Structured ray-wave vector vortex beams in multiple degrees of freedom from a laser

Structured ray-wave vector vortex beams in multiple degrees of freedom from a laser Yijie Shen,1,2,3,* Xilin Yang,4 Darryl Naidoo,5,6 Xing Fu,1,2 AND Andrew Forbes6 Key Laboratory of Photonic Control Technology (TsinghuaUniversity), Ministry of Education, Beijing 100084, China State Key Laboratory of PrecisionMeasurement Technology and Instruments, Department of Precision Instrument, TsinghuaUniversity, Beijing 100084, China Optoelectronics Research Centre &Centre for PhotonicMetamaterials, University of Southampton, Southampton SO17 1BJ, UK School of Optics and Photonics, Beijing Institute of Technology, Beijing 100081, China CSIRNational Laser Centre, POBox 395, Pretoria 0001, South Africa School of Physics, University of theWitwatersrand, Private Bag 3,Wits 2050, South Africa *Corresponding author: Y.Shen@soton.ac.uk


INTRODUCTION
Vector beams (VBs) with spatially nonseparable polarization have attracted great interest and have recently hatched a myriad of applications. In particular, cylindrical VBs [1] (e.g., radial or azimuthal polarization structures) have been widely applied in optical trapping [2], material processing [3], and communication [4]. A popular method for their creation is to exploit spin-orbit coupling to form the vector vortex beams (VVBs) carrying spin angular momentum (SAM) and orbital angular momentum (OAM), finding many applications [5,6]. Moreover, the spacepolarization nonseparable state in VVBs has exotic quantum-like properties [7][8][9][10][11][12], which has also developed fruitful applications. Using the nonseparable states of VVBs mimicking quantum states, quantum channels can be characterized by classical light [13], which can further realize optical communication with improved capacity and stability [14][15][16]. The nonseparable state in the interaction between complex VVBs with matter was studied for developing high-speed kinematic sensing and metrology [17][18][19]. Through generating, characterizing, and measuring more elaborate nonseparable structures in VVBs, in situ detection of a cooperative target was recently realized [20]. To date, there were still rich treasures for exploring more applications in VVBs with this quantum-classical connection.
Therefore, the creation of increasingly structured VBs and VVBs is not only a fundamental scientific endeavor in itself, but is also enabling the way for exciting novel applications. To this end, new structured VBs and VVBs have been designed and tailored, mostly restricted to the higher-order Poincaré sphere [21,22] and hybrid-order Poincaré sphere [23,24], and produced directly from lasers with exotic designs and customized optical elements [25], but more commonly external to the source [6]. These include entanglement-beating VVBs superposed by two general Laguerre-Gaussian (LG) modes with opposite propagation direction to produce an axially varying vector field [26], an SU(2) structured VB as the orthogonally polarized SU(2) coherent states that was generated with an exotic longitudinally varying SAM-OAM coupling [27], and a generalized hyperboloid structured VVB that was proposed to explore the polarization singularities and longitudinal-transverse coupling effects in lasers [28,29]. VVBs with exotic vector vortex lattices were generated, revealing deep fundamental physics of superfluids and Bose-Einstein condensates [30][31][32]. The photonic wheel structure was proposed to unveil the focused VB with both longitudinal and transverse spin [33,34]. A novel skyrmion-like structure was recently generated in a focused VVB field, which can be used to realize deep-subwavelength super-resolution imaging [35].
In this paper, we demonstrate a new laser oscillating mechanism together with an external modulation to create a new class of VVB embedded with more degrees of freedom (DoFs). Our laser cavity exploits ray-wave duality to lase on multiple ray-like trajectories, akin to geometric paths inside the cavity, with differing coherentstate phases and independently tunable polarizations. We achieve these states by precise control of the cavity length and off-axis pump position on the anisotropic crystal, the latter to realize polarization control, inside an otherwise empty cavity. By external transformation through simple geometric optics, we demonstrate a new class of structured light with controllable DoFs in trajectory shape, coherent-state phase, OAM, and polarization, producing exotic states of structured light in four DoFs, surpassing the limit of two-dimensional VVBs. We theoretically describe our structured light as the superposition of SU(2) coherent states, revealing a rich parameter landscape for further exploration. Our theoretical and experimental results demonstrate that the prominent hybrid SU(2) geometry of our new beams provide new DoFs for developing new theories and applications of structured beams.

HYBRID SU(2) GEOMETRIC ORBITS FROM A LASER A. Basic Concepts
We will first outline the basic concept behind our work before placing it on a firm theoretical footing in the sections to follow. It is known that when a laser cavity is operated in a frequencydegenerate state, the mode appears to be the result of a ray-like periodic orbit, forming a closed path that is ever repeating [see Fig. 1(a)]. The shape and number of bounces in this trajectory are determined by the cavity length, the cavity mirror curvatures, and the position of the pump light. The output number of "rays" and their directions are likewise determined by the internal trajectories. Although ray-like, they are coherent modes exhibiting a wave-like behavior-we will show that the formalism of SU(2) coherent states is highly appropriate for their description, teasing out the salient properties. Prior to this study, only single geometric modes (trajectories) with a certain transverse size and oscillating phase have been reported, described by just two DoFs. Here we present a frequency-degenerate laser capable of generating two geometric modes with different transverse sizes and oscillating phases. Crucially, the two modes oscillate in opposing directions, share the same number of rays, and share a trajectory inflection point at the pump spot as shown in Fig. 1. This allows us to control the polarization state of each by a judicious choice of gain crystal. The result is a new vector state of light described by four DoFs and produced directly from a simple laser.

B. Classical Analogy of a SU(2) Coherent State
First, we introduce the property of a quantum coherent state because we will exploit its classical-resembling property that its probability wave packet is coupled with a trajectory of classical movement [36] and expand this property into its classical analogy, i.e., the ray-wave duality of a geometric mode. The coherent state under the SU(2) symmetry group is given by [37] where |K , N represent the set of eigenstates with a constant total boson number of N and φ is termed the coherent-state phase. The SU(2) symmetry applied here is the general geometric symmetry of paraxial light beams, which inspires us to propose this for structured light as a classical analogy to the quantum coherent state. To do so, we replace the quantum eigenstates by free-space eigenmodes: where the eigenmode |ψ n,m,l can be generated by applying SU(2) ladder operators to a fundamental-mode Gaussian beam |ψ 0,0,l : where (a † x , a † x ) are the linear ladder operators to generate various high-order transverse modes, (α, β) are the two parametric angles in a SU(2) unitary matrix, (n, m) are the transverse-mode indices at the (x , y ) plane, and l is the longitudinal-mode index. When β = 0 or β = π , |ψ (α,β) n,m,l is reduced to a Hermite-Gaussian (HG) mode; when α = β = π/2, it becomes a LG mode [38], and so on. For the superposition to be valid (coherent superposition), the two integers P and Q yield that the eigenmodes |ψ (α,β) n+QK,m,l−PK in Eq. (2) should have a constant eigenfrequency. Akin to the quantum-classical coupling in a coherent state, the classical beams of Eq. (2) have the similar interesting property that the beam wave packet is coupled with the SU(2) geometric ray trajectory, namely, ray-wave duality, as a macroscopic manifestation of quantum wave-particle duality.

C. Manipulating SU(2) Geometric Modes
In this section we introduce how to realize these classical SU(2) coherent states as ray-wave geometric beams from a laser. For constructing the well-defined coherent superposition as in Eq. (2), the key issue is the accommodation of a set of eigenmodes with different orders but at the same frequency in a single optical resonator, namely, a frequency-degenerate state. This must satisfy the criterion that the ratio of the transverse-and longitudinal-mode frequency spacing is a simple rational number = f T / f L = P /Q, where P and Q are co-prime integers, and the basic eigenmodes |ψ (α,β) n+QK,m,l−PK construct a frequency-degenerate family (see detailed proof in Section 1 of Supplement 1).
When an optical resonator is close to a frequency-degenerate state, the laser mode and output power would drastically change [39]. Recently, it has been experimentally and theoretically verified that the lasing modes in an off-axis-pumped frequencydegenerate laser cavity have a preference to be localized on periodic ray trajectories. This fulfills the SU(2) re-entrant condition of a two-dimensional coupled oscillator, the trajectories of which for various degenerate states | = P/Q are shown in Fig. 1(a). (a) For a concave mirror with a curvature of radius R and a cavity length of L that is precisely controlled, different periodic oscillating trajectories in a cavity with different round-trip periods Q would emerge at some special cavity lengths corresponding to various frequency-degenerate states | = P /Q . (b) The distribution of frequency-degenerate states can be revealed by the fractal frequency spectrum, i.e., frequency difference ratio ( f n,m,l − f n0,m0,l0 )/ f L as a function of L/R for the range of |n − n 0 | ≤ 12, |m − m 0 | ≤ 12, and |l − l 0 | ≤ 12, where some degeneracy states | = P /Q are marked at corresponding positions. (c) A certain periodic trajectory is determined by not only the degenerate state | = P /Q but also the phase state |φ . For the state | = P /Q , an auxiliary circle with the vertices of an equilateral Q-gon located on it is helpful to determine the starting points of oscillating orbits, where the starting points at the z = 0 plane corresponds to the projection of the vertices on the auxiliary circle. For the two eigenphase states |φ = 0 and |φ = π , there are always coincident projected vertices. Rotating the auxiliary circle for an angle π/Q of one eigen-phase-state will produce another eigen-phase-state. If the two trajectories of the eigen-phase-states are superposed together and sharing at least a pair of coincident projected points, the hybrid SU(2) oscillating orbits can be obtained, where the yellow auxiliary lines connecting the Q-gon vertices on the auxiliary circle contribute possible coincident projections and result in some equilateral star shape. The yellow dots marked in the hybrid trajectories show the shared coincident projections corresponding to the actual positions of a pump spot in the experimental generation.
Based on this ray-wave duality, the ray-represented modes can also be represented by the corresponding wave packet of the SU(2) quantum coherent state, named SU(2) geometric modes [40][41][42]. Without loss of generality, hereinafter we consider a plano-concave cavity with a length of L, a concave spherical mirror with radius of curvature R (as the output coupler), and a flat mirror of high reflectivity for the rear mirror. We set the paraxial propagation direction as that of the z axis and the flat mirror as the (x , y ) plane with the original point located at the center to form the The frequency degeneracy requires that the allowable eigenmodes should have a constant frequency f n 0 ,m 0 ,l 0 , and the SU(2) coherent state just fulfills this condition. The distribution of various degenerate states can also be revealed by topological joints in the fractal spectrum [43], i.e., the frequency difference in the neighborhood of the center indices as shown in Fig. 1(b), where some special degenerate states | = P/Q are marked and coupled with the periodic orbits in the corresponding degenerate cavities in Fig. 1(a). In general, the ray trajectory should have a round-trip period of Q for the | = P/Q state. However, the depicted trajectory is just one instance of an infinite number of possible cases. The shape of a certain trajectory is related not only to the frequency degeneracy but also to the phase of the SU(2) coherent state [44]. Figure 1(c) shows the different trajectories at the two phase states |φ = 0 and |φ = π with opposite topological oscillations. For instance, for the | = 1/4 state, the trajectory is M-shaped when |φ = 0 and W-shaped when |φ = π ; for the | = 1/5 state, the trajectory is WI-shaped when |φ = 0 and IW-shaped when |φ = π , and so on.
Note that the phase state | |φ only specifies the topological shape of the oscillation but not the actual size of it. The actual geometric modes can also have uncertain scales at a certain degeneracy with phase state | |φ , which should be determined by another DoF, i.e., transverse order N [45]. For conveniently describing more detailed geometric modes, we propose the concept of an auxiliary circle for determining the start points of the SU(2) oscillating orbits in a degenerate cavity. The radius of the auxiliary circle √ N determines the scale of the geometric mode, which is also related to the actual off-axis displacement of the pump light (see Section 1 of Supplement 1). For the planar geometric mode in the degenerate state | = P/Q , there are Q auxiliary points on the auxiliary circle with azimuthal angle θ s = 2πs/Q + φ x (s = 0, 1, 2, ... , Q − 1) forming a equilateral Q-gon (polygon with Q sides), while the projected points on the z = 0 plane are just the start points for the Qth periodic orbits as shown in Fig. 1(c). The auxiliary phase φ x is related to the coherent-state phase by φ x = φ/Q. The evolution of this state from φ = 0 to 2π corresponds to the rotation of the auxiliary Q-gon from an angle of 0 to 2π , returning to the initial state. The phase is actually related to the Gouy phase, which can be modulated by astigmatic mode converter systems in experiments [46]. Distinguished from other phase states, the two eigenphase states |φ = 0 and |φ = π have the same trajectory for the positive and negative oscillations; also, there are always coincident projections of auxiliary points. For instance, in | = 1/5 |φ = 0 , there are two kinds of coincident projection, i.e., s = 1 with s = 4 and s = 2 with s = 3; for | = 1/5 |φ = π , s = 0 with s = 4 and s = 1 with s = 3; for | = 1/6 |φ = 0 , s = 1 with s = 5 and s = 2 with s = 4; for | = 1/6 |φ = π , there are three kinds of coincident projection, i.e., s = 0 with s = 5, s = 1 with s = 4, and s = 2 with s = 3 [see the dashed lines marked for various projection pairs in Fig. 1(c)].
We mark all such point pairs by connected lines in Fig. 1(c), which can form exotic star-shaped patterns. Beyond aesthetics, the coincident projected points have a clear physical significance, namely, the inflection point of the bouncing orbit at the flat mirror in the cavity. In an off-axis pumped degenerate cavity, a coincident projected point is usually located at the pump spot rather than the isolated projected point due to the mode matching [44,47]. Therefore, we can adjust the location of the coincident projected point of a geometric mode by off-axis pumping so as to control the transverse scale parameter N. In other words, this mathematical and geometric representation has practical relevance: from it we understand where to pump the crystal in order to excite specific combinations of trajectories with associated sizes. Figure 2(a) illustrates control of an SU(2) oscillation in our experiment with a resonator configured for the | = 1/4 degenerate state, while the pump spot can be adjusted to some desired coincident projected point to adjust the transverse scale. The gain medium was a thin-slice c-cut Nd:YVO 4 crystal with an inner surface antireflective (AR) coating for the laser light and an outer surface AR coated for the pump light and high-reflective coated for the laser light. This allows anisotropic refractive indices and stimulated emission cross sections in (x , z) plane [48]. Therefore, the geometric orbit oscillations in our cavity undergo trajectorydependent polarization modulation in addition to transverse adjustment. To realize vector geometric beams, we exploit the birefringence dependence on the incident angle, shown in Fig. 2(b). Finally, using an astigmatic mode converter constituted by the two cylindrical lenses shown in Fig. 2(c), a planar geometric mode can be converted into a vortex geometric mode. For the vortex geometric modes, the SU(2) rotational symmetry can be directly observed in natural space, which is also a macroscopic manifestation of OAM. Thus, the spatial twisted trajectories can be directly measured via scanning by a charge-coupled device (CCD) camera (see Section 7, Methods, for more details), in contrast to the conventional circular vortex beams where the twisted effect cannot be directly observed and requires other techniques, e.g., interferometry.
In summary, a planar geometric mode in a degenerate cavity should be completely determined by a SU(2) coherent state | |N |φ with three DoFs: frequency-degeneracy , transverse order N, and phase φ. After mode conversion, a planar geometric mode is transformed into vortex mode represented by | | |φ , where the OAM index replaces the transverse order of the planar mode.

D. Hybrid SU(2) Geometric Modes
Next, we demonstrate a hybrid-trajectory state by forcing the laser to oscillate on two different trajectories with orthogonal transverse orders and coherent-state phases simultaneously, to form the new nonseparable state The transverse orders of the two component modes N 1 and N 2 yield a mathematic relationship enabling the two decomposed trajectories to share at least a point of coincident projection, that is, N i /N j = cos 2 (2π/Q)/cos 2 (π/Q) with {i, j } = {1, 2}, while the larger N corresponds to the outer trajectory and the smaller the inner. This state is achieved by locating the pump light at the shared coincident projection of the two trajectories. The hybrid SU(2) orbits can only exist when Q ≥ 5, because you cannot find the shared coincident projection when Q ≤ 4 (see Section 2 of Supplement 1 for detailed derivations). The diagrams in the third column of Fig. 1(c) illustrate the formation of hybrid SU(2) orbits with the help of the auxiliary circle demonstrating the shared coincident projected point. Similar to the pure SU(2) geometric modes, the hybrid SU(2) geometric modes can also be converted into vortex beams carrying OAM. and | = 1/6 , where the first two columns show the pure SU(2) trajectories for the two eigenphase states, and the third column shows the hybrid trajectories. According to SU(2) rotational symmetry, the evolution of phase state |φ for vortex geometric modes just corresponds to the axial rotation with an angle of φ/Q. Yielded by the coincident projection principle, the inner and outer orbits at a propagation plane of a hybrid trajectory just form the equilateral star shape corresponding to the star-shaped auxiliary lines shown in Fig. 1(c). In the experiment, we used a series of Fourier lenses to focus the geometric beams with a beam waist outside the cavity. The lenses were adjusted with a controlled Gouy phase of 0 or π so as to keep the decomposed trajectory at eigenphase state |φ = 0 or |φ = π . Then we moved the CCD camera along the z axis to record the transverse patterns at various propagation distances. Figures 3(b) and 3(c) show the evolution of hybrid SU (2) vortex geometric modes at | = 1/5 and | = 1/6 , respectively. There were two kinds of evolutions of hybrid SU(2) trajectories in our experiment. The first case was observed in the | = 1/5 state, where the outer trajectory was superposed in the process of enlarging the inner trajectory by increasing the pump's off-axis displacement. The second case was in the | = 1/6 state, where the trajectory with phase state |φ can be switched into another eigenphase state |φ + π from the process of increasing the pump's off-axis displacement, and the hybrid trajectory can be obtained by the superposition of the two pure trajectories during the switching. The first-case hybrid trajectory appeared at the region of x from 490 to 610 µm for the | = 1/5 state. There are two regions of the second case where the hybrid trajectory appeared around x = 620 µm and x = 700 µm, respectively. It is proved that the hybrid SU(2) geometric modes can be stably operated in the corresponding hybrid-trajectory regions. The dynamic processes of generating hybrid SU(2) geometric modes were recorded in Visualization 1 and Visualization 2. The theoretical methods for hybrid SU (2) vector geometric modes are demonstrated in Section 3 of Supplement 1. The detailed experimental and theoretical analysis for the polarization and mode evolution for the SU(2) beams are discussed in Section 4 of Supplement 1.

HYBRID SU(2) VECTOR VORTEX BEAMS
Finally, we bring all the tools together, using multiple trajectory control and polarization control to demonstrate a new nonseparable state in multiple DoFs: where |H and |V are the horizontal and vertical linear polarization states. Again, the transverse order can be transformed into OAM, resulting into the hybrid SU(2) VVB: and thus we open up new DoFs for our VVB in contrast to the conventional VVB of | 1 |H + | 2 |V mapped on a 2D Poincaré sphere [21,23].
For | = 1/5 , when x ≤ 480 µm, the bouncing orbits in the cavity have very small inclined angles such that the birefringent modulation is weak; consequently, the horizontal and vertical polarizations are nearly equal, and the output mode is circularly polarized. When x is slightly greater than 480 µm, the outer trajectory is horizontally polarized, while the inner is circularly polarized. This is comprehensible as the two trajectories have different transverse sizes coupled with different birefringent effects in the anisotropic cavity. When x is further increased to 610 µm, the outer trajectory can gradually change into a vertically polarized state. When x is slightly larger than 610 µm, the outer trajectory vanishes with a pure inner trajectory left. The polarization switching phenomenon can still occur for the pure geometric mode without a hybrid trajectory if x is further increased.
Similarly, for | = 1/6 , when x ≤ 480 µm, the bouncing orbits have very small inclined angles, and the components of the horizontal and vertical polarizations are nearly equal. While x is further increased until 610 µm, the geometric mode is gradually transformed into vertical linear polarization. When x is around 620 µm, the hybrid SU(2) trajectory is generated, where the original inner trajectory gradually vanishes and the outer trajectory emerges with the same vertical linear polarization. During 640 µm ≤ x ≤ 680 µm, the output mode is a vertically polarized pure SU(2) geometric mode with a switched phase state rather than the original state. When x is around 700 µm, the hybrid SU(2) trajectory is generated again, where the original outer trajectory gradually vanishes and the inner trajectory emerges with a horizontal linear polarization, which is orthogonal to the polarization of the immediately vanished inner trajectory. The polarization switching phenomenon can still occur in the process of further enlarging of the pure geometric trajectory if x is further increased.
Naturally, we can obtain some exotic VBs and corresponding VVBs with a mode converter in the above process. Same as the pure SU(2) vortex beams, where the OAM effect can be directly observed by their twisted structure with SU(2) rotational symmetry along the propagation axis, the hybrid SU(2) vortex beams also have the twisted structure in their spatial wave packets as shown in Figs. 4(a) and 4(b). Figures 4(c) and 4(d) show our experimental and theoretical results of four typical hybrid SU(2) vortex beams generated at the | = 1/5 state with x = 600 µm and x = 500 µm and at the | = 1/6 state with x = 700 µm and x = 620 µm, which are named beams A, B, C, and D, respectively, for convenient description. The results show that these four hybrid SU(2) vortex beams have different vectorial properties with respect to each other. Beams A, B, and C are three kinds of hybrid SU(2) VVBs, and beam D is a hybrid SU(2) scalar vortex beam. For beam A [ Fig. 4(c)], the inner orbits are circularly polarized, and thus their beam intensities are independent of the orientation of the polarizer; the outer orbits are vertically linearly polarized, and thus their beam intensities vanish for the horizontal orientation of the polarizer with a maximum for the vertical orientation. For beam B [ Fig. 4(d)], the inner orbits are still circularly polarized, while the outer orbits are horizontally linearly polarized. In degenerate state | = 1/6 , we can observe the interesting vector structure where the inner and outer trajectories have orthogonal polarizations as illustrated in beam C. For beam C [ Fig. 4(e)], the inner orbits are horizontally polarized and the outer orbits vertically polarized. Thus, the VB C would be reduced into a horizontally polarized scalar SU(2) mode through a horizontally orientated polarizer and another vertically polarized scalar SU(2) mode with opposite phase state through a vertically orientated polarizer. Besides the hybrid SU(2) VB, the hybrid SU(2) scalar beam can also be obtained as beam D [ Fig. 4(f )], which is a stable vertically polarized scalar vortex beam.

POLARIZATION SINGULARITIES AND TOPOLOGICAL PHASE
Hybrid SU(2) VVBs have exotic properties of polarization singularities and topological phase. For describing the topological properties of the vectorial fields in hybrid SU(2) VVBs, we define two kinds of phase parameters: (1) 1 = arctan(|E x |/|E y |) and (2) 2 = arg(E x /E y ), where E x and E y are the horizontal and vertical components of the electric field. 1 reveals the distribution varying different linear and circular polarization, ranging from 0 to π/2. 1 = 0 and π/2 represent the horizontal and vertical linear polarizations, respectively, and 1 = π/4 the circular polarization. 2 can unravel more vector properties including the orientation and chirality of polarization, ranging from −π to π, which can depict the distribution of polarization singularities of the vectorial field. For a scalar beam, 2 is just reduced to the phase distribution of the light field.   orthogonal polarizations, the polarization singularities are fruitfully distributed over the entire field with a beautiful flower shape. For each of the three beams A-C, the topological phase is induced by a spin-to-orbital coupled effect because it is a superposition of the two trajectories with different SAM and OAM. In the scalar beam D [ Fig. 5(h)], the SAM or polarization is separable, and 2 is just the phase distribution of scalar light with multiple phase singularities, where a large OAM is manifested. The clarification of polarization singularities and topological phase is very helpful for extending these new hybrid SU(2) structured beams in applications such as optical tweezers and assembly and exploration of novel manipulating technologies.

DISCUSSION
If higher-order hybrid SU(2) trajectories with more oscillating periods (Q ≥ 7) could be generated, more intriguing properties would be unveiled. As demonstrated above, a hybrid SU(2) trajectory cannot be composed when Q ≤ 4, and there is only one morphology of a hybrid SU(2) superposed trajectory when Q = 5, 6. However, for higher-order degenerate states with Q ≥ 7, there would be multiple possible cases of hybrid SU(2) trajectories because more inflection points (in the decomposed trajectory) provide more combinations of sharing a pumping spot. Theoretically, there would be two kinds of hybrid SU(2) trajectories for the degenerate states of | = 1/7 and | = 1/8 , and four kinds of hybrid trajectories for | = 1/9 and | = 1/10 , as shown in Figs. 6(a)-6(d), respectively. We presented the theory for obtaining the trajectory-combination number for an arbitrary degenerate state shown in Section 2 of Supplement 1. Moreover, besides the two-trajectory superposition, the hybrid superposition with multiple SU(2) trajectories is also possible to form even higher-order degenerate states, e.g., the three-trajectory-superposed hybrid SU(2) mode shown in Fig. 6(e). Due to the limit of the crystal size in our experiment, it is difficult to control higher-order hybrid SU(2) trajectories when Q ≥ 7. Therefore, there is still a multitude of exotic properties of hybrid SU(2) modes that need to be theoretically and experimentally explored in the future.
The additional DoFs in our hybrid SU(2) VVB allow us to break the two-dimensional limit inherent in conventional VBs, opening up exciting prospects for applied and fundamental studies. For example, while their SU(2) description is complete, we believe it may be possible to use techniques that map structured beams to Poincaré-like spheres [49][50][51] to gain insight into how to elegantly represent this new family of hybrid SU(2) VVBs geometrically. That may yield insight into other relations and dynamics, since multidimensional structured light has remained largely unexplored. The strong trajectory nature of our light may aid studies in trajectory-spin coupled processes in the spin-Hall effect of light [52], while the nonseparability of our beam with more DoFs opens for the first time the possibility to classically simulate multipartite high-dimensional quantum processes, including Greenberger-Horne-Zeilinger (GHZ) states, NOON states, and W states, all of which have been suggested theoretically [8] but not yet demonstrated due to the inability to extend beyond just qubits with vector states of light. Other possibilities include considering the simulation of squeezed state in classical light, a detailed study of polarization singularities and their families in such fields, and propagation dynamics, all of which are logical next steps of study.

CONCLUSION
In conclusion, we proposed a new kind of vectorially structured light in four DoFs and showed how to create it directly from a simple laser cavity. Our hybrid SU(2) geometric mode is constituted by two decomposed pure SU(2) mode states with special superposition conditions, fulfilling a complete ray-trajectory oscillation condition, which largely enriches the general families of both intracavity and free-space modes. We also experimentally generated various examples of SU(2) VVBs in various degenerate states (| = 1/5 and | = 1/6 ). The resulting light was tailored to be exotic star-shaped geometric patterns with complex phase and polarization singularities, which is the first demonstration of such structured light beams to our knowledge. Importantly, our hybrid SU(2) VVBs are represented and controlled in multiple DoFs, surpassing the limit of two-dimensional conventional VVBs. Besides conventional DoFs of amplitude, phase, OAM, and polarization, many new DoFs are hatched such as ray-wave orbits, coherent-state phase, and trajectory combination, offering intriguing prospects of unlimited dimensionality controls in such VVBs. Moreover, the simple and elegant at-the-source generation scheme paves ways to extended applications, including simulations of quantum processes with classical light.

METHODS
The front end was a laser oscillator where the cavity length can be precisely controlled. Through controlling the off-axis pumping position, various SU(2) geometric modes can be generated. A 808 nm fiber-coupled laser diode (LD) (FOCUSLIGHT, FL-FCSE08-7-808-200) was used as the pump source. With a telescope system with magnification of about 1:1 constituted by two identical AR-coated lenses (focal length F = 25 mm), the pump light was focused into a c-cut Nd:YVO 4 slice-like crystal with dopant of 0.5 at.% and thickness of 5 mm, which was wrapped in a copper heat sink and conductively water cooled at 18 • C. The outside surface of the crystal was AR coated at 808 nm and high-reflective coated at 1064 nm, and the inner surface was AR coated at 1064 nm. A plano-concave mirror was used as the output coupler, where radius of curvature was 100 mm, and the transmittance was 10% at 1064 nm for the inner surface and AR for the outside surface. In our whole experiment, the pumping power was fixed at about 5.5 W. For measuring the ray trajectories, we used a CCD camera to capture and scan the transverse pattern at different propagation distances along the z axis. In contrast to the conventional circular vortex beams, the OAM of SU(2) structured beams can be directly unveiled by the spatially twisted trajectories captured by the CCD scanning (see more details in Section 5 of Supplement 1). The CCD camera (Spiricon SP620U) is comprised of 1600 × 1200 pixels with a pixel pitch of 4.4 µm, corresponding to a charge-coupled screen size of 7.04 × 5.28 mm 2 . In the measurement, the beam diameter size within Rayleigh range was confirmed to be less than 5 mm, so that we could scan the full information of the measured beam propagating within the whole Rayleigh range.
Funding. National Natural Science Foundation of China (61975087); Natural Science Foundation of Beijing Municipality (4172030).