Radiant flux

In radiometry, radiant flux or radiant power is the radiant energy emitted, reflected, transmitted, or received per unit time, and spectral flux or spectral power is the radiant flux per unit frequency or wavelength, depending on whether the spectrum is taken as a function of frequency or of wavelength. The SI unit of radiant flux is the watt (W), one joule per second (J/s), while that of spectral flux in frequency is the watt per hertz (W/Hz) and that of spectral flux in wavelength is the watt per metre (W/m)—commonly the watt per nanometre (W/nm). Radiant flux is sometimes called luminosity, especially in astronomy contexts.

Mathematical definitions

Radiant flux

Radiant flux, denoted Φe ('e' for "energetic", to avoid confusion with photometric quantities), is defined as Φ e = d Q e d t Q e = ∫ T ∫ Σ S ⋅ n ^ d A d t {\displaystyle {\begin{aligned}\Phi _{\mathrm {e} }&={\frac {dQ_{\mathrm {e} }}{dt}}\\[2pt]Q_{\mathrm {e} }&=\int _{T}\int _{\Sigma }\mathbf {S} \cdot {\hat {\mathbf {n} }}\,dAdt\end{aligned}}} where

Qe is the radiant energy passing out of a closed surface Σ in time interval T;
t is time;
A is the area of the surface Σ;
S is the Poynting vector, representing the directional flow of energy per unit time, per unit area;
n is the unit normal vector to the differential area element dA.

The rate of energy flow through the surface fluctuates at the frequency of the radiation, but radiation detectors only respond to the average rate of flow. This is represented by replacing the Poynting vector with the time average of its norm, giving Φ e ≈ ∫ Σ ⟨ | S | ⟩ cos ⁡ α d A , {\displaystyle \Phi _{\mathrm {e} }\approx \int _{\Sigma }\langle |\mathbf {S} |\rangle \cos \alpha \ dA,} where ⟨-⟩ is the time average, and α is the angle between n and S.

Spectral flux

Spectral flux in frequency, denoted Φe,ν, is defined as Φ e , ν = ∂ Φ e ∂ ν , {\displaystyle \Phi _{\mathrm {e} ,\nu }={\frac {\partial \Phi _{\mathrm {e} }}{\partial \nu }},} where ν is the frequency.

Spectral flux in wavelength, denoted Φe,λ, is defined as Φ e , λ = ∂ Φ e ∂ λ , {\displaystyle \Phi _{\mathrm {e} ,\lambda }={\frac {\partial \Phi _{\mathrm {e} }}{\partial \lambda }},} where λ is the wavelength.

SI radiometry units

SI radiometry unitsvte
Quantity	Unit	Dimension	Notes
Name	Symbol	Name	Symbol
Radiant energy	Qe	joule	J	M⋅L2⋅T−2	Energy of electromagnetic radiation.
Radiant energy density	we	joule per cubicmetre	J/m3	M⋅L−1⋅T−2	Radiant energy per unit volume.
Radiant flux	Φe	watt	W = J/s	M⋅L2⋅T−3	Radiant energy emitted, reflected, transmitted or received, per unit time. This is sometimes also called "radiant power", and called luminosity in astronomy.
Spectral flux	Φe,ν	watt per hertz	W/Hz	M⋅L2⋅T−2	Radiant flux per unit frequency or wavelength. The latter is commonly measured in W⋅nm−1.
Φe,λ	watt per metre	W/m	M⋅L⋅T−3
Radiant intensity	Ie,Ω	watt per steradian	W/sr	M⋅L2⋅T−3	Radiant flux emitted, reflected, transmitted or received, per unit solid angle. This is a directional quantity.
Spectral intensity	Ie,Ω,ν	watt per steradian per hertz	W⋅sr−1⋅Hz−1	M⋅L2⋅T−2	Radiant intensity per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅nm−1. This is a directional quantity.
Ie,Ω,λ	watt per steradian per metre	W⋅sr−1⋅m−1	M⋅L⋅T−3
Radiance	Le,Ω	watt per steradian per squaremetre	W⋅sr−1⋅m−2	M⋅T−3	Radiant flux emitted, reflected, transmitted or received by a surface, per unit solid angle per unit projected area. This is a directional quantity. This is sometimes also called "intensity".
Spectral radiance Specific intensity	Le,Ω,ν	watt per steradian per squaremetre per hertz	W⋅sr−1⋅m−2⋅Hz−1	M⋅T−2	Radiance of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1. This is a directional quantity. This is sometimes also called "spectral intensity".
Le,Ω,λ	watt per steradian per squaremetre, per metre	W⋅sr−1⋅m−3	M⋅L−1⋅T−3
Irradiance Flux density	Ee	watt per square metre	W/m2	M⋅T−3	Radiant flux received by a surface per unit area. This is sometimes also called "intensity".
Spectral irradiance Spectral flux density	Ee,ν	watt per squaremetre per hertz	W⋅m−2⋅Hz−1	M⋅T−2	Irradiance of a surface per unit frequency or wavelength. This is sometimes also called "spectral intensity". Non-SI units of spectral flux density include jansky (1Jy = 10−26W⋅m−2⋅Hz−1) and solar flux unit (1sfu = 10−22W⋅m−2⋅Hz−1 = 104Jy).
Ee,λ	watt per squaremetre, per metre	W/m3	M⋅L−1⋅T−3
Radiosity	Je	watt per square metre	W/m2	M⋅T−3	Radiant flux leaving (emitted, reflected and transmitted by) a surface per unit area. This is sometimes also called "intensity".
Spectral radiosity	Je,ν	watt per squaremetre per hertz	W⋅m−2⋅Hz−1	M⋅T−2	Radiosity of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅m−2⋅nm−1. This is sometimes also called "spectral intensity".
Je,λ	watt per squaremetre, per metre	W/m3	M⋅L−1⋅T−3
Radiant exitance	Me	watt per squaremetre	W/m2	M⋅T−3	Radiant flux emitted by a surface per unit area. This is the emitted component of radiosity. "Radiant emittance" is an old term for this quantity. This is sometimes also called "intensity".
Spectral exitance	Me,ν	watt per squaremetre per hertz	W⋅m−2⋅Hz−1	M⋅T−2	Radiant exitance of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅m−2⋅nm−1. "Spectral emittance" is an old term for this quantity. This is sometimes also called "spectral intensity".
Me,λ	watt per squaremetre, per metre	W/m3	M⋅L−1⋅T−3
Radiant exposure	He	joule per squaremetre	J/m2	M⋅T−2	Radiant energy received by a surface per unit area, or equivalently irradiance of a surface integrated over time of irradiation. This is sometimes also called "radiant fluence".
Spectral exposure	He,ν	joule per squaremetre per hertz	J⋅m−2⋅Hz−1	M⋅T−1	Radiant exposure of a surface per unit frequency or wavelength. The latter is commonly measured in J⋅m−2⋅nm−1. This is sometimes also called "spectral fluence".
He,λ	joule per squaremetre, per metre	J/m3	M⋅L−1⋅T−2
See also: SI Radiometry Photometry

Radiant flux

Mathematical definitions

Radiant flux

Spectral flux

SI radiometry units

See also

Further reading