Diffraction Gratings — Question 10

Question 10

A square two-dimensional grating is limited by a circle in order space

A square array under normal illumination has periods $d_x=d_y=2\lambda$. Principal directions satisfy $s_x=m\lambda/d_x$ and $s_y=n\lambda/d_y$, where $s_x,s_y$ are transverse direction cosines. A forward nongrazing direction requires $s_x^2+s_y^2<1$.

1. List all allowed integer order pairs and count them.

2. Find the polar angles of the (1,0) and (1,1) orders, and identify the axial grazing orders excluded by the strict inequality.

Question 10 - Solution

A square two-dimensional grating is limited by a circle in order space

(a) Method and calculation

$$s_x=m/2,\ s_y=n/2\Longrightarrow\boxed{m^2+n^2<4}.$$$$\boxed{(m,n)\in\{-1,0,1\}\times\{-1,0,1\}:\ 9\ \text{orders}}.$$

(b) Interpretation and further calculation

$$\sin\theta=\sqrt{s_x^2+s_y^2}\Longrightarrow\boxed{\theta_{10}=30^\circ,\quad\theta_{11}=45^\circ}.$$$$\boxed{(\pm2,0),(0,\pm2)\ \text{are grazing}}.$$

Physical check

Applying separate bounds to m and n is insufficient in general; the combined transverse direction must fit inside the propagation circle.