Radiant flux

Not to be confused withRadiation flux.

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

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

Inradiometry,radiant fluxorradiant poweris theradiant energyemitted, reflected, transmitted, or received per unit time, andspectral fluxorspectral poweris the radiant flux per unitfrequencyorwavelength,depending on whether thespectrumis taken as a function of frequency or of wavelength. TheSI unitof radiant flux is thewatt(W), onejouleper second (J/s), while that of spectral flux in frequency is the watt perhertz(W/Hz) and that of spectral flux in wavelength is the watt per metre (W/m)—commonly the watt per nanometre (W/nm).

Mathematical definitions

Radiant flux

Radiant flux,denotedΦ_e('e' for "energetic", to avoid confusion withphotometricquantities), 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

• tis the time;
• Q_eis theradiant energypassing out of aclosed surfaceΣ;
• Sis thePoynting vector,representing thecurrent densityof radiant energy;
• nis thenormal vectorof a point onΣ;
• Arepresents the area ofΣ;
• Trepresents 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, andαis the angle betweennand$\langle |\mathbf {S} |\rangle.$

Spectral flux

Spectral flux in frequency,denoted Φ_e,ν,is defined as

$$\Phi _{\mathrm {e},\nu }={\frac {\partial \Phi _{\mathrm {e} }}{\partial \nu }},$$

whereνis the frequency.

Spectral flux in wavelength,denotedΦ_e,λ,is defined as

$$\Phi _{\mathrm {e},\lambda }={\frac {\partial \Phi _{\mathrm {e} }}{\partial \lambda }},$$

whereλis 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 calledluminosityin Astronomy.
Spectral flux	Φe,ν	watt perhertz	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 persteradian	W/sr	M⋅L2⋅T−3	Radiant flux emitted, reflected, transmitted or received, per unit solid angle. This is adirectionalquantity.
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 adirectionalquantity.
	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 asurface,per unit solid angle per unit projected area. This is adirectionalquantity. 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 asurfaceper unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1.This is adirectionalquantity. 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 fluxreceivedby asurfaceper 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 asurfaceper unit frequency or wavelength. This is sometimes also confusingly called "spectral intensity". Non-SI units of spectral flux density includejansky(1 Jy=10−26W⋅m−2⋅Hz−1) andsolar flux unit(1 sfu=10−22W⋅m−2⋅Hz−1=104Jy).
	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 fluxleaving(emitted, reflected and transmitted by) asurfaceper 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 asurfaceper 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 fluxemittedby asurfaceper 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 asurfaceper 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 asurfaceper unit area, or equivalently irradiance of asurfaceintegrated 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 asurfaceper 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.ISBN978-0-471-86188-1.