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)