Open Access
Add to BibSonomy
Abstract
We investigate the propagation dynamics of elliptic Gaussian breathers without and with orbital angular momentum (OAM) in thermal nonlocal nonlinear media with both linear and nonlinear isotropy. The beamwidths and intensity of the elliptic breather without OAM, and the beamwidths and rotation angle of the elliptic breather with the critical OAM are analytically discussed in details. In experiment, the OAM-free elliptic beam is observed to oscillate and cannot form an elliptic soliton for any input power. By contrast, the elliptic beam carrying the critical OAM can keep its ellipticity changeless during the propagation and evolve into a spiralling elliptic soliton at the critical power. Specifically, in the vicinity of the critical power, the rotation angle of the spiralling elliptic beam can be controlled linearly by the input power and exceed π/2, while its beamwidths and ellipticity remain unchanged. Such a power-controllable, profile-preserving spiralling elliptic beam may provide potential applications in all-optical manipulation.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Many research efforts have been devoted to the investigation of spatial optical solitons, which can be formed when the self-focusing resulting from the nonlinearity balances the natural diffraction-induced expansion of optical beams [1]. For optical media with both linear and nonlinear isotropy (BLANI), the linear and nonlinear refractive indices along different directions in the transverse possess the same strength. In such media, it is generally believed that only the spatial solitons with cylindrical symmetry can exist, whereas the optical beams with elliptic cross section will undergo periodic oscillations and cannot form elliptic solitons [2–4]. Mighty morphing soliton, whose ellipticity changes with propagation, was theoretically proposed [3] and experimentally corroborated [4] in isotropic saturable nonlinear media. To obtain coherent elliptic solitons, linear anisotropy [5], nonlinear anisotropy [6,7], or both [8] should be introduced, while incoherent elliptic solitons came true by means of the anisotropy of correlation function [9]. In 2010, the orbital angular momentum (OAM) was proposed to stabilize the elliptic solitary wave and make the spiralling elliptic soliton exist in isotropic saturable media [10]. Later, Liang et al. extended this spiralling elliptic soliton to nonlocal nonlinear media without anisotropy [11,12]. They predicted in theory [13] and corroborated in experiment [14] that the OAM-induced anisotropic diffraction makes the elliptic mode exist in isotropic media, where only the circular eigenmode is supposed to exist for a beam without OAM. More generally, when the input power deviates from the critical power, the spiralling elliptic beam will oscillate periodically and evolve as a spiralling elliptic breather, which case will be discussed in this paper.
The controllable rotation of spiralling optical beams have been considered in a variety of applications, such as the all-optical manipulation [15], the excitation of Bose-Einstein condensates [16], and the rotating Doppler effect [17]. Such spiralling beams can form when the OAM involves and the energy flow within the cross section is inhomogeneous. In linear propagation case, Chen et al. introduced a type of shape-invariant rotating beam in linear media with harmonic potentials, revealing that the transverse position of the output beam can be steered by tuning the input parameters [18]. Liang et al. found that the rotating velocity of spiraling elliptic beams can be controlled by both initial OAM and the anisotropy parameter of linear media [19]. The steerable rotation of spatial solitons in nonlinear media exhibits richer features. Rotating solitons possessing complex optical structures, like dipole [20], propeller [21] and azimuthons [22], have been demonstrated in photorefractive, homogeneous, and dissipative media, respectively. Nonlocal nonlinearity can effectively suppress the modulational instability, and subsequently support the propagation of spiralling multi- vortex solitons [23] and revolving optical patterns [24], et al. Off-axis winding beams were investigated in nonlocal nonlinear media, whose trajectories and rotations can be controlled by the optical power and the OAM [25]. Liang et al. discovered that the spiralling elliptic solitons can exist in longitudinally-inhomogeneous nonlocal media with characteristic-length- dependent rotating velocity [26]. The rotation of these spiralling solitons are usually driven by the OAM, and controlled by input beam parameters or propagation-varied parameters of nonlinear media.
In this paper, we analytically and experimentally investigate the propagation properties of the elliptic Gaussian breathers in lead glass, one of the most popular nonlocal nonlinear media. We discuss the periodic oscillation of the beamwidths and intensity of the OAM-free elliptic breather, as well as the beamwidths and rotation angle of the OAM-carrying spiralling elliptic breather. Experimentally, the oscillating dynamics of elliptic breathers and the rotating dynamics of spiralling elliptic breathers are observed at a fixed propagation distance. Interestingly, the rotation of the output spiralling beam can be controlled linearly by the input power in the neighborhood of the critical power, while its beamwidths and beamwidths keep invariant at the meantime.
2. Analytical solution of elliptic breathers
The propagation medium we adopt is cylindrical lead glass that exhibits thermal nonlocal nonlinearity. The lead glass is itself linearly isotropic [6]. On the other hand, its cylindrical boundary guarantees nonlinear isotropy [27] because the far-away boundary conditions significantly affect the nonlinear refractive index under strong nonlocality [6]. The nonlinear propagation behavior of the optical beam in lead glass can be described by the following dimensionless equations [6,14]
Following Snyder’s method [28], which is applied to strongly nonlocal condition, the nonlinear index is expanded in Taylor’s series and only the first two nonzero terms are kept as $G = G{|_{\xi = \eta = 0}} + \gamma ({{\xi^2} + {\eta^2}} )$. The parameter $\gamma $ can be calculated from Eq. (2) as ${\; }\gamma \approx ({1/4} ){\; }({\partial_{\xi \xi }^2 + \partial_{\eta \eta }^2} )G{|_{\xi = \eta = 0}} \approx{-} p/({4\pi {w_{0\xi }}{w_{0\eta }}} )$. Then the coupled Eqs. (1) and (2) can be decoupled to a single evolutional equation, i.e., the well-known Snyder-Mitchell model
2.1 Elliptic breathers without the OAM
To investigate the evolution of the elliptic beam without OAM in nonlocal nonlinear media with BLANI, we set Θ = 0 in Eq. (3), then the expression of the light amplitude at any propagation distance can be obtained
The ellipticity function $\rho (\zeta )= {w_\eta }/{w_\xi }$ is defined as the ratio of the two semi-axes, which is expressed as
Figure 1 demonstrates the evolution of the two semi-axes and the ellipticity for the OAM-free elliptic breather with an initial ellipticity ρ0 = 2.0. We first consider the periodic oscillation of the two semi-axes. Their oscillating period can be obtained from Eqs. (8) and (9) as $T = \pi \sqrt {2\pi {\rho _0}/p} $, which is a decreasing function of the input power p. When $p = \pi /4 = {\; }{p_{c\eta }}$, as shown in Fig. 1(a), ${w_\eta }{\; }$ keeps unchanged during the propagation, while ${w_\xi }{\; }$ first increases and then decreases. On the contrary, when $p = {\; }4\pi = {p_{c\xi }}$, as shown in Fig. 1(c), ${w_\xi }$ keeps unchanged during the propagation, while ${w_\eta }$ first decreases and then increases. Between these two input powers, we can obtain a breather-breather pair [Fig. 1(b)], where ${w_\xi }$ initially increases and then decreases, whereas ${w_\eta }$ initially decreases and then increases. The ratio of ${w_\xi }$ to ${w_{0\xi }}$ vibrates between 1 and $\sqrt {{p_{c\xi }}/p} $, while the ratio of ${w_\eta }$ to ${w_{0\eta }}$ vibrates between 1 and $\sqrt {{p_{c\eta }}/p} $. It can be concluded that the two semi-axes cannot keep invariant simultaneously for any input power, and consequently the OAM-free elliptic breather cannot evolve into an elliptic soliton. From Eq. (11) we can deduce that the ellipticity will vibrate periodically between ${\rho _{min}} = 1/{\rho _0}$ and ${\rho _{max}} = {\rho _0}$ and also cannot keep unchanged for any input power.
Fig. 1. Periodic oscillations of the two semi-axes and the ellipticity for OAM-free elliptic breather with an initial ellipticity ${\rho _0} = 2.0$. (a) $p = \pi /4 = {p_{c\xi }}/16 = {\; }{p_{c\eta }};$ (b) $p = \pi = {p_{c\xi }}/4 = {\; }4{p_{c\eta }};$ (c) $p = 4\pi = {p_{c\xi }} = {\; }16{p_{c\eta }}$. The vertical dashed lines denote the experimental observation distance $\zeta = 4.5$.
And then, we investigate the evolution of the on-axis intensity for the elliptic breather. Obviously, the oscillating period of the on-axis intensity is the same as that of beamwidths and ellipticity, and will decrease with the growth of the input power. Since the optical power is a conservative quantity, we can easily obtain $p = {I_0}\pi {w_{0\xi }}{w_{0\eta }} = I(\zeta )\pi {w_\xi }{w_\eta }$, where I0 is the input on-axis intensity and $I(\zeta )= A_0^2$ is the on-axis intensity. Thus, the normalized intensity $I/{I_0} = ({{w_{0\xi }}/{w_\xi }} )\cdot ({{w_{0\eta }}/{w_\eta }} )$, and vibrates between 1 and $p/\pi $, as shown in the example for p = π/2 and p = 2π in Fig. 2.
Fig. 2. The normalized intensity I / I0 of the OAM-free elliptic breather with an initial ellipticity ${\rho _0} = 2.0$.
2.2 Spiralling elliptic breathers with the critical OAM
As discussed above, the evolution of the elliptic beam in isotropic media is not as fundamental eigenmode. However, when the elliptic beam carries the critical cross phase coefficient and the corresponding critical OAM, respectively reading as [12,14]
it can spiral on the cross section perpendicular to the propagation direction and keep an invariant ellipticity in linearly isotropic media [14]. When the isotropic nonlinearity is introduced, a spiralling elliptic soliton can be formed at the critical power. For the input beam carrying the critical OAM, the light amplitude of the spiralling elliptic beam propagating in the system described by Eq. (5) is found to be [12]The physical quantities of ${A_{NL}}$, ${w_\xi }$, $\; {w_\eta }$, $\; {w_{\xi \eta }}$ are given as
The inclined ellipse, expressed by Eq. (14), can be transformed to its standard elliptic form by a coordinate rotation through the angle
The spiralling elliptic soliton is formed at both the critical OAM given by Eq. (13) and the critical power given by
Figure 3 presents the oscillating behavior of the spiralling elliptic breather with the initial ellipticity ${\rho _0} = 2.0.$ Obviously, the oscillating frequency of the spiralling beam increases with the growth of the normalized input power $p/{p_c}$, where ${p_c} = 25\pi /16$. Since the spiralling beam carries the critical OAM, the ratio of major semi-axis to minor semi-axis remains constant during propagation. The two beamwidths ${w_{maj}}/{w_{0\eta }}$ and ${w_{min}}/{w_{0\eta }}$ vibrate between 1 and $\sqrt {{p_c}/p} $. When the input power is less than the critical power, e.g. $p/{p_c} = 0.5{\; }$ in Fig. 3(a), the beamwidths first increase and then decrease, oscillating synchronously. The minor semiaxis vibrates between 1.0 and $\sqrt 2 $, while the major one vibrates between 2.0 and $2\sqrt 2 $. When the input power is larger than the critical power, e.g., $p/{p_c} = 1.5$ in Fig. 3(b), the beamwidths first decrease and then increase, oscillating synchronously. The minor semiaxis vibrates between $1/\sqrt {1.5} $ and 1.0, while the major one vibrates between $2/\sqrt {1.5} $ and 2.0. Figure 4 presents the rotation angles of spiralling elliptic breathers for different input powers. The rotation angle is a monotonic increasing function of the propagation distance When observed at a fixed distance of $\zeta = 4.0$ (denoted by the vertical dashed line), the rotation angle will increase as the input power increases, which case is confirmed in the following experiment.
Fig. 3. The major-axis ${w_{maj}}$ (blue curve) and minor-axis ${w_{min}}$ (red curve) for the spiralling elliptic breather with ${\rho _0} = 2.0$ at the critical OAM. The vertical dashed line denotes the experimental observation distance $\mathrm{\zeta } = 4.0$.
Fig. 4. The rotation angle for the spiralling elliptic breather with ${\rho _0} = 2.0$ at the critical OAM and different input powers. The vertical dashed line denotes the experimental observation distance $\mathrm{\zeta } = 4.0$.
3. Experimental observation of elliptic breathers
To confirm the aforementioned oscillating property of elliptic breathers without and with OAM, we performed the experiments in cylindrical sample of lead glass. The experimental setup is schematically illustrated in Fig. 5. A linearly polarized Gaussian beam is expanded and collimated by the telescope composed of F1 and F2, and transformed into an elliptic Gaussian beam by using a pair of cylindrical lenses CL1,2. The elliptic beam illuminates on a phase-only spatial light modulator (SLM, Holoeye, PLUTO), modulated with a transverse cross phase. The elliptic beam reflected from the SLM is focused on the front face of the lead glass sample by focusing lens F3. The input beam at the front face of the sample can be expressed as
Fig. 5. Experimental setup. Source, Ti:sapphire solid-state CW laser ($\lambda = 532$nm); F1-F3, 50 mm,100 mm and 200 mm focusing lenses, respectively; CL1, CL2, cylindrical lenses; SLM, spatial light modulator; M, mirror; LG, lead glass, which is thermally contacted by a copper-made heat sink; MO, 5 ${\times} $ microscope objective; CCD, charge-coupled device camera. The insert in the upper right shows the saddle-shaped wavefront of the elliptic beam.
3.1 Oscillating dynamics of elliptic breathers
Figure 6 demonstrates the output intensity distributions [Figs. 6(b)–6(f)], the normalized beamwidths and ellipticity [Fig. 6(g)] of the OAM-free elliptic beam with ρ0 = 2.0 after propagating the distance $\; \zeta = L/kw_{0x}^2 = 4.5$. The abscissa axes in Fig. 6(g) for theoretical and experimental results are normalized by their respective critical powers, i.e., ${p_{c\xi }} = 4\pi $ and ${P_{cx}} = 600$ mW. The two semi-axes and ellipticity in Fig. 1 oscillate periodically with the propagation distance. However, when they are observed at a fixed distance of ζ = 4.5, their oscillations with input power are not periodic in this case, as shown in Fig. 6(g). As the input power increases, the intensity profiles, the output semi-axes, and especially the ellipticity will undergo aperiodic oscillation. It can be concluded that in nonlocal nonlinear media with BLANI, the uniform focusing strength cannot balance the nonuniform expanding strength of an asymmetric beam. Thus, the elliptic beam appears to change its shape constantly and cannot evolve into an elliptic soliton.
Fig. 6. (a) The input beam with initial ellipticity ${\rho _0} = 2.0$ and minor-axis ${w_{0x}} = 24.0\; {\mathrm{\mu} \mathrm{m}}$. (b)-(f) The output elliptic beam at ζ = 4.5 for different input powers given in each of the figures. (g) The output semi-axes and ellipticity as functions of the normalized input power. Solid curves for theoretical semi-axes and ellipticity obtained from Eqs. (8), (9) and (11), and dots for experimental results.
3.2 Rotating dynamics of spiralling elliptic breathers
To obtain elliptic soliton in nonlocal media with BLANI, the input beam should carry the critical OAM, which process can be achieved by loading a phase mask on the SLM in our experiment. Figure 7 demonstrates the intensity distributions of the input beam with ρ0 = 2.0 [Fig. 7(a)] and the output beams for different input powers [Figs. 7(b)–7(h)] after propagating the distance $\zeta = L/kw_{0x}^2 = 4.0$. At relatively low power of P = 1 mW, the output beam diffracts without appreciable nonlinear self-action, and rotates anti-clockwise due to its carrying OAM. As the input power increases, the elliptic beam undergoes stronger and isotropic self-focusing effect. The oblique elliptic spot shrinks visibly and rotates at the meantime, giving rise to a spiralling elliptic soliton with wmin = 25.0 μm at the critical power of Pc = 220 mW [Fig. 7(e)], as compared with the theoretically predicted value of Pc = 215 mW. A further increase of power leads to a monotonous increase of the rotation angle until the final collapse of the elliptic beam. Due to the conservation of angular momentum, the rotation speed is accelerated in the process of beam shrinking. It is worth noting that both the spiralling breathers and the spiralling soliton preserve their invariant ellipticity during this process.
Fig. 7. (a) The input beam with ${\rho _0} = 2.0$ and ${w_{0x}} = 25.3\; \mu \textrm{m}$. (b)-(h) The output elliptic beams at ζ = 4.0 for different input powers given in each of the figures. The arrow indicates the rotating direction, while the white dot marks the major semi-axis of the spiralling elliptic beam.
Figure 8 quantitatively depicts the output beamwidths and rotation angle of the spiralling beam under different input powers ranging from 1 mW to 270 mW. Solid curves in Figs. 8(a) and 8(b) respectively represent the theoretical results obtained from Eqs. (20) and (19), whereas straight line in Fig. 8(b) represents the linear fitting based on the least square method. The abscissa axes for theoretical and experimental results are normalized by their respective critical powers. The theoretically predicted oscillation of the beamwidths and rotation angle varying with the propagation distance in Figs. 3 and 4 is proved by observing the output beamwidths and rotation angle varying with the input power at a fixed distance of ζ = 4.0, as shown in Figs. 8(a) and 8(b), respectively. Generally speaking, at certain propagation distances, the curve of output beamwidth varying with input power presents multi-peak structure. While a small ‘platform’ usually appears around the critical power, especially when the propagation distance is within a few Rayleigh distances. For the present case of ζ = 4.0, the output beamwidth decrease monotonically with the increase of power within a relatively low range. However, in the vicinity of the critical power ${P_c}$= 220 mW, specifically within 0.73 ${P_c}\sim $ 1.23 ${P_c}$ the two beamwidths basically remain unchanged, showing a ‘platform’. On the other hand, within the same power interval, i.e., 160 mW${\sim} $ 270 mW in Fig. 8(b), the rotation of the elliptic beam is accelerated and its rotation angle increases approximately linearly with the input power. Within the power interval of ΔP = 110 mW, the elliptic beam rotates by 95°, and the change rate of rotation angle to the power is 0.9 per milliwatt.
Fig. 8. The semi-axes and ellipticity (a) and the rotation angle (b) of the spiralling elliptic beam with ρ0 = 2.0 as functions of the normalized input power. Solid curves for theoretical semi-axes and rotation angle obtained from Eqs. (20) and (19), and dots for corresponding experimental results. The straight line in (b) is the linear fitting based on the least square method.
To further confirm the propagation properties of the spiralling breather, we observed an analogous behavior for the input beam with another initial ellipticity ρ0 = 1.5. The output beam profiles propagating through the distance ζ = 3.8 are presented in Fig. 9. Here we also set ${w_{0y}} > {w_{0x}}$, but take the opposite value of $\mathrm{\Omega }$, and correspondingly, the opposite value of the critical OAM. As expected, the output beam rotates clockwise on this occasion and remains its ellipticity in the process of increasing power. The spiralling elliptic soliton is formed at the critical power of Pc = 210 mW, as compared with the theoretically predicted value of Pc = 204 mW. As seen in Figs. 10(a) and 10(b), within the power range from 160 mW to 270 mW, the beamwidths basically remains unchanged, whereas the rotation angle increases linearly with the input power. Within the power interval of ΔP = 110 mW, the elliptic beam rotates by 85°, and the change rate of rotation angle is 0.8 for per milliwatt added. In particular, ascribing to the power-controllable rotation, the total rotation angle can exceed 200° after a nonlinear propagation over 3.8 Rayleigh lengths, which vastly exceeds the rotating limit of π/2 in the linear propagation [13,14]. Thus, we can conclude that within the power range of 110 mW near the two critical powers, we realized a power-linearly controlled rotation for profile-preserving elliptic beams, with the change rate of close to 1° for per milliwatt.
Fig. 9. (a) The input beam with ${\rho _0} = 1.5$ and ${w_{0x}} = 26.0\; \mu \textrm{m}$. (b)-(h) The output elliptic beams at $\zeta = 3.8$ for different input powers given in each of the figures.
Fig. 10. The semi-axes and ellipticity (a) and the rotation angle (b) of the spiralling elliptic beam with ${\rho _0} = 1.5$. as functions of the normalized input power. Solid curves for theoretical semi-axes and rotation angle obtained from Eqs. (20) and (19), and dots for corresponding experimental results. The straight line in (b) is the linear fitting based on the least square method.
4. Conclusion
We analytically and experimentally investigated the dynamical properties of the elliptic beam without and with OAM in cylindrical lead glass that exhibits thermally nonlocal nonlinearity. In such a nonlinear medium with BLANI, the beamwidth and intensity of the OAM-free elliptic beam will undergo periodic oscillation and cannot form the elliptic soliton. By contrast, when the elliptic beam carries the critical OAM, it will keep its ellipticity unchanged on the propagation and evolve into a spiralling elliptic soliton at the critical power. Experimentally, the oscillatory behaviors of the elliptic beams without and with OAM are verified by observing the output beam images propagating through lead glass sample. The power-dependent rotation angle of the OAM-carrying spiralling beam can exceed $\pi /2$, which is the upper limit value in the linear propagation. More meaningfully, the rotation can be controlled linearly by the input power in the neighborhood of the critical power, while the ellipticity and beamwidth of the spiralling beam remain unchanged. We expect that such a power-controllable, profile-preserving spiralling elliptic beam can be used as in optical spanners to operate micro- or nanoparticles.
Funding
Natural Science Foundation of Guangdong Province of China (2021A1515012214); Science and Technology Program of Guangzhou (2019050001); Scientific Research and Developed Fund of Zhejiang A&F University (2021LFR014, 2021FR0009).
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.
References
1. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals, (Academic Press: New York, 2003).
2. B. Crosignani and P. Di Porto, “Nonlinear propagation in Kerr media of beams with unequal transverse widths,” Opt. Lett. 18(17), 1394–1396 (1993). [CrossRef]
3. A. W. Snyder and J. D. Mitchell, “Mighty morphing spatial solitons and bullets,” Opt. Lett. 22(1), 16–18 (1997). [CrossRef]
4. V. Tikhonenko, “Observation of mighty morphing spatial solitons,” Opt. Lett. 23(8), 594–596 (1998). [CrossRef]
5. C. Conti, M. Peccianti, and G. Assanto, “Spatial solitons and modulational instability in the presence of large birefringence: The case of highly nonlocal liquid crystals,” Phy. Rev. E 72(6), 066614 (2005). [CrossRef]
6. C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95(21), 213904 (2005). [CrossRef]
7. P. Zhang, J. Zhao, F. Xiao, C. Lou, J. Xu, and Z. Chen, “Elliptical discrete solitons supported by enhanced photorefractive anisotropy,” Opt. Express 16(6), 3865–3870 (2008). [CrossRef]
8. S. V. Polyakov and G. I. Stegeman, “Existence and properties of quadratic solitons in anisotropic media: Variational approach,” Phy Rev. E 66(4), 046622 (2002). [CrossRef]
9. O. Katz, T. Carmon, T. Schwartz, and M. Segev, “Observation of elliptic incoherent spatial solitons,” Opt. Lett. 29(11), 1248–1250 (2004). [CrossRef]
10. A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Suppression of collapse for spiraling elliptic solitons,” Phys. Rev. Lett. 104(5), 053902 (2010). [CrossRef]
11. G. Liang and Q. Guo, “Spiraling elliptic solitons in nonlocal nonlinear media without anisotropy,” Phys. Rev. A 88(4), 043825 (2013). [CrossRef]
12. G. Liang, Q. Guo, W. Cheng, N. Yin, P. Wu, and H. Cao, “Spiraling elliptic beam in nonlocalnonlinear media,” Opt. Express 23(19), 24612–24625 (2015). [CrossRef]
13. G. Liang, Y. Wang, Q. Guo, and H. Zhang, “Anisotropic diffraction induced by orbital angular momentum during propagations of optical beams,” Opt. Express 26(7), 8084–8094 (2018). [CrossRef]
14. G. Liang, H. Zhang, L. Fang, Q. Shou, W. Hu, and Q. Guo, “Influence of transverse crossphases on propagations of optical beams in linear and nonlinear regimes,” Laser Photon. Rev. 14(12), 2000141 (2020). [CrossRef]
15. A. Packer, L. Russell, H. Dalgleish, and M. Häusser, “Simultaneous all-optical manipulation and recording of neural circuit activity with cellular resolution in vivo,” Nat. Methods 12(2), 140–146 (2015). [CrossRef]
16. H. Takeuchi, “Quantum elliptic vortex in a nematic-spin Bose-Einstein condensate,” Phys. Rev.Lett. 126(19), 195302 (2021). [CrossRef]
17. L. Fang, Z. Wan, A. Forbes, and J. Wang, “Vectorial Doppler metrology,” Nat. Commun. 12(1), 4186 (2021). [CrossRef]
18. J. Chen, F. Zhang, K. Bian, C. Jiang, W. Hu, and D. Lu, “Dynamics of shape-invariant rotating beams in linear media with harmonic potentials,” Phys. Rev. A 99(3), 033808 (2019). [CrossRef]
19. G. Liang, T. Jia, and Z. Ren, “Controllable rotations of spiraling elliptic beams in anisotropic linear media,” IEEE Photonics J. 9(5), 6101608 (2017). [CrossRef]
20. T. Carmon, R. Uzdin, C. Pigier, Z. H. Musslimani, M. Segev, and A. Nepomnyashchy, “Rotating propeller solitons,” Phys. Rev. Lett. 87(14), 143901 (2001). [CrossRef]
21. P. Zhang, S. Huang, Y. Hu, D. Hernandez, and Z. Chen, “Generation and nonlinear selftrapping of optical propelling beams,” Opt. Lett. 35(18), 3129–3131 (2010). [CrossRef]
22. C. Ruiz-Jimenez, H. Leblond, M. A. Porras, and B. A. Malomed, “Rotating azimuthons in dissipative Kerr media excited by superpositions of Bessel beams,” Phys. Rev. A 102(6), 063502 (2020). [CrossRef]
23. D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Spiraling multivortex solitons in nonlocal nonlinear media,” Opt. Lett. 33(2), 198–200 (2008). [CrossRef]
24. G. Liang, “Revolving and spinning of optical patterns by two coaxial spiraling elliptic beams in nonlocal nonlinear media,” Opt. Express 27(10), 14667 (2019). [CrossRef]
25. G. Liang and Q. Wang, “Trajectories and rotations controlled off-axis winding beams in nonlocal nonlinear media,” Opt. Express 27(15), 21185–21193 (2019). [CrossRef]
26. G. Liang and Q. Wang, “Rotation controlling of spiraling elliptic beams in inhomogeneous nonlocal media,” New J. Phys. 23(10), 103036 (2021). [CrossRef]
27. Q. Shou, X. Zhang, W. Hu, and Q. Guo, “Large phase shift of spatial solitons in lead glass,” Opt. Lett. 36(21), 4194–4196 (2011). [CrossRef]
28. A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science 276(5318), 1538–1541 (1997). [CrossRef]
29. H. Zhang, M. Chen, L. Yang, B. Tian, C. Chen, Q. Guo, Q. Shou, and W. Hu, “Higher-charge vortex solitons and vector vortex solitons in strongly nonlocal media,” Opt. Lett. 44(12), 3098–3101 (2019). [CrossRef]
