Irradiance

Physical size
Surname	Irradiance
Formula symbol	${\displaystyle E}$, ${\displaystyle E_{\mathrm {e} }}$
Size and unit system unit dimension SI W · m -2 M · T −3

The irradiance ( Engl. : Irradiance , radiant flux density , and the radiation flux density , outdated: radiation current density ) is the term for the entire power of the incoming electromagnetic energy impinging on a surface, based on the size of the area. ${\displaystyle E}$

The photometric equivalent of the irradiance is the illuminance E _v , which also includes the special properties of human perception. To distinguish this, the symbol E _e is often used for the irradiance , whereby the index “e” means that the irradiance is purely energetic , i.e. H. is an objective measurand . In the field of electrical engineering, irradiance is often used synonymously with intensity , but the latter generally refers to waves.

Analogous to the irradiance, there is the specific radiation , which describes the radiant power emanating from an area per area. It should not be confused with irradiation (measured in J⋅m ⁻² ), which describes the accumulated energy per unit area as a time-integrated quantity.

definition

The irradiance is defined as the radiant flux  through the irradiated area  : ${\displaystyle \mathrm {d} \Phi }$ ${\displaystyle \mathrm {d} A}$

${\displaystyle E={\frac {\mathrm {d} \Phi }{\mathrm {d} A}}=\int _{\Omega }L\cos \varepsilon \;\mathrm {d} \Omega }$

With

• ${\displaystyle L}$= Radiance
• ${\displaystyle \varepsilon }$= Angle of the solid angle element to the surface normal . The cosine factor takes into account that in the case of irradiation from any given direction, only the projection of the surface perpendicular to this direction occurs as the effective receiving surface. ${\displaystyle \varepsilon }$ ${\displaystyle \cos \varepsilon \,\mathrm {d} A}$ ${\displaystyle \mathrm {d} A}$
• ${\displaystyle \mathrm {d} \Omega }$= Solid angle element.

General definition in the field

The radiation distribution more generally, i. H. not necessarily collimated , radiation is given by a direction-dependent  radiance ( : spherical coordinates ). In this case, the irradiance in direction ( ) is defined as ${\displaystyle L(\theta ,\varphi )}$ ${\displaystyle \theta ,\varphi }$ ${\displaystyle \theta _{0},\varphi _{0}}$

${\displaystyle {\begin{aligned}E&=\int \limits _{\varphi =0}^{2\pi }\int \limits _{\theta =0}^{\pi }L(\theta ,\varphi )\;{\vec {e}}(\theta _{0},\varphi _{0})\;{\vec {e}}(\theta ,\varphi )\;\sin \theta \;{\mathrm {d} \theta }\;{\mathrm {d} \varphi }\\&=\int _{\Omega }L(\theta ,\varphi )\;{\vec {e}}(\theta _{0},\varphi _{0})\;{\vec {e}}(\theta ,\varphi )\;\mathrm {d} \Omega \end{aligned}}}$

With

• Unit vectors ${\displaystyle {\vec {e}}}$
• the relationship ${\displaystyle \mathrm {d} \Omega =\sin \theta \,\mathrm {d} \theta \,\mathrm {d} \varphi .}$

The following are also defined:

• the scalar irradiance (engl .: scalar irradiance ) that takes into account the beam density regardless of the direction:

${\displaystyle {\begin{aligned}E_{0}&=\int \limits _{\varphi =0}^{2\pi }\int \limits _{\theta =0}^{\pi }L(\theta ,\phi )\;\sin \theta \;{\mathrm {d} \theta }\;{\mathrm {d} \varphi }\\&=\int _{\Omega }L(\theta ,\phi )\;\mathrm {d} \Omega \end{aligned}}}$

• the vectorial irradiance (engl .: vectorial irradiance ) having a net represents irradiance (with direction):

${\displaystyle {\vec {E}}=(E_{x},E_{y},E_{z}),}$

where the components and the irradiance mean in the x , y and z directions. ${\displaystyle E_{x},E_{y}}$ ${\displaystyle E_{z}}$

Gershun equation

The Gershun equation (according to Andre Aleksandrovich Gershun , 1903–1952) relates the scalar and vectorial irradiance to the absorption coefficient  : ${\displaystyle a}$

${\displaystyle \nabla {\vec {E}}=-a\,E_{0}.}$

Since the scattering coefficient does not appear in the relationship , the absorption coefficient can be determined  in any radiation distribution - regardless of the scattering - by determining the two irradiance levels: ${\displaystyle a}$

${\displaystyle \Leftrightarrow a=-\,{\frac {\nabla {\vec {E}}}{E_{0}}}.}$

Spectral irradiance

The spectral irradiance (unit: W m ⁻² Hz ⁻¹ ) indicates which radiation power hits the body at the frequency from the entire half-space per unit area and per unit frequency interval: ${\displaystyle E_{\nu }(\nu )}$ ${\displaystyle \nu }$

${\displaystyle E_{\nu }(\nu )=\int \limits _{\text{Halbraum}}\,L_{\Omega \nu }(\theta ,\varphi ,\nu )\,\cos \theta \,\mathrm {d} \Omega .}$

with the spectral radiance ${\displaystyle L_{\Omega \nu }.}$

Relationship with other radiometric quantities

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

• DIN-Taschenbuch 22. Units and terms for physical quantities . Beuth Verlag, 1999, ISBN 3-410-14463-3
• Erich Helbig: Basics of light measurement technology . 2nd edition, Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1977, DNB 770197817
• Gershun, A. (1936/1939): Svetovoe Pole (English: The Light Field ), Moscow 1936. Translated by P. Moon and G. Timoshenko (1939) in Journal of Mathematics and Physics, 18, 51-151