Luminance
Physical size  

Surname  Luminance  
Formula symbol  

The luminance L _{v} ( English luminance ) provides detailed information about the location and direction dependency of the luminous flux emitted by a light source . The luminance of a surface determines the surface brightness with which the eye perceives the surface and therefore has the most direct relation to optical sensory perception of all photometric quantities . The luminance describes the brightness of extensive, planar light sources; The luminous intensity is more suitable for describing the brightness of point light sources .
definition
introduction
Consider a body serving as a light source (for example an incandescent lamp , an illuminated sheet of paper) which emits a luminous flux (measured in lumens ) into its surroundings. As a rule, different points on the body will emit different amounts of light and it will also emit different amounts of light in different directions. If this characteristic is to be described in detail, the concept of luminance is necessary.
It is not possible to state how many lumens emanate from an infinitely small point on the surface of the body, since the finite number of radiated lumens is distributed over an infinite number of such points and therefore zero lumens are allotted to a single surface point. Instead, one looks at a small area surrounding the point in question, sets the (finite) luminous flux emanating from this area in relation to its (finite) area and allows the area to shrink to zero. Although the emitted luminous flux as well as the radiating surface both approach zero, the ratio of both tends towards a finite limit value ( differential quotient ), the specific light emission of the point, measured in lumens per square meter or equivalent lux .
It is also 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, one looks at a small solid angle surrounding the desired direction , sets the (finite) luminous flux emitted in this solid angle in relation to the (finite) size of the solid angle and lets the solid angle shrink to zero. Again, 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 .
The concept of luminance combines both and in this way describes both the location and the directional dependence of the luminous flux emitted by an infinitely small surface element.
For the definition of the luminance, it is irrelevant whether the light emitted by the surface element is (thermal or nonthermal) selfemission , transmitted or reflected light or a combination thereof.
The luminance is 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.
Mathematical description
If a uniformly luminous flat surface emits a luminous flux with uniform luminous intensity in the solid angle below the beam angle , the luminance is the quotient of the emitted luminous flux and the product of the solid angle irradiated and the area projected in the direction of the beam :
 .
is the angle between the direction of radiation and the surface normal that is perpendicular to the surface element .
If the light intensity varies within the solid angle under consideration and / or the specific light emission within the luminous area , this mathematically simplified formula provides the mean value of the luminance over and . If the variation of the luminance is to be described in detail, the transition to the differential quotient gives:
The definition of the luminance has the special feature that the emitted luminous flux is not related to the radiating surface element , as usual , but to the surface element projected in the direction of emission . The luminous flux emitted in a certain direction depends on the one hand on the (possibly directiondependent) physical radiation properties of the surface and, on the other hand, purely geometrically on the projection of the radiating surface element effective in the radiation direction. The second effect has the effect that the luminous flux emitted at the polar angle is a factor smaller than the luminous flux emitted perpendicularly. Division by the factor calculates this geometric effect, so that only a possible physical directional dependency due to the surface properties (e.g. the luminance coefficient ) remains in the luminance .
Lambertian radiator
Surfaces which no longer show any directional dependence of the luminance after calculating the factor are called diffuse radiators or Lambertian radiators . A Lambertian surface element emits the same luminance in all directions . The luminance is therefore no longer dependent on the angle:
The luminous flux emitted by a Lambertian radiator in a certain direction only varies with the cosine of the radiation angle . Such emitters are therefore particularly easy to handle mathematically:
In particular, when integrating over the solid angle, the now angleindependent luminance can be drawn as a constant in front of the integral, which often considerably simplifies the integration (see below).
An example of a diffusely luminous surface is an illuminated sheet of paper. If you look at it from different directions, the perceived luminance of the surface remains constant, while the total amount of light reaching the viewer (the light intensity) depends on the projected surface and therefore varies with the cosine of the viewing angle.
comment  Luminance (cd / m ^{2} )  

Eye threshold  approx. 3 · 10 ^{−6}  
scotopic vision  pure night vision  3 · 10 ^{−6} to 3… 30 · 10 ^{−3} 
mesopic vision  3… 30 · 10 ^{−3} to 3… 30  
photopic vision  pure daytime vision  over 3… 30 
Cones saturation  Glare  from 10 ^{5} … 10 ^{6} 
Sensitivity of the eyes
The observer perceives the luminance of the surfaces surrounding him directly as their surface brightness. Due to the adaptability of the eye , the perceptible luminance can cover numerous orders of magnitude. The specified values vary from person to person and are also dependent on the frequency of the light.
Examples



Relationship with other photometric quantities
General
Luminance indicates how much light is emitted by a given infinitesimal surface element in a given direction, and thus provides the most detailed description of the luminous properties of the surface in question. Changing the defining equation for the luminance provides the infinitesimal luminous flux , that of the at the point , lying surface element in the solid angle element is irradiated, which in by the angle and is described direction:
If the light emission of a finitely large radiating surface is to be determined in a finitely large solid angle , the following must be integrated via and :
The representation of the solid angle element in spherical coordinates was used:
Because in general the place , on the luminous area and the cleaned directions and may depend, is sometimes encountered a very complicated integral. A significant simplification occurs when the luminous surface is a Lambertian emitter (the luminance is therefore independent of direction) with constant surface properties (the luminance is therefore independent of location). Then the luminance is a constant number and can be drawn in front of the integral:
The remaining integral now only depends on the shape and position of the solid angle and can be solved independently of . In this way, only by the sender and receiver geometry dependent general view factors are determined and ready tabulated.
If, for example, the light emission is considered in the entire halfspace overlooked by the luminous area, the value for the integral results
 ,
and the luminous flux of a homogeneous Lambertian spotlight of the surface in the entire halfspace is simple:
In a similar way, the other photometric quantities can be derived from the luminance by integration over the total area and / or all directions of the halfspace.
Light intensity
If, instead of the radiation of a surface element, one considers the radiation of the entire surface of a body in a given direction, then one has to integrate over the radiation surface , but not over the directions, and one gets the light intensity of the body in this direction:
 ,
the coordinates , the position of the planar element to describe the total area and the angle , the observed beam direction with respect to the surface normal of state. In particular, it is again the angle between the radiation direction considered and the surface normal. The integral is to extend over that part of the entire surface for which is.
The light intensity is, so to speak, the sum of all the luminance of the body surface emitted in a certain direction.
Specific light emission
If, instead of the radiation of the surface element in a certain direction, one considers its radiation in the entire halfspace overlooked by the surface element, then it is not possible to integrate over all directions over the entire surface and one obtains the specific light emission of the surface element:
In the special case of a Lambertian radiator is independent of the angles and and can be pulled outside the integral. As explained above, the remaining integral has the value , and the simple relationship results
(for a Lambertian radiator).
Luminous flux
If one integrates the luminance over all directions of the halfspace and all surface elements of the radiating surface, or the light intensity over all directions, or the specific light emission over all surface elements, one obtains the total luminous flux of the illuminating body:
In the special case of a Lambertian spotlight, the luminous flux can be calculated directly from the luminance:
 , for a Lambertian radiator
If the luminance is also homogeneous in area (i.e. the same over the entire area ), then the integral is simplified to a simple multiplication:
(for a homogeneous Lambertian radiator),
as shown above through a direct integration.
Basic photometric law
The basic photometric law describes the exchange of light between two surfaces. The luminance is a key factor here.
Light emission
Considering a surface element , which with the luminance of a spaced befindliches surface element illuminated so spans from viewed from the solid angle , and follows from the first equation in the previous section:
Here, and are the angles of inclination of the surface elements with respect to the common connecting line.
This is the basic photometric law . By integrating the two surfaces, the total luminous flux flowing from surface 1 to surface 2 is obtained .
Exposure to light
The illumination density is analogous to the luminance, but defined for the case of irradiation. It indicates which luminous flux is received from the direction given by the polar angle and the azimuth angle per projected surface element and per solid angle element . The equations derived so far apply analogously. In particular, the following applies to the luminous flux received on the surface element and emitted by:
This time the solid angle spanned by occurs.
Inference
The luminous flux emitted from to and the luminous flux received from must be identical (unless light is lost through absorption or scattering in a medium lying between the surfaces), and a comparison of the two equations results in:
The luminance emitted by the surface element is identical to the illuminance incident on the surface element .
Please note that the luminance does not decrease with the distance . The total transmitted luminous flux or , on the other hand, decreases as expected with the square of the distance (due to the factor in the denominator of both equations), this is due to the fact that the solid angle spanned by the transmitter surface from the perspective of the receiver surface decreases quadratically with the distance.
Example: If you compare a nearby billboard with an identically illuminated one further away, both appear equally “ bright ” (they have a distanceindependent and therefore identical luminance in both cases). The closer wall, however, takes up a larger solid angle for the observer, so that a larger luminous flux overall reaches the observer from this larger solid angle. The closer wall creates a greater illuminance for the observer ( photometric distance law ).
If the illumination density integrated over the solid angle from which it is derived, then the resulting illuminance called singlebeam light flux area density on the receiver surface in lm / m ^{2} . If the luminance of the transmitter surface emitted in a certain direction is known, then the identical illumination density of the receiver surface from the same direction is known immediately and the illuminance on the receiver surface can be calculated immediately from the luminance distribution of the transmitter surface: ^{}
Example: The sun has a luminance of L _{1} ≈ 1.5 · 10 ^{9} cd / m ^{2} and, when viewed from the earth, appears at a solid angle Ω = 6.8 · 10 ^{−5} sr. Since this solid angle is small, the integration via the solid angle occupied by the solar disk can be reduced to a multiplication with the solid angle. If the sun is at an altitude of 60 ° in summer (i.e. 30 ° deviating from the zenith ), the earth becomes with E _{2} = L _{1} · Ω · cos (30 °) = 89 000 lx irradiated.
units
The SI unit of luminance is candela per square meter (cd / m²).
In Englishspeaking countries, especially in the USA, the designation nit ( unit symbol nt , from Latin nitere = "seem", plural nits ) is used:
The specification of the luminance is used for the brightness of computer screens, which typically have 200 to 300 cd / m². Likewise, the brightness of LED video walls , as used in the event industry, is often specified in nits. Values in the lower to middle fourdigit range are common here. The nit is not a legal entity in Germany and Switzerland.
In addition, the unit Lambert is also used in the USA :
Conversion factors for other units of luminance include: a .:
 Stilb : 1 sb = 1 cd / cm² = 10,000 cd / m²
 Apostilb , Blondel : 1 asb = 1 blondel = 1 / π × 10 ^{−4} sb = 1 / π cd / m²
 Scot : 1 scot = 0.001 blond = 0.001 asb = 10 ^{−3} / π cd / m²
 Footlambert : 1 fL = 1 / π cd / ft² ≈ 3.426 cd / m²
 Candela per square foot : 1 cd / ft² ≈ 10.764 cd / m²
 Candela per square inch : 1 cd / in² ≈ 1550 cd / m².
Comparison of radiometric and photometric quantities
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 
See also
literature
 Hans R. Ris: Lighting technology for practitioners. 2nd edition, VDEVerlag GmbH, Berlin / Offenbach 1997, ISBN 3800721635 .
 Wilhelm Gerster: Modern lighting systems for indoors and outdoors. 1st edition, Compact Verlag, Munich 1997, ISBN 3817423950 .
 Horst Stöcker: Pocket book of physics. 4th edition, Verlag Harry Deutsch, Frankfurt am Main 2000, ISBN 3817116284 .
 Günter Springer: Expertise in electrical engineering. 18th edition, Verlag EuropaLehrmittel, Wuppertal 1989, ISBN 3808530189 .
See also
 Lambert's law
 light
 Light value
 Light intensity (photography)
 Luminance (luminance for monitors)
Individual evidence
 ↑ DIN EN ISO 9288: Heat transfer by radiation  physical quantities and definitions. Beuth Verlag, August 1996, for the analogous case of radiometric radiance .
 ↑ ^{a } ^{b} DIN 5031 radiation physics in the optical field and lighting technology. Part 3: Quantities, symbols and units in lighting technology. Beuth, Berlin 1982.
 ↑ Data sheet xenon radiators ( Memento of the original from March 3, 2016 in the Internet Archive ) Info: The archive link was automatically inserted and not yet checked. Please check the original and archive link according to the instructions and then remove this notice. (PDF; 5.5 MB).
 ↑ Lumitex: Tech Note 11: Light Energy / Brightness Units and Conversions ( Memento from March 17, 2006 in the Internet Archive ) (English).