photon radiance, $$L_{\text{p}}$$

https://doi.org/10.1351/goldbook.P04639
Number of photons (quanta of radiation, $$N_{\text{p}}$$) per time interval (photon flux), $$q_{\text{p}}$$, leaving or passing through a small transparent element of surface in a given direction from the source about the solid angle $$\varOmega$$, divided by the solid angle and by the orthogonally projected area of the element in a plane normal to the given beam direction, $$\text{d}S_{\perp } = \text{d}S\, cos\,\theta$$, with $$\theta$$ the angle between the normal to the surface and the direction of the beam. Equivalent definition: Integral taken over the hemisphere visible from the given point, of the expression $$L_{\text{p}}\, cos\,\theta\,\text{d}\varOmega$$, with $$L_{\text{p}}$$ the photon radiance at the given point in the various directions of the incident beam of solid angle $$\varOmega$$ and $$\theta$$ the angle between any of these beams and the normal to the surface at the given point.
Notes:
1. Mathematical definition: $L_{\text{p}} = \frac{\text{d}^{2}q_{p}}{\text{d}\varOmega \, \text{d}S_{\perp }} = \frac{\text{d}^{2}q_{p}}{\text{d}\varOmega \, \text{d}S\, cos\,\theta}$ for a divergent beam propagating in an elementary cone of the solid @A00346@ $$\varOmega$$ containing the direction $$\theta$$. SI unit is $$\text{m}^{-2}\ \text{s}^{-1}\ \text{sr}^{-1}$$.
2. For a parallel beam it is the number of photons (quanta of radiation, $$N_{\text{p}}$$) per time interval (photon flux), $$q_{\text{p}}$$, leaving or passing through a small element of surface in a given direction from the source divided by the orthogonally projected area of the element in a plane normal to the given direction of the beam, $$\theta$$. Mathematical definition in this case: $$L_{\text{p}} = \text{d}q_{\text{p}}/\left ( \text{d}S\, cos\,\theta \right )$$ If $$q_{\text{p}}$$ is constant over the surface area considered, $$L_{\text{p}} = q_{\text{p}}/\left ( S\, cos\,\theta \right )$$, SI unit is $$\text{m}^{-2}\ \text{s}^{-1}$$.
3. This quantity can be used on a @C01019@ basis by dividing $$L_{\text{p}}$$ by the @A00543@, the symbol then being $$L_{n\text{,p}}$$, the name 'photon @R05037@, amount basis'. For a divergent beam SI unit is $$\text{mol m}^{-2}\ \text{s}^{-1}\ \text{sr}^{-1}$$; common unit is $$\text{einstein m}^{-2}\ \text{s}^{-1}\ \text{sr}^{-1}$$. For a parallel beam SI unit is $$\text{mol m}^{-2}\ \text{s}^{-1}$$; common unit is $$\text{einstein m}^{-2}\ \text{s}^{-1}$$.
Source:
PAC, 2007, 79, 293. 'Glossary of terms used in photochemistry, 3rd edition (IUPAC Recommendations 2006)' on page 396 (https://doi.org/10.1351/pac200779030293)