribute significantly to the impression of brightness, wavelengths outside the visible range do not contribute at all. It is therefore not sufficient to state how many watts of physical radiation power a lamp emits in order to describe the impression of brightness generated by this radiation. Instead, the radiation current measured in watts has to be weighted for each wavelength with the respective spectral photometric radiation equivalent of the eye. The result is the luminous flux measured in lumens, which is a quantitative measure of the light stimulus exerted on the eye.

For the definition of the photometric SI units , it was established in 1979 that monochromatic radiation with a frequency of 540 · 10 12  Hz (corresponds to a wavelength of 555.016 nm in air) and a radiation power of 1 watt is simultaneously a luminous flux of 683 lm. For this wavelength, the spectral photometric radiation equivalent is 683 lm / W. The radiation power at other wavelengths contributes less to the luminous flux.

Spectral photometric radiation equivalent for day vision K (λ) and for night vision K '(λ).

### Day seeing

The spectral photometric radiation equivalent is the quotient of luminous flux and radiant power in the case of monochromatic radiation of the wavelength . It therefore immediately indicates the sensitivity of the eye at the wavelength in question and can be displayed as a curve. Often referred to as ${\ displaystyle K (\ lambda)}$${\ displaystyle \ lambda}$${\ displaystyle K (\ lambda)}$${\ displaystyle K (\ lambda)}$

${\ displaystyle K (\ lambda) = K _ {\ mathrm {m}} \ cdot V (\ lambda)}$

written. This is the so-called "maximum value of the photometric radiation equivalent". Its numerical value follows from the definition of the SI units and is ${\ displaystyle K _ {\ mathrm {m}}}$

${\ displaystyle K _ {\ mathrm {m}} = 683 \ \ mathrm {\ frac {lm} {W}}}$

The wavelength-dependent curve is the " relative brightness sensitivity curve ", which varies between 0 and 1 and describes the course of the sensitivity for different wavelengths relative to the curve maximum at 555 nm. This curve was determined experimentally and is standardized. ${\ displaystyle V (\ lambda)}$

These variables describe the sensitivity of the eye during daytime vision (photopic range).

### Night vision

In night vision (scotopic area) the visual performance is no longer provided by the cones of the retina, but taken over by the rods , which have a higher sensitivity and have the maximum sensitivity at a different wavelength than the cones. In this case the sensitivity of the eye is described by

${\ displaystyle K '(\ lambda) = K' _ {\ mathrm {m}} \ cdot V '(\ lambda)}$

with the scotopic maximum value of the photometric radiation equivalent

${\ displaystyle K '_ {\ mathrm {m}} = 1700 \ \ mathrm {\ frac {lm} {W}}}$

and the scotopic relative light sensitivity curve , the maximum of which is at the wavelength 505 nm (blue-green). ${\ displaystyle V '(\ lambda)}$

Coincidentally, light with a wavelength of 555 nm, the wavelength at which the photopic curve has its maximum and which was chosen as the reference value for determining the photometric SI units, is perceived by the photopic and the scotopic eye with almost the same sensitivity. The deviation is only 3%. Since this difference is so small, it was decided that the same reference should be used for scotopic light perception. By definition, the equivalent of 683 lm gilt 1 W applies equally to photopic and scotopic perception of light of this wavelength: ${\ displaystyle K (\ lambda)}$

${\ displaystyle K (555 \, \ mathrm {nm}) = K '(555 \, \ mathrm {nm}) = 683 \, \ mathrm {lm / W} \,}$.

### Twilight vision

Evaluation functions also exist for the transition area between daytime and nighttime vision (the mesopic area), but these are more complex.

## Calculation of the photometric radiation equivalent

If the spectral (i.e., wavelength-dependent) distribution of a radiometric quantity (e.g. radiant power, radiant intensity, irradiance, etc.) is given, it immediately follows as ${\ displaystyle \ textstyle {\ frac {\ partial X _ {\ mathrm {e}}} {\ partial \ lambda}}}$${\ displaystyle X _ {\ mathrm {e}}}$${\ displaystyle X _ {\ mathrm {e}}}$

${\ displaystyle X _ {\ mathrm {e}} = \ int _ {0} ^ {\ infty} {\ frac {\ partial X _ {\ mathrm {e}}} {\ partial \ lambda}} \, \ mathrm { d} \ lambda}$

The photometric quantity corresponding to the radiometric quantity (e.g. luminous flux, luminous intensity, illuminance etc.) can be derived from the spectrum of . First the spectrum of is determined. This is done by evaluating the spectrum of using the spectral sensitivity curve of the eye. The following applies to every wavelength: ${\ displaystyle X _ {\ mathrm {e}}}$${\ displaystyle X _ {\ mathrm {v}}}$${\ displaystyle X _ {\ mathrm {e}}}$${\ displaystyle X _ {\ mathrm {v}}}$${\ displaystyle X _ {\ mathrm {e}}}$${\ displaystyle K (\ lambda)}$

${\ displaystyle {\ frac {\ partial X _ {\ mathrm {v}}} {\ partial \ lambda}} \, \ mathrm {d} \ lambda = K (\ lambda) {\ frac {\ partial X _ {\ mathrm {e}}} {\ partial \ lambda}} \, \ mathrm {d} \ lambda = K _ {\ mathrm {m}} V (\ lambda) {\ frac {\ partial X _ {\ mathrm {e}}} {\ partial \ lambda}} \, \ mathrm {d} \ lambda}$

It then follows itself as ${\ displaystyle X _ {\ mathrm {v}}}$

${\ displaystyle X _ {\ mathrm {v}} = \ int _ {0} ^ {\ infty} {\ frac {\ partial X _ {\ mathrm {v}}} {\ partial \ lambda}} \, \ mathrm { d} \ lambda = \ int _ {0} ^ {\ infty} \, K (\ lambda) {\ frac {\ partial X _ {\ mathrm {e}}} {\ partial \ lambda}} \, \ mathrm { d} \ lambda = K _ {\ mathrm {m}} \ int _ {0} ^ {\ infty} V (\ lambda) {\ frac {\ partial X _ {\ mathrm {e}}} {\ partial \ lambda} } \, \ mathrm {d} \ lambda}$

The photometric radiation equivalent of the electromagnetic radiation present is the quotient of and : ${\ displaystyle X _ {\ mathrm {v}}}$${\ displaystyle X _ {\ mathrm {e}}}$

${\ displaystyle K = {\ frac {X _ {\ mathrm {v}}} {X _ {\ mathrm {e}}}} = K _ {\ mathrm {m}} {\ frac {\ int _ {0} ^ { \ infty} V (\ lambda) {\ frac {\ partial X _ {\ mathrm {e}}} {\ partial \ lambda}} \, \ mathrm {d} \ lambda} {\ int _ {0} ^ {\ infty} {\ frac {\ partial X _ {\ mathrm {e}}} {\ partial \ lambda}} \, \ mathrm {d} \ lambda}}}$

The photometric radiation equivalent can also be determined from other pairs of photo- or radiometric quantities, not just luminous flux and radiation power.

## Examples

The maximum possible value of the photometric radiation equivalent is available for monochromatic radiation with a wavelength of 555 nm and is 683 lm / W. It is smaller for all other wavelengths and for mixed wavelengths. The light of a frequency-doubled Nd: YAG laser reaches 604 lm / W at 532 nm, while that of a helium-neon laser only achieves 160 lm / W at 633 nm.

Photometric radiation equivalent for Planck emitters as a function of temperature. The right scale normalizes the value to the maximum possible photometric radiation equivalent, i.e. indicates K / K m .

If the wavelength mixture has the spectrum of a Planck radiator , its photometric radiation equivalent K depends on the temperature of the radiator. At low temperatures, almost all of the radiation is emitted in the infrared and it is K  ≈ 0. With the onset of red heat , part of the radiation is perceived as visible light, but is still at the red wavelengths, to which the eye is less sensitive. With rising temperature and the associated shift of the radiation maximum to shorter wavelengths , an ever larger proportion of the radiation reaches the wavelength ranges to which the eye is particularly sensitive.

At a temperature of 2800  K (the filament temperature of an incandescent lamp ), the Planckian radiator has a radiation equivalent of 15 lm / W, with 6% of the radiation being emitted in the visible range from 400 to 700 nm. (Real incandescent lamps are somewhat more efficient and reach 15 lm / W at around 2500 K because they are not ideal Planckian radiators and emit comparatively less radiation in the infrared.)

At a temperature of 5778 K ( corresponding to the surface temperature of the sun ), the Planckian radiator has a radiation equivalent of 93 lm / W, and 37% of its radiation falls within the visible range of 400 to 700 nm.

At a temperature of 6640 K, the Planck radiator reaches 96.1 lm / W, the maximum possible photometric radiation equivalent for Planck radiation. With a further increase in temperature, ever larger proportions of the radiation shift into the invisible ultraviolet and the photometric radiation equivalent decreases again.

### White light

Mixtures of wavelengths that are perceived as “white” and have no components outside of the visible spectral range achieve photometric radiation equivalents between about 250 and 370 lm / W , depending on the desired color temperature and the color rendering index .

### Artificial light sources

The following laboratory results are exemplary for modern, economical light sources:

A compact fluorescent lamp (16 watts, 900 lumens, luminous efficacy 56 lm / W) achieved a photometric radiation equivalent of 283 lm / W immediately after being switched on and 349 lm / W when it was warm. A comparison of the light yield with the radiation equivalent shows that in this case only 56/349 = 16% of the electrical power consumed was converted into electromagnetic radiation power.

Two LED lamps with color temperatures of 3000 K and 6500 K had radiation equivalents of 341 lm / W and 287 lm / W, respectively. The white background lighting of the displays of two laptops, which used fluorescent tubes or LEDs as a light source, was 317 lm / W and 293 lm / W, respectively.

These artificial sources essentially limit their spectra to the visible range (in contrast to incandescent lamps) and therefore generally achieve photometric radiation equivalents of around 250 to 350 lm / W, although their light spectra can sometimes differ significantly in detail.

### Natural light sources

The wavelength mixture of daylight (without direct solar radiation) has a photometric radiation equivalent of about 125 lm / W, that of the sun is between almost 20 lm / W (sun at a low position) and about 100 lm / W (sun at its zenith). The photometric radiation equivalent of sunlight outside the earth's atmosphere is 98 lm / W. Light with the spectrum of the standard light type D 65 , which is similar to daylight , reaches 110 lm / W.

So far, the eye has been considered as the receiver and the light sensitivity curves of the eye as evaluation functions. However, there are also other “receivers” that react to light with their own sensitivity curves. Something like that

• photographic films that react with darkness
• Skin that reacts to UV light with sunburn
• Plants that use light for photosynthesis. The “maximum value of the phytophotometric radiation equivalent” is K m = 247 lm / W.

## Other measures of efficiency

### Optical efficiency

The optical useful effect of a radiation is the quotient of the radiation power emitted in the visible range to the total radiation power: ${\ displaystyle O}$

${\ displaystyle O = {\ frac {\ int _ {\ mathrm {380 \ nm}} ^ {\ mathrm {780 \ nm}} {\ frac {\ partial \ Phi _ {\ mathrm {e}}} {\ partial \ lambda}} \, \ mathrm {d} \ lambda} {\ int _ {0} ^ {\ infty} {\ frac {\ partial \ Phi _ {\ mathrm {e}}} {\ partial \ lambda} } \, \ mathrm {d} \ lambda}}}$

### Visual benefit

The visual efficiency results from the optical efficiency by evaluating the radiation power in the visible range with the relative light sensitivity curve : ${\ displaystyle V (\ lambda)}$

${\ displaystyle V = {\ frac {\ int _ {0} ^ {\ infty} V (\ lambda) {\ frac {\ partial \ Phi _ {\ mathrm {e}}} {\ partial \ lambda}} \ , \ mathrm {d} \ lambda} {\ int _ {0} ^ {\ infty} {\ frac {\ partial \ Phi _ {\ mathrm {e}}} {\ partial \ lambda}} \, \ mathrm { d} \ lambda}} = {\ frac {{\ frac {1} {K _ {\ mathrm {m}}}} \ Phi _ {\ mathrm {v}}} {\ Phi _ {\ mathrm {e}} }} = {\ frac {K} {K _ {\ mathrm {m}}}}}$

The quantities and are quotients of two performance quantities and therefore represent dimensionless efficiencies that can be given in percent . The photometric radiation equivalent, on the other hand, is the quotient of a photometric and a radiometric quantity and therefore no efficiency. ${\ displaystyle O}$${\ displaystyle V}$

### Light output

The light output of a lamp is the quotient of the luminous flux emitted by the lamp and the power it consumes : ${\ displaystyle \ eta _ {\ mathrm {v}}}$${\ displaystyle \ Phi _ {\ mathrm {v}}}$ ${\ displaystyle P}$

${\ displaystyle \ eta _ {\ mathrm {v}} = {\ frac {K_ {m} \ int _ {0} ^ {\ infty} V (\ lambda) {\ frac {\ partial \ Phi _ {\ mathrm {e}}} {\ partial \ lambda}} \, \ mathrm {d} \ lambda} {P}} = {\ frac {\ Phi _ {\ mathrm {v}}} {P}}}$

## Remarks

1. a b c Strictly speaking, one would have to distinguish between three wavelengths:
a) λ m , the wavelength at which K has its maximum ( K m = 683.0016 lm / W). This wavelength was defined by the CIPM to be exactly 555 nm;
b) λ cd , the wavelength that was chosen as the reference for the definition of the SI units and which  corresponds to the frequency 540 · 10 12 Hz (λ cd = 555.016 nm in air) - here Kcd ) and K 'cd ) by definition the value 683 lm / W;
c) the wavelength at which photopic and scotopic vision have the same sensitivity (555.80 nm).
These three wavelengths hardly differ from one another.
2. The perception of this objectifiable physical light stimulus as a subjective sensation of brightness with its adaptation, contrast and other physiological effects is no longer the subject of photometry.
3. a b Since the revision of the International System of Units on May 20, 2019, this stipulation applies directly; previously it was formulated indirectly via the definition of the basic unit candela . The numerical value of 683 was chosen so that the unit candela remained as unchanged as possible when it was redefined in 1979.

## Individual evidence

1. International Electrotechnical Commission (IEC): International Electrotechnical Vocabulary , ref. 845-01-56, Luminous efficacy of radiation, (accessed February 20, 2015)
2. a b c The International System of Units (SI) . German translation of the BIPM brochure "Le Système international d'unités / The International System of Units (8e édition, 2006)". In: PTB-Mitteilungen . tape  117 , no. 2 , 2007 ( Online [PDF; 1.4 MB ]). Footnote on page 17: “[…] the spectral light yield of monochromatic radiation is only independent of the degree of adaptation at the frequency 540 × 10 12 Hertz and is defined there as 683 lumens per watt. The value at the maximum photopic evaluation is K m = 683.0016 lm / W […] The corresponding value at the maximum scotopic evaluation is K ' m = 1700.06 lm / W […] "
3. BIPM: Principes Régissant la photometry. Pavillon de Breteuil 1983 ( PDF 1.06 MB ), tableau 1: Efficacité lumineuse relative spectrale V (λ) pour la vision photopique: V (510 nm) = 0.503, V (610 nm) = 0.503.
4. BIPM: Principes Régissant la photometry. Pavillon de Breteuil 1983 ( PDF 1.06 MB ), Tableau 1: Efficacité lumineuse relative spectrale V (λ) pour la vision photopique: V (665 nm) = 0.04458
5. a b 26th CGPM (2018) - Resolutions adopted / Résolutions adoptées. (PDF; 1.2 MB) Versailles 13–16 November 2018. In: bipm.org. Bureau International des Poids et Mesures, November 19, 2018, accessed September 8, 2019 (English, French). ; see also SI brochure , chapter 2.2
6. ^ Minutes of the 16th General Conference on Weights and Measures , 1979, pp. 57–58, accessed March 28, 2020, in French
7. Comité International des Poids et Mesures - Procès verbaux des séances . 66 e session. 2 e série, 1977, p.  130 f . ( bipm.org [PDF]). (7.4 MB): “The nouvelle définition est conçue pour assurer la continuité de l'unité pour les grandeurs photopiques; si l'on veut qu'elle s'applique also à l'unité pour les mesures scotopiques, comme c'est le cas de la définition actuelle, alors l'échelle de mesure scotopique changera d'environ 3%. [...] Toutefois, on accepte finalement que la valeur de 683 lm / W s'applique indifféremment aux grandeurs photopiques, scotopiques et mésopiques .. “The CCPR (Comité Consultatif de Photométrie et Radiométrie) is the competent advisory body of the International Committee for Dimensions and weight (CIPM) .
8. ^ WR Blevin, B. Steiner: Redefinition of the Candela and the Lumen. In: Metrologia . 11, 1975, pp. 97-104 doi: 10.1088 / 0026-1394 / 11/3/001 .
9. a b H.-J. Hentschel: Light and Lighting - Theory and Practice of Lighting Technology. 4th edition, Hüthig Buch, Heidelberg 1994, ISBN 3-7785-2184-5 , pp. 27ff.
10. T.W. Murphy, Jr .: Maximum Spectral Luminous Efficacy of White Light. In: Journal of Applied Physics. 111, 2012, 104909, doi: 10.1063 / 1.4721897 .
11. DIN 5034 daylight indoors. Part 2 Basics of Beuth Verlag, Berlin 1985.
12. S. Darula, R. Kittler, CA Gueymard: Reference luminous solar constant and solar luminance for illuminance calculations. In: Solar Energy. Volume 79, Issue 5, November 2005, pp. 559-565 doi: 10.1016 / j.solener.2005.01.004 . For the standard light sensitivity curve V (λ): 97.6019325 lm / W, for the 1988 modified light sensitivity curve V M (λ): 98.1685089 lm / W.
13. ^ DL MacAdam: Color Measurement - Theme and Variations. 2nd ed., Springer, Berlin / Heidelberg 1985, ISBN 978-3-540-15573-7 , p. 105. It is assumed that the radiation power for wavelengths longer than 830 nm (up to that point the spectrum is defined) is zero.
14. a b H.-J. Hentschel: Light and Lighting - Theory and Practice of Lighting Technology. 4th edition, Hüthig Buch, Heidelberg 1994, ISBN 3-7785-2184-5 , p. 39ff.
15. a b H.-J. Hentschel: Light and Lighting - Theory and Practice of Lighting Technology. 4th edition, Hüthig Buch, Heidelberg 1994, ISBN 3-7785-2184-5 , p. 38.
16. International Electrotechnical Commission (IEC): International Electrotechnical Vocabulary , ref. 845-01-57, Luminous efficiency (of radiation), (accessed March 3, 2015).
17. H.-J. Hentschel: Light and Lighting - Theory and Practice of Lighting Technology. 4th edition, Hüthig Buch, Heidelberg 1994, ISBN 3-7785-2184-5 , p. 37.
18. International Electrotechnical Commission (IEC): International Electrotechnical Vocabulary , ref. 845-01-55, Luminous efficacy of a source, (accessed March 3, 2015).