Longitudinal modulation on light field polarization

Abstract

The polarization of light, a fundamental property governing light-matter interactions, has historically been engineered in the transverse plane perpendicular to its propagation direction and produced a various spatially structured light fields, namely, vector beams, exhibiting intriguing phenomena and effects in focusing and light-matter interaction. With the increasing demand for light field manipulation and multiplexing in more dimensions, the polarization modulations along the propagation direction have unveiled the potential of spatially varying polarization states along the optical axis, with significant propagation properties such as self-activity. This review synthesizes recent research on the longitudinal polarization engineering, emphasizing its theoretical foundations, generation methodologies, and transformative implications. We begin by outlining the polarization evolution dynamics of structured light fields during propagation, highlighting the three-dimensional (3D) variation of state of polarizations. Key techniques for realizing the longitudinal engineering of polarization without changing of transverse intensity profile are discussed. Finally, we discuss the prospect and challenges in longitudinal modulation of polarization such as achieving precise spatiotemporal control and dynamic reconfigurability.

Graphical abstract

Keywords

polarization / longitudinal modulation / vector beams / phase modulation

1 Introduction

Polarization, as one of the basic properties of light field, describing the vibration orientation of the electric field vector and playing a crucial role in manipulating light-matter interaction, has found extensive applications in fields such as polarization imaging, data storage, optical sensing and metrology [1−6]. These applications mostly use scalar light fields with the homogeneous polarization at each point in the transverse plane of the beam, which in turn brings the limitations such as mode mismatch. In contrast, vector light fields, characterized by spatially variant polarization states, have revolutionized photonics by enabling unprecedented control over light-matter interactions. Their unique polarization states and propagation dynamics underpin diverse applications spanning optical trapping, single-molecule imaging, nonlinear optics, super-resolution imaging, and quantum information [7−12].

The research of vector beams can be traced back to 1961, when the basic concept of vector light field was proposed by Snitzer [13]. In 1972, Pohl and Mushiake [14, 15] experimentally obtained the two most classical vector beams, radial polarized and angular polarized light fields, respectively. The polarization of these two typical vector beams at different positions in the transverse plane are all linearly polarized, and the polarization directions are along the radial and angular directions, respectively. Early research on vector beams progressed slowly, mainly focusing on the generation and theoretical foundations of vector beams. Not until 2000 did Youngworth demonstrated [16] that tightly focused radially and angularly polarized light fields could generate strong longitudinal electric and magnetic fields, respectively. This was the first time that the unique optical properties brought by the polarized structure of the light field were discovered, and since then, the research of vector beam has been set off.

The modulation of polarization in the transverse plane is no longer limited to localized linear polarization distributions. Hybrid polarization vector beams, Poincaré beams, singular vector beams, non-cylindrically symmetric vector beams, and polarization Möbius strips have been successively developed [17−32]. With the gradual enrichment of transverse plane modulation techniques, researchers have realized that polarization control should not be confined to two-dimensional (2D) planes (i.e., the x−y plane), but should further explore the propagation and evolution of polarization in 3D space. Light fields with longitudinally varying polarization have been purposely proposed, exhibiting novel phenomena. This can further expand the applications of vector light fields, making them have broad application prospects in fields such as high-resolution microscopic imaging, optical micromachining, optical detection and metrology, optical information storage, and high-capacity optical communication [33−51].

Fig.1 illustrates the overview of modulations on light field polarization. In this review, we report the recent research progress on the longitudinal modulation of vector beams, referring to the 3D and z-dependent modulations. In Section 2, we introduce generalized longitudinal modulation of polarization, which considers the evolutions of transverse polarization structures upon vector beams propagation, as well as their 3D modulation. In Section 3, we report z-dependent polarization modulations, that is, longitudinal engineering of polarizations without changing of transverse intensity profiles. Finally, we briefly summarize and outlook on the development trends and application prospects of longitudinal polarization modulation.

2 Generalized longitudinal modulation of polarization

In general, the polarization of light field does not change when it propagates in free space. However, for vector beams formed by the coherent superposition of different eigenmodes, their intensity, phase, and polarization structures vary during propagation because of the mode dispersion. On the other hand, this provides inversely a strategy for manipulating the evolution of polarization state in different lateral planes, as well as 3D polarization structures. Generalized longitudinal modulation of polarization refers to the significant changes in the light field structure as the polarization evolves along the longitudinal direction.

2.1 Spin separations in three dimensions

Whether generating vector beams through the coherent superposition of orthogonal polarization modes or utilizing polarization transition elements, the underlying principle of polarization modulation can be understood by decomposing the incident light fields into left- (LCP) and right-handed circular polarization (RCP) components. As illustrated in Fig.2(a), an incident scalar light field E_A can be described by its LCP and RCP, denoted as E_L,R, and e_L,R, respectively. Here, E_L,R represent the complex amplitude distributions of the two spin states of the incident light field, while e_L,R denote the unit vectors corresponding to the LCP and RCP, respectively. After passing through the polarization modulation system, the output light field is E_B = M·E_A = E_Le^i2φe_R + E_Re^−i2φe_L. During the polarization transition process, the two spin states exhibit distinct responses to the polarization transition system, acquiring phases of 2φ(x, y) and −2φ(x, y), respectively. This phase is a typical geometric phase, also known as the Pancharatnam−Berry (PB) phase, which is commonly described by the solid angle corresponding to the polarization transition trajectory represented on the Poincaré sphere, as illustrated in the bottom left of Fig.2(a). Utilizing the PB phases not only enables wavefront shaping of the two spin components of the light field but also facilitates the modulation of their propagations.

When the PB phase is 1D and linear, e.g., 2φ(x, y) = k_xx, where k_x represents the wave vector component of the plane wave along the x-direction, the incident linearly polarized light, after passing through the polarization manipulation systems such as 1D liquid crystal or metasurface polarization grating, generates a vector light field with horizontally varying polarization orientation. In this process, the LCP and RCP components acquire tilted phases k_xx and −k_xx, respectively. Beyond the output plane, under the linear PB phase effects, these two spin components undergo transverse separation during propagation, generating a phenomenon known as the photonic spin Hall effect [55, 56]. In 2002, Hasman et al. [57] proposed spatially varying subwavelength gratings and discovered their polarization selectivity, wherein circularly polarized beams of opposite handedness are diffracted in different directions. In 2013, Zhang et al. [58] utilized an optical metasurface to achieve an enhanced photonic spin Hall effect. The metasurface consisted of V-shaped gold nanoantennas with a transverse periodic variation, and incident linear polarized beams acquired opposite transverse phase gradients for their spin components, leading to transverse separation. In 2015, Luo et al. [59] realized a significantly enhanced photonic spin Hall effect using periodic curved nanogratings induced in glass by femtosecond laser. If the incident linearly polarized light field is chosen to be a vortex beam, Hermite−Gaussian beam, Airy beam, or even a beam with an arbitrary intensity distribution, the output light field will split into two beams with identical complex amplitude distributions but opposite circular polarization, i.e., LCP and RCP [60].

When the PB phase is 2D and quadratic, e.g., 2φ(x, y) = αr², where α is a constant controlling the divergence and convergence of the light field, the LCP and RCP components of the beam acquire (are multiplied by) the focusing phases exp(iαr²) and exp(−iαr²), respectively. This results in the LCP component diverging (for α>0) while the RCP component converges to a focal point at f(α) = k/{2[α + (αω⁴)⁻¹]}, where k is the wave vector magnitude (wavenumber) and ω is the radius of the incident Gaussian beam. Consequently, the two components form spin-dependent focusing. In 2003, Hasman et al. [61] utilized radially varying nanogratings to realize a polarization-dependent bifocal lens with focal lengths of ±f. In 2014, Cui et al. [62] discovered the dual-focusing phenomenon in radially polarization-varying light fields. In 2016, our research group modulated the polarization transition characteristics of the beam using a Sagnac interferometer [52]. By controlling the PB phases, we achieved not only transverse and longitudinal separation of the two spin states but also arbitrary 3D spatial separation of the spin states through their combination. For example, when the PB phase is set to 2φ(x, y) = k_xx + k_yy + αr², the two spin states, after passing through the polarization transition system, focus at coordinates (x_R, y_R, z_R) and (x_L, y_L, z_L), respectively, where z_R,L = f(α₀ ± α), x_R,L = k_xz_R,L/k, and y_R,L = k_yz_R,L/k, as illustrated in Fig.2(b).

Luo et al. [63] proposed a photonic spin filter by leveraging the spin-dependent focusing and defocusing effect. This filter consists of two confocal metalenses. Under the action of the first lens, the incident light field induces converging and diverging behaviors for the two spin states, respectively. A low-pass filter is applied at the spectral plane using a pinhole to eliminate the diverging spin state, and the spin state is then restored by the second metalens. Zhang et al. [64] achieved spin-dependent focusing and separation using a metasurface and further utilized this capability to realize polarization-selective imaging. Furthermore, Chen et al. [53] achieved a multifocal lens based on the PB phase using a dielectric metasurface with partitioned regions. Fig.2(c) schematically illustrates the working principle of a longitudinal multi-focal lens. To realize multi-focal points, the metasurface is divided into three annular regions with equal area but distinct PB phases, thereby focusing the incident fields onto multiple longitudinal focal points. The experimental results are shown in Fig.2(d).

Metasurfaces have recently gained widespread application in light field modulation due to their ability to simultaneously and independently control multiple wavefront parameters, including amplitude, phase, and polarization [65−69]. In 2019, Li et al. [54] achieved independent modulation of spin-dependent focal points by controlling the PB phase and the dynamic phase within a dielectric metasurface. The top part of Fig.2(e) schematically illustrates the simultaneous control of two spin states in both transverse and longitudinal directions. The bottom part of Fig.2(e) presents the dynamic phase (left), the PB phase (middle), and the combined modulation phase (right) used for the independent modulation of different spin states. When linear polarized light is normally incident on the metasurface, the LCP component acquires both the PB phase and the dynamic phase, while the RCP component acquires the conjugate PB phase and the same dynamic phase. After passing through the metasurface, the two circularly polarized components experience different transverse deflection angles and focus at distinct longitudinal positions, as shown in Fig.2(f). Zang et al. [70] further employed the joint modulation of PB phase and dynamic phase within a terahertz metasurface, to realize the polarization rotation of linearly polarized dual foci. The rotation angle of the polarization was correlated with the rotation angle of the nanostructures on the metasurface. As illustrated in Fig.3(a), the metasurface enables distinct polarization rotations of incident light along the longitudinal direction. Fig.3(b) shows the y-polarized and x-polarized intensity distributions of the polarization-rotated bifocal points in the x−z plane. Li et al. [71] achieved arbitrary polarization rotation and wavefront manipulation in the terahertz band based on the PB phase of a quaternary metasurface. Tian et al. [72] introduced a spin-dependent dual-focus phase and achieved longitudinal spin-dependent separation by combining the modulation of PB phase and dynamic phase, as depicted in Fig.3(c). When the ellipticity of the incident light was modulated, the longitudinal densities of the two spin states changed accordingly, as demonstrated in Fig.3(d).

In 2021, Song et al. [74] developed a multifunctional metasurface based on the phase transition material (VO₂) and the PB phase. By employing dual modulation of polarization and temperature, they achieved dynamic tunability in both the intensity and position of spin separation. Wang et al. [73] utilized a two-layer cascaded metasurface, where the combined modulation of the PB phase and the dynamic phase imparted distinct phase profiles to the LCP and RCP components. When the two-layer cascaded metasurface was transverse displaced, the focal points associated with different spin states were continuously tuned along the longitudinal direction. Fig.3(e) illustrates the schematic of the bifocal metalens. When arbitrarily polarized light passes through the two-layer cascaded metasurface, the output light retains the same polarization as the incident light, as shown in Fig.3(f). As the two-layer cascaded metasurface is continuously displaced by a distance d in the transverse direction, the focal points corresponding to different spin states are continuously longitudinal displaced, as depicted in Fig.3(g).

In 2023, Wang et al. [75] efficiently designed a bifocal metalens capable of longitudinal spin separation by combining deep learning and genetic algorithms, significantly reducing the design time. The intensity ratio of the two spin components could be continuously modulated by tuning the ellipticity of the incident beam. In 2024, Zhou et al. [76] achieved longitudinal spin separation modulation through the modulation of the PB phase in a bilayer liquid crystal structure, with the densities of the two spin components dynamically controllable. When circularly polarized light passed through the first liquid crystal layer, a portion was converted into cross-polarized components via PB phase modulation, while the remainder remained unconverted. As the two orthogonal circularly polarized components traversed the second liquid crystal layer, they were simultaneously modulated by the PB phase, thereby realizing spin separation control. Leveraging the electrically tunable polarization conversion efficiency of the liquid crystal, the densities of the two spin components could be dynamically modulated by applying an external voltage to the first liquid crystal layer.

2.2 3D vectorial light fields

Beyond the longitudinal polarization evolution arising from discrete intensity variations induced by multiple spin-dependent foci, the continuous longitudinal modulation of complex amplitude corresponding to orthogonal polarizations enables precise manipulation of 3D polarization configurations, namely, 3D vectorial light fields. As illustrated in Fig.4(a) and (b), the method of interference superposition can be combined with the phase modulation principle to control the 3D position of the focal point, enabling the generation of discrete 3D vectorial light fields with elaborate intensity and polarization distributions [77]. In Fig.4(c) and (d), by combining the Fourier only-phase encoding, the macro-pixel encoding, and the interference superposition, multiple perfect vortex vector beams (PVVBs) with controllable shapes can be simultaneously and independently constructed in 3D space without the need for spatial filtering [78]. Holography constitutes a technique for recording and reconstructing light field information. Holography is a technique for recording and reconstructing light field information. Our group combined holography with the spin-decoupled complex amplitude modulation of a tetratomic metasurface to achieve a polarization-switchable stereoscopic holographic scene, as shown in Fig.4(e) and (f) [79].

On the other hand, by reversely designing the trajectories of orthogonal basis, vector light fields with polarization varying along 3D curves can be produced, as shown in Fig.5(a) and (b) [80]. Specially, 3D PVVBs, of which customized intensity profiles are independent on the polarization orders, can be generated in different planes nearby the focus. This offers significant potential for applications such as optical trapping and optical encryption. Extending PVVBs from 2D to 3D would provide an additional spatial control dimension and enhance information capacity for applications like 3D modulation of particles and higher security information processing. To this end, Liu et al. [81] proposed a theoretical model for constructing 3D PVVBs, using a silicon carbide metasurface, which achieved the coaxial superposition of two orthogonal circularly polarized 3D perfect vortex beams, enabling the generation of arbitrary 3D PVVBs with identical spatial trajectories but distinct polarization orders, as shown in Fig.5(c) and (d) [81]. Fig.5(e) shows the design principle and experimental demonstration of optical encoding and image encryption for a 3D PVVB array based on a metasurface. It should be noted that light fields focused on different focal planes generate background crosstalk, which affects the local polarization purity of 3D vector fields. The quantization performance varies according to the generation method employed.

Jones matrix provides a mathematical framework for characterizing light field polarization. In 2023, Zhao et al. [82] combined Jones matrix and holography by proposing a longitudinal polarization transition method based on stereo Jones matrix holography. Through metasurface design, they demonstrated continuous 3D polarization modulation of the light field. As depicted in Fig.5(f), the target light field undergoes decomposition into multiple transverse planes, with corresponding Jones matrices computed for each plane. Utilizing backpropagation and iterative optimization, the researchers engineered the metasurface structure to achieve precise phase and amplitude modulation. Notably, Li and Wang proposed and experimentally demonstrated holographic metalenses capable of generating arbitrarily shaped focal curves with predefined polarization distributions, as shown in Fig.5(g). Their work successfully produces longitudinally variable 3D polarization knots, whose polarization profiles can be dynamically modulated by manipulating the incident light’s linear polarization direction. This approach, which combines polarization, color, and longitudinal control in 3D space, provides additional degrees of freedom for engineering complex vector beams [83−88]. Nan et al. [89] achieved polarization control at arbitrary trajectory points in the terahertz band by modulation the amplitude and phase difference of orthogonally polarized beams using a three-layer metasurface. This enabled continuous variations such as transitioning from 15° linear polarization to 75° linear polarization and from 15° linear polarization to RCP and then to 75° linear polarization along a helical trajectory. Furthermore, substantial research has focused on harnessing the polarization distribution of tightly focused fields, demonstrating significant advancements in controlling the 3D polarization distribution [42, 90−99]. In fact, the creation of 3D vector light fields is principally accomplished through modulation of both intensity and polarization across various longitudinal positions. The engineered 3D vector light fields exhibit considerable potential for applications in optical information encryption, data storage, and micromanipulation.

3 z-dependent polarization variation

During propagation of the aforementioned light field, the polarization distribution undergoes continuous transformation due to longitudinal spin intensity changes. Concurrently, the intensity distributions of these light fields exhibit significant structural modifications. The generation of a specialized light field featuring stable mode characteristics and a longitudinally continuous polarization variation would substantially broaden polarization applications. Consequently, this has led to the conceptualization of narrow-meaning longitudinal polarization modulation, wherein polarization varies along the propagation axis while maintaining near-constant field intensity structure. The Bessel beams, canonical non-diffracting beams characterized by linearly varying Gouy phases, have emerged as the optimal solution for achieving z-dependent polarization state control.

3.1 Polarization modulation based on transverse-longitudinal mapping

Within the framework of the linear focusing approach for Bessel beam generation, transverse-longitudinal mapping has been initially proposed as a method for creating longitudinally modulated nondiffracting optical fields featuring linearly rotating polarization structures during propagation [100, 101]. As depicted in Fig.6(a), a conical birefringent prism was implemented to induce radially varying phase retardation, thereby producing radially varying polarization. Subsequently, by exploiting the transverse-longitudinal mapping relationship inherent in Bessel beam generation via an axicon, the polarization was systematically controlled at varying propagation distances, achieving scalar nondiffracting light fields with longitudinally varying polarization. Fig.6(b) demonstrates the experimental results, revealing constant intensity throughout beam propagation while exhibiting polarization variation at the beam center. Additionally, through the integration of a polarization transition system with an axicon, radially linear phase variations with distinct slopes (interpretable as PB phases) were introduced for both two spin components, as shown in Fig.6(c). This configuration generated vector Bessel beams with z-dependent polarization and polarization order variations, as illustrated in Fig.6(d) [102]. Jankowski et al. [103] proposed a composite axicon device capable of generating Bessel beams with longitudinally varying polarization states. The composite axicon consists of a refractive axicon and a liquid crystal (LC) tunable axicon, where the LC axicon influences the polarization component of the incident beam parallel to the LC director. When linearly polarized light is incident on the composite axicon, two Bessel beams with different transverse wavenumbers are generated and coaxially superimposed, resulting in a periodic variation of the polarization state along the longitudinal direction. The period of this variation can be dynamically controlled by applying an electric field to tune the LC axicon.

In addition, Zheng et al. [105] utilized the modulation of PB phase and dynamic phase to integrate two long depth of focus vortex beams with different focal lengths into orthogonal circularly polarized components. Through the superimposition of these orthogonal modes, they generated vectorial vortex beams exhibiting longitudinally varying polarization. Subsequently, He et al. [104] employed a spin-decoupled spatial segmentation method to design a displacement between regions of opposite spin states, enabling independent control of LCP and RCP components, thereby producing vector modes with arbitrary tunability along the propagation direction. As illustrated in Fig.6(e), the schematic depicts the generation of high-order vectorial vortex beams using a metasurface. Fig.6(f) shows the distribution of the intensity and purity of the mode polarization components of the generated high-order vector beams within the corresponding longitudinal range, demonstrating that this method effectively suppresses intermodal crosstalk.

Wang et al. [106] developed a dual-mode light field featuring distinct focal depths by implementing partitioned joint modulation of both PB phase and dynamic phase on a metasurface, as shown in Fig.7(a) and (b). This configuration enabled longitudinal polarization evolution capable of transitioning between scalar-to-vector or vector-to-vector. Subsequently, Yang et al. [107] demonstrated continuous polarization state control of Bessel beams along the propagation axis. By leveraging the transverse-longitudinal mapping relationship, the metasurface incorporated a radially dependent phase gradient in the transverse plane, producing a longitudinally modulated phase difference between LCP and RCP components. This manipulation yielded Bessel beams exhibiting longitudinal periodic polarization state variations, as illustrated in Fig.7(c) and (d). The researchers further established a metasurface-based methodology for customizing light transmission trajectories, polarization, and orbital angular momentum (OAM) , as demonstrated in Fig.7(e) and (f) [108]. By partitioning phase design, they achieved z-dependent OAM variation along the propagation axis. A radially dependent phase profile introduced longitudinally tunable phase difference between LCP and RCP components. Additionally, phase compensation design enabled precise control over the beam’s longitudinal propagation path.

3.2 Polarization modulation based on longitudinal amplitude control

According to the Jones vector formalism, a light field exhibiting z-dependent polarization while maintaining a nearly constant field intensity profile can be described as the represented as the superposition of two Bessel beams with engineered z-dependent complex amplitudes. Thus, the electric field distribution of the vector field can be expressed as

(1)E\left(r,z\right)=\exp \left(-{\displaystyle \frac{r^2}{{\omega }_0^2}}\right)J_0\left(k_{r}r\right)\exp \left(\mathrm{i}{k}_{z}z\right)\left(\begin{array}{c}{E}_{\mathrm{H}}\left(z\right){\mathrm{e}}^{\mathrm{i}{\delta }_{\mathrm{H}}\left(z\right)}\\ E_{\mathrm{V}}\left(z\right){\mathrm{e}}^{\mathrm{i}{\delta }_{\mathrm{V}}\left(z\right)}\end{array}\right),

where E_H,V(z) represents the normalized longitudinal amplitude envelope functions of the horizontal and vertical polarization components, respectively, and δ_H,V(z) denotes the phase retardation. Based on the relationship between the Jones vector and the polarization ellipse, the aforementioned polarization can be expressed in terms of the ellipticity angle χ and the azimuthal angle φ as

(2)\begin{array}{rl}& \sin \left(2\chi \right)={\displaystyle \frac{2E_{\mathrm{H}}{E}_{\mathrm{V}}\sin \left({\delta }_{\mathrm{H}}-{\delta }_{\mathrm{V}}\right)}{E_{\mathrm{H}}^2+E_{\mathrm{V}}^2}},\\ & \tan \left(2\phi \right)={\displaystyle \frac{2E_{\mathrm{H}}{E}_{\mathrm{V}}\cos \left({\delta }_{\mathrm{H}}-{\delta }_{\mathrm{V}}\right)}{E_{\mathrm{H}}^2-E_{\mathrm{V}}^2}}.\end{array}

Equation (2) indicates that longitudinal continuous polarization variation can be achieved through two distinct mechanisms: (i) modulation of the on-axis amplitude ratio between two polarization components (E_H(z)/E_V(z) ≠ const) via longitudinal intensity envelop functions, i.e., the longitudinal intensity envelope functions [109]; and (ii) modulation of the inter-component phase difference [δ_H(z) − δ_V(z) ≠ const] [110, 111].

In this principle, our group proposed a nondiffracting light field featuring longitudinally modulated polarization, achieved through spatial spectral modulation to control the amplitude relationship between two orthogonal polarization components along the propagation axis [112]. Fig.8(a) presents a schematic of the axial amplitude envelope modulation principle. By independently modulating the complex amplitudes of the two orthogonal polarization components in the spectral domain, intensity envelopes exhibiting linear axial variations (either increasing or decreasing) are achieved in real space. The superimposed field retains nondiffracting characteristics due to the implementation of Bessel spectra. Fig.8(b) displays the experimentally measured intensity variations of both polarization components as a function of propagation distance. The nonlinear variation in the amplitude ratio between the orthogonal components induces a nonlinear polarization shift, highlighting the method’s versatility in controlling polarization evolution. Furthermore, this light field can exhibit polarization evolution along arbitrary trajectory on the Poincaré sphere during propagation. Within the Poincaré sphere representation of polarization, each point (2φ, 2χ) corresponding to a unique polarization that can be decomposed into any pair of orthogonal polarization components symmetric about the sphere’s center. Specially, the coherent superposition of two orthogonal polarization components with arbitrary amplitude ratios and phase differences can yield the polarization represented on the Poincaré sphere.

Another method for controlling the on-axial amplitude envelope of light fields is frozen waves, which was first introduced by Zamboni-Rached et al. [113, 114]. This longitudinally modulated light field is formed by the coaxial superposition of a series of Bessel beams with the same order but different transverse wavevectors. Since any linear combination of Bessel functions can serve as a solution to the wave equation, optical frozen waves can also be regarded as a new class of nondiffracting solutions that remain stable within a specific range. The characteristic feature of frozen waves is that the group velocity of the wave packet is zero (v = 0), and the longitudinal structure of the field intensity can be arbitrarily modulated, including intensity profile and even OAM mode. Fig.8(c) shows the principle of constructing a polarization oscillating beam by realizing amplitude modulation based on the frozen wave [115]. Assuming the two frozen waves have a constant phase difference of π/2 and carry orbital angular momenta (OAM) of mℏ and nℏ, respectively. The synthesized field exhibits periodic polarization mode changes along the red trajectory on the hybrid Poincaré sphere during propagation. Fig.8(d) displays the experimental results of the polarization oscillating beams, showing both the intensity profiles of the two frozen waves and the transverse polarization distributions of the synthesized field at equidistant planes. The experimental results demonstrate that the predefined intensity distributions of the frozen waves generate longitudinal polarization variations.

The third method for modulating the axial amplitude of light field involves multifocal combination [116]. As depicted in Fig.8(e) (top), when linearly polarized light impinges upon the metasurface, it undergoes conversion into two distinct beams exhibiting longitudinal polarization evolution. The first beam demonstrates a polarization transition from vertical to horizontal along the propagation axis, while the second beam evolves from LCP to RCP. By encoding both the desired focusing phases and initial phases into the incident LCP and RCP components, the target functionality can be realized. The requisite phase distributions are as follows:

(3)\begin{array}{rl}& {\phi }_{\mathrm{L}\mathrm{C}\mathrm{P}}^n=\sum _{i=1}^n\left[\frac{2\pi }{\lambda }\sqrt{{\left(x-x_i\right)}^2+{\left(y-y_i\right)}^2+f_i^2}-f_i\right],\\ & {\phi }_{\mathrm{R}\mathrm{C}\mathrm{P}}^m=\sum _{j=1}^n\left[\frac{2\pi }{\lambda }\sqrt{{\left(x-x_j\right)}^2+{\left(y-y_j\right)}^2+f_j^2}-f_j\right],\\ & {\phi }_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}=\arg \left[\exp \left(\mathrm{i}{\phi }_{\mathrm{L}\mathrm{C}\mathrm{P}}^n+{\alpha }_n\right)+\exp \left(\mathrm{i}{\phi }_{\mathrm{R}\mathrm{C}\mathrm{P}}^m+{\alpha }_m\right)\right],\end{array}

where {\phi }_{\mathrm{L}\mathrm{C}\mathrm{P}}^n and {\phi }_{\mathrm{R}\mathrm{C}\mathrm{P}}^m focus the incident LCP and RCP components into n and m focal points, respectively, while φ_total denotes the metasurface phase requirement for generating multiple Bessel-like beams with longitudinally varying polarization. Fig.8(e) (bottom) schematically illustrates the polarization evolution imaging process, where the imaging sample is positioned at different transverse planes of the Bessel-like beam to demonstrate polarization-switching imaging capabilites. Fig.8(f) presents the experimental results of the longitudinally varying polarization beams and longitudinal polarization evolution imaging.

3.3 Polarization modulation based on longitudinal phase control

According to the second mechanism presented in Eq. (2), the abundant Laguerre−Gaussian (LG) and Bessel−Gaussian (BG) modes give rise to modulate the mode dispersion and inter-component phase difference, thereby enabling flexible manipulation of longitudinal polarization transitions. In 2013, Cardano et al. [117] demonstrated the generation of Poincaré beams exhibiting polarization rotation along the propagation axis through coaxial superposition of orthogonally polarized Gaussian beams and LG beams. The distinct mode characteristics of these beam types produce a propagation-dependent Gouy phase shift. Within the Rayleigh range, this phase difference induces a continuous rotation of beam’s polarization structure, resulting in spatial rotation of lemon, star, and spiral topological structures. However, owing to the gradual variation of Gouy phase shift, the polarization structure completes only a π/2 rotation between −z_R and z_R. In 2017, we demonstrated longitudinally varying polarization in zero-order Bessel beams through the PB phase modulation, as depicted in Fig.9(a) and (b) [118]. These 3D polarization variations also exhibit self-healing properties, as shown in Fig.9(a) (bottom).

By exploiting the Gouy phase shift characteristics of radial higher-order LG and BG modes, we further achieved a nonlinear polarization transformation along the longitudinal direction [119]. Our investigation revealed that radial higher-order LG mode not only demonstrate mode distributions remarkably similar to Bessel modes of corresponding orders but also exhibit comparable nondiffracting properties, thereby ensuring optical mode propagation stability. The Gouy phase shift of LG beams manifests as a nonlinear function of the propagation distance, whereas Bessel beam exhibit a linear Gouy phase shift dependence. By configuring the beam’s orthogonal circular polarization components as Bessel and corresponding LG beams, respectively, we observed a nonlinear variation in their phase difference, as illustrated in Fig.9(c). Fig.9(d) shows the evolution of the total beam intensity alongside its orthogonal polarization components. Fig.9(e) presents the theoretical predictions and experimental measurements of polarization direction (top) and the ellipticity (bottom) as a function of propagation distance, confirming nonlinear on-axis polarization rotation. To assess polarization stability within the beam’s paraxial region, we measured ellipticity versus propagation distance. The results indicate near-linear polarization maintenance throughout propagation. Furthermore, Lü et al. [121] achieved controlled polarization rotation direction and rate in Bessel-like beams through amplitude and phase modulation, adjusting radial parameters and intensity ratios of superimposed beams. Li et al. [122] achieved continuous variation of vector optical fields along the propagation path and independent control on multiple planes by introducing continuous and discrete phase differences for the LCP and RCP components, respectively, and using long focal depth and multifocal metalenses in the terahertz band. Subsequently, they achieved the manipulation of terahertz linear polarization along the propagation path, and proposed a refractive index sensing application. By introducing an additional medium into the propagation path, the polarization direction at the output plane rotates as the refractive index of the medium changes, thus facilitating the measurement of the refractive index of an unknown medium [123].

The distinct Gouy phase shift between the two orthogonal circular polarization components results in differential phase velocities during propagation, analogous to spin mode birefringence in free space. PB phase elements can impose opposing phase modulations on the polarization components of Bessel beams, thereby modifying their respective Gouy phase shifts. This mechanism enables the realization of optical-activity-like effects for Bessel beams in free space. Our group designed a liquid crystal PB phase element capable of generating conical phase distribution (termed a conical-wave plate, CWP), which splits the wave vector of a Bessel beam into two orthogonally circular polarizations with distinct transverse wave vector components, as shown in Fig.9(f) [120]. Consequently, two Bessel beams with differing Gouy phase shift are produced, inducing polarization rotation during propagation. Fig.9(g) demonstrates a polarization rotator constructed using this principle, where beam polarization rotates between two oppositely oriented CWPs. By modulating the inter-CWP distance, the output light’s polarization rotation angle can be precisely controlled. This device enables the fabrication of medium-free optical isolators and circulators. The intrinsic nature of the Gouy phase shift indicates that this optical-activity-like phenomenon is generalizable to diverse wave-based physical systems.

3.4 Polarization modulation based on Jones matrix optics

Among the aforementioned methods, longitudinal variations in polarization state exhibit a strong dependence on the incident polarization state. In contrast, Capasso et al. [124] proposed a Jones matrix metasurface optics to decouple the output response from the input polarization, thereby enabling z-dependent polarization transformation independent of the incident polarization, as shown in Fig.10(a). Furthermore, this device allows the response simultaneously to all incident polarizations. Since the Jones matrix can be used to describe any polarization device, in order to decouple the output response from the input polarization, according to the required polarization transformation, a Jones matrix function \tilde{F}(z) that varies along the optical axis is first constructed. Moreover, \tilde{F}(z) can be arbitrarily selected, such as a longitudinally varying polarizer, and wave plate. Then, the coefficients \tilde{A}(z) of the 2 × 2 matrix values of 2N+1 Bessel functions are calculated based on \tilde{F}(z), as shown in Fig.10(b). Finally, according to the frozen wave theory, a longitudinally variable response can be constructed over a long-distance range. Fig.10(c) shows the measured and simulated on axis intensity variations of the light field when the metasurface device functions as a linear polarizer under illumination with incident light of different polarization (black arrows). The red arrows depict the actual optical axis orientations of the metasurface polarizer at five distinct z planes. The results demonstrate that a single integrated metasurface device achieves universal polarization control, enabling dynamic tuning of the polarization with a static metasurface.

4 Outlook and perspective

Since the introduction of radially and azimuthally polarized light fields in the early 1970s, extensive research has been dedicated to investigating the polarization structures of light fields and their propagation characteristics. Different vector fields report in the literature have found widespread applications in polarization imaging, super-resolution imaging, and particle manipulation [34−36, 125−128]. Throughout this process, the principles, methods, and devices for the longitudinal modulation of light field polarization structures have continuously evolved. Longitudinal modulations of polarization, including spin component separation and spin-selective focusing, have been extensively achieved through tailored spin-dependent phase control. Diverse z-axis polarization variations and 3D polarization structures have been engineered via precise manipulation of light field intensity, phase, and spatial positioning. Emerging innovations in polarization structured light fields and their design methods have been discussed and reviewed from two architectures.

Remarkable advancements have been achieved in the longitudinal control of polarization within the light field, demonstrating significant potential across diverse application scenarios. Since polarization critically governs light−matter interactions, extending polarization manipulation from the transverse plane to 3D space enables expands the dimensionality for optical storage [46, 129], particle manipulation [130, 131], etc. For example, based on the direction-dependent rotator shown in Fig.9(g), Liu et al. [120] demonstrated a possibility to form the isolator and circulator, as illustrated by Fig.11(a), which are essential components used to manage signal flow in communication systems. The 3D polarization structure, which combines transverse and longitudinal manipulation, introduces a novel degree of freedom for optical information processing. This advancement offers precise control over depth information and longitudinal resolution in optical imaging by correlating polarization changes with point spread functions [132, 133]. Ren et al. [134] proposed 3D vectorial holography where an arbitrary 3D vectorial field distribution on a wavefront can be precisely reconstructed, as shown in Fig.11(b). This 3D vectorial holography allows the lensless reconstruction of a 3D vectorial holographic image with an ultrawide viewing angle and a high diffraction efficiency [134]. Moreover, 3D polarization variation facilitates the breaking of space-polarization mode inseparability, thereby creating new possibilities for quantum information transmission. For instance, Otte et al. [135] demonstrated longitudinally variant non-separable state enable z-dependent dynamic entanglement, namely, entanglement beating in free space, as shown in Fig.11(c) [135]. However, more work is needed to address the remaining challenges in practice application such as longitudinally controllable range and accuracy of polarization variation. Because achieving more precise longitudinal variations in polarization, such as compressing the longitudinal change period to the wavelength scale, is critical for advancing the application of such optical fields in processing, measurement, and manipulation.

Deep integration of longitudinal polarization modulation technology with emerging fields will bring unprecedented development opportunities. The combination with artificial intelligence will inaugurate a new era of intelligent control of light fields. Artificial intelligence algorithms, especially machine learning and deep learning techniques, possess powerful data processing and pattern recognition capabilities. These algorithms can be utilized to analyze and learn from a large amount of light field data, establishing a complex mapping relationship between the polarization of the light field and application requirements. For instance, in the field of optical information encryption, complex polarization encoding keys can be generated through deep learning algorithms to achieve more secure information transmission [136]. Meanwhile, artificial intelligence can be employed to monitor real time changes in the light field and automatically adjust modulation parameters, realizing adaptive control of the light field polarization structure and enhancing the efficiency and stability of light field applications [137, 138].

The research and development of novel optical devices also represent a crucial direction for future studies. As this review demonstrates, polarization modulation optics have evolved from the cumbersome coherent interferometer configurations to compact and multifunctional metasurfaces, with substantial enhancements in both efficiency and stability. At present, most optical devices rely on the polarization of the incident light field and the characteristics of the propagating wave, which restricts their utilization in complex environments and diverse application scenarios [124, 139]. In the future, it is necessary to explore new optical principles and materials that are independent of these traditional conditions. Furthermore, optical modulators based on phase variation materials can be developed. By applying external stimuli (such as light, electricity, or heat), phase variation in the materials can be induced, thereby enabling dynamic control of the polarization of the light field [74, 76]. This provides the possibility for the realization of reconfigurable and multi-functional optical devices.

References

[1] N. A. Rubin , G. D'Aversa , P. Chevalier , Z. Shi , W. T. Chen , and F. Capasso , Matrix Fourier optics enables a compact full-Stokes polarization camera, Science 365(6448), eaax1839 (2019)

[2] J. Hong , J. Van De Groep , N. Lee , S. J. Kim , P. Lalanne , P. G. Kik , and M. L. Brongersma , Nonlocal metasurface for circularly polarized light detection, Optica 10(1), 134 (2023)

[3] J. H. Song , J. van de Groep , S. J. Kim , and M. L. Brongersma , Non-local metasurfaces for spectrally decoupled wavefront manipulation and eye tracking, Nat. Nanotechnol. 16(11), 1224 (2021)

[4] C. X. Huang , J. Zhang , Q. Cheng , and T. J. Cui , Polarization modulation for wireless communications based on metasurfaces, Adv. Funct. Mater. 31(36), 2103379 (2021)

[5] H. Zang , Z. Xi , Z. Zhang , Y. Lu , and P. Wang , Ultrasensitive and long-range transverse displacement metrology with polarization-encoded metasurface, Sci. Adv. 8(41), eadd1973 (2022)

[6] F. Yesilkoy , E. R. Arvelo , Y. Jahani , M. Liu , A. Tittl , V. Cevher , Y. Kivshar , and H. Altug , Ultrasensitive hyperspectral imaging and biodetection enabled by dielectric metasurfaces, Nat. Photonics 13(6), 390 (2019)

[7] A. Ashkin , J. M. Dziedzic , J. E. Bjorkholm , and S. Chu , Observation of a single-beam gradient force optical trap for dielectric particles, Opt. Lett. 11(5), 288 (1986)

[8] Y. Roichman , B. Sun , Y. Roichman , J. Amato-Grill , and D. G. Grier , Optical forces arising from phase gradients, Phys. Rev. Lett. 100(1), 013602 (2008)

[9] L. Novotny , M. R. Beversluis , K. S. Youngworth , and T. G. Brown , Longitudinal field modes probed by single molecules, Phys. Rev. Lett. 86(23), 5251 (2001)

[10] A. Ciattoni , B. Crosignani , P. Di Porto , and A. Yariv , Azimuthally polarized spatial dark solitons: Exact solutions of Maxwell’s equations in a Kerr medium, Phys. Rev. Lett. 94(7), 073902 (2005)

[11] X. Li , P. Venugopalan , H. Ren , M. Hong , and M. Gu , Super-resolved pure-transverse focal fields with an enhanced energy density through focus of an azimuthally polarized first-order vortex beam, Opt. Lett. 39(20), 5961 (2014)

[12] C. Gabriel , A. Aiello , W. Zhong , T. G. Euser , N. Y. Joly , P. Banzer , M. Förtsch , D. Elser , U. L. Andersen , C. Marquardt , P. S. J. Russell , and G. Leuchs , Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes, Phys. Rev. Lett. 106(6), 060502 (2011)

[13] E. Snitzer , Cylindrical dielectric waveguide modes, J. Opt. Soc. Am. 51(5), 491 (1961)

[14] Y. Mushiake , N. Nakajima , and K. Matsumur , Generation of radially polarized optical beam mode by laser oscillation, Proc. Inst. Mech. Eng. 60(9), 1107 (1972)

[15] D. Pohl , Operation of a Ruby laser in the purely transverse electric mode TE₀₁, Appl. Phys. Lett. 20(7), 266 (1972)

[16] K. S. Youngworth and T. G. Brown , Focusing of high numerical aperture cylindrical-vector beams, Opt. Express 7(2), 77 (2000)

[17] H. T. Wang,X. L. Wang,Y. Li,J. Chen,C. S. Guo,J. Ding, A new type of vector fields with hybrid states of polarization, Opt. Express 18(10), 10786 (2010)

[18] G. M. Lerman , L. Stern , and U. Levy , Generation and tight focusing of hybridly polarized vector beams, Opt. Express 18(26), 27650 (2010)

[19] A. M. Beckley , T. G. Brown , and M. A. Alonso , Full Poincare beams, Opt. Express 18(10), 10777 (2010)

[20] A. M. Beckley , T. G. Brown , and M. A. Alonso , Full Poincare beams II: Partial polarization, Opt. Express 20(9), 9357 (2012)

[21] G. Arora , Ruchi , and P. Senthilkumaran , Full Poincare beam with all the Stokes vortices, Opt. Lett. 44(22), 5638 (2019)

[22] Z. H. Jiang,L. Kang,T. Yue,H. X. Xu,Y. Yang,Z. Jin,C. Yu,W. Hong,D. H. Werner,C. W. Qiu, A single noninterleaved metasurface for high-capacity and flexible mode multiplexing of higher-order Poincare sphere beams, Adv. Mater. 32(6), 1903983 (2020)

[23] E. J. Galvez , S. Khadka , W. H. Schubert , and S. Nomoto , Poincare-beam patterns produced by nonseparable superpositions of Laguerre-Gauss and polarization modes of light, Appl. Opt. 51(15), 2925 (2012)

[24] S. K. Pal and P. Senthilkumaran , Cultivation of lemon fields, Opt. Express 24(24), 28008 (2016)

[25] S. K. P. Ruchi and P. Senthilkumaran , Generation of V-point polarization singularity lattices, Opt. Express 25(16), 19326 (2017)

[26] S. K. Pal and P. Senthilkumaran , Lattice of C points at intensity nulls, Opt. Lett. 43(6), 1259 (2018)

[27] Y. Pan , Y. Li , S. M. Li , Z. C. Ren , Y. Si , C. Tu , and H. T. Wang , Vector optical fields with bipolar symmetry of linear polarization, Opt. Lett. 38(18), 3700 (2013)

[28] Y. Pan , Y. Li , Z. C. Ren , Y. Si , C. Tu , and H. T. Wang , Parabolic-symmetry vector optical fields and their tightly focusing properties, Phys. Rev. A 89(3), 035801 (2014)

[29] X. Z. Gao , Y. Pan , M. Q. Cai , Y. Li , C. Tu , and H. T. Wang , Hyperbolic-symmetry vector fields, Opt. Express 23(25), 32238 (2015)

[30] Y. Pan , Y. Li , S. M. Li , Z. C. Ren , L. J. Kong , C. Tu , and H. T. Wang , Elliptic-symmetry vector optical fields, Opt. Express 22(16), 19302 (2014)

[31] T. Bauer , P. Banzer , E. Karimi , S. Orlov , A. Rubano , L. Marrucci , E. Santamato , R. W. Boyd , and G. Leuchs , Observation of optical polarization Mobius strips, Science 347(6225), 964 (2015)

[32] T. Bauer , S. N. Khonina , I. Golub , G. Leuchs , and P. Banzer , Ultrafast spinning twisted ribbons of confined electric fields, Optica 7(10), 1228 (2020)

[33] Y. Kozawa , D. Matsunaga , and S. Sato , Superresolution imaging via superoscillation focusing of a radially polarized beam, Optica 5(2), 86 (2018)

[34] S. Segawa , Y. Kozawa , and S. Sato , Resolution enhancement of confocal microscopy by subtraction method with vector beams, Opt. Lett. 39(11), 3118 (2014)

[35] X. Xie , Y. Chen , K. Yang , and J. Zhou , Harnessing the point-spread function for high-resolution far-field optical microscopy, Phys. Rev. Lett. 113(26), 263901 (2014)

[36] S. Segawa , Y. Kozawa , and S. Sato , Demonstration of subtraction imaging in confocal microscopy with vector beams, Opt. Lett. 39(15), 4529 (2014)

[37] T. F. Scott , B. A. Kowalski , A. C. Sullivan , C. N. Bowman , and R. R. McLeod , Two-color single-photon photoinitiation and photoinhibition for subdiffraction photolithography, Science 324(5929), 913 (2009)

[38] Z. Gan , Y. Cao , R. A. Evans , and M. Gu , Three-dimensional deep sub-diffraction optical beam lithography with 9 nm feature size, Nat. Commun. 4(1), 2061 (2013)

[39] Q. W. Zhan and J. R. Leger , Microellipsometer with radial symmetry, Appl. Opt. 41(22), 4630 (2002)

[40] Q. W. Zhan and J. R. Leger , Measurement of surface features beyond the diffraction limit with an imaging ellipsometer, Opt. Lett. 27(10), 821 (2002)

[41] Q. W. Zhan and J. R. Leger , High-resolution imaging ellipsometer, Appl. Opt. 41(22), 4443 (2002)

[42] X. Li , T. H. Lan , C. H. Tien , and M. Gu , Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam, Nat. Commun. 3(1), 998 (2012)

[43] H. Ren , X. Li , and M. Gu , Polarization-multiplexed multifocal arrays by a π-phase-step-modulated azimuthally polarized beam, Opt. Lett. 39(24), 6771 (2014)

[44] M. Gu , X. Li , and Y. Cao , Optical storage arrays: A perspective for future big data storage, Light Sci. Appl. 3(5), e177 (2014)

[45] X. Li , Q. Zhang , X. Chen , and M. Gu , Giant refractive-index modulation by two-photon reduction of fluorescent graphene oxides for multimode optical recording, Sci. Rep. 3(1), 2819 (2013)

[46] P. Zijlstra , J. W. M. Chon , and M. Gu , Five-dimensional optical recording mediated by surface plasmons in gold nanorods, Nature 459(7245), 410 (2009)

[47] P. R. Dolan , X. Li , J. Storteboom , and M. Gu , Complete determination of the orientation of NV centers with radially polarized beams, Opt. Express 22(4), 4379 (2014)

[48] G. Gibson , J. Courtial , M. J. Padgett , M. Vasnetsov , V. Pas’ko , S. M. Barnett , and S. Franke-Arnold , Free-space information transfer using light beams carrying orbital angular momentum, Opt. Express 12(22), 5448 (2004)

[49] A. E. Willner , H. Huang , Y. Yan , Y. Ren , N. Ahmed , G. Xie , C. Bao , L. Li , Y. Cao , Z. Zhao , J. Wang , M. P. J. Lavery , M. Tur , S. Ramachandran , A. F. Molisch , N. Ashrafi , and S. Ashrafi , Optical communications using orbital angular momentum beams, Adv. Opt. Photonics 7(1), 66 (2015)

[50] W. Cheng , J. W. Haus , and Q. Zhan , Propagation of vector vortex beams through a turbulent atmosphere, Opt. Express 17(20), 17829 (2009)

[51] J. Wang , J. Y. Yang , I. M. Fazal , N. Ahmed , Y. Yan , H. Huang , Y. Ren , Y. Yue , S. Dolinar , M. Tur , and A. E. Willner , Terabit free-space data transmission employing orbital angular momentum multiplexing, Nat. Photonics 6(7), 488 (2012)

[52] S. Liu , P. Li , Y. Zhang , X. Gan , M. Wang , and J. Zhao , Longitudinal spin separation of light and its performance in three-dimensionally controllable spin-dependent focal shift, Sci. Rep. 6(1), 20774 (2016)

[53] X. Chen , M. Chen , M. Q. Mehmood , D. Wen , F. Yue , C. W. Qiu , and S. Zhang , Longitudinal multi-foci metalens for circularly polarized light, Adv. Opt. Mater. 3(9), 1201 (2015)

[54] S. Li , X. Li , G. Wang , S. Liu , L. Zhang , C. Zeng , L. Wang , Q. Sun , W. Zhao , and W. Zhang , Multidimensional manipulation of photonic spin Hall effect with a single-layer dielectric metasurface, Adv. Opt. Mater. 7(5), 1801365 (2019)

[55] M. Onoda , S. Murakami , and N. Nagaosa , Hall effect of light, Phys. Rev. Lett. 93(8), 083901 (2004)

[56] O. Hosten and P. Kwiat , Observation of the spin Hall effect of light via weak measurements, Science 319(5864), 787 (2008)

[57] Z. Bomzon , G. Biener , V. Kleiner , and E. Hasman , Space-variant Pancharatnam-Berry phase optical elements with computer-generated subwavelength gratings, Opt. Lett. 27(13), 1141 (2002)

[58] X. Yin , Z. Ye , J. Rho , Y. Wang , and X. Zhang , Photonic spin Hall effect at metasurfaces, Science 339(6126), 1405 (2013)

[59] X. Ling , X. Zhou , X. Yi , W. Shu , Y. Liu , S. Chen , H. Luo , S. Wen , and D. Fan , Giant photonic spin Hall effect in momentum space in a structured metamaterial with spatially varying birefringence, Light Sci. Appl. 4(5), e290 (2015)

[60] Y. Ke , Y. Liu , Y. He , J. Zhou , H. Luo , and S. Wen , Realization of spin-dependent splitting with arbitrary intensity patterns based on all-dielectric metasurfaces, Appl. Phys. Lett. 107(4), 041107 (2015)

[61] E. Hasman , V. Kleiner , G. Biener , and A. Niv , Polarization dependent focusing lens by use of quantized Pancharatnam-Berry phase diffractive optics, Appl. Phys. Lett. 82(3), 328 (2003)

[62] B. Gu , Y. Pan , J. L. Wu , and Y. Cui , Manipulation of radial-variant polarization for creating tunable bifocusing spots, J. Opt. Soc. Am. A 31(2), 253 (2014)

[63] Y. Ke , Y. Liu , J. Zhou , Y. Liu , H. Luo , and S. Wen , Photonic spin filter with dielectric metasurfaces, Opt. Express 23(26), 33079 (2015)

[64] S. Wang , X. Wang , Q. Kan , J. Ye , S. Feng , W. Sun , P. Han , S. Qu , and Y. Zhang , Spin-selected focusing and imaging based on metasurface lens, Opt. Express 23(20), 26434 (2015)

[65] X. Fan , P. Li , X. Guo , B. Li , Y. Li , S. Liu , Y. Zhang , and J. Zhao , Axially tailored light field by means of a dielectric metalens, Phys. Rev. Appl. 14(2), 024035 (2020)

[66] X. Guo , J. Zhong , B. Li , S. Qi , Y. Li , P. Li , D. Wen , S. Liu , B. Wei , and J. Zhao , Full-color holographic display and encryption with full-polarization degree of freedom, Adv. Mater. 34(3), 2103192 (2022)

[67] X. Guo , P. Li , J. Zhong , D. Wen , B. Wei , S. Liu , S. Qi , and J. Zhao , Stokes meta-hologram toward optical cryptography, Nat. Commun. 13(1), 6687 (2022)

[68] X. Bi , X. Wu , X. Fan , C. Zhao , D. Wen , S. Liu , X. Gan , J. Zhao , and P. Li , Concurrent image differentiation and integration processings enabled by polarization-multiplexed metasurface, Laser Photonics Rev. 19(4), 2400718 (2025)

[69] X. Bi , X. Guo , X. Wu , X. Fan , B. Wei , D. Wen , S. Liu , J. Zhao , and P. Li , Wideband optical edge detection based on dielectric metasurface, Appl. Phys. Lett. 123(6), 061702 (2023)

[70] X. Zang,H. Ding,Y. Intaravanne,L. Chen,Y. Peng,J. Xie,Q. Ke,A. V. Balakin,A. P. Shkurinov,X. Chen,Y. Zhu,S. Zhuang, A multi-foci metalens with polarization-rotated focal points, Laser Photonics Rev. 13(12), 1900182 (2019)

[71] J. Li , X. Lu , H. Li , C. Song , Q. Tan , Y. He , J. Liu , L. Luo , T. Tang , T. Liu , H. Xu , S. Xiao , W. Huang , Y. Shen , Y. Zhang , Y. Zhang , and J. Yao , Racemic dielectric metasurfaces for arbitrary terahertz polarization rotation and wavefront manipulation, Opto-Electron. Adv. 7(10), 240075 (2024)

[72] S. Tian , H. Guo , J. Hu , and S. Zhuang , Dielectric longitudinal bifocal metalens with adjustable intensity and high focusing efficiency, Opt. Express 27(2), 680 (2019)

[73] C. Wang , Y. Sun , Q. Zhang , Z. Yu , C. Tao , J. Zhang , F. Wu , R. Wu , and Z. Zheng , Continuous-zoom bifocal metalens by mutual motion of cascaded bilayer metasurfaces in the visible, Opt. Express 29(17), 26569 (2021)

[74] Y. Song , W. Liu , X. Wang , F. Wang , Z. Wei , H. Meng , N. Lin , and H. Zhang , Multifunctional metasurface lens with tunable focus based on phase transition material, Front. Phys. (Lausanne) 9, 651898 (2021)

[75] F. Wang and X. Shu , Design of a bifocal metalens with tunable intensity based on deep-learning-forward genetic algorithm, J. Phys. D Appl. Phys. 56(9), 095101 (2023)

[76] Y. Zhou , X. Ye , Z. Li , D. Tang , and F. Fan , Bifocal lenses with adjustable intensities enabled by bilayer liquid crystal structures, Opt. Lett. 49(19), 5495 (2024)

[77] T. Zeng , C. Chang , Z. Chen , H. T. Wang , and J. Ding , Three-dimensional vectorial multifocal arrays created by pseudo-period encoding, J. Opt. 20(6), 065605 (2018)

[78] P. Li , X. Fan , D. Wu , X. Guo , Y. Li , S. Liu , and J. Zhao , Shaping vector fields in three dimensions by random Fourier phase-only encoding, Opt. Express 27(21), 30009 (2019)

[79] X. Guo , J. Zhong , P. Li , D. Wen , S. Liu , B. Wei , S. Qi , and J. Zhao , Metasurface-assisted multidimensional manipulation of a light wave based on spin-decoupled complex amplitude modulation, Opt. Lett. 47(2), 353 (2022)

[80] C. Chang , Y. Gao , J. Xia , S. Nie , and J. Ding , Shaping of optical vector beams in three dimensions, Opt. Lett. 42(19), 3884 (2017)

[81] M. Liu , P. Lin , P. Huo , H. Qi , R. Jin , H. Zhang , Y. Ren , M. Song , Y. Lu , and T. Xu , Monolithic silicon carbide metasurfaces for engineering arbitrary 3D perfect vector vortex beams, Nat. Commun. 16(1), 3994 (2025)

[82] R. Zhao , Q. Wei , Y. Li , X. Li , G. Geng , X. Li , J. Li , Y. Wang , and L. Huang , Stereo Jones matrix holography with longitudinal polarization transformation, Laser Photonics Rev. 17(8), 2200982 (2023)

[83] Y. Li , M. A. Ansari , H. Ahmed , R. Wang , G. Wang , and X. Chen , Longitudinally variable 3D optical polarization structures, Sci. Adv. 9(47), eadj6675 (2023)

[84] Y. Intaravanne , R. Wang , H. Ahmed , Y. Ming , Y. Zheng , Z. K. Zhou , Z. Li , S. Chen , S. Zhang , and X. Chen , Color-selective three-dimensional polarization structures, Light Sci. Appl. 11(1), 302 (2022)

[85] R. Wang , Y. Intaravanne , S. Li , J. Han , S. Chen , J. Liu , S. Zhang , L. Li , and X. Chen , Metalens for generating a customized vectorial focal curve, Nano Lett. 21(5), 2081 (2021)

[86] Y. Li , M. A. Ansari , H. Ahmed , R. Wang , G. Wang , Q. Yu , C. Zhang , S. Chen , and X. Chen , Multiple multicolored 3D polarization knots arranged along light propagation, ACS Photonics 11(10), 4380 (2024)

[87] M. A. Ansari , H. Ahmed , Y. Li , G. Wang , J. E. Callaghan , R. Wang , J. Downing , and X. Chen , Multifaceted control of focal points along an arbitrary 3D curved trajectory, Light Sci. Appl. 13(1), 224 (2024)

[88] Y. Li , Q. Yu , S. Zhao , Q. Yuan , F. Zhong , X. Chen , and S. Wang , Metasurface-enabled cascaded multispectral polarization structures along the longitudinal direction, Adv. Opt. Mater. 13(11), 2403093 (2025)

[89] T. Nan , H. Zhao , J. Guo , X. Wang , H. Tian , and Y. Zhang , Generation of structured light beams with polarization variation along arbitrary spatial trajectories using tri-layer metasurfaces, Opto-Electron. Sci. 3(5), 230052 (2024)

[90] S. Wang , J. Luo , Z. Zhu , Y. Cao , H. Wang , C. Xie , and X. Li , All-optical generation of magnetization with arbitrary three-dimensional orientations, Opt. Lett. 43(22), 5551 (2018)

[91] W. Yan , Z. Nie , X. Liu , X. Zhang , Y. Wang , and Y. Song , Arbitrarily spin-orientated and super-resolved focal spot, Opt. Lett. 43(16), 3826 (2018)

[92] W. Chen and Q. Zhan , Diffraction limited focusing with controllable arbitrary three-dimensional polarization, J. Opt. 12(4), 045707 (2010)

[93] S. Zhang , S. Fu , H. Zhang , X. Ge , Z. Bai , Y. Lyu , R. Zhao , and Z. Man , Fully controlled photonic spin in highly confined optical field, Opt. Express 27(23), 33621 (2019)

[94] Q. Lu , S. Wen , H. Luo , and W. Shu , Photonic spin rings formed by toroidal spin momentum, Opt. Lett. 50(4), 1333 (2025)

[95] W. Chen and Q. Zhan , Three dimensional polarization control in 4Pi microscopy, Opt. Commun. 284(1), 52 (2011)

[96] A. F. Abouraddy and K. C. Toussaint , Three-dimensional polarization control in microscopy, Phys. Rev. Lett. 96(15), 153901 (2006)

[97] C. Yan , X. Li , Y. Cai , and Y. Chen , Three-dimensional polarization state and spin structure of a tightly focused radially polarized Gaussian Schell-model beam, Phys. Rev. A 106(6), 063522 (2022)

[98] J. Chen , C. Wan , L. J. Kong , and Q. Zhan , Tightly focused optical field with controllable photonic spin orientation, Opt. Express 25(16), 19517 (2017)

[99] Y. J. Wu , J. H. Zhuang , P. P. Yu , Y. F. Liu , Z. Q. Wang , Y. M. Li , C. W. Qiu , and L. Gong , Time-varying 3D optical torque via a single beam, Nat. Commun. 16(1), 593 (2025)

[100] I. Moreno , J. A. Davis , M. M. Sanchez-Lopez , K. Badham , and D. M. Cottrell , Nondiffracting Bessel beams with polarization state that varies with propagation distance, Opt. Lett. 40(23), 5451 (2015)

[101] W. Y. Wang , J. X. Guo , S. Liu , J. Q. Lü , and Z. W. Lu , Manipulating the variation of polarization during propagation along arbitrary circular trajectory of the Poincaré sphere, Appl. Phys. Lett. 121(16), 161102 (2022)

[102] J. A. Davis , I. Moreno , K. Badham , M. M. Sanchez-Lopez , and D. M. Cottrell , Nondiffracting vector beams where the charge and the polarization state vary with propagation distance, Opt. Lett. 41(10), 2270 (2016)

[103] T. Jankowski , N. Bennis , A. Spadlo , J. F. Algorri , M. M. Sánchez-López , and I. Moreno , Liquid crystal anisotropic axicon for the generation of non-diffracting Bessel beams with longitudinally varying polarization, Opt. Laser Technol. 170, 110255 (2024)

[104] X. He , H. Bao , F. Zhang , T. Kang , M. Pu , Y. Chen , Y. Guo , J. Gong , M. Xu , and X. Luo , Longitudinally continuous varying high-order cylindrical vector fields enabled by spin-decoupled metasurfaces, Nanophotonics 13(9), 1657 (2024)

[105] C. Zheng , J. Li , J. Liu , J. Li , Z. Yue , H. Li , F. Yang , Y. Zhang , Y. Zhang , and J. Yao , Creating longitudinally varying vector vortex beams with an all-dielectric metasurface, Laser Photonics Rev. 16(10), 2200236 (2022)

[106] W. Wang , J. Wang , Q. Wu , T. Zhang , J. Lin , P. Jin , S. Liu , and K. Zhou , Generation of scalar and vector beams with longitudinally varying polarization based on all-silicon metasurfaces, Phys. Scr. 99(11), 115530 (2024)

[107] J. Yang , R. Zhao , Y. Li , H. Xiong , Y. Li , X. Li , J. Li , Y. Wang , and L. Huang , Generation of the Bessel beam of longitudinally varied polarization with dielectric metasurfaces, Adv. Opt. Mater. 11(10), 2202896 (2023)

[108] J. Yang , R. Zhao , Y. Li , C. Tian , X. Ji , X. Li , J. Li , Y. Wang , and L. Huang , Transformation of longitudinally customizable curved vector vortex beams using dielectric metasurface, Laser Photonics Rev. 18(10), 2400226 (2024)

[109] Z. Qing , W. Yan , X. Long , Z. Yuan , Z. C. Ren , X. L. Wang , J. Ding , and H. T. Wang , Longitudinal manipulation of local nonseparability in vector beams, Opt. Lett. 49(10), 2557 (2024)

[110] A. H. Dorrah , M. Tamagnone , N. A. Rubin , A. Zaidi , and F. Capasso , Introducing Berry phase gradients along the optical path via propagation-dependent polarization transformations, Nanophotonics 11(4), 713 (2022)

[111] R. Droop , E. Asché , E. Otte , and C. Denz , Shaping light in 3D space by counter-propagation, Sci. Rep. 11(1), 18019 (2021)

[112] P. Li , Y. Zhang , S. Liu , L. Han , H. Cheng , F. Yu , and J. Zhao , Quasi-Bessel beams with longitudinally varying polarization state generated by employing spectrum engineering, Opt. Lett. 41(20), 4811 (2016)

[113] M. Zamboni-Rached , Stationary optical wave fields with arbitrary longitudinal shape by superposing equal frequency Bessel beams: Frozen waves, Opt. Express 12(17), 4001 (2004)

[114] M. Corato-Zanarella , A. H. Dorrah , M. Zamboni-Rached , and M. Mojahedi , Arbitrary control of polarization and intensity profiles of diffraction-attenuation-resistant beams along the propagation direction, Phys. Rev. Appl. 9(2), 024013 (2018)

[115] P. Li , D. Wu , Y. Zhang , S. Liu , Y. Li , S. Qi , and J. Zhao , Polarization oscillating beams constructed by copropagating optical frozen waves, Photon. Res. 6(7), 756 (2018)

[116] F. Tang , T. Zhang , Z. Tan , X. Zang , Y. Zhu , and S. Zhuang , Metasurfaces for multiplexing pencil-like beams with longitudinally inhomogeneous versatile polarization states and polarization-evolutive imaging, Photon. Res. 12(12), 3012 (2024)

[117] F. Cardano , E. Karimi , L. Marrucci , C. de Lisio , and E. Santamato , Generation and dynamics of optical beams with polarization singularities, Opt. Express 21(7), 8815 (2013)

[118] P. Li , Y. Zhang , S. Liu , H. Cheng , L. Han , D. Wu , and J. Zhao , Generation and self-healing of vector Bessel-Gauss beams with variant state of polarizations upon propagation, Opt. Express 25(5), 5821 (2017)

[119] P. Li , X. Fan , D. Wu , S. Liu , Y. Li , and J. Zhao , Self-accelerated optical activity in free space induced by the Gouy phase, Photon. Res. 8(4), 475 (2020)

[120] S. Liu , S. Qi , P. Li , B. Wei , P. Chen , W. Hu , Y. Zhang , X. Gan , P. Zhang , Y. Lu , Z. Chen , and J. Zhao , Analogous optical activity in free space using a single Pancharatnam-Berry phase element, Laser Photonics Rev. 16(1), 2100291 (2022)

[121] J. Q. Lü , X. L. Wang , G. L. Zhang , C. Tu , Y. Li , and H. T. Wang , Bessel-like beams with controllable rotating local linear polarization during propagation, Opt. Lett. 45(7), 1738 (2020)

[122] J. Li , J. Li , Z. Yue , C. Zheng , G. Wang , J. Liu , H. Xu , C. Song , F. Yang , H. Li , F. Li , T. Tang , Y. Zhang , Y. Zhang , and J. Yao , Structured vector field manipulation of terahertz wave along the propagation direction based on dielectric metasurfaces, Laser Photonics Rev. 16(12), 2200325 (2022)

[123] J. Li , J. Liu , Z. Yue , J. Li , C. Zheng , F. Yang , H. Li , Y. Zhang , Y. Zhang , and J. Yao , Polarization variable terahertz metasurface along the propagation path, Fundam. Res. 5(1), 124 (2025)

[124] A. H. Dorrah , N. A. Rubin , A. Zaidi , M. Tamagnone , and F. Capasso , Metasurface optics for on-demand polarization transformations along the optical path, Nat. Photonics 15(4), 287 (2021)

[125] X. Wu , K. Pan , X. Wu , X. Fan , L. Zhou , C. Zhao , D. Wen , S. Liu , X. Gan , P. Li , and J. Zhao , Wavelength-insensitive snapshot Stokes polarimetric imaging based on cascaded metasurfaces, Adv. Photonics 7(1), 016008 (2025)

[126] V. Shvedov,A. R. Davoyan,C. Hnatovsky,N. Engheta,W. Krolikowski, A long-range polarization-controlled optical tractor beam, Nat. Photonics 8(11), 846 (2014)

[127] M. Li , S. Yan , Y. Liang , P. Zhang , and B. Yao , Transverse spinning of particles in highly focused vector vortex beams, Phys. Rev. A 95(5), 053802 (2017)

[128] Q. W. Zhan , Trapping metallic Rayleigh particles with radial polarization, Opt. Express 12(15), 3377 (2004)

[129] X. Ouyang , Y. Xu , Z. Feng , W. Tang , Y. Cao , and X. Li , Polychromatic and polarized multilevel optical data storage, Nanoscale 11(5), 2447 (2019)

[130] V. Garcés-Chávez , D. McGloin , H. Melville , W. Sibbett , and K. Dholakia , Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam, Nature 419(6903), 145 (2002)

[131] H. Y. Kuo , S. Vyas , C. H. Chu , M. K. Chen , X. Shi , H. Misawa , Y. J. Lu , Y. Luo , and D. P. Tsai , Cubic-phase metasurface for three-dimensional optical manipulation, Nanomaterials (Basel) 11(7), 1730 (2021)

[132] Z. Shen , F. Zhao , C. Jin , S. Wang , L. Cao , and Y. Yang , Monocular metasurface camera for passive single-shot 4D imaging, Nat. Commun. 14(1), 1035 (2023)

[133] H. Hao , H. Wang , X. Wang , X. Ding , S. Zhang , C. F. Pan , M. A. Rahman , T. Ling , H. Li , J. Tan , J. K. W. Yang , W. Lu , J. Liu , and G. Hu , Single-shot 3D imaging meta-microscope, Nano Lett. 24(42), 13364 (2024)

[134] H. Ren , W. Shao , Y. Li , F. Salim , and M. Gu , Three-dimensional vectorial holography based on machine learning inverse design, Sci. Adv. 6(16), eaaz4261 (2020)

[135] E. Otte , C. Rosales-Guzman , B. Ndagano , C. Denz , and A. Forbes , Entanglement beating in free space through spin-orbit coupling, Light Sci. Appl. 7(5), 18009 (2018)

[136] J. Guo , Y. Zhang , H. Ye , L. Wang , P. Chen , D. Mao , C. Xie , Z. Chen , X. Wu , M. Xiao , and Y. Zhang , Spatially structured-mode multiplexing holography for high-capacity security encryption, ACS Photonics 10(3), 757 (2023)

[137] S. Song , Q. He , Y. Liu , P. Wu , Q. Sun , L. Zhao , T. Wu , and L. Guo , Personalized federated learning based adaptive optical compensation for atmospheric turbulence, J. Lightwave Technol. 43(1), 177 (2025)

[138] Y. Luo , S. Yan , H. Li , P. Lai , and Y. Zheng , Towards smart optical focusing: Deep learning-empowered dynamic wavefront shaping through nonstationary scattering media, Photon. Res. 9(8), B262 (2021)

[139] X. Jin , Z. Wang , M. Liu , J. Li , Z. Zhu , Y. Wang , L. Li , Y. Chen , W. Pan , S. Ma , Q. He , L. Zhou , and S. Sun , Vectorial beams with longitudinally varying polarizations generated by surface-wave metasurfaces, Laser Photonics Rev. 19(12), 2402298 (2025)

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