The vortex beam, initially proposed by Allen in 1992 [1], is characterized by its unique spiral wavefront. Due to its ability to carry orbital angular momentum with a discrete value of lℏ per photon [where l is the topological charge (TC), and ℏ is the reduced Planck constant], it has found applications in several fields, such as optical tweezers [2], fiber-optic communication [3], imaging [4], anti-turbulence communication [5], object motion detection [6], and encryption [7]. The Laguerre-Gaussian beam, is a typical vortex beam with a radius that increases with the TC [8], which limits its applications in areas such as optical tweezers and fiber-optic communication. To address this, Ostrovsky et al. proposed the concept of a perfect vortex beam that features a narrow bright ring whose size is not affected by the TC [9–12]. To maximize the space utilization and modulation dimensions, multi-ring perfect vortex beams have been proposed [13–15]. They can be generated by obtaining the Fourier transforms of a superimposed Bessel beam with axicon phase [13] or directly using holograms [15]. The dark ring distribution between the double-bright rings can be used to trap metal particles [16]. Furthermore, it has also been used in quantum secret sharing and anti-turbulence propagation [17,18].

The rotational Doppler effect, used as a remote-sensing scheme for detecting the velocity of a rotating object, was initially demonstrated using the conjugate superposition of vortex beams [6]. In addition to the visible light domain, the rotational Doppler effect can be used in the terahertz domain [19]. The modulation effect of objects has been studied [20]. To fulfill more extensive requirements, additional modulation has been implemented on the light source [21–28]. To solve the problem of misalignment between the rotational axes and optical axes [21], the rotational axis can be pre-measured using symmetry-broken light fields [22,23]. In contrast, multi-ring vortex beams can improve misalignment situations and increase the signal-to-noise ratio (SNR) [24], particularly for multi-ring Laguerre-Gaussian and perfect vortex beams. However, a maximum of only four rings has been demonstrated because of interference between adjacent rings. More rings have been investigated in high-order Laguerre-Gaussian vortex beams; however, their TCs were fixed [25]. As another degree of freedom in optical spatial modulation, polarization modulation has also been used in Doppler frequency detection, such as generating frequency shifts using cylindrical vector polarization fields [26]. By combining the Doppler frequency shift induced by the conjugate vortex phases and polarization modulation, the Doppler metrology scheme can be used to detect both the longitudinal speed and rotational direction [27,28].

In this study, we propose a scheme to generate an ultra-dense vector perfect vortex beam by cross-polarization superposition and explore its wide range of application scenarios in the rotational Doppler effect. The ultra-dense vector perfect vortex beam has an intensity distribution of concentric rings whose radius, beam axis, ring waist, intensity, and TC can be independently controlled. Owing to the orthogonality of the cross-polarization, up to 12 spatially distinguishable vortex rings have been experimentally generated in a radius of 3 mm, which can be further increased by modulating the beam parameters. The density of the rings increased compared with the scalar case, and the SNR of the Doppler frequency shift also improved. We theoretically and experimentally demonstrated the generation of a frequency comb and its application in perceiving a rotating object, including the rotation radius and spatial size. Owing to the ultra-dense spatial distribution, the object’s velocity and rotation axis can also be detected.

The transverse distribution of the perfect vortex can be written as [9] E(r,φ)=δ(r−r0)exp(ilφ),(1)where (r,φ) indicates the transverse position, r0 is the ring radius, l is the TC of the beam, and δ is the Dirac function. In contrast to the widely used Laguerre-Gaussian beam, whose amplitude and phase terms both contain l, the amplitude term of the perfect vortex beam is independent of l, which means that the amplitude and phase terms can be modulated independently. As it is difficult to obtain the Dirac function experimentally, Eq. (2) can be approximated as follows [29]: E(r,φ)=exp[(r−r0)2ω2]exp(ilφ),(2)where ω determines the width of each ring. Subsequently, the expression for the scalar multi-ring perfect vortex beam isE(r,φ)=∑NAnexp[(r−rn)2ωn2]exp(ilnφ),(3)where N represents the total number of rings, subscript n means the nth ring, and An indicates the amplitude coefficient of the nth ring. For convenience, the innermost ring is termed the first ring.

In practical application scenarios, the beam waist always has an upper limit, implying that N cannot be increased to infinity. In this study, the radius of the outermost ring was set as 3 mm. However, within a finite range, interference becomes inevitable as the rings become closer to each other. To increase the number of rings in a limited space, adjacent rings were set in different polarization states, such that the x- and y-polarized rings alternated along the radial direction. The corresponding vector form, namely, the ultra-dense vector perfect vortex beam, can be represented as E(r,φ)=∑NxAnxexp[(r−rnx)2ωnx2]exp(ilnxφ)e^x+∑NyAnyexp[(r−rny)2ωny2]exp(ilnyφ)e^y,(4)where e^x and e^y are the unit vectors for the x- and y-direction polarization states, respectively, in the Cartesian coordinate system. r(n−1)y<rnx<rny and rnx<rny<r(n+1)x, which is necessary to ensure that the adjacent rings possess different polarization states to reduce interference.

In the following analysis, we set N to be an even number, which means that rings with different polarization directions appear in pairs. The ring waists ωnx and ωny are set to the same value ω0, as well as the ring interval Δr and amplitude coefficients Anx and Any. Then, E(r,φ)=∑n=1N2{exp{[r−(2n−1)⋅Δr]2ω02}exp(ilnφ)e^x+exp[(r−2n⋅Δr)2ω02]exp(ilnφ)e^y},(5)where Anx=Any=1. If N is odd, that is, if the numbers of x-polarized and y-polarized rings are not equal, it is also feasible.

According to the principle of the rotational Doppler effect, when a petal-like beam, which is coherently superimposed by conjugate vortex beams with TC=l and −l, hits a rotating scattering object vertically, an additional frequency shift Δf is introduced into the scattered light. After applying the fast Fourier transform on the collected scattered light, the frequency shift Δf can be determined as Δf=lΩπ,(6)where Ω is the angular velocity of the rotating object. Because the TC of each ring can be modulated separately, frequency combs can be generated in the frequency spectrum, from which more information can be obtained about the rotating target. In contrast, if the angular velocity of the rotating target is determined, by modulating the TCs of each ring, a tunable frequency comb was obtained where each peak corresponds to a unique ln.

Figure 1 shows a schematic of the generation of an ultra-dense vector perfect vortex beam. A typical set of x-polarized, y-polarized, and superimposed intensities is shown in Figs. 1(a1)–1(a3), respectively. Figures 1(b)–1(d) show several simulated intensities of the ultra-dense vector perfect vortex beams for different TCs. In Fig. 1(c), the beam consists of 12 rings; the inner four rings are the conjugate superposition of four rings with l=5 and l=−5, the middle four rings are the conjugate superposition of four rings with l=8 and l=−8, and the outer four rings are the conjugate superposition of four rings with l=12 and l=−12. The ring waist ω0 of each ring is set as 0.2 mm and the ring interval is Δr=0.25mm. Figures 1(e)–1(g) show three diagrams indicating a wide range of application scenarios for perceiving the rotating target by identifying the quadrant of the rotating axis, measuring the radius of rotation, and measuring the size of the object. In Fig. 1(f), the beam consists of 12 rings; the innermost two rings are the conjugate superposition of four rings with l=2 and l=−2, the third and fourth rings are the conjugate superposition of four rings with l=4 and l=−4, and the outermost two rings are the conjugate superposition of four rings with l=12 and l=−12, which is the meaning of l=±(2:2:12).

Figure 1(a) is an experimental diagram for the light source generation and detection of the Doppler frequency shift. A 532 nm laser beam with linear polarization was emitted from a solid-state laser and expanded (omitted in the figure for simplicity). A computer-generated hologram comprising two orthogonal electric fields with different gratings was loaded onto a spatial light modulator (SLM), which modulated the amplitude and phase of the beam. A 4-f system consisting of two lenses (f=250mm) and a filter made of two hard-edged apertures was then used to filter out the two diffractive orders required for polarization superposition. Immediately behind the filter plane, we placed two linear polarizers with orthogonal polarization directions to create x- and y-polarization paths. The SLM was located on the input plane of the 4-f system. On the back focal plane of the lens (L2), a Ronchi grating was used to combine the two cross-polarized components into a single beam to produce an ultra-dense vector perfect vortex beam.

Figure 2 depicts the experimentally captured light intensities and their corresponding crosslines. The outermost radius was set to 3 mm and the ring waist ω0 of each ring was set to 0.2 mm. Figures 2(a1)–2(f1) show scalar multi-ring perfect vortex beams with N=2, 4, 6, 8, 10, and 12. The TCs of all the rings were the same, l=6. Figures 2(a3)–2(f3) show the corresponding vector results. Figures 2(a2)–2(f2) and 2(a4)–2(f4) are the crosslines in Figs. 2(a1)–2(f1) and 2(a3)–2(f3), respectively. The crosslines show that with the same value of N, the vector beams show more obvious valleys between rings than the scalar beams, indicating less interference between adjacent rings. When N=10, the rings in the scalar beams are hardly distinguishable. The boundaries between the rings were blurred and cross-stripe interference occurred. However, the rings in the vector beams with N=12 could still be distinguished. More quantitatively, the concept of the Rayleigh diffraction limit was applied to define the distinguishable limit of the rings. We calculated the average peak-to-valley value of the crosslines, expressed as D=P¯/V¯, where P¯ and V¯ are the average intensities of all the peak and valley values, respectively.

In Rayleigh diffraction, the saddle intensity between two peaks is 0.735 of the maximum value, which produces the threshold value D=1.36. For Figs. 2(c1)–2(f1), D=7.01, 1.89, 1.35, and 1.34, respectively. Correspondingly, as shown in Figs. 2(c3)–2(f3), D=11.95, 3.07, 1.78, and 1.43, respectively. The scalar superposed rings suffered greatly from interference, which was more severe as the number of rings increased; however, in the cross-polarization superposed intensity, almost no interference was observed. In Figs. 2(e1) and 2(f1), the intensities were severely influenced by the fork grating caused by interference; thus, the D value was not reliable. Finally, 12 rings each with a ring waist of 0.2 mm were generated in an outermost radius of 3 mm. In this case, vector nested beam is more advantageous, so we will use the 12 rings with 0.2 mm waist in a later experiment. More rings can be generated with smaller ring waists and larger outermost radii.

We further investigated the maximum numbers of rings of ultra-dense vector perfect vortex beams with different ring waists, as shown in Fig. 3. Even and odd numbers of rings were all considered. Figures 3(a1)–3(c1) show the simulated intensities of ultra-dense vector perfect vortex beams with ω=0.15, 0.10, and 0.05 mm, respectively. The TCs of all the rings were the same, l=6. Figures 3(a2)–3(c2) show the corresponding crosslines. The maximum numbers of rings are marked in the upper-right corner of the intensity charts. The illustrations in Figs. 3(b1) and 3(c1) correspond to four-fold enlarged views of the intensity. Notably, when ω=0.05mm, the beam encompasses up to 43 rings with an interval of 0.07 mm. The experimental results corresponding to these simulations are shown in Figs. 3(d1)–3(f2). The pixel size of SLM used is 8 μm. In Fig. 3(f1), the beam waist ω=0.05mm is comparable to the SLM pixel size, leading to slight pixelation of the intensity. This ultra-dense vector perfect vortex beam may have potential applications in fields such as large-capacity fiber-optic communications, multiparticle optical manipulation, and microfluidic viscosity measurements.

In the rotational Doppler effect measurements, the conjugate superposition of ultra-dense vector perfect vortex beams was used to generate a petal-like intensity distribution. The superimposed beam hit the rotating scattering object vertically. The scattered light was then collected by the photodetector, and a Fourier transform was applied to the collected intensity to obtain the frequency information. In Fig. 4, the rotational object is an 8mm×0.2mm rectangular object rotating at 50 r/s (round per second). Viewed along the light, the object looks like a long rod rotating around its center. Figures 4(a)–4(e) show the frequency-domain spectra for N=2, 4, 6, 8, and 12, respectively. Before vortex modulation, the light source remained unchanged, the TCs were fixed at six, and the outermost radius was fixed at 3 mm. The ring intervals for Figs. 4(a)–4(e) are then 1.5, 0.75, 0.5, 0.375, and 0.25 mm, respectively. The illustrations in each figure represent the corresponding experimental intensities on the plane of the object. The height of the peak increased with the number of rings. The fundamental physical mechanism behind this is obvious; as the number of rings increases, the variation in reflected light intensity increases, resulting in an increase in the peak value of the Doppler signal. For each N, we collected 20 sets of data and calculated the mean peak value and error bar, as shown in Fig. 4(f), which shows that not only the SNR is improved, but also the error bar is reduced meaning that the peak signal is becoming increasingly robust. In experiment, we maintain the same source power before SLM, so energy utilization and SNR can be improved by simply increasing the number of rings with the same TC.

As shown in Fig. 5, the frequency combs are generated using ultra-dense vector perfect vortex beams with different TC combinations. Figures 5(a1)–5(c1) show the intensity distributions after conjugate superposition of three preset TC combinations, and Figs. 5(a2)–5(c2) show the corresponding time-domain signals at up to 0.1 s. By taking the fast Fourier transform of these signals, we obtain the corresponding Doppler frequency shift shown in Figs. 5(a3)–5(c3). There is only a single peak in Fig. 5(a3) at 600 Hz corresponding to l=±6. In Fig. 5(b3), two frequency peaks show up at 500 and 800 Hz corresponding to l=±(4,8), respectively. Similarly, Fig. 5(c3) contains three peaks at 500, 800, and 1200 Hz corresponding to l=±(5,8,12), respectively. In Fig. 5, we observe that multiple peaks in the frequency-domain signals lead to a lower SNR for each peak. As the number of topological charges increases, the related power of each topological charge decreases, resulting in a further reduction of the SNR of each peak within acceptable range. From Fig. 6(f), we can see that all six frequency peaks can be produced simultaneously; although they are relatively low, this does not hinder our experimental detection of object size. This frequency comb is expected to find application in the velocity detection of complex targets with inhomogeneous rotational speeds, such as fluids and atmospheric vortices [30,31]. In Ref. [30], the authors measured velocity at different radii by varying the beam size, obtaining the gradient distribution of velocity across the entire plane after multiple measurements. Due to the advantage of multiple TC carrying ability of ultra-dense vector perfect vortex beams, these velocity values can be obtained in a single measurement. By analyzing these discrete velocity positions, we can derive the velocity distribution.

If the ultra-dense vector perfect vortex beam is sufficiently dense, the size and radius perception of the objects can be realized beyond traditional rotation speed measurement by further increasing the number of TCs. As previously mentioned, the interval between the rings can be set very small owing to the cross-polarization superposition. Figure 6 shows the simulation and experimental results for the measurement of the rotation radius and radial dimension length. Figures 6(a) and 6(b) are the simulation and experimental petal-like intensities, respectively, with l=±(2:2:12) and radius interval Δr=0.25mm. Each TC occupies two rings. As a result, the radius measurement accuracy is the width of the two rings, which is 0.5 mm. The white ball in Fig. 6(a) shows a schematic of the perception of the rotational radius. It was a 0.5 mm diameter ball with a rotation velocity of 50 r/s, simulated by a digital micromirror device (DMD). Figure 6(c) shows the simulated results with the location of the ball set at r=0.375, 0.875, 1.375, 1.875, 2.375, and 2.875 mm, corresponding to the 1st-2nd, 3rd-4th, 5th-6th, 7th-8th, 9th-10th, and 11th-12th group of rings, respectively. Figure 6(d) shows the corresponding experimental results, which are consistent with the simulation results. For example, when the ball center is at a radius of 1.375 mm, the location corresponds to the fifth and sixth rings of l=±6; there is a single peak at a frequency of 600 Hz, shown in yellow in Figs. 6(c) and 6(d). If the beam consists of 12 rings with 12 different TCs, the measurement accuracy can reach 0.25 mm. The measurement accuracy can be further improved by increasing the ring number and decreasing the ring interval.

The purple rectangle in Fig. 6(a) was loaded on a DMD with a width of 0.2 mm and variable length, which was used to verify the size detection ability. Figure 6(e) is the simulated results with the length set as 0.5, 1.0, 1. 5, 2.0, 2.5, and 3.0 mm. As the length increased, more rings were included in the trajectory, correspondingly increasing the number of frequency peaks. Figure 6(f) shows the corresponding experimental results, which are consistent with the simulations. For example, when the radical length is 1.0 mm, the light from the fifth to eighth rings is reflected, and two frequency peaks are obtained at 600 and 800 Hz in the frequency spectrum, shown in green in Figs. 6(e) and 6(f). As we only discuss the 12 rings’ beams with 0.2 mm waist, the resolution in Fig. 6 is limited to 0.5 mm. For a smaller beam waist, a beam with more rings and more different TCs can be constructed. For instance, if the beam waist is reduced to 0.1 mm, the resolution is expected to be increased to 0.25 mm. However, as the resolution increases, the signal-to-noise ratio of the signal will decrease.

With the help of a tight focusing system, the measurement resolution can reach the micron level. Although the experiment here is conducted under the premise that the centers of the target and the beam are aligned, the alignment can also be achieved through our experimental scheme. For example, if the axes are misaligned, the frequency peak will broaden, necessitating an adjustment of the beam’s axis to align it with the target. Once the broadening effect disappears, both the location of the axes and the desired object parameters can be identified simultaneously. In practical applications, this technology can be utilized for monitoring industrial bearing wear and measuring particle speed using optical tweezers.

These rings in an ultra-dense vector perfect vortex beam can be non-coaxially nested, providing tolerance for misalignment issues during detection, as well as feedback of the axis direction. Qiu et al. reported that when the rotor is not perfectly coaxial with the beam, the frequency-shift peak broadens [32]. The misaligned frequency shift can be calculated by the following formula: Δfma=lΩ2π·(1+dcosφr),(7)where d is the misalignment distance between the object and beam axes, and r is the beam radius. If d=0, there is no peak broadening; otherwise, the broadening increases with increasing misalignment distance. In simulations and experiments, the broadening Δfma appears as discrete peaks, and the peak interval is equal to the rotational speed. For the non-coaxial nested rings, the misalignment of each ring is different. Thus, by setting TCs to different values, a frequency comb and its broadening can help determine the general direction of the rotational axis.

As an example, Fig. 7 shows the simulation results of a four-ring perfect vortex beam. Figures 7(a) and 7(b) are frequency combs of coaxial nested ultra-dense vector perfect vortex beams for alignment and misalignment issues (0.15 mm along the positive x-direction), l=±(6,9,12,15) with r=1.2, 1.8, 2.4, and 3.0 mm. The ring waists are all set to 0.2 mm. As a reference, the coaxial beam and alignment results are shown in Fig. 7(a), where the centers of the four rings are on the optical axis and aligned with the axis of the rotating object. The corresponding frequency spectrum shows four sharp peaks at 600, 900, 1200, and 1500 Hz. In contrast, when the target is misaligned by 0.15 mm along the positive x-direction, Fig. 7(b) shows a clear broadening of all peaks. The broadening of each peak is calculated using Eq. (7). The broadening of each peak is the same owing to the fixed ratio of the TC to the corresponding radius.

However, for a non-coaxial nested four-ring perfect vortex beam, as shown in Figs. 7(c)–7(f), the beam axes of the four rings are different. From the inner to the outer rings, the shift is 0.1 mm in the positive x-, positive y-, negative x-, and negative y-directions, respectively. Different misalignment situations are then tested using a non-coaxial beam. The rotation axis is diagonally shifted 0.052mm to the first, second, third, and fourth quadrants, respectively. Taking Fig. 7(c) as an example, the rotation axis is set at (0.05 mm, 0.05 mm) in Cartesian coordinates. The misalignment distances of the object and beam axes are calculated as d=0.052mm for the first and second rings, and d=0.0510mm for the third and fourth rings. This relationship with d is depicted in the frequency spectrum; the peaks at 600 and 900 Hz show weak broadening, whereas the peaks at 1200 and 1500 Hz broaden severely. According to the degree of broadening, we can determine which ring center is closest to the rotation axis, thus inferring which quadrant the axis is in. This can be extended to a beam with more nested rings and multiple shifts so that the detection of the axis direction can be more accurate. At the same time, it poses higher challenges for experimental measurements.

The experimental error is discussed in Fig. 8. The parameters were set the same as Fig. 4(e). The incident light consisted of 12 rings with l=±6, ω=0.2mm and the rotational object was an 8mm×0.2mm rectangular object rotating at 50 r/s. We sampled for 100 s at a sampling rate of 10,000 per second and the measured frequency f=599.90Hz was taken as the ground truth, as shown in Fig. 8(a). Then the effect of sampling time on the frequency error margin is discussed in Fig. 8(b). We reduced the real sampling time while maintaining the frequency resolution at 0.01 Hz by zero-padding. For different sampling times, we cut 10 sets of data and calculated the average peak value and error bar. As shown in Fig. 8(b), longer sampling times improved accuracy and reduced the error bars. For sampling times longer than 4 s, the difference between the ground truth and the average peak value is less than 0.01 Hz. For sampling times shorter than 0.5 s, the signal peak may be submerged by noise. The illustration in Fig. 8(c) shows the average error exceeding 0.1 Hz for the first time when the real sampling time is reduced to 0.7 s. In Fig. 8(c), the noise level is stronger than in Fig. 8(a), demonstrating the advantages of longer sampling times.

As a restriction of the current model for perfect vortex beams, they only exist around a fixed plane. For our ultra-dense vector perfect vortex beam, it is only realized at the source plane with large divergence. In remote target applications, one can use a laser radar to determine the target’s position, and then modulate the imaging system to achieve the long-distance detection. Recently, research on more perfect vortices is emerging [33], which means that even after simulated propagation at 1000 m, perfect vortices will maintain their “perfect” vortex structure. It is expected to be combined with our design of ultra-dense vector perfect vortex beams to address the divergence problems.

In this study, we constructed an ultra-dense vector perfect vortex beam and demonstrated its application in rotational object perception. From both the theoretical and experimental results, the interference was clearly reduced owing to the cross-polarization superposition using x- and y- polarized rings, which enabled perfect vortex rings to be nested within a limited region. Experimentally, 12 rings with ring waists of 0.2 mm were generated with an outermost radius of 3 mm. More rings can be generated if the ring waist is reduced or the outermost radius is increased. In simulations, up to 43 distinguishable rings each with a ring waist of 0.05 mm were generated within an outermost radius of 3 mm. Therefore, beyond the velocity measurement, the detections of the radial size and location are also feasible. Due to the control flexibility of TCs, a frequency comb was generated and used to address misalignment issues by non-coaxial nesting. More applications are expected for rotating object identification, such as rotational direction perception, velocity gradient measurement, and polarization selection response.