Q:

Physical meaning of angular spectrum

I have been going through the paper 

Diffraction from an irregular screen with applications to ionospheric problems. H.G. Booker, J.A. Ratcliffe and D.H. Shinn. Philos. Trans. Royal Soc. A 242, 579 (1950).

The authors state that

it is possible to represent a wave emerging from a diffracting screen in terms of an angular spectrum of plane waves, and it can be related to the Fourier analysis of the wave disturbance over the aperture.

Is this 'angular spectrum' a mathematical quantity by itself, or does it have something to with the electromagnetic spectrum we know of?

A:

It's quite frequent to use the word 'spectrum' whenever a given quantity $f(q,t)$ (where $q$ is some generalized coordinate and $t$ is time) can be decomposed as a linear combination of 'simpler' functions $g_\kappa(q,t)$,
$$
f(q,t) = \int A(\kappa) g_\kappa(q,t)  \mathrm d\kappa,
$$
in which case $A(\kappa)$ is known as the 'spectrum' of $f(q,t)$. (Though not generically ─ it's always used with a qualifier unless the context makes that qualifier redundant.)

The standard electromagnetic spectrum is specifically the frequency spectrum, which corresponds to the temporal Fourier transform,
$$f(q,t) = \int A(q,\omega) e^{-i\omega t}\mathrm d\omega.$$
The angular spectrum is generally defined only for monochromatic waves, and it corresponds to decomposing a complex-valued amplitude function of position $f(\mathbf r)$ as a linear combination of plane waves propagating along different directions $\hat{\mathbf n}$, i.e.
$$ f(\mathbf r) = \iint_{\mathbb S^2} A(\hat{\mathbf n}) e^{i\hat{\mathbf n}\cdot \mathbf r} \mathrm d\Omega,$$
with $\mathrm d\Omega$ the solid-angle differential at unit vector $\hat{\mathbf n}$ taken over the entire unit sphere $\mathbb S^2$.

