Radiant flux

Mathematical definitions

Radiant flux

Radiant flux, denoted Φ_e ('e' for "energetic", to avoid confusion with photometric quantities), is defined as $${\displaystyle \begin{array}{cc}{\mathrm{\Phi }}_{\mathrm{e}}& =\frac{d{Q}_{\mathrm{e}}}{dt}\\ Q_{\mathrm{e}}& ={\int }_T{\int }_{\mathrm{\Sigma }}\mathbf{S}\cdot \hat{\mathbf{n}}dAdt\end{array}}$$ where

• Q_e 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 $${\displaystyle {\mathrm{\Phi }}_{\mathrm{e}}\approx {\int }_{\mathrm{\Sigma }}⟨|\mathbf{S}|⟩\cos \alpha dA,}$$ where ⟨-⟩ is the time average, and α is the angle between n and S.

Spectral flux

Spectral flux in frequency, denoted Φ_e,ν, is defined as $${\displaystyle {\mathrm{\Phi }}_{\mathrm{e},\nu }=\frac{\mathrm{\partial }{\mathrm{\Phi }}_{\mathrm{e}}}{\mathrm{\partial }\nu },}$$ where ν is the frequency.

Spectral flux in wavelength, denoted Φ_e,λ, is defined as $${\displaystyle {\mathrm{\Phi }}_{\mathrm{e},\lambda }=\frac{\mathrm{\partial }{\mathrm{\Phi }}_{\mathrm{e}}}{\mathrm{\partial }\lambda },}$$ where λ is the wavelength.