Intensity in Single-Slit Diffraction, Qualitatively — Question 4

Question 4

Adjacent diffraction lobes can have opposite field signs

For a centered uniformly illuminated slit, choose the far-field amplitude $F(\beta)=F_c\sin\beta/\beta$ real, with positive $F_c$. Compare $\beta=\pi/2$ and $\beta=3\pi/2$. A weak coherent reference can reveal the field sign.

1. Find the signed field at both points and their relative phase.

2. Find the ratio of their intensities and explain what an intensity-only image misses.

Question 4 - Solution

Adjacent diffraction lobes can have opposite field signs

(a) Method and calculation

$$\boxed{F(\pi/2)=2F_c/\pi,\quad F(3\pi/2)=-2F_c/(3\pi)}.$$$$\boxed{\Delta\arg F=\pi\pmod{2\pi}}.$$

(b) Interpretation and further calculation

$$\boxed{I(3\pi/2)/I(\pi/2)=1/9}.$$$$|F|^2=|-F|^2\Longrightarrow\boxed{\text{intensity alone does not retain the field sign}}.$$

Physical check

Crossing a simple amplitude zero reverses the field sign even though the recorded intensity remains nonnegative.