# Optics aperture modelling rect functions.

**I am trying to model a shape T aperture through 2D rect functions. Both the horizontal and vertical ‘lines’ have length b and width a, and do not overlap. The origin should be taken to be the centre of the vertical line.**

**The question has hinted at the function describing the aperture to be cartesian seperable.**

My issues, I think, seem to stem from not knowing how the rect functions can be combined through operations together…

**Here are my thoughts on the vertical ‘line’ :**

The RHS of it (to the right of the origin) I believe is: [itex]\frac{a}{2}[/itex] rect [itex]\frac{y}{b}[/itex] [1]

Similarly the LHS i think is : -[itex]\frac{a}{2}[/itex]rect[itex]\frac{y}{b}[/itex]. [2]

My problem is then to express these two together. I am not sure how you define [1] + [2].

Would this be zero?

Perhaps you should multiply them, in which case I get -[itex]\frac{a^{2}}{4}[/itex] rect[itex]^{2}[/itex] [itex]\frac{y}{b}[/itex]; so to me it then makes more sense to look at [itex]\frac{a}{2}[/itex] rect[itex]^{2}[/itex] [itex]\frac{y}{b}[/itex]

Again I’m not sure how you would define a rect[itex]^{2}[/itex] function.

**Here are my thoughts on the horizontal ‘line’ :**

First of all, it can not be a single rect function as either the top or bottom line would then be missing.

I think [itex]\frac{b}{2}[/itex] **e[itex]_{2}[/itex]** – [itex]\frac{a}{2}[/itex]rect[itex]\frac{x}{b}[/itex] for the bottom half, and ([itex]\frac{b}{2}[/itex] + [itex]\frac{a}{2}[/itex] )**e[itex]_{2}[/itex]** + [itex]\frac{a}{2}[/itex]rect [itex]\frac{x}{b}[/itex] for the top half.

*BUT as said above, the question hints towards the function being Cartesian separable, but in describing the horizontal ‘line’ I have introduced e[itex]_{2}[/itex] – the unit vector in the y direction. This also doesn’t look right in general, as isn’t rect a scalar ?*

**
Many thanks to anyone who can help shed some light .** !

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