Young's Interference Experiment — Question 6

Question 6

Two colors share bright fringes at common path differences

Two mutually incoherent monochromatic components of wavelengths $450\,\mathrm{nm}$ and $600\,\mathrm{nm}$ illuminate the same slit pair normally. The slit spacing is $d=0.30\,\mathrm{mm}$ and screen distance $L=1.50\,\mathrm m$. Use paraxial geometry.

1. Find the smallest positive screen height where both colors have bright fringes and identify both orders.

2. Find the number of intervals of each color between the central maximum and this coincidence, and determine whether the midpoint is bright for either color.

Question 6 - Solution

Two colors share bright fringes at common path differences

(a) Method and calculation

$$m(450)=q(600)\Longrightarrow3m=4q;\quad\boxed{m=4,\ q=3,\ D=1800\,\mathrm{nm}}.$$$$\boxed{y=LD/d=9.00\,\mathrm{mm}}.$$

(b) Interpretation and further calculation

$$\boxed{N_{450}=4,\quad N_{600}=3};\quad D_{\mathrm{mid}}=900\,\mathrm{nm}.$$$$D_{\mathrm{mid}}/450=2,\quad D_{\mathrm{mid}}/600=3/2\Rightarrow\boxed{450\,\mathrm{nm}:\ \text{bright};\ 600\,\mathrm{nm}:\ \text{dark}}.$$

Physical check

The two colors add intensities, but each color independently interferes between its two slit paths.