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Simple diffraction

  1. Aug 8, 2005 #1
    Hello, I've recently been studying simple diffraction for an upcoming proficiency test, and in the lecture notes for the class, a simple equation regarding single-slit Fraunhoffer diffraction was derived. The derivation was a little weird, so as an exercise, I just decided to go ahead and use phasors to derive the formula myself. Anyway, when I was finished, I nearly had the exact same result as the notes, except for one thing: a factor of a, the width of the slit. Because I couldn't find my mistake anywhere, I decided to look up the result on Wikipedia, which I suspected would do it the same way that I did. Anyway, in the step where the factor of a in the denominator was introduced, there seems to be an integration error. Am I just missing something here?

    [tex]= C \int_{-\frac{a}{2}}^{\frac{a}{2}}e^\frac{ikxx^\prime}{z} \,dx^\prime
    =C \frac{\left(e^\frac{ikax}{2z} - e^\frac{-ikax}{2z}\right)}{\frac{2ikax}{2z}}[/tex]
     
  2. jcsd
  3. Aug 8, 2005 #2
    where do you think there is a mistake ?

    marlon
     
  4. Aug 8, 2005 #3

    Doc Al

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    Staff: Mentor

    Looks to me like they snuck in that factor of a in the denominator where it doesn't belong.
     
  5. Aug 8, 2005 #4
    you are right Doc Al

    i did not even see that...i must be getting delirious again

    marlon
     
  6. Aug 8, 2005 #5

    Claude Bile

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    Science Advisor

    In my notes, the LHS should be;

    [tex]C \int_{-\frac{a}{2}}^{\frac{a}{2}}e^{ikxsin\theta} dx[/tex]

    Which is just a Fourier integral ([itex] u = xsin\theta [/itex] is the conjugate variable), whereby the correct result is the sinc function given.

    It would appear that you should be integrating with respect to x rather than x-prime.

    Claude.
     
    Last edited: Aug 8, 2005
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