Radiant flux

Not to be confused with Radiation flux.

"Spectral power" redirects here. Not to be confused with Spectral power density.

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).

A flow chart describing the relationship of various physical quantities, including radiant flux and exitance.

Mathematical definitions

Radiant flux

Radiant flux, denoted Φ_e ('e' for "energetic", to avoid confusion with photometric quantities), is defined as $${\begin{aligned}\Phi _{\mathrm {e} }&={\frac {dQ_{\mathrm {e} }}{dt}}\\[2pt]Q_{\mathrm {e} }&=\int _{T}\int _{\Sigma }\mathbf {S} \cdot {\hat {\mathbf {n} }}\,dAdt\end{aligned}}$$ where

• t is the time;
• Q_e is the radiant energy passing out of a closed surface Σ;
• S is the Poynting vector, representing the current density of radiant energy;
• n is the normal vector of a point on Σ;
• A represents the area of Σ;
• T represents the time period.

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 $$\Phi _{\mathrm {e} }\approx \int _{\Sigma }\langle |\mathbf {S} |\rangle \cos \alpha \ dA,$$ where ⟨-⟩ is the time average, and α is the angle between n and $\langle |\mathbf {S} |\rangle .$

Spectral flux

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

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

SI radiometry units

Comparison of photometric and radiometric quantities

SI radiometry units

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 cubic metre	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 square metre	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 confusingly called "intensity".
Spectral radiance Specific intensity	Le,Ω,ν	watt per steradian per square metre 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 confusingly called "spectral intensity".
	Le,Ω,λ	watt per steradian per square metre, 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 confusingly called "intensity".
Spectral irradiance Spectral flux density	Ee,ν	watt per square metre per hertz	W⋅m−2⋅Hz−1	M⋅T−2	Irradiance of a surface per unit frequency or wavelength. This is sometimes also confusingly called "spectral intensity". Non-SI units of spectral flux density include jansky (1 Jy = 10−26 W⋅m−2⋅Hz−1) and solar flux unit (1 sfu = 10−22 W⋅m−2⋅Hz−1 = 104 Jy).
	Ee,λ	watt per square metre, 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 confusingly called "intensity".
Spectral radiosity	Je,ν	watt per square metre 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 confusingly called "spectral intensity".
	Je,λ	watt per square metre, per metre	W/m3	M⋅L−1⋅T−3
Radiant exitance	Me	watt per square metre	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 confusingly called "intensity".
Spectral exitance	Me,ν	watt per square metre 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 confusingly called "spectral intensity".
	Me,λ	watt per square metre, per metre	W/m3	M⋅L−1⋅T−3
Radiant exposure	He	joule per square metre	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 square metre 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 square metre, per metre	J/m3	M⋅L−1⋅T−2
See also: SI Radiometry Photometry

See also

• Luminous flux
• Heat flux
• Power (physics)
• Radiosity (heat transfer)

Further reading

• Boyd, Robert (1983). Radiometry and the Detection of Optical Radiation (Pure & Applied Optics Series). Wiley-Interscience. ISBN 978-0-471-86188-1.