Rays from an isotropic light source inside a medium with refractive index $n_1$ hit a planar surface of refractive index $n_2$. Only rays inside a cone (where the light source is placed at the tip and the surface is the ground plane of the cone) with opening angle $\theta_0$ are regarded. I want to calculate how big the transmittance of all rays in the cone is. Therefore I use the Fresnel equation, which gives the reflectance of a surface as $R(\theta)$. As this is a quite complicated function, I'm probing the reflection for numerous rays inside the cone and average over them (kind of monte-carlo raytracing). I'm struggling with the distribution of the rays inside the cone to get a even sampling. As the problem has a rotational symmetry about the central ray, I figured there has to be a posibilty to probe only rays in one plane and weight them differently in the averaging sum, to account for the circular intersection between light cone and plane.
I feel, that the problem is somewhat related to finding a distribution of rays, defined with opening angles $\theta_0$ in respect to the central ray, that equally sample the inside of a cone.