As is well known, the angular momentum of a photon comes in two forms: the spin angular momentum that corresponds to the circular polarization [1–3], and the orbital angular momentum (OAM) that determines the helical wavefront of the beam [4]. In 1992, Allen et al. proposed a kind of special optical beam that carries OAM with a helical phase factor eilφ, where l is the topological charge and φ is the azimuthal angle [5]. This OAM-carrying beam with the characteristics of a central phase singularity and the phase varying with the azimuthal angle is called a vortex beam [6,7]. A Laguerre–Gaussian (LG) beam and a Bessel–Gaussian beam are typical vortex beams [8,9]. In contrast to spin angular momentum, the OAM mode is theoretically infinite and orthogonal [10]. Therefore, how to make full use of this feature of OAM has become a topic of interest for researchers [11–17].

The rotational Doppler effect (RDE) refers to the frequency shift of angular momentum carrying beams caused by the scattering of a rotating object whose surface is perpendicular to the propagation direction of the beam. In 2013, Lavery et al. used a beam with a superposition-mode optical vortex to illuminate the surface of a rotating disk whose rotation axis is precisely aligned to the axis of the beam [18]. By analyzing the Doppler frequency spectrum of the scattered light, they obtained a single-peak RDE signal. Thereafter, they observed the RDE with white light backscattered from a rotating object and created an illumination beam comprising several OAM modes to illuminate the objects, resulting in a cluster of peaks on the Fourier spectrum [19]. Since then, researchers have explored and elucidated the principle of the RDE from different perspectives [20–22]. Many new techniques have been proposed that greatly enriched the detection methods and analyses under nonideal conditions based on the RDE [23–28]. In terms of the applied research aspects of RDE, many subsequent studies used a superimposed vortex beam or coaxial superposition of OAM modes as the probe beam to obtain more information [29–34]. For example, Zhu et al. recently used a spliced superposed optical vortex to measure the rotating axis of an object [35]. Previous research also shows that a noncoaxial LG beam or LG beam under atmospheric turbulence can be represented as a combination of coaxial LG beams by mode decomposition [36–38]. The mode decomposition also improves the description of the rotating objects and has great advantages in the analysis of object characteristics [39,40].

In this work, composite vortex beams (CVBs) with tailored OAM spectra, which can be generated by using a spatial light modulator (SLM), are used as the probe beams. The RDE frequency spectrum of the CVB with a tailored OAM spectrum is theoretically analyzed and consistent results are obtained in experiments. It is proven that the RDE frequency spectrum can be calculated from the product of the autocorrelation function of the OAM spectrum of a probe beam and the autocorrelation function of the spiral spectrum of the rotating object. Several CVBs are designed as probe beams, such as the recently proposed OAM combs [41,42]. Their RDE frequency spectra are analyzed to reveal the factors that cause variations of the RDE frequency spectrum. Furthermore, we demonstrate the recovery of a single-peak RDE spectrum under misalignment illumination by tailoring the OAM spectrum of the probe beam. RDE analysis from the perspective of the OAM spectrum is conducive to make full use of the diversity of OAM and may help to extend the application scenarios of RDE in remote sensing and many others.

The typical vortex beam is usually expressed as [43] EVB(r,φ)=al(r)eilφ,(1)where l is the topological charge and al(r) is the field amplitude. The CVB refers to a coherent superposition of vortex beams with different OAM modes, so ECVB(r,φ)=∑lal(r)eilφ.(2)

Through varying the complex value of al(r), we can modulate both the amplitude and relative phase of each OAM mode in the structured beam, corresponding to the tailoring of the OAM amplitude spectrum and OAM phase spectrum, respectively [44]. Specifically, the OAM complex spectrum, defined as the normalized complex amplitude of the lth-order OAM mode at radius r, can be expressed as Al(r)=al(r)∑l|al(r)|2.(3)

In this paper, the CVBs are generated using a phase-only spatial light modulator (SLM). In terms of hologram design, we use a complex amplitude modulation method to flexibly modulate both the phase and amplitude [45]. The generated CVB contains coaxial superposition of several vortex beams with different OAM modes.

When the beam illuminates on a rotating object with a rough surface, its phase is modulated [46] in the scattered light. The modulation function of the rough surface can be expressed as [20,39,47] M(r,φ)=eiϕ(r,φ)=∑kBk(r)eikφ,(4)where Bk(r) is a complex amplitude of the kth order modulation factor and in general its amplitude decreases with increasing |k|. Bk(r) is also the normalized complex amplitude of kth order spiral mode in the spiral spectrum of the object. When the object rotates at an angular frequency Ω, the modulation function acquires an additional time-varying term and becomes M(r,φ,t)=∑kBk(r)eikφe−ikΩt.(5)

The scattered light field is expressed as the product of the illuminating light field amplitude and the modulation function of a rotating object, written as $$\begin{gathered} Es(r,φ,t)=M(r,φ,t)ECVB′(r,φ) \\ ⁢∑k∑lAl(r)Bk(r)ei(−l+k)φe−ikΩt, \end{gathered}$$(6)where ECVB′(r,φ)=∑lAl(r)e−ilφ is the normalized illuminating light field. The minus sign ahead of l is due to the reflection of illuminating light, which reverses the winding direction of helical phases and results in a flipped OAM spectrum [48].

Furthermore, we obtain the intensity of the RDE frequency spectrum at frequency mΩ (see Appendix A for details) as Pm=4π2|∫∑l|Al(r)Al+m*(r)|e−iΦl,l+m⁢∑k|Bk(r)Bk+m*(r)|e−iΦk,k+mrdr|,(7)where Φl,l+m=angle{Al(r)Al+m∗(r)}, Φk,k+m=angle{Bk(r)Bk+m∗(r)}, and m is a positive integer. Equation (7) is one of the main results in this work. It indicates that the RDE frequency spectrum contains the contribution from not only the spiral spectrum of rotating object Bk(r) but also the OAM spectrum of probe beam Al(r). Specifically, as will be demonstrated in the following sections, the OAM amplitude spectrum and OAM phase spectrum collectively determine the distribution and relative magnitude of the peaks in the RDE frequency spectrum. In some situations, the influence of relative phase among OAM modes can be neglected. For example, the probe beam is usually designed on demand, so if there is no special need, we can modulate the beam so that there is no relative phase among the OAM modes. And, when the phase of each OAM mode is linearly correlated with l, the RDE frequency spectrum is exactly the same as that of CVB without relative phase difference between OAM modes. Therefore, we can neglect the influence of relative phase among OAM modes when the phase among OAM modes takes this particular form. Therefore, the intensity of RDE spectrum at frequency mΩ can be expressed in a simpler form as Pm=4π2∫∑l|Al(r)||Al+m(r)|∑k|Bk(r)||Bk+m(r)|rdr.(8)

It can be seen that, apart from the effect of a radial integral, the RDE frequency spectrum of the scattered light is the product of the autocorrelation function of the OAM spectrum of the probe beam and the autocorrelation function of the spiral spectrum of the rotating object.

Meanwhile, in the literature [20,31,48] the scattered light is spatially filtered to get a certain OAM mode of light (l=0 as usual) received by the detector, which is different from the unfiltered multimode reception scheme used in this work. Therefore, in the same situations their results of intensity of the RDE frequency spectrum at frequency mΩ are different from Eq. (8) and should be expressed as Pm′=4π2∫∑l|Al(r)||Al+m(r)||Bl(r)||Bl+m(r)|rdr.(9)

According to Eqs. (8) and (9), it is obvious that Pm>Pm′ and they are not linearly correlated. Instead of the product of two autocorrelation functions, Eq. (9) can be seen as the autocorrelation of the product of OAM spectrum |Al(r)| and spiral spectrum |Bl(r)|. The physical meaning lies in that the single-mode reception through spatially filtering the scattered light reduces the cross-coupling effect and retains only the partial RDE signal. In contrast, in the case of unfiltered multimode reception, as shown in Eq. (8), much richer information can be inferred from the RDE spectrum and it is more suitable for rotating objects with special spiral spectra due to the completeness of all the cross-coupling terms (see Appendix D). It also helps to reduce the influence on the RDE frequency spectrum due to the vibration of the receiving system.

We have numerically simulated the RDE spectrum of a CVB with a tailored OAM spectrum comprising four OAM modes, as shown in Fig. 1. The physical process of RDE is shown in Fig. 1(a). The beam with OAM is irradiated on the rotating object, and the scattered light is focused by the lens, received by the detector, and converted into the electrical signal to do the Fourier transform to obtain the frequency spectrum shown in Fig. 1(b). According to Eq. (8), the simulated process of the RDE frequency spectrum can be represented, as shown in Fig. 1(c). The probe beam and the surface of the rotating object are represented as an OAM spectrum and a spiral spectrum by mode decomposition. The autocorrelation of the OAM amplitude spectrum and spiral spectrum is calculated separately, and the simulated RDE frequency spectrum is obtained by the Hadamard product. The tailored OAM spectrum in Fig. 1 consists of four OAM modes (l=±10,±12), and the corresponding autocorrelation has four peaks at the modal indices m=2,20,22,24. The proportion of autocorrelation of the spiral spectrum decreases with increasing m because of the high percentage on the low frequency in the spiral spectrum of the rotating object. The autocorrelation function of the OAM spectrum of the beam and spiral spectrum of the object together determines the relative intensity of each peak in the RDE spectrum.

To verify the theoretical analysis above on the correlation between the OAM spectrum and RDE frequency spectrum, a proof-of-principle experiment is performed, as shown in Fig. 2. The light beam generated by a He–Ne laser is polarized by a horizontal linear polarizer (P), and expanded and collimated by two lenses (L1, L2). The spatial structure of the beam is modulated on an SLM, where different holograms are loaded to generate CVBs with different OAM spectra. The generated CVB is successively filtered through two lenses (L3, L4) and an aperture (AP) to select the desired first-order diffracted light. The 4-f system consisting of SLM, L3, L4, and DMD enables holograms loaded on an SLM to be accurately projected onto a DMD without changing the relative phase between OAM modes due to propagation. The size of generated CVB also can be adjusted by the 4-f system consisting of L3 and L4. The beam is divided into two paths by a beam splitter (BS); the beam on one path is captured by a charge-coupled device (CCD), and the other is irradiated on the rotating target. After being modulated by the rotating target, the scattered light is captured by a photoelectric detector (PD) through the focusing lens (L5). A PD converts the intensity of the light into an electrical signal and transmits it to a computer for fast Fourier transform (FFT) processing. The result of the FFT is the experimental RDE frequency spectrum. The position of the rotating target can be precisely controlled through a translation stage and the rotation speed is regulated by a controller.

The rotating target to be detected is also emulated by a digital micromirror device (DMD) [49]. The rotating image is loaded on the DMD and its distribution of spiral modes is shown in Fig. 2. As shown in previous studies [38,46], the distribution of the spiral spectrum approximates a Gaussian-like distribution. Therefore, we think using an image with a similar OAM distribution is a better fit for the practical object than an image emulating a rotating particle.

We generate a series of CVBs to probe the rotating object. The rotational speed of the rotating object emulated by DMD is 27.78 Hz. By adjusting the position of the rotating target, the center of the CVB spot is coaxial with the center of the rotating object. We keep the rotational speed of the rotating object constant and change the OAM spectra of the probe beam. The measured RDE frequency spectra are shown in Figs. 3–6, where the rotating object is emulated by O1. It is ready to obtain the rotational speed from the RDE spectrum and the error of the rotational speed measured by experiment is less than 1.2%.

As shown in Fig. 3, the experimental results are basically consistent with the simulation analysis based on theoretical calculation. It can be found that upon interacting with the same rotating object, the RDE frequency spectra of scattered light are diverse for several probe beams with different OAM amplitude spectra. Figure 3(a) proves that the RDE frequency spectrum of the OAM comb is approximately triangular in shape, and the interval of each frequency peak is determined by the OAM mode interval of the OAM comb. When an OAM comb lacks a certain OAM mode, its RDE frequency spectrum also deforms, as shown in Fig. 3(b). Compared to the equal proportion of all OAM modes in Fig. 3(b), the V-shaped OAM mode distribution causes the RDE spectrum to produce a bulge in shape between 5Ω and 10Ω, as shown in Fig. 3(c). As shown in Fig. 3(d), the addition of another isolated OAM mode to an OAM comb appears as the addition of a series of approximately even peaks in the resulting RDE frequency spectrum.

However, the OAM amplitude spectrum is not the only factor that affects the RDE frequency spectrum of a CVB. In the proof-of-concept experiment, to completely describe the effect of the CVB on the RDE frequency spectrum, the other variables are discussed. Simulated and experimental results for three CVBs with identical OAM amplitude spectrum but different OAM phase spectra are illustrated in Fig. 4. Changing the relative phase of individual OAM modes affects the intensity distribution of the peaks in the RDE frequency spectrum, as shown in Figs. 4(b) and 4(c). However, it is worth noting that when the phase of each OAM mode is linearly correlated with l, the RDE frequency spectrum is exactly the same as that of CVB without a relative phase difference between OAM modes. This is because a linear phase spectrum in the OAM mode domain corresponds to an angular displacement in real space, as clearly shown in Figs. 4(b) and 4(d). Therefore, the RDE frequency spectrum of a CVB consisting of LG beams of the same sign as l does not change with the accumulated Gouy phase shift during propagation [50,51].

The RDE also depends on the radial distribution of different OAM modes. The light field at different radial positions cannot interfere with each other to generate an RDE beat frequency, so a CVB composed of two coaxially superimposed vortex beams with different radii and minimal overlap has only two peaks on the RDE frequency spectrum, as shown in Fig. 5. If the influence of the spiral mode distribution of the rotating object is ignored, the intensity ratio is simply equal to the intensity ratio of the two superimposed vortex beams.

In practical scenarios, the misalignment of an optical axis is an influential factor for RDE related applications. Figure 6 shows a more realistic OAM spectrum. Figure 6(a1) is the simulated OAM spectrum of the original superimposed LG beam (l=±5). Figure 6(b1) is the simulated OAM spectrum of a misaligned beam with lateral displacement d=0.05w, where w is the beam radius [36]. It can be seen from Figs. 6(b3) and 6(b4) that the RDE spectrum exhibits significant broadening with several sub-peaks around the primary frequency peak. Figure 6(c) is the simulated OAM spectrum of the tailored CVB to compensate for the effect of misalignment. By adjusting the OAM spectrum, the intensity and phase distributions of the petal-like patterns in the beam spot are changed so that the azimuthal distributions of each petal with respect to the rotation axis are approximately symmetric. From Figs. 6(f) and 6(g) it can be seen that the intensity and phase of the tailored CVB are asymmetric about the center line of the spot. The simulation and experiment results show that using this tailored CVB as the probe beam can almost eliminate the misalignment-induced broadening of the RDE frequency spectrum and improve the distinguishability of the primary frequency peak, which is of some significance for application of this method to further improve the probe beam under complex conditions. The lateral displacement for this experiment is relatively small; if larger displacements or more complex scenarios are considered, the radial distribution spectrum is a factor that must be taken into account (see Appendix D for details). This makes compensation more complicated. It is more meaningful to combine intelligent algorithms such as deep reinforcement learning [52] with the theory in this work to obtain a closed-loop control strategy for compensating misalignment or turbulence.

In conclusion, we have investigated the mechanism of the RDE of CVBs with tailored OAM spectra. The results show that the RDE frequency spectrum is related to the product of the two autocorrelation functions of (i) the OAM spectrum of the beam and (ii) the spiral spectrum of the object. As for the signal detection scheme, the difference between unfiltered multimode reception and single-mode reception is analyzed. The experimental results of RDE spectra using various CVBs or misaligned vortex beams also have a good correspondence with the theoretical results. Our theoretical model enables us to better understand the physical processes and frequency spectrum characteristics of RDE in extended detection conditions. It can provide theoretical guidance for the research of specifically tailored structured beams as the probe. It may also lay the foundation for all-optical computation of autocorrelation and cross-correlation. Additionally, inversely analyzing the properties of the surface of a rotating object based on the RDE frequency spectrum also may become possible.