Diffraction Gratings — Question 2

Question 2

Groove density determines the set of propagating diffraction orders

A transmission grating has 600 lines per millimeter and is illuminated normally at $\lambda=500\,\mathrm{nm}$. Treat the slits as narrow and require $|\theta|<90^\circ$.

1. Find the first and third positive principal-order angles.

2. List all allowed principal orders and explain why the fourth is absent.

Question 2 - Solution

Groove density determines the set of propagating diffraction orders

(a) Method and calculation

$$d=(600\,\mathrm{mm^{-1}})^{-1}=1.66667\,\mu\mathrm m;\quad\sin\theta_m=m\lambda/d=0.300m.$$$$\boxed{\theta_1=17.4576^\circ,\quad\theta_3=64.1581^\circ}.$$

(b) Interpretation and further calculation

$$|0.300m|<1\Longrightarrow\boxed{m=-3,-2,-1,0,1,2,3}.$$$$m=4:\ \sin\theta_4=1.20>1\Longrightarrow\boxed{\text{no propagating fourth order}}.$$

Physical check

The existence of an order follows from the grating equation; its brightness also depends on slit width and groove structure.