Diffraction by a Single Slit: Locating the Minima — Question 2

Question 2

Exact screen positions are nonlinear functions of minimum order

A slit of width $a=2.50\,\mu\mathrm m$ is illuminated at $\lambda=500\,\mathrm{nm}$. A planar screen is $L=1.20\,\mathrm m$ away. Retain exact trigonometry for the far-field minimum direction.

1. Find the second minimum angle and its screen coordinate.

2. Determine the largest positive minimum order visible at a finite screen coordinate, excluding grazing directions.

Question 2 - Solution

Exact screen positions are nonlinear functions of minimum order

(a) Method and calculation

$$\sin\theta_2=2\lambda/a=0.400\Longrightarrow\boxed{\theta_2=23.5782^\circ}.$$$$\boxed{y_2=L\tan\theta_2=\frac{0.480}{\sqrt{0.84}}\,\mathrm m=0.523723\,\mathrm m}.$$

(b) Interpretation and further calculation

$$|m\lambda/a|<1\Longrightarrow0<m<5.$$$$\boxed{m_{\max}=4};\quad m=5\Rightarrow\theta=90^\circ,\ |y|\to\infty.$$

Physical check

The equality case reaches grazing propagation and is not a finite point on a screen perpendicular to the axis.