Objective The optical vortex is a spatially structured optical beam, the photon of which carries orbital angular momentum (OAM). When this beam illuminates the surface of a rotating object, the frequency of scattered light shifts. This phenomenon is called the optical rotational Doppler effect (RDE). The value of the frequency shift of the scattered light is related to the topological charge of the optical vortex and the rotational speed of the object. In engineering applications, the quality of the optical vortex is not ideal. For example, the mode of the optical vortex is usually not pure due to atmospheric turbulence. Additionally, the optical axis and the rotation axis do not always coincide with each other, and the optical beam may be partially cut out. All the circumstances above will make the scattered light from a rotating object have more than one frequency shift value, which causes many peaks in the frequency spectrum of the scattered light. In that case, it is difficult to distinguish the frequency peak related to the topological charge and the rotational speed, and thus, the amount of measurement errors increases. An optimized signal processing method is urgently required. To raise the measurement accuracy of the rotational speed, we analyze the characteristic of the broadened frequency spectrum on the basis of OAM decomposition and present dual Fourier analysis to transform the broadened frequency spectrum into a spectrum with a single peak related to the rotational speed. Methods The broadened frequency spectrum of the scattered light is essentially related to the OAM- mode broadening. A standard Laguerre-Gaussian (LG) mode is a solution to the paraxial wave equation in the cylindrical coordinate system and carries a single OAM. All of the standard LG modes make up a complete set of orthogonal vectors, and any optical beam can be represented as a superposition of standard LG modes. When the mode of the optical vortex is not pure, a lateral displacement exists, or the beam is not intact, and the LG modes that constitute the illuminating beam are not single, which leads to more than one rotational Doppler shift value. Due to the quantization of the OAM, the interval between topological charges of the LG modes that constitute the illuminating beam is one ( Fig. 2) when near the original topological charge of the illuminating optical vortex. Hence, the interval between the rotational Doppler shift values is the rotational frequency of the rotating object. When the rotational speed is fixed, the frequency interval between the peaks in the frequency spectrum of scattered light is the same. Therefore, we consider the frequency spectrum as a periodic function of frequency, whose period is the rotational frequency. By performing Fourier transform again to the frequency spectrum, we can obtain the secondary frequency spectrum with a single peak whose frequency value is the reciprocal value of the rotational frequency. For easy distinction, the original frequency spectrum is called the primary frequency spectrum. As we perform Fourier transform twice on the scattered light signal, we call this method dual Fourier analysis. Results and Discussions We design an experiment of RDE using an LG vortex with an impure mode or a lateral displacement and a half- covered LG vortex ( Fig. 3). After the Fourier transform on the signal received by the photodetector (PD), we can obtain the primary frequency spectrum (Fig. 6). The frequency values of the peaks in the primary frequency spectrum have an interval of the given rotational frequency (Fig. 6). As the intensity of LG modes constituting the beam is different, the amplitude of these peaks varies. Thus, we consider that the primary spectrum is a periodic function modified by an amplitude modulation function. Then, we take a logarithm of the primary spectrum function and perform Fourier transform again to obtain the secondary spectrum with a single peak whose frequency value is the reciprocal value of the rotational frequency (Fig. 6). In situations of the impure LG mode, beam misalignment, and incomplete beam, we can acquire the same result using this method, and the rotational speed can be always measured from the secondary frequency spectrum. By this means, we separate the rotational speed from the broadened primary frequency spectrum, and the rotational speed measured from the secondary frequency spectrum is relatively accurate. Conclusions By analyzing the characteristic of the broadened frequency spectrum based on the mechanism of the OAM decomposition, we present a method called dual Fourier analysis to transform the broadened frequency spectrum into a spectrum with a single peak. Accurate rotational speed can be measured from the second frequency spectrum. This method simplifies the demand of beam quality and incidence conditions, and hence, it can obtain the rotational speed of the object in a more convenient and clearer manner. The method also promotes the application of RDE in practical projects.
