Polarization and phase are two intrinsic properties of light^[1]. Light with left-handed circular polarization (LCP) or right-handed circular polarization (RCP) has a spin angular momentum of $\sigma \hslash$ per photon, respectively, where $\sigma =\pm 1$ implies the sign of photon spin angular momentum projection to the positive propagation direction^[2]. In addition to its spin angular momentum, light with a helical phase dependence of $\exp (il\phi )$ possesses an orbital angular momentum of $l\hslash$ per photon, where $l$ is an integer number representing the topological charge of singularity, and $\phi =\arctan (y/x)$ is called the azimuthal angle in the polar coordinate system $(r,\phi )$^[3]. Spin angular momentum and orbital angular momentum constitute the total angular momentum per photon of $J=\sigma \hslash +l\hslash$^[4].

The hybrid-order Poincaré sphere^[5], which generalizes the Poincaré sphere^[6,7] and high-order Poincaré sphere^[8,9], has been demonstrated to describe the evolution of the phase and polarization under paraxial light interacting with inhomogeneous anisotropic media. Beams on the hybrid-order Poincaré sphere exhibit the features of possessing space-variant polarization states as vector beams^[10] and carrying helical wavefronts as vortex beams^[3,11], which were therefore called vector vortex beams. Such beams provide more degrees of freedom for light shaping and, therefore, can be developed into irreplaceable applications in particle acceleration^[12], high-resolution imaging^[13], quantum information processing^[14–16], and the photonics spin-Hall effect^[17–19].

Plasmonic or dielectric metasurfaces^[20–24], spatial light modulators^[25–27], and liquid crystal elements^[28–31] have been proposed to tailor the intensities, phases, and polarization states of light fields. Among these approaches, transmission-type liquid crystal polarization holographic elements with an azimuthally distributed orientation angle and half-wave birefringence phase retardation are widely used relying on the spin-orbital angular momentum interaction process^[32]. Even better, liquid crystal also shows the electric driving ability, which can facilitate the modulation of the beam state^[28–30]. More importantly, the reflective spin to the orbital angular momentum conversion process and even the multidimensional regulation of light from cholesteric liquid crystal (CLC) systems were investigated ever since the revelation of the reflective Bragg–Berry phase from chiral anisotropic optical media^[33–39]. Compared to transmissive liquid crystal Berry phase optical elements, CLCs have more advantages because they do not need to satisfy the half-wave birefringence retardation and also are capable of operating light in the Bragg polychromatic spectral regime^[40,41]. However, CLC planar optical elements, unlike the transmissive-type Berry phase optical elements, cannot lead to the Berry phase reversal when flipping the incident circular polarization to the opposite handedness, which prevents the generation of vector vortex beams from a single CLC with monotonous chirality. More recently, effective proposals by adding a rear mirror^[42] or an opposite-handed CLC device with the same surface pattern^[43] have been presented; nevertheless, these proposals cannot generate the generalized vector vortex beams on the hybrid-order Poincaré sphere.

In this work, we aim to utilize CLCs in an attempt to generate arbitrary vector vortex beams on the hybrid-order Poincaré sphere. The beam is considered as a superposition of two bases with orthogonal handedness of circular polarization and different topological charges of orbital angular momentum. Two opposite-handed CLC planar optical elements with different surface patterns can independently generate two bases on the north and south poles of the hybrid-order Poincaré sphere. Due to its polarization selectivity characteristic, vector vortex beams, therefore, can be generated via combining two CLC elements together. Moreover, arbitrary vector vortex beams on the hybrid-order Poincaré sphere can be mapped through adjusting the directions of the polarizer and the quarter-wave plate in front of CLCs. Based on this strategy, the proof-of-concept experiment is successfully performed to generate typical vector vortex beams. The experimentally measured intensity and polarization profiles of beams agree well with the simulation results.

A generalized vector vortex beam can be considered as a linear combination of two orthogonal circularly polarized vortices $|N_l⟩$ and $|S_m⟩$ with coefficients ${\psi }_N^l$ and ${\psi }_S^m$, respectively. In the paraxial approximation, it can be formulated as $$|\psi ⟩={\psi }_N^l|N_l⟩+{\psi }_S^m|S_m⟩,$$(1)with $|N_l⟩=[({\mathbf{e}}_x+i{\mathbf{e}}_y)/\sqrt{2}]\exp (il\phi -i{\phi }_0/2),|S_m⟩=[({\mathbf{e}}_x-i{\mathbf{e}}_y)/\sqrt{2}]\exp (im\phi +i{\phi }_0/2)$, where $l$ and $m$ are the corresponding topological charges of helical phases, ${\mathbf{e}}_x$ and ${\mathbf{e}}_y$ are the unit vectors along $x$ and $y$ axes in the Cartesian coordinate system, and ${\phi }_0$ is a constant phase.

As depicted in Fig. 1, any vector vortex state with inhomogeneous polarization states and phases can be mapped to a point $(\theta ,\mathrm{\Phi })$ at the sphere coordinate on the hybrid-order Poincaré sphere of order $(l,m)$. Here, $\mathrm{\Phi }=\arg ({\psi }_N^l)-\arg ({\psi }_S^m)$ is the relative phase difference between them, and $\tan (\theta /2)=|{\psi }_N^l/{\psi }_S^m|$ refers to the amplitude ratio of ${\psi }_N^l$ to ${\psi }_S^m$. The corresponding Stokes parameters projecting in the spherical Cartesian coordinates can be formulated as $$S_0^{l,m}={|{\psi }_N^l|}^2+{|{\psi }_S^m|}^2,$$(2)$$S_1^{l,m}=2|{\psi }_N^l||{\psi }_S^m|\cos \mathrm{\Phi },$$(3)$$S_2^{l,m}=2|{\psi }_N^l||{\psi }_S^m|\sin \mathrm{\Phi },$$(4)$$S_3^{l,m}={|{\psi }_N^l|}^2-{|{\psi }_S^m|}^2,$$(5)where ${|{\psi }_N^l|}^2$ and ${|{\psi }_S^m|}^2$ are the intensities of $|N_l⟩$ and $|S_m⟩$, respectively.

In this paper, we use two CLC planar optical elements to generate two bases on the north and south poles of the hybrid-order Poincaré sphere. The molecular directors of CLCs can be characterized by assembling the chiral superstructures along the helical axis ($z$-axis) and the space-variant alignment pattern at the front surface $(\alpha =2\pi \chi z/p+{\alpha }_{\mathrm{surf}})$, where $\chi =\pm 1$ depends on the chirality of helical superstructures, and $p$ denotes the helical pitch. As schematically represented in Figs. 2(a) and 2(b), these two CLC elements have a right-handed chiral helix (RCLC, ${\chi }_1=1$) and a left-handed chiral helix (LCLC, ${\chi }_2=-1$), respectively. Besides, the front surface alignment pattern of RCLC is imprinted with an azimuthally variant structure as the $q$-plate $({\alpha }_{\mathrm{surf}1}=q\phi ,q=1)$, whereas the surface of LCLC is imprinted with a homogeneous pattern parallel to the $x$-axis $({\alpha }_{\mathrm{surf}2}=0)$.

The complex electric field of a $\sigma$-handed circularly polarized light propagating toward the $z$-axis can be expressed as ${\mathit{E}}_{\text{in}}=E_0[({\mathbf{e}}_x+i\sigma {\mathbf{e}}_y)/\sqrt{2}]\exp (ikz-i\omega t)$, where $E_0$ is the amplitude of the electric field, $k$ is the free-space wave vector, and $\omega$ is the angular frequency. Based on the coupled-mode theory, the optical performance of CLC interacting with light propagating along the helical axis can be analyzed as follows. On the one hand, circularly polarized light having the same chirality as that of CLCs $(\sigma =-\chi )$ experiences a polarization-preserving Bragg-reflection in the wavelength region of $n_{o}p<\lambda <n_{e}p$, where $n_e$ and $n_o$ refer to the extraordinary and ordinary refractive indices of liquid crystal materials. The reflected light can be expressed as ${\mathit{E}}_r=E_0[({\mathbf{e}}_x-i\sigma {\mathbf{e}}_y)/\sqrt{2}]\exp (2i\sigma {\alpha }_{\mathrm{surf}})\exp (-ikz-i\omega t)$, which indicates that a Berry phase of ${\mathrm{\Phi }}_B=2\sigma {\alpha }_{\mathrm{surf}}$ is additionally encoded to the reflected light. On the other hand, the circularly polarized light having opposite chirality $(\sigma =\chi )$ transmits through CLCs, which maintains its polarization and cannot generate the Berry phase. Based on the circular Bragg reflection phenomenon, input light in the wavelength range $n_{o}p<\lambda <n_{e}p$ interacting sequentially with RCLC $({\alpha }_{\mathrm{surf}1}=\phi )$ and LCLC $({\alpha }_{\mathrm{surf}2}=0)$ in this paper can be analyzed as follows. On the one hand, as illustrated in Fig. 3(a), RCP light $(\sigma =-1)$ is directly reflected from RCLC while preserving the circular polarization handedness. The spiral Berry phase with ${\varphi }_{B1}=-2{\alpha }_{\mathrm{surf}1}=-2\phi$ is encoded in the reflected light, demonstrated through carrying orbital angular momentum with topological charge $l=-2$. On the other hand, LCP light $(\sigma =1)$ illustrated in Fig. 3(b) transmits through RCLC media, and afterward it is reflected from LCLC. Similarly, light in the backward direction preserves the circular polarization handedness and then passes through RCLC. The Berry phase of reflected light can be expressed as ${\varphi }_{B2}=2{\alpha }_{\mathrm{surf}2}=0$, which confirms a plane wavefront with $m=0$ caused by the homogeneous surface pattern on LCLC. As a result, the states of these two beams reflected from RCLC and LCLC correspond to the bases $|N_l⟩$ with $l=-2$ and $|S_m⟩$ with $m=0$ in Eq. (1), respectively. Here, ${\phi }_0$ in Eq. (1) denotes the dynamical phase difference caused by the optical path difference between these two bases.

An input light with a homogeneous polarization state on the Poincaré sphere can be considered as a composition of LCP and RCP lights, that is, ${\mathit{E}}_{\text{in}}=E_0[a({\mathbf{e}}_x+i{\mathbf{e}}_y)/\sqrt{2}+b({\mathbf{e}}_x-i{\mathbf{e}}_y)/\sqrt{2}]\exp (ikz-i\omega t)$. Based on the analysis above, the reflected light, thus, can be derived as ${\mathit{E}}_r=E_0[a\exp (2i{\alpha }_{\mathrm{surf}2}+i{\phi }_0/2)({\mathbf{e}}_x-i{\mathbf{e}}_y)/\sqrt{2}+b\exp (-2i{\alpha }_{\mathrm{surf}1}-i{\phi }_0/2)({\mathbf{e}}_x+i{\mathbf{e}}_y)/\sqrt{2}]\exp i(-kz-\omega t)$, which corresponds to a single state of the vector vortex beam after interacting with customized RCLC and LCLC. It also indicates that any state in the polarization Poincaré sphere can be mapped onto the hybrid-order Poincaré sphere $(l=-2,m=0)$ through changing the input homogeneous polarization. More generally speaking, by designing the appropriate surface patterns for the RCLC with ${\alpha }_{\mathrm{surf}1}=q_1\phi$ and LCLC with ${\alpha }_{\mathrm{surf}2}=q_2\phi$, arbitrary vector vortex beams on the hybrid-order Poincaré sphere with topological charges $l=-2q_1$ and $m=2q_2$ can also be generated during the spin-to-orbital angular momentum conversion processes.

In our experiment, the CLC planar optical elements are fabricated as polymer films because of their high resolution and high stability. The azobenzene dye brilliant yellow is first dissolved in dimethylformamide (DMF) at a concentration of 0.4% by weight. Then, the solution is spin-coated onto a clean glass substrate at 800 rpm (r/min) for 5 s and then 3000 rpm for 30 s to form a photoalignment layer. In order to record the polarization hologram for the $q$-plate structure, the substrates with the alignment layer are exposed to the polarization rotation system^[44]. After that, the left/right-handed CLC materials are also prepared as a mixture of 93% reactive mesogen RM257, 5% photo-initiator Irgacure 184, and 2.12% left/right-handed chiral dopant S5011/R5011. Next, the CLC mixture is diluted into toluene at a weight ratio of 12:88 and spin-coated onto the exposed substrates 10 times at a speed of 1800 rpm for 30 s. After each coating, the samples are cured through the illumination of the UV lamp in a nitrogen environment. Finally, the CLC layer of samples is fabricated with a thickness of about 4 $\mathrm{\mu m}$.

We characterize the transmissive spectra of RCLC and LCLC ranging from 480 to 780 nm, which correspond to Figs. 4(a) and 4(b). The transmittance spectra under RCP and LCP light source illumination are plotted as red and blue curves, respectively, which illustrate the capability of polarization selection, that is, light reflects circular polarization with the same handedness of helix structure and transmits circular polarization with opposite handedness of helix structure in the Bragg bandgap. Here, the samples in this experiment provide Bragg reflection spectra centering around 633 nm and maximum reflectivity of 0.85. In practice, the Bragg reflection region can be shifted effectively by adjusting the proportion of chiral dopant. The maximum reflectivity can be continuously increased by promoting the thickness of the CLC layers. Figures 4(c) and 4(d) are the captured photos of RCLC and LCLC under a reflection-mode polarization optical microscopy. Moreover, in order to identify the topological charge of the phase singularities, Mach–Zehnder interference experiments are also implemented. Figures 4(e) and 4(f) show the measured intensity patterns of reflected light from RCLC and LCLC when interfering with a plane wave, respectively, which manifests a topological charge of $l=-2$ and $m=0$.

To generate and analyze arbitrary vector vortex beams on the hybrid-order Poincaré sphere, an optical setup shown in Fig. 5 is established. A helium-neon gas laser with operational wavelength $\lambda =632.8\mathrm{nm}$ and beam waist size ${\omega }_0=0.48\mathrm{mm}$ was utilized as the source. A polarizer (P1) and a QWP (QWP1) in part (I) is first inserted to transform the input laser into an arbitrary elliptical polarization on the fundamental Poincaré sphere. Here, the polarization state of the laser beam after passing through P1 and QWP1 is investigated, which can be described by the orientation angle $\eta$ and the elliptical angle $\delta$ of the polarization ellipse, that is, $\eta$ is equal to the angle between the optical axis of QWP1 and the vertical direction, and $\delta$ is equal to the angle of the optical axis between P1 and QWP1. Next, the laser with elliptical polarization consecutively interacts with RCLC and LCLC in part (II), resulting in the generation of arbitrary vector vortex beams on the hybrid-order Poincaré sphere. It should be noted that LCLC and RCLC are stacked together in our experiment to ensure that the beam-size difference between RCP and LCP is small enough and, therefore, can be neglected, so we can only consider the evolution of polarization and phase. Here, any state of the vector vortex beam can be mapped into a point $(\theta ,\mathrm{\Phi })$ on the hybrid-order Poincaré sphere through rotating the angle of $\eta$ and $\delta$. A relationship between $(\eta ,\delta )$ and $(\theta ,\mathrm{\Phi })$ can be formulated as $\eta =\mathrm{\Phi }/2$ and $\delta =(2\theta -\pi )/4$. Obviously, rotating the optical axes of the P1 and QWP1 can, respectively, conduct the evolution of the state point in the longitude and latitude degrees on the surface of the hybrid-order Poincaré sphere. Furthermore, arbitrary vector vortex beams on the sphere can also be generated by switching the positions of RCLC and LCLC or integrating them together. Eventually, the beams reflected from CLC elements are deflected by the beam splitter (BS) and captured by the CCD camera.

Here, a QWP (QWP2) and a polarizer (P2) in part (III) are additionally inserted in front of the CCD camera to further depict the polarization state distribution of vector vortex beams. The Stokes parameters $S_1$, $S_2$, and $S_3$ of laser beams at each pixel in the cross-section of the laser beam, therefore, can be derived and expressed as^[45]$$S_1=\frac{I(0°,0°)-I(90°,90°)}{I(0°,0°)+I(90°,90°)},$$(6)$$S_2=\frac{I(45°,45°)-I(135°,135°)}{I(45°,45°)+I(135°,135°)},$$(7)$$S_3=\frac{I(-45°,0°)-I(45°,0°)}{I(-45°,0°)+I(45°,0°)},$$(8)where $I({\beta }_1,{\beta }_2)$ stands for the intensity of the laser beam measured by the CCD camera, and ${\beta }_1$ and ${\beta }_2$ represent the direction angles of the optical axes of QWP2 and P2 with regard to the vertical direction, respectively. Figure 6 experimentally and theoretically shows the intensity and polarization profiles of some typical vector vortex beams on the hybrid-order Poincaré sphere. The theoretical results of vector vortex beams on the north pole (0, 0,1), south pole $(0,0,-1)$, the intermediate point between the equator and north pole $(0,-\sqrt{2}/2,\sqrt{2}/2)$, and the intermediate point between the equator and south pole $(0,\sqrt{2}/2,-\sqrt{2}/2)$ are orderly plotted in the first row from left to right. Their experimental results are also exhibited in the second row, respectively. Here, the lasers on the north and south poles demonstrate a Laguerre–Gaussian mode laser with RCP $(|N_{l=-2}⟩)$ and a fundamental-mode Gaussian laser with LCP $(|S_{m=0}⟩)$, respectively. The vector vortex beams on the points of $(0,-\sqrt{2}/2,\sqrt{2}/2)$ and $(0,\sqrt{2}/2,-\sqrt{2}/2)$ are composed of the composition of $|N_{l=-2}⟩$ and $|S_{m=0}⟩$ in the intensity ratios of 5.83:1 and 1:5.83, respectively. Similarly, the theoretical results at four specified points $|D_{0,-2}⟩$, $|H_{0,-2}⟩$, $|A_{0,-2}⟩$, and $|V_{0,-2}⟩$, namely (0, 1, 0), (1, 0, 0), $(0,-1,0)$, and $(-1,0,0)$ on the equator, are orderly plotted in the third row from left to right. Their corresponding experimental results are also exhibited in the fourth row, respectively. The equatorial points on the hybrid-order Poincaré sphere represent the vector vortex beam states that have a superposition of two orthogonal bases $|N_{l=-2}⟩$ and $|S_{m=0}⟩$ with equal intensities.

In order to further verify the distributions of polarization states, radially and azimuthally polarized vector vortex beams, corresponding to $|H_{0,-2}⟩$ and $|V_{0,-2}⟩$ on the equator, are chosen to analyze their intensity patterns after passing through a polarizer with different directions (white arrows), as displayed in Fig. 7. The first row and second row, respectively, present the theoretical and experimental results of the radially polarized vector vortex beam. The third row and fourth row, respectively, present the theoretical and experimental results of the azimuthally polarized vector vortex beam. After passing through a polarizer, the beam exhibits a typical s-shaped pattern that rotates with the same direction and angle speed as the polarizer, manifesting a polarization order with $P=1$.

In summary, we first demonstrated the generation of arbitrary vector vortex beams on the hybrid-order Poincaré sphere with $l=-2$ and $m=0$ from CLCs. In our proposal, the scheme first used a polarizer and a quarter-wave plate to realize the generation and evolution of arbitrary polarization states on the fundamental Poincaré sphere. Afterward, benefitting from the polarization selectivity characteristic and Berry phase effect under Bragg reflection by chiral anisotropic optical media, two CLC planar optical elements with different surface patterns and opposite handedness were, therefore, utilized to generate two bases on the hybrid-order Poincaré sphere. Through the spin-to-orbital angular momentum conversion process conducted by RCLC and LCLC, the laser beam with an arbitrarily homogeneous polarization state can be transformed into a vector vortex beam, whose state can be mapped on the surface of the hybrid-order Poincaré sphere. Here, arbitrary vector vortex beams on the hybrid-order Poincaré sphere can be evolved through adjusting the directions of the polarizer and the quarter-wave plate. Furthermore, we numerically calculated the intensity profiles and polarization distributions of some typical vector vortex beams and then established the setup to record the experimental results. Comparisons of the theoretical and experimental results show good agreement. Our proposal can also make it possible to generate and modulate the states of vector vortex beams on arbitrary hybrid-order Poincaré spheres with other parameters $l$ and $m$ by changing the surface patterns of CLCs. In conclusion, our approach provides a robust and direct way to generate arbitrary vector vortex beams in the polychromatic spectral region. The proposal can also be utilized to tailor the intensities, phases, and polarization states of other beams through designing the surface patterns of CLCs. On the other hand, the demonstrated experimental configuration can also exhibit broader possibilities in switchable spiral phase contrast imaging^[46] and other applications.