12.4 A Radiosity Formulation for Luminescent Reflection

In this section we review a formulation of the radiosity method for the case of luminescent reflection [4,121]. This pertains to low energy particles and is needed in the simulations in following sections.

The flux reaching the (straight) surface elements obtained in the surface extraction step may be written as a vector, i.e.,

$$\mathrm{Flux} = \beta_0 I_S + \beta \Psi L I_R$$

$$= \frac{\beta-\beta_0}{1-\beta}I_S +\frac{\beta(1-\beta_0)}{1-\beta} \underbrace{L^{-1} (L^{-1} - (1-\beta)\Psi)^{-1}}_{T:=} I_S.$$

Here $I_S$ is the vector of fluxes coming from the sources to the surface elements, $I_R$ is the vector of fluxes that arrive because of reflections, $\beta_0$ the sticking coefficient for particles coming directly from the source, $\beta$ the one for secondary bounces, $L$ the diagonal matrix containing the lengths of the surface elements, and

$$\Psi_{ij} = \frac{n_i\cdot(t_j-t_i) n_j\cdot(t_i-t_j)}{\pi \vert t_j-t_i\vert^3} [$$$$\mbox{$i$\ visible $j$}$$$$],$$

where $t_i$ are the centroids of the surface elements, $n_i$ their unit normal vectors, and $[$$\mbox{$i$\ visible $j$}$$]$ is $1$ or 0 if surface element $j$ is visible from $i$ respectively not.

The second line in the equation above is obtained from the first line and the relationship $I_R= (1-\beta_0)I_S + (1-\beta) \Psi L I_R$ after some straightforward algebraic manipulations.

In the case of multiple, low energy species the calculation of the visibility matrix and the inverse $T$ only depends on topographic information and thus does not have to be repeated for each species.