Search light in billiard tables

We investigate whether a search light, $S$, illuminating a tiny angle ("cone") with vertex $A$ inside a bounded region $Q \in \mathbb{R}^2$ with the mirror boundary $\partial Q$, will eventually illuminate the entire region $Q$. It is assumed that light rays hitting the corners of $Q$ terminate. We prove that: $(\mathbf{I})$ if $Q =$ a circle or an ellipse, then either the entire $Q$ or an annulus between two concentric circles/confocal ellipses (one of which is $\partial Q$) or the region between two confocal hyperbolas will be illuminated; $(\mathbf{II})$ if $Q =$ a square, or $(\mathbf{III})$ if $Q =$ a dispersing (Sinai) or semidespirsing billiards, then the entire region $Q$ is will be illuminated.