  The direct determination of the complete transversal state of an
electromagnetic field and accompanying mode indices is essential for the proper
quantification of all light-matter interactions. In particular light fields
with cylindrical symmetry such as Laguerre-Gaussian beams can possess orbital
angular momentum, central to a wide range of emergent applications in quantum
cryptography, manipulation, astrophysics and microscopy. A wide array of
diffractive structures such as arrays of pinholes, triangular apertures, slits,
and holograms have all recently been used to measure the azimuthal index L of
individual LG beams. However, all these approaches measure only one single
degree of freedom of LG beams, neglecting the radial component or P index and
are thus not applicable for a priori unknown beams. Furthermore, it is unclear
which is the optimal aperture and scheme needed to determine simultaneously the
azimuthal and radial indices nor the extent with which such an aperture can
tolerate deviations in beam parameters. Here, we demonstrate a powerful
approach to simultaneously measure the radial and azimuthal indices of both
pure and mixed LG light fields. We show that the shape of the diffracting
element used to measure the mode indices is in fact of little importance and
the crucial step is training any diffracting optical system and transforming
the observed pattern into uncorrelated variables. Modest fluctuations in beam
parameters such as waist size and alignment variations can be tolerated in our
scheme. Importantly, beam superpositions can be experimentally decomposed
delivering the intensity of each mode and their relative phases. Our results
demonstrate the complete characterisation of LG beams. The approach can be
expanded to other families of beams and represents a powerful method for
characterising the optical multi-dimensional Hilbert space.
