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The direct determination of the complete transversal state of anelectromagnetic field and accompanying mode indices is essential for the properquantification of all light-matter interactions. In particular light fieldswith cylindrical symmetry such as Laguerre-Gaussian beams can possess orbitalangular momentum, central to a wide range of emergent applications in quantumcryptography, manipulation, astrophysics and microscopy. A wide array ofdiffractive structures such as arrays of pinholes, triangular apertures, slits,and holograms have all recently been used to measure the azimuthal index L ofindividual LG beams. However, all these approaches measure only one singledegree of freedom of LG beams, neglecting the radial component or P index andare thus not applicable for a priori unknown beams. Furthermore, it is unclearwhich is the optimal aperture and scheme needed to determine simultaneously theazimuthal and radial indices nor the extent with which such an aperture cantolerate deviations in beam parameters. Here, we demonstrate a powerfulapproach to simultaneously measure the radial and azimuthal indices of bothpure and mixed LG light fields. We show that the shape of the diffractingelement used to measure the mode indices is in fact of little importance andthe crucial step is training any diffracting optical system and transformingthe observed pattern into uncorrelated variables. Modest fluctuations in beamparameters such as waist size and alignment variations can be tolerated in ourscheme. Importantly, beam superpositions can be experimentally decomposeddelivering the intensity of each mode and their relative phases. Our resultsdemonstrate the complete characterisation of LG beams. The approach can beexpanded to other families of beams and represents a powerful method forcharacterising the optical multi-dimensional Hilbert space.

Complete characterisation of the azimuthal and radial indices of light fields carrying orbital angular momentum