Diffraction Gratings — Question 5

Question 5

A one-third open grating suppresses every third nonzero principal order

An ideal scalar amplitude grating has transparent width $a=d/3$ in each period. Its normalized Fourier coefficient is $c_m=f\operatorname{sinc}(\pi mf)$, with open fraction $f=a/d$. In this model the power fraction in order m is $\eta_m=|c_m|^2$. Ignore obliquity corrections.

1. Find the missing principal orders.

2. Calculate the zero-order and each first-order power fraction relative to incident power.

Question 5 - Solution

A one-third open grating suppresses every third nonzero principal order

(a) Method and calculation

$$\pi mf=m\pi/3;\quad\sin(m\pi/3)=0\ \text{for}\ m=3p,\ p\ne0.$$$$\boxed{m=\pm3,\pm6,\ldots\ \text{are suppressed}}.$$

(b) Interpretation and further calculation

$$\boxed{\eta_0=f^2=1/9=0.111111}.$$$$\boxed{\eta_{+1}=\eta_{-1}=\left[\frac{\sin(\pi/3)}\pi\right]^2=\frac3{4\pi^2}=0.0759909}.$$

Physical check

An open fraction of one third does not put one third of the incident power into the zero order; the transmitted field is distributed among orders.