We propose and generate a class of structured light fulfilling the mathematical form of a SU(2) coherent state based on a set of circular Airy vortex modes. Such wave packets possess strong focus with both radial and angular self-accelerations, which exploit more general 3D inhomogeneous velocity control with global spatial symmetry of multilayer rotation akin to galactic kinematics, termed galaxy waves. Galaxy waves are endowed with higher degrees of freedom to control strong focusing and acceleration, which opens a direction of multi-dimensional accelerating of 3D structured light field, promising numerous applications in optical trapping, manufacturing, and nonlinear optics.

Airy beams have received continuous attention in recent years.^{1} Due to their unique self-accelerating characteristic and curved parabolic geometry,^{2} Airy beams have been widely used in filamentation,^{3,4} particle manipulation,^{5,6} and imaging.^{7} In particular, circular Airy beams (CABs) evolve along the curved track radially and, thus, are also termed auto-focusing Airy beams.^{8} Vortex beams are also concerned closely for their rotating wavefronts and orbital angular momenta (OAM), which benefit a myriad of applications such as particle trapping, optical manufacturing, communications, imaging, and quantum entanglement.^{9–12,29–34} Introducing vortex phase into CABs, circular Airy vortex beams (CAVBs) have both rotating and self-accelerating characteristics for more flexible applications.^{13} Specially, energy of CAVBs converges to the center upon propagation before the focal region, and evolutions of their inner and outer phase distributions are inhomogeneous.^{14,15} Nonetheless, like most self-accelerating structured light,^{1,16–18} the CAVB only shows concentric-circle intensity distribution and lacks general angular rotation and geometry control for extended applications, especially in particle manipulation. To deal with this, the tornado wave is proposed to break the uniformity of angular intensity distribution, which brings rotating angular velocity features to self-accelerating structured light.^{19–22} However, the rotating angular velocity of this wave packet changes only upon propagation in one dimension. Is it possible to create and control accelerating structured light in 3D? In fact, in order to control 3D geometric modes, quantum-analog coherent states with general SU(2) symmetry have been widely used to design complex spatial wave packets with stable patterns and controllable trajectory.^{23–25} Many exotic structured beams such as SU(2) coherent states but using diverse bases have been designed for customized geometric patterns, based on Hermit–Laguerre–Gaussian modes,^{23} Bessel modes,^{26} and Ince–Gaussian modes.^{27} Nevertheless, extant modes designed by this method cannot achieve self-acceleration and lack inhomogeneous angular velocity evolution in 3D space, which limits the extension of applications.

In this Letter, we design a class of self-accelerating structured light whose wave packets have 3D inhomogeneous angular velocity evolution, in terms of the propagation dimension and transverse two dimensions characterizing multilayer rotations of intensity peaks in diverse angular velocities, akin to galactic kinematics, thus termed galaxy waves. The galaxy wave is crafted by CAVB superposition fulfilling the mathematical form of the SU(2) coherent state. SU(2) in mathematics provides rich tunable parameters for elaborate mode control, which enable us to tune both inhomogeneity of angular velocities of layered rotations and global symmetry of galaxy waves. Experimental control of galaxy waves largely increases the tunable dimensions of light shaping and accelerating so that offers more insight in applications such as optical trapping, manufacturing, and nonlinear optics.

First, it is necessary to introduce CAVBs briefly. CAVBs have both self-accelerating characteristics of CABs and rotating phase characteristics of vortex beams, which can be expressed as^{19}

where $Ai(\xb7)$ represents Airy function, $(r,\theta )$ are the polar coordinates in transverse plane, *a* is the exponential truncation factor, *r*_{0} is the radius of the primary Airy ring, *m* is the OAM topological charge, and *R* is a fixed scaling factor. Here, a special parameter $\alpha m,n$ is introduced for greater flexibility in tuning the radial light field distribution, which represents the $(n+1)th$ zero point (without the original point) of the $mth$-order Bessel function. It is noted that the Bessel function introduced here is to create tunable multiple indices to match the mathematical form of SU(2) coherent state well. When this parameter is set to a fixed number which is irrelevant to *m* and *n*, the expression degenerates to the traditional one. CAVBs have intriguing properties, one of which is autofocusing. Specifically, energy flows to the center with evolution so that the intensity of center regions increases sharply within a certain propagation distance. Figure 1(a) clearly shows the *x*–*z* longitudinal section of intensity evolution distribution of the CAVB with $(m,n)=(6,0)$. Moreover, CAVBs also have a peculiar phase distribution, especially inhomogeneous phase evolution of the inner and outer light field regions, as shown in the right part of Fig. 1(a). The two insets show phase distributions of the CAVB with $(m,n)=(6,0)$ at *z*_{0} and *z*_{1}, corresponding to the marked positions on the left intensity figure. Differences between phase evolutions of the inner and outer parts are clearly shown in the below zoom-in panels. The phase evolution is caused by the vortex phase in the CAVBs and would lead to the multilayer inhomogeneous rotation of intensity pattern evolution.

With the introduced parameter $\alpha m,n$, a CAVB can be expanded to a set of CAVBs under the control of *m* and *n*, which provides larger potential for mode superposition. Here, we propose a structured light family, termed as the galaxy waves, which fulfill the mathematical superposition rule of SU(2) coherent state and leverage CAVB modes as base elements. It should be noted that in the previous works, SU(2) coherent state modes are constructed based on eigenmodes,^{23,26,27} but considering that the mathematical form of SU(2) coherent state endows us with rich adjustable parameters, it shows great potential to be extended to non-eigenmodes to create nontrivial superposition beams, so as to realize more flexible mode design and achieve more interesting beam spatial characteristics, as studied in this paper. A galaxy wave is expressed as follows:

where $\varphi $ is the phase factor in the SU(2) coherent state, and $\Phi m0+qk,n0+pk$ represents a CAVB mode with $m=m0+qk$ and $n=n0+pk$ in Eq. (1). *m*_{0} and *n*_{0} are initial mode orders, *p* and *q* are interval numbers, and *N* is related to the number of superposed modes. Here, we take $m0=6,\u2009n0=0$, *p* = 1, *q* = 3, *N* = 5 as an example, which includes six modes. By superposing these CAVBs shown in Fig. 1(b), generated galaxy wave is shown on the right, in which the inner and outer radial bi-layer peak regions follow angular threefold symmetry. The symmetry is decided by the *q* index.

The galaxy wave is experimentally generated based on digital hologram method using a spatial light modulator (SLM, resolution of 1280 × 1024, pixel size of 12.5 *μ*m). The experimental setup is shown in Fig. 2(a). A 532 nm beam from the laser is expanded to 12.8 mm diameter on the SLM and illuminates the loaded phase holograms to carry information of the designed modes, as seen in the zoom-in mask in the red box next to the SLM in Fig. 2(a). The hologram is generated by multiplying the amplitude pattern of the on-demand mode and the phase component together. The phase component is the superposition of the phase distribution of the on-demand mode and the blazed grating. After modulation by the holograms and filtering, the $+1st$-order diffracted beam becomes the galaxy wave and is recorded by a charged-coupled device (CCD, resolution of 1280 × 1024, pixel size of 5.3 *μ*m) at different propagation distance *z*. (The focal plane of L2 is seen as the plane with *z* = 0 mm.)

Experimental intensity distributions from *z* = 0 to 128 mm are shown in Fig. 2(b), which match well with the simulated results below. It can be seen that peak regions of the inner and outer light fields rotate in different angular velocities upon propagation, which can be credited to the inhomogeneous phase distribution evolution of CAVBs. Intensity peak regions are extracted as shown in Fig. 3(a) to clearly showcase the self-accelerating evolution and rotation features upon propagation. In Fig. 3(a), energy of light field flows to the center twice when propagation distance *z* increases, resulting in intensity bi-peak distribution upon propagation. The calculated Poynting vector is further shown in Visualization 1 with the range $z\u22650$, and intensity evolution of the other side of *z* = 0 is symmetrical to this side. In addition, from Fig. 3(a), inner peaks rotate faster than outer peaks, and the galaxy wave is with inhomogeneous angular velocities at different longitudinal distances. We demonstrate this property by analyzing specific angular variations of the inner and outer peak regions with the propagation distance shown in Fig. 3(b), where the angles of inner peak regions vary between 0 and 0.919 rad seen from the blue line and the angles of outer peak regions vary between 0 and 0.2857 rad seen from the red line in the simulation. The angular variation range of inner peak regions is much larger than that of the outer. Corresponding angular positions in experiment at different propagation distance *z* are shown as error bars around simulated lines evaluating the errors to find the average angular positions of peak regions. Those blue and red error bars represent angular positions of 0.0002% maximum intensity regions of the inner and outer layers, respectively, at different *z*. Figure 3(c) shows corresponding angular rotating velocities of the inner and outer peak regions upon propagation, respectively, as blue and red simulated lines, and experimental results are shown as error bars around simulated lines evaluating the errors to find average angular velocities of peaks. The maximum angular velocity of inner peak regions is 23.47 rad/m and that of outer peak regions is nearly 3 rad/m. From the large difference of maximum angular velocities, we can conclude that galaxy waves provide a great tunable range for inhomogeneity of bi-layer angular velocities which varies from 0.61 to 21.66 rad/m.

Moreover, galaxy waves have rich tunable parameters for mode control. With different initial index *m*_{0} or parameter related to the number of superimposed CAVBs *N*, the rotating characteristics of the inner and outer intensity peak regions are tunable. When $m0=6$ and *N* is tuned from 3 to 6, angular variations of the inner and outer peak regions upon propagation are shown in Fig. 4(a) as blue lines and red lines in different linestyles, respectively. Corresponding slopes *k _{i}*, $i=1,2,\u2026,8$, which represent the angular rotating velocities of peak regions, are shown in the inset. It is noticed that with the increase in

*N*, the angular variation ranges expand and angular rotating velocities increase. Moreover, when

*N*= 5 and

*m*

_{0}is tuned from 4 to 7, angular variations of the inner and outer peak regions are shown in Fig. 4(b) as blue and red lines in different linestyles, respectively. Corresponding angular rotating velocities are also shown as slopes

*k*, $i=1,2,\u2026,8$ in the inset. It is found that with the increase in

_{i}*m*

_{0}, the angular variation ranges also expand and angular rotating velocities increase, but not that much as the case while

*N*increases. Related light fields at

*z*= 148 mm are shown in Figs. 4(c) and 4(d) to indicate the evolution differences caused by

*N*and

*m*

_{0}, respectively. Therefore, angular velocities of the inner and outer peak regions can be tunable with the designed indices, which increases the flexibility of modes in applications.

In addition to the dynamic characteristics on the angular variation of peak regions, global symmetry of the light field distribution can be well controlled by parameter *q*. For example, intensity and phase distributions of the galaxy wave with *q* = 4 at $z=0,20,80,140$ mm are shown in Figs. 5(a) and 5(b), respectively, and modes with *q* = 5 are shown in Figs. 5(c) and 5(d), respectively. (More detailed evolutions of the intensity distributions with parameter *q* = 4, 5 are shown in Visualization 2, 3, respectively.)

Based on the peculiar distribution evolution of peak regions, the intensity gradient of galaxy waves can induce nontrivial gradient force distribution. To demonstrate this claim, we here simulate the gradient force and potential well distribution at different propagation distances with a Rayleigh particle example. Note that for simplicity, we normalize the gradient force and potential well distribution and focus on their relative changes upon propagation in Fig. 6. Particles with 0.01 *μ*m radius and 1.59 refractive index are captured by the galaxy wave (*q* = 3) in a liquid with 1.33 refractive index. Particles are trapped at the bottom of potential wells with the combined effect of scattering force *F _{s}* and gradient force

*F*, respectively, shown as follows:

_{g}^{28}

where *n _{m}* is the refractive index of the liquid,

*I*

_{0}is the light intensity,

*c*is the speed of light, and

*σ*is the scattering cross section. $\sigma =8/3\pi (ka\u2032)4a\u20322(m2\u22121m2+2)2$ and $\beta =\pi \epsilon 0nm2m2\u22121m2+2a\u20323$, where $k=2\pi /\lambda $ (

*λ*is the wavelength), $a\u2032$ is the radius of particles,

*m*is the refractive index ratio of the particle and the liquid, and

*ε*

_{0}is the permittivity of vacuum.

Scattering force and gradient force are proportional to $a\u20326$ and $a\u20323$, respectively. Thus, gradient force can be designed to be much larger than scattering force by choosing the appropriate radius of particles. Transverse gradient force distribution at *z* = 0 is shown in Fig. 6(a), based on Eq. (3). White arrows point to the inner and outer peak regions, and their lengths represent the relative magnitude of the force. Figure 6(b) shows the corresponding potential well of this galaxy wave in the simulation, whose deep hollows can trap particles. Transverse gradient force distribution at *z* = 18 mm is shown in Fig. 6(c), and the corresponding potential well is shown in Fig. 6(d). The angular variation of the deep hollows reflects the ability of manipulating particles rotating in the transverse dimensions, and the depth variation of the potential well hollows shows the particle manipulative ability along the propagation dimension. Comparing Figs. 6(b) and 6(d), the galaxy wave shows great application potential in particle manipulation in 3D. (The entire evolution is shown in Visualization 4.)

In conclusion, we design a nontrivial self-accelerating structured light named galaxy wave, which is angularly symmetric and has radially multilayer intensity peak regions with inhomogeneous rotation angular velocities. Moreover, the peak intensity has a bi-peak distribution upon propagation. The rotating angular velocities of the inner and outer intensity peak regions and the mode symmetry can be tuned by adjusting superposition components elaborately. The galaxy wave provides peculiar potential wells for particle manipulation in 3D, due to the transverse inhomogeneous rotating distribution and the longitudinal propagating evolution. For the brilliant flexibility, galaxy waves show huge application potential, especially in optical trapping, manufacturing, and nonlinear optics.

See the supplementary material for Visualizations 1, 2, 3, 4.

This work was funded by the National Natural Science Foundation of China (Grant No. 61975087).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Jing Pan:** Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **Hao Wang:** Investigation (supporting); Visualization (supporting); Writing – original draft (supporting); Writing – review & editing (equal). **Yijie Shen:** Conceptualization (supporting); Investigation (supporting); Methodology (supporting); Visualization (supporting); Writing – original draft (equal); Writing – review & editing (equal). **Xing Fu:** Writing – original draft (supporting); Writing – review & editing (equal). **Qiang Liu:** Funding acquisition (lead); Project administration (lead); Resources (lead); Writing – review & editing (lead).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

## References

_{4}lasers with external mode converters

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