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The optical rotational Doppler effect (RDE) is closely related to the unique orbital angular momentum (OAM) carried by optical vortex, whose topological charge means the mode of OAM. Compared with the coaxial incidence, the rotational Doppler frequency shift spectrum of a misaligned optical vortex (misaligned RDE) widens according to a certain law. In this paper, an OAM modal decomposition method of the misaligned optical RDE is proposed and the relative intensity of different OAM modes, namely the OAM spectrum, is derived based on an inner product computation. Analyses show that lateral displacements and angular deflections change the distribution of OAM modes relative to the rotation axis of the object. A misaligned Laguerre-Gaussian (LG) vortex can be represented as a specific combination of coaxial LG modes, and the difference between the topological charge of two adjacent modes is 1 or 2 with lateral displacements or angular deflections respectively. An experiment of misaligned optical RDE using a superimposed LG vortex is executed, and the obtained frequency shift spectrum with misaligned incidence expands into a set of discrete signals, which agrees well with the theoretical results. Moreover, we can get the rotation frequency of the object from an expanded frequency spectrum more quickly and accurately based on the difference between two adjacent signal peaks. The proposed method contributes to analyze the misaligned optical RDE comprehensively, which is significant in remote sensing and optical metrology.

the generation and measurement of optical vortex (OV) are the basis for a variety of related applications. However, the special case of high-order OVs has not been sufficient addressed yet. Herein, a generation and measurement method of high-order OV via utilizing the CP is investigated. In the experiment, we generate OVs with 60 l  , 20 p  and successfully measure OVs with 200 l  , 0 p  , where experimental results agree well with simulation outcome. On this basis, the intensity distributions of LG and HG beams (corresponding to the generation and measurement) versus waist radius of initial light beams is discussed. This work provides an alternative method to generate or measure high-order OV, which will facilitate applications in optical micro-manipulation, quantum entanglement and rotation speed detection.

Increasing demand for practical applications is forcing more in-depth research into optical vortices (OVs): from the generation and measurement to the shaping and multi-singularity manipulation of OVs. Herein, we propose a new type of phase structure called the high-order cross-phase (HOCP), which can be employed to modulate OVs to implement polygonal shaping and multi-singularity manipulation simultaneously at far-field. Theoretically, we investigate the propagation characteristics of OVs modulated by a HOCP. In experiments, we achieve the polygonal shaping and multi-singularity manipulation of OVs via HOCPs. On this basis, we discuss the relationship between shapes and the orders of HOCPs, where we find that the order of a HOCP is exactly equal to the number of sides of polygonal OVs. This work provides a novel method to achieve polygonal shaping and multi-singularity manipulation simultaneously, which facilitate applications in optical micro-manipulation and optical communication.