Diffraction Gratings — Question 4

Question 4

The first-zero width of a principal grating peak scales inversely with slit count

A grating has N=400 illuminated slits, spacing $d=2.00\,\mu\mathrm m$, and wavelength $600\,\mathrm{nm}$. Consider principal order m=2. The nearest array zeros occur at phase offsets $\Delta\delta=\pm2\pi/N$.

1. Derive the angular half-width from the principal peak to a nearest zero and evaluate it.

2. Find the full zero-to-zero width and its change when the illuminated slit count doubles.

Question 4 - Solution

The first-zero width of a principal grating peak scales inversely with slit count

(a) Method and calculation

$$\delta=2\pi d\sin\theta/\lambda\Longrightarrow\Delta\delta\simeq(2\pi d/\lambda)\cos\theta_m\Delta\theta.$$$$\sin\theta_2=0.600,\quad\boxed{\Delta\theta_{\mathrm{half}}=\lambda/(Nd\cos\theta_2)=9.375\times10^{-4}\,\mathrm{rad}}.$$

(b) Interpretation and further calculation

$$\boxed{W_\theta=2\Delta\theta_{\mathrm{half}}=1.875\times10^{-3}\,\mathrm{rad}}.$$$$N\to800\Longrightarrow\boxed{W_\theta\to9.375\times10^{-4}\,\mathrm{rad}}.$$

Physical check

The width is evaluated locally in angle, so the cosine factor must be retained away from the normal.