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Abstract
In this paper, the circular Airy derivative beams carrying rotationally-symmetric power-exponent-phase vortices are proposed for the first time, whose evolutionary properties are explored by theoretical analysis as well as experimental verification. The intensity and phase distributions of this kind of beam can be flexibly modulated by controlling its parameters such as derivative order, topological charge, and power order. Intriguingly, the evolution of such beams with different fractional topological charges is also investigated in detail by means of phase singularities distribution. In addition, from the perspective of transverse gradient force, the proposed beam is capable of efficiently protecting the trapped Rayleigh particles located at the beam center from the impact by surrounding particles. Furthermore, the beam width, orbital angular momentum (OAM) density and spiral spectrum are also compared and analyzed under different beam parameters. The proposed beams are expected to be useful for extending applications of optical vortices and autofocusing beam, especially for multi-regional particle gathering and central particle protecting.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Circular Airy beams (CABs) have attracted many attentions due to its remarkable autofocusing ability without external optical elements since it was introduced in 2010 [1,2]. In free space, its propagation can maintain a low intensity profile over several Rayleigh lengths until an abruptly focusing takes place right before focus, where the beam intensity is suddenly enhanced by several orders of magnitude. CABs are found to highlight several potential applications in the biomedical treatment, light bullet generation, and microparticles manipulation, etc. [1,3–5]. Very recently, a novel kind of CABs-like namely circular Airy derivative beams (CADBs) was proposed through a combination of theoretical and experimental studies. As an extension of CABs, the CADBs have a radial profile that is described by derivative of the Airy function and exhibit stronger abruptly autofocusing ability than the CABs under the same conditions [6]. With this advantage, the CADBs could be more beneficial for particle trapping and manipulation in the field of optical tweezers to realize strong trapping stiffness. Various new types of CADBs have been proposed and investigated for modulating the abruptly autofocusing property or studying new phenomenon. For example, Zhou et al. imposed first- and second-order chirped factors on the CADB for improving K-value [7]. Meanwhile, to further enhance the autofocusing ability, they also built and studied a new class of beams called ring Airyprime beams array [8]. Anita et al. numerically and experimentally studied the autofocusing and self-healing of partially blocked CADBs [9]. In addition, Ge et al. found that the CADB splits into the left-handed circularly polarized (LCP) and the right-handed circularly polarized (RCP) beams with different propagating trajectories in the chiral medium [10]. Under the nonparaxial propagation, Wang et al. investigated the propagation dynamics of the chirped CADBs based on vector angular spectrum method and demonstrated that the derivative order and chirp factor of the beam can be used to coarse and fine tune the capture effect of Rayleigh microsphere, respectively [11].
As another structured light beam, commonly known as optical vortices (OVs), have been an active area of optical research since the pioneering work of Nye and Berry on phase singularities in optical fields [12]. OVs are often characterized by spiral phase structure, doughnut intensity distribution, and intriguing orbital angular momentum (OAM), which leads to a plenty of applications in particle trapping, quantum information processing, and free space optical communications [13–15]. Generally, OVs are classified into many different types, the most common of which is canonical OV with phase term of $\textrm{exp} (il\varphi ).$ l is the topological charge around the azimuthal angle φ of the field. This type of OV carries a spiral phase varying uniformly with azimuthal angle. Introducing canonical OVs into the autofocusing beam has been reported to present new optical properties [16–18].
On the other hand, to explore novel features and applications of OAM, several types of noncanonical OVs have been derived in recent decades, such as helico-conical OVs, nonsymmetric OVs, fractional OVs, Mathieu OVs, and power-exponent-phase vortex [19–23]. Especially, the power-exponent-phase vortex has two main forms, one is rotationally-asymmetric power-exponent-phase vortex with the phase term of $\textrm{exp} [il2\pi {({\varphi / {2\pi }})^n}]$[23] and the other is rotationally-symmetric power-exponent-phase vortex with the phase term of $\textrm{exp} \{ i2\pi {[{{rem(l\varphi ,2\pi )} / {2\pi }}]^n}\} $[24–27], where the parameter n denotes power order. When l = 1, the rotationally-symmetric power-exponent-phase vortex evolves into rotationally-asymmetric power-exponent-phase vortex. In the case of n = 1, both of them degenerate into canonical OV.
Nowadays, it is an inevitable trend to introduce noncanonical OV into autofocusing beams in the field of light manipulations. For example, Zhao et al. have theoretically and experimentally investigated spiral autofocusing properties of the CABs carrying a rotationally-asymmetric power-exponent-phase vortex [23]. As a new type of autofocusing beam with higher K-value, the propagation characteristics of CADBs superimposed OVs have also attracted more attention [17,28]. In fact, the propagation dynamics of such kind of beam embedded with single and multiple canonical OVs in free space have been investigated in detail through theoretical analysis and experimental verification. The most outstanding advantage of the CADB with OVs is that it possesses the abruptly autofocusing ability, hollow focal intensity profile and carries OAM, which have promising applications in particles rotating as an optical spanner and others involving OAM free space optical communication. However, to our best knowledge, the propagation characteristics of CADBs superimposed noncanonical OVs have not been studied before. Thus, the CADBs carrying rotationally-symmetric power-exponent-phase vortices are introduced in this paper, whose evolutionary properties are explored by theoretical analysis as well as experimental verification. Such a special class of beams opens new possibilities in multi-regional particle gathering, central particle protecting, and creates a new platform for further study.
2. CADBs carrying rotationally-symmetric power-exponent-phase vortices
It is assumed that the electric field of the CADBs carrying the rotationally-symmetric power-exponent-phase vortices in the source plane has a form of
In the situation of paraxial approximation, the electric field in arbitrary transverse plane through an ABCD paraxial optical system can be solved with the help of the generalized Collins formula [30]
3. Experimental generation and simulation analysis
3.1 Experimental setup
In this section, we experimentally generate the CADBs carrying rotationally-symmetric power-exponent-phase vortices by using a phase-only spatial light modulator (SLM), and then measure the intensity distribution at different observation planes. Figure 1 shows our experimental setup, where the Gaussian beam (λ = 632.8 nm) is generated firstly from He-Ne laser. Subsequently, the beam is regulated by an attenuator. A beam expander (BE) is used to expand the beam for completely irradiating the response region of the SLM (1024 × 768 liquid crystal pixel array with each pixel size 9 × 9 µm). A polarization controller is inserted between the SLM and the beam expander for changing the polarization state of the incident light. Here, a computer-generated hologram (CGH) is displayed on the SLM, with its intensity profiles computed by the complex amplitude modulation method mentioned in Ref. [32]. After going through the SLM, the modulated beam is sent through the 4f filter system which is consisted of two lenses (L1, L2) and a circular aperture (CA). The CA is placed before the L2 to block unwanted diffractive light generated by the hologram and the SLM, and only allows through the positive first-order diffractive light. The intensity distribution of CADBs carrying the rotationally-symmetric power-exponent-phase vortices can be obtained at the focal plane of the L2. Finally, placing the charge-coupled device (CCD) at different distances, we can observe the evolutionary properties of the proposed beam propagation.
Fig. 1. Experimental setup. BE: beam expander; SLM: spatial light modulator; CA: circular aperture; CCD: charge-coupled device; L1, L2: lens (focal length = 150 mm); The insets depict the hologram (top right) and the corresponding output intensity distribution at the initial plane (bottom left).
3.2 Evolutionary properties under different beam parameters
In the experiment and simulation, the beam parameters are set as follows: a = 0.2, w0 = 80 µm, r0 = 1 mm. m, l, n represent the derivative order of Airy function, the number of topological charges, and the power order of the spiral phase, respectively. Next, the effect of different m, l, and n values on the evolution properties of the beam are studied in detail. To investigate the modulation effect of derivative order m, the topological charge l and power order n are first fixed to 6 and 4, respectively. Figure 2 shows the intensity distributions of the CADBs carrying rotationally-symmetric power-exponent-phase vortices at different propagation distance z through numerical simulation and experimental measurement, where the derivative orders are set to m = 0, 1 and 2, respectively. We can clearly see that the simulation results on the left are consistent with the experimental measurement results on the right. As seen from the Figs. 2(a1), (a4) and (a7), the energy distribution of the beam at the initial plane gradually spreads into the outer ring with the increase of derivative order m. When the beam propagates forward, the multiple-ring profiles gradually evolve into a helical fan-like shaped profiles, which resulted from the rotationally-symmetric power-exponent-phase factor. In addition, the energy of the beam also converges towards the center as the propagation distance increases due to the existence of autofocusing property. It can be seen from the simulation and measurement results at the plane of z = 320 mm and z = 420 mm, the higher the derivative order, the better the autofocusing effect of the beam.
Fig. 2. Simulation (a1-a9) and experimental (b1-b9) results of the intensity distributions of the CADBs carrying the rotationally-symmetric power-exponent-phase vortices at different propagation distance.
Figure 3 shows the simulated phase contours at different propagation distance z, corresponding to the simulated intensity distributions in Fig. 2. It can be seen that at the initial plane the beam phase has a rotational symmetry, which can be regarded as a combination of the phase of the CADB and the rotationally-symmetric power-exponent-vortex-phase. Unlike the rotationally-asymmetric power-exponent-phase vortices in [23], the rotationally-symmetric properties of the phase contours dominate the rotational symmetry of the beam intensity pattern. Due to the impact of power order n, the phase contours change unevenly with the azimuthal angle, which leads to the phase eventually evolving into a fan-like blade structure with the increase of propagation distance. It also shows the original single-phase singularity at the initial plane will be split into multiple singularities equal to the topological charge. In essence, the splitting evolution of the phase singularity also reveals the origin of the distribution of light intensity from a multi-ring to a fan-like shaped.
Fig. 3. The phase contours of the CADBs carrying the rotationally-symmetric power-exponent-phase vortices at different propagation distance for different derivative order m.
For studying the effect of topological charge l, we need to concentrate on Fig. 4, where m = 1, n = 2, and l is set 5, 6, 7, 8 and -8, respectively. Figures 4(a1)-(a5) show the simulated intensity distributions at the plane of z = 390 mm. It can be noticed that the CADBs carrying the rotationally-symmetric power-exponent-phase vortices have l dark spots, which are resulted from the split of the singularity as previously described. In each subgraph, all dark spots are indicated with small white circles. With the increase of l, the dark area in the middle of the beam is bigger, and the trails of the fan-like blades become longer. Especially, when the topological charge is opposite (e.g., l = 8 and -8), the intensity distributions remain consistent, whereas the rotation direction of the beam trails is exactly opposite. Figures 4(b1)-(b5) show the images of intensity distributions obtained from the experiment, which are basically consistent with the simulation results. Figures 4(c1)-(c5) show the corresponding phase contours, and it can be seen that as l increases, the singularity number correspondingly increases. Moreover, the singularities stay away from the optical axis, which corresponds to the expansion of the dark area in the intensity distributions. It is worth noting that the rotation directions of the phase structure with positive and negative topological charges are mirrored to each other, which also explains that the rotation directions of the beam trails with positive and negative topological charges are mirrored to each other.
Fig. 4. The intensity and phase distributions of the CADBs carrying the rotationally-symmetric power-exponent-phase vortices for different l value, under the condition of m = 1, n = 2, z = 390 mm. (a1)–(a5): simulation results of the intensity distribution; (b1)-(b5): experimental results of the intensity distribution; (c1)-(c5): the corresponding phase contours obtained by simulations.
Intriguingly, in this paper we also study the effect of fractional topological charge for the beam intensity distributions. As seen from Figs. 5(a1)-(a5), under the condition of m = 1, n = 3, z = 390 mm, if l varies from 6.0 to 7.0, the number of the fan-like blade will be gradually evolved from 6 to 7 lobes. The most obvious region of the evolution has been marked in each subgraph with a different color circle, where the single lobe eventually splits into two lobes with almost equal intensity. The physical mechanism can be interpreted by the evolution characteristics of beam phase singularities. Each subgraph in Figs. 5(c1)-(c5) presents the evolution process of the new phase singularity. Compared with Fig. 5(c1), the blade marked by the white circle in Fig. 5(c2) has been changed. When l = 6.5, the new phase singularity initially forms, which can interpret the formation of the new blade. As l increases to 7.0, the new phase singularity has been completely formed. Figures 5(b1)-(b5) show the corresponding experimental results, where the evolution properties of beam intensity are agree with the simulation results.
Fig. 5. The intensity and phase distributions of the CADBs carrying the rotationally-symmetric power-exponent-phase vortices for different fractional topological charge (m = 1, n = 3, z = 390 mm). (a1)–(a5): simulation results of the intensity distribution; (b1)-(b5): experimental results of the intensity distribution; (c1)-(c5): the corresponding phase contours obtained by simulations.
To investigate the effect of power order n, here the intensity distributions at the plane of z = 390 mm are obtained by the simulation and experiment with beam parameters: m = 1, l = 5, and n = 3, 5, 6, 8 and 10, respectively. It is obvious that from Fig. 6 a feeble blade is parasitized next to each main blade, and the intensity of the parasitical blade will be enhanced with the increase of n. The enhancement of the parasitical blade causes the beam intensity distribution to tend to be more enclosed. When n takes an extreme large value, the phase term in Eq. (1) would approach to 0 except the case that φ is very close to 2π/l. Under this condition, the phase function would be almost a constant, thus the beam’s profile would take the intensity form of CADBs with derivative order m = 1. The corresponding phase distributions are illustrated in Figs. 6(c1)-(c5). It can be noticed that with the increase of n, the phase distributions near the optical axis change from a shape of spiral blades to a pentagon. We can predict that the phase profile will eventually approach a circular distribution if n takes an extreme large value.
Fig. 6. The intensity and phase distributions of the CADBs carrying the rotationally-symmetric power-exponent-phase vortices for different power order n value (m = 1, l = 5, z = 390 mm). (a1)–(a5): simulation results of the intensity distribution; (b1)-(b5): experimental results of the intensity distribution; (c1)-(c5): the corresponding phase contours obtained by simulations.
Owing to the peculiar distribution evolution of fan-like blades, the intensity gradient of the proposed beam can induce the unique transverse gradient force distributions. In the approximation of the radiant force, the scattering force of the longitudinal component does not play a crucial role in the resultant optical phenomena [33,34]. Therefore, we just consider the gradient force of the transverse component to describe trapping ability. Taking the trapping of Rayleigh particle as an example, the gradient force distributions under different beam parameters are shown in Fig. 7 to analyze the application prospect of the novel beam in the field of optical tweezers. The gradient force Fg can be expressed as [33]
Fig. 7. The transverse gradient force distributions of the CADBs carrying the rotationally-symmetric power-exponent-phase vortices with different parameter l and n at z = 440 mm.
In simulations, we assume that np = 1.592, nm = 1.332, R = 5 nm, the derivative order m = 1, the propagation distance z = 440 mm, and laser power P = 0.5 W. The transverse gradient force distributions with different parameter l and n are presented in Figs. 7(a1)-(a3) and (b1)-(b3), respectively. The direction of blue arrows represents the direction of the gradient force, and their length represents the relative magnitude of the gradient force in each subgraph. The background denotes the numerical value of the beam intensity. The magnitude of the transverse gradient force corresponding to the white dotted line are shown in Figs. 7(a4)-(a6) and (b4)-(b6). As expected, the transverse gradient force distributions do correlate with the intensity distributions of the fan-like blades. When l is changed, there will be some differences in the number and location of the regions in which the Rayleigh particle can be trapped. The larger the topological charge is, the more regions the particles can be trapped. It is worth noting that there is always a fixed trapping region in the center of the beam. Similarly, we can see from Figs. 7(b1)-(b6) that under the same topological charge (l = 5), with the increase of n, the energy of the light spot gradually shrinks, and the energy of the fan-like blades transfers to the central bright spot, which also leads to the enhancement of the gradient force at the central bright spot. According to the arrow direction of gradient force, we can find that the particles in the central bright spot can be stably trapped, and the gradient force formed in the outer part of the center bright spot can protect the trapped particles from external interference.
3.3 Beam width, OAM density and spiral spectrum
To quantify the capability of the beam focus to change with propagation distance, the beam width in the x and y directions are introduced as [35]
Due to the rotational symmetry of the beam, the beam width in the x and y directions is theoretically equal. As seen from Fig. 8(a), here we only plot the beam width along the x direction, for (l = 1, n = 1; l = 6, n = 1; l = 6, n = 4) and m is equal to 0, 1, 2 respectively. The beam width under different beam parameters is indicated by different colored lines combined with arrows. The derivative order m obviously affects the beam width, for instance, for m = 0, the beam width decreases slowly and then increases inconspicuously. For m = 1, 2, the beam width obviously decreases first and then increases with the increase of z, reflecting the process of the beam first autofocusing and then defocusing. Figure 8(a) also shows that a higher derivative order can improve the autofocusing ability at the focal plane. In addition, for canonical OVs, the autofocusing ability of l = 6 is weaker than that of l = 1. For the same m and l, the autofocusing ability of n = 4 is weaker than that of n = 1.
To further explore the effect of l and n on the autofocusing ability, Figs. 8(b) and (c) show the beam width curves under the condition of (m = 1, n = 2, l = 1, 4, 5, 6, 8, 9) and (m = 1, l = 5, and n = 1, 4, 8, 15, 40, 80), respectively. These curves in Fig. 8(b) show that increasing l will weaken the autofocusing ability of the beam, which can be attributed to the enlarging of the dark region in the center of the beam due to the increase of topological charge. As indicated in Fig. 8(c), when n gradually increases from 1 to 8, beam width at the focal plane increases slowly. When n is between 8 and 15, the minimum value of beam width almost coincides, indicating that the autofocusing ability hardly changes with n value. Nevertheless, with the further increase of n until 80, it can be seen that the minimum value of beam width gradually decreases, indicating that the autofocusing ability of the beam increases. Thus, we can flexibly choose the beam parameters to regulate the autofocusing ability of the beam.
The proposed beams also carry OAM owing to the spiral wavefront. Here the OAM density and spiral spectrum need to be quantitatively calculated to elucidate the beam properties. The OAM density around the z axis in free space can be expressed as [36]
Figures 9(a1)-(a4) display the OAM density distributions of the beam with different topological charges (l = 5, 6, 7, 8) in the case of derivative order m = 1 and power order n = 2. As seen, all the OAM density distributions are rotationally symmetric, and their rotation centers are on the optical axis. The angle between the fan-like blades in each subgraph is equal to 2π/l. Figures 9(b1)-(b4) show the OAM density distributions in the case of m = 1, l = 5 and n = 3, 5, 8, 10, respectively, at propagation distance z = 390 mm. When the topological charge l is the same, as n increases, the OAM density becomes more dispersed, and the proportion of parasitical blades will be larger. It reveals that the topological charge and power order can effectively adjust the OAM density distributions.
Fig. 9. The OAM density distributions of the CADBs carrying the rotationally-symmetric power-exponent-phase vortices at the plane of z = 390 mm, under the condition of different topological charges and power orders.
Due to the introduction of the power order, the characteristics of the proposed beam differ from the beam carrying canonical OVs, which is closely related to the spiral spectrum. Any beam can be expressed as a superposition of spiral harmonics [37]
The energy of each harmonic mode k can be described using ${C_k} = \int_0^\infty {{{|{{a_k}} |}^2}\rho d\rho } .$ The power weight of each OAM state for the light field can be given by
Figures 10(a1)-(a3) show the spiral spectrum of the beam at the initial plane, with n = 1, 2, 3 in the case of l = 3 and m = 1. It can be seen from Fig. 10(a1) that when CADBs carry canonical optical vortices (l = 3, n = 1), there is a single mode peak, corresponding to the vortex topological charge. However, when the power order n increases, the mode distribution of spiral spectrum differs from the case of the beam carrying canonical optical vortices. As shown in Figs. 10(a2) and (a3), their spiral spectrums are dispersed into multiple adjacent modes. Figures 10(b1)-(b3) show the spiral spectrum of the beam at the initial plane, with different topological charges (l = 1, 2, 3) in the case of n = 4 and m = 1. It is obvious that when l = 1, spiral spectrum disperses into other harmonic modes at an interval of 1. However, when l = 2 and 3, the intervals become 2 and 3, respectively. These spectral properties are of great significance for OAM optical communication in free space.
Fig. 10. Spiral spectrum of the CADBs carrying the rotationally-symmetric power-exponent-phase vortices with different beam parameters at the initial plane.
4. Conclusion
In summary, the circular Airy derivative beams carrying rotationally-symmetric power-exponent-phase vortices have been studied through theory and experiment, under different beam parameters. Its evolutionary properties such as the intensity, phase and singularity distributions can be flexibly modulated by controlling derivative order, topological charge, and power order. Due to these unique properties, the proposed beam can realize multi-regional particle gathering and central particle protecting. Furthermore, the beam width, OAM density and spiral spectrum have also been compared and analyzed under different beam parameters. Our results will be helpful in the use of this new beam in optical particle manipulation and free space optical communication applications.
Funding
Chongqing University of Posts and Telecommunications (E011A2022310); Natural Science Foundation of Chongqing (cstc2021jcyj-msxmX1119); National Natural Science Foundation of China (62105049); Science and Technology Research Project of Chongqing Education Commission (KJQN202100618); Chongqing Postdoctoral Research Fund Project (D63012022051); Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education and Guangdong Province (GD202102).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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