Transmittance Function

cahill8

1. The problem statement, all variables and given/known data
I have to find the transmittance function T(x,y) of two narrow slits of length L and separated by a distance 2d.


2. Relevant equations

This is what I'm talking about. In my case the x-axis runs in between the slits from the top to bottom (or bottom to top), the slits have a length L and are a distance d from the x-axis.
[tex]
\mathrm{rect}(\frac{t}{H}) = \begin{cases} 0 & \mbox{if } |t| > \frac{H}{2} \\ 1 & \mbox{if } |t| < \frac{H}{2}. \\ \end{cases}
[/tex]


3. The attempt at a solution
[tex]
T(x,y)=1 \mbox{ if } -\frac{L}{2} \leq x \leq \frac{L}{2} \mbox{ and } y=|d|
[/tex]
[tex]
\mbox{Otherwise } T=0
[/tex]
I need to put this into a nicer form, one which will be useful for Fourier transforms. I can reproduce a single slit using step functions, but I'm not sure how to get the two slits at the same time.

cahill8

I figured out the case when the slit has some width: [tex]T(x,y)=rect\left(\frac{x}{L}\right) \left[rect\left(\frac{y+d}{t}\right)+rect\left(\frac{y-d}{t}\right)\right][/tex] where t is the thickness. However I'm looking for the case where t is "infinitely narrow"

What I want is to replace [tex]rect\left(\frac{y-d}{t}\right)[/tex] with something like [tex]\delta(y-d)[/tex], however I need it to equal 1 (or something finite) instead of infinity