Article · Wikipedia archive · Last revised May 17, 2026

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), one joule per second, while that of spectral flux in frequency is the watt per hertz and that of spectral flux in wavelength is the watt per metre —commonly the watt per nanometre. Radiant flux is sometimes called luminosity, especially in astronomy contexts.

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

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). Radiant flux is sometimes called luminosity, especially in astronomy contexts.

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

  • Qe is the radiant energy passing out of a closed surface Σ in time interval T;
  • t is time;
  • A is the area of the surface Σ;
  • S is the Poynting vector, representing the directional flow of energy per unit time, per unit area;
  • n is the unit normal vector to the differential area element dA.

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.

Spectral flux

Spectral flux in frequency, denoted Φe,ν, is defined as1 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 as1 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

Comparison of photometric and radiometric quantities source ↗

SI radiometry units
Quantity Unit Dimension Notes
Name Symbolnb 1 Name Symbol
Radiant energy Qenb 2 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 Φenb 2 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,νnb 3 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,λnb 4 watt per metre W/m M⋅L⋅T−3
Radiant intensity Ie,Ωnb 5 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,νnb 3 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,λnb 4 watt per steradian per metre W⋅sr−1⋅m−1 M⋅L⋅T−3
Radiance Le,Ωnb 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 called "intensity".
Spectral radiance
Specific intensity 		Le,Ω,νnb 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 called "spectral intensity".
Le,Ω,λnb 4 watt per steradian per square metre, per metre W⋅sr−1⋅m−3 M⋅L−1⋅T−3
Irradiance
Flux density 		Eenb 2 watt per square metre W/m2 M⋅T−3 Radiant flux received by a surface per unit area. This is sometimes also called "intensity".
Spectral irradiance
Spectral flux density 		Ee,νnb 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 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,λnb 4 watt per square metre, per metre W/m3 M⋅L−1⋅T−3
Radiosity Jenb 2 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 called "intensity".
Spectral radiosity Je,νnb 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 called "spectral intensity".
Je,λnb 4 watt per square metre, per metre W/m3 M⋅L−1⋅T−3
Radiant exitance Menb 2 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 called "intensity".
Spectral exitance Me,νnb 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 called "spectral intensity".
Me,λnb 4 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,νnb 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,λnb 4 joule per square metre, per metre J/m3 M⋅L−1⋅T−2
See also:
  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 letter nu, not to be confused with a letter "v", indicating a photometric quantity.)
  4. Spectral quantities given per unit wavelength are denoted with suffix "λ".
  5. Directional quantities are denoted with suffix "Ω".
See also

See also

References

References

Further reading

Further reading