Intensity in Single-Slit Diffraction, Quantitatively — Question 10

Question 10

An intensity ratio at doubled spatial frequency removes the unknown peak level

A uniform slit is measured at spatial frequencies q and 2q, with $q=1.00\times10^4\,\mathrm{m^{-1}}$. The readings are $I(q)=9.00$ and $I(2q)=2.25$ in common units. Let $\beta=qa/2$ and assume $0<\beta<\pi/2$. There is no background.

1. Derive a simple ratio using the double-angle identity and infer the slit width.

2. Find the unknown on-axis intensity.

Question 10 - Solution

An intensity ratio at doubled spatial frequency removes the unknown peak level

(a) Method and calculation

$$\frac{I(2q)}{I(q)}=\left[\frac{\sin2\beta}{2\sin\beta}\right]^2=\boxed{\cos^2\beta=1/4}.$$$$\beta=\pi/3\Longrightarrow\boxed{a=2\beta/q=209.440\,\mu\mathrm m}.$$

(b) Interpretation and further calculation

$$I(q)/I_c=(\sin(\pi/3)/(\pi/3))^2=27/(4\pi^2).$$$$\boxed{I_c=9(4\pi^2/27)=4\pi^2/3=13.1595}.$$

Physical check

The ratio eliminates the peak calibration, while the specified phase interval selects the physically intended inverse solution.