Combination of a Lens and a Mirror

A lens can also be combined with a mirror, most commonly by silvering (coating with a thin reflective layer) one of its two curved faces, so that light entering through the un-silvered face is refracted by the lens, reflected back by the silvered face acting as a mirror, and refracted by the lens a second time on its way back out -- two refractions through the lens and one reflection at the silvered surface, all in a single pass. Such a silvered-lens system behaves, overall, exactly like a single equivalent mirror, and its equivalent focal length $F$ is found by adding the effective power contributed by each of the three optical events -- refraction through the lens, reflection at the silvered surface (treated as a mirror whose own focal length is set by the radius of curvature of that silvered face), and refraction back through the lens a second time. Since the light passes through the lens twice for every one reflection at the mirror, the powers combine as $\frac{1}{F}=\frac{2}{f_{\text{lens}}}+\frac{1}{f_{\text{mirror}}},$ where $f_{\text{mirror}}$ is the focal length the silvered surface would have if it were an ordinary mirror of that same radius of curvature (for a plane surface that has been silvered, $f_{\text{mirror}}=\infty$, so that term simply drops out, since a plane mirror has zero power). Whatever the actual shape of the original lens, the silvered combination as a whole always behaves as either an equivalent concave mirror (converging, positive equivalent power) or an equivalent convex mirror (diverging, negative equivalent power …