Intensity in Single-Slit Diffraction, Quantitatively — Question 7

Question 7

A rectangular aperture factorizes into two independent sinc amplitudes

A uniform rectangular aperture has widths $a_x,a_y$. Its far-field amplitude is the double integral of $A_0e^{-i(q_xu+q_yv)}$ over the rectangle. Define $\beta_x=q_xa_x/2$, $\beta_y=q_ya_y/2$.

1. Factor the integral and derive the normalized intensity.

2. Evaluate the intensity at $\beta_x=\pi/2$, $\beta_y=\pi/3$, and describe what happens along $\beta_x=\pi$ for arbitrary $\beta_y$.

Question 7 - Solution

A rectangular aperture factorizes into two independent sinc amplitudes

(a) Method and calculation

$$F=A_0\left(\int_{-a_x/2}^{a_x/2}e^{-iq_xu}du\right)\left(\int_{-a_y/2}^{a_y/2}e^{-iq_yv}dv\right).$$$$\boxed{I/I_c=\left(\frac{\sin\beta_x}{\beta_x}\right)^2\left(\frac{\sin\beta_y}{\beta_y}\right)^2}.$$

(b) Interpretation and further calculation

$$\boxed{I/I_c=\frac4{\pi^2}\frac{27}{4\pi^2}=\frac{27}{\pi^4}=0.277182}.$$$$\beta_x=\pi\Longrightarrow\boxed{I=0\ \text{for every }\beta_y}.$$

Physical check

Independent aperture dimensions generate dark lines, not merely isolated dark points, in the ideal separable model.