Simultaneous determination of the constituent azimuthal and radial mode indices for light fields possessing orbital angular momentum

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 of individual Laguerre-Gaussian beams. Here, we demonstrate a powerful approach to simultaneously measure both the radial and azimuthal indices of pure Laguerre-Gaussian light fields using the method of principal component analysis. We find 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. The method is generic and may be extended to other families of light fields such as Bessel or Hermite-Gaussian beams.