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Homework Help: Far field intensity distribution

  1. Nov 24, 2012 #1
    1. The problem statement, all variables and given/known data
    Describe the far-field intensity distribution if

    [itex]f(x',y') = circ(\frac{\rho}{D}) cos(\frac{k \theta x'}{2}[/itex]

    where [itex]\rho = \sqrt{x^{2} + y^{2}}[/itex].

    2. Relevant equations
    The fourier transform of the circ function was given to us earlier:
    [itex]\textbf{F} [ circ(\frac{\rho}{D}) ] = \frac{\pi D^{2}}{4} Jinc(\frac{k_{\rho} D}{2})[/itex]
    where Jinc is a bessel function

    I know that the fourier transform of a product of two functions is the convolution of the two fourier transforms. (Convolution theorem)

    The far field intensity is given by [itex]\textbf{I}^{(z)} = \frac{\textbf{I}_{0}}{\lambda^{2} z^{2}} |\textbf{F}[f(x',y')]|^{2} [/itex]

    Also [itex]k_{x} = \frac{k x}{z} = k \theta [/itex]

    3. The attempt at a solution

    so the fourier transform reads: [itex]\frac{\pi D^{2}}{4} Jinc(\frac{k_{\rho} D}{2}) \otimes \delta ( k_{x} - \frac{k \theta}{2} ) + \delta ( k_{x} + \frac{k \theta}{2} )[/itex]

    then the [itex]\delta ( k_{x} - \frac{k \theta}{2} ) [/itex] simplifies to [itex] \delta (k_{x}/2 ) [/itex] and similar for the other one.
    So there would be two bessel functions on the far field plane.
    Is this right?
  2. jcsd
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