The radiant power or radiant flux or is that differential amount of energy ( is the radiant energy ) that is transported by electromagnetic waves per period of time : ${\ displaystyle \ Phi}$${\ displaystyle \ Phi _ {\ mathrm {e}}}$ ${\ displaystyle \ mathrm {d} Q}$${\ displaystyle Q}$ ${\ displaystyle \ mathrm {d} t}$

${\ displaystyle \ Phi = {\ frac {\ mathrm {d} Q} {\ mathrm {d} t}}}$

Your unit is W ( watt ).

In astronomy , the radiant power of astronomical objects is called luminosity .

## Photons

From the photon stream (number of photons per unit of time) , the radiation power for monochromatic light results as: ${\ displaystyle \ phi = {\ tfrac {\ mathrm {d} N} {\ mathrm {d} t}}}$

${\ displaystyle \ Phi = h \ cdot \ phi \ cdot \ nu}$

With

• ${\ displaystyle h}$the Planck's constant
• ${\ displaystyle \ nu}$the light frequency .

For electromagnetic radiation with a frequency of 540 THz (green light with a wavelength of 555 nm), a photon current of 2 corresponds .79e18 s −1 of a radiation power of 1 W.

For polychromatic light one has to form the integral over all frequencies:

${\ displaystyle \ Phi = h \ cdot \ int _ {0} ^ {\ infty} {\ frac {\ mathrm {d} \ phi} {\ mathrm {d} \ nu}} \ cdot \ nu \ cdot \ mathrm {d} \ nu}$.

## Basic photometric law

Explanatory graphic for the basic photometric law

In order to determine the dependence of the radiant power on a surface element of a radiator surface of the luminance of a Lambert radiator (constant surface brightness) on a surface element located at a distance , the so-called fundamental photometric law can be used, which combines Lambert's law of cosines and the law of photometric distance . ${\ displaystyle \ mathrm {d} ^ {2} \ Phi _ {\ mathrm {1 \ rightarrow 2}}}$${\ displaystyle \ mathrm {d} A_ {1}}$${\ displaystyle A_ {1}}$${\ displaystyle L_ {1}}$${\ displaystyle r_ {12}}$${\ displaystyle \ mathrm {d} A_ {2}}$

${\ displaystyle \ mathrm {d} ^ {2} \ Phi _ {\ mathrm {1 \ rightarrow 2}} = L_ {1} \ cdot {\ frac {\ mathrm {d} A_ {1} \ cos \ beta _ {1} \ cdot \ mathrm {d} A_ {2} \ cos \ beta _ {2}} {r_ {12} ^ {2}}}}$

This depends, among other things, on the mutual position of the two surfaces in space, which is taken into account by the angle and between the beam direction and the surface normals. ${\ displaystyle \ beta _ {1}}$${\ displaystyle \ beta _ {2}}$

## Relation to other sizes

If the radiation output is related to the size of the irradiated area, the irradiance is obtained (unit: W / m²): ${\ displaystyle E}$

${\ displaystyle E = {\ frac {\ mathrm {d} \ Phi} {\ mathrm {d} A}}}$.

If, on the other hand, it is related to the solid angle into which a light beam emanating from a light source falls, then the radiation intensity is obtained${\ displaystyle \ Omega}$

${\ displaystyle I = {\ frac {\ mathrm {d} \ Phi} {\ mathrm {d} \ Omega}}}$

with the unit W / sr.

In photometry (lighting technology) the corresponding measurand is the luminous flux , measured in the unit lumen . While the radiant power (mostly written in this context ) is an energetic , i.e. objective, measured variable, the spectral sensitivity of the human eye is included in the luminous flux ( V-lambda curve ). The link between the two quantities is the photometric radiation equivalent of the light source ${\ displaystyle \ Phi _ {\ mathrm {v}}}$${\ displaystyle \ Phi _ {\ mathrm {e}}}$ ${\ displaystyle K}$

${\ displaystyle K \, = \, {\ frac {\ Phi _ {\ mathrm {v}}} {\ Phi _ {\ mathrm {e}}}}}$,

which depends on their wavelength spectrum.

The following table gives an overview of quantities and units in radiometry and photometry:

 radiometric quantity Symbol a) SI unit description photometric equivalent b) symbol SI unit Radiant flux radiant power, radiant flux, radiant power ${\ displaystyle \ Phi _ {\ mathrm {e}}}$ W ( watt ) Radiant energy through time Luminous flux luminous flux, luminous power ${\ displaystyle \ Phi _ {\ mathrm {v}}}$ lm ( lumens ) Radiant intensity irradiance, radiant intensity ${\ displaystyle I _ {\ mathrm {e}}}$ W / sr Radiation flux through solid angles Luminous intensity luminous intensity ${\ displaystyle I _ {\ mathrm {v}}}$ cd = lm / sr ( candela ) Irradiance irradiance ${\ displaystyle E _ {\ mathrm {e}}}$ W / m 2 Radiation flux through the receiver surface Illuminance illuminance ${\ displaystyle E _ {\ mathrm {v}}}$ lx = lm / m 2 ( lux ) Specific radiation emission current density, radiant exitance ${\ displaystyle M _ {\ mathrm {e}}}$ W / m 2 Radiation flux through the transmitter surface Specific light emission luminous exitance ${\ displaystyle M _ {\ mathrm {v}}}$ lm / m 2 Radiance radiance, radiance, radiance ${\ displaystyle L _ {\ mathrm {e}}}$ W / m 2 sr Radiant intensity through effective transmitter area Luminance luminance ${\ displaystyle L _ {\ mathrm {v}}}$ cd / m 2 Radiant energy amount of radiation, radiant energy ${\ displaystyle Q _ {\ mathrm {e}}}$ J ( joules ) by radiation transmitted energy Amount of light luminous energy, quantity of light ${\ displaystyle Q _ {\ mathrm {v}}}$ lm · s Irradiation irradiation, radiant exposure ${\ displaystyle H _ {\ mathrm {e}}}$ J / m 2 Radiant energy through the receiver surface Exposure luminous exposure ${\ displaystyle H _ {\ mathrm {v}}}$ lx s Radiation yield radiant efficiency ${\ displaystyle \ eta _ {\ mathrm {e}}}$ 1 Radiation flux through absorbed (mostly electrical) power Luminous efficiency (overall) luminous efficacy ${\ displaystyle \ eta _ {\ mathrm {v}}}$ lm / W
a)The index "e" is used to distinguish it from the photometric quantities. It can be omitted.
b)The photometric quantities are the radiometric quantities, weighted with the photometric radiation equivalent K , which indicates the sensitivity of the human eye.

## literature

• F. Pedrotti, L. Pedrotti, W. Bausch, H. Schmidt: Optics for engineers: Fundamentals . 2nd Edition. Springer, Berlin 2001, ISBN 3-540-67379-2 .

## Individual evidence

1. electropedia , International Electrotechnical Dictionary (IEV) of the International Electrotechnical Commission : Entry 845-01-24 (area of ​​lighting) is synonymous with: radiant flux = radiant power = "radiation power" = "radiation flux"