Geometrical optics. Photometry. Quantum nature of light > Rectilinear propagation and reflection of light > Problem 13.1.12

Statement

$13.1.12.$
From the base of a hollow cone of height h with a small angle at the top, a
small ring was cut off and placed in a parallel beam of light, with a wide part
in the direction of the beam. At what distance from the ring will the reflected
light rays focus?

Solution

The ring is cut from the base of a hollow cone of height h and semi-angle$ \alpha \ll 1$. Its radius is approximately $R = h \tan\alpha \approx h\alpha. $
When placed with the wide part facing the beam, light hits the inner conical surface parallel to the axis.

Each ray arrives with an angle of incidence$ \alpha$ with respect to the normal to the surface. Reflection deflects the ray by an angle$ 2\alpha $towards the axis. The distance f from the point of incidence to the focus on the axis satisfies$ R = f \tan 2\alpha$
Therefore,

$f = \frac{R}{\tan 2\alpha} = \frac{h\tan\alpha}{\tan 2\alpha} = \frac{h}{2}\bigl(1 - \tan^2\alpha\bigr)$

Since the angle at the vertex is small, $\tan^2\alpha \ll 1$, and the focus is located at a distance

$\boxed{\dfrac{h}{2}}$

from the plane of the ring.

Answer

$\boxed{\dfrac{h}{2}}$