Etendue

Remarks

In the coordinates of Hamiltonian optics, with $S$ a plane $z = \text{const}$, one has $n^2 \cos\theta \, d\Omega \, dA = dx \, dy \, dp \, dq$. So $G$ is the phase-space volume of the beam, and it is conserved by Liouville's theorem for Hamiltonian flows.

Example: a flat source of area $A$ in air that emits into a cone of half-angle $\alpha$ about its normal. Check: $\int \cos\theta \, d\Omega = \int_0^{2\pi} \int_0^\alpha \cos\theta \sin\theta \, d\theta \, d\phi = \pi \sin^2 \alpha$, so $G = \pi A \sin^2\alpha$.

Non-example: the product of the area and the solid angle, $A \cdot 2\pi(1 - \cos\alpha)$, is not the etendue. It lacks the factor $\cos\theta$ and the factor $n^2$, and it is not conserved by refraction.