# 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]

where

- ∂ is the partial derivative symbol;
*Q*_{e}is the radiant energy emitted, reflected, transmitted or received;*t*is the time.

## 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":

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:

where < • > is the time-average.

## SI radiometry units

Quantity | Unit | Dimension | Notes | |||||
---|---|---|---|---|---|---|---|---|

Name | Symbol[nb 1] | Name | Symbol | Symbol | ||||

Radiant energy | Q_{e}[nb 2] |
joule | J | M⋅L^{2}⋅T^{−2} |
Energy of electromagnetic radiation. | |||

Radiant energy density | w_{e} |
joule per cubic metre | J/m^{3} |
M⋅L^{−1}⋅T^{−2} |
Radiant energy per unit volume. | |||

Radiant flux | Φ_{e}[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 | Φ_{e,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}. | |||

Φ_{e,λ}[nb 4] |
watt per metre | W/m | M⋅L⋅T^{−3} | |||||

Radiant intensity | I_{e,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 | I_{e,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. | |||

I_{e,Ω,λ}[nb 4] |
watt per steradian per metre | W⋅sr^{−1}⋅m^{−1} |
M⋅L⋅T^{−3} | |||||

Radiance | L_{e,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 | L_{e,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". | |||

L_{e,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 |
E_{e}[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 |
E_{e,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). | |||

E_{e,λ}[nb 4] |
watt per square metre, per metre | W/m^{3} |
M⋅L^{−1}⋅T^{−3} | |||||

Radiosity | J_{e}[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 | J_{e,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". | |||

J_{e,λ}[nb 4] |
watt per square metre, per metre | W/m^{3} |
M⋅L^{−1}⋅T^{−3} | |||||

Radiant exitance | M_{e}[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 | M_{e,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". | |||

M_{e,λ}[nb 4] |
watt per square metre, per metre | W/m^{3} |
M⋅L^{−1}⋅T^{−3} | |||||

Radiant exposure | H_{e} |
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 | H_{e,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". | |||

H_{e,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 | ε_{ν}orε_{λ} |
N/A | 1 |
Spectral exitance of a surface, divided by that of a black body at the same temperature as that surface. | ||||

Directional emissivity | ε_{Ω} |
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 | ε_{Ω,ν}orε_{Ω,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 | A_{ν}orA_{λ} |
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 | A_{Ω} |
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 | A_{Ω,ν}orA_{Ω,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 | R_{ν}orR_{λ} |
N/A | 1 |
Spectral flux reflected by a surface, divided by that received by that surface. | ||||

Directional reflectance | R_{Ω} |
N/A | 1 |
Radiance reflected by a surface, divided by that received by that surface. | ||||

Spectral directional reflectance | R_{Ω,ν}orR_{Ω,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 | T_{ν}orT_{λ} |
N/A | 1 |
Spectral flux transmitted by a surface, divided by that received by that surface. | ||||

Directional transmittance | T_{Ω} |
N/A | 1 |
Radiance transmitted by a surface, divided by that received by that surface. | ||||

Spectral directional transmittance | T_{Ω,ν}orT_{Ω,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 | μ_{ν}orμ_{λ} |
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 | μ_{Ω} |
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 | μ_{Ω,ν}orμ_{Ω,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 |

- Standards organizations recommend that radiometric quantities should be denoted with suffix "e" (for "energetic") to avoid confusion with photometric or photon quantities.
- Alternative symbols sometimes seen:
*W*or*E*for radiant energy,*P*or*F*for radiant flux,*I*for irradiance,*W*for radiant exitance. - 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.
- Spectral quantities given per unit wavelength are denoted with suffix "λ" (Greek).
- Directional quantities are denoted with suffix "Ω" (Greek).

## References

- "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.