## Abstract

We present a unified and extended perspective of Bessel beams, irrespective of their orbital angular momentum (OAM)—zero, integer or noninteger—and mode—scalar or vectorial, and LSE/LSM or TE/TM in the latter case. The unification is based on the integral superposition of constituent waves along the angular-spectrum cone of the beam, and enables us to describe, compute, relate, and implement all Bessel beams, and even other types of beams, in a universal fashion. We first establish the integral superposition theory. Then, we demonstrate the existence of noninteger-OAM TE/TM Bessel beams, compare the LSE/LSM and TE/TM modes, and establish useful mathematical relations between them. We also provide an original description of the position of the noninteger-OAM singularity in terms of the initial phase of the constituent waves. Finally, we introduce a general technique for generating Bessel beams using an adequate superposition of properly tuned sources. This global perspective and theoretical extension may be useful in applications such as spectroscopy, microscopy, and optical/quantum force manipulations.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

Electromagnetic Bessel beams represent a fundamental form of structured light. They are *localized waves* [1,2] with transverse Bessel function profiles that carry *orbital angular momentum (OAM)* along their propagation axis. Localized waves were first reported as soliton-like waves by Bateman [3], next derived for the Bessel case as the TE/TM solutions to the cylindrical wave equation by Stratton [4], and then generalized as solutions to a class of equations admitting “waves without distortion” as solution by Courant and Hilbert [5]. They are waves that propagate without spatial dispersion (or diffraction) and without temporal (or chromatic) dispersion. Their energy is, thus, uniformly confined and invariant perpendicular to and along, respectively, their direction of propagation. OAM is a beam property whose macroscopic manifestation is an isophase surface that has the form of a vortex along the axis of the beam. It may be integer or noninteger. In the former case, the wave has the transverse phase dependence ${{e}^{{in\phi}}}$ ($n \in {\mathbb Z}$), corresponding to an OAM of $n\hbar$ per photon [6], while in the latter case the wave is made of a superposition of integer-OAM waves that combine so as to produce noninteger OAM per photon [7]. This property association of localization and OAM confers to Bessel beams specific capabilities for manipulating light that may be exploited in diverse applications, such as nanoparticle guiding [8], orbiting and spinning [9], trapping [10,11] and tracting [12–14], spectroscopy [15], microscopy [16], and quantum key distribution [17]. Figure 1 shows a general classification of Bessel beams that pertain to the sequel of the paper.

Bessel beams are the simplest form of light OAM after the Laguerre–Gauss beams and the most studied localized waves. They are monochromatic beams with a transverse amplitude pattern that follows Bessel functions of the first kind, ${J_n}(\alpha \rho)$, multiplied by the phase function ${{e}^{{in\phi}}}$ ($n \in {\mathbb Z}$) or combinations of such waves in the noninteger-OAM case. Their simplest forms are the *scalar* Bessel beams, also existing in acoustics and restricted to the paraxial approximation in optics. Such beams were first, to the best of our knowledge, experimentally showed by Durmin [18] for $n = 0$ (no-OAM), while their integer-OAM carrying versions were reported shortly thereafter [19]. Noninteger-OAM scalar Bessel beams were reported only 15 years later [20]. In the case of electromagnetic waves, such as light, Bessel beams are *vectorial* and may be either longitudinal section electric (LSE)/longitudinal section magnetic (LSM) or TE/TM, as mentioned in Fig. 1. Vectorial Bessel beams, the only exact forms of electromagnetic Bessel beams, were introduced in LSE/LSM [21]. Integer-OAM forms in [22] (51 years after their TE/TM form introduction by Stratton [4]) and is generalized to the noninteger case in [23]. Finally, Bessel beams of various complexities have been generated by an axicon illuminated with a Laguerre–Gaussian beam [24], by a single helical axicon (azimuthal phase plate) [25], by a spatial light modulator [20,26], by a leaky-wave antenna [27], and by a metasurface [28].

This paper fills up some fundamental gaps existing in the literature on Bessel beams. Specifically, it reports on (1) a unified representation of Bessel beams, (2) a demonstration of the existence of noninteger TE/TM OAM Bessel beams and their detailed characterization, and (3) a generic and efficient approach for the practical implementation of Bessel beams.

#### A. Unified Representation of Bessel Beams

Bessel beams have been described either in the spatial direct (${\textbf r}$) domain [4,19,22,23] or in the spatial inverse (${\textbf k}$)—or spectral—domain [18,20,29]. While the spatial approach immediately describes the nature of the beam, it is restricted to the beam in question. In contrast, the spectral approach, although less explicit, allows generalizations and provides insight into simple generation techniques. The spectral approach is therefore more powerful. However, it has only been applied to the scalar case in the context of Bessel beams. This paper presents a general electromagnetic vectorial spectral formulation that applies to all Bessel beams (scalar and vectorial, with integer and noninteger OAM, LSE/LSM and TE/TM, and all related combinations) and even to other types of conical (e.g., Mathieu, Weber) and nonconical beams (e.g., Gauss–Laguerre and hypergeometric Gaussian beams), hence providing novel perspectives and possibilities.

#### B. Existence and Characterization of TE/TM with Noninteger OAMs

Whereas vectorial TE/TM Bessel beams [4,19,27,30] and scalar Bessel beams with arbitrary OAM [20,31] have been separately described in the literature, we report here for the first time, to the best of our knowledge, the existence of TE/TM Bessel beams with noninteger OAMs and present a detailed description of these new modes, showing superior focusing capabilities [32] than their LSE/LSM counterparts in addition to richer optical force opportunities.

#### C. Generic and Efficient Practical Implementation

Both scalar and vectorial Bessel beams have been experimentally generated using diverse technologies, all suffering from some drawbacks. The technologies for scalar beams include the helical axicon [25], which has the disadvantage of being a bulky lens, and spatial light modulators [20,26], which have restricted polarization flexibility due to their inherent phase-only and magnitude-only restriction. The technologies for vectorial beams include leaky-wave antennas [27], which can produce specific low and integer-order OAM TE/TM beams, and metasurfaces [28], which are the most attractive technology but have been conceived in the direct domain and have, hence, not yet benefited from the generality of the vectorial spectral-domain formulation [33] presented in this paper. Based on this approach, we present here an efficient and generic approach for the practical implementation of any Bessel beam (and other conical or nonconical beams).

## 2. SCALAR SOLUTION

A Bessel beam may be generally described in a linear system by the scalar function of space ($x$, $y$, $z$) and time ($t$),

where ${\psi _\nu}({\phi _{G}})$ represents a continuous set of waves that propagate at the angle ${\phi _{G}}$ along a circular cone towards its apex, so as to form a transverse interference pattern in the form of a Bessel function, as illustrated in Fig. 2, and where $\nu$ is equal to the OAM of the beam when it is integer or half-integer (see Supplement 1). These waves have the mathematical form with the oblique wave vectorIn these relations, $w(\xi)$ is the transverse apodization of each wave with respect to its propagation direction, ${\textbf k}({\phi _{G}})$, with $\xi$ being the radial variable of the corresponding local coordinate system, ${\rm frac}(\cdot)$ is the fractional part function [34], and ${\phi _{{G},0}}$ is related to the position of the phase singularity for $\nu \in {\mathbb R}\backslash {\mathbb Z}$, as will be seen later. Note that Eq. (2d) represents a sawtooth function of ${\phi _{G}}$ that reduces to ${\gamma _\nu}({\phi _{G}}) = \nu {\phi _{G}}$ for ${\phi _{G}} \in [0,2\pi [$ and ${\phi _{{G},0}} = 0$. Also note that possible loss may be simply accounted for by making the ${\textbf k}$ vector complex. Finally, note that the cone in Fig. 2 corresponds to the angular spectrum of the formulation in Eq. (1) with Eq. (2) [35].

Practically, the waves ${\psi _\nu}({\phi _{G}})$ must be spatially limited [e.g., Gaussian cross-sectional apodization $w(\xi)$], but they will be initially considered as plane waves [$w(\xi) = {\rm const.}$]. [36] because, as we shall see, an analytical solution can be derived for Eq. (1) in this case. Moreover, their azimuthal separation ($\Delta {\phi _{G}}$) is infinitesimally small, so that they effectively merge into the azimuthal continuum corresponding to Eq. (1). Finally, these waves may be temporarily considered as a scalar, but they will later be seen to represent any of the components of fully vectorial electromagnetic waves.

The Bessel beam superposition in Eq. (1) is plotted in Fig. 3 using 20 equi-spaced plane waves for ${\psi _\nu}({\phi _{G}})$ in Eq. (2), with the top row showing the superposition of the maxima and minima, and the bottom row plotting the superposition of the actual waves with continuous sinusoidal gradients. This figure shows that a perfectly smooth Bessel pattern is obtained around the axis of the beam with a restricted number of constituent waves [37] and illustrates the increasing structural complexity of the Bessel beam without OAM ($\nu = 0$), with integer OAM ($\nu = 1$), and with noninteger OAM ($\nu = 1.5$).

Substituting Eq. (2) with $w(\xi) = {A^{{\rm PW}}}$ (const.) into Eq. (1), simplifying the resulting integral (see Supplement 1, Section 3), and decomposing the field into its transverse-dependence and longitudinal/temporal-dependence parts as ${U_\nu}(\rho ,\phi ,z,t) = {U_{\nu , \bot}}(\rho ,\phi){{e}^{i(\beta z - \omega t)}}$, then yields

For $\nu = n \in {\mathbb Z}$, the integral in Eq. (3) has a tabulated closed-form primitive [38], leading to ${U_{n, \bot}} = 2\pi {i^{- n}}{{e}^{- in{\phi _{{G},0}}}}{A^{{\rm PW}}}{J_n}(\alpha \rho){{e}^{{in\phi}}}$, which is the conventional integer-OAM Bessel solution for circular cylindrical problems. In contrast, for $\nu \in {\mathbb R}\backslash {\mathbb Z}$, the integral does not admit a simple primitive, and we must devise a strategy to lift this restriction so as to find the most general beam solution. This can be accomplished by the following procedure. First, we replace the generally non-periodic ($\nu \in {\mathbb R}\backslash {\mathbb Z}$) complex exponential function ${{e}^{i\nu \phi ^\prime}}$ in Eq. (3) by its expansion in terms of the complete and orthogonal set of *periodic* ($\nu = m \in {\mathbb Z}$) complex exponential functions ${{e}^{im\phi ^\prime}}$ (see Supplement 1) [39], i.e.,

Then, we substitute Eq. (4) into Eq. (3), which leads to

Finally, we eliminate the integral by applying the Bessel identity $\int_0^{2\pi} {{e}^{- i\alpha \rho \cos (\phi - \phi ^\prime)}}{{e}^{im\phi ^\prime}}{\rm d}\phi ^\prime = 2\pi {i^{- m}}{J_m}(\alpha \rho){{e}^{{im\phi}}}$ [40] and find the general solution to Eq. (5), with the singularity phase parameter ${\phi _{{G},0}}$ appended to the corresponding expression given in [20], in the closed form

with the complex weighting distributionEquation (6) contains the integer-OAM Bessel solution, since $\nu = n \in {\mathbb Z}$ transforms Eq. (6b) into $A_m^{{\rm BB}} = 2\pi {i^{- m}}{{e}^{- in{\phi _{{G},0}}}}{A^{{\rm PW}}}{\delta _{{mn}}}$, which reduces the sum in Eq. (6a) to the single term ${U_{n, \bot}} = 2\pi {i^{- n}}{{e}^{- in{\phi _{{G},0}}}}{A^{{\rm PW}}}{J_n}(\alpha \rho){e^{{in\phi}}}$. But, it also contains noninteger-OAM ($\nu \in {\mathbb R}\backslash {\mathbb Z}$) solutions, where the satisfaction of the circular periodic boundary condition is realized by a *superposition* of integer-OAM Bessel waves with proper phase (${{e}^{{im\phi}}}$) and weighting coefficients [$A_m^{{\rm BB}}(\nu ,{\phi _{{G},0}})$]. In this noninteger-OAM case, the sum in Eq. (6a) must be practically truncated to an integer $m = \pm M$ that is large enough to provide a satisfactory approximation of the Bessel beam.

Equation (6a) reveals that the parameter $\alpha$ of the cone in Fig. 2 corresponds to the spatial frequency of the Bessel pattern. Since this parameter is proportional to the axicon angle, $\delta$, according to Eq. (2c), we find that increasing the aperture of the cone in the integral construction of Eq. (1) compresses the Bessel ring pattern towards the axis of the beam.

Figure 4 plots the magnitude and phase of the Bessel beam given by Eq. (6) for different integer and noninteger OAMs [41]. The cases $\nu = 0$, 1, and 1.5 correspond to the instantaneous field plots in Fig. 3. The OAM-less beam $\nu = 0$ has simultaneously azimuthally symmetric magnitude and phase. The integer beams $\nu = n \in {\mathbb N}$ have azimuthally symmetric magnitudes but asymmetric phase (OAM). The beams $\nu \in {\mathbb R}\backslash {\mathbb Z}$ have simultaneously asymmetric magnitude and phase. The parameter ${\phi _{{G},0}}$, which is zero here, corresponds to a dummy initial phase of the integer OAM and the position of the discontinuity of the noninteger OAM in the individual waves, with increasing ${\phi _{{G},0}}$ clockwise rotating the asymmetric magnitude of the noninteger-OAM pattern (see Supplement 1).

## 3. VECTORIAL SOLUTION CONSTRUCTION

The general [42] scalar Bessel solutions described above are restricted to acoustic waves, quantum waves, and vectorial waves under special conditions such as the paraxial condition ($\delta \ll \pi /2$) in the electromagnetic case. On the other hand, they fail to describe Bessel beams with a large axicon aperture (angle $\delta$ in Fig. 2), which are relevant to applications such as microscopy and optical force manipulations. Therefore, we extend here the previous scalar generalization to the vectorial case.

For the scalar case, we have established two alternative solutions: the integral solution of Eq. (1) and the analytical solution of Eq. (6). In the present extension to the vectorial case, we shall restrict our treatment to the integral approach [43], because it provides more insight into the physical nature of the beam, and because it will constitute the basis for the practical implementation to be discussed later. We shall still assume a plane wave construction [$w(\xi) = {A^{{\rm PW}}}$ (const.) in Eq. (2b)] for simplicity.

As indicated in Fig. 1, the vectorial Bessel beams can be ${{\rm LSE}_i}/{{\rm LSM}_i}$ with $i = \{x,y\}$ or ${{\rm TE}_z}/{{\rm TM}_z}$, where the subscript denotes the field component that is zero. In the former case, the electric/magnetic transverse field component that is nonzero is set as the scalar Bessel beam solution ${U_\nu}$ in Eq. (1), while, in the latter case, it is the magnetic/electric longitudinal component of the field that is set as ${U_\nu}$, and the other components are found from Eq. (2) via Maxwell equations (see Supplement 1).

The ${{\rm LSE}_y}$ (${E_y} = 0$, ${E_x} = {U_\nu}$) and ${{\rm TM}_z}$ (${H_z} = 0$, ${E_z} = {U_\nu}$) solutions are, respectively, given by (see Supplement 1)

A detailed investigation of these solutions reveals that the axicon angle ($\delta$) distinctly affects the LSE/LSM and TE/TM modes (see Supplement 1). In both cases, increasing $\delta$ compresses the Bessel ring pattern towards the axis of the beam; however, this variation also breaks the symmetry of the transverse LSE/LSM patterns, even for $\nu = 0$, whereas it leaves the TE/TM pattern azimuthally symmetric. It is also interesting to note that, for a small axicon angle, i.e., $\delta \ll \pi /2$, the ${{\rm LSE}_y}$ modes essentially reduce to their ${E_x}$ and ${H_y}$ components, similar to the scalar form.

Figures 5 and 6 depict the ${{\rm LSE}_y}$ and ${{\rm TM}_z}$ Bessel beams of global order $\nu = 1.5$ corresponding to the solutions of Eqs. (7) and (8), respectively. The vectorial field distributions plotted in the panels (a) and (b) of the Figs. 5 and 6 represent samples of the constituent waves of the integral construction of the beam (Fig. 2). Their strong vectorial nature starkly contrasts with the configuration of the scalar solution, except for the ${{\rm LSE}_y}$ case in the aforementioned axicon limit ($\delta \ll \pi /2$). Note that the electric field of the constituent waves of the ${{\rm LSE}_y}$ mode is linearly polarized in the $x$ direction, while that of the ${{\rm TM}_z}$ modes is radially polarized. Nonzero and noninteger $\nu$ vectorial modes are obtained from their fundamental counterpart by simply setting the $\nu$ parameter in the initial phase of the constituent waves, i.e., ${\gamma _\nu}({\phi _{G}})$ in Eq. (2d), to the desired OAM.

Figure 7 plots time-average Poynting vectors of integer and noninteger ${{\rm LSE}_y}$ and ${{\rm TM}_y}$ Bessel beams. Interestingly, whereas the maxima of the ${{\rm LSE}_y}$ transverse Poynting vector components are superimposed with those of the longitudinal Poynting vector component, the ${{\rm TM}_z}$ transverse maxima are not overlapping the longitudinal maxima. Also notice that the ${{\rm TM}_z}$ longitudinal Poynting vector component for $\nu = n = 1$ does *not* exhibit a null on the beam axis, contrarily to the case of all nonzero-OAM scalar solutions; this is allowed by the fact that the polarization singularities associated with the radial configuration of the constituent plane waves of the TM/TE modes cancel out the phase singularities in this particular case of $\nu = 1$. These various results, with the complementariness of the LSE/LSM–TE/TM modes, and their extension to higher OAMs, illustrate the structural diversity of the vectorial Bessel beams, including horizontal/vertical/right-circular/left-circular LSE/LSM polarization (see Visualization 1 and Visualization 2) and azimuthal/radial/hybrid TE/TM polarization (see Visualization 3), and suggest that they may lead to a wealth of still unexplored opportunities for the optical force manipulation of nanoparticles.

The LSE/LSM and TM/TE electromagnetic vectorial Bessel beams are related by the following relations, which may be easily verified upon comparing Eqs. (7) and (8):

## 4. PHYSICAL IMPLEMENTATION

Several techniques have been proposed for generating Bessel beams experimentally. The main ones are axicon lenses illuminated by a Laguerre–Gauss beam [24], spatial light modulators [26], open circular waveguides with selectively excited modes [44], antenna arrays with a proper phase feeding network [45], metasurfaces illuminated by plane waves [28], and two-dimensional (2D) circular leaky-wave antennas [46]. Unfortunately, these techniques are restricted to simple beams, excessively complex to implement, bulky and expensive, or suffering from poor efficiency.

The unified integral formulation presented in this paper [Fig. 2 with Eq. (1) for the scalar case and Eqs. (7) and (8) for the vectorial case] naturally points to a generation technique that is immune to these issues and that offers in addition a universal implementation framework. Indeed, circularly distributing a set of sources with the phases, amplitudes, and polarizations of the derived modal field solutions (e.g., top panels of Figs. 5 and 6) would exactly and efficiently produce the corresponding Bessel beams, irrespective to their order or complexity.

Specifically, the integral-formulation generation technique consists of the following design steps: (1) select a sufficient number of sources ($N$) to properly sample the desired OAM according to the Nyquist criterion, (2) determine an appropriate beam apodization [$w(\xi)$] for each of the constitutent waves to be radiated by these sources, and (3) adequately set the phase, magnitude, and polarization of each of the sources and orient them so as to launch the constituent waves along a cone with the selected axicon angle ($\delta$). This is mathematically expressed by the formula

where $w({\phi _{G}})$ is the apodization of the constituent waves, ${{\textbf E}^{{\rm PW}}}({\phi _{G}})$ is their plane-wave modal field solution [e.g., Eqs. (7) or (8)], and $\Delta {\phi _{G}} = 2\pi /N$. In the case of a (typical) Gaussian apodization, we have where ${w_0}$ is the waist of the beam, and $({x_ \circ}({\phi _{G}}),{y_ \circ}({\phi _{G}}))$ represents the local conical coordinatesNote that apodization of the plane wave ${{\textbf E}^{{\rm PW}}}$ by the function $w(\xi)$ results in a localization of the beam in a restricted of extent $L = {w_0}/\sin (\delta)$ about the center of the cone at ($z = 0$). Moreover, the discretization of the integral induces a distortion of the Bessel pattern, which grows with the distance from the axis of the beam, as previously explained, so that $N$ may have to be increased to provide a satisfactory beam approximation across the transverse area of interest.

Figure 8 depicts the experimental implementation of the integral-formulation Bessel beam generation. Figure 8(a) represents a direct incarnation of this formulation, which consists of a circular array of laser beams with proper magnitudes, phases, and polarizations, as illustrated in Fig. 8(b). Such an implementation, involving $N$ independent lasers with respective magnitude, phase, and polarization controls, is quite complex and cumbersome. Fortunately, recent advances in metasurface technology suggest the much more practical implementation shown in Fig. 8(c). Indeed, this metasurface-based Bessel beam generator requires only one laser source, while being ideally compact and inexpensive.

The metasurface required in the implementation of the Bessel beam generator depicted in Fig. 8(c) can be easily realized using the latest metasurface synthesis techniques [47]. The simplest implementation strategy would consist of cascading metasurfaces that separately tailor the amplitude, phase, inclination, and polarization of the incident wave in the transverse plane of the system. Specifically, assuming a linearly polarized incident wave, such a design would then consist of three cascaded metasurfaces. Two of these metasurfaces would be common to the LSE/LSM and TE/TM cases, with one metasurface providing the required azimuthal phase distribution via azimuthal sectors made of particles inducing progressive transmission delays and the other providing the required conical inclination via a constant radial phase gradient. In contrast, the third metasurface would be different for the LSE/LSM and TE/TM cases. In the former case, given the linear transverse polarization [Eq. (7) and Fig. 5], there is no required polarization processing, and the third metasurface needs to provide the proper transverse magnitude distribution, which can be accomplished with dissipative particles, while, in the latter case, given the constant transverse magnitude [Eq. (8) and Fig. 6], there is no required magnitude processing, and the third metasurface needs to provide the proper transverse polarization distribution, which can be accomplished with birefringent particles.

## 5. CONCLUSION

We have presented a unified perspective of Bessel beams of arbitrary OAM (zero, integer, and noninteger) and nature (scalar, LSE/LSM, and TE/TM) based on an integral formulation and deduced from this formulation as a universal and efficient generation technique. The proposed formulation may be extended to other conical beams, such as Mathieu [48] and Weber [49] beams, upon simply adjusting the amplitude modulation function $w(\xi)$ of Eq. (2a) as in [50,51] and to nonconical beams, such as the Gauss–Laguerre and hypergeometric Gaussian beams, upon nesting a corresponding extra integral for the proper spectrum in Eq. (1). This formulation increases the insight into their characteristics and facilitates their generation for non-complex spectra as conical ones. This global perspective opens up new horizons in structured light for a variety of applications, such as spectroscopy, microscopy, and optical/quantum force manipulations.

## Disclosures

The authors declare no conflicts of interest.

## Supplemental document

See Supplement 1 for supporting content.

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**37. **The top row of Fig. 3 shows that this radial effect is due to the radially decreasing density of the constituent waves.

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**41. **The fractional superposition of Bessel waves produces novel beam properties, such as internal vortices [9] and negative wave propagation [56], which are beyond the scope of this paper.

**42. **By the term ‘general’, we refer both to the arbitrariness of $\nu$ ($\nu = n \in {\mathbb N}$ or $\nu \in {\mathbb R}\backslash {\mathbb N}$) and to the alternative constructions of Eqs. (1) (integral) and (3) (analytical), with the discontinuity parameter ${\phi _{{G},0}}$.

**43. **The vectorialization of the analytical solution is no more complicated than that of the integral solution, and it is formally equivalent to the auxiliary potential vector method [57].

**44. **M. A. Salem, A. H. Kamel, and E. Niver, “Microwave Bessel beams generation using guided modes,” IEEE Trans. Antennas Propag. **59**, 2241–2247 (2011). [CrossRef]

**45. **P. Lemaître-Auger, S. Abielmona, and C. Caloz, “Generation of Bessel beams by two-dimensional antenna arrays using sub-sampled distributions,” IEEE Trans. Antennas Propag. **61**, 1838–1849 (2012). [CrossRef]

**46. **W. Fuscaldo, G. Valerio, A. Galli, R. Sauleau, A. Grbic, and M. Ettorre, “Higher-order leaky-mode Bessel-beam launcher,” IEEE Trans. Antennas Propag. **64**, 904–913 (2015). [CrossRef]

**47. **K. Achouri and C. Caloz, “Design, concepts, and applications of electromagnetic metasurfaces,” Nanophotonics **7**, 1095–1116 (2018). [CrossRef]

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**50. **In the case of Mathieu beams, we need to add the coefficient ${A}({\phi _{G}})$ as ${A_{e}}({\phi _{G}}) = {{\rm ce}_m}({\phi _{G}},q)$ and ${A_{o}}({\phi _{G}}) = {{\rm se}_m}({\phi _{G}},q)$ in Eq. (2a), where ${{\rm ce}_m}(\cdot)$ and ${{\rm se}_m}(\cdot)$ are the even (subscript ‘$e$’) and odd (subscript ‘$o$’) angular Mathieu functions of order $m$ and ellipticity $q$ [58], and we set $w(\xi) = 1$ and ${\gamma _\nu}({\phi _{G}}) = 0$ in Eq. (2a).

**51. **In the case of Weber beams, we need to add the coefficient ${A}({\phi _{G}})$ as ${A_{e}}({\phi _{G}}) = {e^{ia{\rm ln}| {\tan ({\phi _{G}}/2)} |}}/({2\sqrt {\pi | {\sin ({\phi _{G}})} |}})$ or ${A_{o}}({\phi _{G}}) = - i{A_{e}}({\phi _{G}})$ for $0 \le {\phi _{G}} \le \pi$ and ${A_{o}}({\phi _{G}}) = i{A_{e}}({\phi _{G}})$ for $\pi \lt {\phi _{G}} \le 2\pi$, with $a$ being a parameter, and we set $w(\xi) = 1$ and ${\gamma _\nu}({\phi _{G}}) = 0$ in Eq. (2a).

**52. **M. Soskin, V. Gorshkov, M. Vasnetsov, J. Malos, and N. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A **56**, 4064–4075 (1997). [CrossRef]

**53. **C. A. Balanis, *Antenna Theory: Analysis and Design* (Wiley, 2016).

**54. **Y. Wang, W. Dou, and H. Meng, “Vector analyses of linearly and circularly polarized Bessel beams using Hertz vector potentials,” Opt. Express **22**, 7821–7830 (2014). [CrossRef]

**55. **A. Chafiq and A. Belafhal, “Optical Fourier transform of pseudo-nondiffracting beams,” J. Quant. Spectrosc. Radiat. Transf. **258**, 107357 (2020). [CrossRef]

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**57. **A. Ishimaru, *Electromagnetic Wave Propagation, Radiation, and Scattering from Fundamentals to Applications* (Wiley, 2017).

**58. **J. C. Gutiérrez-Vega, “Formal analysis of the propagation of invariant optical fields in elliptic coordinates,” Ph.D. thesis (Instituto Nacional de Astrofísica Óptica y Electrónica, 2000).