Intensity in Single-Slit Diffraction, Qualitatively — Question 1

Question 1

The phasor arc explains a broad central maximum and exact dark directions

For a uniformly illuminated slit, infinitesimal field contributions form a circular phasor arc of total angle $2\beta$. Its arc length is the on-axis field magnitude $E_c$, and its radius is $\rho=E_c/(2\beta)$ for $\beta>0$.

1. Use the chord length to derive the field and intensity ratios.

2. Interpret the limits $\beta\to0$, $\beta=\pi/2$, and $\beta=\pi$ geometrically.

Question 1 - Solution

The phasor arc explains a broad central maximum and exact dark directions

(a) Method and calculation

$$|E|=2\rho|\sin\beta|\Longrightarrow\boxed{|E|/E_c=|\sin\beta/\beta|}.$$$$\boxed{I/I_c=(\sin\beta/\beta)^2}.$$

(b) Interpretation and further calculation

$$\beta\to0:\ \boxed{I/I_c\to1};\quad\beta=\pi/2:\ \boxed{I/I_c=4/\pi^2}.$$$$\beta=\pi:\quad\text{closed full circle}\Longrightarrow\boxed{E=0}.$$

Physical check

The central limit corresponds to almost parallel contributions, not to a singular physical field.