GATE 2022 PH – Question 64
A gas laser has bandwidth $\Delta\nu=(2\nu/c)\sqrt{\alpha/A}$ with $\alpha=3.44\times10^6$ square meters per square second and atomic mass number A. For a helium-4/neon-20 laser at 633 nm, find n in $\Delta\nu=n\times10^9$ Hz to one decimal.
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Correct answer: 1.29 to 1.31
Explanation
The Doppler-broadened bandwidth of a gas laser is
$$\Delta\nu=\frac{2\nu}{c}\sqrt{\frac{\alpha}{A}},$$
where $A$ is the mass number of the **lasing atom**. In a helium-neon laser, the helium atoms only transfer energy to the neon atoms, and the lasing transition (633 nm) is in **neon**, so $A=20$.
**Frequency:** $\dfrac{\nu}{c}=\dfrac1\lambda$, so
$$\Delta\nu=\frac{2}{\lambda}\sqrt{\frac{\alpha}{A}}=\frac{2}{633\times10^{-9}}\sqrt{\frac{3.44\times10^6}{20}}.$$
**Numbers.** $\sqrt{3.44\times10^6/20}=\sqrt{1.72\times10^5}=414.7$ m/s, and $\dfrac{2}{633\times10^{-9}}=3.16\times10^6$ m⁻¹:
$$\Delta\nu=3.16\times10^6\times414.7=1.31\times10^9\text{ Hz}.$$
So $n=\mathbf{1.3}$.