  The transverse structure of light is recognized as a resource that can be
used to encode information onto photons and has been shown to be useful to
enhance communication capacity as well as resolve point sources in
superresolution imaging. The Laguerre-Gaussian (LG) modes form a complete and
orthonormal basis set and are described by a radial index p and an orbital
angular momentum (OAM) index l. Earlier works have shown how to build a sorter
for the radial index p or/and the OAM index l of LG modes, but a scalable and
dedicated LG mode sorter which simultaneous determinate p and l is immature.
Here we propose and experimentally demonstrate a scheme to accomplish complete
LG mode sorting, which consists of a novel, robust radial mode sorter that can
be used to couple radial modes to polarizations, an l-dependent phase shifter
and an OAM mode sorter. Our scheme is in principle efficient, scalable, and
crosstalk-free, and therefore has potential for applications in optical
communications, quantum information technology, superresolution imaging, and
fiber optics.
