  The function to measure orbital angular momentum (OAM) distribution of vortex
light is essential for OAM applications. Although there are lots of works to
measure OAM modes, it is difficult to measure the power distribution of
different OAM modes quantitatively and instantaneously, let alone measure the
phase distribution among them. In this work, we demonstrate an OAM complex
spectrum analyzer, which enables to measure the power and phase distribution of
OAM modes simultaneously by employing rotational Doppler Effect. The original
OAM mode distribution is mapped to electrical spectrum of beating signals with
a photodetector. The power distribution and phase distribution of superimposed
OAM beams are successfully retrieved by analyzing the electrical spectrum. We
also extend the measurement to other spatial modes, such as linear polarization
modes. These results represent a new landmark of spatial mode analysis and show
great potentials in optical communication and OAM quantum state tomography.
