Analyzing the study of propagation characteristics of a vortex beam in a chiral medium

1. Introduction

Light structuring is the process of fine-tuning the amplitude and phase front of an optical beam to look for interesting patterns (Angelsky et al., 2020). One type of structured light that has caught the attention of optical researchers is beams that contain orbital angular momentum (OAM) (Fatkhiev et al., 2021). Such vortex beams feature a ring amplitude structure and a null central beam intensity (Kotlyar et al., 2018; Kotlyar et al., 2025a). Moreover, as the beam propagates, the phase profile twists in a helical shape. The direction of the twist (clockwise or counterclockwise) depends on the value of the OAM or topological charge $\left(l\right)$ that goes with the propagation of the vortex beam (Biton et al., 2021).

The vortex beams have unique properties that make them useful in many areas, such as optical manipulation, optical tweezers, electromagnetic scattering, imaging, and trapping (Arnold et al., 2015; Kotlyar et al., 2025b). The Lorentz-Gauss vortex beam (LGVB) is one of the most famous vortex beams. The production and transformation of mechanical torque into microparticles signifies a prospective application of LGVBs in the investigation of optical manipulation. It is possible to control the optical features of LGVBs by adjusting the beam configuration factors. Beam intensity distributions of varying types are produced by this technique.

LGVBs have created laser beam technology, which has led to new possibilities in various fields, including optical communications, remote sensing, and environmental optics. LGVB can be produced by solving the homogeneous linear wave equation. Its mathematical representation embraces the expansion of the Lorentz distribution in terms of a linear superposition of Hermite–Gaussian (HG) functions. The helicity of LGVB is regulated by a parameter indicated as l, referred to as the beam topological charge. LGVB is produced when radiation originating from a single mode of a diode laser passes via a spiral phase plate (SPP). The helical wave front of LGVB can be controlled using SPP. The built-in characteristics of LGVB, i.e., zero intensity in the central zone as well as the spiral phase front parameter, make it superior to that of LGB.

OAM is a relatively new field, with the concept of vortex beams conveying OAM, and it came into the limelight through the 1992 breakthrough research paper of Allen and his co-researchers (Allen et al., 1992). As the optical researchers have investigated the creation, propagation, and detection of OAM in vortex light beams, the number of research publications in the field has increased dramatically (Padgett, 2014; Padgett and Bowman, 2011; Yao and Padgett, 2011). Potential applications owing to the unique features of spatially structured light have also sparked the interest of optical researchers (Arfan et al., 2022; Arfan et al., 2023c, Arfan et al., 2023d).

As light control and OAM reception and separation capabilities improve, OAM-based laser systems will surely bring about a major technological revolution. Chiral materials are getting better at controlling and changing light, especially when they get and separate OAM. Consequently, researchers investigated the transmission and propagation characteristics of vortex beams in relation to their OAM within metamaterials (Qusailah et al., 2023). The interaction of vortex light beams with chiral materials during their propagation across free space renders the subject intriguing in the context of materials optics (Qiu and Liu, 2024).

It is essential to make a clear distinction between the LGVB that we explored in our manuscript and other vortex beams that have already been analyzed, such as the well-known Laguerre-Gauss Vortex beam, Gaussian Vortex beam (GVB), or Bessel vortex beams (BVBs), etc.

The use of the LGVB model is vital and offers a unique advantage. The LGVB is considered a better way to designate the output from some real-world avenues, specifically high-power single-mode diode lasers, which have a very angular spread that the basic Gaussian model doesn’t comply with. The LGVB’s profile comprised two functions, i.e., Lorentzian and Gaussian. It means that the decay rate of LGVB’s is quite slower in the far zone than the LG beams. This difference is considered an important factor for various applications, which comprise propagation at long distances, in turbulent media, and in complex media such as that of chiral medium, where the model of LGVB represents the improved beam’s characteristics (Zhou and Ru, 2013).

The intensity profile of a Laguerre-Gaussian vortex beam is constructed by a Gaussian function multiplied by a Laguerre polynomial, which results in a curve of bell shape for radial mode $p=0$ or a ring type for $l\ne 0$ whose end decays swiftly. For $p=0=l$, the Laguerre-Gaussian vortex beam transforms into a fundamental Gaussian beam (Arfan et al., 2023b).

A Gaussian vortex light beam (GVLB) is characterized by a Gaussian intensity profile and a helical wave front owing to dependence on an $\exp \left(il\phi \right)$ azimuthal phase. Its orbital angular momentum (OAM) characteristics are modeled by the helical wave front (Kovalev et al., 2020). Bessel vortex beams (BVBs) are a specific category of Bessel beam that carries OAM (Sahin et al., 2015). Its vortex features are maintained by the $\exp \left(il\phi \right)$ term. Its intensity profile is modeled by the Bessel function of the first kind.

Optical researchers are interested in chiral materials because of their possible optical properties (Liu and Zhao, 2014). They possess distinctive characteristics owing to their significant interaction with electromagnetic fields, a property absent in naturally occurring materials. Taking into account the basic shapes and uneven properties of materials in electromagnetics and optics, this leads to a very interesting phenomenon called chirality. Chirality leads to the creation of chiral media composed of asymmetric entities exhibiting anisotropic properties. As a light beam moves across chiral media, its left circularly polarized (LCP) and right circularly polarized (RCP) parts split.

Chiral medium materials respond differentially to right-handed and left-handed CP light (Wang et al., 2009). When a light beam hits the chiral material, its natural optical anisotropy shows optical activity. Chiral mediums are important in many areas of nonlinear optics because they have special properties include polarization rotation, negative index of refraction, and circular dichroism (CD). The characteristics of various vortex beams in various media, including chiral medium, gradient index, atmospheric turbulence, turbid medium, aerosol particles, chiral scatterer, dispersive medium, and plasma, have been the subject of several investigations (Razzaz and Arfan, 2025; Qusailah et al., 2024; Liu et al., 2018; Khanom et al., 2024; Arfan et al., 2023a; Woźniak et al., 2019; Borda-Hernández et al., 2015; Lin et al., 2018).

The propagation characteristics of several fundamental and structured-shaped light beams have been broadly inspected in free space and traditional dielectric media. To our comprehensive understanding, this is certainly the primary document comprised of analytical as well as numerical studies on the propagation features of the LGVB within a chiral medium. Furthermore, less consideration has been given to the electromagnetic interaction of intricate beam profiles with unusual electromagnetic materials, which are considered among metamaterials. Unambiguously, preceding investigations have mainly focused on studying the characteristics of simpler beams, like that of basic Gaussian and Laguerre-Gaussian beams, within chiral media characterized by the medium’s handedness and induced coupling between electromagnetic fields. To our understanding, the propagation features of the LGVB, which possesses OAM, remain fully unmapped within this exceptional environment of electromagnetics in a chiral medium. The findings of the work not only close this gap but also establish a framework to understand the combined effects of the beam’s non-traditional amplitude and the chirality on beam divergence, field structure, and evolution.

Nevertheless, to the author’s comprehensive knowledge, up to now, there has been no documentation on the propagation characteristics of LGVB within a chiral medium. Subsequently, exploring the LGVB interaction with chiral mediums becomes quite important for expressing the scattering, propagation, and transmission of various vortex beams through diverse scattering media. On combining the model of the LGVB field with the chiral medium and the use of the extended HFDI for an analytical solution characterizes a noteworthy, previously unmapped work. This novelty validates the proposed work and is quite indispensable to exactly capture the interaction of LGVB within chiral media.

This article examines the propagation of LGVBs in chiral media because of their increasing optical capabilities and is categorized as follows. The theoretical model and interaction of LGVBs with the chiral medium are described in Section 2. Moreover, the normalized LGVB intensity is calculated. In Section 3, the numerical results regarding the propagation of LGVBs by a chiral medium are computed and discussed. Finally, Section 4 presents a comprehensive conclusion.

2. Theoretical Analysis

The development of LGVBs has been an important milestone in laser technology, allowing for unprecedented capabilities in diverse optical fields (Ni and Zhou, 2012). In optics, the LGVBs take the shape of a vortex beam by combining two different kinds of beams, i.e., the Lorentz and the Gaussian. LGVBs have zero intensity and a twisted phase profile. The potential applications of LGVB make it a promising candidate in optics and electromagnetics. The distribution of field amplitudes in the source plane, considering the LGVB, can be expressed as (Zhou, 2014), represented in Eq. (1)

where ${\rho }_0=\sqrt{x_0{}^2+y_0{}^2}$, with $x_0$ & $y_0$ express transverse Cartesian coordinates.

The l parameter indicates the beam topological charge or OAM mode index. This factor produces twisting in the LGVB. The beam waist is denoted by $w_0$. The parameters $w_{0x}$ and $w_{0y}$ represent the beam waist radius of the Lorentzian part in the x-direction and y-direction, respectively.

The Lorentzian part, which interprets the high angular spread, constructs the LGVB as an additional physically realistic and real-world beam model for precisely indicating the propagation features of diode laser beams in intricate systems such as the chiral medium, where precise boundary conditions (BCs) and beam shapes are key factors for precise investigation using the extended HFDI formula and ABCD matrix system (Zhou, 2009).

The intensity distribution and phase of the LGVBs for various values of OAM are shown in Fig. 1. It is established that when the OAM mode index is zero, total intensity at the beam center is nonzero, while no result appears for the phase distribution. For $l=0$, the LGVB transforms into basic LGB. Its angular dependence vanishes, and LGVB loses its vortex characteristics. Furthermore, its ring-type structure does not hold. However, for $l\ne 0$, the intensity pattern and phase show variations in their behavior. At $l=1$ or 2, LGVB maintains its doughnut profile, and also its intensity distribution gains symmetry about to the optical axis of the beam’s propagation. The selection of the OAM mode index mainly determines the degree of symmetry in the distribution of LGVB intensity and phase. An increase in the l parameter causes the annulus to expand and move outside. By augmenting the OAM mode index of an LGVB, the phase singularity enhances, consequently causing the intensity of the vortex beam to redistribute itself into a wider slice-shaped pattern. Changing the OAM mode number flips the phase profile while vortex beam intensity remains unchanged. The coupling of OAM with the chiral medium makes the phase profile much more sensitive toward dispersion and optical characteristics, which consequently intensifies the polarization rotation and field attenuation between the LCP and RCP wave components.

Close

Fig. 1.

(a-f) Plots of field intensity (top) and phase profile (bottom) of LGVB: l=0,1, and 2 (left to right) showing the screening of the noncircular rings and its influence on the phase pattern.

Export to PPT

In Fig. 2, the interaction of the LGVB with its propagation about the z direction within the chiral medium is sketched. The chirality of the chiral medium impacts the LGVB field. The shape of the response function—whether it’s exponential, Gaussian, or any other profile, has a significant impact on wave propagation in chiral media. In a chiral medium, the RCP beam and the LCP beam characterize the entire LGVB field. These two parts of the light beam travel at different phase velocities and follow different trajectories.

Close

Fig. 2.

Schematic illustration of the propagation model of LGVB through a chiral medium.

Export to PPT

As illustrated in Fig. 2, LGVB propagates in a chiral medium along the z-axis as its propagation direction. Consequently, it is appropriate to employ the extended HFDI formula besides the optical ABCD matrix system to modify its propagation dynamics. It is expressed as (Yang et al., 2020), represented in Eq. (2)

Here A, B, and D indicate various elements of the transfer matrix. Consequently, the total ABCD matrix considering propagation of LGVB through a chiral medium can be expressed as (Zhuang et al., 2012), represented in Eqs (3 and 4)

Here, $n_R=\frac{n_0}{1-n_{0}k\gamma }$ and $n_L=\frac{n_0}{1+n_{0}k\gamma }$. These represent the refractive indices of the right-handed and left-handed chiral polarized LGVBs. $n_0$ indicates the original refractive index regarding the chiral medium, while $\gamma$ represents chirality.

Implementing both the extended HFDI formula and the optical ABCD matrix system approach to investigate the propagation features of an LGVB in a chiral medium results in the combination of the vortex characteristics and chirality coupling. This contributes toward the findings of the present work. The HFDI formula and the optical ABCD matrix system are used together since they play harmonizing roles. The ABCD matrices offer a compact and paraxial description of the transformation of the beam’s configuration parameters over optical elements, while the extended HFDI formula aids in propagating the field of LGVB for inhomogeneous media as well as free space. The LGVB treated in this study remains within the paraxial regime appropriate to optical and chiral-medium studies. Thus, the ABCD matrix system provides the correct paraxial mapping of the LGVB, and the extended HFDI formula then applies that mapped field, yielding an accurate field evolution that preserves the vortex characteristics.

The HFDI formula and the ABCD optical matrix formulation are inferred under the paraxial approximation. Whereas LGVBs can show non-paraxial characteristics, the use of this united model undertakes that the beam’s components chiefly meet the paraxial approximation. This statement is standard regarding analytical work, including tangible structure-shaped beams propagating over larger distances. However, if there are any non-paraxial effects present, then these can be treated as higher-order effects. This mode is acceptable, as it is merely an existing theoretical framework that can manipulate both the intricate LGVB profile and the general optical matrix system.

By using the given integral, represented in Eq. (5)

and by plucking Eqs. (3-4) into Eq. (1) and then substituting this modified expression into Eq. (2), yields an expression concerning LGVB by using an ABCD optical system matrix for the chiral medium. It can be expressed as Eq. (6)

${\sigma }_{2m}$ and ${\sigma }_{2n}$ denote the beam expanded coefficients.

also Eq. (7)

and Eq. (8)

where

$\begin{array}{c}{a}_{x,c}=\left(\frac{1}{2w_{0x}^2}+\frac{1}{w_0^2}+\frac{ikA}{2B_c}\right),\\ a_{y,c}=\left(\frac{1}{2w_{0y}^2}+\frac{1}{w_0^2}+\frac{ikA}{2B_c}\right),\text{and}c=L\text{or}R.\end{array}$

The chirality of the medium causes the initial LGVB field to divide into two components, i.e., RCP and LCP. The phase velocities of the RCP and LCP waves are different because their refractive indices are different. Thereby, the total field intensity of the LGVB within a chiral medium having unique propagation characteristics can be denoted as Eq. (9)

where $I_{interference}$ expresses the term for interference effect. It specifies the left and right beam fields in the chiral medium. It can be written as Eq. (10)

Here, complex conjugates of beam fields are denoted by * terms.

Eq. (6) and Eqs. (9)–(10) characterize the analytical theory, which can suitably be employed to develop the LGVB beam intensity for propagation within the chiral medium. It comprises a dependent function of the LGVB configuration parameters and the chirality of the medium.

The proposed model offers substantial fresh theoretical insights that have not been exposed by preceding research. The profile of LGVB presents a more diverse distribution of angular spectrum than that of a basic Gaussian or Laguerre-Gauss vortex beam. This exceptional signature dictates a tangible and noticeable interaction with the two states produced in the chiral medium (LCP and RCP waves). The model of the work also explains the transfer effects of the topological charge. The chirality parameter $\gamma$, besides the Lorentzian factor, tunes the rotational effects in chiral media. The medium’s handedness characterizes the beam’s twisting and splitting.

3. Results Analysis

This section depicts and investigates the numerical results to describe the pattern of propagation characteristics taking various yet specific values of OAM mode number, propagation distance, beam waist radius, and chirality parameter. The incident beam operating wavelength is $\lambda =632.8 nm$. The rest of the parameters regarding computation are considered as $w_0= 1.0 mm$, $w_{0x}=w_{0y}=2.0 mm$, $\gamma =\frac{0.16}{k}$, and $l=1$(unless otherwise mentioned). All the numerical results have been analyzed for the intensity distributions of LGVB in the presence of chiral material.

Fig. 3 depicts the influence of the beam topological charge l on the intensity of LGVB for $(l=0,1,2,\&3)$. The beam intensity associated with LGVB decreases owing to the interaction of the beam’s topological charge within the chiral medium. At the beam center of the Lorentz-Gauss beam in the focus plane, the intensity along the optical axis vanishes during its propagation for higher values of the beam’s topological charge l. As l augments, the beam’s intensity during its propagation at its focal plane center represses. The transverse width of LGVB also elevates in relation to the l parameter. The LGVB transforms into a typical LGB as the beam mode number becomes 0. The intensity of LGVB decreases with the elevation of the beam topological charge. It is also noted that by enhancing the beam’s topological charge (l), the peaks of the scattered field slightly shift downward. Moreover, for the zeroth-order beam mode index, the size of the central lobe for the scattered field becomes significant. The intensity for LGVB with a high-order OAM mode number is noticeably smaller than that for LGB, because the beam is illuminated on the optical axis, and the beam intensity becomes null at the center of the OAM beam. Consequently, LGVB’s penetration ability would be better than LGB’s in the same chiral medium.

Close

Fig. 3.

Intensity profiles of LGVB propagation in a chiral medium by varying the beam OAM, i.e., (l=0…2) for LCP, RCP, and total intensity distributions of the light beam for two chiral factors (a1, b1, c1) chirality=0.16/k, and (a2, b2, c2) chirality=0.32/k.

Export to PPT

Fig. 4 displays the effect of the waist radius on the propagation of LGVB within chiral media. Due to the increase in the magnitude of the waist radius, the amplitude of the intensity distribution increases gradually. The waist radius is varied as $w_0=1.00 mm, 1.25 mm, 1.50 mm$, respectively. Generally, increasing the beam waist radius enhances the beam width, which successively boosts the optical effects (scattering + absorption). Increasing the waist radius of the LGVB results in enhancing the beam to be wider and less divergent. Its output then augments the LGVB intensity within the chiral medium. A good match between the beam width and the wavelength of the incoming beam improves the transmission properties. The beam waist radius regulates the pattern of beam intensity, therefore choosing the right one makes propagation more stable and well-adjusted. When the beam waist increases, the light becomes less focused, which stops the beam from spreading out and bending. LGVB propagation becomes optimum when the beam waist radius becomes equal to that of the operational wavelength of LGVB. The OAM of the LGVB, as well as its waist radius, can tune the beam structure and energy distribution. Optimizing the configuration source parameters of the beam sustains vortex characteristics over extended distances. Additionally, adjusting the aforementioned factors can enhance the vortex beam’s potential for self-healing characteristics.

Close

Fig. 4.

Intensity profiles of an LGVB propagation in a chiral medium by varying the beam waist radius, i.e., (1.00 mm,1.25 mm,1.50 mm) for LCP, RCP, and total intensity distributions of the light beam for two sets of chiral factors (a1, b1, c1) chirality=0.16/k and (a2, b2, c2) chirality=0.32/k.

Export to PPT

To visualize the intensity distribution of the chiral medium with chirality parameters, the intensity distribution with different $\left(\gamma \right)$ values is depicted in Figs. 5(a-c), and 5(d), respectively. The intensity distribution reduces as the chirality of the chiral medium increases. The variation of the chirality parameter includes chiral effects in the scattering phenomena, and these are distinct from the plane wave or LGB. The characteristic pattern shows that the chiral medium can be used to tune the intensity distribution for the LGVB. Increasing the chirality parameter influences the field distributions, which contributes toward impacting the scattering rate. Furthermore, augmenting the chirality within the chiral medium boosts the optical activity, which causes enhanced coupling between the left and right components of the electric field of the LGVB. This results in stronger phase shift and polarization rotation during vortex beam propagation; it reduces the transmitted intensity of LGVB owing to amplified energy exchange with the chiral medium and conceivable CD losses. Within the realm of a chiral medium, escalating the chirality factor reinforces the coupling of asymmetric type between the OAM of LGVB’s and the electromagnetic response of the handedness of the chiral material. Thereby, this augmented interaction between the OAM field and chirality generates amplified phase modulation and polarization rotation in addition to the field attenuation due to CD. Therefore, a part of the energy of the vortex beam transforms into non-propagating modes, decreasing the intensity of LGVB during its forward propagation as well as intensifying the chiral effects. As the chirality parameter increases, the intensity distributions transform into four boosted petals that are centered.

Close

Fig. 5.

Total intensity distribution profiles of a LGVB propagation in a chiral medium by varying the chirality as follows: (a) chirality= 0.16/k, (b) chirality=0.28/k, (c) chirality= 0.35/k, and (d) chirality= 0.45/k.

Export to PPT

Fig. 6 depicts the influence of the LGVB’s propagation distance on the chiral medium’s intensity. On varying the propagation distance (z) for LGVB in a chiral medium, the total intensity distribution as a sum for various beam polarization types, such as RCP, LCP, and interference expression, is displayed in Fig. 6. The intensity distribution of the LGVB at various z parameters depend on the field distribution of its different categories, such as left and right. For the source plane, consequently, the intensity profiles of RCP, LCP, interference expression, and total LGVB remain the same and preserve their profile. However, as propagation distance increases $(z>0)$, the LGVB profile changes and keeps its intensity maximum at the beam center, bounded with an intense ring during its propagation within the chiral medium. Here, $z_R=\frac{1}{2}\left(k{w}_0^2\right)$ represents the Rayleigh length.

Close

Fig. 6.

Total intensity distribution profiles of a LGVB propagation in a chiral medium by varying the propagation distance z, i.e., (z=2z_R,3z_R,4z_R) for two sets of chiral factors (a1, b1, c1) chirality=0.16/k and (a2, b2, c2) chirality=0.28/k.

Export to PPT

In Fig. 7, the intensity distribution of propagation of LGVB for the beam waist radius of the Lorentzian part in both the x-direction and y-direction through a chiral medium is shown. Increasing the Lorentzian waist $\left(w_{0x} \& w_{0y}\right)$ affects the widths and central intensity of LGVB. Augmenting $w_{0x}$ and $w_{0y}$ results in broadening the intensity profile along both axes, i.e., $\left(x \& y\right),$ but weakening the total intensity of LGVB in a chiral medium. The profile of LGVB indicates that the total profile of intensity is focused in the middle due to the chirality parameter. In addition to chirality and waist radius, dispersion effects in a chiral medium also significantly influence the propagation of vortex light beams. Augmenting the chirality of the medium amplifies dispersion; thus, it causes the different OAM-based polarization components of LGVB to achieve unique phase velocities. The overall effect is less coherent forward intensity and more interaction between the chiral medium and OAM-based LGVB. Here, numerical results depict that the intensity distribution is greatly affected by the $w_{0x}$,$w_{0y}$, and chirality of the chiral medium. Numerical studies also reveal that chirality and Lorentzian waist size can broadly regulate beam form.

Close

Fig. 7.

Total intensity distribution profiles of a LGVB propagation in a chiral medium by varying the Lorentzian waist, i.e., (w_0x=w_0y=1.0 mm,2.0 mm,3.0 mm) for two sets of chiral factors (a1, b1, c1) chirality=0.16/k and (a2, b2, c2) chirality=0.28/k.

Export to PPT

Numerical findings demonstrate that the intensity of LGVB is highly sensitive to the source beam characteristics in addition to the chirality of the medium. The size and intensities of the central region can be adjusted to regulate the LGVB profile. This model will also be helpful to understand the evolution of the beam waist, OAM integrity, and beam’s divergence when the beam propagates over a chiral environment. This provides a theoretical and physical insight into optical systems designed for various applications, i.e., chiral sensing and OAM-based communication systems.

4. Conclusions

This research emphasizes on examining the intensity distribution of the LGVB, which propagates across ABCD optical systems rooted in the extended HFDI principle within a chiral medium. Furthermore, numerical simulations are performed to demonstrate the impact of the LGVB setup parameters on the intensity distribution within the chiral medium. Numerical results demonstrate the propagation characteristics of the LGVB in a chiral media. This article offers an extensive analysis of the pattern of field intensity distributions within a chiral material. Numerical findings indicate that beam parameters and chirality dictate the features of the LGVB during its propagation in the chiral medium. The characteristics of chiral scatterers can be tuned by varying the LGVB configuration parameters, such as beam topological charge l, propagation distance z, and beam waist width $w_0$, and chirality $\gamma$. As LGVB propagates, its Gaussian characteristics seem to decay. When the vortex LG beam mode index l is set to zero, the LGVB is transformed into a Lorentz-Gauss beam with a Gaussian profile. The numerical results indicate that the intensity of the chiral material medium for the LGVB increases as the OAM mode number increases. By adjusting the configuration parameters of LGVB, i.e., $l \& w_0$ as well as the chirality $\gamma$, the beam intensity for the chiral medium can be tuned very well. Varying the LGVB parameters results in unique characteristics toward field intensity. Augmenting the beam waist radius leads to an enhancement in the beam intensity; conversely, this results in a decrease in the beam chirality parameter. LGVB exhibits various ring-shaped intensity structures and a spiral profile for phase, which notably influences beam intensity during its propagation for chiral medium. The obtained results will provide a favorable avenue to better investigate and understand the interaction of vortex beams with various chiral material structures. The results of the work exceed previous studies in the following ways: It offers an analytical framework considering LGVBs within chiral media. It also provides fresh chirality-induced effects on the dynamics of LGVB. It will be advantageous to study and analyze the characteristics of metamaterial structures for various shaped vortex beams.