Spherical Mirrors — Question 1

Question 1

A source at the center of a sphere sends every reflected ray back along itself

A concave spherical mirror is part of a sphere of radius R. A point source is exactly at the center of curvature C. The surrounding medium is homogeneous with light speed v.

1. Use the local surface normal to find the reflected direction for every ray reaching the mirror.

2. Find the round-trip optical travel time to C and its spread between rays striking different mirror heights.

Question 1 - Solution

A source at the center of a sphere sends every reflected ray back along itself

(a) Method and calculation

$$\hat n=\frac{\mathbf r_{\mathrm{hit}}-\mathbf r_C}{R},\quad\hat k_i=\hat n.$$$$\hat k_r=\hat k_i-2(\hat k_i\cdot\hat n)\hat n=\boxed{-\hat k_i}.$$

(b) Interpretation and further calculation

$$\boxed{\ell_{\mathrm{round}}=2R,\quad t_{\mathrm{round}}=2R/v}.$$$$\boxed{\Delta t=0}\quad\text{for the ideal spherical surface and point source}.$$

Physical check

This exact return property uses radial incidence and does not rely on the paraxial approximation.