PX275 - I6b - rectangular aperture

considering a rectangular hole of width, $2 w$ , and height, $2 H$ , centered at the origin
the aperture function:

a (y_{1}, y_{2}) = {\begin{cases} 1 & if - w < y_{1} < w and - H < y_{2} < H, \\ 0 & otherwise . \end{cases}

\begin{aligned} \tilde{a} (\vec{k}) & = \int_{- w}^{w} \int_{- H}^{H} \exp (- \frac{i k (x_{1} y_{1} + x_{2} y_{2})}{D}) d y_{1} d y_{2} \\ = {[- \frac{D}{i k x_{1}} \exp (- \frac{i k x_{1} y_{1}}{D})]}_{- w}^{w} {[- \frac{D}{i k x_{2}} \exp (- \frac{i k x_{2} y_{2}}{D})]}_{- H}^{H} \\ = \dots \\ = 2 w sinc (\frac{k x_{1} w}{D}) 2 H sinc (\frac{k x_{2} H}{D}) \end{aligned}

I (\vec{x}) \propto {sinc}^{2} (\frac{k x_{1} w}{D}) {sinc}^{2} (\frac{k x_{2} H}{D})

\begin{matrix} \frac{k x_{1} w}{D} = n π ⟹ Δ x_{1} = \frac{π D}{k w} = \frac{D λ}{2 w} \\ \frac{k x_{2} H}{D} = n π ⟹ Δ x_{2} = \frac{π D}{k H} = \frac{D λ}{2 H} \end{matrix}