Intensity in Single-Slit Diffraction, Quantitatively — Question 9

Question 9

A triangular amplitude profile broadens the central lobe and reduces tails

A triangular aperture amplitude is $A(u)=A_0(1-2|u|/a)$ for $|u|\leq a/2$. It is a scaled convolution of two equal rectangles of width $a/2$. Use the convolution theorem and define $\beta=qa/2$.

1. Find the normalized far-field amplitude and intensity.

2. Compare its first-zero location with a uniformly illuminated slit of the same full width and evaluate its normalized intensity at $\beta=3\pi$.

Question 9 - Solution

A triangular amplitude profile broadens the central lobe and reduces tails

(a) Method and calculation

$$\boxed{F(0)=A_0a/2,\quad F(q)/F(0)=\left[\frac{\sin(\beta/2)}{\beta/2}\right]^2}.$$$$\boxed{I/I_c=\left[\frac{\sin(\beta/2)}{\beta/2}\right]^4}.$$

(b) Interpretation and further calculation

$$\boxed{\beta_{1,\mathrm{tri}}=2\pi=2\beta_{1,\mathrm{rect}}}.$$$$\beta=3\pi:\quad\boxed{I/I_c=\frac{16}{81\pi^4}=0.00202785}.$$

Physical check

Smoother illumination suppresses distant light at the cost of a broader central lobe; the comparison uses separately normalized axial intensities.