PX275 - I6c - multiple slits

considering the young's double slit interference pattern
two rectangular slits, $2 w \times 2 H$ , centred at $y_{1} = \pm l$ from the origin
shifted versions of the single slit function can be combined to describe the two slits
reminder: convolution of a function with a $δ$ function shifts the function to $δ f_{n}$
the aperture function:

\begin{aligned} a (\vec{y}) & = \int a_{s l i t} ({\vec{y}}^{'}) [δ ({\vec{y}}^{'} - (\vec{y} + l {\hat{y}}_{1})) + δ ({\vec{y}}^{'} - (\vec{y} - l {\hat{y}}_{1}))] d y^{'} \\ = a_{s l i t} * g (\vec{y}) \\ a_{s l i t} * g (x^{'}) & = \int a_{s l i t} (x^{'}) g (x - x^{'}) d x^{'} \end{aligned}

\tilde{a} (\vec{k}) = F (a (\vec{y})) = {\tilde{a}}_{s l i t} (\vec{k}) \tilde{g} (\vec{k})

it is known that:

{\tilde{a}}_{s l i t} \propto sinc (\frac{k x_{1} w}{D}) sinc (\frac{k x_{2} H}{D})

\begin{aligned} \tilde{g} (k) & = \int \exp (- i \vec{k} \cdot \vec{y}) [δ (\vec{y} + l {\hat{y}}_{1}) + δ (\vec{y} - l {\hat{y}}_{1})] d \vec{y} \\ = \exp (i \vec{k} \cdot l {\hat{y}}_{1}) + \exp (- i \vec{k} \cdot l {\hat{y}}_{1}) \\ = \exp (i k_{1} l) + \exp (- i k_{1} l) \\ = 2 \cos (k_{1} l) \end{aligned}

the intensity:

I (\vec{x}) \propto {sinc}^{2} (k_{1} w) {sinc}^{2} (k_{2} H) 2 \cos (k_{1} l)

where, $k_{1} = k x_{1} / D$ and $k_{2} = k x_{2} / D$

the fringe spacing:
- large rectangular pattern:

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

- smaller fringes:

Δ x_{1} = \frac{π D}{k l}