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I have a little problem in wave optics: I have a wave function \psi_{ap} that depends on some geometric parameters, but that has no units itself (as one would expect). But unfortunately when I calculate the Fourier transform of this wave function the Fourier transform has a unit.

Now I'd like to explain my problem a bit more in detail: The wave function \psi_{ap}, that is to be transformed, is a plane wave traveling in z-direction through two circular apertures that lie in the xy-plane and that are displaced from the origin of this plane by d_i along the x-axis. Additionally I assume a phase shift \varphi_i for each aperture.

For convenience I will have only a look on the wave functions along the x-axis with y=0.

\begin{align}

A(x) =& \left\{

\begin{array}{lcc}

1 & \mathrm{for} & x\le 1\\

0 && \mathrm{else}

\end{array}

\right.\\

\psi_i =& A\left(\frac{x-d_i}{R}\right)*e^{-i\varphi_i}\\

\psi_{ap} =& \psi_1+\psi_2\\

\mathrm{with} :& d_1=-33.2\mu m,\,d_2=33.2\mu m,\,R=29\mu m,\, \varphi_1=0,\,\varphi_1=\pi/2\nonumber

\end{align}

\psi_ap is the wave function to be Fourier transformed and A is the function describing an aperture. Using the linearity of the Fourier transform I can calculate the transforms for each aperture separately. As the phase shift in each aperture is constant I can put it in front of the Fourier transform.

\begin{align}

\psi_{sp} =& \mathcal{F}(\psi_{ap}) = \mathcal{F}(\psi_1)+\mathcal{F}(\psi_2)\\

\mathcal{F}(\psi_i) =& e^{-i\varphi_i}*\mathcal{F}\left(A\left(\frac{x-d_i}{R}\right)\right)

\end{align}

This far everything is fine, but now I have (referring to the remarks on the Wikipedia article) a shift in the "time" and "frequency" domain (Equations 102 and 104 in the tables of the article). By using first 104 and then 102 my Fourier Transform looks like this:

\begin{align}

\mathcal{F}\left(A\left(\frac{x-d_i}{R}\right)\right) =& R*e^{-2\pi i d_iq}\frac{\mathrm{J}_1(2\pi Rq)}{Rq}

\end{align}

where q is the coordinate in Fourier space, that by definition should have a unit of "1/m".

My problem now with this expression is that

- this expression has a unit of "m"
- this expression -- except for the exponential term -- looks quite different than expected (especially the leading R); I would have expected J_1(2\pi Rq)/R/q

I would appeciate any remarks.

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# Fourier Transform of non-centered circular aperture

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