Light intensity (photometry)
Physical size  

Surname  Light intensity  
Formula symbol  

The luminous intensity ( English luminous intensity , symbol I _{v} ) indicates the luminous flux related to the solid angle . It is a basic quantity in the SI system of units . Its SI unit is the candela (cd).
The index v (for visual) characterizes the light intensity as a photometric quantity that includes the physiological influence of the light sensitivity curve in addition to the radiometric aspect .
The definition of the light intensity with reference to the solid angle presupposes a light beam that diverges or converges with respect to a point . The corresponding variable in relation to a lit or flowed through area is the illuminance or luminous flux density.
definition
Explanation
To assess a light source, it is not only of interest which luminous flux (measured in lumens ) the source emits as a whole, it will also emit different amounts of light in different directions . If this directional characteristic is to be described in detail, the concept of light intensity is necessary.
It is namely not possible to specify how many lumens are emitted in a certain direction, since the finite number of radiated lumens is distributed over an infinite number of possible directions and therefore zero lumens are allotted to each individual direction. Instead, a small solid angle surrounding the desired direction is considered (measured in steradian ), the (finite) luminous flux emitted in this solid angle is related to the (finite) size of the solid angle and the solid angle is reduced to zero. Both the solid angle and the emitted luminous flux contained in it tend towards zero, but their ratio towards a finite limit value , the luminous intensity emitted in the relevant direction (measured in lumens per steradian or candela ).
This limit crossing corresponds, for example, to that which is used to determine the mass density : The mass contained in an infinitely small volume is zero, but the quotient of the mass contained in a volume and the volume tends towards a finite limit value when the volume is reduced Mass density of the material under consideration.
Exact definition
According to the explanation above, the light intensity in a given direction is the solid angle density of the light flux in this direction. If the source emits the differential luminous flux in a differential solid angle which contains the direction in question, the luminous intensity in this direction is the quotient of the two differential quantities:
.
Simplified definition
If the luminous intensity in all directions that fall in a finite solid angle has the same value, then the limit value observation is unnecessary and the differential definition changes over to the following simplified definition: The luminous intensity for all directions that fall in the solid angle is the quotient from the luminous flux emitted by the light source in this solid angle and the solid angle radiated through :
If the light intensity is not constant within the considered finite solid angle, the simplified definition can still be used. The result of the formation of the quotient is then the arithmetic mean value of the light intensities falling in the solid angle, which is formed over the relevant solid angle.
Example 1: spherical emitter
A spherical radiator is an ideal light source that emits its light evenly in all directions of the room. (An incandescent lamp hanging freely in the room without a lamp housing is a good approximation.) If this spherical radiator is located in the center of a fictitious sphere with a radius of r = 1 m and emits a luminous flux of 1256 lumens, the luminous flux ( proportion) of 100 lumens. The light intensity I _{v} = 100 cd can thus be assigned to the spherical emitter for any direction in the room . The formulas for this example:
Example 2: Headlights with 1sr illumination
In the case of a fictitious light source that emits all of its light in a solid angle of size Ω = 1 sr, the room within the solid angle Ω is irradiated evenly, the room outside the bright solid angle remains completely dark. (A headlight could be a good first approximation.) If such a spotlight with Φ _{v} = 100 lm emits its entire luminous flux into the illuminated solid angle Ω = 1 sr, the light intensity within the bright solid angle is by definition I _{v} = 100 cd. Outside the illuminated solid angle, the light intensity is I _{v} = 0 cd. The formulas for this example:
Example 3: Spotlight
If you reduce the solid angle of the headlight (from example 2) without changing the value of the generated luminous flux Φ _{v} = 100 lm, the value of the luminous intensity must assume higher values. (In the case of small solid angles, a comparison with a spotlight is useful.) If the luminous flux Φ _{v} = 100 lm is concentrated on the solid angle of Ω = 0.1 sr, the result is a luminous intensity of I _{v} = 1000 cd. The formulas for this example:
unit
In the past, different light sources were used as reference values, in Germany the Hefner candle and from 1937 internationally a cavity radiator ( black radiator ).
Since 1979 the unit candela has been traced back to the derived SI unit watt . The wording of the definition was updated in 2019 and is now:
“The candela, unit symbol cd, is the SI unit of light intensity in a certain direction. It is defined by setting the numerical value 683 for the photometric radiation equivalent K _{cd of} the monochromatic radiation of frequency 540 · 10 ^{12} Hz, expressed in the unit lm W ^{−1} , which is cd sr W ^{−1} or cd sr kg ^{−1} m ^{−2} s ^{3} , where the kilogram, the meter and the second are defined by h , c and Δν _{Cs} . "
The specified frequency corresponds to a wavelength of approx. 555 nm (green light). It happens to be near the maximum brightness sensitivity of the 2 ° standard observer as well as at the intersection of the two brightness curves defined by the CIE and published by the BIPM for photopic vision (daytime vision) and scotopic vision (night vision). Therefore the unit candela is valid for both excited states of the human eye.
Luminous intensity measurement
The description of a simple procedure for determining the light intensity also uses the fictitious sphere with a radius of one meter, in the center of which a lamp generates light. From the surface of this sphere, a solid angle with the size of one steradian, by definition, cuts exactly the area A of one square meter. The light intensity can be elegantly linked with the illuminance via this one square meter spherical surface, the definition of which also uses the reference area of one square meter. At a distance of exactly one meter from the light source, the luminous flux is distributed inside the solid angle over an area A of one square meter. By definition, the quotient of luminous flux and area A gives the illuminance .
At a distance of exactly one meter from the light source, every illuminance measurement can be interpreted as a luminance measurement because the reference areas (one square meter each) are identical.
When driving around the light source (at a distance of one meter!), The numerical value (without unit of measure!) Of the light intensity can be measured with any commercially available lux meter and the dependency of the light intensity on spatial coordinates can be easily checked.
In addition to the luminous flux, the luminous intensity is an essential characteristic of light sources and, not least because of the simple measuring process, forms one of the basic quantities of the SI system of units.
properties
 The luminous intensity is the photometric equivalent of the radiometric radiant intensity . If the radiation intensity of the electromagnetic radiation emitted by a source in a certain direction is known, the corresponding light intensity can be determined therefrom by weighting the contributions of the individual wavelengths with the respective spectral light sensitivity of the eye. The light intensity of an infrared radiation source of any radiation intensity, for example, is zero because it is invisible to the human eye.
 The choice of luminous intensity as a photometric basic parameter initially appears to be difficult to understand, since from a modern point of view, for example, luminous flux or luminance would be viewed as more fundamental parameters. In the early days of photometry, however, when the focus was on the visual comparison of light sources, the light intensity was the property of the sources that was easiest to compare and which was therefore introduced as the fundamental photometric quantity.
 The light intensity is a property of the light source and does not depend on the distance of an observer. The light emitted by the source "dilutes" as it moves away from the source, but the luminous flux emitted into a given solid angle cone always remains the same: A cross section through the solid angle cone serving as a test area increases with distance to the same extent of area as the luminous flux “dilutes”. The luminous flux shining through the entire surface is therefore the same at all distances and thus also the quotient of the luminous flux and the solid angle.
 This does not contradict the fact that an observer perceives the light source as "weaker" with increasing distance. The illuminance on the pupil surface of the observer's eye is decisive for the perception of brightness , i.e. the ratio of the luminous flux radiating through the surface to the radiating surface. If the eye moves away from the source, the size of the receiving surface remains the same, but the luminous flux radiating through the surface decreases due to its increasing “dilution”. The relevant calculation formulas are explained in the → Illuminance section .
 For a point source, the luminous intensity describes the luminous flux transported in a given direction with a "light beam". For a twodimensional source, the light intensity describes the totality of all "light rays" that emanate from all points of the source radiating in this direction and  running parallel to each other  transport the luminous flux in the given direction.
 The light intensity is not only defined for selfilluminating light sources, but also for reflecting or transmitting light sources. It is even defined at every point in the room where there is light. Instead of a lightemitting surface element, imagine a fictitious surface element in the room through which light shines.
 The brightness of a light source perceived by the eye only partially corresponds to the physical light intensity. The contrast with the environment influences the physiological perception. A light source with a small surface is perceived as brighter (or more dazzling) than a light source with the same physical light intensity but a larger surface. This impression can be observed, for example, with car headlights of different sizes or when the moon or the sun rise or set.
Relationship with other photometric quantities
For the sake of brevity, the simplified, nondifferential terms are mainly used in the following. If the necessary prerequisites are not given (i.e. if the light intensity is not constant over the solid angle considered), the corresponding differential quotients or integrals must be applied.
Luminous flux
Opening angle α  Solid angle Ω 

360 °  (4π =) 12.566 sr 
270 °  10,726 sr 
180 °  (2π =) 6.283 sr 
120 °  (π =) 3.142 sr 
90 °  1,840 sr 
65.541 °  1 sr 
60 °  0.842 sr 
45 °  0.478 sr 
30 °  0.214 sr 
10 °  0.0239 sr 
5 °  0.00598 sr 
2 °  0.000957 sr 
1 °  0.000239 sr 
Solving the equation of definition for the luminous flux provides
The luminous flux passing through the solid angle is therefore the product of the light intensity directed into the solid angle and the solid angle. If the luminous intensity varies within the solid angle considered, the result for the luminous flux remains exact if the arithmetic mean of the luminous intensities is used.
The shape of the solid angle is arbitrary. A situation that is particularly easy to handle arithmetically occurs when a solid angle spanned by a circular surface is considered. The light rays passing through such a solid angle form a circular cone with the opening angle . The solid angle corresponding to this cone is
 Steradian.
This formula can be used to calculate the solid angle radiated through for light sources with conical emission characteristics. The adjacent table contains some exemplary numerical values.
The exact equation for calculating the luminous flux from the distribution of the luminous intensities in a solid angle is:
 .
Illuminance
The illuminance is the quotient of the luminous flux hitting a surface and the surface area . According to the photometric law of distance, the following applies
,
where is the angle between the surface normal and the radiation direction.
The illuminance generated by the light source on the surface therefore decreases with the square of the distance , although the light intensity emitted by the source in the direction of the surface is independent of the distance.
Luminance
The definition of luminance essentially corresponds to that of luminous intensity. While the light intensity directed in a certain direction includes all light rays sent by the light source in this direction, the luminance only takes into account the rays emitted in this direction and by a certain surface element. If the surface sends light in a given direction at the angle of radiation , its luminance in this direction is equal to the quotient of the light intensity of the light source in this direction and the radiation surface projected in the beam direction:
.
In the exact formulation, the relationship is:
.
Comparative overview
radiometric quantity  Symbol ^{a)}  SI unit  description  photometric equivalent ^{b)}  symbol  SI unit 
Radiant flux radiant power, radiant flux, radiant power 
W ( watt ) 
Radiant energy through time 
Luminous flux luminous flux, luminous power 
lm ( lumens ) 

Radiant intensity irradiance, radiant intensity 
W / sr  Radiation flux through solid angles 
Luminous intensity luminous intensity 
cd = lm / sr ( candela ) 

Irradiance irradiance 
W / m ^{2}  Radiation flux through the receiver surface 
Illuminance illuminance 
lx = lm / m ^{2} ( lux ) 

Specific radiation emission current density, radiant exitance 
W / m ^{2}  Radiation flux through the transmitter surface 
Specific light emission luminous exitance 
lm / m ^{2}  
Radiance radiance, radiance, radiance 
W / m ^{2} sr  Radiant intensity through effective transmitter area 
Luminance luminance 
cd / m ^{2}  
Radiant energy amount of radiation, radiant energy 
J ( joules ) 
by radiation transmitted energy 
Amount of light luminous energy, quantity of light 
lm · s  
Irradiation irradiation, radiant exposure 
J / m ^{2}  Radiant energy through the receiver surface 
Exposure luminous exposure 
lx s  
Radiation yield radiant efficiency 
1  Radiation flux through absorbed (mostly electrical) power 
Luminous efficiency (overall) luminous efficacy 
lm / W 
Sample calculations
example 1
A light source generates a luminous flux of 12 lumens and emits this light isotropically , i. H. the light intensity is the same in all directions. What is the light intensity of the source in any given direction?
Due to the assumed constancy of the light intensity, the simplified formula can be used. Dissolve after supplies . The full solid angle surrounding the light source is 4π steradians. The luminous flux of 12 lumens is evenly distributed over the solid angle of 4π steradian, the light intensity is therefore 12 lumens per 4π steradian ≈ 1 lumen per steradian = 1 candela.
Such a light source corresponds approximately to a freestanding household candle if the shadowing of the flame by the candle body downwards and its reflector effect upwards, as well as the flickering of the intensity are neglected.
The same light source is now provided with a spotlightlike focusing device, so that all of the light generated is emitted evenly within a circular cone with an opening angle α of 5 °. What is the light intensity of the source in a direction within the radiation cone?
The cone spans the solid angle 0.006 steradian (see table above). Since the light intensity is assumed to be constant within this solid angle, the simplified formula can be used again. The light intensity for all directions within the cone is therefore 12 lumens per 0.006 steradian, i.e. 2000 lumens per steradian or 2000 candela. For all other directions it is zero candela. By focusing, the light intensity of the source could be increased in certain directions (and at the expense of other directions).
If the light is not emitted evenly within the cone, the light intensities for the directions falling into the cone are different and either the exact formulas must be used, or the simplified formulas are used to determine an average value.
Example 2
A Lambert radiator (also: diffuse radiator) is a radiator that emits light with the same luminance in all directions.
A flat radiation surface of 1 m² is given, which radiates uniformly over the entire surface and in all directions with a luminance of 1000 cd / m². With what light intensity does it emit in a vertical direction (0 °)? How big is the light intensity in the beam directions 45 ° and 90 °?
Since the luminance is assumed to be constant over the area, the simplified formula can be used. Since the luminance is assumed to be identical in all directions, it can be treated as a directionindependent constant. Solving the equation for yields:
Inserting the numerical values gives:
90 °  1000 cd / m²  1 m²  1000 cd 
45 °  1000 cd / m²  0.707 m²  707 cd 
0 °  1000 cd / m²  0 m²  0 cd 
Such a diffuse radiator could, for. B. be represented by an appropriately illuminated white sheet of paper. The eye perceives the luminance as surface brightness. The fact that the paper radiates diffusely, i.e. emits the same luminance in all directions, means for the viewer that it has the same surface brightness when viewed from all directions. However, since it appears shortened by the projection factor cos (ε) when viewed from an angle (i.e. it occupies a smaller solid angle), the viewer receives a smaller amount of light despite the surface brightness remaining the same: the light intensity in this direction is lower.
A light source is referred to as anisotropic if its light intensity depends on the viewing direction. In the present case, the anisotropy results solely from the projection effect while the luminance remains the same. With most light sources, the luminance also depends on the direction. The dependence of the light intensity on the viewing angle is also referred to as the radiation characteristic.
The radiation characteristic of the isotropic house candle flame is spherical and that of the Lambert radiator follows the mathematical function . The exact radiation characteristics of real light sources (e.g. flashlights) are partially provided by the manufacturer.
Examples of luminous intensities of different light sources
Light source  Light intensity 

Firefly  0.0002 cd 
Candle (in all directions)  approx. 1 cd 
100 W incandescent lamp (in all directions)  approx. 100 cd 
Aviation obstacle marking Obstacle lights  red continuously illuminated Hazard lights  red flashing white flashing medium white flashing high 
10 to 32 cd 2000 cd 2 · 10 ^{4} cd 2 · 10 ^{5} cd 
Green laser pointer , 532 nm, 5 mW, 1.0 mrad divergence (in beam direction) 
3.8 x 10 ^{6} cd 
Beacon Helgoland (in the direction of the beam) 
40 x 10 ^{6} cd 
Sun (in all directions)  3.0 x 10 ^{27} cd 
Obsolete units
Light intensity units that were common in the past were:
 the old light unit, defined by a wax candle weighing 83 g with a flame height of 42 mm
 Vereinsparaffinkerze (1868) the unit of the German Association of Gas and Water Experts , DVGW, defined by a paraffin candle with a diameter of 20 mm and a flame height of 50 mm
 the Berlin light unit , defined by a Walrat candle with a flame height of 44.5 mm and a consumption of 7.77 g per hour
 the Violle unit, named after the French physicist Jules Violle , defined in 1889 as the light intensity of one square centimeter of platinum at a solidification temperature of 2042 Kelvin, it replaced the Carcel unit , which had been valid since 1842 .
 the " Bougie Décimale ", defined as ^{1} ⁄ _{20 of} the Violle unit, was a unit of light intensity in France before 1901 and was adopted in 1909 as the "International Candle" (IK) by Great Britain and the USA.
 English normal candle, Candlepower (equivalent to the Berlin light unit) a pure Walrat candle introduced in 1860 to measure the light intensity.
 1candle pentane lamp (1877), 1candle pentane wick lamp (1887), 10candle pentane gas lamp (1898), are lamps invented by Augustus Harcourt with pentane as fuel which successively replaced the English normal candle.
 from 1896 the Hefner candle (HK) was used in Germany
All units were replaced in 1942 by the "New Candle" (NK), which was renamed Candela in 1948 and has since been the SI base unit for light intensity.
New candle (NK) / Candela 
Hefner candle (HK)  International Candle (IK) 
Berlin LE  DVGW candle  Violle 

1  1.1074  0.98  0.9014  0.9225  0.04907 
0.9030  1  0.8860  0.8140  0.8330  0.04433 
1.0190  1.1280  1  0.9187  0.9402  0.05000 
1.11  1.2278  1.0885  1  1.0233  0.05442 
1.08  1.1998  1.0636  0.9772  1  0.05318 
20.38  22.5600  20.0000  18.3747  18.8036  1 
See also
literature
 Wilhelm von Zahn : About the photometric comparison of different colored light sources. In: Meeting reports of the Natural Research Society in Leipzig . 1st year, 1874, W. Engelmann, Leipzig 1875, pp. 2529.
Web links
 Photometry applet  illustration of photometric quantities
 Converter for candela, lumen, lux, beam angle, distance
Remarks
 ↑ The luminous flux is an integral quantity from which all other photometric quantities can be derived through differentiation. Luminance is a differential quantity from which all other photometric quantities can be derived through integration.
 ↑ More precisely: the illuminance on the retina, which, however, is determined by the illuminance on the pupil surface for a given pupil opening and a given degree of transmission of the eye media, cf. DIN 5031: Radiation physics in the optical field and lighting technology. Part 6: Pupillary light intensity as a measure of the retinal illumination. , BeuthVerlag, Berlin 1982. See also → Troland .
 ↑ What is meant is the "full" opening angle counted from one side of the cone jacket to the opposite side, not the "half" opening angle between the cone axis and the cone jacket.
Individual evidence
 ↑ ^{a } ^{b} H.J. Hentschel: Light and Lighting  Theory and Practice of Lighting Technology. 4th edition, Hüthig Buch, Heidelberg 1994, ISBN 3778521845 , p. 24.
 ↑ ^{a } ^{b } ^{c } ^{d } ^{e } ^{f} DIN 5031 Radiation physics in the optical field and lighting technology , Part 3: Quantities, symbols and units in lighting technology. Beuth, Berlin 1982.
 ↑ Directive (EU) 2019/1258  official German translation from the SI brochure from 2019 (9th edition)
 ^ WR Blevin, B. Steiner: Redefinition of the Candela and the Lumen . In: Metrologia . tape 11 , no. 3 , July 1975, p. 97 , doi : 10.1088 / 00261394 / 11/3/001 .
 ^ M. Minnaert: Light and Color in the Outdoors. Springer, New York 1993, ISBN 9780387944135 , Chapter 13: Luminous Plants, Animals, and Stones , p. 371 doi : 10.1007 / 9781461227229_13 (limited preview)
 ↑ Assumed as an isotropic source: 1380 lm / (4π) ≈ 100 cd
 ^ R. Bishop: The Intensity, Luminance, and Illuminance of GLPs. (accessed on March 4, 2015). I _{v} = Φ _{v} / Ω = Φ _{e} · K _{m} · V (λ) / Ω = 0.005 W · 683 lm / W · 0.88 / (7.85 × 10 ^{7} sr) = 3 lm / (7, 85 · 10 ^{−7} sr) = 3.8 · 10 ^{6} cd. For Φ _{e} and K _{m} · V (λ) see → Photometric radiation equivalent .
 ↑ H.J. Hentschel: Light and Lighting  Theory and Practice of Lighting Technology. 4th edition, Hüthig Buch, Heidelberg 1994, ISBN 3778521845 , p. 207.
 ↑ 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. 559565 doi: 10.1016 / j.solener.2005.01.004 . For the standard brightness sensitivity curve V (λ): 2.9839513 · 10 ^{27} cd, for the brightness sensitivity curve V _{M} (λ) modified in 1988 : 3.0012730 · 10 ^{27} cd.
 ↑ Clarence Herzog, Clarence Feldmann: Handbook of electric lighting. 3rd edition, Springer 1907, ISBN 9783642506888 , p. 27.
 ^ Wilhelm H. Westphal: Physical Dictionary Springer, 1952, ISBN 9783662127070 , pp. 792 f.