Young's Interference Experiment — Question 7

Question 7

A controlled source phase creates a fractional fringe translation

Two narrow slits have equal amplitudes and an imposed relative phase. At screen coordinate y the total phase is $\delta(y)=2\pi dy/(\lambda L)+\pi/2$. Let $d=0.25\,\mathrm{mm}$, $\lambda=500\,\mathrm{nm}$, and $L=1.00\,\mathrm m$.

1. Find the bright fringe closest to the origin and the two neighboring dark fringes.

2. Find the phase change needed to move that bright fringe upward by $0.20\,\mathrm{mm}$ without relabeling its order.

Question 7 - Solution

A controlled source phase creates a fractional fringe translation

(a) Method and calculation

$$\Delta y=\lambda L/d=2.00\,\mathrm{mm};\quad y_m=(m-1/4)\Delta y.$$$$\boxed{y_{\mathrm{bright}}=-0.500\,\mathrm{mm}};\quad y_{\mathrm{dark}}=(m+1/4)\Delta y\Rightarrow\boxed{-1.500,\ +0.500\,\mathrm{mm}}.$$

(b) Interpretation and further calculation

$$\delta\phi_0+\frac{2\pi}{\Delta y}\delta y=0.$$$$\boxed{\delta\phi_0=-2\pi(0.20/2.00)=-0.20\pi\ \mathrm{rad}}.$$

Physical check

The direction of translation depends on the explicitly defined phase convention; holding the same order fixes the sign.