Intensity in Single-Slit Diffraction, Quantitatively — Question 5

Question 5

A calibrated central-lobe intensity can determine an unknown slit width

At a direction with $\sin\theta=0.00400$, a uniform slit gives $I/I_c=4/\pi^2$. The wavelength is $600\,\mathrm{nm}$, and the point is known to lie in the positive central lobe. Use the monotonic decrease of sinc-squared on $0<\beta<\pi$.

1. Find the unique phase parameter and infer the slit width.

2. Find the sine of the first-zero angle and the ratio of its paraxial screen coordinate to that of the measured point.

Question 5 - Solution

A calibrated central-lobe intensity can determine an unknown slit width

(a) Method and calculation

$$(\sin\beta/\beta)^2=4/\pi^2\Longrightarrow\boxed{\beta=\pi/2}\quad(0<\beta<\pi).$$$$\pi a\sin\theta/\lambda=\pi/2\Longrightarrow\boxed{a=\lambda/(2\sin\theta)=75.0\,\mu\mathrm m}.$$

(b) Interpretation and further calculation

$$\boxed{\sin\theta_1=\lambda/a=0.00800}.$$$$\theta\ll1:\quad\boxed{y_1/y_{\mathrm{measured}}\simeq\theta_1/\theta\simeq2}.$$

Physical check

The central-lobe restriction removes phase-parameter ambiguity; the small-angle coordinate ratio is separate from the exact sine relation.