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Homework Statement
a light source a(x) is defined by
a(x) = Acos(pi*x/a)[theta(x+(a/2)) -theta(x-(a/2))]
calculate the diffraction pattern I(X)
Homework Equations
I(X)=2pi|a~((2pi/(LAMBDA*d))*X)|2
this is the equation for a diffraction pattern on a screen at distance d from a 1D souce
of light with wavelength LAMBDA, where a~(k) is the Fourier transform of a(x)
The Attempt at a Solution
using the Fourier transform
a~(k) = A/sqrt(2pi)(integrate)[exp^(-ikx) * cos((pi*x)/a)]
(integrating over the range -a/2 to a/2)
i then use euler's method to change cosine term into exponential terms, then simplify to get
= A/2*sqrt(2pi)(integrate)[exp^((i*pi*x)/a)-(ikx)) +exp^(-(i*pi*x)/a)-(ikx))]
(integrating over the range -a/2)
the answer i get once i have simplified won't simplify down to trig terms and so makes no sense when i put the value of a~(k) into the equation for I(X). how can this integral be performed elegantly to give a decent looking answer? any help would be great