Malus-Dupin theorem

Pass the pencil of rays through an arbitrary amount of reflections and refractions, then let it emerge in some other homogenous medium.

The theorem states that the resulting pencil of light rays is still perpendicular to some other surface.

Suppose there is, then draw an infinitesimal square in the x-y plane, and follow the path along the surface.

Similarly, one can nest a sequence of circular pencils, each making up a hyperboloid of one-sheet.

[5] Pass the pencil of rays through an arbitrary system of reflecting and refracting material.