Light as a Wave — Question 9

Question 9

A moving boundary between two media tunes optical phase

A straight path of fixed length L contains lengths x and L-x of indices $n_1=1.60$ and $n_2=1.30$. A movable boundary increases x at rate $u=2.0\,\mu\mathrm{m/s}$. Use monochromatic vacuum wavelength $600\,\mathrm{nm}$ and quasistatic propagation; neglect Doppler corrections to the material indices.

1. Find the optical phase rate relative to an unchanged reference path.

2. Find the time for one phase cycle and the boundary travel in that time.

Question 9 - Solution

A moving boundary between two media tunes optical phase

(a) Method and calculation

$$\mathcal L=n_1x+n_2(L-x),\quad\dot{\mathcal L}=(n_1-n_2)u=0.60\,\mu\mathrm{m/s}.$$$$\boxed{\dot\phi=\frac{2\pi}{\lambda_0}\dot{\mathcal L}=2\pi\,\mathrm{rad/s}}.$$

(b) Interpretation and further calculation

$$\boxed{T=2\pi/\dot\phi=1.00\,\mathrm s}.$$$$\boxed{\Delta x=uT=2.00\,\mu\mathrm m=\lambda_0/(n_1-n_2)}.$$

Physical check

Translating a boundary between identical indices would produce no phase tuning, regardless of the geometric motion.