PX275 - I6a - square aperture

considering a square hole with sides, $2 w$ , centred at the origin
the aperture function:

a (y_{1}, y_{2}) = {\begin{cases} 1 & for | y_{1} | < w, | y_{2} | < w, \\ 0 & otherwise \end{cases}

the fourier transform:

\begin{aligned} \tilde{a} (\vec{k}) & = \iint_{- \infty}^{\infty} a (\vec{y}) \exp (- i \vec{k} \cdot \vec{y}) d y_{1} d y_{2} \\ = \int_{- w}^{w} \int_{- w}^{w} \exp (- i \vec{k} \cdot \vec{y}) d y_{1} d y_{2} \\ = \int_{- w}^{w} \int_{- w}^{w} \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})]}_{- w}^{w} \\ = - \frac{D}{i k x_{1}} [\exp (- \frac{i k x_{1} w}{D}) - \exp (\frac{i k x_{1} w}{D})] (- \frac{D}{i k x_{2}}) [\exp (- \frac{i k x_{2} w}{D}) - \exp (\frac{i k x_{2} w}{D})] \\ = \frac{2 D}{k x_{1}} \sin (\frac{k x_{1} w}{D}) \frac{2 D}{k x_{2}} \sin (\frac{k x_{2} w}{D}) \\ = 2 w sinc (\frac{k x_{1} w}{D}) 2 w sinc (\frac{k x_{2} w}{D}) \end{aligned}

where, $sinc θ = \frac{\sin θ}{θ}$

the observed intensity:

\begin{matrix} I \propto | u |^{2} \propto | \tilde{a} (\vec{k}) |^{2} \\ ∴ I (x_{1}, x_{2}) \propto {sinc}^{2} (\frac{k x_{1} w}{D}) {sinc}^{2} (\frac{k x_{2} w}{D}) \end{matrix}

zeroes when $k w x_{1} / D$ or $k w x_{2} / D = n π$
separation of fringes: $Δ x_{1} = Δ x_{2} = π D / k w$