Hi there,(adsbygoogle = window.adsbygoogle || []).push({});

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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Fourier Transform of non-centered circular aperture

**Physics Forums | Science Articles, Homework Help, Discussion**