Spherical Mirrors — Question 10

Question 10

Immersion changes travel time without changing mirror focal length

A spherical concave mirror of radius R is first used in air and then fully immersed in a uniform transparent liquid of index n. A parallel beam is already propagating inside the liquid before it reaches the mirror. Ignore interfaces elsewhere and coating changes.

1. Find the paraxial focal length in both media.

2. Compare travel times from the mirror vertex plane to the focal plane along the axis and the wavelengths at fixed frequency.

Question 10 - Solution

Immersion changes travel time without changing mirror focal length

(a) Method and calculation

$$\theta_r=\theta_i\quad\text{in either medium}\Rightarrow\boxed{f_{\mathrm{air}}=f_{\mathrm{liquid}}=R/2}.$$$$v_{\mathrm{air}}\simeq c,\quad v_{\mathrm{liquid}}=c/n.$$

(b) Interpretation and further calculation

$$\boxed{t_{\mathrm{liquid}}/t_{\mathrm{air}}=n}.$$$$\boxed{\lambda_{\mathrm{liquid}}/\lambda_{\mathrm{air}}=1/n}\quad\text{at fixed frequency}.$$

Physical check

The reflection geometry is index-independent in a homogeneous medium; this differs from refraction by a lens.