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Fourier transform question (optics)

  1. Dec 2, 2008 #1
    1. The problem statement, all variables and given/known data

    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)


    2. Relevant 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)



    3. 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 wont 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
     
  2. jcsd
  3. Dec 3, 2008 #2
    anyone?
     
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