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 @[email protected] \(\varOmega\), divided by the solid @[email protected] 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 @[email protected] between the normal to the surface and the direction of the beam. Equivalent definition: Integral taken over the hemisphere @[email protected] from the given point, of the expression \(L_{\text{p}}\, cos\,\theta\,\text{d}\varOmega\), with \(L_{\text{p}}\) the photon @[email protected] at the given point in the various directions of the incident beam of solid @[email protected] \(\varOmega\) and \(\theta\) the @[email protected] 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 @[email protected] \(\varOmega\) 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 @[email protected] basis by dividing \(L_{\text{p}}\) by the @[email protected], the symbol then being \(L_{n\text{,p}}\), the name 'photon @[email protected], 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 [Terms] [Paper]