Intensity in Single-Slit Diffraction, Qualitatively — Question 6

Question 6

Smoothing an aperture edge suppresses distant diffraction tails

For $F(q)=\int A(u)e^{-iqu}du$, integration by parts gives $F=[A e^{-iqu}/(-iq)]_{\mathrm{edges}}+(1/iq)\int A^\prime e^{-iqu}du$. Compare a hard rectangular amplitude edge with a triangular profile that is continuous and zero at its outer edges but has slope jumps. Consider generic large q away from isolated zeros.

1. Determine the leading amplitude and intensity decay powers for the rectangular profile.

2. Use a second integration by parts to infer the triangular profile tail and compare the effect of doubling q.

Question 6 - Solution

Smoothing an aperture edge suppresses distant diffraction tails

(a) Method and calculation

$$A\ \text{jumps at an edge}\Longrightarrow\boxed{|F_{\mathrm{rect}}|=O(q^{-1})}.$$$$\boxed{I_{\mathrm{rect}}=O(q^{-2}),\quad I(2q)/I(q)\ \text{has envelope factor }1/4}.$$

(b) Interpretation and further calculation

$$A=0\ \text{at outer edges; }A^\prime\ \text{has jumps}\Longrightarrow\boxed{|F_{\mathrm{tri}}|=O(q^{-2})}.$$$$\boxed{I_{\mathrm{tri}}=O(q^{-4}),\quad\text{envelope factor at }2q:\ 1/16}.$$

Physical check

The statements concern the decaying envelope; oscillatory zeros prevent a universal pointwise ratio at every q.