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 as
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 as
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 as
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
See also Luminous flux Heat flux Power (physics) Radiosity (heat transfer)
References
Further reading Boyd, Robert (1983). Radiometry and the Detection of Optical Radiation (Pure & Applied Optics Series). Wiley-Interscience. ISBN 978-0-471-86188-1.