Diffraction Gratings — Question 8

Question 8

The Littrow condition selects a blaze wavelength for a reflection grating

For a reflection grating used in Littrow geometry, the incident and diffracted beams retrace at angle $\theta_B$ from the grating normal, and $2d\sin\theta_B=m\lambda_B$. Assume groove facets are oriented so this direction is their specular blaze direction. The groove density is 600 per millimeter.

1. Find the blaze angle for $\lambda_B=500\,\mathrm{nm}$ in order m=2.

2. At the same geometry, find the blaze wavelength in first order and explain what the blaze condition does not determine.

Question 8 - Solution

The Littrow condition selects a blaze wavelength for a reflection grating

(a) Method and calculation

$$d=1.66667\,\mu\mathrm m,\quad\sin\theta_B=\frac{2(500\,\mathrm{nm})}{2d}=0.300.$$$$\boxed{\theta_B=17.4576^\circ}.$$

(b) Interpretation and further calculation

$$m\lambda_B=2d\sin\theta_B=1000\,\mathrm{nm}\Longrightarrow\boxed{\lambda_{B,m=1}=1000\,\mathrm{nm}}.$$$$\boxed{\text{The angular condition alone does not specify absolute diffraction efficiency.}}$$

Physical check

Blazing aligns a preferred groove reflection direction with a grating order; efficiency still depends on material, polarization, and groove details.