Vector Vortex Beams: Theory, Generation, and Detection of Laguerre–Gaussian and Bessel–Gaussian Types

Abstract

A vector vortex beam (VVB) combines the phase singularity of a vortex beam (VB) with the anisotropic polarization of a vector beam, enabling the transmission of complex optical information and offering broad application prospects in optical sensing, high-capacity communication, and high-resolution imaging. In this work, we present a detailed theoretical analysis of the generation and detection of VVBs with Laguerre–Gaussian (LG) and Bessel–Gaussian (BG) forms. Particular emphasis is placed on the polarization characteristics of VVBs, the evolution of beam profiles after passing through polarizers with different orientations, and the interference features arising from the coaxial superposition of a VVB with a circularly polarized divergent spherical wave. To validate the theoretical analysis, LGVVBs were experimentally generated using a Mach–Zehnder interferometer by superposing two vortex beams with opposite topological charges and orthogonal circular polarizations. Furthermore, the introduction of an axicon enabled the direct conversion of LGVVBs into BGVVBs. The excellent agreement between theoretical predictions and experimental observations lays a solid foundation for beginners to systematically understand VVB characteristics and advance future research.

1. Introduction

Vortex beams (VBs) are a type of ring-shaped beam characterized by a helical phase distribution [1,2,3]. Because of phase singularity at their center, the intensity drops to zero, creating a doughnut structure with a central dark spot in the field distribution [4]. VBs can be categorized into several types, such as Laguerre–Gaussian VBs (LGVBs) [5], Bessel–Gaussian vortex beams (BGVBs) [6], perfect VBs [7], and composite VBs [8]. In 1992, Allen [9] et al. first proposed that VBs could carry orbital angular momentum (OAM). The expression for the optical field of a VB includes a phase factor, exp (ilφ), where φ is the azimuthal coordinate, and l is the topological charge (TC) of the VB. Typically, a VB characterized by a TC of l possesses an OAM of lħ per photon. This significant discovery has spurred the development of OAM applications in fields such as free-space optical communication [10,11,12,13], optical microscopy [14,15], optical manipulation [16,17], and rotational target detection [18,19]. Vector beams are a class of light beams characterized by a spatially varying polarization state, meaning that different points across the same wavefront exhibit distinct polarization orientations [20]. Unlike conventional scalar beams—such as linearly or circularly polarized beams, which maintain a uniform polarization state—vector beams possess a spatially non-uniform polarization distribution and typically display anisotropic polarization characteristics [21,22,23]. Locally, the vector optical field may display various polarization states, including linear, circular, and elliptical types. The complex variations in polarization states make vector beams highly promising for applications such as three-dimensional focal field manipulation [24,25,26,27], optical data storage [28,29], optical information transmission [30,31], and second-harmonic generation [32,33,34,35]. Advancements in optical field modulation technologies have brought vector vortex beam (VVB) into the spotlight. This novel class of beam merges the phase vortex traits of scalar VB with the spatially anisotropic polarization of vector beam [36]. Typically, their intensity profiles display a “doughnut” shape, endowed with both OAM and SAM, and this singularity remains consistently preserved throughout propagation. The distinctive spatial structure and optical field distribution of VVBs enable them to transmit more complex optical information [37]. This capability makes them highly valuable for applications in optical sensing [38,39], ultra-high-capacity optical communication [40,41], light-matter interactions [42], spin–orbit interactions [43] and quantum optics [44,45]. In recent years, the study of controllable non-uniform elliptical polarization fields has garnered significant attention. Related research has investigated the formation mechanisms and evolution of spatially varying elliptical polarization within focused structured beams [46], as well as the phase distribution and resulting non-uniform polarization patterns of tightly focused elliptical polarization vortex beams [47,48]. These studies have demonstrated remarkable advantages in laser material processing—particularly in the fabrication of laser-induced periodic surface structures [49,50,51,52,53]—revealing substantial potential for practical applications. Meanwhile, higher-dimensional VVBs further extend the degrees of freedom of light beams [54], enabling them to serve as carriers for ultra-capacity optical information [55] and offering new possibilities for applications in information processing and optical communications [56].

With growing research interest in VVBs, there has been widespread attention on how to generate VVBs simply and efficiently. To date, researchers worldwide have proposed various methods for this purpose. These methods can generally be categorized into two main types: intra-cavity generation and extra-cavity generation. Intra-cavity generation relies on specialized optical components within the laser resonator to manipulate the beam mode through optical field modulation. For example, Naidoo et al. [57] demonstrated this in 2016 by using a wave-plate and non-homogeneous polarization optic inside a laser cavity for geometric phase control. They mapped SAM to OAM, creating a novel laser that produces VVBs with specific polarization states carrying OAM. In 2020, Sroor et al. [58] used a novel dielectric J-plates metasurface. This device can convert any two orthogonal polarization states into OAM spiral modes with arbitrary TCs. Integrating the J-plates into the laser cavity, along with infrared pumping, a polarizer, and a nonlinear crystal, enabled them to create a metasurface-enhanced laser generating asymmetric VVBs. 2023, Oh et al. [59] ingeniously employed S-waveplate in a telescopic laser resonator for complex VVB output. By 2025, Wang et al. [60] had leveraged Pancharatnam–Berry phase modulation in a spatial loop of a fiber laser, enabling mutual conversion between Pancharatnam TCs and polarization TCs, and thus generating VVBs at arbitrary points on the hybrid Poincaré sphere. Unlike the intracavity generation method, the extracavity approach does not necessitate modifying the internal structure of the resonator. Instead, it relies on converting the beam using optical components situated outside the resonator. Commonly used components for this purpose include spiral phase plate (SPP), spatial light modulator (SLM), digital micromirror device (DMD), q-plate and metasurface. Kumar et al. [61] in 2023 introduced a novel method to generate VVBs by modulating orthogonal light components via a single-phase pattern displayed on an SLM. This technique simplifies the production of light beams with V-point polarization singularities. In 2024, Li et al. [62] introduced a method for generating VVBs using a DMD. After mitigating the DMD’s inherent limitations, they developed a binary multiplexed hologram. This significantly improves the quality of VB generation by adjusting the ellipticity of the VB’s amplitude and phase distribution. In 2024, Guo et al. [63] proposed and experimentally demonstrated a spin-independent programmable metasurface for generating arbitrary VVBs through dynamic, independent control of the TCs, initial phases, and amplitude coefficients of the reflected left-handed circular polarized (LHCP) and right-handed circular polarized (RHCP) vortex components. In 2025, Yao et al. [64] explored a method for generating VVBs featuring both helical phase distributions and polarization state distributions using a q-plate. The conversion of polarization states primarily relies on two quarter-wave plates (QWP). This facilitates rapid and flexible VVB generation. Concurrently in 2025, Harrison et al. [65] employed an improved Mach–Zehnder interferometer (MZI) for arbitrary modal phase and amplitude control of VVBs over the Poincaré sphere, thereby enabling the generation of arbitrary cylindrical VVBs. However, the types of VVB investigated in all these studies were restricted to LG beam. Experimental generation of BGVVB has been rarely reported. High-order BGVVBs possess some unique features, including non-diffracting propagation [66], self-healing properties [67,68], and cross-sectional anisotropic vortex polarization coupled with helical phase. These properties make them particularly attractive for applications in information transmission, imaging, anti-interference, and particle manipulation. Despite this potential, only a limited number of works have explored the generation and properties of BGVVBs. For instance, Dudley et al. [69] generated nondiffracting BGVVBs by transforming arbitrary scalar fields into vector fields through the combined use of a spatial light modulator (SLM) and a q-plate. In 2021, Baltrukonis et al. presented the BGVVB generation using two geometric phase-based optical elements, a high-order S-wave plate and an axicon in combination with simple optical elements, such as lenses, wave plates, and polarizers [70].

The detection of VVBs has emerged as a multifaceted research domain, encompassing diverse foci such as the measurement and analysis of TC characteristics, the detection and control of polarization states, and the exploration of spatial structural properties—including phase distribution, intensity profiles, and mode composition. Over the past decade, significant advancements have been made in both the generation and detection of these beams, driven by innovative optical techniques and device engineering. In 2011, Zhou et al. [71] demonstrated the conversion of a circularly polarized Gaussian beam into a radially polarized vortex beam using a grating device. The polarization distribution of the generated beam was analyzed with a combination of a quarter-wave plate and a polarizer, while its focused field intensity was measured via a confocal microscopy system. Two years later, Dudley et al. [69] detected the polarization and azimuthal components of BGVVBs using polarization-selective modal decomposition based on a phase grating and a second SLM. Also in 2013, Moreno et al. [72] utilized a vortex grating to decompose and analyze the components of VVBs. In 2015, Wakayama et al. [73] extended the application of vectorial vortex analysis to the terahertz regime, measuring the polarization states of arbitrarily polarized beams within the 0.1–1.6 THz range using achromatic axially symmetric wave plates made of polytetrafluoroethylene, which provided a phase retardance of 163°. Subsequently, He et al. [74] in 2017 analyzed the polarization order of cylindrical VVBs with a Glan laser polarizer and determined their TC by interfering the beams with left- or right-handed circularly polarized (LHCP/RHCP) reference light. Recent years have witnessed the emergence of more sophisticated and nondestructive detection strategies. In 2019, Wang et al. [75] introduced a structured gold film composed of parallel and V-shaped slits, enabling simultaneous detection of both OAM and SAM via plasmonic interferences. In 2022, Zhang et al. [76] proposed and theoretically analyzed an integrated microscale chiral plasmonic lens for cylindrical vector beam detection, while Qi et al. [77] employed coaxial and small-angle interference with a Gaussian plane beam to detect the topological order of vortex and cylindrical vector beams. That same year, Kumar et al. [78] utilized a self-referenced interferometric approach based on a modified MZI to characterize VVB properties through fringe pattern analysis. Collectively, these studies reflect the rapid evolution and growing sophistication of VVB detection methodologies, underscoring the field’s dynamic and interdisciplinary nature. However, most existing studies predominantly emphasize experimental investigations, with comparatively limited attention devoted to the systematic theoretical derivation of their generation mechanisms and detection methods. Therefore, a more detailed and comprehensive theoretical analysis is urgently required.

In this paper, we employ the Jones matrix formalism to theoretically analyze the formation mechanism of VVBs, their transmitted beam patterns through a linear polarizer, and the interference fringes resulting from the superposition of LG (BG) VVBs with a circularly polarized divergent spherical wave. Building on this theoretical framework, we implement an efficient and stable method for generating VVBs using an MZI. In this scheme, two LGVBs with distinct TCs and mutually orthogonal polarizations (vertical and horizontal) are non-coherently superimposed and subsequently modulated by a quarter-wave plate to generate a LGVVB. Furthermore, by incorporating an axicon, the LGVVB is converted into a BGVVB. The paper is organized as follows: Section 2 presents the theoretical models for generating all types of VVBs and characterizing their properties. Section 3 details the experimental setup and generation procedures. In Section 4, we compare experimental results with theoretical predictions for the generated LGVVBs and BGVVBs, including beam profiles before and after transmission through a linear polarizer at various angles, as well as interferograms formed by coaxial interference between the generated VVBs and a divergent spherical wave (left- or right-circularly polarized). Finally, conclusions are drawn in Section 5. This work offers a more comprehensive and rigorous approach to the analysis and generation of VVBs.

2. Theoretical Analysis

2.1. Theoretical Models for Generating LGVB and BGVB

The LGVB is a typical representative of VBs. When a fundamental mode (TEM₀₀) Gaussian beam is incident on a SPP, the thickness variation in the SPP induces a phase modulation on the TEM₀₀ beam that varies with the azimuthal angle φ. This phase modulation enables the input TEM₀₀ Gaussian beam to acquire an exp(ilφ) term. The resulting beam can be considered a good approximation of an LG_0lVB. The amplitude expression for the optical field of the resulting LG_0lVB in cylindrical coordinates (r, φ, z) is given as [79,80,81]:

$$E_{\mathrm{LG}}(r,\phi ,z)=A_{\mathrm{LG}}{\left(\sqrt{2}r/w_g(z)\right)}^{\left|l\right|}\exp \left(-r^2/w_g^2(z)\right)\exp \left. [i{\varphi }_{\mathrm{LG}}+ik{r}^2/2R(z)+ikz+il\phi \right].$$

(1)

where $A_{\mathrm{LG}}=\sqrt{P_{\mathrm{LG}}/\left. [\pi n_{0}c{\epsilon }_0\left|l\right|!w_g^2(z)\right]}$ denotes the normalization constant of the LGVB. P_LG represents the power of the LGVB, and l is its TC. n₀ is the refractive index of the transmission medium (n₀ ≈ 1 in air), ε₀ is the permittivity of vacuum, and c is the light speed in vacuum. $w_g(z)=w_g\sqrt{1+{\left(z/z_R\right)}^2}$ and $w_g$ indicate the beam radius and waist radius of the input fundamental mode Gaussian beam. The Gouy phase for the LG_0lVB is given by ${\varphi }_{LG}=-\left(\left|l\right|+1\right)\arctan \left(z/z_R\right)$. $z_R=k{w}_g^2/2=\pi w_g^2/\lambda$ is the Rayleigh range, $R\left(z\right)=z+z_R^2/z$ is the radius of wavefront curvature, and $k=2\pi n/\lambda$ is the module of wave vector.

Using an axicon, the LGVB can be converted into a BGVB. The BGVB exhibits transmission characteristics similar to those of a Bessel beam within a finite propagation distance. After passing through an axicon positioned at z = 0, the LGVVB undergoes radial linear phase modulation, and upon free-space propagation and interference, forms a BGVVB, achieving the conversion from the LGVVB mode to the BGVVB mode. [82,83,84]. The optical field of an lth-order BGVB in cylindrical coordinates (r, φ, z) can be represented as

$${\mathrm{E}}_{\mathrm{BG}}(r,\phi ,z)=A_{\mathrm{BG}}{J}_l(k_{r}r)\exp \left. [-r^2/w_g^2(z)\right]\exp (i{k}_{z}z+il\phi ).$$

(2)

where $A_{\mathrm{BG}}=\sqrt{P_{\mathrm{BG}}/\left. [\pi n_{0}c{\epsilon }_0{w}_g^2(z)\right]}$ denotes the normalization constant of the BGVB, $P_{\mathrm{BG}}$ denotes the power of the BGVB. $J_l(\varsigma )$ is the lth-order Bessel function of the first kind, which can be express as: $J_l(\varsigma )={\displaystyle \sum _{m=0}^{\infty }\frac{{(-1)}^m}{m!\mathsf{\Gamma }(m+l+1)}}{({\displaystyle \frac{\varsigma }{2}})}^{2m+l}$, When l is odd, $J_{-l}(k_{r}r)=-J_l(k_{r}r)$; when l is even, $J_{-l}(k_{r}r)=J_l(k_{r}r)$. $k_r$ and $k_z$ are the wave numbers in the radial and beam propagation directions, respectively. They satisfy the relationship: $k_r^2+k_z^2=k^2=4{\pi }^2/{\lambda }^2$. Meanwhile, $k_r/k=\sin =[(n-1)\gamma ]$. Here, n is the axicon’s refractive index and γ represents the base angle of the axicon. In contrast to the ideal Bessel beam, the BG beam diverges after propagating to a certain distance. This distance is known as the Rayleigh range (diffraction-free range), and its expression is $Z_{\max }=w_g/(n-1)\gamma$ [85].

2.2. Generation of VVBs

This section focuses on the generation of VVBs through the superposition of orthogonal circularly polarized beams (LHCP and RHCP) based on theoretical analysis, with particular emphasis on the evolution of their polarization characteristics during the generation process.

A VVB generated by the superposition of two orthogonal circularly polarized VBs (with TCs of l₁ and l₂, l₁ ≠ l₂) can be expressed as

$$E_{\mathrm{VVB}}={\displaystyle \frac{1}{\sqrt{2}}}\left\{{\displaystyle \frac{A_{l_1}}{\sqrt{2}}}\left. [\begin{array}{l}1\\ i\end{array}\right]e^{i{l}_1\phi }+{\displaystyle \frac{A_{l_2}}{\sqrt{2}}}\left. [\begin{array}{l}1\\ -i\end{array}\right]e^{i{l}_2\phi }{e}^{i\delta }\right\},$$

(3)

where $A_l=A_{\mathrm{LG}}{\left. [\sqrt{2}r/w_g(z)\right]}^{\left|l\right|}\exp \left. [-r^2/w_g^2(z)\right]$ for a collimated LGVB, and $A_l=A_{\mathrm{BG}}{J}_l(k_{r}r)\exp \left. [-r^2/w_g^2(z)\right]$ for a BGVB. l represents l₁ or l₂.

For |l₁| ≠ |l₂|, the resulting VVB is classified as a hybrid-order VVB (HVVB) [86,87]. Unlike conventional vector beams with uniform polarization distributions, HVVBs exhibit spatially varying elliptical polarization states, wherein both the ellipticity and orientation of the polarization ellipse vary continuously across the beam cross-section. This complex polarization structure arises from the superposition of two orthogonally polarized components, each carrying a distinct TC. The interference and coupling between these components—due to the TC mismatch—produce a highly nonuniform polarization distribution. As a result, the local polarization state is predominantly elliptical, rather than linear, and displays pronounced spatial modulation in both the major axis orientation and ellipticity. In contrast to traditional radially or azimuthally polarized vector beams, HVVBs are not constrained to linear polarization states. Instead, they exhibit richer spatial features and greater structural complexity, making them particularly valuable for advanced applications in optical manipulation, communication, and imaging.

For |l₁| = |l₂|, the VVB has linear polarization with spatially varying polarization directions. Here, we focus on a detailed discussion of this type of LGVVBs. The optical field for the LGVVB can be simplified as

$$E_{{\left(\mathrm{LGVVB}\right)}_1}={\displaystyle \frac{1}{\sqrt{2}}}\left\{{\displaystyle \frac{A}{\sqrt{2}}}\left. [\begin{array}{l}1\\ i\end{array}\right]e^{i{l}_1\phi }+{\displaystyle \frac{A}{\sqrt{2}}}\left. [\begin{array}{l}1\\ -i\end{array}\right]e^{i{l}_2\phi }{e}^{i\delta }\right\}={\displaystyle \frac{A}{2}}\left. [\begin{array}{c}{e}^{i{l}_1\phi }+e^{i{l}_2\phi }{e}^{i\delta }\\ i{e}^{i{l}_1\phi }-i{e}^{i{l}_2\phi }{e}^{i\delta }\end{array}\right].$$

(4)

Assume l₁ = −l₂ = −l (l is a positive integer), The LGVVB shown in Equation (4) can be expressed as

$$E_{{\left(\mathrm{LGVVB}\right)}_2}={\displaystyle \frac{A}{2}}\left. [\begin{array}{c}{e}^{-il\phi }+e^{il\phi }{e}^{i\delta }\\ i{e}^{-il\phi }-i{e}^{il\phi }{e}^{i\delta }\end{array}\right].$$

(5)

For δ = π, it becomes

$$E_{\mathrm{LGVVB}}^{\mathrm{I}}=iA\left. [\begin{array}{c}-\sin \left(l\phi \right)\\ \cos \left(l\phi \right)\end{array}\right]$$

(6)

Swapping the TCs of the LHCP and RHCP VBs (such as l₁ = −l₂ = l) yields another type of VVB, which we designate as type II LGVVB [88], and it can be expressed as

$$E_{\mathrm{LGVVB}}^{\mathrm{II}}=iA\left. [\begin{array}{c}\sin \left(l\phi \right)\\ \cos \left(l\phi \right)\end{array}\right].$$

(7)

For δ = 0 and l₁ = −l₂ = −l, Equation (4) becomes

$$E_{\mathrm{LGVVB}}^{\mathrm{III}}=A\left. [\begin{array}{c}\cos \left(l\phi \right)\\ \sin \left(l\phi \right)\end{array}\right].$$

(8)

It is denoted as type III LGVVB [88].

For δ = 0 and l₁ = −l₂ = l, we obtain

$$E_{\mathrm{LGVVB}}^{\mathrm{IV}}=A\left. [\begin{array}{c}\cos \left(l\phi \right)\\ -\sin \left(l\phi \right)\end{array}\right].$$

(9)

It is denoted as type IV LGVVB [88].

For |l| = 1, Type I and Type III exhibit azimuthal and radial symmetries, respectively. Hence, Type I is designated as the azimuthally polarized LGVVB, whereas Type III is referred to as the radially polarized LGVVB. Type II and Type IV are referred to as quasi-azimuthal polarization and quasi-radial polarization [89,90,91], these four types of VVBs differ not only in their polarization distributions but also exhibit distinct longitudinal components after tight focusing, a distinction that provides significant advantages for laser applications in material processing. [92,93]. Moreover, for |l| > 1, all four types are collectively classified as high-order cylindrical LGVVBs.

Due to the oscillatory behavior of the Bessel function in the radial direction of the BG beam’s complex amplitude expression, an additional phase of π is introduced when the TC is an odd number. Under these conditions, different combinations of TCs and phase differences δ give rise to four types of VVBs: Type I VVB is generated for l₁ = −l₂ = −l and δ = 0; Type II VVB is generated for l₁ = −l₂ = l and δ = 0; Type III VVB is generated for l₁ = −l₂ = −l and δ = π; and Type IV VVB is generated for l₁ = −l₂ = l and δ = π. The detailed derivation is presented in Appendix A. When l is even, for the same values of l₁, l₂, and δ, the four types of BGVVBs are identical to that of the corresponding LGVVBs.

2.3. Characterizing the Properties of the VVB

Here, we primarily analyze the beam patterns of LGVVBs after passing through polarizers oriented along different transmission directions, as well as the characteristics of the interferograms generated by the coaxial interference between LGVVBs and circularly polarized diverging spherical waves.

The Jones matrix for the polarizer can be expressed as

$$M=\left. [\begin{array}{cc}{\sin }^2\theta & -\sin \theta \cos \theta \\ -\sin \theta \cos \theta & {\cos }^2\theta \end{array}\right].$$

(10)

Here θ denotes the angle between the transmission axis and the vertical direction. After passing through the polarizer, the optical field of the LGVVB is given by

$$\begin{array}{l}{E}_1=M\cdot E_{{\left(\mathrm{LGVVB}\right)}_1}={\displaystyle \frac{A}{2}}\left. [\begin{array}{cc}{\sin }^2\theta & -\sin \theta \cos \theta \\ -\sin \theta \cos \theta & {\cos }^2\theta \end{array}\right]\left. [\begin{array}{c}{e}^{i{l}_1\phi }+e^{i{l}_2\phi }{e}^{i\delta }\\ i{e}^{i{l}_1\phi }-i{e}^{i{l}_2\phi }{e}^{i\delta }\end{array}\right]\\ =A\left. [\begin{array}{l}\sin \theta \sin \left(\theta +{\displaystyle \frac{\left(l_1-l_2\right)\phi -\delta }{2}}\right)\\ -\cos \theta \sin \left(\theta +{\displaystyle \frac{\left(l_1-l_2\right)\phi -\delta }{2}}\right)\end{array}\right]\exp \left(i{\displaystyle \frac{\left(l_1+l_2\right)\phi +\delta }{2}}\right)\end{array}.$$

(11)

Its intensity is given by

$$I_1\propto |E_1{|}^2\propto 1-\cos \left. [2\theta +(l_1-l_2)\phi -\delta \right].$$

(12)

From Equation (12), the intensity expression is a periodic function. Therefore, the transmitted beam exhibits a petal pattern of bright and dark regions, whose position is determined by l₁, l₂ and δ. The period of the intensity in the azimuthal direction is $2\pi /\left|l_1-l_2\right|$, and the number of petals is $N=2\pi /(2\pi /\left|l_1-l_2\right|)=\left|l_1-l_2\right|$.

Bright petal positions satisfy the following condition:

$$2\theta +(l_1-l_2)\phi -\delta =\left(2m+1\right)\pi (m=0,\pm 1,\pm 2\cdots ),$$

(13)

and the angular coordinate φ of the bright petal is

$$\phi =\left. [\left(2m+1\right)\pi +\delta -2\theta \right]/\left(l_1-l_2\right)(m=0,\pm 1,\pm 2\cdots ).$$

(14)

Changing θ to θ + Δθ results in the change in φ from φ to φ + Δφ, where Δφ is given by

$$\mathsf{\Delta }\phi =-2\mathsf{\Delta }\theta /\left(l_1-l_2\right).$$

(15)

For Type I LGVVBs and even-order BGVVBs (δ = π and l₁ = −l₂ = −l), φ = − (mπ − θ)/l and Δφ = Δθ/l. The bright spots of the petal pattern for θ = 0° are located at angular positions φ = mπ/l (m = 0, 1, 2, … 2l − 1). The petal pattern rotates in the same direction as the polarizer; however, the angle it rotates through is 1/l times the angle rotated by the polarizer.

For Type II LGVVB and even-order BGVVBs (δ = π and l₁ = −l₂ = l), φ = (mπ − θ)/l and Δφ =−Δθ/l. The bright spots of the petal pattern for θ = 0° are located at angular positions φ = mπ/l (m = 0, 1, 2, …2l − 1). The petal pattern rotates in the opposite direction as the polarizer; however, the angle it rotates through is 1/l times the angle rotated by the polarizer.

For Type III LGVVB and even-order BGVVBs (δ = 0 and l₁ = −l₂= −l), φ = − [(m + 1/2) π − θ]/l and Δφ = Δθ/l. The bright spots of the petal pattern for θ = 0° located at angular positions φ = mπ/l − π/2l (m = 1, 2, …2l). The petal pattern rotates in the same direction as the polarizer; however, the angle it rotates through is 1/l times the angle rotated by the polarizer.

For Type IV LGVVB and even-order BGVVBs (δ = 0 and l₁ = −l₂ = l), φ = [(m + 1/2) π − θ]/l and Δφ = −Δθ/l. The bright spots of the petal pattern for θ = 0° located at angular positions φ = mπ/l − π/2l (m = 1, 2, … 2l). The rotation direction of the petal pattern is opposite to that of the polarizer.

For odd-order BGVVBs, the presence of an additional phase term makes their form not directly equivalent to that of LGVVBs or even-order BGVVBs. By eliminating this extra phase difference, the odd-order BGVVBs can be transformed into the same four types of VVBs as those generated by LGVVBs and even-order BGVVBs. Specifically, by setting δ = 0, 0, π, and π for Type I, Type II, Type III, and Type IV, respectively, the resulting beams become identical to the corresponding LGVVBs and even-order BGVVBs.

An interferogram, generated by coaxially superimposing the LGVVB with a circularly polarized (LHCP/RHCP) divergent spherical wave, shows the TCs related to the LHCP/RHCP components of the LGVVB.

The superposition field formed by coaxial superposition between the LGVVB and LHCP (RHCP) divergent spherical wave can be expressed as

$$E_2=E_{{\left(\mathrm{LGVVB}\right)}_1}+E_L={\displaystyle \frac{A}{2}}\left. [\begin{array}{c}{e}^{i{l}_1\phi }+e^{i{l}_2\phi }{e}^{i\delta }\\ i{e}^{i{l}_1\phi }-i{e}^{i{l}_2\phi }{e}^{i\delta }\end{array}\right]+{\displaystyle \frac{B}{\sqrt{2}}}\left. [\begin{array}{l}1\\ i\end{array}\right]e^{{\displaystyle \frac{ik{r}^2}{2R}}}={\displaystyle \frac{1}{2}}\left. [\begin{array}{c}A(e^{i{l}_1\phi }+e^{i{l}_2\phi }{e}^{i\delta })+\sqrt{2}B{e}^{{\displaystyle \frac{ik{r}^2}{2R}}}\\ iA(e^{i{l}_1\phi }-e^{i{l}_2\phi }{e}^{i\delta })+\sqrt{2}iB{e}^{{\displaystyle \frac{ik{r}^2}{2R}}}\end{array}\right],$$

(16)

or

$$E_3=E_{{\left(\mathrm{LGVVB}\right)}_1}+E_R={\displaystyle \frac{A}{2}}\left. [\begin{array}{c}{e}^{i{l}_1\phi }+e^{i{l}_2\phi }{e}^{i\delta }\\ i{e}^{i{l}_1\phi }-i{e}^{i{l}_2\phi }{e}^{i\delta }\end{array}\right]+{\displaystyle \frac{B}{\sqrt{2}}}\left. [\begin{array}{l}1\\ -i\end{array}\right]e^{{\displaystyle \frac{ik{r}^2}{2R}}}={\displaystyle \frac{1}{2}}\left. [\begin{array}{c}A(e^{i{l}_1\phi }+e^{i{l}_2\phi }{e}^{i\delta })+\sqrt{2}B{e}^{{\displaystyle \frac{ik{r}^2}{2R}}}\\ i\left. [A(e^{i{l}_2\phi }-e^{i{l}_1\phi }{e}^{i\delta })-\sqrt{2}B{e}^{{\displaystyle \frac{ik{r}^2}{2R}}}\right]\end{array}\right].$$

(17)

Here, $B=\left. [A_{\mathrm{sph}}/\sqrt{1+{\left(z/z_R\right)}^2}\right]\exp \left. [-r^2/w_g^2(z)\right]$, and A_sph is the amplitude of the spherical wave.

The intensity of the interference beam shown in Equations (16) and (17) is given by

$$I_2=|E_2{|}^2\propto |A{|}^2+|B{|}^2+\sqrt{2}|A\left|\right|B|\cos (l_1 \phi -k{r}^2/2R),$$

(18)

or

$$I_3=|E_3{|}^2\propto |A{|}^2+|B{|}^2+\sqrt{2}|A\left|\right|B|cos(l_2 \phi +\delta -k{r}^2/2R).$$

(19)

The bright fringes in the interferograms satisfy the following relationship:

$$l_1 \phi -{\displaystyle \frac{k{r}^2}{2R}}=2m\pi \to \phi ={\displaystyle \frac{k{r}^2}{2R{l}_1}}+{\displaystyle \frac{2m\pi }{l_1}},$$

(20)

or

$$l_2 \phi +\delta -{\displaystyle \frac{k{r}^2}{2R}}=2m\pi \to \phi ={\displaystyle \frac{k{r}^2}{2R{l}_2}}+{\displaystyle \frac{2m\pi -\delta }{l_2}}.$$

(21)

The interferogram exhibits a spiral fringe structure when l₁ (l₂) ≠ 0. The number of spiral lobes depends on the TC value of l₁ (l₂), while the twist direction depends on the sign of the TC l₁ (l₂): clockwise for a negative sign and counterclockwise for a positive sign.

Similarly, by adjusting the relative phase difference δ between the two vortex components (LHCP and RHCP) in odd-order BGVVBs, the additional phase term can be effectively eliminated. As a result, the spiral interference structures formed by odd-order BGVVBs with divergent spherical waves (LHCP/RHCP) become identical to those obtained with LGVVBs and even-order BGVVBs. Moreover, when an LHCP wave interferes with a VVB, the resulting interference pattern reveals the TC information of the LHCP component, whereas interference with an RHCP wave encodes the TC of the RHCP component. The characteristics of the coaxial interference pattern formed between a BGVVB and a divergent circularly polarized spherical wave are analogous to those observed with LGVVBs; thus, they will not be discussed further.

3. Experimental Setup

The experimental setup for generating VVBs and analyzing their properties is schematically shown in Figure 1. A continuous-wave, high-power (up to 15 W), single-frequency (<100 kHz linewidth), linearly polarized Yb-doped fiber laser and amplifier (Koheras Y10, NKT Photonics, Birkerød, Denmark) operating at 1064 nm serves as the laser source. An optical isolator is used to prevent back-reflections from reaching the laser. Additionally, a power-control system comprising a half-wave plate (HWP₂) and polarizing beam splitters (PBS₁) is incorporated to adjust the laser power in the subsequent optical path. A lens (f₁ = 500 mm) collimates the laser beam for the subsequent experiment. The HWP₃ and PBS₂ combination splits the main beam path into two branches. The s-polarized beam reflected by PBS₂ is converted into a divergent circularly polarized (LHCP/RHCP) spherical wave after passing through a QWP and a lens, in sequence. The result spherical wave will interfere with the VVB, thereby revealing the TC information carried by the LHCP/RHCP VB components that constitute the VVB. To improve the beam quality and achieve a more circular and symmetrical spot, a 100 µm diameter circular pinhole diffracts the p-polarized beam transmitted by PBS₂. Subsequently, a circular diaphragm is used to select the central Airy spot, which will serve as the standard fundamental Gaussian beam for subsequent experiments. A lens with 150 mm focal length is then employed to collimate this fundamental Gaussian beam.

The MZI, which consists of mirrors M₃ and M₄, PBS₃, and PBS₄, is used to non-coherently combine two LGVBs with different TCs and orthogonal polarizations (horizontal and vertical). By rotating HWP₄, the power ratio between the two arms of the MZI can be adjusted. The SPP (UPO Labs) is inserted into the MZI to generate LGVBs with different TCs required. The position of mirror M₃, which is mounted on a translation stage, is carefully adjusted to control the relative phase difference δ between two VBs. After passing through QWP₂, the horizontal and vertical LGVBs are converted into LHCP and RHCP LGVBs, respectively, and an LGVVB is created. This LGVVB is subsequently modulated by an axicon (LBTEK, base angle γ = 0.5°), resulting in the generation of the BGVVB. A CMOS camera (Duderstadt, Germany, CINOGY, CinCam, CMOS-1202) is used to separately capture the following patterns: the beam pattern of the VVB, the polarization pattern after passing through the polarizer with its transmission direction set at 0°, 45°, 90°, and 135° from the vertical direction, and the interferogram resulting from the coaxial interference between the VVB and the LHCP/RHCP divergent spherical wave, respectively.

4. Experimental Results and Discussion

In experiments, we studied VVBs created by incoherently superimposing LHCP and RHCP VBs with opposite TCs, in both LG and BG forms.

Figure 2 illustrates beam patterns (polarization distributions) of LGVVBs with TCs of ±1, polarization patterns at 0°, 45°, 90°, 135° rotation of linear polarizer, as well as the interferograms between the LGVVB and the LHCP/RHCP divergent spherical wave. As shown in column 1 of Figure 2, there are four types of LGVVBs: Type I (azimuthally polarized), Type II (quasi-azimuthal polarization), Type III (radially polarized), and Type IV (quasi-radial polarization. The four LGVVB types share the same intensity profile with a single-ring structure but have unique polarization distributions, as determined by the polarization patterns at 0°, 45°, 90°, and 135° rotations of a linear polarizer (Figure 2, Columns 3–6). Here, the polarization pattern exhibits a two-lobe structure. When the transmission axis of the linear polarizer is oriented vertically (θ = 0°), the lobes are along the angular φ = 0 and π (horizontally) for type I and II LGVVBs, and along the angular φ = π/2 and 3π/2 (vertically) for type III and IV LGVVBs. Upon rotation of the linear polarizer, a corresponding rotation is observed in the polarization pattern. The polarization pattern rotates in the same (opposite) direction as the polarizer for type I and III (II and IV) LGVVBs, and the angle it rotates is the same as the angle rotated by the polarizer. These results indicate that these four types of LGVVBs can be distinguished by two key characteristics: the polarization pattern transmitted through a polarizer at the same transmission axis, and the rotation direction of the polarization pattern as the polarizer is rotated. As shown in columns 7 and 8 of Figure 2, the interferograms between the LGVVB and LHCP/RHCP divergent spherical wave exhibit spiral fringes with one spiral lobe. The TC l₁ is −1 in type I (III) LGVVB, and +1 in type II (IV) LGVVB. The TC l₂ is +1 in type I (III) LGVVB, and −1 in type II (IV) LGVVB.

The experimental results for the LGVVB formed by superposing two orthogonal circular LGVBs with TCs of ±2 are shown in Figure 3. As shown in columns 3–6 of Figure 3, Here, the polarization pattern exhibits a four-lobe structure. When the transmission axis of the linear polarizer is oriented vertically (θ = 0°), the lobes are along the angular φ = 0, π/2, π, and 3π/2 for type I and II LGVVBs, and along the angular φ = π/4, 3π/4, 5π/4, and 7π/4 for type III and IV LGVVBs. As the linear polarizer is rotated, the polarization pattern rotates accordingly, with its rotation angle being half that of the polarizer. For type I and III LGVVBs, the polarization pattern rotates in the same direction as the polarizer, whereas for type II and IV LGVVBs, it rotates in the opposite direction. As shown in columns 7 and 8 of Figure 3, the interferograms between the LGVVB and a divergent spherical wave (LHCP/RHCP) exhibit spiral fringes with two spiral lobes. The TCs in the two components of LGVVB are determined as follows: for type I (III) LGVVBs, the TCs are l₁ = −2 and l₂ = +2, whereas for type II (IV) LGVVBs, they are l₁ = +2 and l₂ = −2.

These experimental results are in excellent agreement with our theoretical derivation of LGVVBs. The number of petals is ∣l₁ − l₂∣. When the linear polarizer (Δθ) is rotated clockwisely, the rotation angle of the petal pattern satisfies Δφ = −2Δθ/(l₁ − l₂); specifically, for l₁ − l₂ > 0, the petals rotate counterclockwise, while for l₁ − l₂ < 0, they rotate clockwise. Moreover, when a generated VVB interferes with a divergent spherical wave (LHCP or RHCP), the number of spiral fringes is governed by the TC (l₁ or l₂). Specifically, for an LHCP spherical wave, the spiral fringes reflect the LHCP component of the VVB: a negative TC induces clockwise rotation, while a positive one in the LHCP component leads to counterclockwise rotation. Conversely, with an RHCP spherical wave, the fringes correspond to the RHCP component of the LGVVB.

An axicon will convert LGVVBs shown above to the BGVVBs. In the following section, the properties of the experimentally generated BGVVB will be discussed.

Figure 4 depicts beam patterns (polarization distributions) of BGVVBs with TCs of ±1, polarization patterns at 0°, 45°, 90°, 135° rotation of linear polarizer, as well as the interferograms between the BGVVB and the LHCP/RHCP divergent spherical wave. As shown in Column 1 of Figure 4, there are four types of BGVVBs: Type I, Type II, Type III, and Type IV. The four BGVVB types share the same intensity profile with a multi-ring structure but have unique polarization distributions, as determined by the polarization patterns at 0°, 45°, 90°, and 135° rotations of a linear polarizer (Figure 4, Columns 3–6). Here, the polarization pattern exhibits a multi-ring two-lobe structure. For a vertically oriented linear polarizer (θ = 0°), the lobes of type I and II BGVVBs are oriented along the horizontal axis (φ = 0 and π), while those of type III and IV VVBs are oriented along the vertical axis (φ = π/2 and 3π/2). The polarization pattern of type I and III BGVVBs rotates in the same direction and by the same angle as the polarizer, whereas for type II and IV BGVVBs, it rotates in the opposite direction but by the same angle as the polarizer. These results demonstrate that four types of BGVVBs are uniquely identified by two properties: the specific polarization pattern for a given polarizer angle, and the handedness of the transmitted pattern’s rotation with respect to the polarizer’s rotation. As illustrated in columns 7 and 8 of Figure 4, the interferograms resulting from coaxial interference between the BGVVB and the LHCP/RHCP divergent spherical wave present spiral fringes with a single spiral lobe. The TC l₁ is −1 in type I (III) BGVVB, and +1 in type II (IV) BGVVB. The TC l₂ is +1 in type I (III) BGVVB, and −1 in type II (IV) BGVVB.

The experimental results for the BGVVB formed by superposing two orthogonal circular BGVBs with TCs of ±2 are shown in Figure 5. As shown in columns 3–6 of Figure 5, Here, the polarization pattern exhibits a multi-ring four-lobe structure. When the transmission axis of the linear polarizer is oriented vertically (θ = 0°), the lobes of type I and II BGVVBs are along the angular φ = 0, π/2, π, and 3π/2. In contrast, for type III and IV BGVVBs, the lobes are oriented at φ = π/4, 3π/4, 5π/4, and 7π/4. As the linear polarizer is rotated, the polarization pattern of the BGVVB rotates correspondingly, with its rotation angle being half that of the polarizer’s rotation angle. Notably, for type I and III BGVVBs, the polarization pattern rotates in the same direction as the polarizer, whereas for type II and IV BGVVBs, it rotates in the opposite direction. As illustrated in columns 7 and 8 of Figure 5, the interferograms between the BGVVB and a divergent spherical wave with LHCP/RHCP exhibit spiral fringes characterized by two distinct spiral lobes. The TCs in the two components of BGVVB are determined as follows: for type I (III) BGVVBs, the TC are l₁ = −2 and l₂ = +2, whereas for type II (IV) BGVVBs, they are l₁ = +2 and l₂ = −2. These experimental findings effectively validate the corresponding theoretical predictions.

5. Conclusions

In conclusion, this study theoretically predicted the generation of VVBs, their evolution upon transmission through a linear polarizer, and the resulting interference patterns with divergent left- or right-handed circularly polarized spherical waves. To validate the theoretical model, we experimentally generated LGVVBs by superposing two LGVBs with opposite TCs and orthogonal circular polarizations using an MZI. Subsequently, LGVVBs were converted into BGVVBs via an axicon. Experimental results demonstrate excellent agreement with theoretical predictions. We show that VVBs can be classified into four distinct types (I, II, III, and IV). When a VVB is formed by superposing two VBs with TCs ±l, it is termed a cylindrical VVB for l = ±1. Specifically, types I, II, III, and IV correspond to azimuthally polarized, quasi-azimuthally polarized, radially polarized, and quasi-radially polarized VVBs, respectively. Although these four types exhibit identical intensity distributions, they possess distinct polarization structures. Upon transmission through a linear polarizer, the polarization pattern of a VVB forms a 2l-lobed petal structure. As the polarizer rotates, the pattern rotates by an angle 1/l times that of the polarizer. For type I and III VVBs, the polarization pattern rotates in the same direction as the polarizer, whereas for type II and IV VVBs, it rotates in the opposite direction.

Our findings demonstrate that the four VVB types can be distinguished by two key features: the polarization pattern under a fixed polarizer axis and its rotational response to polarizer rotation. By having a VVB interfere with a divergent LHCP (RHCP) spherical wave, the TC of its constituent VB (l₁ or l₂) can be determined. The number of spiral lobes in the interferogram equals the magnitude of the TC, while the sign is indicated by the handedness of the spiral fringes: clockwise for negative and counterclockwise for positive.

Crucially, the theoretical derivations presented in this study are systematic and detailed, providing a robust framework for understanding the generation, evolution, and characteristics of VVBs, and laying a solid foundation for future experimental investigations. For newcomers to the field, these derivations serve as an invaluable resource, fostering a deeper and more comprehensive understanding of the fundamental concepts and physical mechanisms of VVBs, thereby offering strong support for further research and potential applications.