Light as a Wave — Question 2

Question 2

A layered arm is balanced by changing a different material

An interferometer replaces $t_A=20\,\mu\mathrm m$ of air in arm A by material $n_A=1.40$. Arm B contains a variable thickness $t_B$ of material $n_B=1.60$, also replacing air. The remaining air paths give equal total geometric lengths. Ignore reflections.

1. Find the positive thickness that gives equal optical paths without an integer-order ambiguity.

2. For vacuum wavelength $640\,\mathrm{nm}$, find the phase error if arm B is made $0.20\,\mu\mathrm m$ too thick; define phase difference as B minus A.

Question 2 - Solution

A layered arm is balanced by changing a different material

(a) Method and calculation

$$\mathcal L_B-\mathcal L_A=(n_B-1)t_B-(n_A-1)t_A.$$$$\Delta\mathcal L=0\Longrightarrow\boxed{t_B=\frac{0.40}{0.60}(20)=13.333\,\mu\mathrm m}.$$

(b) Interpretation and further calculation

$$\delta\mathcal L=0.60(0.20\,\mu\mathrm m)=0.120\,\mu\mathrm m.$$$$\boxed{\delta\phi=2\pi\frac{0.120}{0.640}=0.375\pi\ \mathrm{rad}\simeq1.178\,\mathrm{rad}}.$$

Physical check

The index contrast with the displaced air, rather than the material index alone, controls the compensation.