Slit diffraction with a beam of spatially distributed intensity

[shyree]

Homework Statement

I have the classic slit-diffraction experiment, where light is incident on two narrow slits on a screen and is diffracted to create a pattern of maxima and minima on the far surface. It is usually assumed that the light incident on the slits is uniform. In practice, the light generally varies a little bit, i.e. you have a spatial distribution of intensity rather than one intensity, so the intensity of the incident light through each slit is a little bit different.

The Attempt at a Solution

I started by summing the diffracted bits of electric field:

E = E(0)e^i(kr - wt) + E(0)e^i(k(r + dsin(theta)) - wt) + E(0)e^i(k(r + 2dsin(theta)) - wt)...

If all the E(0)'s are the same, they can be pulled out from the exponential, and then the exponential can be summed alone and a trig identity used to change it into an equation with sines instead of e^1(thing)'s. If they're all different...

E = Ae^i(kr - wt) + Be^i(k(r + dsin(theta)) - wt) + Ce^i(k(r + (N-1)dsin(theta)) - wt)

let (phi) = kdsin(theta)

E = e^i(kr - wt)*(A + Be^i(phi) + Ce^i2(phi))

and I can't get it to simplify any more than that. Is there a way to get this into a form with sines and cosines? Failing that, is there a way to graph this as-is to get the distribution pattern? I haven't been able to find any graphing software that'll play nice with e^i factors.

 

[turin]

Don't you need to make those r's into (r-r0)'s (vectorwise)? You refer to a 2-slit problem in your problem statement, but in your solution you are dealing with multiple slits. You should take the real part of the result, which will be a sum of products of sines and cosines.