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).
Mathematical definitions
Radiant flux
Radiant flux, denoted Φe ('e' for "energetic", to avoid confusion with photometric quantities), is defined as1 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
- t is the time;
- Qe 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 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 | ⟩ . {\displaystyle \langle |\mathbf {S} |\rangle .}
Spectral flux
Spectral flux in frequency, denoted Φe,ν, is defined as2 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 as3 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 units- v
- t
- e
| Quantity | Unit | Dimension | Notes | ||
|---|---|---|---|---|---|
| Name | Symbol4 | Name | Symbol | ||
| Radiant energy | Qe5 | 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 | Φe6 | 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,ν7 | 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,λ8 | watt per metre | W/m | M⋅L⋅T−3 | ||
| Radiant intensity | Ie,Ω9 | 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,ν10 | 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,λ11 | watt per steradian per metre | W⋅sr−1⋅m−1 | M⋅L⋅T−3 | ||
| Radiance | Le,Ω12 | 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 radianceSpecific intensity | Le,Ω,ν13 | 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,λ14 | watt per steradian per square metre, per metre | W⋅sr−1⋅m−3 | M⋅L−1⋅T−3 | ||
| IrradianceFlux density | Ee15 | 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 irradianceSpectral flux density | Ee,ν16 | 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,λ17 | watt per square metre, per metre | W/m3 | M⋅L−1⋅T−3 | ||
| Radiosity | Je18 | 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,ν19 | 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,λ20 | watt per square metre, per metre | W/m3 | M⋅L−1⋅T−3 | ||
| Radiant exitance | Me21 | 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,ν22 | 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,λ23 | 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,ν24 | 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,λ25 | joule per square metre, per metre | J/m3 | M⋅L−1⋅T−2 | ||
| See also: | |||||
See also
Further reading
- Boyd, Robert (1983). Radiometry and the Detection of Optical Radiation (Pure & Applied Optics Series). Wiley-Interscience. ISBN 978-0-471-86188-1.
References
"Thermal insulation — Heat transfer by radiation — Physical quantities and definitions". ISO 9288:1989. ISO catalogue. 1989. Retrieved 2015-03-15. http://www.iso.org/iso/home/store/catalogue_tc/catalogue_detail.htm?csnumber=16943 ↩
"Thermal insulation — Heat transfer by radiation — Physical quantities and definitions". ISO 9288:1989. ISO catalogue. 1989. Retrieved 2015-03-15. http://www.iso.org/iso/home/store/catalogue_tc/catalogue_detail.htm?csnumber=16943 ↩
"Thermal insulation — Heat transfer by radiation — Physical quantities and definitions". ISO 9288:1989. ISO catalogue. 1989. Retrieved 2015-03-15. http://www.iso.org/iso/home/store/catalogue_tc/catalogue_detail.htm?csnumber=16943 ↩
Standards organizations recommend that radiometric quantities should be denoted with suffix "e" (for "energetic") to avoid confusion with photometric or photon quantities. /wiki/Standards_organization ↩
Alternative symbols sometimes seen: W or E for radiant energy, P or F for radiant flux, I for irradiance, W for radiant exitance. ↩
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 letter nu, not to be confused with a letter "v", indicating a photometric quantity.) /wiki/Frequency ↩
Spectral quantities given per unit wavelength are denoted with suffix "λ". /wiki/Wavelength ↩
Directional quantities are denoted with suffix "Ω". /wiki/%CE%A9 ↩
Spectral quantities given per unit frequency are denoted with suffix "ν" (Greek letter nu, not to be confused with a letter "v", indicating a photometric quantity.) /wiki/Frequency ↩
Spectral quantities given per unit wavelength are denoted with suffix "λ". /wiki/Wavelength ↩
Directional quantities are denoted with suffix "Ω". /wiki/%CE%A9 ↩
Spectral quantities given per unit frequency are denoted with suffix "ν" (Greek letter nu, not to be confused with a letter "v", indicating a photometric quantity.) /wiki/Frequency ↩
Spectral quantities given per unit wavelength are denoted with suffix "λ". /wiki/Wavelength ↩
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 letter nu, not to be confused with a letter "v", indicating a photometric quantity.) /wiki/Frequency ↩
Spectral quantities given per unit wavelength are denoted with suffix "λ". /wiki/Wavelength ↩
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 letter nu, not to be confused with a letter "v", indicating a photometric quantity.) /wiki/Frequency ↩
Spectral quantities given per unit wavelength are denoted with suffix "λ". /wiki/Wavelength ↩
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 letter nu, not to be confused with a letter "v", indicating a photometric quantity.) /wiki/Frequency ↩
Spectral quantities given per unit wavelength are denoted with suffix "λ". /wiki/Wavelength ↩
Spectral quantities given per unit frequency are denoted with suffix "ν" (Greek letter nu, not to be confused with a letter "v", indicating a photometric quantity.) /wiki/Frequency ↩
Spectral quantities given per unit wavelength are denoted with suffix "λ". /wiki/Wavelength ↩