Michelson's Interferometer — Question 9

Question 9

Matching carrier phase does not necessarily match pulse arrival time

In a Michelson interferometer, arm A contains a plate of thickness $t_A=1.00\,\mathrm{mm}$ with phase index $n_A=1.50$ and group index $n_{gA}=1.54$. Arm B contains thickness $t_B$ of another material with $n_B=1.60$ and $n_{gB}=1.62$. Each plate replaces air on both passes; all other geometric paths match. Use air indices equal to 1 and $c=3.00\times10^8\,\mathrm{m/s}$.

1. Find $t_B$ for exact equality of excess carrier optical path.

2. Calculate the remaining group-delay difference A minus B and determine which pulse arrives later.

Question 9 - Solution

Matching carrier phase does not necessarily match pulse arrival time

(a) Method and calculation

$$2(n_A-1)t_A=2(n_B-1)t_B\Longrightarrow\boxed{t_B=\frac{0.50}{0.60}(1.00\,\mathrm{mm})=0.833333\,\mathrm{mm}}.$$$$\boxed{\Delta\mathcal L_{\mathrm{carrier}}=2[0.50t_A-0.60t_B]=0}.$$

(b) Interpretation and further calculation

$$\Delta\tau_g=\frac2c[(n_{gA}-1)t_A-(n_{gB}-1)t_B]=\frac2c\left[0.540-0.516667\right]\mathrm{mm}.$$$$\boxed{\Delta\tau_g=+0.155556\,\mathrm{ps};\quad\text{arm A arrives later}}.$$

Physical check

Broadband pulse overlap requires group-delay matching as well as carrier-phase alignment; the two materials have different dispersion.