Physically Based Rendering, 3rd Edition

15.5.3: Monopole solution

Consider an infinite homogeneous medium with a point light source of unit power (a monopole) located at its origin. The emitted radiance function L_e for this setup is given by

$L_{\mathrm{e}}\left(\mathrm{p},\omega \right)=\frac{1}{4\pi }\delta \left(\mathrm{p}\right)\text{,}$

and the corresponding moments are equal to

$Q_i\left(\mathrm{p}\right)=\left\{\begin{array}{ll}\delta \left(\mathrm{p}\right)\hfill & i=0,\hfill \\ 0,\hfill & i=1\text{.}\hfill \end{array}\right.$

The fluence rate due to this type of source has a simple analytic expression:

${\varphi }_{\mathrm{M}}\left(r\right)=\frac{1}{4\pi D}\frac{{\mathrm{e}}^{-{\sigma }_{\mathrm{t}\mathrm{r}}r}}{r}\text{,}$

  [15.20]

where r is the distance from the light source. The constant ${\sigma }_{\mathrm{t}\mathrm{r}}=\sqrt{{\sigma }_{\mathrm{a}}/D}$ is called ...