Gratings: Dispersion and Resolving Power — Question 10

Question 10

Independent coherent patches limit a grating to the patch size

A grating has N=12000 illuminated slits, divided into four equal contiguous groups of M=3000. Slits within a group are fully coherent; relative phases between groups average randomly. Every slit alone contributes intensity $I_1$. The envelope is broad and the groups share the same principal-order directions.

1. Find the principal peak intensity and compare with full coherence over N slits.

2. Under this model, find the effective resolving power and wavelength separation in order m=2 at $600\,\mathrm{nm}$.

Question 10 - Solution

Independent coherent patches limit a grating to the patch size

(a) Method and calculation

$$I_{\mathrm{peak}}=4M^2I_1=NM I_1,\quad I_{\mathrm{fully\ coherent}}=N^2I_1.$$$$\boxed{I_{\mathrm{peak}}/I_{\mathrm{fully\ coherent}}=M/N=1/4}.$$

(b) Interpretation and further calculation

$$\text{Incoherent sum of identical M-slit patterns:}\quad\boxed{\mathcal R=mM=6000}.$$$$\boxed{\Delta\lambda=600\,\mathrm{nm}/6000=0.100\,\mathrm{nm}}.$$

Physical check

Adding independently phased groups increases collected power and peak signal but does not narrow the individual group’s diffraction peaks.