PX275 - I6d - circular aperture

considering a circular hole with radius, $R$
using polar coordinates:

\begin{matrix} y_{1} = r \cos θ \\ y_{2} = r \cos θ \\ y_{3} = 0 \\ x_{1} = ρ \cos ϕ \\ x_{2} = ρ \cos ϕ \\ x_{3} = D \end{matrix}

in the far field regime: $\vec{k} = k \frac{\vec{x}}{| \vec{x} |} = \frac{k}{D} (\begin{matrix} ρ \cos ϕ \\ ρ \sin ϕ \\ D \end{matrix})$
using the jacobian integrals:

\begin{matrix} d y_{1} d y_{2} = r d r d θ \\ d x_{1} d x_{2} = ρ d ρ d ϕ \end{matrix}

the aperture can be written as:

a (r, θ) = {\begin{cases} 1 & if r < R, \\ 0 & otherwise \end{cases}

the fourier transform:

\begin{aligned} \tilde{a} (\vec{k}) & = \int_{0}^{2 π} \int_{0}^{\infty} a e^{- i \vec{k} \cdot \vec{y}} r d r d θ \\ = \int_{0}^{2 π} \int_{0}^{R} e^{- i \vec{k} \cdot \vec{y}} r d r d θ \\ = \int_{0}^{2 π} \int_{0}^{R} \exp (- \frac{i k r ρ}{D} (\cos ϕ \cos θ + \sin ϕ \sin θ)) d ρ d θ \\ = \int_{0}^{2 π} \int_{0}^{R} \exp (- \frac{i k r ρ}{D} (\cos (θ - ϕ)) d ρ d θ \\ = \int_{0}^{2 π} \int_{0}^{R} \exp (- \frac{i k r ρ}{D} \cos θ) d ρ d θ^{'} \end{aligned}

here, bessel functions come in handy

J_{n} (x) = \frac{i^{n}}{2 π} \int_{0}^{2 π} \exp (- i n θ - i x \cos θ) d θ

here, taking $n = 0$ and $x = k ρ r / D :$

J_{0} (x) = \frac{1}{2 π} \int_{0}^{2 π} \exp (- i x \cos θ) d θ

the recurrence relation is a convenient property of bessel functions:

\frac{d}{d x} (x^{n} J_{n} (x)) = x^{n} J_{n - 1} (x)

this can be rewritten as:

\int_{a}^{b} x^{1} J_{0} (x) d x = [x^{1} J_{1} (x)]_{a}^{b}

going back to the fourier transform:

\begin{aligned} \tilde{a} (\vec{k}) & = 2 π \int_{0}^{R} r J_{0} (\frac{k ρ r}{D}) d r \\ = 2 π \int_{0}^{k ρ R / D} \frac{D}{k ρ} x J_{0} (x) \frac{D}{k ρ} d x \\ = 2 π {(\frac{D}{k ρ})}^{2} \int_{0}^{k ρ R / D} x J_{0} (x) d x \\ = 2 π {(\frac{D}{k ρ})}^{2} [x J_{1} (x)]_{0}^{k ρ R / D} \\ = 2 π {(\frac{D}{k ρ})}^{2} (\frac{k ρ R}{D} J_{1} (\frac{k ρ R}{D})) - 0 \\ = 2 π R (\frac{D}{k ρ}) J_{1} (\frac{k ρ R}{D}) \\ = 2 π R^{2} \frac{J (\frac{k ρ R}{D})}{\frac{k ρ R}{D}} \end{aligned}

this is similar to the $sinc$ function
therefore, it can be concluded that:

I \propto | u |^{2} \propto | \tilde{a} (\vec{k}) |^{2} \propto {(\frac{J (\frac{k ρ R}{D})}{\frac{k ρ R}{D}})}^{2}

this is known as the airy pattern