Radiance

Physical size
Surname	Radiance
Formula symbol	${\displaystyle L}$
Size and unit system unit dimension SI Watt  (W) per square meter (m²) and per steradian (sr) M · T −3

The radiation density or radiation density  L (also specific intensity , English radiance ) provides detailed information about the location and direction dependency of the radiation emitted by a transmitting surface.

definition

Introduction and consideration of limit values

Most objects emit different amounts of radiation power from different points on their surface.

Consider a body (for example an incandescent lamp , a light emitting diode ) which emits radiation (measured for example in watts ) into its surroundings. As a rule, each point on the body will emit different amounts of radiation in different directions. If this characteristic is to be described in detail, the concept of radiance is necessary.

It is not possible to state how many watts emanate from an infinitely small point on the surface of the body, since the finite radiation power is distributed over an infinite number of such points and therefore zero watts are allotted to a single surface point. Instead, one looks at a small area around the point in question, sets the (finite) radiation power emanating from this area in relation to its (finite) area and lets the area shrink to zero. Although the radiated power and the radiating area both approach zero, the ratio of both tends towards a finite limit value ( differential quotient ), the area output or specific radiation of the point, measured in watts per square meter. ${\displaystyle M=\mathrm {d} \Phi /\mathrm {d} A}$

Most objects emit different amounts of radiation power in different directions.

In the same way, it is not possible to specify which power is emitted in a certain direction, since the finite radiation power is distributed over an infinite number of possible directions and therefore zero watts are allotted to each individual direction. Instead, one considers a small solid angle surrounding the desired direction , sets the (finite) power output 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 radiated power contained therein tend towards zero, but their ratio towards a finite limit value, the radiant intensity emitted in the relevant direction , measured in watts per steradian . ${\displaystyle I=\mathrm {d} \Phi /\mathrm {d} \Omega }$

The radiation density combines both and in this way describes both the location and the directional dependence of the radiation emitted by an infinitely small surface element.

Radiance

The radiation density indicates which radiation power is emitted from a given point of the radiation source in the direction given by the polar angle and the azimuth angle per projected surface element and per solid angle element . ${\displaystyle L_{\Omega }(\beta ,\varphi )}$ ${\displaystyle \mathrm {d} ^{2}\Phi }$ ${\displaystyle \beta }$ ${\displaystyle \varphi }$ ${\displaystyle \cos(\beta )\mathrm {d} A}$ ${\displaystyle \mathrm {d} \Omega }$

${\displaystyle L_{\Omega }(\beta ,\varphi )={\frac {\mathrm {d} ^{2}\Phi }{\cos(\beta )\mathrm {d} A\ \cdot \mathrm {d} \Omega }}}$

${\displaystyle \beta }$is the angle between the radiation direction and the surface normal .

The definition of the radiance has the special feature that the emitted radiant power is not related to the radiating surface element , as usual , but to the surface element projected in the direction of emission . The radiation power emitted in a certain direction depends on the one hand on the (possibly direction-dependent) 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 radiant power emitted at the polar angle is lower by a factor than the power emitted perpendicularly. The division by the factor calculates this geometric effect, so that only a possible physical directional dependency due to the surface properties remains. ${\displaystyle \mathrm {d} A}$ ${\displaystyle \cos(\beta )\mathrm {d} A}$ ${\displaystyle \beta }$ ${\displaystyle \cos(\beta )}$ ${\displaystyle \cos(\beta )}$

Surfaces that no longer show any directional dependence of the radiance after the factor has been calculated are called diffuse emitters or Lambertian emitters . A Lambertian surface element emits the same radiance in all directions: ${\displaystyle \cos }$

${\displaystyle L_{\Omega }(\beta ,\varphi )=L_{\Omega }={\text{const.}}}$

The radiation power emitted by it in a certain direction only varies with the cosine of the radiation angle; Such emitters are therefore particularly easy to handle mathematically:

${\displaystyle \mathrm {d} ^{2}\Phi =L_{\Omega }\cos(\beta )\mathrm {d} A\ \cdot \mathrm {d} \Omega }$

In particular, when integrating over the solid angle, the now angle-independent radiance can be drawn as a constant in front of the integral (see below). ${\displaystyle L_{\Omega }}$

For the definition of the radiation density, it is irrelevant whether the radiation emitted by the surface element is (thermal or non-thermal) self-emission , transmitted or reflected radiation, or a combination thereof.

Radiance is defined at every point in space where radiation is present. Instead of a radiating surface element, think of a fictitious radiating surface element in space.

The photometric equivalent of the radiance is the luminance , which can therefore be used for illustration: The luminance is a measure of the brightness with which a surface is perceived. If you look at a diffusely luminous surface, e.g. B. a sheet of paper, from different directions, the perceived luminance of the area remains constant, while the total amount of light reaching the viewer depends on the projected area and therefore varies with the cosine of the viewing angle. Similarly, the radiance of a diffuse radiator is the same in all directions, but the radiant power emitted in a certain direction also depends on the beam area projected in the relevant direction.

Spectral radiance

The spectral radiance (engl. Spectral radiance ) (unit: W · m ^-2 · Hz ^-1 · sr ^-1 ) of a body indicates which radiation power of the body at the frequency in the by the polar angle and the azimuth angle of projected given direction per Area, per solid angle and per frequency width. ${\displaystyle L_{\Omega \nu }(\theta ,\varphi ,\nu )}$ ${\displaystyle \nu }$ ${\displaystyle \theta }$ ${\displaystyle \varphi }$

The spectral radiance is also given as (unit: W · m ⁻³ · sr ⁻¹ ) related to the unit wavelength interval . ${\displaystyle L_{\Omega \lambda }(\beta ,\varphi ,\lambda )}$

The spectral radiance provides the most detailed representation of the radiation properties of a radiator. It explicitly describes the directional dependence and the frequency (or wavelength) dependence of the emitted radiation. The other radiation quantities can be derived from the spectral radiance by integrating them over the directions and / or frequencies. Integration over the relevant frequency or wavelength interval in particular again provides the radiance, which is therefore also called the total radiance if it has to be distinguished from the spectral radiance.

Black body

A black body is an idealized body that completely absorbs all electromagnetic radiation that hits it. For thermodynamic reasons, the thermal radiation emitted by such a body has a universal spectrum and it must necessarily be a Lambertian radiator. Real emitters never fully achieve these ideal properties, but can come close to them. The radiation properties of a black body can therefore often be used as a good approximation for a real body.

The deviation of a real radiator from the black ideal can be recorded by an emissivity . Since a real emitter cannot radiate more strongly than a black body at the same temperature at a given wavelength, the emissivity must always be less than 1. The emissivity can depend on the wavelength and, if the real emitter is not a Lambertian emitter, also depending on the direction. The emissivities are determined by comparing the radiance or the spectral radiance of the real and black body.

Spectral distribution of the intensity of black body radiation in a double-logarithmic plot

Spectral radiance of a blackbody

Derivation

According to Planck, the following applies to the spectral energy density of a black body:

${\displaystyle U_{\nu }^{o}(\nu ,T)\,\mathrm {d} \nu =8\pi {\frac {h\nu ^{3}}{c^{3}(e^{\frac {h\nu }{kT}}-1)}}\,\mathrm {d} \nu }$

This gives the spectral radiance:

${\displaystyle L_{\Omega \nu }^{o}(\nu ,T)\,\mathrm {d} \nu ={\frac {c}{4\pi }}\cdot U_{\nu }^{o}(\nu ,T)\,\mathrm {d} \nu }$

The factor can be understood in such a way that the radiation spreads with the speed in the entire solid angle . ${\displaystyle c}$ ${\displaystyle \Omega =4\pi }$

Inference

For the spectral radiance of a black body the absolute temperature applies according to Planck ${\displaystyle L_{\Omega \nu }^{o}}$ ${\displaystyle T}$

in the frequency display:

${\displaystyle L_{\Omega \nu }^{o}(\nu ,T)\,\cos(\beta )\,\mathrm {d} A\,\mathrm {d} \nu \,\mathrm {d} \Omega ={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{\left({\frac {h\nu }{kT}}\right)}-1}}\cos(\beta )\,\mathrm {d} A\,\mathrm {d} \nu \,\mathrm {d} \Omega }$

With

${\displaystyle L_{\Omega \nu }^{o}(\nu ,T)}$	:	spectral radiance of the blackbody,	W m −2  Hz −1  sr −1
${\displaystyle \nu }$	:	Frequency,	Hz

and in the wavelength display:

${\displaystyle L_{\Omega \lambda }^{o}(\lambda ,T)\,\cos(\beta )\,\mathrm {d} A\,\mathrm {d} \lambda \,\mathrm {d} \Omega ={\frac {2hc^{2}}{\lambda ^{5}}}{\frac {1}{e^{\left({\frac {hc}{\lambda kT}}\right)}-1}}\cos(\beta )\,\mathrm {d} A\,\mathrm {d} \lambda \,\mathrm {d} \Omega }$

With

${\displaystyle L_{\Omega \lambda }^{o}(\lambda ,T)}$	:	spectral radiance of the blackbody,	W m −2  μm −1  sr −1
${\displaystyle \lambda }$	:	Wavelength,	m, µm
${\displaystyle T}$	:	absolute temperature,	K
${\displaystyle h}$,	:	Planck's quantum of action ,	Js
${\displaystyle c}$	:	Speed ​​of light ,	m / s
${\displaystyle k}$	:	Boltzmann constant ,	J / K

${\displaystyle L_{\Omega \nu }^{o}(\nu ,T)\,\cos(\beta )\,\mathrm {d} A\,\mathrm {d} \nu \,\mathrm {d} \Omega }$is the radiant power from the surface element in the frequency range and in that between the azimuth angles and and the polar angles and space spanned angle element is radiated. The directional dependency of this radiation power is only due to the geometric factor; the spectral radiance itself is independent of direction. ${\displaystyle \mathrm {d} A}$ ${\displaystyle \nu }$ ${\displaystyle \nu +\mathrm {d} \nu }$ ${\displaystyle \varphi }$ ${\displaystyle \varphi +\mathrm {d} \varphi }$ ${\displaystyle \beta }$ ${\displaystyle \beta +\mathrm {d} \beta }$ ${\displaystyle \mathrm {d} \Omega }$ ${\displaystyle \cos }$

When converting between frequency and wavelength representation, it should be noted that because of

${\displaystyle \lambda ={\frac {c}{\nu }}}$

applies:

${\displaystyle |\mathrm {d} \lambda |={\frac {c}{\nu ^{2}}}|\mathrm {d} \nu |\quad {\text{und}}\quad |\mathrm {d} \nu |={\frac {c}{\lambda ^{2}}}|\mathrm {d} \lambda |}$

The ratio of the spectral radiance of a surface element of a given radiator emitted in a certain direction and observed at a certain wavelength to the spectral radiance of a black body of the same temperature observed at the same wavelength is the directional spectral emissivity of the surface element.

If one integrates the spectral radiance of the black body over all directions of the half-space in which the surface element radiates, one obtains the spectral specific radiation of the black body. The integral provides an additional factor . For the formula see the article " Planck's law of radiation ". ${\displaystyle \pi }$

Total radiance of a blackbody

If the spectral radiance is integrated over all frequencies or wavelengths, the total radiance is calculated : ${\displaystyle L_{\Omega }^{o}(T)}$

${\displaystyle L_{\Omega }^{o}(T)\cos(\beta )\,\mathrm {d} A\,\mathrm {d} \Omega =\int _{\nu =0}^{\infty }L_{\Omega \nu }^{o}(\nu ,T)\cos(\beta )\,\mathrm {d} A\,\mathrm {d} \nu \,\mathrm {d} \Omega }$

The evaluation of the integral yields because of : ${\displaystyle \int _{0}^{\infty }{\frac {x^{3}}{e^{x}-1}}\,\mathrm {d} x={\frac {\pi ^{4}}{15}}}$

${\displaystyle L_{\Omega }^{o}(T)\cos(\beta )\,\mathrm {d} A\,\mathrm {d} \Omega ={\frac {2\pi ^{4}k^{4}}{15h^{3}c^{2}}}T^{4}\cos(\beta )\,\mathrm {d} A\,\mathrm {d} \Omega }$

With

${\displaystyle L_{\Omega }^{o}(T)}$: Total radiance of the black body, W m ⁻² sr ⁻¹ .

${\displaystyle L_{\Omega }^{o}(T)\cos(\beta )\,\mathrm {d} A\,\mathrm {d} \Omega }$is the radiant power that is radiated from the surface element at all frequencies into the solid angle element in that direction . ${\displaystyle \mathrm {d} A}$ ${\displaystyle \beta }$ ${\displaystyle \mathrm {d} \Omega }$

The ratio of the total radiation density emitted in a certain direction of a surface element of a given radiator to the total radiation density of a black body of the same temperature is the directed total emissivity of the surface element.

If one integrates the total radiation density of the black body over all directions of the half-space in which the surface element radiates, one obtains the specific radiation of the black body. The integral provides an additional factor . For the formula, see the article “ Stefan Boltzmann Law ”. ${\displaystyle \pi }$

application

Changing the definition equation for the radiance provides the radiant power that is radiated from the surface element into the solid angle element , which lies in the direction described by the angles and : ${\displaystyle \mathrm {d} A}$ ${\displaystyle \mathrm {d} \Omega }$ ${\displaystyle \beta }$ ${\displaystyle \phi }$

${\displaystyle \mathrm {d} ^{2}\Phi (\beta ,\varphi )=L_{\Omega }(\beta ,\varphi )\cdot \cos(\beta )\mathrm {d} A\cdot \mathrm {d} \Omega }$

If the radiation of a finitely large radiating surface is to be determined in a finitely large solid angle , then integrate via and : ${\displaystyle A}$ ${\displaystyle \Omega }$ ${\displaystyle \mathrm {d} A}$ ${\displaystyle \mathrm {d} \Omega }$

${\displaystyle \Phi =\int _{\Omega }\int _{A}L_{\Omega }(\beta ,\varphi )\cdot \cos(\beta )\mathrm {d} A\cdot \mathrm {d} \Omega =\int _{\Delta \beta }\int _{\Delta \varphi }\int _{A}L_{\Omega }(\beta ,\varphi )\cdot \cos(\beta )\sin(\beta )\cdot \mathrm {d} A\,\mathrm {d} \beta \,\mathrm {d} \varphi }$

The representation of the solid angle element in spherical coordinates was used:

${\displaystyle \mathrm {d} \Omega =\sin(\beta )\,\mathrm {d} \beta \,\mathrm {d} \varphi }$

Since it can generally depend on the location on the beam surface and the directions swept over, a very complicated integral may result. A significant simplification occurs when the radiant surface is a Lambertian radiator (the radiance is independent of direction) with constant surface properties (the radiance is independent of location). Then the radiance is a constant number and can be drawn in front of the integral: ${\displaystyle L_{\Omega }}$ ${\displaystyle A}$ ${\displaystyle L}$

${\displaystyle \Phi =A\cdot L\int _{\Omega }\cos(\beta )\ \mathrm {d} \,\Omega }$

The 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. ${\displaystyle \Omega }$ ${\displaystyle L}$

If, for example, the radiation in the entire half-space overlooked by the beam surface is considered, the value for the integral results and the radiation of a Lambertian radiator of the surface in the entire half-space is simple: ${\displaystyle \pi }$ ${\displaystyle A}$

${\displaystyle \Phi =\pi \,A\,L}$ (Radiant power of a Lambertian radiator in the half space)

If the radiation surface is a black body of temperature , the required radiation density can be calculated immediately using Planck's law of radiation (see formulas above). If it is a gray body , the Planck radiance has to be reduced in order to reduce the emissivity . A possible location and direction dependency of the emissivity as well as possible reflections can make the integration more difficult. ${\displaystyle T}$

Basic photometric law

The basic photometric law states that the luminance remains unchanged on the way from the light source to the illuminated surface. In radiometry, this applies analogously: The radiance at the location of the transmitter in the direction of the receiver is equal to the radiation density at the location of the receiver from the direction of the transmitter. For a detailed description see Luminance # Photometric Basic Law .

Relation to other radiometric quantities and to 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

• HD Baehr, K. Stephan: Heat and mass transfer. 5th edition. Springer, Berlin 2006, ISBN 978-3-540-32334-1 , chap. 5: thermal radiation.

Individual evidence

1. ↑ ^{a b} electropedia , International Electrotechnical Dictionary (IEV) of the International Electrotechnical Commission : Entry 845-01-30 (area of ​​lighting) has the translation: radiance = "radiation density"
2. ↑ DIN EN ISO 9288: Heat transfer by radiation - physical quantities and definitions. Beuth Verlag, August 1996