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Abstract
Optical vortex, typically characterized by a helical phase front, results in a possession of orbital angular momentum. In recent years, teleportation of the vortex mode using novel beams with peculiar features has gained great interest. Here, we experimentally demonstrate the propagation dynamics for a new class of the auto-focusing vortex circular Pearcey beam (VCPB), which is theoretically described by delivering the coaxial or off-axial spiral phases into the circular Pearcey beam (CPB), forming the crescent or bottle-like focal structure with self-rotation. Notably, such a hybrid beam with various types is experimentally obtained through a digital micromirror device (DMD) with the binary amplitude holography, and this DMD-based modulation scheme combined with controllable vortex modes enables dynamic switching among the VCPBs. We also measure the topological phase by interferometry and we explain the beam property on the basis of Poynting vector, showing a good agreement with the simulations. Further, the number, location and mode of embedded vortices could offer multiple dimensions of flexibility for target beam modulation, thus the experimentally controllable VCPBs will bring potential to high-speed optical communications and particle manipulations that require dynamic shaping.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Vortices exist in many natural phenomena. For instance, a vortex is a region in a fluid where the flow rotates around an axis line, which may be straight or curved. Besides the vortex appears in various natural phenomena, it could also be observed in some physical forms, ranging from the surface plasmon [1], electron [2], neutron [3], acoustics to optics [4]. Specifically in optics, vortex beams consist of spiraling wavefronts and phase singularities, giving rise to the orbital angular momentum (OAM) around its propagation direction. Moreover, the OAM related to the topological structure of the light beam, is determined by the topological charge (TC) of twisting a cycle back to the original state [5]. Each twisted photon carries an OAM [6], providing a degree of freedom for optical information encoding and the photon entanglement. Due to its own property, the vortex beam has fulfilled the applications in various areas, including the particle manipulation [7], multiplex free-space communication and high-dimensional quantum cryptography [8].
The earliest studies on vortex beams have been focused in Laguerre-Gaussian mode [9]. Later, the vortices naturally existed in the forms of diffraction-free beams were presented, such as Mathieu and high-order Bessel beams [10,11]; various types of vortex beams have been discussed and experimentally realized both in linear and nonlinear media [12,13]. Notably, the optical vortex intentionally delivered into self-accelerating Airy beams that follow curved paths were explored both in theory and experiment [14–18], and the interest in this kind of hybrid beam was induced since it offered an innovative way in beam steering. Meanwhile, such hybrid types were further expanded through embedding vortices into a special form of autofocusing beams, whose analytical expressions were based on the modulated Airy functions and share naturally self-focusing property without utilizing any optical elements [17]. The vortices introduced could dynamically shape the energy distribution of autofocusing waves, so that this mode of vortex beams could better satisfy the demands of the applications ranging from optical communications [15], micromanipulation to biomedical surgery [19,20]. As a result, embedding optical vortex into variant modes of autofocusing beams are still intensively explored.
As a type of autofocusing waves, circular Pearcey beam (CPB), whose analytical expression arises from transforming Pearcey function to the cylindrical coordinate, has a stronger peak intensity distribution than other kinds of autofocusing beams [21–24]. Here, we experimentally discuss the significance to deliver the topological phase into the autofocusing CPB termed as vortex circular Pearcey beam (VCPB), and also demonstrate various VCPBs embedded with a coaxial or off-axial vortex phase mode in a dynamically shaped fashion, where the rapidly-refreshing DMD combined with binary holography are employed [25,26]. The on-axis topological phases enable a bottle-like hollow channel formed, while the off-axis counterparts assist the beam to dynamically shape its energy distribution. Moreover, such beam with adjustable focal depth leads to a more evident self-focusing effect than previously reported autofocusing beams, so that it will induce a stronger strength in applications aforementioned. Besides, we also measure the exact phase of VCPBs by recording the interference patterns between VCPBs and planar wave, and the traits of VCPB could be explained by the Poynting vector calculated in the transverse planes at various locations. The experimentally controllable VCPBs will find applications that need dynamic modulation, such as particle manipulation in a rapidly-changing circumstances, or high-speed optical communications under strong scattering.
2. Theory of vortex circular Pearcey beam
A detailed insight into the theoretical origin and characteristics of VCPB is presented. In theory, the electric field of VCPB can be described by the product of a Pearcey function with an optical vortex (OV) factor [21], defined as an integral of the complex exponential function with a polynomial argument,
${A_0}$ is a normalized constant amplitude of the initial spatial field; p is the spatial distribution factors which controls the intensity distribution of the input beams, and it determinate the focal length as well (The focal length $L = 2{p^2}$) [28]. $\mathrm{\alpha}$ and $\mathrm{\beta}$ are factors used to adjust amplitude distribution of the initial electric field at the range $r < {r_0}$; ${r_k}$ and $\varphi $ denote the radial location and the orientation of the vortices; m and n represents the topological charge for each optical vortex. In theory, VCPBs should have infinite energy [22], which is impossible to realize in experiments. Here, we apply a simple cut-off term $q(r )$ in Eq. (2) to ensure the VCPBs own finite energy. It’s worth noting that Eq. (4) is a phase modulation term without adding any effect on the amplitude. For the propagation of VCPBs in free space, it is more convenient to describe (2 + 1) dimensional normalized potential-free Schrödinger equation in polar coordinates, which could be rewritten as [29]:
Where $E$ stands for the beam envelope; ${\; }\theta = arctan({y/x} )$ is the azimuthal angle; r corresponds to the radial coordinate, normalized and scaled by an arbitrary transverse width $\; {x_0}$, and $\textrm{z}$ represents the normalized propagation distance with the corresponding Rayleigh range ${z_0} = k{x_0}^2$, $k = 2\pi /\lambda $ ($\lambda $ is the center wavelength in the free space) being the wavenumber. Further, the solution of Eq. (5) can be expressed in terms of the Fresnel integral,
Although the analytical solution for Eq. (6) is hard to obtain, numerical simulations of beam propagation could be realized by using the angular spectrum method [30]. Throughout this paper, the parameters are set as ${x_0} = 1\textrm{mm}$, $\lambda = 632\textrm{nm}$ and ${A_0} = 1$, $p = 0.1$, $\alpha = 0.1$, $\beta = 1$, ${r_0} = 1.5$, ${\; }\varphi = \pi /2$, m and n are adjustable; the coaxial or off-axial VCPBs will be obtained by flexibly changing m and n. While $\textrm{m}$ and $\textrm{n}$ are both set as 0, a typical CPB without vortex imposed is obtained.
In Fig. 1, we discuss the general propagation properties of the typical CPB with $m = 0\; $ and ${\; }n = 0$. Initially, the circular Pearcey beam rendered in a circularly symmetric profile with concentric rings, focuses till it reaches the focal plane, then defocuses displayed by the propagation simulation. Inset (a) is the 3D slice plot of the beam profile, and slices at various positions demonstrate the transverse profiles at the discrete locations. For better visualization, we present 2D intensity profiles that exhibit a bottle-like structure in (b)-(e). As compared to autofocusing Airy beam, the CPB shares an enhanced peak intensity contrast and more easily adjustable focal structure [24,31].
Fig. 1. Transversal beam profiles for the CPB at various propagation distance. (a) Seven cross-sectional profiles of various planes in free space; (b)–(e) represent the transversal intensity profiles that correspond to the certain planes shown in the (a); each position marked by two dashed lines with ${\; }z = {\; }0,{\; }0.013,{\; }0.026$, and 0.04 respectively, and parameters are set as $m = n = 0$.
In contrast to the common CPB, the on-axis VCPB autofocuses to a bottle-like focal structure, as confirmed by the beam propagation simulation for the coaxial VCPB with $m = 2$, $n = 0$ and ${r_k} = 0$, shown in Fig. 2. We model the side-view propagation progress of the VCPB embedded with the coaxial vortex mode, in order to reveal the interplay between optical vortex and CPB in the free space. In Fig. 2(b1)–(b4), the transverse intensity patterns are taken at the planes marked by the dashed lines in Fig. 2(a). The on-axis VCPB inherits the autofocusing property from CPB, but focuses into a doughnut (b2) and forms a dark hollow channel. Due to its vortex structure that exhibits a strong axial binding force, the autofocused doughnut beam is suitable for simultaneous trapping and rotating micro-particles with either high or low refractive indices [32]. More importantly, the focus of the coaxial VCPB has a much lower power density and thus causes less optically-induced thermal damage on an object in the beam center [32,33].
Fig. 2. Numerical demonstrations of an on-axis VCPB with $TC = 2$ propagating in free space. (a) numerically simulated side-view propagation of the VCPB; (b1)- (b4) snapshots of the transverse intensity patterns taken at the planes marked by the dashed lines in (a); (c1)- (c4) the corresponding phase distributions at different planes marked in (a). Parameters are set as $m = 2$, $n = 0$, and ${r_k} = 0$.
We also investigate how the off-axis vortex interacts with the CPB. As a demonstration, the parameters are set as $m = 2$, $n = 0$ and ${r_k} = 0.5$; the numerically simulated side-view intensity profiles of the off-axis VCPB at different propagation distances are depicted in Fig. 3(a). As it propagates, the crescent-like intensity profile gets rotated with $\frac{\pi }{2}$ degree anticlockwise at focal plane in Fig. 3(b2), while the annular profile rotates with $\mathrm{\pi }$ degree counterclockwise (Fig. 3(b4)). The number and position of the embedded off-axis vortices could enable the beam reshaped flexibly, since asymmetric intensity profile and strong rotation result in uneven distribution of its energy and focal plane. Worth noting that the autofocus position of the VCPB with an off-axis vortex is still unchanged, which proves parameter p instead of vortex that determines the position of the focal plane.
Fig. 3. Numerical demonstrations of an off-axis VCPB propagating in free space. (a) numerically simulated side-view propagation of the VCPB; (b1)-(b4) snapshots of the transverse intensity patterns taken at the planes marked by the dashed lines in (a); (c1)-(c4) the corresponding phase distributions at different planes marked in (a). Parameters are set as $m = 2$, $n = 0$, ${r_k} = 0.5.$
Further, the propagation traits of the VCPB with the imposed off-axis vortex pair are also explored and discussed. We can see clearly from Fig. 4 that two vortices are placed symmetrically on two beam sides, resulting in double moving potential wells during propagation. Due to the effect of vortex pair delivered, the entire beam shape gets more rotated and tailored than that of the off-axis VCPB. Thus, multiple off-axis vortices discretely embedded may better facilitate the applications that require beams steered in a flexibly controllable fashion.
Fig. 4. The intensity profiles of the VCPB with an off-axis vortex pair propagating at different propagation distances (a)$z{\; } = {\; }0.2L$, (b)$z{\; } = {\; }0.5L$, (c)$z{\; } = {\; }0.8L$, (d)$z{\; } = {\; }L$, (e)$z{\; } = 1.3L$, (f) $z\; = 1.9L$. $m{\; } = n\; = {\; }1$. Other parameters are the same as those in Figs. 3.
3. Experimental method and result
As a proof of concept, we demonstrate the experimental realization of the VCPB, implemented by employing a high refreshing-rate and reconfigurable digital micromirror device (DMD) [34], and the binary amplitude holograms projected on the DMD are generated from super-pixel method [35,36], where the setup is shown in Fig. 5. The DMD-based scheme facilitates switching among the generated VCPB types, and such experimental scheme ensures the VCPBs tailored in a rapidly shaped way, making it possible to achieve controllable beam modes while maintaining a high speed.
Fig. 5. Schematic of the experimental setup. Insets near the DMD and camera show the displayed binary amplitude hologram and the experimental beam profile of an on-axis VCPB ($m = n = 1$).
Fig. 6. Experimental demonstration of on-axis VCPBs. (a) a side view of an on-axis VCPB ($TC = 2$); (a1)-(a4) transverse beam profiles at the planes $z = 0.005{z_0}$, 0.02${z_0}$, 0.028${z_0}$, and 0.038${z_0}$ (${z_0} = 11.8{\; m}$), marked by the dashed lines in (a), respectively; (b) the side view of an on-axis VCPB ($TC = 4$) and the corresponding snapshots (b1-b4) at the planes $z = 0.005{z_0}$, 0.02${z_0}$, 0.028${z_0}$, and 0.038${z_0}$, respectively. The scale bars for all figures are the same as shown in (a).
A single-mode He-Ne laser (Thorlabs, 633nm) is expanded and collimated through two lenses before modulated by DMD (DLP Discovery 4100, Texas Instruments, 1024×768). A flat mirror is exploited to adjust the direction of the incident beam to ensure the incident angle to be 24° with respect to the normal direction [37]. To remove the high frequencies, we adopt a 4-f system composed of two lenses with a pinhole in the Fourier plane. A Fourier lens L3 (focal length 20cm) collects the modulated light diffracting from the DMD and focuses it to the spectrum space at the back focal plane, where a pinhole is used to select only the 1st diffraction order. The produced intensity patterns are further projected by lens L4 (focal length 10 cm) and captured by a CMOS camera (DS-CFM300-H, resolution 2048 × 1536, pixel size 3.2μm) [34]. Two neutral density filters are utilized to attenuate the beam power before the CMOS camera in order to prevent the camera from being saturated, and an optical dump is used to collect undesired background light.
The propagation dynamics of the on-axis VCPBs with $TC = 2$ and 4 are experimentally verified by mapping the 3D beam intensity (Fig. 6). Compared with common CPB (Fig. 1), we can see that the spiral phase superimposed into the autofocusing region enables the CPB to generate a cylindrically hollow optical channel, which shows good consistency with the theoretical prediction. For VCPB with coaxial vortices, the hollow region in the center would persist even at the focal point or after the focal point, forming a bottle-like structure. This phenomenon becomes more evident, as topological charge increases. More importantly, this controllability of the light capsules could be exploited in the transportation of multiple particles with different absorbing properties or refractive index. Such bottle-like beam may be beneficial for biomedical areas due to the unique features in applications of diagnosis, in vivo imaging, and drug delivery [38–40].
Finally, we show the VCPBs with vortex pairs through delivering multiple off-axis vortices into the single or various positions, and their energy distribution and intensity profile could be dynamically tailored, shown in Fig. 7 (a1)-(a3) and (b1)-(b3) respectively. The beam also experiences an apparent strong self-focusing, while the self-rotation is also evident, confirming its numerical simulation. The number, location and mode of embedded vortices provide multiple dimensions of flexibilities for target beam modulation; the controllable VCPBs and experimental scheme could facilitate applications that demand dynamic wavefront shaping, such as particle manipulation in a rapidly-changing and complex circumstance, or high-speed optical communications under strong scattering effect.
Fig. 7. The experimental intensity and phase distribution of the VCPBs with off-axis vortices at different propagation distances. Other parameters are the same as those in Fig. 4, except $m = 2,n = 0$ in (b1)–(b4); (a4) and (b4) show the experimentally recovered phase distribution of VCPBs at their initial plane in (a1) and (b1), respectively.
Meanwhile, to verify the topological phase distributions of VCPB with multiple off-axis vortices, we recover them through recording the interference pattern between VCPBs and plane-wave beams [41]. The optical setup is shown in Fig. 5, where a Mach-Zehnder interferometer are adopted [42]. Two beam splitters (BS) separate and interfere the object beam (VCPB) and reference beam into the CMOS camera. In Fig. 7(a4) and (b4), we recover the topological phase of VCPBs with two types of off-axis vortices in experiment respectively, and the positions of vortices are marked by the white circles.
4. Discussion
Intuitively, the self-focusing and rotation property of the VCPBs could be explained by studying the local energy flow of the beam, which can be expressed in terms of Poynting vector [43]. In the Lorenz gauge, the time-averaged Poynting vector is,
The Poynting vector could be used to explain the beam physics behind the self-rotation phenomenon and energy distributions of different VCPBs. Figure 8 show the energy flowing map superimposed on the intensity patterns in the transverse planes with ascending propagation distances for on or off-axis VCPBs ($TC = 2$). The white arrows show the dynamically change of the direction and magnitude of the energy flow in the transverse planes in propagation. Before the focus, for both coaxial and off-axial VCPBs ($TC = 2$), the beam energy flows to the center with anti-clockwise rotation; at the focusing plane, the beam only demonstrates the rotating energy distribution (a2-c2); while defocusing, the energy flows to the outer, and the rotation effect gets more obvious with increase of propagation distance. Hence, the beam orientation and its energy distribution get influenced by the number, position and mode of vortices imposed.
Fig. 8. Numerical demonstrations of the energy flow (white arrows) superimposed on the transversal intensity maps for VCPBs at $z{\; } = {\; L}/2$, L, 3*L/2, respectively. (a1)-(a3) the Poynting vectors for on-axis VCPB with $TC = 2$ ($m\; = n{\; } = {\; }1$); (b1)-(b3) the Poynting vectors for off-axis VCPB with $TC = 2$($m\; = n{\; } = {\; }1$), and (c1)-(c3) the Poynting vectors for off-axis VCPB with $TC = 2$($m = 2$, $n = 0$).
5. Conclusion
In summary, we have demonstrated a kind of VCPBs that exhibit both autofocusing property and topological phase structure; propagation dynamics for VCPBs at different distances are also investigated both in theory and experiment. Notably, we experimentally observed the dynamic switching among various VCPB modes with a DMD-based modulation scheme, including on and off-axial vortex embedded forms. The coaxial vortices enable a hollow channel formed, and also rotate the beam profile during propagation; the single or multiple off-axis vortices help to cause an evident self-rotation phenomenon and modulated intensity distribution, and phase distributions of VCPB are recovered experimentally to better confirm its topological structure. Moreover, the Poynting vector is calculated to offer a more intuitive explanation of the beam dynamics, and discuss the behaviors of the coaxial or off-axial vortices imposed into the CPB. The number, location and mode of imposed vortices combined provide multiple dimensions of flexibilities for target beam modulation, and such controllable VCPB modes with the experimental method could offer more possibilities to applications where dynamic wavefront shaping is required, such as particle manipulation in a rapidly-changing atmosphere, or high-speed optical communications under strong scattering effect.
Funding
Popularization of Science Foundation of Chinese Academy of Sciences (KP2015C10); the Strategic Priority Research Program (C) of the CAS (XDC07040200); National Key Research and Development Program of China (2018AAA0100301); Fundamental Research Funds for the Central Universities (WK2480000006, WK9100000001); National Natural Science Foundation of China (60974038); National Natural Science Foundation of China (31670866); Natural Science Foundation of Anhui Province (1708085MF143).
Acknowledgment
This work is sponsored by the Anhui Natural Science Foundation (Grant No. 1708085MF143), the National Natural Science Foundation of China (Grant No. 31670866, 60974038), the Popularization of Science Foundation of Chinese Academy of Sciences (KP2015C10), the Fundamental Research Funds for Central Universities (WK2480000006, WK9100000001), the National Key R & D Program of China (2018AAA0100301), and the Strategic Priority Research Program (C) of the CAS (XDC07040200).
Disclosures
The authors declare no conflicts of interest
References
1. D. Garoli, P. Zilio, Y. Gorodetski, F. Tantussi, and F. De Angelis, “Beaming of Helical Light from Plasmonic Vortices via Adiabatically Tapered Nanotip,” Nano Lett. 16(10), 6636–6643 (2016). [CrossRef]
2. L. Clark, A. Beche, G. Guzzinati, A. Lubk, M. Mazilu, R. Van Boxem, and J. Verbeeck, “Exploiting Lens Aberrations to Create Electron-Vortex Beams,” Phys Rev Lett 111 (2013).
3. C. W. Clark, R. Barankov, M. G. Huber, M. Arif, D. G. Cory, and D. A. Pushin, “Controlling neutron orbital angular momentum,” Nature 525(7570), 504–506 (2015). [CrossRef]
4. X. Jiang, Y. Li, B. Liang, J. C. Cheng, and L. K. Zhang, “Convert Acoustic Resonances to Orbital Angular Momentum,” Phys Rev Lett 117 (2016).
5. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital Angular-Momentum of Light and the Transformation of Laguerre-Gaussian Laser Modes,” Phys. Rev. A: At., Mol., Opt. Phys. 45(11), 8185–8189 (1992). [CrossRef]
6. L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999). [CrossRef]
7. X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys Rev Lett 105 (2010).
8. M. Mirhosseini, O. S. Magana-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J Phys 17(2015).
9. J. Chen, F. S. Zhang, K. Bian, C. J. Jiang, W. Hu, and D. Q. Lu, “Dynamics of shape-invariant rotating beams in linear media with harmonic potentials,” Phys. Rev. A: At., Mol., Opt. Phys. 99 (2019).
10. I. Zeylikovich and A. Nikitin, “The formation and propagation of Mathieu-Gauss type spatial structures on diffraction of a multi segment-shaped laser beam,” Eur Phys J D 74 (2020).
11. X. X. Chu, Q. Sun, J. Wang, P. Lu, W. K. Xie, and X. J. Xu, “Generating a Bessel-Gaussian beam for the application in optical engineering,” Sci Rep-Uk, 5 (2015).
12. N. Gupta and S. Kumar, “Self-focusing of multi-Gaussian laser beams in nonlinear optical media as a Kepler’s central force problem,” Opt Quant Electron 52 (2020).
13. Y. Chen, Z. X. Fang, Y. X. Ren, L. Gong, and R. D. Lu, “Generation and characterization of a perfect vortex beam with a large topological charge through a digital micromirror device,” Appl. Opt. 54(27), 8030–8035 (2015). [CrossRef]
14. Z. X. Fang, Y. Chen, Y. X. Ren, L. Gong, R. D. Lu, A. Q. Zhang, H. Z. Zhao, and P. Wang, “Interplay between topological phase and self-acceleration in a vortex symmetric Airy beam,” Opt. Express 26(6), 7324–7335 (2018). [CrossRef]
15. K. Kitamura, M. Kitazawa, and S. Noda, “Generation of optical vortex beam by surface-processed photonic-crystal surface-emitting lasers,” Opt. Express 27(2), 1045–1050 (2019). [CrossRef]
16. R. P. Chen, Z. Z. Chen, K. H. Chew, P. G. Li, Z. L. Yu, J. P. Ding, and S. L. He, “Structured caustic vector vortex optical field: manipulating optical angular momentum flux and polarization rotation,” Sci Rep-Uk 5 (2015).
17. Y. F. Jiang, K. K. Huang, and X. H. Lu, “Propagation dynamics of abruptly autofocusing Airy beams with optical vortices,” Opt. Express 20(17), 18579–18584 (2012). [CrossRef]
18. B. Y. Wei, S. Liu, P. Chen, S. X. Qi, Y. Zhang, W. Hu, Y. Q. Lu, and J. L. Zhao, “Vortex Airy beams directly generated via liquid crystal q-Airy-plates,” Appl Phys Lett 112 (2018).
19. W. M. Lee, X. C. Yuan, and W. C. Cheong, “Optical vortex beam shaping by use of highly efficient irregular spiral phase plates for optical micromanipulation,” Opt. Lett. 29(15), 1796–1798 (2004). [CrossRef]
20. M. Xu, C. Y. Sun, Y. C. Zang, J. Zhu, B. X. Xue, and W. Tao, “The feasibility and safety of photoselective vaporization for prostate using a 180-W XPS Greenlight laser in day-surgery pattern in China,” Laser Med Sci (2020).
21. J. D. Ring, J. Lindberg, A. Mourka, M. Mazilu, K. Dholakia, and M. R. Dennis, “Auto-focusing and self-healing of Pearcey beams,” Opt. Express 20(17), 18955–18966 (2012). [CrossRef]
22. X. Y. Chen, D. M. Deng, G. H. Wang, X. B. Yang, and H. Z. Liu, “Abruptly autofocused and rotated circular chirp Pearcey Gaussian vortex beams,” Opt. Lett. 44(4), 955–958 (2019). [CrossRef]
23. X. Y. Chen, D. M. Deng, J. L. Zhuang, X. Peng, D. D. Li, L. P. Zhang, F. Zhao, X. B. Yang, H. Z. Liu, and G. H. Wang, “Focusing properties of circle Pearcey beams,” Opt. Lett. 43(15), 3626–3629 (2018). [CrossRef]
24. C. Sun, D. M. Deng, X. B. Yang, and G. H. Wang, “Propagation dynamics of autofocusing circle Pearcey Gaussian vortex beams in a harmonic potential,” Opt. Express 28(1), 325–333 (2020). [CrossRef]
25. Y. X. Ren, Z. X. Fang, L. Gong, K. Huang, Y. Chen, and R. D. Lu, “Dynamic generation of Ince-Gaussian modes with a digital micromirror device,” J Appl Phys 117 (2015).
26. Y. X. Ren, Z. X. Fang, L. Gong, K. Huang, Y. Chen, and R. D. Lu, “Digital generation and control of Hermite-Gaussian modes with an amplitude digital micromirror device,” J Optics-Uk 17 (2015).
27. D. Deng, C. Chen, X. Zhao, B. Chen, X. Peng, and Y. Zheng, “Virtual source of a Pearcey beam,” Opt. Lett. 39(9), 2703–2706 (2014). [CrossRef]
28. X. Chen, D. Deng, J. Zhuang, X. Peng, D. Li, L. Zhang, F. Zhao, X. Yang, H. Liu, and G. Wang, “Focusing properties of circle Pearcey beams,” Opt. Lett. 43(15), 3626–3629 (2018).
29. Y. Q. Zhang, X. Liu, M. R. Belic, W. P. Zhong, F. Wen, and Y. P. Zhang, “Anharmonic propagation of two-dimensional beams carrying orbital angular momentum in a harmonic potential,” Opt. Lett. 40(16), 3786–3789 (2015). [CrossRef]
30. I. Chremmos, P. Zhang, J. Prakash, N. K. Efremidis, D. N. Christodoulides, and Z. G. Chen, “Fourier-space generation of abruptly autofocusing beams and optical bottle beams,” Opt. Lett. 36(18), 3675–3677 (2011). [CrossRef]
31. X. X. Chu, “Propagation of an Airy beam with a spiral phase,” Opt. Lett. 37(24), 5202–5204 (2012). [CrossRef]
32. K. T. Gahagan and G. A. Swartzlander, “Simultaneous trapping of low-index and high-index microparticles observed with an optical-vortex trap,” J. Opt. Soc. Am. B 16(4), 533–537 (1999). [CrossRef]
33. C. L. Zhao and Y. J. Cai, “Trapping two types of particles using a focused partially coherent elegant Laguerre-Gaussian beam,” Opt. Lett. 36(12), 2251–2253 (2011). [CrossRef]
34. X. Y. Ding, Y. X. Ren, L. Gong, Z. X. Fang, and R. D. Lu, “Microscopic lithography with pixelate diffraction of a digital micro-mirror device for micro-lens fabrication,” Appl. Opt. 53(24), 5307–5311 (2014). [CrossRef]
35. Y. X. Ren, R. D. Lu, and L. Gong, “Tailoring light with a digital micromirror device,” Ann. Phys. (Berlin, Ger.) 527(7-8), 447–470 (2015). [CrossRef]
36. Z. X. Fang, H. Z. Zhao, Y. Chen, R. D. Lu, L. Q. He, and P. Wang, “Accelerating polygon beam with peculiar features (vol 8, 8593, 2018),” Sci Rep-Uk 9 (2019).
37. V. Lerner, D. Shwa, Y. Drori, and N. Katz, “Shaping Laguerre-Gaussian laser modes with binary gratings using a digital micromirror device,” Opt. Lett. 37(23), 4826–4828 (2012). [CrossRef]
38. C. Sun, J. S. H. Lee, and M. Q. Zhang, “Magnetic nanoparticles in MR imaging and drug delivery,” Adv. Drug Delivery Rev. 60(11), 1252–1265 (2008). [CrossRef]
39. C. J. Xu, L. Y. Mu, I. Roes, D. Miranda-Nieves, M. Nahrendorf, J. A. Ankrum, W. A. Zhao, and J. M. Karp, “Nanoparticle-based monitoring of cell therapy,” Nanotechnology 22 (2011).
40. H. Lee, E. Sun, D. Ham, and R. Weissleder, “Chip-NMR biosensor for detection and molecular analysis of cells,” Nat. Med. 14(8), 869–874 (2008). [CrossRef]
41. A. Porfirev and A. Kuchmizhak, “Non-ring perfect optical vortices with p-th order symmetry generated using composite diffractive optical elements,” Applied Physics Letters 113 (2018).
42. D. Naylor, “Recent developments in the measurement of convective heat transfer rates by laser interferometry,” Int. J. Heat Fluid Flow 24(3), 345–355 (2003). [CrossRef]
43. H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express 16(13), 9411–9416 (2008). [CrossRef]
