Vortex beams, featuring helical wavefronts and central phase singularities, demonstrate characteristic annular intensity profiles with null central irradiance. Since the seminal work of Allen et al. in 1992^[1], which demonstrated that Laguerre–Gaussian (LG) beams with a complex amplitude containing the phase term exp(ilφ) carry an orbital angular momentum (OAM) of lℏ per photon—where l is the angular quantum number, also referred to as the topological charge (TC), φ is the azimuthal angle, and ℏ is the reduced Planck constant—vortex beams have garnered significant attention due to their unique properties and wide-ranging applications^[2]. These structured light fields have enabled transformative applications, such as optical tweezers^[3–5], optical communications^[6–8], total angular momentum manipulation^[9], photonic lattices^[10,11], optical imaging^[12], beam shaping^[13] and ultrafast laser material processing^[14], catalyzing synergistic developments in photonic theory and applied technologies. However, traditional vortex beams face a fundamental limitation: their beam radius increases with the TC, which restricts their utility in applications such as particle manipulation and optical communication. To address this issue, the concept of perfect vortex beams was introduced in 2013^[15]. Unlike conventional vortex beams, a perfect vortex beam exhibits a special feature that its beam radius is independent of the TC, thereby overcoming the limitations associated with traditional vortex beams. Despite this advancement, a single-ring perfect vortex beam is only effective at trapping particles with a higher refractive index than the surrounding medium, leaving particles with a lower refractive index unaddressed. This limitation drove the development of dual-ring perfect vortex beams, which utilize the dark region between the rings to trap low-refractive-index particles^[16–18]. Building on this progress, multi-ring perfect vortex beams have been proposed, which further enable more applications such as quantum secret sharing and rotation detection^[19,20].

Rotation is a fundamental and ubiquitous phenomenon in nature. Quantitative characterization of rotational motion has therefore become a cornerstone of research for astrophysics^[21,22], atomic cooling^[23], and fluid dynamics^[24–26]. A particularly fascinating phenomenon arises when vortex beams carrying OAM are incident perpendicularly onto a rotating surface: the scattered waves exhibit a Doppler shift^[27,28]. To enhance the observability of this Doppler shift, researchers have employed petal-like beams, which are coherently superimposed by conjugate vortex beams with TC=l and −l^[29]. At this point, the Doppler shift is directly linked to the TC of the beam and the rotational speed of the object. Conventional structured beams, such as LG beams^[30], Airy beams^[31], fractional-order vortex beams^[32], perfect vortex beams^[33], and vector vortex beams^[34], have been widely utilized in studies of the rotational Doppler effect. Additionally, partially coherent light sources have also shown unique advantages in Doppler measurements, enabling non-aligned rotational speed detection of sub-Rayleigh objects through specialized coherence structures^[35]. Nevertheless, current Doppler measurement systems continue to face significant challenges in both achieving multi-task rotating detection of microscopic objects and enhancing signal-to-noise ratios (SNRs) during dynamic monitoring applications.

To address these challenges, we proposed a scheme to construct multi-ring vector perfect vortex beams employing the Bessel beam kinoform method and investigated the wide-ranging applications of this approach. In contrast to the direct spatial light modulator (SLM) modulation method, the Bessel beam kinoform approach exhibits significant advantages in generating beams with higher energy and extremely fine ring half-width, which can be effectively used for the multi-task and high-SNR detection of rotating microscopic objects. Additionally, the generated multi-ring vector perfect vortex beams are also applicable to detecting velocity gradients.

The electrical field of a multi-ring vector perfect vortex beam can be written as^[36]$$\begin{gathered} E(r,θ)=∑nxexp(ilnxθ)exp[(r−rnx)2ω0nx2]e^x \\ +∑nyexp(ilnyθ)exp[−(r−rny)2ω0ny2]e^y, \end{gathered}$$(1)where (r,θ) indicates the transverse position, lnx and lny are the TCs of the x- and y-polarized beams, respectively, and e^x and e^y represent the unit vectors for the x- and y-polarization states. The ring half-width of the nth x-polarized perfect vortex beam is denoted as ω0nx and rnx represents the ring radius of the nth x-polarized perfect vortex beam. To simplify the model, the amplitude coefficient of each ring can be set as unity. This approach is referred to as the direct SLM modulation method. However, the ring half-width generated by this method is limited to the SLM pixel size (e.g.,  8 µm), leading to pixelation of the intensity.

The single-ring perfect vortex beam can be generated via the Fourier transform of a Bessel–Gaussian beam. In cylindrical coordinates, the complex amplitude of the single-ring Bessel–Gaussian beam at the source plane can be expressed as^[37]E(ρ,φ)=exp(−ρ2ωg2)Jl(krρ)exp(ilφ),(2)where (ρ,φ) indicates the transverse position, Jl(·) is the lth order Bessel function of the first kind, ωg is the radius of the Bessel–Gaussian profile, and kr is the beam radial wave number. Consequently, the generation of a multi-ring perfect vortex beam could be realized by the Fourier transform of multiple Bessel–Gaussian beams, formulated as $$\begin{gathered} E(ρ,φ)=∑nxAnxexp(−ρ2ωgnx2)Jlnx(krnxρ)exp(ilnxφ)e^x \\ +∑nyAnyexp(−ρ2ωgny2)Jlny(krnyρ)exp(ilnyφ)e^y. \end{gathered}$$(3)

Here, to achieve minimal spacing between adjacent rings and to avoid significant interference due to small spacing, orthogonal polarization states are set to adjacent rings. In addition, the values of krnx and krny are constrained by the following conditions: kr(n−1)y<krnx<krny and kr(n−1)x<krny<krnx. Anx and Any are used to modulate the amplitude of the nth ring in the x and y directions, respectively. The ring half-widths of the Bessel–Gaussian profile associated with the nth ring in the x and y polarizations are represented by ωgnx and ωgny, respectively. In the following simulation and experiment, a thin convex lens with focal length f was used to perform the Fourier transform, which converts E(ρ,φ) to $$\begin{gathered} E(r,θ)=∑nxCnxexp(ilnxθ)exp[−(r−rnx)2ω0nx2]e^x \\ +∑nyCnyexp(ilnyθ)exp[−(r−rny)2ω0ny2]e^y. \end{gathered}$$(4)

Here, the coefficients Cnx=Anxωgnxilnx−1/ω0nx and Cny=Anyωgnyilny−1/ω0ny are the amplitude coefficients of the nth ring in the x and y directions, respectively. In the following simulations and experiments, Anx and Any were modulated to generate uniform multi-ring perfect vortex beams. The nth x-polarized ring half-width is denoted as ω0nx=fλ/πωgnx, where λ denotes the wavelength. Here, rnx=krnxf/k represents the radius of the nxth ring of the perfect vortex beam, and k=2π/λ is the wave number^[38].

This method is referred to as the Bessel beam kinoform method^[39]. Since this method relies on the Fourier transform principles, the impact of discrete source pixels on the generated ring size can be considered negligible provided that the sampling interval satisfies the Nyquist criterion. The minimal achievable focal spot size after Fourier-based focusing is fundamentally constrained by the Abbe diffraction limit, given as 1.22λ/NA, under ideal conditions of aberration-free optics and coherent illumination. Here, NA is the numerical aperture. Therefore, an extremely fine ring half-width can be generated with the Bessel beam kinoform method.

According to the Doppler effect, when a rotating object is illuminated by conjugate vortex beams with TC=l and −l, the scattered light undergoes intensity modulation. The frequency of this intensity modulation can be extracted from the signal spectrum and is given by^[29]fmod=2|l|Ω2π,(5)where Ω represents the angular velocity of the rotating object. The resulting frequency shift, though significantly smaller in magnitude compared to the optical frequency, is experimentally observable. This principle can be widely applied to the measurement of rotational speeds. By leveraging this principle, the simultaneous determination of the velocities of multiple objects can be achieved using the conjugate superposition of multi-ring vector perfect vortex beams. By independently modulating the TC of each ring, a tunable frequency comb can be constructed, where each peak corresponds to a unique TC |l|. This scheme can be utilized to measure the velocity gradient.

Through numerical simulations, we investigated the generation of multi-ring vector perfect vortex beams with conjugate superposition using the two aforementioned methods under varying ring half-widths. Figure 1 illustrates the intensity results, along with their corresponding crosslines along the horizontal direction. Figures 1(a)–1(c) depict the intensity distributions of the direct SLM modulation method; the ring half-width decreases from left to right. It is evident that for ring half-widths greater than 8 µm, the generated beams exhibit good quality, with the crosslines displaying complete and uniform structures. However, as the ring half-width is reduced to ω0nx<8µm, the beam quality deteriorates significantly, with noticeable pixelation effects. Additionally, the crosslines become increasingly inhomogeneous. This phenomenon is due to the limitations imposed by the SLM. As a result, when the ring half-width becomes smaller than the pixel size, pixelation effects inevitably arise, compromising the overall beam quality.

In contrast, Figs. 1(d)–1(f) demonstrate the results for the proposed Bessel beam kinoform method, which exhibits a markedly different behavior. As the ring half-width decreases from left to right, the vortex beams by this method consistently maintain high quality, with uniform intensity distributions and the corresponding crosslines remain highly complete. This observation highlights the advantages of the Bessel beam kinoform method, which is not constrained by the pixel size of the phase-modulating device. The Bessel beam kinoform method does not rely on the discrete pixel sizes of the input plane, thereby maintaining the integrity of the vortex beam even with a fine ring half-width. Furthermore, due to the tunable amplitude coefficients inherent to this approach, the generated crosslines exhibit a highly uniform intensity distribution.

Figure 2 presents a schematic of the experimental setup used to generate multi-ring vector perfect vortex beams via the Bessel beam kinoform method. In the experiment, a solid-state laser emitting a wavelength of 532 nm was used to generate a horizontally polarized laser beam. After passing through a half-wave plate, the beam was transformed into a vertically polarized beam that matches the SLM (HDSLM85T, with a resolution of 1920×1080 and a pixel size of 0.0085 mm). This vertically polarized beam was then expanded by a beam expander. As shown in Fig. 2(a1), the SLM was loaded with a computer-generated hologram composed of electric fields with two different gratings. As shown in Figs. 2(a2) and 2(a3), the ±1 diffraction orders through the SLM generate Bessel–Gaussian beams with different radii, respectively. Bessel–Gaussian beams exhibit petal-like profiles due to the coherent superposition of the conjugated beams. To filter out the desired ±1 diffraction orders, two lenses (f=250mm) and two apertures were used. To ensure that adjacent rings have orthogonal polarization states to minimize interference, two linear polarizers oriented at perpendicular angles were placed behind the apertures. At the back focal plane of lens L2, a Ronchi grating was employed to combine the two cross-polarized components into a single beam, thereby producing multi-ring Bessel–Gaussian vector beams, as shown in Fig. 2(b). Following this, the beam passed through lens L3 and was focused onto the digital micromirror device (DMD) (F4320 DDR 0.95 1080P, with a resolution of 1920×1080 and a pixel size of 0.0108 mm) located on the back focal plane of lens L3. At this plane, the multi-ring vector perfect vortex beam was generated.

As illustrated in Fig. 3, we experimentally generated eight-ring vector perfect vortex beams with l=±5 and a ring half-width of 6 µm, which then illuminated a rectangular object (with a length of 9.72 mm and a width of 0.1 mm) loaded on the DMD, as depicted in Fig. 2(c). To ensure the rectangular object rotates at a constant speed, we pre-generated 360 frames using MATLAB, covering the complete rotation of the rectangle from 0° to 359°, with each frame corresponding to a 1° angular interval. As the object is displayed on the DMD at a speed of 50 r/s, the frame rate of the corresponding image sequence is 18000 frames per second (50 r/s multiplied by 360 frames per revolution). According to Eq. (5), the peak should theoretically be observed at a frequency of 500 Hz.

Here, we employed two different methods to generate multi-ring vector perfect vortex beams for velocity measurement. Figures 3(a)–3(d) show the results obtained from the Bessel beam kinoform method, and Figs. 3(e)–3(h) present the results from the direct SLM modulation method. A ring half-width of ω0<8µm (the limit of SLM pixel size) was chosen, e.g.,  ω0=6µm, to highlight the advantages of the Bessel kinoform method. The intensity plots indicate that the beams generated by the direct SLM modulation method exhibit noticeable pixelation effects, and the noise influence on the beam quality is significant. In addition, as the beam radius is reduced, the ring spacing becomes increasingly blurred, compromising the precision of the generated vortex beam. In contrast, the Bessel beam kinoform method does not suffer from such issues. The uniformity of the rings is maintained at high quality.

The spectral analysis further reveals that the direct SLM modulation method yields a relatively low SNR, and the signal is increasingly influenced by noise as the beam radius and beam quality decrease. Nevertheless, for the Bessel beam kinoform method, as the beam radius decreases, the signal intensity increases. This is attributed to the improvement of beam quality and the concentration of light intensity during the Fourier transform. For larger Bessel beams, the resulting perfect vortex beam leads to a smaller beam radius, thus concentrating more energy into the beam and enhancing the SNR.

Generating a fine ring radius is essential in improving SNR for microscopic object detection. Due to the Bessel beam kinoform method being pixelation-free, the generated multi-ring vector perfect vortex beam with ultra-thin rings proves to be significantly more effective for extracting velocity information when measuring micro-scale objects. In Fig. 4, a microscopic object with a radius of 16 µm rotating at 50 r/s is irradiated by multi-ring vector perfect vortex beams, which reside at the second ring of the beam. When the perfect vortex beam in Fig. 4(a) irradiates the microscopic object (outlined with a white circle and shown in a local magnification within the figure), the corresponding Doppler signal is shown in Fig. 4(b). Similarly, Fig. 4(c) corresponds to Fig. 4(d), and Fig. 4(e) corresponds to Fig. 4(f). As shown in Fig. 4(b), due to the disproportionately large beam radius, the response of the object (i.e., scattered signals) acts as noise, making the frequency shift information induced by its rotational motion easily drowned out by the noise, resulting in the failure of rotation detection. Thus, for measuring microscopic objects, multi-ring vector perfect vortex beams with smaller beam radii and finer ring sizes are required. However, as shown in Figs. 4(c) and 4(d), when using direct SLM modulation, poor-quality beams can easily overwhelm frequency shift information with noise.

In contrast, the Bessel beam kinoform method, as illustrated in Fig. 4(f), not only achieves a higher SNR but also enables accurate determination of the object’s rotational velocity. This superior performance can be attributed to the method’s ability to avoid the limitation of the SLM pixel size. Consequently, the Bessel beam kinoform method emerges as a robust and reliable technique for measuring the rotational dynamics of micro-scale objects, offering significant advantages in both accuracy and signal fidelity.

The multiple rings also enable multi-task applications, such as velocity gradients of multiple objects. The rotational motion of eight objects was simulated with the DMD [Fig. 5(a)]. An eight-ring vector perfect vortex beam generated by the Bessel beam kinoform method was employed as the incident beam with the beam radius of 1.3 mm and a ring waist of 0.03 mm. The corresponding rotational Doppler shift spectrum comb indicates the information of the velocity gradient, as shown in Figs. 5(a)–5(c). Experimentally, we first measured the rotational speeds of these objects, all of which were rotating at 50 r/s. The corresponding rotational Doppler shift spectrum [Fig. 5(d)] is used as the reference line for measuring the velocity gradient. Since the topological charges corresponding to the beams at different radii are known, the relationship between the radius and the rotational speed can also be plotted based on the spectrum and Eq. (5), as shown in Fig. 5(g).

Then, we investigated three distinct rotational scenarios, including an approximation of the Rankine vortex^[26], an increasing velocity gradient, and a random distribution. Since the three objects closest to the central point rotation speed were identical, to prevent crosstalk between the velocities at different radii, the TCs in Fig. 5 were configured. Simultaneously, we can determine which ring generates the spectral line by the distance between the reference line and the spectral line, thereby obtaining the corresponding radius-versus-speed relationship. In the first scenario, illustrated in Fig. 5(a), the three objects closest to the central point rotated at the same speed, while the rotational speed decreased with increasing distance from the central point, with a speed difference of 1 r/s between adjacent objects. The measured frequency comb for this case is shown in Fig. 5(b), and the corresponding radius-versus-speed relationship is presented in Fig. 5(c). In the second scenario, the three objects closest to the central point again rotated at the same speed, but the rotational speed increased with distance from the central point, with a speed difference of 1 r/s between adjacent objects. The frequency comb for this case is displayed in Fig. 5(e), and the corresponding radius-versus-speed relationship is shown in Fig. 5(h). The third scenario, depicted in Fig. 5(f), involved the three objects closest to the central point rotating at the same speed, while the rotational speeds of the objects farther from the central point exhibited irregular variations. The corresponding frequency comb and radius-versus-speed relationship are illustrated in Figs. 5(f) and 5(i), respectively. In all Doppler frequency spectra, the orange dashed line represents the reference spectral line, corresponding to a rotational speed of 50 r/s. This reference line serves as a benchmark for comparing the measured spectra under different rotational conditions. Accordingly, this method could be utilized to measure the variation of a fluid’s rotational speed with respect to radius, thereby determining its velocity gradient.

In this work, an approach was proposed for measuring the rotational speed by generating multi-ring vector perfect vortex beams with conjugate superposition using the Bessel beam kinoform method. The comparison with the direct SLM modulation method shows the superiority of the Bessel beam kinoform method in solving pixelation issues and increasing SNRs. Due to the generation of ultra-thin rings, the proposed method shows advantages and effectiveness in the dynamic detection of microscopic objects. Furthermore, the independent modulation of each ring enables the multi-task rotational measurement, such as the perception of locations and velocity gradients. Therefore, the multi-ring vector perfect vortex beams generated via the Bessel beam kinoform method open up new possibilities for future applications in more complicated and microscopic fluid dynamics.