Coordinates in 1970 Labeyrie article I'm reading the 1970 article by Labeyrie on speckles in astronomy and I encountered a small problem. The author uses two sets of coordinates, $(\alpha, \beta)$ and $(x, y)$, connected by Fourier transform. Conventionally $\alpha$ and $\beta$ should be angles, while $x$ and $y$ positions, but this makes no sense. Since the variables are connected through Fourier transform, their units of measure should completely cancel out when I multiply e.g. $x \cdot \alpha$, and this is not the case. Can someone clarify this for me?
 A: The variables in the frequency domain are spatial frequencies and indeed if one would multiply a spatial frequency with a spatial coordinate the result is dimensionless. So where do the angles then comes from.
First some context. The reason why the Fourier transform is relevent in this situation is because the far field is related to the near field by Fraunhofer diffraction, which is formally the same as a Fourier transform. However, in the far field the propagation distance becomes irrelevant. If a distance $z$ is considered to be in the far field then $z+\Delta z$ is still in the far field. For that reason it is only the angles that matters. (The spatial spectrum of a light beam is often called an angular spectrum.)
The spatial frequencies $\{a_x,a_y\}$ in the Fraunhofer diffraction pattern are related to the spatial coordinates in the far field $\{u,v\}$ by 
$$ a_x = \frac{u}{\lambda z} ~~~~ a_v = \frac{v}{\lambda z} , $$
where $\lambda$ is the wavelength. One can also add a $z$-component for the spatial frequency vector so that
$$ |{\bf a}|^2 = a_x^2+a_y^2+a_z^2=\frac{1}{\lambda^2} . $$
Then one can convert the three-dimensional vector into spherical coordinates. This would give expressions for $u$ and $v$ in terms of two angles $\alpha$ and $\beta$. Then we throw away the magnitude (because it is irrelevant in the farfield) and retain only the angles.
Basically it gives a sequences going from angular spectrum to farfield in terms of different sets of variables:
$$ F(a_x,a_y) \rightarrow F'(u,v) \rightarrow F''(\alpha,\beta) . $$
