Diffraction by a Single Slit: Locating the Minima — Question 9

Question 9

A common dark direction for two colors must satisfy both integer conditions

A slit of width $a=2.00\,\mu\mathrm m$ is illuminated by mutually incoherent wavelengths $500\,\mathrm{nm}$ and $600\,\mathrm{nm}$. A perfectly dark direction would need to be a single-slit minimum for both colors. Assume scalar uniform illumination.

1. Find the smallest positive common value of $a\sin\theta$ required by the two minimum conditions.

2. Decide whether a common nonzero dark direction exists and find the least slit width that permits one at a nongrazing angle.

Question 9 - Solution

A common dark direction for two colors must satisfy both integer conditions

(a) Method and calculation

$$a\sin\theta=m(500\,\mathrm{nm})=q(600\,\mathrm{nm})\Longrightarrow5m=6q.$$$$m=6j,\ q=5j\Longrightarrow\boxed{(a\sin\theta)_{\min}=3.00\,\mu\mathrm m}.$$

(b) Interpretation and further calculation

$$a=2.00\,\mu\mathrm m:\quad\sin\theta=1.50\Longrightarrow\boxed{\text{no common nonzero minimum}}.$$$$|\sin\theta|<1\Longrightarrow\boxed{a>3.00\,\mu\mathrm m};\quad a=3.00\,\mu\mathrm m\ \text{is grazing only}.$$

Physical check

Because incoherent intensities are nonnegative and add, both colors must vanish separately to make the combined pattern exactly dark.