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), that is the joule per second (J/s) in SI base units, 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).

Mathematical definitions

Radiant flux

Radiant flux, denoted Φ_e ("e" for "energetic", to avoid confusion with photometric quantities), is defined as[1]

${\displaystyle \Phi _{\mathrm {e} }={\frac {\partial Q_{\mathrm {e} }}{\partial t}},}$

where

• ∂ is the partial derivative symbol;
• Q_e is the radiant energy emitted, reflected, transmitted or received;
• t is the time.

Spectral flux

Spectral flux in frequency, denoted Φ_e,ν, is defined as[1]

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

where ν is the frequency.

Spectral flux in wavelength, denoted Φ_e,λ, is defined as[1]

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

where λ is the wavelength.

Relationship with the Poynting vector

One can show that the radiant flux of a surface is the flux of the Poynting vector through this surface, hence the name "radiant flux":

${\displaystyle \Phi _{\mathrm {e} }=\oint _{\Sigma }\mathbf {S} \cdot \mathbf {\hat {n}} \,\mathrm {d} A=\oint _{\Sigma }|\mathbf {S} |\cos \alpha \,\mathrm {d} A,}$

where

• Σ is the surface;
• S is the Poynting vector;
• n is a unit normal vector to that surface;
• A is the area of that surface;
• α is the angle between n and S.

But the time-average of the norm of the Poynting vector is used instead, because in radiometry it is the only quantity that radiation detectors are able to measure:

${\displaystyle \Phi _{\mathrm {e} }=\oint _{\Sigma }\langle |\mathbf {S} |\rangle \cos \alpha \,\mathrm {d} A,}$

where < • > is the time-average.

SI radiometry units

SI radiometry units

Quantity		Unit		Dimension	Notes
Name	Symbol[nb 1]	Name	Symbol	Symbol
Radiant energy	Qe[nb 2]	joule	J	M⋅L^2⋅T^−2	Energy of electromagnetic radiation.
Radiant energy density	we	joule per cubic metre	J/m^3	M⋅L^−1⋅T^−2	Radiant energy per unit volume.
Radiant flux	Φe[nb 2]	watt	W = J/s	M⋅L^2⋅T^−3	Radiant energy emitted, reflected, transmitted or received, per unit time. This is sometimes also called "radiant power".
Spectral flux	Φe,ν[nb 3]	watt per hertz	W/Hz	M⋅L^2⋅T^−2	Radiant flux per unit frequency or wavelength. The latter is commonly measured in W⋅nm^−1.
	Φe,λ[nb 4]	watt per metre	W/m	M⋅L⋅T^−3
Radiant intensity	Ie,Ω[nb 5]	watt per steradian	W/sr	M⋅L^2⋅T^−3	Radiant flux emitted, reflected, transmitted or received, per unit solid angle. This is a directional quantity.
Spectral intensity	Ie,Ω,ν[nb 3]	watt per steradian per hertz	W⋅sr^−1⋅Hz^−1	M⋅L^2⋅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[nb 4]	watt per steradian per metre	W⋅sr^−1⋅m^−1	M⋅L⋅T^−3
Radiance	Le,Ω[nb 5]	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	Le,Ω,ν[nb 3]	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[nb 4]	watt per steradian per square metre, per metre	W⋅sr^−1⋅m^−3	M⋅L^−1⋅T^−3
Irradiance Flux density	Ee[nb 2]	watt per square metre	W/m^2	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[nb 3]	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 = 10^4 Jy).
	Ee,λ[nb 4]	watt per square metre, per metre	W/m^3	M⋅L^−1⋅T^−3
Radiosity	Je[nb 2]	watt per square metre	W/m^2	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[nb 3]	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[nb 4]	watt per square metre, per metre	W/m^3	M⋅L^−1⋅T^−3
Radiant exitance	Me[nb 2]	watt per square metre	W/m^2	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[nb 3]	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[nb 4]	watt per square metre, per metre	W/m^3	M⋅L^−1⋅T^−3
Radiant exposure	He	joule per square metre	J/m^2	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[nb 3]	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[nb 4]	joule per square metre, per metre	J/m^3	M⋅L^−1⋅T^−2
Hemispherical emissivity	ε	N/A		1	Radiant exitance of a surface, divided by that of a black body at the same temperature as that surface.
Spectral hemispherical emissivity	εν  or ελ	N/A		1	Spectral exitance of a surface, divided by that of a black body at the same temperature as that surface.
Directional emissivity	εΩ	N/A		1	Radiance emitted by a surface, divided by that emitted by a black body at the same temperature as that surface.
Spectral directional emissivity	εΩ,ν  or εΩ,λ	N/A		1	Spectral radiance emitted by a surface, divided by that of a black body at the same temperature as that surface.
Hemispherical absorptance	A	N/A		1	Radiant flux absorbed by a surface, divided by that received by that surface. This should not be confused with "absorbance".
Spectral hemispherical absorptance	Aν  or Aλ	N/A		1	Spectral flux absorbed by a surface, divided by that received by that surface. This should not be confused with "spectral absorbance".
Directional absorptance	AΩ	N/A		1	Radiance absorbed by a surface, divided by the radiance incident onto that surface. This should not be confused with "absorbance".
Spectral directional absorptance	AΩ,ν  or AΩ,λ	N/A		1	Spectral radiance absorbed by a surface, divided by the spectral radiance incident onto that surface. This should not be confused with "spectral absorbance".
Hemispherical reflectance	R	N/A		1	Radiant flux reflected by a surface, divided by that received by that surface.
Spectral hemispherical reflectance	Rν  or Rλ	N/A		1	Spectral flux reflected by a surface, divided by that received by that surface.
Directional reflectance	RΩ	N/A		1	Radiance reflected by a surface, divided by that received by that surface.
Spectral directional reflectance	RΩ,ν  or RΩ,λ	N/A		1	Spectral radiance reflected by a surface, divided by that received by that surface.
Hemispherical transmittance	T	N/A		1	Radiant flux transmitted by a surface, divided by that received by that surface.
Spectral hemispherical transmittance	Tν  or Tλ	N/A		1	Spectral flux transmitted by a surface, divided by that received by that surface.
Directional transmittance	TΩ	N/A		1	Radiance transmitted by a surface, divided by that received by that surface.
Spectral directional transmittance	TΩ,ν  or TΩ,λ	N/A		1	Spectral radiance transmitted by a surface, divided by that received by that surface.
Hemispherical attenuation coefficient	μ	reciprocal metre	m^−1	L^−1	Radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume.
Spectral hemispherical attenuation coefficient	μν  or μλ	reciprocal metre	m^−1	L^−1	Spectral radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume.
Directional attenuation coefficient	μΩ	reciprocal metre	m^−1	L^−1	Radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.
Spectral directional attenuation coefficient	μΩ,ν  or μΩ,λ	reciprocal metre	m^−1	L^−1	Spectral radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.
See also: SI · Radiometry · Photometry

1. Standards organizations recommend that radiometric quantities should be denoted with suffix "e" (for "energetic") to avoid confusion with photometric or photon quantities.
2. Alternative symbols sometimes seen: W or E for radiant energy, P or F for radiant flux, I for irradiance, W for radiant exitance.
3. Spectral quantities given per unit frequency are denoted with suffix "ν" (Greek)—not to be confused with suffix "v" (for "visual") indicating a photometric quantity.
4. Spectral quantities given per unit wavelength are denoted with suffix "λ" (Greek).
5. Directional quantities are denoted with suffix "Ω" (Greek).

See also

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

References

1. "Thermal insulation — Heat transfer by radiation — Physical quantities and definitions". ISO 9288:1989. ISO catalogue. 1989. Retrieved 2015-03-15.

Further reading

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