Universal translation operator for Laguerre-Gaussian mode sorting

Laguerre-Gaussian (LG) mode is widely adopted in various applications due to carrying orbital angular momentum (OAM). In practice, it is challenging to sort Laguerre-Gaussian mode according to various features. Traditional fork grating and optical geometric transformations are regarded as convenient methods for sorting LG mode. However, these methods have to destroy the OAM carried by the beam during the sorting process. Here, we demonstrate a nondestructive beam sorting mechanism for LG mode carrying orbital angular momentums based on a translation operator. The numerically solved operator can sort the input beams to pre-defined positions with faithful preservation of all the beam features, which shows good agreement with the experimental results. We believe the proposed translation operator will benefit potential applications in information processing, quantum optics, and optical communications.