Physical foundations of vector beams and applications in micro-nano optics: A review

Abstract

In view of the fast development of vector beams and their broad application prospects, the systematical review about the characterizations, generation methods, propagation mechanisms and diverse micro-nano optical applications of various vector beams are provided. Four mathematical characterizations of vector beams are generalized and the relation of different characterization methods is uncovered. The direct and indirect generation methods including some innovation techniques are summarized. The propagation fields of vector beams in different circumstances are analyzed and the related phenomena are explained. The diverse applications in various fields are enumerated. The detailed description of vector beams provides a theoretical basis for deeply understanding of the polarization properties of vector beams. The comparison of several generation methods brings the researchers the purposeful choices in practice works. The analysis about propagation mechanisms of vector beam lays the foundation for predicting the unique phenomena and utilizing reasonably vector beams. The presented applications confirm the significance of vector beams in many fields. We expect this review will bring more help for the deep studies about foundational and applications of vector beams.

Graphical abstract

Keywords

polarisation / Poincaré beam / micro-nano optics / metasurface / generation / focus / nonlinear effect / transmission

1 Introduction

Polarization is natural property of light, and it has been applied in optical measuring [1], high resolution imaging [2], radar image detection [3, 4], cosmic birefringence observations [5], magnetic field analysis [6] because of high sensitivity to the environment. Light beams are usually divided into scalar beams and vector beams in terms of their polarization distributions in the cross-section perpendicular to the propagation direction. Scalar beams point to the light beams with spatially uniform polarization including linearly polarized light, circularly polarized light and elliptically polarized light. Vector beams point to the light beams with spatially non-uniform polarization. Compared to scalar beams, vector beams exhibit some special phenomena, and they have aroused widespread interest and their applications have covered many fields.

In practice, vector beams can be categorized into different types, like linearly polarized vector beams (LPVBs) [7], elliptically polarized vector beams (EPVBs) [7] and hybrid polarized vector beams (HPVBs) [8]. LPVBs are composed of linear states of polarization along different directions, and the common radial (RVB) and azimuthal vector beam (AVB) are two special cases. EPVBs have elliptical states of polarization with spatially varying ellipticity and orientation. Due to the symmetric polarization distributions with respect to the propagation axis, LPVBs and EPVBs are called as cylindrical vector beams (CVBs), and they can be expressed by superimposing two orthogonal circularly polarized light [9] and visualized intuitively by Poincaré sphere [10, 11]. HPVBs may carry spatially varying phases and their polarization distributions may lack axial symmetry [8].

Specific polarization and phase topological properties of vector beams provide such properties as tight focusing and more modulation degrees of freedom during the propagation. As is well known, vector beam can realize the tight focusing with focusing sizes at the order of sub-wavelengths [12], which takes on impressive advantages for achieving more stable particle capture [13], higher precision laser processing [14] and higher contrast imaging [15] in nanometer scale. Focusing characteristics of vector beams in nonlinear media also show important applications in high-resolution nonlinear imaging [16]. Vector beams exhibit high stability during the propagation in turbulence, which helps the beam to avoid loss of optical information [17].

Early researches on vector beams concentrate on the focusing of CVBs and their applications [18]. RVB and AVB can realize the punching in thin and thick plates with high precision [14]. Moreover, RVB and AVB are also used to trap metal nanoparticles [19] and carbon nanotubes [20]. The tight focusing of CVB make it exceptional in microscopy imaging [21].

In recent years, the application of vector beams continuously deepens towards the nanoscale. The focusing of vector beams can break through the diffraction limit [22], and it enables higher machining accuracy [23] and higher imaging resolution [24]. The advancement of vector field modulation techniques, like the utilization of metasurfaces composed of nanounits, may bring great development for the applications of vector beams. The capture of particles can extend from one-dimensional to three-dimensional, from single-particle capture to multiple-particle simultaneous capture [25] and from regular spherical particles to elliptical particles [26]. And as expected, the advanced metasurface technologies have provided powerful tools for vector beam multiplexing and multi-channel vector holographic encryption [27].

With the in-depth study of vector beams, some challenges appear on the scientific and technological level. The special spatial modes of vector beams are desired for high-resolution imaging [28]. Complex optical field patterns of vector beams may increase the difficulty of image processing and analysis [29]. Unstable beam modes influence fabrication precision of sample [23]. Complex control systems to control vector beams hinder the lithographic efficiency and precision [30]. Local optimization issues limit the micro-manipulation performance [31]. Stringent precision requirements for equipment and beam crosstalk are accompanied by the high information capacity of vector beams in optical communication [32]. The studies about the optimization of vector beams and the high-quality applications still need to be continuously conducted.

The generation of new-type vector beams must speed the applications of vector beams. Recently, the integrated or compound vector beams with the different polarization distributions are generated [33] and vector beams with the polarization varying in three-dimensional space are generated [34]. Besides the traditional methods based on the intra-cavity modulation and the diffraction and interference, some new techniques using photonic crystals [35] and compact metasurfaces [36] are utilized to generate vector beams. The fundamental and application studies of vector beams still take on vigorous development trend. The review about the studies of vector beams especially in recent years are necessary though this kind of works were done in 2009 [37] and 2018 [38, 39].

This article aims to present recent developments in vector beams and overview their main applications in micro-nano optics. For systematic and completeness, this work carries out from the characterization of vector beams, the generation of vector beams, the propagation rules and the applications. First, the mathematical description of generalized vector beams and the representation in several polarization bases are presented. Then, the generation methods of vector fields using laser intra-cavity, gratings, q-plates, interferometers, spatial light modulators, photonic crystals and metasurfaces are enumerated. The phenomena of vector fields during propagating in different transmission media or meeting some obstacles are provided. These preliminaries provide the foundation for the potential applications of vector beams. In the end, the recent applications in micro-nano optics like laser processing, optical trapping, and optical imaging are summarized.

This work provides the researchers with a systematic review about the studies about vector beams. It can help them comprehensively understand the foundation of vector beams including the characterization of vector beams and the generation methods. Lots of applications may arouse great interests in various fields and provide the guidance with wider ideas about the applications of vector beams in different fields such as optical physics, quantum physics, electromagnetism, biomedicine and materials.

2 Characterization of vector beams

The characterization of vector beams is the start to generate vector beams and utilize vector beams. Here, four kinds of methods are provided to characterize vector beams and the related descriptions are as detailed as possible.

2.1 Mathematical expressions of vector beams

Vector beam with certain state of polarization can be expressed in different coordinate basis. In the Cartesian coordinate basis, vector beam can be expressed as

(1)\mathit{E}(x,y)=A(\cos \alpha {\mathit{e}}_x+\sin \alpha {\mathrm{e}}^{\mathrm{i}{\delta }_{xy}}{\mathit{e}}_y),

where A denotes the amplitude of vector beam, and the angles of α and δ_xy determine the polarization type of vector beam. As the angle of α satisfies α = nφ + ϕ, where n is an integer, ϕ is a constant and φ is the spatial position angle, the vector beam exhibits the axis-symmetric polarization distribution and it can be written as

(2)\mathit{E}(x,y)=A[\cos (n\phi +\varphi ){\mathit{e}}_x+\sin (n\phi +\varphi ){\mathrm{e}}^{\mathrm{i}{\delta }_{xy}}{\mathit{e}}_y].

As δ_xy = 0 or π, the vector beam is called as the LPVB, specially, and as n = 1 and ϕ = 0 or π/2, the vector beam corresponds to the common RVB and AVB [40]. Moreover, as δ_xy = π/2 or 3π/2, the vector beam is the EPVB. Fig.1 gives the polarization distributions of vector beams as δ_xy takes 0 and π/2, n takes 1 and 2, and ϕ takes 0, π/4, and π/2, as the result given by Ref. [41].

It can be seen that as δ_xy = 0, the polarization of vector beam everywhere in the space is linear polarization, and as δ_xy = π/2, the polarization direction and ellipticity for the latter cases change with the spatial position. The value of n determines the degree of symmetry of vector beam. Certainly, the polarization distribution is different as the angle of ϕ takes different value. In addition to the above cases with the angle of α varying with angular position and the phase difference of δ_xy taking the constant, the polarization distributions with the angle of α varying with radial direction and the phase of δ_xy changing with spatial position can be presented in terms of Eq. (2).

One vector beam can be represented by the circular polarization basis consisting of two orthogonal circular polarization vectors, and similarly, it can be expressed by [42]

(3)\mathit{E}(x,y)=A(\cos \beta {\mathit{e}}_{\mathrm{R}}+\sin \beta {\mathrm{e}}^{\mathrm{i}{\delta }_{RL}}{\mathit{e}}_{\mathrm{L}}),

where δ_RL denotes the phase difference of two polarization components, and e_L = 2^−0.5(e_x − ie_y) and e_R = 2^−0.5(e_x + ie_y) represent two orthogonal basis vectors along left- and right-handed circular polarizations. Naturally, the polarization of vector beam depends on the values of β and δ_RL. As β = 0 or π, the polarization is RCP, and as β = π/2 or 3π/2, the polarization is LCP. Fig.2 gives the polarization distributions of several vector beams expressed in circular polarization basis as the phase of δ_RL varies with the spatial position and satisfies δ_RL = nφ + ϕ, where β takes π/6, π/4 and π/3, n takes 1 and 2, and takes 0 and π/4, as the result given by Ref. [42].

For the certain angle of β, the polarization type of vector beam at any position is the same and the polarization direction changes with spatial position and other parameters. The ellipticity of vector beam depends on the angle of β. When β = π/4 the vector beam is the LPVB with identical contributions of left- and right-handed circularly polarized components. Vector beams with β taking other values are the EPVB. The difference from the first case, the polarization distribution of vector beam is not always cylindrically symmetric. In addition to the above cases, the phase of δ_RL may change with the radial direction and the angle of β may also changes with the spatial position. The polarization distribution of vector beam with different parameters is different.

Vector beam can be also represented in polar polarization basis consisting of radial and azimuthal polarization vectors, and any vector beam can be represented as

(4)\mathit{E}(x,y)=A(\mathrm{c}\mathrm{o}\mathrm{s}\gamma {\mathit{e}}_{\mathit{r}}+\mathrm{s}\mathrm{i}\mathrm{n}\gamma {\mathrm{e}}^{\mathrm{i}{\delta }_{r\phi }}{\mathit{e}}_{\mathit{\phi }}),

where δ_rφ is the phase difference between the radial and azimuthal polarization components of vector beam, and e_r = cosφe_x + sinφe_y and e_φ = −sinφe_x + cosφe_y are radial and azimuthal polarization vectors in the polar coordinate basis. The type of polarization of vector beam depends on the values of γ and δ_rφ. Fig.3 gives the polarization distributions of vector beams in polar polarization basis as δ_rφ satisfies δ_rφ = nφ + ϕ, where γ takes π/6, π/4 and π/3, n takes 1 and 2, and ϕ takes 0 and π/4, as the result given by Ref. [41].

From the polarization distributions on the same line, one can see that the angle of γ does not affect the polarization handedness of vector beam at any position, but affects the ellipticity of vector beam. The polarization distributions on the same row show that the angle of ϕ makes the polarization distribution of vector beam rotate. The value of n determines the rotational symmetry of polarization distribution. Besides the above cases, the phase of δ_rφ may vary with radial coordinate and the angle of γ may also change with spatial position. Certainly, the polarization distribution of vector beam is different as these parameters take different values.

2.2 Jones matrices of vector beams

Jones vector was first proposed by Jones to represent the state of polarization of optical field [43]. Traditionally, one 1 × 2 or 2 × 1 Jones vector is comprised of two orthogonal components of optical field in Cartesian coordinate basis. The vector beam denoted by Eq. (1) can be written in the following form [41]:

(5)\mathit{E}=A\left(\begin{array}{c}\cos \alpha \\ \sin \alpha {\mathrm{e}}^{\mathrm{i}{\delta }_{xy}}\end{array}\right).

The x- and y-polarized base vectors can be expressed as (1, 0) and (0, 1). The normalized RVB and AVB are (cosφ, sinφ) and (−sinφ, cosφ) in Cartesian coordinate basis. Similarly, Eqs. (3) and (4) can be expressed in Jones vectors. The orthogonal base vectors in circular polarization basis are (1, 0) and (0, 1), and the orthogonal base vectors in polar polarization basis are also (1, 0) and (0, 1).

For a certain vector beam, its Jones vector in different polarization basis can be transformed. The relationship of Jones vectors can be expressed as

(6)\left(\begin{array}{c}{E}_{b1}\\ E_{b2}\end{array}\right)=\left(\begin{array}{c}{T}_{11}{T}_{12}\\ T_{21}{T}_{22}\end{array}\right)\left(\begin{array}{c}{E}_{a1}\\ E_{a2}\end{array}\right).

Here, (E_a1, E_a2) represents the vector beam in the a-polarization basis and (E_b1, E_b2) represents the vector beam in the b-polarization basis. The 2 × 2 matrix represents the evolution matrix for the transformation between different Jones vectors [44]. The evolution matrix for the transformation from Cartesian coordinate basis to circular polarization basis is

(7){\mathit{T}}_{\mathit{d}\mathbf{-}\mathit{c}}=\left(\begin{array}{c}{T}_{11}{T}_{12}\\ T_{21}{T}_{22}\end{array}\right)={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{c}1-j\\ 1j\end{array}\right).

The evolution matrix for the transformation from Cartesian coordinate basis to polar polarization basis is

(8){\mathit{T}}_{\mathit{d}\mathbf{-}\mathit{p}}=\left(\begin{array}{c}{T}_{11}{T}_{12}\\ T_{21}{T}_{22}\end{array}\right)=\left(\begin{array}{c}\cos \phi \sin \phi \\ -\sin \phi \cos \phi \end{array}\right).

Furthermore, the evolution matrices for the transformations from circular and polar polarization bases back to Cartesian coordinate basis are the inverse matrixes of two above cases.

2.3 Stokes parameter characterization of vector beams

Stokes parameters including S₀, S₁, S₂ and S₃ were first introduced by G. G. Stokes to characterize the state of polarization of optical field [45]. In terms of the expression of vector beam in Cartesian coordinate basis, Stokes parameters are defined by

(9)\left(\begin{array}{c}{S}_0\\ S_1\\ S_2\\ S_3\end{array}\right)=\left(\begin{array}{c}{I}_0\\ I_0\cos 2\alpha \\ I_0\sin 2\alpha \cos {\delta }_{xy}\\ I_0\sin 2\alpha \sin {\delta }_{xy}\end{array}\right),

where S₀ = I₀ represents the total intensity of vector beam at any position, and it can be also expressed as the sum of two intensity components along two orthogonal basis vectors in different polarization basis, such as the horizontal and vertical directions, diagonal and anti-diagonal directions, radial and azimuthal directions, and RCP and LCP basis vectors. Usually, the above expression can be rewritten as

(10)\left(\begin{array}{c}{S}_0\\ S_1\\ S_2\\ S_3\end{array}\right)=\left(\begin{array}{c}{I}_0\\ I_0\cos 2\theta \cos 2\psi \\ I_0\cos 2\theta \sin 2\psi \\ I_0\sin 2\theta \end{array}\right),

where the angle of ψ (0 ≤ ψ ≤ π) denotes the elliptical polarization direction and the value of tanθ (−π/4 ≤ θ ≤ π/4) represents the ratio of ellipticity, and they satisy tan2ψ = tan2αcosδ_xy and sin2θ = sin2α sinδ_xy. In terms of the expression of vector beam in circular polarization basis, the Stokes parameters can be expressed by

(11)\left(\begin{array}{c}{S}_0\\ S_1\\ S_2\\ S_3\end{array}\right)=\left(\begin{array}{c}{I}_0\\ I_0\cos 2\beta \\ I_0\sin 2\beta \cos {\delta }_{RL}\\ I_0\sin 2\beta \sin {\delta }_{RL}\end{array}\right).

In practice, Stokes parameters in circular polarization basis may be also expressed by the form of Eq. (10) with the angles of ψ and θ satisfying tan2ψ = tan2βcosδ_RL and sin2θ = sin2βsinδ_RL. Similarly, the Stokes parameters in polar polarization basis can be expressed by

(12)\left(\begin{array}{c}{S}_0\\ S_1\\ S_2\\ S_3\end{array}\right)=\left(\begin{array}{c}{I}_0\\ I_0\cos 2\gamma \\ I_0\sin 2\gamma \cos {\delta }_{r\phi }\\ I_0\sin 2\gamma \sin {\delta }_{r\phi }\end{array}\right).

Stokes parameters in polar polarization basis may be also expressed by the form of Eq. (10) with the angles of ψ and θ satisfying tan2ψ = tan2γcosδ_rφ and sin2θ = sin2γsinδ_rφ.

For one certain vector beam, its Stokes parameters in different polarization bases can be transformed, and their relationship can be expressed in following form

(13)\left(\begin{array}{c}{S}_{b0}\\ S_{b1}\\ S_{b2}\\ S_{b3}\end{array}\right)=\left(\begin{array}{c}{M}_{11}{M}_{12}{M}_{13}{M}_{14}\\ M_{21}{M}_{22}{M}_{23}{M}_{24}\\ M_{31}{M}_{32}{M}_{33}{M}_{34}\\ M_{41}{M}_{42}{M}_{43}{M}_{44}\end{array}\right)\left(\begin{array}{c}{S}_{a0}\\ S_{a1}\\ S_{a2}\\ S_{a3}\end{array}\right),

where S_ai represents the Stokes parameter in the a-polarization basis, S_bi represents the Stokes parameter in the b-polarization basis, and the 4 × 4 matrix represents the evolution matrix between different polarization bases. The measurement of Stokes parameters in any polarization basis can reflect the polarization distribution of vector beam.

2.4 Poincaré sphere representation of vector beams

Poincaré sphere proposed by Poincaré can reflect the state of polarization using the point on the surface of unit sphere [46]. The radius of Poincaré sphere is S₀ and three axes correspond to the parameters of S₁, S₂ and S₃, as shown in Fig.4(a). In fact, any point on Poincaré sphere corresponds to a set of Stokes parameters expressed in Cartesian coordinate basis. Two poles on S₃ axis with S₃ = 1 and −1 represent RCP (denoted by red circles) and LCP (denoted by blue circles), two poles on S₁ axis with S₁ = 1 and −1 represent horizontal and vertical linearly polarization (denoted by black short lines), and two poles on S₂ axis with S₂ = 1 and −1 represent diagonal and anti-diagonal linearly polarization. The points on the upper hemisphere denote right-handed elliptical polarization and the points on the lower hemisphere denote left-handed elliptical polarization. In fact, any polarization states on Poincaré sphere can be taken as the superposition of two orthogonal circularly polarized beams with the amplitudes of cos(π/4−θ) and sin(π/4−θ) and the phases of −ϕ and ϕ, namely, E = cos(π/4−θ)e^−iϕe_R + sin(π/4−θ)e^iϕe_L, where the angles of θ and ϕ satisfy 2θ = arcsin⁻¹(S₃/S₀) and 2ϕ = arctan⁻¹(S₂/S₁).

With the help of Poincaré sphere, the vector beam in circular polarization basis can be also expressed as E = cos(π/4−θ)e^{−i(nφ + ϕ)}e_R + sin(π/4−θ)e^{i(nφ + ϕ)}e_L with β = π/4−θ and ψ = nφ + ϕ. When n = 1, such vector beams are called Poincaré beams, and for other integer values of n, these vector beams are collectively called higher-order Poincaré beams [10]. Fig.4(b) illustrates the higher order Poincaré beam corresponding to n = 2. Fig.4(b) illustrates the higher order Poincaré beam corresponding to n = 2. Where the points on the pole on S₃ represent circular polarization states carrying two opposite two-order spiral phases. The points on the equator of Poincaré sphere correspond to the LPVBs, and the points on the upper and lower hemisphere correspond to the EPVBs. The similar result can be found in Ref. [41].

In order to illustrate intuitively vector beams shown in Fig.1 and Fig.3, one can construct Poincaré-like spheres, as shown in Fig.5. Fig.5(a) shows Poincaré-like sphere for the Poincaré-like beam of E = cos(π/4−θ)e^{−i(φ + ϕ)}e_x + sin(π/4−θ)e^{i(φ + ϕ)}e_y. Where the points on two poles of S₃ axis represent horizontal and vertical linear polarization and the points on upper and lower half sphere represent the EPVBs. Generally, for one Poincaré-like beam consisting of two orthogonal polarization vectors, which can expressed by E = cos(π/4−θ)e^{−i(φ + ϕ)}e₁ + sin(π/4−θ)e^{i(φ + ϕ)}e₂ with e₁ and e₂ denoting two orthogonal polarization base vectors like the radial and azimuthal polarization vectors [42], one can also construct Poincaré-like sphere to represent this kind of vector beams. Even for the n-order Poincaré-like beam of E = cos(π/4−θ)e^{−i(nφ + ϕ)}e₁ + sin(π/4−θ)e^{i(nφ + ϕ)}e₂, the Poincaré-like sphere can be still constructed similarly to Fig.5(a).

The above-mentioned Poincaré beams or Poincaré-like beams have two orthogonal components carrying opposite spiral phase. More generally, the phases of two orthogonal components may be expressed by ψ₁ = −(nφ + ϕ) and ψ₂ = mφ + ϕ with m ≠ n. This kind of vector beams are called as hybrid Poincaré beams or HPVBs, and they can be expressed by hybrid Poincaré-like sphere. And they are the commonly vector vortex beams. Fig.5(b) illustrates hybrid Poincaré-like sphere and hybrid Poincaré beam superposed by radially and azimuthally polarized beams with n = 2 and m = 3. The points on two poles of S₃ axis of hybrid Poincaré sphere shown in Fig.5(b) represent radial and athimuthal polarization carrying the spiral phases with the topological charge taking -2 and 3, respectively. The points at the poles of S₁ and S₂ axes are the HPVBs [41]. The polarization distributions at the poles of S₁ are symmetric along y axis, and the polarization distributions at the poles of S₂ are symmetric along x axis. Further discussions can be found in reference [11].

Four methods are provided in this section to characterize vector beams using vector formulas, visual matrices, specific parameters and intuitive diagrams. These methods have inherent relevance and they mutually confirm to reflect the polarization distribution characteristics of vector beams. This comprehensive representation for vector beams not only offers one powerful framework for understanding vector beams in depth, but also serves as important references for generating vector beams in the following.

3 Vector beam generation

Vector beam generation schemes and mechanisms have been extensively explored, and here, the related advances for vector beam generation are summarized, where traditional devices such as birefringent elements, gratings, q-plates, interferometers and spatial light modulators and novel devices like metasurfaces and photonic crystals are used. With these techniques, vector beams are either directly converted or indirectly produced.

3.1 Intra-cavity modulation

We know that the common laser do not directly radiate vector beam. To output the vector beam, anisotropic elements need to be introduced into the cavity of laser. These anisotropic elements modulate the resonant modes in the cavity and ensure the gain of beam with the required polarization. Thereby, vector beam with controllable polarization distribution is actively produced through intra-cavity modulation.

Some birefringent devices may be inserted into laser cavities to generate vector beams, such as double conical prisms [47] and birefringent crystal [48]. Vector beam can be generated through polarization choice and the cavity length is adjusted to determine the type of output beam [48]. Variable vector beam generation may bring great convenience to the applications of vector beams. The combination of wave plates and vortex plates with different topological charges may be placed in resonant cavity so as to generate vector beam with changeable order and initial angle [49].

Compared with traditional lasers, fiber lasers have the advantages of compact, stable, small size and low cost. The vector beam is generated with the help of offset splicing spot, long-period fiber grating and mode-selective coupler. The fiber splicing can convert the LP₀₁ mode into the LP₁₁ mode and the RVB generation is achieved by controlling the amount of fiber offset and selective reflection of the quad-mode fiber as shown in Fig.6(a) [50]. Where adjusting the polarization controller optimizes the polarization state of the different modes to ensure LP₁₁ mode generation and the reflection wavelength of the whole fiber grating ensures high mode purity output of RVB.

The fiber gratings with special structures [51] or working under Bragg condition [52] or combining with active phase modulator [53] are also utilized to generate high purity CVBs. Except for the single wavelength output, the utilization of Sagnac loop filter and long-period fiber grating in random laser realizes multi-wavelength vector beam generation [54]. Nonlinear gain modulation in cavity [55] and the combination of two lasers with fiber grating [56] are also utilized to generate dual-wavelength CVBs. In addition, the acoustic-optic effect to modulate the refractive index of fibers is utilized to generate the CVBs [57]. The adjustment of the incident polarization state and the acoustic frequency in spiral core ring fiber [58] and multi-core fiber [59] can cause the conversion of the fundamental mode to high-order vector beams.

Recently, the generation of temporal vector beams attracts the attention. The fiber chirped-pulse amplification and second harmonic generation technology are used to produce the femto-second CVBs [60]. Briefly, the intra-cavity active modulation methods offer high efficient technique for vector beam generation. However, the modulation process for any vector-mode laser is much more complex. The following passive modulation methods with optical elements inserted outside the cavity show simpler operation though the sacrifice of stability and efficiency is inevitable.

3.2 Grating modulation

Grating is commonly used to generate vector beam based on its spatial anisotropy, multiple order diffraction and the additional reciprocal vector [61]. Physically, gratings refer to one-dimensional or two-dimensional periodic structures, and the periodic structures with some deformations are also categorized as gratings, like radially curved gratings [62] and forked gratings [63]. The modulation of amplitude, phase and polarization by grating enables the production of vector beam.

Fork grating points to periodic amplitude or phase distribution formed by the interference between plane wave and vortex beam. The ±1st order diffractions of forked grating carry opposite spiral phases [64]. One can generate the pure RVB beam using one vortex beam with the help of concentric ring grating [64], or controlling the superposition of two vortex beams output from forked grating [61].

Sub-wavelength grating can be functionally equivalent to uniaxial crystals and the groove orientation of sub-wavelength grating controls the polarization direction of the transmitted field [65]. Radial subwavelength gratings can convert linear polarization into RVB [66]. In practice, the central region of grating with the period radially varying is limited by the fabrication technique and the nonuniform period of grating with period azimuthally varying is also limited by the conversion efficiency. Two identical mirror-reversed azimuthal gratings are proposed to replace one azimuthal grating so as to improve the conversion efficiency of vector beams [67], where the phase difference of the beams transmitted by two parts should be compensated. Furthermore, several segmented azimuthally varying sub-wavelength gratings are proposed to generate RVBs [68], where the local transmission polarization is controlled by the phase difference between adjacent segments [69].

In recent years, with the development of nanometer manufacturing technology, metasurface gratings with the nanometer thickness have been used to generate vector beams. Metasurface gratings not only modulate the state of polarization, but integrate multiple functionalities to enhance compactness of optical system. Moreover, the combination of metasurface grating with spatial light modulator enables the generation of various vector beams due to the dynamic modulation. This further enhances the flexibility of the vector beam generation.

3.3 q-plate modulation

q-plate points to one polarization device composed of anisotropic units with the fast axis rotating azimuthally, and each unit equivalent to a half-wave plate rotates q times of the position angle. The q-plate can convert LCP light to RCP light carrying the spiral phase of 2qφ or RCP light to LCP light carrying the spiral phase of −2qφ. Then the q-order LPVB of A[e^i(2qφ−ϕ)e_R, e^{−i(2qφ−ϕ)}e_L] or A[cos(2qφ−ϕ), sin(2qφ−ϕ)] generates as the linearly polarized light illuminates with ϕ denoting the incident polarization angle.

To obtain controllable vector beam, the combination of q-plate, polarizer and quarter-wave plate (QWP) is used to control the polarization states of two orthogonal polarized beams, as shown in Fig.6(b) [70]. The modulated beam is converted into arbitrary HPVB. The q-plate determines the helical phase carried by the beams at the south and north poles of hybrid Poincaré sphere. The rotation angles of QWP2 and QWP3 can flexibly modulate the phase and polarization of the HPVB.

Generally, the q-value of q-plate is fixed, which limits the order of the generated vector beams. Low-order q-plates are more commonly used in the practical applications, and higher-order vector beams are less utilized because they are difficult to generate. To solve this problem, different q-values of q-plates may be combined to achieve a larger equivalent q-value. Specifically, one half-wave plate placed between two q-plates may obtain q-values of q₁ + q₂, while one half-wave plate placed ahead of two q-plates may form q-values of q₁ − q₂ [71]. This operation extends the order range of the generated vector beam. It should be noted that this q-value operator is only applicable to half-wave q-plates with a phase delay of π [72].

The phase delay of common q-plate strongly depends on the working wavelength [73]. Yet, q-plate at non-operating wavelength is also studied to generate vector beams. It is demonstrated that q-plate with a delay of π/2 under wavelength modulation can generate the HPVB. The polarization state and phase of the fully Poincaré beam are obtained under circularly polarized illumination [74, 75]. Additionally, the combination of electronically controlled liquid crystal q-plate and electronically controlled wave plate is also used to generate arbitrary HPVB on the hybrid Poincaré sphere, where the q-plate controls the ellipticity of beam and the waveplate controls the initial phase of beam [76].

3.4 Interferometer modulation

Since two optical paths of interferometer can be independently controlled, it is favorable for the generation of vector beams through adjusting path difference and states of polarization of two arms. Mach−Zehnder interferometer [77], Sagnac interferometer [79] and two-element interferometer [80] are proposed to generate vector beams.

Fig.6(c) shows the schematic of Mach−Zehnder interferometer for the vector beam generation [77]. The vector beam is generated through adjusting individually beams on two arms of Mach−Zehnder interferometer, where many additional elements are used. The spiral phase plate is used to generate the vortex beam, the quarter-wave plates, prism and the phase compensator are used to form the desired phase and polarization. The superposition of two orthogonal polarized beams with controllable phases forms any Poincaré beams. The generation mechanism of this kind of vector beams can be seen in Section 2.4.

Sagnac interferometer with two beams transmitting along opposite directions in one loop-shaped path is also used to generate vector beam [81]. In Sagnac interferometer, the incident linearly polarized beam is separated into two orthogonally polarized components by a beam splitter. Two beams transmit along the clockwise and counterclockwise directions of the ring-shaped optical path. Two beams, with equal optical paths, pass through a holographic phase plate forward and backward, respectively, which are loaded with opposite spiral phases, and finally return to the beam splitter and produce vector beam. To simplify, the holographic phase plate being projected with reversed beams can be replaced with reflective spatial light modulator in practical applications. In addition, a sagnac-like interferometer with a spatial light modulator placed outside the ring path is proposed to generate vector beams [82].

Additionally, a two-element interferometer is proposed to generate arbitrary vector beams on higher-order Poincaré spheres [80]. In two-element interferometer, one vortex beam passes through a beam displacer, and its s-polarized and p-polarized components are separated. The transmitted s-polarized and reflected p-polarized components with π/2 phase shift pass through a quarter-wave plate, and ultimately form a cylindrical vector beam because of the superposition of two beams with opposite topological charges and orthogonal circular polarization states. By adjusting the polarization state of input beam, the vector beam can be flexibly controlled to generate arbitrary vector beams on the higher-order Poincaré sphere.

3.5 Spatial light modulator modulation

Spatial light modulator is a programmable diffraction device which consists of individually addressable pixels. The phase and intensity of the reflected or transmitted field can be statically or dynamically controlled through computer programming. Therefore, spatial light modulators have been widely used to manipulate optical fields and generate vector beams.

Fig.6(d) shows the utilization of spatial light modulator in vector beam generation [78], where two beams carrying opposite helical phases output from spatial light modulator are converted into two orthogonal circularly polarized beams in the spectral plane of 4f imaging system. The diffraction of grating realizes the superposition of two orthogonally polarized beams. Besides the CVBs, the defaulted vector beams can be generated in terms of the diffraction of spatial light modulator, the polarization conversion of wave plate and the beam superposition of combiner [83]. Furthermore, the arrayed vector beams can be produced when the information of Dammann vortex grating is loaded on the spatial light modulator [84].

Recent developments about spatial light modulator-based vector beam generation have focused on improving efficiency and simplifying optical configurations. Efficient on-axis SLM engineering employing two liquid-crystal on silicon spatial light modulators in a common path architecture has been demonstrated [85]. This approach significantly reduces system complexity since it avoids the use of beam-splitters. Modified off-axis interferometric holography combined with liquid crystal spatial light modulator has demonstrated remarkable capability to generate vector beam [86]. Furthermore, the liquid crystal SLM may generate hybrid vector beam without the help of interference devices [87]. This technique represents a significant step forward in creating complex structured light field with a single device.

Two beams may also be directly modulated by the slit-screen spatial light modulator [88]. However, the generation of vector beams still needs the help of polarization beam splitter and combiner. It is obvious that the split screen of spatial light modulator must lead to relatively lower working efficiency. Thus, the distances between different polarization components need to exactly adjust so as to obtain high efficiency [89], and the utilization of special polarization beam splitter may cause the efficiency of the generated vector beams to increase threefold [90].

3.6 Metasurface modulation

Metasurfaces composed of periodic nanoscaterers exhibit strong electromagnetic control abilities [91] and the control of electromagnetic field rely on local response of nanoscaterers not optical path accumulation effect and this makes metasurfaces extremely superior for device miniaturization and integration [92]. The anisotropic nanoscaterers with appropriate shape, size, material and spatial arrangement may be equivalent to half-wave plate [40, 93], quarter-wave plate [94] or polarizer [95], and the different nanoscatterer or rotated nanoscatterer may introduce additional phase delay. Therefore, vector beams can be generated using the local polarization effect of anisotropic nanoscaterers or the phase and polarization modulation of metasurface [96, 97].

The nanoscaterers equivalent to quarter- and half-wave plates and arranged along special trajectories may generate CVBs [44, 98]. Fig.7(a) show two metasurface structures and the intensity and polarization distributions of the generated CVBs [44], where the equivalent quarter-wave plates function with circularly polarized light illumination and the equivalent half-wave plates work with linearly polarized light illumination. During the polarization conversion, the unexpected additional phase needs to be deleted [99].

Fig.7(b) shows the RVB generation based on the superposition of the LCP light and the converted LCP carrying different spiral phases reflected by metasurface [100], and Fig.7(c) shows the AVB generation from the superposition of two orthogonal circularly polarized beams carrying opposite spiral phases transmitted by metasurface [101]. In addition, the linear phase gradient introduced by the metasurface may allow off-axis superposition of two orthogonal circularly polarized beams to generate the vector beam [102].

Two orthogonal polarized beams simultaneously focused by metasurface may generate vector beam [41]. Fig.7(d) and (e) provide the schematic diagrams for the generation of Poincaré beams, where Fig.7(d) shows the metal metasurface [42] and Fig.7(e) exhibits the dielectric metasurface [36]. Certainly, the polarization distribution of vector beam changes with the weights of two orthogonal polarization components and the spiral phases carried by two orthogonal polarization components. In this way, any HPVBs can generate.

The arrayed vector beams may be generated using the holographic metasurface, and the holographic metasurface can be constructed based on multiple phase function superposition [103]. Vector beams with different orders are produced at different transmission positions [104]. Certainly, the number of multiplexed channels of vector beams may change with the status of metasurface units [105], and the well-designed metasurface can realize the variation from one vector beam to another vector beam in space [106]. With help of the wavelength dependence of polarization response, wavelength multiplexing vector beams may be generated by metasurface [107]. In brief, as the metasurface guides two orthogonal polarized beams to overlap in space, the defaulted vector beam generates at the over-lapping position [108].

3.7 Nonlinear crystal modulation

The above-mentioned methods belong to linear optical domain, where the working wavelength remains unchanged. Recently, nonlinear optical effects of crystal, such as second-harmonic generation [109], third-harmonic generation [110], Kerr effect [111], stimulated Raman scattering and stimulated Brillouin scattering [112], have been applied to generate vector beams since the larger additional phase can be introduced during polarization conversion. The crystal materials include BBO [113], KDP [114], ZnTe [35] and ithiumniobate [63]. The output beam with different frequency can be obtained through solving the coupled wave equations with specified boundary conditions, and the phase-matching condition should be satisfied in order to achieve high efficiency.

Fig.7(f) shows the generation of second-harmonic vector beams using ithiumniobate crystal [63]. The second harmonic vector beam is generated using half-wave plate and nonlinear fork grating. The second harmonic linear polarization vortex beams satisfy Type I phase matching condition and the polarization combiner guide two beams to superpose. Fig.7(g) shows the generation of second-harmonic Poincaré sphere beam using a KTP crystal [109]. Where the fundamental vortex beam generates based on the spatial light modulator. The KTP crystal used to generate the second-harmonic vector beam functions under Type II phase matching.

When the incident laser power is excessively high, third-order or even higher order nonlinear effect occurs. Graphene is proposed to generate higher harmonic vector beams [115], where the topological structure of monolayer graphene preserves the Poincaré index invariance of vector beams. Moreover, the four wave mixing process in Rb vapor is also studied based on the nonlinear magneto-optical rotation mechanism and third-order nonlinear vector beams are generated [116].

Besides nonlinear crystals under phase-matching condition utilize to obtain nonlinear vector beams, periodic crystals under quasi-phase matching conditions are employed to generate non-linear vector beams. In this case, the reciprocal lattice vector of periodic crystal compensates for phase mismatch and the quasi-phase matching condition is satisfied. The combination of a quasi-periodically QPPKTP crystal and Sagnac interferometer to achieve third-harmonic generation of vector field [110]. Where vortex beams modulated by the SLM are adjusted by the polarization prism to form vector fundamental beams. Two orthogonal linearly polarized beams in the Sagnac interferometer pass through the QPPKTP crystal in opposite directions and third-harmonic vector field generates.

Multipolar Mie-type resonances between all-dielectric nanoparticles may amplify the nonlinear effects, which do not require phase matching. The generation of a third-harmonic vector beam induced by a-Si nanoparticle oligomers with vector beams is proposed [117], where the vector beams are adjusted by liquid crystal polarization converter and strong THG signal results from the collective response of the nanoparticles. This approach avoids complex setup and enhances nonlinear effect.

Overall, different methods for vector beam generation take on distinct characteristics and the qualities of the generated vector beams also vary. Actively controlled vector light sources offer high efficiency and stable modes, but the setup is complex and lack of flexibility. Methods based on q-plates and gratings require more additional components, and the generated vector beams are fixed. Spatial light modulator allows for dynamic control of optical field, but the electric computer and other additional elements are required. Metasurfaces can generate one or more vector beams, and yet the working efficiency and beam quality need to be improved. Nonlinear effect is still weaker than linear effect though it brings more sensitive modulation. Anyway, these methods provide more choice for the generation of desired vector beams.

4 Propagation of vector beam

4.1 Vector diffraction theory

The vector field of E(x, y, z) at any point in space follows vector Helmholtz equation of \mathrm{\nabla } × (\mathrm{\nabla } × E) + k²E = 0, where k is the propagation constant of vector field and the symbol of \mathrm{\nabla } denotes the gradient operator. In Cartesian coordinates, three components of vector field satisfy the following differential relations [118]:

(14)\begin{array}{rl}& -\frac{{\mathrm{\partial }}^2{E}_x}{\mathrm{\partial }{y}^2}-\frac{{\mathrm{\partial }}^2{E}_x}{\mathrm{\partial }{z}^2}+\frac{{\mathrm{\partial }}^2{E}_y}{\mathrm{\partial }x\mathrm{\partial }y}+\frac{{\mathrm{\partial }}^2{E}_z}{\mathrm{\partial }x\mathrm{\partial }z}+k^2{E}_x=0,\\ & -\frac{{\mathrm{\partial }}^2{E}_y}{\mathrm{\partial }{z}^2}-\frac{{\mathrm{\partial }}^2{E}_y}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^2{E}_x}{\mathrm{\partial }x\mathrm{\partial }y}+\frac{{\mathrm{\partial }}^2{E}_z}{\mathrm{\partial }y\mathrm{\partial }z}+k^2{E}_y=0,\\ & -\frac{{\mathrm{\partial }}^2{E}_z}{\mathrm{\partial }{x}^2}-\frac{{\mathrm{\partial }}^2{E}_z}{\mathrm{\partial }{y}^2}+\frac{{\mathrm{\partial }}^2{E}_x}{\mathrm{\partial }x\mathrm{\partial }z}+\frac{{\mathrm{\partial }}^2{E}_y}{\mathrm{\partial }y\mathrm{\partial }z}+k^2{E}_z=0.\end{array}

Suppose one vector beam with a slowly varying longitudinal component E_z propagates along z axis and the second derivation of E_z is neglected, the third equation in Eq. (14) can be approximated as E_z = −[∂²E_x/(∂x∂z) + ∂²E_y/(∂y∂z)]/k². This indicates the longitudinal component of vector field at any position can be expressed by two transverse components. In terms of the Fresnel−Kirchhoff diffraction, three components of vector field in free space can be expressed as [119]

(15)\begin{array}{rl}& E_x(x_1,y_1,z)=\iint \frac{z}{\mathrm{i}\lambda R^2}{E}_{0x}(x,y,0){\mathrm{e}}^{\mathrm{i}kR}\mathrm{d}x\mathrm{d}y,\\ & E_y(x_1,y_1,z)=\iint \frac{z}{\mathrm{i}\lambda R^2}{E}_{0y}(x,y,0){\mathrm{e}}^{\mathrm{i}kR}\mathrm{d}x\mathrm{d}y,\\ & E_z(x_1,y_1,z)=\iint [\mathrm{\Delta }x{E}_{0x}(x,y,0)+\mathrm{\Delta }y{E}_{0y}(x,y,0)]A(R)\mathrm{d}x\mathrm{d}y,\end{array}

where Δx = x₁ − x, Δy = y₁ − y, R = [(x₁ − x)² + (y₁ − y)² + z²]^1/2 represents the distance between two points with the coordinates of (x, y, 0) and (x₁, y₁, z). The term A(R) may be approximated as A(R) = (ik−2/R)e^ikR/(iλR³k²). If the vector beam satisfies the paraxial approximation during propagation, the tilt factor z/R can be neglected and A(R) and R is simplified to A(R) = e^ikR/(λz³k) and R = z + [(x₁ − x)² + (y₁ − y)²]/2z.

Similarly, in cylindrical coordinates, the diffraction field for three components of vector beam can be expressed as

(16)\begin{array}{rl}& E_r(r,\phi ,z)=\frac{1}{\mathrm{i}\lambda z}\iint [E_{\rho }(\rho ,\theta ,0)\cos (\theta -\phi )-E_{\theta }(\rho ,\theta ,0)\sin (\theta -\phi )]{\mathrm{e}}^{\mathrm{i}kR}\rho \mathrm{d}\rho \mathrm{d}\theta ,\\ & E_{\phi }(r,\phi ,z)=\frac{1}{\mathrm{i}\lambda z}\iint [E_{\rho }(\rho ,\theta ,0)\sin (\theta -\phi )+E_{\theta }(\rho ,\theta ,0)\cos (\theta -\phi )]{\mathrm{e}}^{\mathrm{i}kR}\rho \mathrm{d}\rho \mathrm{d}\theta ,\\ & E_z(r,\phi ,z)=\frac{1}{\lambda z^{3}k}\iint \{E_{\rho }(\rho ,\theta ,0)[\rho \cos (\theta -\phi )-\rho ]+E_{\theta }(\rho ,\theta ,0)r\sin (\theta -\phi )\}{\mathrm{e}}^{\mathrm{i}kR}\rho \mathrm{d}\rho \mathrm{d}\theta .\end{array}

In the above equation, R² = r² + ρ² + z²−2rρcos(φ−θ) represents the distance between two points with coordinates (r, φ, 0) and (ρ, θ, z), where x = ρcosθ, y = ρsinθ, x₁ = rcosφ, and y₁ = rsinφ.

Besides the above representations, the propagation of vector beam can be also analyzed using vector angular spectrum method [120], vector Rayleigh diffraction method [121] or phasing method [122]. In addition, the strong focusing of vector beam is usually described by Richards−Wolf vector diffraction theory [123, 124].

4.2 Diffraction of vector beam by small aperture

Non-paraxial propagation of vector beam through an aperture is accurately described in terms of rigorous vector diffraction methods. Based on Eq. (16), the diffraction of vector beams through one small aperture can be expressed as [125]

(17)\begin{array}{rl}{E}_x(x_1,y_1,z)=& \iint \frac{z}{\mathrm{i}\lambda R^2}{E}_{0x}(x,y,0)P(x,y){\mathrm{e}}^{\mathrm{i}kR}\mathrm{d}x\mathrm{d}y,\\ E_y(x_1,y_1,z)=& \iint \frac{z}{\mathrm{i}\lambda R^2}{E}_{0y}(x,y,0)P(x,y){\mathrm{e}}^{\mathrm{i}kR}\mathrm{d}x\mathrm{d}y,\\ E_z(x_1,y_1,z)=& \iint [(x_1-x)E_{0x}(x,y,0)\\ & +(y_1-y)E_{0y}(x,y,0)]P(x,y)A(R)\mathrm{d}x\mathrm{d}y,\end{array}

where P(x, y) represents the pupil function of aperture. For the RVB with Laguerre−Gaussian mode, its x and y components can be expressed as [119]

(18)\begin{array}{rl}& E_{0x}(x,y,0)=\sqrt{2}{E}_0\frac{x}{w_0}{L}_n^1(\frac{2{\rho }^2}{w_0^2}){\mathrm{e}}^{-{(\frac{\rho }{w_0})}^2},\\ & E_{0y}(x,y,0)=\sqrt{2}{E}_0\frac{y}{w_0}{L}_n^1(\frac{2{\rho }^2}{w_0^2}){\mathrm{e}}^{-{(\frac{\rho }{w_0})}^2},\end{array}

where ρ² = x² + y², E₀ is the amplitude of optical field, w₀ is the beam waist, and L_n¹ denotes Laguerre polynomials. Substituting this equation into Eq. (17), one can obtain the non-paraxial diffraction of RVB. The studied result show the diffraction energy of RVB varies with the order of n, waist width of w₀ and aperture radius of a. Similarly, the diffraction of RVB through annular aperture can be studied [126]. The increase of the order n leads to the increase of diffraction intensity, the increase of beam waist w₀ results in the higher peak moving to the optical axis. The inner radius b causes the peak intensity to decrease and shift towards the optical axis.

For higher-order CVB with two components denoted by the following expression [127]:

(19)\begin{array}{rl}& E_x(\rho ,\theta ,0)=\sqrt{2}{E}_1\frac{\rho }{w_0}{\mathrm{e}}^{-{(\frac{\rho }{w_0})}^2}\cos (n\theta +\varphi ),\\ & E_y(\rho ,\theta ,0)=\sqrt{2}{E}_1\frac{\rho }{w_0}{\mathrm{e}}^{-{(\frac{\rho }{w_0})}^2}\sin (n\theta +\varphi ),\end{array}

its diffraction distribution through an annular aperture can be obtained through inserting Eq. (19) into Eq. (17). The study shows that the transverse polarization remains unchanged during the propagation, while the longitudinal component varies with the azimuthal angle. The diffraction intensity distribution takes on multiple concentric rings, and the longitudinal intensity and far-field divergence angle are influenced by the order, the inner and outer radii of aperture and the beam waist.

Besides the investigation about the variation of vector field with the aperture plane perpendicular to the optical axis, the diffraction of vector beam through an aperture inclined with relative to the optical axis is also studied [128]. The results show the slight offset between the beam and aperture center may cause two polarization components of CVB to separate along the direction perpendicular to the displacement, and thus, the diffraction pattern of vector beam becomes asymmetric.

4.3 Focusing of vector beam by lens

Strongly focused light fields cannot be analyzed using our familiar Kirchhoff-Fresnel diffraction theory or vector angular spectrum method since the longitudinal component has strong influence on the diffraction field. And here, strongly focused diffraction theory proposed by B. Richards and E. Wolf may be utilized [124]. On basis of this theory, the diffraction of RVB and AVB [18] and even generally CVBs [37, 129] by an objective with high numerical aperture can be explored.

Suppose one CVB is focused by a lens with the focal length of f, and the incident plane can be represented in Cartesian coordinates as (x₀, y₀, 0) or cylindrical coordinates as (ρ, ϕ, 0). The focal plane can be represented in Cartesian coordinates as (x, y, f) or cylindrical coordinates as (r, φ, f), as illustrated in Fig.8.

The optical field around the focal plane of lens can be expressed as the superposition of plane waves from the exit pupil plane of lens,

(20)\mathit{E}(r,\phi ,z)={\displaystyle \iint A(\theta )P(\theta )\mathit{a}(\theta ,\varphi ){\mathrm{e}}^{\mathrm{i}k\mathit{s}\cdot \mathit{r}}\sin \theta \mathrm{d}\theta \mathrm{d}\varphi ,}

where A(θ) is amplitude factor, which satisfies energy conservation E₀²2πρdρ = A²(θ)2πf²sinθdθ and sinusoidal condition ρ = fsinθ, P(θ) is pupil function, and a(θ,ϕ) is the unit vector of optical field on the exit surface of pupil. s_x = sinθcosϕ, s_y = sinθsinϕ and s_z = cosθ are unit vectors for the outgoing rays along the radial, angular and longitudinal directions, and x = rcosφ and y = rsinφ are coordinates of the observation point on the focal plane. The dot product of s·r can be simplified as follows:

(21)\mathit{s}\cdot \mathit{r}=z\cos \theta +r\sin \theta \cos (\varphi -\phi ).

For the LPVB of cos(nϕ + ϕ₀)e_x + sin(nϕ + ϕ₀)e_y, the angular polarization component of the incident light at any position on the pupil or the exit plane is always vertical to the meridian plane composed of propagation axis and optical axis. The unit vector of LPVB may be also written in the cylindrical coordinates as a = cos[(n−1)ϕ + ϕ₀]e_ρ + sin[(n−1)ϕ + ϕ₀]e_ϕ, where e_ρ = cosϕe_x + sinϕe_y and e_ϕ = −sinϕe_x + cosϕe_y. Then, the unit vector on the exit plane has the cylindrical coordinates of e_ρ′ = cosθe_ρ + sinθe_z and e_ϕ′ = e_ϕ, and the Cartesian components of the unit vector a(θ, ϕ) can be expressed as

(22)\begin{array}{rl}& a_x=\cos [(n-1)\varphi +{\varphi }_0]\cos \theta \cos \varphi -\sin [(n-1)\varphi +{\varphi }_0]\sin \varphi ,\\ & a_y=\cos [(n-1)\varphi +{\varphi }_0]\cos \theta \sin \varphi +\sin [(n-1)\varphi +{\varphi }_0]\cos \varphi ,\\ & a_z=\cos [(n-1)\varphi +{\varphi }_0]\sin \theta .\end{array}

Insert Eqs. (21) and (22) into Eq. (20), one can rewrite the optical field around the focal plane of lens as

(23)\begin{array}{rl}& E_x(r,\phi ,z)=C\mathrm{i}{\int }_0^{{\theta }_{max}}P(\theta )\sqrt{\cos \theta }\sin \theta {\mathrm{e}}^{\mathrm{i}kz\cos \theta }F(\theta )\mathrm{d}\theta ,\\ & E_y(r,\phi ,z)=C\mathrm{i}{\int }_0^{{\theta }_{max}}P(\theta )\sqrt{\cos \theta }\sin \theta {\mathrm{e}}^{\mathrm{i}kz\cos \theta }G(\theta )\mathrm{d}\theta ,\\ & E_z(r,\phi ,z)=2C{\int }_0^{{\theta }_{max}}P(\theta )\sqrt{\cos \theta }\sin \theta {\mathrm{e}}^{\mathrm{i}kz\cos \theta }H(\theta )\mathrm{d}\theta ,\end{array}

where C is a constant, θ_max is the maximum field angle of optical pupil, F(θ) = iⁿ{(cosθ−1)cos(nφ + ϕ₀)J_n(krsinθ) − (cosθ + 1)cos[(n−2)φ +ϕ₀]J_n−2(krsinθ)}, G(θ) = iⁿ{(cosθ + 1)sin(nφ + ϕ₀)J_n(krsinθ) + (cosθ−1)sin[(n−2)φ + ϕ₀]J_n−2(krsinθ)}, H(θ) = iⁿ⁻¹sinθcos[(n−1)φ + ϕ₀]J_n−1 (krsinθ), and J_l(x) denotes the first type of lth-order Bessel function. The upper limit of the integration corresponding to the maximum field angle, and as n = 1 and ϕ₀ = 0, namely, the RVB, F(θ) = 2icosθcosφJ₁(l_θ), G(θ) = 2icosθsinφJ₁(l_θ), H(θ) = sinθJ₁(l_θ), the distribution of the vector field near the focal plane is [37]

(24)\begin{array}{rl}& E_x(r,\phi ,z)=\mathrm{i}C\cos \phi {\int }_0^{{\theta }_{max}}\sqrt{\cos \theta }P(\theta )\sin 2\theta J_1(l_{\theta }){\mathrm{e}}^{\mathrm{i}kz\cos \theta }\mathrm{d}\theta ,\\ & E_y(r,\phi ,z)=\mathrm{i}C\sin \phi {\int }_0^{{\theta }_{max}}\sqrt{\cos \theta }P(\theta )\sin 2\theta J_1(l_{\theta }){\mathrm{e}}^{\mathrm{i}kz\cos \theta }\mathrm{d}\theta ,\\ & E_z(r,\phi ,z)=2C{\int }_0^{{\theta }_{max}}\sqrt{\cos \theta }P(\theta ){\sin }^2\theta J_0(l_{\theta }){\mathrm{e}}^{\mathrm{i}kz\cos \theta }\mathrm{d}\theta ,\end{array}

where l_θ = krsinθ. Furthermore, the light field around the focal plane is often expressed in cylindrical coordinates [130]:

(25)\begin{array}{rl}& E_r(r,\phi ,z)=\mathrm{i}C{\int }_0^{{\theta }_{max}}{\cos }^{1/\phantom{12}2}(\theta )P(\theta )\sin 2\theta J_1(l_{\theta }){\mathrm{e}}^{\mathrm{i}kz\cos \theta }\mathrm{d}\theta ,\\ & E_{\phi }(r,\phi ,z)=0,\\ & E_z(r,\phi ,z)=2C{\int }_0^{{\theta }_{max}}{\cos }^{1/\phantom{12}2}(\theta )P(\theta ){\sin }^2\theta J_0(l_{\theta }){\mathrm{e}}^{\mathrm{i}kz\cos \theta }\mathrm{d}\theta .\end{array}

Obviously, the focusing field of RVB only contains the radial and longitudinal components. Fig.9(a) shows the longitudinal and transverse fields of RVB at the focusing plane of lens [129], and the longitudinal field is stronger than the transverse field.

For the RVB with the Laguerre−Gaussian mode, the equivalent pupil function can be expressed as A(β₀, θ)exp[−A(β₀, θ)sinθ]L_p¹[2A(β₀, θ)sinθ] [133, 134] with A(β₀, θ) = β₀²sinθ/sin²θ_max and β₀ = a/w₀ denoting the pupil filling factor. The parameter of β₀ may adjust the energy distribution of the RVB, like eliminate the longitudinal component and obtain an optical cage with only radial component. The size of optical cage can be regulated through changing the radial phase [135]. Furthermore, optical cage arrays can also be generated through superposition of RVBs and AVBs, as shown in Fig.9(b), where the number and position of optical cages can be modulated through adjusting the phase of RVBs and AVBs [131].

As n = 1, ϕ₀ = 90°, namely, the AVB, F(θ) = −2isinφJ₁(l_θ), G(θ) = 2isinφJ₁(l_θ) and H(θ) = 0. Insert F(θ), G(θ) and H(θ) into Eq. (24), the focusing field in cylindrical coordinates is [134]

(26)\begin{array}{rl}& E_r(r,\phi ,z)=0,\\ & E_{\phi }(r,\phi ,z)=2\mathrm{i}C\cos \phi {\int }_0^{{\theta }_{max}}\sqrt{\cos \theta }P(\theta )\sin \theta J_1(l){\mathrm{e}}^{\mathrm{i}kz\cos \theta }\mathrm{d}\theta ,\\ & E_z(r,\phi ,z)=0.\end{array}

The focused field of AVB only contains the azimuthal component. Consequently, the optical field at the focal plane exhibits the doughnut shape. For the Laguerre−Gaussian AVB, the intensity distribution and polarization direction of the focusing field also change with the focusing angle of the inner ring [134]. Furthermore, when Poincaré beams are used for tight focusing, the focusing intensity changes with the initial phase of Poincaré beams, as shown in Fig.9(c) [132]. The transverse field varies from weak to strong and then to weak, and the axial field varies from strong to weak and then to strong.

4.4 Scattering of vector beam in random medium

The scattering of random medium or any object may cause the depolarization effect. To address this, various techniques have been developed to explore the polarization change, including transmission matrix method [136], optical phase conjugation technique [137], iterative optimization method [138] and deep learning model [139]. When vector beams propagate through random scattering media such as atmospheric turbulence, oceanic turbulence and or biological tissues, the statistical theories should be utilized to analyze the change of polarization.

The scattering of a monochromatic scalar field can be typically analyzed using the scattering potential function of medium and Helmholtz equation. Under weak scattering conditions, the scattered field satisfies the first-order Born approximation and Helmholtz equation simplifies to (\mathrm{\nabla }² + k²)U^s(r) = −4πFU^s(r). In this case, the scattered field can be expressed as

(27)U^s(\mathit{r})={\displaystyle {\int }_{D}F(\mathit{\rho })U^i(\mathit{\rho })G(|\mathit{r}-\mathit{\rho }|)\mathrm{d}\mathit{\rho },}

where Uⁱ(r) denotes the incident field and G(|r−ρ|) is the Green’s function. F(ρ) is the scattering potential of medium, and for the scattering medium with the refractive index of n, F(ρ) = k²[n²−1]/(4π). The statistical average of the scattered field may be represented by the cross-spectral density function [140]:

(28)\begin{array}{rl}{W}^s({\mathit{r}}_1,{\mathit{r}}_2)=& {\iint }_D{W}^i({\mathit{\rho }}_1,{\mathit{\rho }}_2)⟨F({\mathit{\rho }}_1)F({\mathit{\rho }}_2)⟩\\ & \times G^{\ast }(|{\mathit{r}}_1-{\mathit{\rho }}_1|)G(|{\mathit{r}}_2-{\mathit{\rho }}_2|)\mathrm{d}{\mathit{\rho }}_1\mathrm{d}{\mathit{\rho }}_2.\end{array}

Here, W^i\left({\mathit{\rho }}_1,{\mathit{\rho }}_2\right)=⟨U^{i^{\ast }}\left({\mathit{\rho }}_1\right)U^i\left({\mathit{\rho }}_2\right)⟩ and W^s\left({\mathit{r}}_1,{\mathit{r}}_2\right)=⟨U^{s^{\ast }}\left({\mathit{r}}_1\right)U^s\left({\mathit{r}}_2\right)⟩ are the cross-spectral density functions of the incident and scattered fields, ⟨\cdot ⟩ denotes the statistical average and the superscript of ^* denotes the complex conjugate. ⟨F\left({\mathit{\rho }}_1\right)\left({\mathit{\rho }}_2\right)⟩ is also known as the scattering potential correlation function for random media, and for different scattering media, it has different expression.

For a monochromatic vector field Eⁱ(ρ)exp(iks∙ρ) propagating along the direction with unit vector of s, the vector scattered field is E^s(r) = \mathrm{\nabla } × \mathrm{\nabla } × [∫F(ρ)Eⁱ(ρ)G(|r−ρ|)/k²]dρ [141]. Under weak scattering and the first-order Born approximation conditions, the Green’s function in the far field can be represented as exp(ikr)exp(−iks∙ρ)/(iλr), where λ is the incident wavelength and the scattered field can be simplified. The cross-spectral density matrix of vector scattered field is obtained in terms of W^s(r₁,r₂) = [⟨{\mathit{E}}_j^{s^{\ast }}\left({\mathit{r}}_1\right)\cdot {\mathit{E}}_j^{s^{\mathrm{\prime }}}\left({\mathit{r}}_2\right)⟩] can be obtained and the intensity, degree of polarization and coherence of vector scattered field can be expressed as [142]

(29)\begin{array}{rl}& S^s(\mathit{r})=Tr\{W^s(\mathit{r},\mathit{r})\},\\ & P^s(\mathit{r})=\sqrt{1-\frac{4Det\{W^s(\mathit{r},\mathit{r})\}}{Tr\{W^s(\mathit{r},\mathit{r})\}}},\\ & {\mathrm{\Gamma }}^s({\mathit{r}}_1,{\mathit{r}}_2)=\frac{Tr\{W^s({\mathit{r}}_1,{\mathit{r}}_2)\}}{\sqrt{Tr\{W^s({\mathit{r}}_1,{\mathit{r}}_1)\}Tr\{W^s({\mathit{r}}_2,{\mathit{r}}_2)\}}}.\end{array}

Tr{∙} and Det{∙} in the above equation denote the trace and determinant operation of the cross-spectral density matrix, respectively. The coherence matrix and its transmission principle are proposed to study the propagation of Gaussian vector beams through atmospheric turbulence under the Kolmogorov spectrum and the second-order approximation of the Rytov phase structure function [143].

For atmospheric turbulence, the refractive index of scattering medium exhibits random fluctuation in space and time. The correlation function of scattering potential can be expressed as ⟨F\left({\mathit{\rho }}_1\right)\left({\mathit{\rho }}_2\right)⟩ = exp[−(ρ₁−ρ₂)²/ξ²], where ξ relates to the structure constant of atmospheric turbulence. The scattered vector beam at a transmission distance z can be expressed as [142]

(30)\begin{array}{rl}& {\mathrm{\Gamma }}_{\alpha \beta }^s(r,z)\\ & ={\displaystyle \frac{1}{{\lambda }^2{z}^2}}{\displaystyle \iiint {\mathrm{\Gamma }}_{\alpha \beta }^i({\rho }_1,{\rho }_2){\mathrm{e}}^{-\frac{\mathrm{i}k}{2z}[{({\rho }_1-r)}^2{({\rho }_2-r)}^2]-\frac{1}{{\xi }^2}{({\rho }_1-{\rho }_2)}^2}\mathrm{d}{\rho }_1\mathrm{d}{\rho }_2.}\end{array}

Thus, the intensity and degree of polarization of the scattered vector beam can be determined. The study results show though the beam in scattering media is always vector beam, the polarization structure [144] and the intensity distribution change with the propagation.

Based on the above principle, the scattering of RVB in atmospheric turbulence limited by a circular aperture is analyzed [145]. As the aperture gets larger and it causes the change of beam shape along the z-axis. The aperture mitigates the impact of atmospheric turbulence on radial polarization. Moreover, the degree of polarization for focusing RVB can be derived in terms of the extended Huygens−Fresnel integral formula and the coherence matrix of focused vector field [142]. The increase of the structure constant of atmospheric turbulence may accelerate the evolution of the beam profile. In addition, relative to the collimated vector laser beam, the focused vector beam exhibits faster coherence degradation in turbulence.

4.5 Propagation of vector beam in nonlinear media

When vector beam propagates through nonlinear media, the nonlinear response of media causes different phenomena from the linear case. Generally, the nonlinear response can be classified into local and nonlocal nonlinear responses. When the characteristic length of response function is smaller than the width of incident light, the nonlinear response is local, and contrarily, it is a nonlocal. In local nonlinear media, the coupled propagation of beams can be described using coupled wave equation. For nonlocal nonlinear media like lead glass and liquid crystals, the coupled propagation of beams needs to be described using the nonlinear Schrödinger equation and Poisson equation.

During the propagation in nonlinear media, vector beam experiences the incoherent coupling of two orthogonal polarization components. Thus, under the conditions of slow-variable envelope and paraxial approximation, the transmission of vector beam can be described using the nonlinear Schrödinger equation [146]:

(31)\begin{array}{rl}& \mathrm{i}k\frac{\mathrm{\partial }{\mathit{E}}_1}{\mathrm{\partial }z}=\frac{1}{2}{\mathrm{\nabla }}_{\mathrm{\perp }}^2{\mathit{E}}_1+\frac{2n_2{k}^2}{3n_0}({\left|{\mathit{E}}_1\right|}^2+2{\left|{\mathit{E}}_2\right|}^2){\mathit{E}}_1,\\ & \mathrm{i}k\frac{\mathrm{\partial }{\mathit{E}}_2}{\mathrm{\partial }z}=\frac{1}{2}{\mathrm{\nabla }}_{\mathrm{\perp }}^2{\mathit{E}}_2+\frac{2n_2{k}^2}{3n_0}({\left|{\mathit{E}}_2\right|}^2+2{\left|{\mathit{E}}_1\right|}^2){\mathit{E}}_2,\end{array}

where E₁ and E₂ represent two orthogonal polarization components of vector beam, k is linear wave number, n₀ and n₂ denote the linear and nonlinear refractive indices, and \mathrm{\nabla }² represents the Laplacian operator. For Kerr nonlinear media, the second terms on the right of two equations represent the focusing behavior of vector beam. When Kerr effect and the diffraction effect cancel each other, the beam propagates stably in the nonlinear medium. And since the vector beam is subjected to self- and cross-phase modulation, the state of polarization changes during the propagation and the collapse or filamentation phenomenon appears.

As one CVB with the power exceeding the critical power propagates through one Kerr medium, two orthogonal polarization components couple and the vector beam splits, which is known as beam collapse. The inhomogeneous polarization distribution of vector beam can suppress the beam collapse and the weights of orthogonal polarization components of vector beam determine the propagation stability. Polarization distributions of LPVBs and HPVBs through hot Rb atomic vapor are also studied [147]. The results show that polarization distributions of LPVBs are almost unchanged. And yet the polarization distributions of HPVBs rotate during the propagation, and Fig.10(a) shows the simulated results for the variation of polarization rotation [148]. The polarization rotation of HPVB is also influenced by the spiral phases of two orthogonal components and the nonlinear refractive index.

Additionally, the EPVB collapse results in the filaments and the number of filaments is related to the spiral phases carried with orthogonal components [149]. And the EPVB propagating through hot Rb atomic vapor uncovers the relationship of the position of collapse and the state of polarization. Fig.10(b) displays the collapse process of EPVB in hot Rb atomic vapor with and without considering atomic absorption [150]. The results show that the filaments ultimately locate in the circularly polarized position due to the atomic vapor absorption. In the transparent medium [151], where the atomic absorption is not considered, the energy accumulates on the linear polarization and it leads to the collapse position due to the larger nonlinear refractive index caused by the linear polarization.

The polarization distribution of EPVB soliton in CS2 planar waveguide also rotates during the propagation [152], and the state of polarization of the soliton ultimately tends to stabilize at larger propagation distance. Yet for two vector solitons in birefringent fiber, the polarization distributions of two vector solitons change in different ways and two vector solitons relatively move. Two solitons do not collide but gradually separate through energy exchanging [153]. The coupling of orthogonal polarization components in photonic crystal fiber also influences the stability and bifurcation of vector solitons [154]. The adjustment of nonlinear coefficient in nonlinear fiber can effectively control the intensity and spatial distribution of vector soliton [155].

In nonlocal nonlinear medium, the propagation characteristics of vector beam depend not only on the local medium but also the close medium even covering a larger region. This long-range nonlinear response endows vector beam with unique modulation properties. For simplicity, the nonlocal nonlinear Schrödinger equation is often degraded to the Snyder−Michell model. In this case, the far-field beam is represented by the nonlocal response function R(r−r_a), and Eq. (30) should be rewritten as [157]

(32)\begin{array}{rl}& \frac{\mathrm{\partial }{\mathit{E}}_1}{\mathrm{\partial }z}=\frac{\mathrm{i}}{2k}{\mathrm{\nabla }}_{\mathrm{\perp }}^2{\mathit{E}}_1+\frac{2\mathrm{i}{n}_2{k}^2{\mathit{E}}_1}{n_0}\int R(\mathit{r}-{\mathit{r}}_{\mathit{a}})[I_1({\mathit{r}}_{\mathit{a}})+I_2({\mathit{r}}_{\mathit{a}})]\mathrm{d}{\mathit{r}}_{\mathit{a}},\\ & \frac{\mathrm{\partial }{\mathit{E}}_2}{\mathrm{\partial }z}=\frac{\mathrm{i}}{2k}{\mathrm{\nabla }}_{\mathrm{\perp }}^2{\mathit{E}}_2+\frac{2\mathrm{i}{n}_2{k}^2{\mathit{E}}_2}{n_0}\int R(\mathit{r}-{\mathit{r}}_{\mathit{a}})[I_2({\mathit{r}}_{\mathit{a}})+I_1({\mathit{r}}_{\mathit{a}})]\mathrm{d}{\mathit{r}}_{\mathit{a}},\end{array}

where I_j = |E_j|², and two integrations represent the normalized spatial response function of nonlinear medium.

Unstable vector solitons in local nonlinear media can change into stable bound states in nonlocal nonlinear media. These bound states exhibit multi-soliton structures in each polarization component [158]. The studies about vector Laguerre-Gaussian solitons in local nonlinear media, general nonlocal nonlinear media and strongly nonlocal nonlinear media demonstrate the strong nonlocality of medium can maintain the stability of vector structure [159]. And for strong nonlocal nonlinear media, vector vortex solitons exhibit stability during the propagation [160]. The propagation of hybrid Poincaré sphere beams in lead glass verifies that the stable propagation of vector beam needs some specific conditions, like the beam width and power ratio of two orthogonal polarization components of vector beam [161].

Moreover, the self-focusing of vector beam in nonlinear medium is also different from that in linear medium. The self-focusing of RVB in Kerr medium shows the longer self-focusing length and Fig.10(c) gives the transverse and longitudinal intensity distributions of RVB and uniformly polarized Gaussian beam with the same power [156]. The results show that the self-focusing length of RVB is several times of that of scalar light. Moreover, the self-focusing length of RVB decreases with increase of power and it has relation to the order of vector beam.

The propagation of dark vector soliton in nonlinear medium may be different from the case of bright vector soliton. The dark vector solitons in nonlocal nematic liquid crystals may stably propagate under the weak nonlocality case but unsteadily propagate under the strong nonlocality case [162]. It is just opposite to the case of bright solitons. This is because bright solitons with the intensity concentrated in the central region require strong nonlocal nonlinear to stabilize the beam and suppress collapse. In contrast, dark solitons characterized by an intensity dip surrounded by high-intensity regions are less sensitive to collapse. Weak nonlocality is sufficient to balance the nonlinear effect and diffraction effect and maintain the stability of solitons.

5 Applications of vector beams in micro-nano optics

Vector beams have attracted extensive attention due to their spatially varying polarization characteristics, and their applications have entered various fields including optics [163], electrics [164], magnetics [165], material science [166] and biomedicine [28]. Here, the applications of vector beams in micro-nano optics are surveyed, including micro-nano laser processing, optical micro-manipulation, optical storage, and optical imaging.

5.1 Micro-nano laser processing

Structured intensity distributions induced by vector beams are benefit to laser processing at the surface or inside the structure with high precision nano-fabrication. Femto-second vector beam is usually utilized to realize nano-fabrication, and the minimum width of fabricated metal nanoantennas can be up to the order of 100 nm [167]. Since the focus spot of vector beam depends on the deflection angle of vector beam, this means the line-width of nanostructure may be further optimized. Fig.11(a) shows the optimized results, and the line-width of 63 nm is achieved under the LPVB illumination with 24° deflection angle [168].

The polarization distribution of vector beam also influences the material surface patterning [169]. Fig.11(b) shows the oriented growth of spiral nanostructures guided by the transverse polarization distribution of vector beams [170]. In addition, hybrid periodic nanostructures with parallel and vertically arranged on the 4H-SiC surface form under the guidance of the AVB pulses, with structural periods taking 1306 nm at high frequencies and 500 nm at low frequencies [171]. Compared to the holes ablated by AVB, the nanopores ablated by the strong longitudinal field using focused RVB have smaller diameters, and Fig.11(c) demonstrates the structure of nanopores ablated by RVB on a sapphire substrate by single-pulse ablation [23].

Vector beams can achieve high-precision micro-groove processing on brittle and hard materials, which are often applied in microfluidic and semiconductor manufacturing. The RVB and AVB are used to ablate the micro-grooves with the help of laser-induced microjetassisted ablation. The shape of groove changes with the polarization of vector beam, and the focusing RVB is used to produce W-shaped grooves and AVB is used to produce U-shaped grooves [172]. The non-uniform polarization of vector beam makes surface scattering more uniform through preventing an increase of surface roughness. The AVB is utilized to ablate gentle and non-sharp grooves on brittle materials such as silicon carbide, and it effectively reduces micro-cracks and avoids large groove taper angles [173]. Meanwhile, the CVB is also used to ablate rusted steel and titanium surfaces, and the surface roughness of ripples is relatively lower [174]. The CVB in femtosecond laser ablation produces deeper one- and two-dimensional patterns with lower roughness.

In addition to surface laser processing, vector beams can also be used for the fabrication of nano-structures within transparent materials. High-order vector Bessel beams is used to achieve three dimensional structure in transparent media with the help of femtosecond laser-induced nonlinear effect [175], and it is also suitable for the fabrication of complex volume devices. Certainly, with the increase of processing accuracy and the realization of nonlinear processing, vector beams will provide one powerful tool for future micro-nano fabrication and precision manufacturing.

5.2 Optical micro-manipulation

One micro-nano particle illuminated by one focusing beam experiences the scattering force and absorption force as well as gradient forces. The gradient force originates from the interaction between the induced electric dipole in the particle and the electromagnetic field, and it causes the particle to move toward the center of focal spot. The scattering force and absorption force arise from momentum exchange between the particle and photons, and they drive the particle outward from the focal spot. When the gradient force exceeds the others, the particle is trapped at the center of focal spot. This is optical tweezing or optical trapping. Vector beams offer high-precision manipulation of particles.

Quantitative analysis of optical forces exerted on particles can be performed using the Richards−Wolf vector diffraction theory. The RVB may create deep optical potential well that produces a strong longitudinal gradient force. This enables efficient trapping of particles, like metallic Rayleigh particles [19]. Fig.11(d) illustrates the glass sphere displaced from the focal spot of vector beam is drawn back to equilibrium under the action of optical forces [163]. In contrast, the AVB exerts optical torque on trapped particles, which confines them to an annular region and causes them to rotate along the optical axis [176]. Thus, three-dimensional manipulation of particles can be achieved through simultaneously utilizing RVB and AVB. The optical cage created by dual mode CVBs enables the trapping of low-refractive-index particles such as cold atoms [177].

For the birefringent particle, the interaction between the vector beam and the particle may cause a non-uniform spin current. Fig.11(e) shows that one large particle is stably trapped by the focusing RVB and one small particle is induced to rotate in clockwise direction [178]. Conversely, the particle cannot be trapped by the focusing AVB, but rotates instead. For the intermediate scale Rayleigh particle, the transverse trapping caused by focused vector beam depends on the particle radius and the incident wavelength [19]. For metallic particles with the size reaching the micrometer scale, the gravitational force becomes comparable to the optical force, and the manipulation of particles is different from the above cases. With the help of gravity, three-dimensional manipulation of micrometer-sized particle can be realized and Fig.11(f) shows dynamic manipulation of micrometer-sized gold particle under the effect of the optical force with opposite direction to gravity [179].

Furthermore, the flat-top vector beam is proposed to trap and control micrometer-sized particles, and the power obviously influences the motion trajectories and the probability density distribution of particles [180]. Hybrid vector beam carrying spiral phase provides multiple degrees of freedom for particle manipulation, and the studies show the trapping position of particle relates to the topological charge of incident light. Besides, one hybrid vector beam can realize the trapping of multiple Rayleigh particles [181]. Hybrid vector beam carrying fractional spiral phase can trap the chiral nano-particles, and the adjustment of the topological charge also enables simultaneous trapping and manipulation of multiple particles [25].

As for the issues of efficiency and stability of optical trapping, the theoretic and experiment works are performed. The particle size, trapping depth and numerical aperture influence the efficiency of optical trapping, which may be analyzed using generalized Lorenz-Mie theory [182]. Traditional RVBs induce scattering forces on the trapped particles, and they reduce the stability of optical trapping. The utilization of double axicon in experiment may reduce scattering force so as to enhance the trapping stiffness [183].

5.3 Optical information processing

The rapid growth of data volume has driven an increasing demand for information processing with high capacity and strong security using the integrated devices. Optical information processing with its advantages of high density, low energy consumption and long-term preservation is a key technology for future information transmission. Vector beams with complex polarization carry high-dimensional information and they can significantly enhance the capacity and performance of optical systems. Through modulating the polarization distribution, phase distribution and topological structure of vector beams, multidimensional information storage and retrieval can be achieved within the same medium [184].

The inseparability of the spatial and polarization degrees of freedom of vector beams enabling the 2-bit information encoding [185]. The non-orthogonal states of polarization of vector beams are used to encode and decode information. The optimized convolution neural network can distinguish smaller polarization angular differences, and it improves the information encoding of vector beams up to 8-bit, which enhances the encoding capacity and propagation accuracy in optical systems [186].

Vector beam may realize the high-density multidimensional information storage through encoding state of polarization, phase and amplitude, and it allows more information to be stored in one single medium. Image information can be integrated into different states of polarization through designing the structure of metasurface. Fig.12(a) verifies the availability of multidimensional optical storage based on vectorial metasurface consisting of GAN nanopillars with 300 nm period, where the image information is stored in the intensity, ellipticity, azimuth angle through modulating amplitude and relative phases of its right and left circularly polarized components [187].

In addition, the parallel information processing ability can be further increased by combining wavelength multiplexing and angular multiplexing. The utilization of segmented CVBs combined with wavelength multiplexing is proposed. Each segmant of CVB independently encodes different information, and 24-channel information may be stored in a 30 μm² area through 8 states of polarization and 3 wavelengths [188]. Fig.12(b) shows the reading results of image information with different polarization states based on the designed metasurface, which are achieved using the polarization and angular multiplexing [189].

In addition to direct information storage, vector beams are also employed in encrypted storage in order to increase data security. The stored information may be encoded in vector beams by modulating spiral phase and phase difference of left- and right-handed circularly polarized components. Fig.12(c) illustrates the encryption process and decoding process using four different vector beams [190], where the metasurface is formed by combining phase information from stored images with certain vector polarization. Only the defined vector beam can obtain the correct image, while incorrect vector beams result in a blurred and undifferentiable image.

In addition to the utilization of states of polarization for selectively reading information, the orbital angular momentums carried by the polarization components of vector beam are also used to achieve information encryption and protection. Fig.12(d) demonstrates the information anti-counterfeiting using states of polarization and orbital angular momentum modes with the help of the designed multichannel metasurface [191], where the anti-counterfeiting image is stored in the non-uniformly distributed polarization, and the other two image information stored in the OAM beams. This provides more possibilities of high-density data anti-counterfeiting.

5.4 Microscopic imaging

In order to overcome the diffraction limit and improve spatial resolution in optical microscopy, several advanced imaging methods including subtraction imaging, nonlinear imaging, super-oscillation imaging and stimulated emission depletion microscopy have been developed. Vector beams can offer high resolution and large contrast for optical imaging [193], and the combination of vector beams and microscopy will bring remarkable progress for microscopic imaging.

RVB exhibits the tight focusing effect and AVB provides dark focusing spot with smaller size. The utilization of two beams may realize the super-resolution imaging. Confocal images of RVB and negative confocal images of AVB are proposed to enhance imaging quality [194]. Compared to conventional confocal microscopy and fluorescence emission difference using scalar beams, this method achieves higher resolution and low background noise. Furthermore, the deconvolution of fluorescence emission difference images with the CVB illumination further enhances the resolution.

Fig.12(e) illustrates the super-oscillatory focusing of the RVB [192]. Where the side lobe is excellently suppressed, and the transverse resolution approaches 100nm by adjusting the beam parameters, radial polarization order and confocal pinhole size. For further suppressing the intensity of side lobes, a π-phase-ring mask is introduced into the confocal microscopy system. The RVB passing through this confocal system can distinguish two close fluorescent molecules [195].

Vector beams have also been applied to stimulated emission depletion microscopy, and the super-resolution imaging of arbitrarily oriented molecules is realized with the EPVB illumination in stimulated emission depletion microscopy. Fig.12(f) shows the resolution analysis of microscope imaging with the EPVB illumination, two orientation independent fluorescent single molecules in the longitudinal and transverse planes are distinguished by the deflection angle and the intensity of the petal threshold, and the transverse resolution gets improved under the optimal threshold condition [29]. In addition, the addition of a pair of focused CVBs enhances the axial resolution [192, 194].

6 Summary and outlook

Due to the unique polarization distributions, vector beams exhibit many amazing properties during the propagation and they have attracted much attention in various fields. Recently, the attention to vector beams no longer concentrates only on the CVBs with symmetric polarization distributions and their applications, but the studies about vector beams have also been extended to the ones with asymmetric polarization distributions and even carrying the spiral phase. Thus, this review tries to systemically summarize the fundamentals and applications of diverse vector beams.

With comparison to the existing reviews about vector beams, this review describes the mathematical representations of vector beams in different polarization bases, and the intuitive polarization patterns of various vector beams are provided. This offers the foundation for the advancement of generation methods of various vector beams. Especially, this review provides the theoretical description about the propagation of vector beams in different situations and the propagation rules of vector beams passing through different media uncover the relation of light field, vector beam and the media.

There are undoubtedly many important works that have not been systematically covered in addition to those included in this review. Partially coherent vector beams show more robust against complex environmental disturbances and offer new directions for the applications in complex environments like optical encryption [196]. Furthermore, the other works are about high-power vector beams, which solve the issue about insufficient power in vector beam laser processing and can realize significant applications in micro-nano fabrication and high-precision material cutting [197]. The quantum entangled vector lights can cause super high-resolution and fast electromagnetic field dynamics [198], and they will bring new applications in the construction of quantum information networks [199] and ghost imaging [200].

Besides the applications of vector beams summarized in this review, more applications may emerge in the future in pace with the deeper studies. The vector beams have high sensitivity on media and they can be utilized to develop the molecule type and concentration detection of chiral media [201]. This technique can further be expanded to monitoring chiral compound synthesis, quantifying organic compound concentrations in the atmosphere and biomedical detection. The absorption of vector beams in atomic media can be used for vector magnetic field measurements because of the sensitive magnetic response [202]. This monitoring technique holds great potential in geomagnetic field surveys and the detection of weak magnetic signals from biological tissues. The association with parity enables the vector beams in parity-demultiplexed CVB-encoded holography [203].

Furthermore, the potential applications may emerge along with the appearance of new-type vector beams. The vector skyrmions can create with higher degrees of freedom [204], and the manipulation of vector skyrmions will realize efficient and durable information storage and enhanced electronic detection. The metasurface technique provides effective tool for integrated optical devices [205] and these integrated optical systems with smaller size, lower power consumption and higher efficiency make vector beams widely applicable in portable devices. The vector beams may be integrated into many frontier fields such as nanobots, magnetic field surveying, micro-sensors, and augmented and virtual reality.

In conclusion, this paper provides a comprehensive review of the research about vector beams, from the fundamental concepts, generation methods, transmission characteristics to typical applications in micro-nano optics. It partly solves the problems about how to describe vector beams, how to generate vector beams, how to analyze the propagation of vector beams in different conditions and how to apply vector beams in different fields. We hope it will provide the beginners some enlightenment and the researchers some help to enter the studies about vector beams. We believe that vector beam will play an indispensable role in the following development of science and technology, and the applications of vector beams will emerge in wider fields.

References

[1] X. Zhang , J. Chen , and C. Sun , Depth-resolved stress field measurement of multilayered system using swept-source polarization-sensitive optical coherence tomography, Opt. Lasers Eng. 178, 108215 (2024)

[2] S. Wen , Y. Zheng , and F. Lu , Polarization guided specular reflection separation, IEEE Trans. Image Process. 30, 7280 (2021)

[3] X. Zhang , G. Gao , and S. W. Chen , Polarimetric autocorrelation matrix: A new tool for joint characterizing of target polarization and doppler scattering mechanism, IEEE Trans. Geosci. Remote Sens. 62, 1 (2024)

[4] H. Guan , J. Huang , L. Li , X. Li , S. Miao , W. Su , Y. Ma , Q. Niu , and H. Huang , Improved Gaussian mixture model to map the flooded crops of VV and VH polarization data, Remote Sens. Environ. 295, 113714 (2023)

[5] J. R. Eskilt and E. Komatsu , Improved constraints on cosmic birefringence from the WMAP and Planck cosmic microwave background polarization data, Phys. Rev. D 106(6), 063503 (2022)

[6] K. Akiyama , A. Alberdi , W. Alef , . First sagittarius A* event horizon telescope results. VII. Polarization of the ring, Astrophys. J. 964(2), L25 (2024)

[7] Y. Pan , Z. C. Ren , S. X. Qian , X. Z. Gao , Y. Li , C. Tu , and H. T. Wang , Uniformly elliptically-polarized vector optical fields, J. Opt. 17, 035616 (2015)

[8] X. L. Wang,Y. Li,J. Chen, ., A new type of vector fields with hybrid states of polarization, Opt. Express 18(10), 10786 (2010)

[9] X. L. Wang , J. Ding , W. J. Ni , C. S. Guo , and H. T. Wang , Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement, Opt. Lett. 32(24), 3549 (2007)

[10] M. Giovanni , H. I. Sztul , D. A. Nolan , . Higher-order Poincaré sphere, stokes parameters, and the angular momentum of light, Phys. Rev. Lett. 107(5), 053601 (2011)

[11] X. Yi , Y. Liu , X. Ling , X. Zhou , Y. Ke , H. Luo , S. Wen , and D. Fan , Hybrid-order Poincaré sphere, Phys. Rev. A 91(2), 023801 (2015)

[12] R. Dorn , S. Quabis , and G. Leuchs , Sharper focus for a radially polarized light beam, Phys. Rev. Lett. 91(23), 233901 (2003)

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

[14] M. Kraus , M. A. Ahmed , A. Michalowski , A. Voss , R. Weber , and T. Graf , Microdrilling in steel using ultrashort pulsed laser beams with radial and azimuthal polarization, Opt. Express 18(21), 22305 (2010)

[15] G. Bautista , M. J. Huttunen , J. Makitalo , J. M. Kontio , J. Simonen , and M. Kauranen , Second-harmonic generation imaging of metal nano-objects with cylindrical vector beams, Nano Lett. 12(6), 3207 (2012)

[16] G. Bautista , J. P. Kakko , V. Dhaka , X. Zang , L. Karvonen , H. Jiang , E. Kauppinen , H. Lipsanen , and M. Kauranen , Nonlinear microscopy using cylindrical vector beams: Applications to three-dimensional imaging of nanostructures, Opt. Express 25(11), 12463 (2017)

[17] Z. Zhu , M. Janasik , A. Fyffe , D. Hay , Y. Zhou , B. Kantor , T. Winder , R. W. Boyd , G. Leuchs , and Z. Shi , Compensation-free high-dimensional free-space optical communication using turbulence-resilient vector beams, Nat. Commun. 12(1), 1666 (2021)

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

[19] J. Q. Qin,X. L. Wang,D. Jia,J. Chen,Y. X. Fan,J. Ding,H. T. Wang, FDTD approach to optical forces of tightly focused vector beams on metal particles, Opt. Express 17(10), 8407 (2009)

[20] M. G. Donato , S. Vasi , R. Sayed , P. H. Jones , F. Bonaccorso , A. C. Ferrari , P. G. Gucciardi , and O. M. Maragò , Optical trapping of nanotubes with cylindrical vector beams, Opt. Lett. 37(16), 3381 (2012)

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

[22] X. Zhu,P. Zheng,X. Xie, Super-diffraction focusing based on Bessel beam controlled by composited light field, Laser & Optoelectr. Prog. 59(21), 2105001 (2022)

[23] Z. Li , O. Allegre , and L. Li , Realising high aspect ratio 10 nm feature size in laser materials processing in air at 800 nm wavelength in the far-field by creating a high purity longitudinal light field at focus, Light Sci. Appl. 11(1), 339 (2022)

[24] Y. Liu , Q. Chen , H. Zhang , Z. Feng , G. Zou , and D. Zhang , Cascaded momentum-space polarization filters enabled label-free black-field microscopy for single nanoparticles analysis, Proc. Natl. Acad. Sci. USA 121(13), e2321825121 (2024)

[25] X. Xiong , Y. Zhou , Y. Lu , and Z. Shen , Enantioselective optical trapping of chiral nanoparticles by tightly focused fractional vector beams, Opt. Express 32(20), 34639 (2024)

[26] X. Zhang and X. Ning , Three-dimensional orientation of ellipsoidal particles with arbitrarily polarized vector beams, Acta Opt. Sin. 45(5), 0526001 (2025)

[27] H. He , M. Peng , G. Cao , Y. Li , H. Liu , and H. Yang , Higher-order Poincaré sphere multiplexed metasurface holography for optical information encryption, Opt. Laser Technol. 180, 111555 (2025)

[28] M. Yoshida , Y. Kozawa , and S. Sato , Subtraction imaging by the combination of higher-order vector beams for enhanced spatial resolution, Opt. Lett. 44(4), 883 (2019)

[29] S. Boichenko , Toward super-resolution fluorescent microscopy of arbitrarily oriented single molecules, Phys. Rev. A 101(4), 043823 (2020)

[30] V. V. Belloni , M. Hassan , L. Furfaro , R. Giust , A. M. Seydoux-Guillaume , S. Sao-Joao , and F. Courvoisier , Single shot generation of high-aspect-ratio nano-rods from sapphire by ultrafast first order Bessel beam, Laser Photonics Rev. 18(3), 2300687 (2024)

[31] A. Kritzinger , A. Forbes , and P. B. C. Forbes , Optical trapping and fluorescence control with vectorial structured light, Sci. Rep. 12(1), 17690 (2022)

[32] C. Xie,L. Huang,T. Zuo,C. Wang,J. Ren,L. Nong,J. Liu,J. Xiao,S. Chen, LDPC-coded cylindrical vector beam multiplexing for improved communication performance, Opt. Laser Technol. 170, 110267 (2024)

[33] Z. Yue , J. Xu , P. Lu , and S. Teng , Versatile integrated polarizers based on geometric metasurfaces, Nanomaterials (Basel) 12(16), 2816 (2022)

[34] F. Zhang , M. Pu , Y. Guo , X. L. Ma , X. Li , P. Gao , and X. G. Luo , Synthetic vector optical fields with spatial and temporal tunability, Sci. China Phys. Mech. Astron. 65(5), 254211 (2022)

[35] H. Iwase and S. Ohno , Direct generation of a terahertz vector beam from a ZnTe crystal excited by a focused circular polarized pulse, Opt. Express 31(16), 26923 (2023)

[36] M. Gu , C. Cheng , Z. Zhan , Z. Zhang , G. Cui , Y. Zhou , X. Zeng , S. Gao , D. Y. Choi , C. Cheng , and C. Liu , Dielectric supercell metasurfaces for generating focused higher-order Poincaré beams with the residual copolarization component eliminated, ACS Photonics 11(1), 204 (2024)

[37] Q. Zhan , Cylindrical vector beams: From mathematical concepts to applications, Adv. Opt. Photonics 1(1), 1 (2009)

[38] J. Chen , C. Wan , and Q. Zhan , Vectorial optical fields: Recent advances and future prospects, Sci. Bull. (Beijing) 63(1), 54 (2018)

[39] C. Rosales-Guzmán,B. Ndagano,A. Forbes, A review of complex vector light fields and their applications, J. Opt. 20(12), 123001 (2018)

[40] H. Lv , X. Lu , Y. Han , Z. Mou , C. Zhou , S. Wang , and S. Teng , Metasurface cylindrical vector light generators based on nanometer holes, New J. Phys. 21(12), 123047 (2019)

[41] D. Liu , C. Zhou , P. Lu , J. Xu , Z. Yue , and S. Teng , Generation of vector beams with different polarization singularities based on metasurfaces, New J. Phys. 24(4), 043022 (2022)

[42] Z. Yue , P. Lu , J. Xu , Z. Li , and S. Teng , Vector beams encoded by diverse orthogonal polarization states and their generation based on metasurfaces, New J. Phys. 25(1), 013018 (2023)

[43] R. C. Jones, A new calculus for the treatment of optical systems. I. Description and discussion of the calculus, J. Opt. Soc. Am. 31(7), 488 (1941)

[44] S. Teng , Q. Zhang , H. Wang , L. Liu , and H. Lv , Conversion between polarization states based on a metasurface, Photon. Res. 7(3), 246 (2019)

[45] G. G. Stokes, Mathematical and Physical Papers, Cambridge: Cambridge University Press, 1901

[46] H. Poincaré, Théorie mathématique de la lumière II.: Nouvelles études sur la diffraction. --Théorie de la dispersion de Helmholtz. Leçons professées pendant le premier semestre, G. Carré, 1889

[47] J. Li , K. i. Ueda , M. Musha , A. Shirakawa , and L. Zhong , Generation of radially polarized mode in Yb fiber laser by using a dual conical prism, Opt. Lett. 31(20), 2969 (2006)

[48] S. Chen , J. Li , and K. I. Ueda , Cylindrical vector beam rotary Nd: YAG disk laser with birefringent crystal, Chin. Opt. Lett. 18(12), 121401 (2020)

[49] L. Cao , M. Zhang , J. Dou , J. Zhao , Y. Hu , and B. Li , Controlled generation of order-switchable cylindrical vector beams from a Nd: YAG laser, Chin. Opt. Lett. 21(10), 101401 (2023)

[50] Y. Zhou , J. Lin , X. Zhang , L. Xu , C. Gu , B. Sun , A. Wang , and Q. Zhan , Self-starting passively mode-locked all fiber laser based on carbon nanotubes with radially polarized emission, Photon. Res. 4(6), 327 (2016)

[51] S. Ramachandran , P. Kristensen , and M. F. Yan , Generation and propagation of radially polarized beams in optical fibers, Opt. Lett. 34(16), 2525 (2009)

[52] B. Sun , A. Wang , L. Xu , C. Gu , Z. Lin , H. Ming , and Q. Zhan , Low-threshold single-wavelength all-fiber laser generating cylindrical vector beams using a few-mode fiber Bragg grating, Opt. Lett. 37(4), 464 (2012)

[53] S. Liu , H. Liu , X. Qi , W. Peng , Y. Feng , L. Chen , Z. Li , Y. Sun , Y. Ma , Z. Zhao , Q. Gao , Z. Liu , and C. Tang , Coherent beam combining of cylindrical vector beams for power scaling, Opt. Lett. 48(19), 5121 (2023)

[54] X. Liu , J. Luo , J. Lv , Y. Chen , P. Yao , C. Gu , and L. Xu , Generation of multiwavelength cylindrical vector beams from a random fiber laser assisted with nonlinear amplifying loop mirror, Infrared Phys. Technol. 140, 105416 (2024)

[55] C. Lv , X. Ding , B. Yin , Y. Lv , and S. Wu , Switchable dual-wavelength mode-locked cylindrical vector beam fiber laser based on unpumped EDF saturable absorber and nonlinear polarization rotation, Infrared Phys. Technol. 138, 105271 (2024)

[56] J. Zhang , J. Wang , J. Yao , X. Jiang , L. Gui , S. Zhang , Q. Zhan , and A. Wang , Tunable spectrally-flat multi-wavelength cylindrical vector beam fiber laser by using a re-circulating frequency shifter loop with two external injections, Opt. Laser Technol. 170, 110310 (2024)

[57] M. Xie , J. Xu , J. Wang , H. Zhao , Y. Liu , J. Wen , F. Pang , J. Sun , and X. Zeng , Generating broadband cylindrical vector modes based on polarization-dependent acoustically induced fiber gratings using the dispersion turning point, Photon. Res. 12(9), 1907 (2024)

[58] W. Zhang , X. Li , L. Zhang , L. Huang , F. Lu , and F. Gao , Generation of cylindrical vector beams and optical vortex in a solid-core ring fiber based on an acoustically induced fiber grating, IEEE J. Sel. Top. Quantum Electron. 26(4), 1 (2020)

[59] D. Lin , J. Carpenter , Y. Feng , S. Jain , Y. Jung , Y. Feng , M. N. Zervas , and D. J. Richardson , Reconfigurable structured light generation in a multicore fibre amplifier, Nat. Commun. 11(1), 3986 (2020)

[60] H. Chen , J. Yin , M. Zhang , Y. Yu , W. Wan , F. He , J. Yang , and P. Yan , High-power, femtosecond vortex beams generation in the visible and near-infrared region, J. Lumin. 248, 118949 (2022)

[61] J. A. Davis , I. Moreno , M. del Mar Sanchez-Lopez , and D. M. Cottrell , Vector beams generated with a Fourier transform processor with geometric phase grating, Opt. Eng. 62(1), 013104 (2023)

[62] Z. Bomzon , V. Kleiner , and E. Hasman , Formation of radially and azimuthally polarized light using space-variant subwavelength metal stripe gratings, Appl. Phys. Lett. 79(11), 1587 (2001)

[63] Q. Yang , Y. Yang , H. Li , H. Liu , and X. Chen , Nonlinear generation of vector beams by using a compact nonlinear fork grating, Photon. Res. 12(5), 1036 (2024)

[64] Z. Xie , J. He , X. Wang , S. Feng , and Y. Zhang , Generation of terahertz vector beams with a concentric ring metal grating and photo-generated carriers, Opt. Lett. 40(3), 359 (2015)

[65] Z. E. Bomzon , G. Biener , V. Kleiner , and E. Hasman , Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings, Opt. Lett. 27(5), 285 (2002)

[66] T. Ye , J. Wang , J. Liu , P. Qi , L. Huang , X. Yuan , Y. Zhang , Y. Yang , X. Lin , Z. Huang , and X. Tan , Generation of vector beams based on diffraction characteristics of a linear polarization hologram in coaxial recording, Opt. Express 31(5), 7764 (2023)

[67] U. Levy , C. H. Tsai , L. Pang , and Y. Fainman , Engineering space-variant inhomogeneous media for polarization control, Opt. Lett. 29(15), 1718 (2004)

[68] M. Abdou Ahmed , F. Beirow , A. Loescher , T. Dietrich , D. Bashir , D. Didychenko , A. Savchenko , C. Pruss , M. Fetisova , F. Li , P. Karvinen , M. Kuittinen , and T. Graf , High-power thin-disk lasers emitting beams with axially-symmetric polarizations, Nanophotonics 11(4), 835 (2022)

[69] G. M. Lerman and U. Levy , Generation of a radially polarized light beam using space-variant subwavelength gratings at 1064 nm, Opt. Lett. 33(23), 2782 (2008)

[70] J. Yao , S. Wu , X. Li , J. Liu , Q. Zhan , and A. Wang , Generation of arbitrary vector vortex beam using a single Q-plate, Laser Photonics Rev. 19(12), 2402290 (2025)

[71] S. Zheng , L. Zha , J. Lu , X. Zeng , Y. Li , S. Xu , and D. Fan , Liquid crystal-based order electrically controlled q-plate system, Opt. Express 27(11), 16103 (2019)

[72] J. Piłka , M. Kwasny , M. Czerniewicz , M. Karpierz , and U. Laudyn , Multi-wavelength Q-plate arithmetic in an all-liquid-crystal modular setup, Opt. Laser Technol. 169, 110019 (2024)

[73] M. M. Sánchez-López , I. Abella , D. Puerto-García , J. A. Davis , and I. Moreno , Spectral performance of a zero-order liquid-crystal polymer commercial q-plate for the generation of vector beams at different wavelengths, Opt. Laser Technol. 106, 168 (2018)

[74] Z. Liu , Y. Liu , Y. Ke , Y. Liu , W. Shu , H. Luo , and S. Wen , Generation of arbitrary vector vortex beams on hybrid-order Poincare sphere, Photon. Res. 5(1), 15 (2017)

[75] W. Shu , X. Ling , X. Fu , Y. Liu , Y. Ke , and H. Luo , Polarization evolution of vector beams generated by q-plates, Photon. Res. 5(2), 64 (2017)

[76] S. Lou , Y. Zhou , Y. Yuan , T. Lin , F. Fan , X. Wang , H. Huang , and S. Wen , Generation of arbitrary vector vortex beams on hybrid-order Poincaré sphere based on liquid crystal device, Opt. Express 27(6), 8596 (2019)

[77] C. Chen , Y. Zhang , L. Ma , Y. Zhang , Z. Li , R. Zhang , X. Zeng , Z. Zhan , C. He , X. Ren , C. Cheng , and C. Liu , Flexible generation of higher-order Poincaré beams with high efficiency by manipulating the two eigenstates of polarized optical vortices, Opt. Express 28(7), 10618 (2020)

[78] J. Wang , M. Pu , J. Jin , F. Zhang , L. Liu , W. Kong , X. Li , Y. Guo , and X. Luo , Generation of a space-variant vector beam with catenary-shaped polarization states, Materials (Basel) 15(8), 2940 (2022)

[79] C. Acevedo , A. Guzmán , Y. Torres Moreno , and A. Dogariu , Sagnac interferometer based generation of controllable cylindrical vector beams, Int. J. Opt. 2016, 1 (2016)

[80] J. Mendoza-Hernández , M. F. Ferrer-Garcia , J. Arturo Rojas-Santana , and D. Lopez-Mago , Cylindrical vector beam generator using a two-element interferometer, Opt. Express 27(22), 31810 (2019)

[81] S. Liu , P. Li , T. Peng , and J. Zhao , Generation of arbitrary spatially variant polarization beams with a trapezoid Sagnac interferometer, Opt. Express 20(19), 21715 (2012)

[82] T. Wang,S. Fu,S. Zhang,C. Gao,F. He, A Sagnac-like interferometer for the generation of vector beams, Appl. Phys. B 122(9), 231 (2016)

[83] H. Chen , J. Hao , B. F. Zhang , J. Xu , J. Ding , and H. T. Wang , Generation of vector beam with space-variant distribution of both polarization and phase, Opt. Lett. 36(16), 3179 (2011)

[84] L. Guo , Z. Feng , Y. Fu , and C. Min , Generation of vector beams array with a single spatial light modulator, Opt. Commun. 490, 126915 (2021)

[85] P. García-Martínez , D. Marco , J. L. Martínez-Fuentes , M. M. Sánchez-López , and I. Moreno , Efficient on-axis SLM engineering of optical vector modes, Opt. Lasers Eng. 125, 105859 (2020)

[86] W. Zhu , F. Gao , Q. Fu , X. Zhou , Y. Xie , B. Zhang , and S. Kumar , Multi-mode vector light field generation using modified off-axis interferometric holography and liquid crystal spatial light modulators, Photonics 11(1), 33 (2024)

[87] P. Kumar , A. S. Rao , and T. Omatsu , Generation of V-point polarization singularity using single phase encoding with a spatial light modulator, Sci. Rep. 13(1), 315 (2023)

[88] S. Liu , S. Qi , Y. Zhang , P. Li , D. Wu , L. Han , and J. Zhao , Highly efficient generation of arbitrary vector beams with tunable polarization, phase, and amplitude, Photon. Res. 6(4), 228 (2018)

[89] Y. Gao , Z. Chen , J. Ding , and H. T. Wang , Single ultra-high-definition spatial light modulator enabling highly efficient generation of fully structured vector beams, Appl. Opt. 58(24), 6591 (2019)

[90] B. Qian , T. Zeng , Z. Chen , and J. Ding , Generation of vector beams using a Wollaston prism and a spatial light modulator, Optik (Stuttg. ) 148, 312 (2017)

[91] Y. Liu , C. Li , B. Fang , L. Chen , J. Xu , F. Liu , L. Xu , X. Li , Y. Tang , Z. Hong , and X. Jing , Research progress in tunable metamaterial absorbers, Adv. Photon. Res. 5(2), 2300258 (2024)

[92] C. D. Zhou , Z. Mou , R. Bao , Z. Li , and S. Y. Teng , Compound plasmonic vortex generation based on spiral nanoslits, Front. Phys. (Beijing) 16(3), 33503 (2021)

[93] Z. Mou , C. Zhou , P. Lu , Q. Yue , S. Wang , and S. Teng , Structured vortices generated by metasurface holography, Photon. Res. 9(10), 2125 (2021)

[94] Q. Zhang , P. Li , Y. Li , H. Wang , L. Liu , Y. He , and S. Teng , Vector beam generation based on the nanometer-scale rectangular holes, Opt. Express 25(26), 33480 (2017)

[95] H. Wang , L. Liu , X. Lu , H. Lü , Y. Han , S. Wang , and S. Teng , Spatial multiplexing plasmonic metalenses based on nanometer cross holes, New J. Phys. 20(12), 123009 (2018)

[96] Y. Hu , X. Wang , X. Luo , X. Ou , L. Li , Y. Chen , Ping Yang , S. Wang , and H. Duan , All-dielectric metasurfaces for polarization manipulation: Principles and emerging applications, Nanophotonics 9(12), 3755 (2020)

[97] H. He , H. Yang , Z. Xie , and X. Yuan , Dynamic polarization rotation and vector field steering based on phase change metasurface, Front. Phys. (Beijing) 18(1), 12303 (2023)

[98] C. Li , Z. Zheng , L. Ke , B. Fang , Z. Hong , M. Yu , and X. Jing , Terahertz vector beams generated by rectangular multilayer transmission metasurface, Opt. Laser Technol. 169, 110143 (2024)

[99] J. Wen , H. Feng , J. Chen , K. Wang , Y. Lv , Y. Zhong , and D. Zhang , Plasmonic holographic metasurfaces for generation of vector optical beams, IEEE Photonics J. 9(1), 1 (2017)

[100] F. Yue , D. Wen , J. Xin , B. D. Gerardot , J. Li , and X. Chen , Vector vortex beam generation with a single plasmonic metasurface, ACS Photonics 3(9), 1558 (2016)

[101] Y. He , Y. Li , J. Liu , X. Zhang , Y. Cai , Y. Chen , S. Chen , and D. Fan , Switchable phase and polarization singular beams generation using dielectric metasurfaces, Sci. Rep. 7(1), 6814 (2017)

[102] Y. Zhang , X. Yang , and J. Gao , Generation of polarization singularities with geometric metasurfaces, Sci. Rep. 9(1), 19656 (2019)

[103] F. Yue , D. Wen , C. Zhang , B. D. Gerardot , W. Wang , S. Zhang , and X. Chen , Multichannel polarization-controllable superpositions of orbital angular momentum states, Adv. Mater. 29(15), 1603838 (2017)

[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] 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 Poincaré sphere beams, Adv. Mater. 32(6), 1903983 (2020)

[106] 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)

[107] L. Shao , D. Zhang , J. Liu , J. Zhang , Z. Li , X. Wang , and W. Zhu , Single-layer noninterleaved metasurface for arbitrary vector beam conversion in triple bands, ACS Appl. Electron. Mater. 4(1), 443 (2022)

[108] T. Zhou , Q. Liu , Y. Liu , and X. Zang , Spin-independent metalens for helicity-multiplexing of converged vortices and cylindrical vector beams, Opt. Lett. 45(21), 5941 (2020)

[109] L. Zhang , X. Qiu , F. Li , H. Liu , X. Chen , and L. Chen , Second harmonic generation with full Poincaré beams, Opt. Express 26(9), 11678 (2018)

[110] Y. C. Lou , Z. M. Cheng , Z. H. Liu , Y. X. Yang , Z. C. Ren , J. Ding , X. L. Wang , and H. T. Wang , Third-harmonic generation of spatially structured light in a quasi-periodically poled crystal, Optica 9(2), 183 (2022)

[111] B. Gu , B. Wen , G. Rui , Y. Xue , Q. Zhan , and Y. Cui , Varying polarization and spin angular momentum flux of radially polarized beams by anisotropic Kerr media, Opt. Lett. 41(7), 1566 (2016)

[112] X. Zhu , Y. Miao , Q. Shi , H. He , and J. Dong , Cylindrical vector beams with topological charges up to 6 generated in a Raman microchip laser, J. Lightwave Technol. 42(15), 5286 (2024)

[113] L. Zhang , F. Lin , X. Qiu , and L. Chen , Full vectorial feature of second-harmonic generation with full Poincaré beams, Chin. Opt. Lett. 17(9), 091901 (2019)

[114] H. Li , H. Liu , Y. Yang , R. Lu , and X. Chen , Nonlinear generation of perfect vector beams in ultraviolet wavebands, Chin. Phys. Lett. 39(3), 034201 (2022)

[115] A. García-Cabrera , R. Boyero-García , Ó. Zurrón-Cifuentes , J. Serrano , J. S. Román , L. Plaja , and C. Hernández-García , Topological high-harmonic spectroscopy, Commun. Phys. 7(1), 28 (2024)

[116] C. Pan , C. Yang , H. Hu , J. Wang , Y. Zhang , Y. Qin , D. Wei , H. Chen , H. Gao , and F. Li , Trans-spectral vector beam nonlinear conversion via parametric four-wave mixing in alkali vapor, Opt. Lett. 46(22), 5579 (2021)

[117] M. K. Kroychuk , A. S. Shorokhov , D. F. Yagudin , D. A. Shilkin , D. A. Smirnova , I. Volkovskaya , M. R. Shcherbakov , G. Shvets , and A. A. Fedyanin , Enhanced nonlinear light generation in oligomers of silicon nanoparticles under vector beam illumination, Nano Lett. 20(5), 3471 (2020)

[118] D. G. Hall , Vector-beam solutions of Maxwell’s wave equation, Opt. Lett. 21(1), 9 (1996)

[119] X. Jia , Y. Wang , and B. Li , Nonparaxial analyses of radially polarized beams diffracted at a circular aperture, Opt. Express 18(7), 7064 (2010)

[120] X. Jia and J. Jia , Effect of evanescent wave on vector vortex beams on higher-order Poincaré sphere, Laser Phys. 34(10), 105404 (2024)

[121] L. N. Guo , Z. L. Tang , and J. Wang , Non-paraxial propagation of cylindrical vector vortex beams in the far-field, Chin. Phys. Lett. 31(7), 074101 (2014)

[122] X. J. Jia , B. Li , and Y. Wang , Far-field properties of a cylindrically polarized vector beam and its beam quality beyond the paraxial approximation, Chin. Opt. Lett. 8, 517 (2010)

[123] E. Wolf , Electromagnetic diffraction in optical systems-I. An integral representation of the image field, Proc. R. Soc. Lond. 253(1274), 349 (1959)

[124] B. Richards and E. Wolf , Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system, Proc. R. Soc. Lond. A 253(1274), 358 (1959)

[125] P. S. Liu , B. Lü , and K. L. Duan , Propagation of vectorial nonparaxial Gaussian beams through an annular aperture, Opt. Laser Technol. 38(3), 133 (2006)

[126] J. N. Chen , Nonparaxial propagation of a radially polarized beam diffracted by an annular aperture, Chin. Phys. Lett. 28(12), 124202 (2011)

[127] B. Gu , D. Xu , Y. Pan , and Y. Cui , Nonparaxial propagation and focusing properties of azimuthal-variant vector fields diffracted by an annular aperture, J. Opt. Soc. Am. A 31(7), 1657 (2014)

[128] G. Remesh , B. S. Athira , S. Gucchait , . Diffraction of an off-axis vector-beam by a tilted aperture, J. Opt. 24(10), 105602 (2022)

[129] Q. Zhan and R. L. James , Focus shaping using cylindrical vector beams, Opt. Express 10(7), 324 (2002)

[130] Y. Kozawa and S. Sato , Focusing property of a double-ring-shaped radially polarized beam, Opt. Lett. 31(6), 820 (2006)

[131] X. Weng , L. Du , P. Shi , and X. Yuan , Tunable optical cage array generated by Dammann vector beam, Opt. Express 25(8), 9039 (2017)

[132] S. K. Pal and L. A. Rusch , Tailoring focal plane component intensities of polarization singular fields in a tight focusing system, Sci. Rep. 14(1), 13565 (2024)

[133] Y. Kozawa and S. Sato , Sharper focal spot formed by higher-order radially polarized laser beams, J. Opt. Soc. Am. A 24(6), 1793 (2007)

[134] B. Tian and J. Pu , Tight focusing of a double-ring-shaped, azimuthally polarized beam, Opt. Lett. 36(11), 2014 (2011)

[135] B. Pant , B. K. Mishra , S. Singh , . Mode transformation and dark spot formation of cylindrical vector beams by thin dielectric film, Opt. Laser Technol. 180, 111539 (2025)

[136] Q. Zhao , S. Tu , Q. Lei , C. Guo , Q. Zhan , and Y. Cai , Creation of cylindrical vector beams through highly anisotropic scattering media with a single scalar transmission matrix calibration, Photon. Res. 10(7), 1617 (2022)

[137] J. Dou , Y. Mai , W. Jiang , K. Wang , L. Zhong , J. Di , and Y. Qin , Vector modulation of fully-polarized phase conjugate light field through scattering media, Opt. Laser Technol. 181, 111987 (2025)

[138] H. Li , C. M. Woo , T. Zhong , Z. Yu , Y. Luo , Y. Zheng , X. Yang , H. Hui , and P. Lai , Adaptive optical focusing through perturbed scattering media with a dynamic mutation algorithm, Photon. Res. 9(2), 202 (2021)

[139] 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)

[140] E. Wolf, Introduction to the Theory of Coherence and Polarization of Light, Cambridge: Cambridge University Press, 2007

[141] Z. Tong and O. Korotkova , Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media, Phys. Rev. A 82(3), 033836 (2010)

[142] R. Tao , X. Wang , L. Si , P. Zhou , and Z. Liu , Propagation of focused vector laser beams in turbulent atmosphere, Opt. Laser Technol. 54, 62 (2013)

[143] Y. Cai , Q. Lin , H. T. Eyyuboğlu , and Y. Baykal , Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere, Opt. Express 16(11), 7665 (2008)

[144] I. Nape , K. Singh , A. Klug , W. Buono , C. Rosales-Guzman , A. McWilliam , S. Franke-Arnold , A. Kritzinger , P. Forbes , A. Dudley , and A. Forbes , Revealing the invariance of vectorial structured light in complex media, Nat. Photonics 16(7), 538 (2022)

[145] X. Jia and Y. Yang , Propagation of radially polarized beams diffracted at a circular aperture in turbulent atmosphere, Opt. Lasers Eng. 50(12), 1683 (2012)

[146] X. M. Wang and X. X. Hu , Controllable vector soliton in (2+1)-dimensional coupled nonlinear Schrödinger equations with varying coefficients, Nonlinear Dyn. 112(5), 3785 (2024)

[147] F. Bouchard , H. Larocque , A. M. Yao , C. Travis , I. De Leon , A. Rubano , E. Karimi , G. L. Oppo , and R. W. Boyd , Polarization Shaping for Control of Nonlinear Propagation, Phys. Rev. Lett. 117(23), 233903 (2016)

[148] C. J. Gibson , P. Bevington , G. L. Oppo , and A. M. Yao , Control of polarization rotation in nonlinear propagation of fully structured light, Phys. Rev. A 97(3), 033832 (2018)

[149] R. P. Chen , L. X. Zhong , K. H. Chew , T. Y. Zhao , and X. Zhang , Collapse dynamics of a vector vortex optical field with inhomogeneous states of polarization, Laser Phys. 25(7), 075401 (2015)

[150] H. Hu , D. Luo , C. Pan , Y. Qin , Y. Zhang , D. Wei , H. Chen , H. Gao , and F. Li , Collapse of hybrid vector beam in Rb atomic vapor, Opt. Lett. 46(11), 2614 (2021)

[151] S. M. Li , Y. Li , X. L. Wang , L. J. Kong , K. Lou , C. Tu , Y. Tian , and H. T. Wang , Taming the collapse of optical fields, Sci. Rep. 2(1), 1007 (2012)

[152] M. Delqué , G. Fanjoux , and T. Sylvestre , Polarization dynamics of the fundamental vector soliton of isotropic Kerr media, Phys. Rev. E 75(1), 016611 (2007)

[153] P. Balla and G. P. Agrawal , Nonlinear interaction of vector solitons inside birefringent optical fibers, Phys. Rev. A 98(2), 023822 (2018)

[154] J. R. Salgueiro , Vector-vortex solitons in nonlinear photonic crystal fibers, J. Opt. 18(7), 074004 (2016)

[155] C. Dai , Y. Zhang , R. Tao , J. Lin , P. Yao , L. Xu , C. Gu , and Q. Zhan , The nonlinear propagation of cylindrical vector beams in an optical fiber, Opt. Laser Technol. 130, 106336 (2020)

[156] L. Lu , Z. Wang , and Y. Cai , Self-focusing propagation characteristics of a radially-polarized beam in nonlinear media, Opt. Express 30(10), 15905 (2022)

[157] L. Wu , L. Ge , Q. Kong , . Nonlocal two-color vector dark solitons, Results Phys. 48, 106433 (2023)

[158] Z. Xu , Y. V. Kartashov , and L. Torner , Stabilization of vector soliton complexes in nonlocal nonlinear media, Phys. Rev. E 73(5), 055601 (2006)

[159] Q. Wang and J. Z. Li , Vector Laguerre-Gaussian soliton in strong nonlocal nonlinear media, Opt. Commun. 354, 174 (2015)

[160] Z. Xu , N. F. Smyth , A. A. Minzoni , and Y. S. Kivshar , Vector vortex solitons in nematic liquid crystals, Opt. Lett. 34(9), 1414 (2009)

[161] J. J. Li and H. C. Zhang , Propagation dynamics of hybrid-order Poincare beams in thermal nonlocal media, Chaos Solitons Fractals 171, 113444 (2023)

[162] L. Wu , L. Ge , Q. Kong , and M. Shen , Nonlocal two-color vector dark solitons, Results Phys. 48, 106433 (2023)

[163] J. L. Zhu , R. C. Jin , L. L. Tang , Z. G. Dong , J. Q. Li , and J. Wang , Multidimensional trapping by dual-focusing cylindrical vector beams with all-silicon metalens, Photon. Res. 10(5), 1162 (2022)

[164] M. Montagnac , Y. Brule , A. Cuche , J. M. Poumirol , S. J. Weber , J. Müller , G. Larrieu , V. Larrey , F. Fournel , O. Boisron , B. Masenelli , G. Colas des Francs , G. Agez , and V. Paillard , Control of light emission of quantum emitters coupled to silicon nanoantenna using cylindrical vector beams, Light Sci. Appl. 12(1), 239 (2023)

[165] K. Tian , W. Ding , and Z. Wang , Dead-zone-free atomic magnetometer based on hybrid Poincaré beams, Photon. Res. 12(5), 1093 (2024)

[166] N. Livakas , E. Skoulas , and E. Stratakis , Omnidirectional iridescence via cylindrically-polarized femtosecond laser processing, Opto-Electron. Adv. 3(5), 190035 (2020)

[167] L. Zhao , L. Huang , J. Huang , K. Xu , M. Wang , S. Xu , and X. Wang , Far-field parallel direct writing of sub-diffraction-limit metallic nanowires by spatially modulated femtosecond vector beam, Adv. Mater. Technol. 7(9), 2200125 (2022)

[168] G. Chen , D. Mo , J. Chen , and Q. Zhan , Self-aligned dual-beam superresolution laser direct writing with a polarization-engineered depletion beam, Photon. Res. 12(6), 1194 (2024)

[169] J. Zheng , J. Huang , and S. Xu , Multiscale micro-/nano-structures on single crystalline SiC fabricated by hybridly polarized femtosecond laser, Opt. Lasers Eng. 127, 105940 (2020)

[170] X. Lu , X. Wang , S. Wang , and T. Ding , Polarization-directed growth of spiral nanostructures by laser direct writing with vector beams, Nat. Commun. 14(1), 1422 (2023)

[171] Y. E. Li , J. F. Kuo , C. W. Cheng , and A. C. Lee , Efficient fabrication of high-quality hybrid periodic nanostructures on 4H-SiC using a single-step vector femtosecond laser processing, Opt. Laser Technol. 179, 111363 (2024)

[172] Y. Guo , P. Qiu , and S. Xu , Combined effects of polarization and secondary ablation on precision machining of microgrooves by laser-induced microjet-assisted ablation, Opt. Express 30(25), 44665 (2022)

[173] B. Dong , Z. Xu , C. Shi , K. Zhang , Y. Zhang , R. Hua , W. Zhao , and J. Wang , High-quality micro/nano structures of 4H-SiC patterning by vector femtosecond laser, Opt. Laser Technol. 163, 109338 (2023)

[174] D. Pallarés-Aldeiturriaga , A. A. Khalil , J. P. Colombier , . Ultrafast cylindrical vector beams for improved energy feedthrough and low roughness surface ablation of metals, Materials (Basel) 16(1), 176 (2023)

[175] J. Baltrukonis , O. Ulčinas , S. Orlov , and V. Jukna , High-order vector bessel-gauss beams for laser micromachining of transparent materials, Phys. Rev. Appl. 16(3), 034001 (2021)

[176] L. Guo , Y. Liang , X. Yun , M. He , T. Zhao , S. Wang , and M. Lei , Annular optical trapping of metallic nanoparticles using the azimuthally polarized beam, Opt. Lett. 49(15), 4206 (2024)

[177] X. L. Wang , J. Ding , J. Q. Qin , J. Chen , Y. X. Fan , and H. T. Wang , Configurable three-dimensional optical cage generated from cylindrical vector beams, Opt. Commun. 282(17), 3421 (2009)

[178] R. N. Kumar , A. D. Ranjan , S. Roy , S. D. Gupta , N. Ghosh , and A. Banerjee , Inhomogeneous-spin-momentum-induced orbital motion of birefringent particles in tight focusing of vector beams in optical tweezers, Phys. Rev. A 110(2), 023512 (2024)

[179] Y. Zhang , X. Dou , Y. Dai , X. Wang , C. Min , and X. Yuan , All-optical manipulation of micrometer-sized metallic particles, Photon. Res. 6(2), 66 (2018)

[180] A. Kritzinger , F. Andrew , and B. C. F. Patricia , Optical trapping and fluorescence control with vectorial structured light, Sci. Rep. 12(1), 17690 (2022)

[181] L. Zhang , X. Qiu , L. Zeng , and L. Chen , Multiple trapping using a focused hybrid vector beam, Chin. Phys. B 28(9), 094202 (2019)

[182] H. Moradi , V. Shahabadi , E. Madadi , E. Karimi , and F. Hajizadeh , Efficient optical trapping with cylindrical vector beams, Opt. Express 27(5), 7266 (2019)

[183] X. Wang , Y. Zhang , Y. Dai , C. Min , and X. Yuan , Enhancing plasmonic trapping with a perfect radially polarized beam, Photon. Res. 6(9), 847 (2018)

[184] J. Xu , Z. Yue , P. Lu , R. Wu , K. Jiang , X. Jiang , and S. Teng , Wavelength and polarization dual-multiplexed imaging based on holographic metasurfaces, Chin. Opt. Lett. 21(10), 100501 (2023)

[185] G. Milione , T. A. Nguyen , J. Leach , D. A. Nolan , and R. R. Alfano , Using the nonseparability of vector beams to encode information for optical communication, Opt. Lett. 40(21), 4887 (2015)

[186] S. Fang , Y. Cai , D. Xu , H. Guo , W. Zhang , and L. Chen , Non-orthogonal polarization encoding/decoding assisted by structured optical pattern recognition, Opt. Express 30(23), 42026 (2022)

[187] Q. Song , A. Baroni , P. C. Wu , S. Chenot , V. Brandli , S. Vézian , B. Damilano , P. de Mierry , S. Khadir , P. Ferrand , and P. Genevet , Broadband decoupling of intensity and polarization with vectorial Fourier metasurfaces, Nat. Commun. 12(1), 3631 (2021)

[188] M. Xian , Y. Xu , X. Ouyang , Y. Cao , S. Lan , and X. Li , Segmented cylindrical vector beams for massively-encoded optical data storage, Sci. Bull. (Beijing) 65(24), 2072 (2020)

[189] E. Wang , J. Niu , Y. Liang , H. L. Li , Y. Hua , L. Shi , and C. Xie , Complete control of multichannel, angle-multiplexed, and arbitrary spatially varying polarization fields, Adv. Opt. Mater. 8(6), 1901674 (2020)

[190] Z. Xie , Z. Liang , H. Wu , . Cylindrical vector beam multiplexing holography employing spin-decoupled phase modulation metasurface, Nanophotonics 13(4), 529 (2024)

[191] C. Zhang , F. Dong , Y. Intaravanne , . Multichannel metasurfaces for anticounterfeiting, Phys. Rev. Appl. 12, 034028 (2019)

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

[193] M. Liu , Y. Lei , L. Yu , X. Fang , Y. Ma , L. Liu , J. Zheng , and P. Gao , Super-resolution optical microscopy using cylindrical vector beams, Nanophotonics 11(15), 3395 (2022)

[194] Z. Rong , C. Kuang , Y. Fang , G. Zhao , Y. Xu , and X. Liu , Super-resolution microscopy based on fluorescence emission difference of cylindrical vector beams, Opt. Commun. 354, 71 (2015)

[195] I. Iglesias and J. M. Filiu , Phase-only mask for superoscillatory enhanced resolution in confocal microscopy, Opt. Express 32(20), 35395 (2024)

[196] Y. Liu , Z. Dong , Y. Zhu , H. Wang , F. Wang , Y. Chen , and Y. Cai , Three-channel robust optical encryption via engineering coherence Stokes vector of partially coherent light, PhotoniX 5(1), 8 (2024)

[197] Y. Wu , C. Zhang , Z. Nie , M. Sinclair , A. Farrell , K. A. Marsh , E. P. Alves , F. Tsung , W. B. Mori , and C. Joshi , Efficient generation and amplification of intense vortex and vector laser pulses via strongly-coupled stimulated Brillouin scattering in plasmas, Commun. Phys. 7(1), 18 (2024)

[198] S. Sederberg , F. Kong , F. Hufnagel , C. Zhang , E. Karimi , and P. B. Corkum , Vectorized optoelectronic control and metrology in a semiconductor, Nat. Photonics 14(11), 680 (2020)

[199] Y. H. Zhang , S. J. Liu , P. Chen , D. Zhu , W. Chen , S. J. Ge , Y. Wang , Z. F. Zhang , and Y. Q. Lu , Logical rotation of non-separable states via uniformly self-assembled chiral superstructures, Nat. Commun. 15(1), 1108 (2024)

[200] Z. X. Li , D. Zhu , J. S. Tang , M. Y. Chen , Y. P. Ruan , J. J. Liu , H. Zhang , P. Chen , K. Y. Xia , L. J. Zhang , and Y. Q. Lu , Quantum ghost imaging of a vector field, Opt. Express 32(4), 5898 (2024)

[201] X. B. Hu , C. X. Liu , Y. C. Zhu , R. P. Chen , B. Zhao , F. M. Wu , and C. Rosales-Guzmán , Inhomogeneous enantiomeric solution concentration measurement harnessing vectorial structured light, ACS Photonics 11(11), 4533 (2024)

[202] J. Chen , L. Jiang , X. Zhao , J. Liu , Y. Chai , M. Tian , and Z. Lu , Single-beam vector atomic magnetometer with high dynamic range based on magnetic field modulation, Adv. Quantum Technol. 7(12), 2400289 (2024)

[203] C. Zhou , W. Liang , Z. Xie , J. Ma , H. Yang , X. Yang , Y. Hu , H. Duan , and X. Yuan , Optical vectorial-mode parity Hall effect: a case study with cylindrical vector beams, Nat. Commun. 15(1), 4022 (2024)

[204] J. Ma , Z. Xie , and X. Yuan , Tailoring arrays of optical stokes skyrmions in tightly focused beams, Laser Photonics Rev. 19(3), 2401113 (2024)

[205] S. He , Y. Tian , H. Zhou , M. Zhu , C. Li , B. Fang , Z. Hong , and X. Jing , Review for micro-nano processing technology of microstructures and metadevices, Adv. Funct. Mater. 35(24), 2420369 (2025)

RIGHTS & PERMISSIONS

Higher Education Press