Spatial mode analysis of optical beams carrying monstar disclinations

Asymmetric polarization disclinations, such as monstars, can be generated in two distinct ways: (a) by an inseparable superposition of three spatial modes bearing optical vortices with circular polarization states; (b) by using a modulated Poincaré beam, consisting of an inseparable superposition of a circularly-polarized fundamental Gaussian beam TEM00 and a second beam exhibiting an azimuthally-modulated vortex with an m-fold rotational symmetry and the opposite circular polarization. Based on the analysis of the spatial modes indirectly involved into the superposition through the latter method, we investigate its capability of spanning as many disclinations as possible, as well as its capability of enabling effective predictions about the generated patterns, such as relevant geometric features, already at the design stage.


Introduction
The development of increasingly effective methods for mastering polarization singularities (PSs) has become compulsory in multiple fields of both fundamental and applied optics, thanks to the capability of jointly manipulating both spatial and polarization degrees of freedom [1]. PSs-which arise within spatially inhomogeneous polarization transverse distributions-are points in which one of the parameters specifying the polarization of the light is undefined. For example, a C-point PS corresponds to an isolated (singular) point in * Author to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. the plane transverse to the wave propagation where the state of polarization is circular, hence with an undefined 'axes orientation', surrounded by a non-uniform pattern of elliptical polarization states, whose ellipse axes rotate clockwise or counterclockwise about it. They are characterized by an integer or half-integer index I C , indicating that the azimuthal coordinate on the Poincaré sphere rotates 2|I C | times per turn about the singularity. These polarization patterns occurring around C-points are further named as 'lemon', 'star', or 'monstar', depending on the sign and value of I C and on other geometric features. Likewise, in vector-vortex fields, which are characterized by patterns of linear polarization having nonuniform orientation, isolated points at which this orientation is indeterminate are named V-points [2,3].
A distinctive feature of PSs is their structural inseparability-the so-called classical entanglementunderpinning the spin-dependent phase and spatial structures of the optical beams within which they are woven. Generating PSs requires superposing two orthogonally-polarized beams with different wavefronts, at least one of which exhibits a screw dislocation. This can be achieved by splitting an input beam into two orthogonally polarized components, whose wavefronts are independently reshaped, and finally recombined through an interferometric setup. Alternatively PSs can be created by exploiting Spin-orbit Interactions (SOIs) of light [4], i.e. optical phenomena in which the spin affects and controls the spatial degrees of freedom of light. PSs can be, in fact, obtained also by spin-to-vortex conversion of a paraxial beam propagating through optical fibres [5] or anisotropic crystals [6][7][8]. If a circularly polarized beam with a definite helicity is fed to a multimode fiber with a stepped index profile and a circularly polarized beam with opposite helicity is extracted at the output, the transmitted beam will exhibit an isolated wavefront dislocation, and a sign change in the helicity at the fiber input together with a corresponding sign change in the helicity of the selected output will result in a change in the sign of the screw dislocation [5]. Highly efficient control of the polarization degrees of freedom, as well as controllable shaping of the intensity and phase distributions have been recently demonstrated by combining dielectric media anisotropy and inhomogeneities [2,9]. The propagation of the optical fields bearing PSs is associated with some novel phenomena, in particular in 3D space. Therefore, developing increasingly effective methods for polarization sculpting is not only interesting per se, but might be useful in material analysis, as well as material processing and micromanipulation. The focusing behavior of PSs, in particular, reveals non-trivial aspects having diverse potential applications. For example, it has been recently demonstrated, on theoretical grounds, that an in-phase superposition of a radially and an azimuthally polarized beam of equal magnitude yields a vector field having a non-trivial azimuthal imaginary Poynting momentum density [10]. The mechanical effect of such momentum could be potentially exploited to continuously rotate objects that are neither inhomogeneous nor anisotropic, since the mechanism underpinning such an optically-induced rotation is fully alternative to the optical spin or orbital angular momenta. Morevover, the optical field in the focal plane of a tightly focused beam with a non-vanishing radially polarized component has a strong longitudinal component leading to many new phenomena. The latter include, for instance, the generation of pure transverse spin for a superposition of two optical fields carrying different singularities [11] or the generation of photonic skyrmions in a confined electromagnetic field [12,13], which has important potential applications in Bose-Einstein condensates and Quantum Spin-Hall effect as a form of topological protection. In light-matter interactions, both in linear and nonlinear regimes, PSs have been shown to play a relevant role. By letting a PS propagate through a linear birefringent crystal, for instance, the polarization state of the light beam can be driven adiabatically from the equator toward the poles of the higher-order Poincaré sphere [14]. This leads to the spin-orbit optical Hall effect-a further manifestation of SOI effects-consisting in the separation of the spin and orbital components of the light beam and the related capability of controlling the orbital angular momentum (OAM) flow for particle manipulations. In nonlinear regime, on the other hand, the nonlinear dipole moment of the medium undergoes spin-orbit coupling effects whenever the driving light field and/or the generated field are inhomogeneously polarized [15,16]. By tightly focusing light beams with a nonvanishing azimuthally polarized component, a sub-diffractionlimited pure longitudinal magnetization can be generated by inverse Faraday effect [17][18][19]. The magneto-optical effects related to PSs can be potentially beneficial to spintronics, highdensity magneto-optical memory, multipole atoms trapping, and light-induced magneto-lithography.
Compared to phase singularities, the generation of PSs typically requires a larger set of control parameters and tailoring optical beams with a specified topological structure is a difficult task. We have recently introduced a general method for engineering inhomogeneously polarized beams (IPBs) in which the rotation rate of the local polarization azimuth around a C-or a V-point can be set as desired [2,9], based on a geometric approach that enables shaping the wavefronts of the spatial modes required to generate the target polarization map without passing through the direct manipulation of the OAM spectrum. In a recent paper, Free-Form Hollow (FFH) beams have been introduced [9]. An FFH beam carries non-circular screw phase dislocations generated by imparting to a TEM 00 mode, in the beam waist, a periodical purely-azimuthal phase that reflects the symmetry properties of a suitable generating plane curve. In other words, an FFH, unlike a helical mode, exhibits an oscillatory rather than linear azimuthal phase dependence in the near field. The inseparable superposition of two orthogonal circularly polarized FFH modes gives birth to an IPB, specifically dubbed Modulated Poincaré mode (MPB). Importantly, the latter can be advantageously generated through a liquid-crystal-based wavePlate with uniform retardation and Spatially Varying optic Axis (SVAP) [2]. For suitable tuning of the retardation, an SVAP-whose operation principle is based on the Pancharatnam-Berry phase-has the remarkable capability of converting a circularly-polarized input beam into a linear-polarization pattern that just replicates its optic axis distribution. In practice, any arbitrary disclination pattern can be thus imparted or changed over the transverse wavefront of an input beam, by accordingly arranging the optic axis distribution of the SVAP. In a recent paper [20], we have adopted such approach to generate monstar patterns of any index I C . Unlike other well-established methods [3], our approach is not based on the direct preparation of a suitable mix of helical modes, but rather on the general geometrical definition of monstars introduced in [3].
In the present manuscript, we specifically analyze the OAM mode decompositions of the monstar disclinations presented in [20] and generated in the form of MPBs. The modal analysis is carried out on three patterns deduced from three distinct curves that are however homeomorphic to each other, i.e. that can be continuously deformed into each other. The deformation of the curves is reflected into the reshaping of the generated polarization patterns and, consequently, into an OAM modal remix. This suggests that relevant geometric features can be deduced directly from the symmetry properties of the generating curve. The indirectly-manipulated OAM spectrum of the optical field-involving a given number of OAM modes, with specified relative intensities and intermodal phases-suggests that our geometric approach tends to include a wide class of both symmetric and asymmetric disclinations, a feature that may help us acquire a deeper understanding of the link between the OAM spectrum and the geometric properties of both the generating curve and the ensuing polarization pattern.

Methods for generating asymmetric monstar disclinations
A monstar asymmetric disclination was first identified in the late 70 s as a polarization disclination pattern having a topological index I C = +1/2, like for a lemon, and three 'radial lines' -straight lines, radiating from the singularity, on which the polarization ellipse axes are all parallel to the line-like for a star, hence the name [21][22][23][24]. Here, by 'asymmetric' we mean that the ellipse axes in the polarization pattern do not rotate at a uniform rate when circling around the PS, and by 'degree of asymmetry' we denote the extent to which the axes rotation rate varies. Indeed, a strict correlation between the I C index and the number N of radial lines is peculiar of symmetric disclinations only [25], i.e.
for lemons (I C > 0, I C ̸ = +1) and stars (I C < 0). For I C = +1, N = ∞. The systematic exploration of asymmetric disclinations of any order has led to the conclusion that monstars may occur for any value of I C , whether the latter be negative or positive. For asymmetric PSs, N in equation (1) gives the minimum number of radial lines. Indeed, the monstars' distinctive feature [3], for any order I C , is the presence of a number of radial lines N m > N, with the extra N m − N lines bounding portions of the wavefront containing curved lines that radiate from the singularity (parabolic sectors).
In [3], monstar polarization disclinations were observed in optical fields generated by using three spatial modes containing optical vortices in non-separable superpositions with circular polarizations. This was achieved by applying to the TEM 00 mode, in the beam waist, the spin-orbit factor where |R⟩ and |L⟩ denote the right-and left-circular polarization states respectively, ϕ is the azimuthal angle around the polarization singularity located on the beam axis, the indices l 1 , −l 1 and l 2 represent the topological indices of the involved helical modes. The modes l 1 , −l 1 are superimposed in the same polarization state with coefficients cos β and sin βe −iη . The angular parameter β takes any value in the range 0 ⩽ β ⩽ π/2, β ̸ = π/4, and controls the degree of asymmetry of the disclination. The latter turns into an actual monstar only when β is larger than a threshold value β th . This modal approach enables generating monstars within a wide class and some of their fundamental properties can be inferred. However, equation (2) is likely to describe only the most basic disclinations and going beyond would require adding even more modes to the mix. In [20], in order to encompass a larger variety of disclinations, possibly having a higher degree of asymmetry, a different approach has been proposed in which a light beam embedding a complex disclination pattern can be obtained by superposing the fundamental TEM 00 mode, in a circular polarization state, with a Free-From Helical (FFH) mode of integer order m in an orthogonal circular polarization state [9] which belongs to the recently introduced class of the Modulated Poincaré Beams (MPBs) [2,9]. An FFH m q mode of order m is defined as where r is the radial coordinate in the transverse plane, w the beam waist, 2q the net topological charge of the FFH mode-possibly zero-and the azimuthal phaseψ(ϕ) is a periodic function of ϕ with period 2π/m. The topological index of the polarization disclination embedded into equation (3) is in which By varying the parameters m, n 1 , n 2 , n 3 , a, and b, equation (6) describes, in polar coordinates ρ, ϕ, several families of plane curves γ m (a, b, n 1 , n 2 , n 3 ). The positive real numbers a and b parameterize the radii of the circumferences respectively inscribed and circumscribed to the curve and the three real numbers n 1 , n 2 and n 3 control the local radius of curvature. When a = b and n 2 = n 3 , γ m exhibits an m-fold rotational symmetry C m . While varying all the free parameters in equation (6), the generated curves can be very diverse and possibly even develop cusps. For a fixed value of I C , there is a 1:1 correspondence between the phase modulation functionψ(ϕ) and the curve γ m described by equation (6). In [2], it has been demonstrated that the near-field polarization pattern described by equation (3) obeys the rule θ(ϕ) being the angle providing the orientation of local unit normal to the curve γ m . In particular, Ψ(ϕ) = θ(ϕ), when q = 1. Equation (7) was the foundation for our geometric approach (g) theoretical monstar pattern generated through SVAP Q 2 ; (h) experimental monstar pattern generated through SVAP Q 2 . The rainbow color code represents the angular deviation of the local polarization from the radial direction. Reproduced with permission from [20].
to engineering asymmetric disclinations presented in [20]. In essence, complex monstar disclinations have been generated not by direct manipulation of the OAM spectrum of the spinorbit factor applied to the input TEM 00 Gaussian mode, but rather through manipulation of the curve γ m to cause the formation of additional radial lines bounding parabolic sectors. An example of this procedure is sketched in figure 1.
Besides, to avoid flying blind, a criterion is required to select more modes and their relative phases enriching the mix aiming at a specified target. No doubt a geometric approach greatly facilitates the task. On top of that, Modulated Poincaré Beams can be easily produced by using electrically tunable liquid crystals-based spatially varying axis plates (SVAPs), based on Pancharatnam-Berry phase. It has been shown [2] that an SVAP with retardation δ = π/2-quarter waveplate operation-transfers its optic axis distribution to the nearfield polarization distribution of an impinging TEM 00 Gaussian mode. Therefore, an SVAP can be designed so to have an optic axis distribution coincident with the required polarization pattern. Besides, SVAPs have already been used successfully for generating monstar disclination patterns [26], by introducing a high enough degree of astigmatism into an otherwise circularly symmetric optic axis distribution.

Modal analysis of optical beams bearing monstar disclinations: theoretical predictions
To get a deeper insight into the properties of asymmetric polarization disclinations or to employ them for some specific applications, a criterion is needed for tailoring designed disclination patterns. The modal approach requires adding into the superposition more modes with specified relative phases aiming at a specified target. However, as clarified below, it is unlikely that a general selection criterion can be found extending the one described in equation (2) to a combination of more than three modes. We think that the geometric approach proposed in [20] could greatly facilitate this task. We here analyze the OAM modal decomposition of several optical fields bearing monstar patterns generated through this geometric method.
The optical field in equation (3), besides the TEM 00 mode in the left-circular polarization state, includes the helical modes contained in the 2q-charged m-order FFH mode, i.e. in the phase factor in equation (4), consisting, in turn, of the modulation phase factor e i qψ(ϕ) and the phase factor e i 2q ϕ imprinting the 2q ℏ average OAM per photon. The overall azimuthal mode expansion can be written as follows [2], Then, the OAM modal expansion of the involved FFH mode includes only the components with indices l = 2q ± |p| m, p being any positive or negative integer and the integer number m, as above mentioned, is the folding order of the rotational symmetry characterizing the curve γ m . The specific weights of the involved modes and their relative phases are provided by the expansion coefficients c l and are therefore determined by the detailed shape of the generating curve γ m . The polarization disclinations presented in [20] have been generated employing equation (3) in six cases: FFH 6 ±1/2 (a = b, n 1 = 2/3, n 2 = n 3 = 2.3); FFH 6 ±1/2 (a = b, n 1 = 3/10, n 2 = n 3 = 2.3); FFH 6 ±1/2 (a = b, n 1 = 1/6, n 2 = n 3 = 2.3). The generating curves-the primroses-differ from one another only in their petals depth, which increases as n 1 decreases (figures 1(a) and (e)). For q = +1/2 and n 1 > n th 1 (q = +1/2) = 2/5, the degree of asymmetry is below threshold and no parabolic sector arises. For q = −1/2, n th 1 (q = −1/2) ≳ 1. In figures 3(a)-(c) the theoretical spectra of the three FFH 6 +1/2 modes only are displayed, as their counterparts for q = −1/2 can be obtained by simply flipping the horizontal axis. The overall picture of the theoretical predictions is the following: • n 1 = 2/3, the squared-amplitude of the OAM mode l = +1 exceeds that of both the modes l = −5 and the mode l = +7 by about 95%: the disclination exhibits a lemon-like pattern for q = +1/2 and a monstar pattern with six narrow parabolic sectors and three hyperbolic sectors. • n 1 = 3/10, the squared-amplitude of the OAM mode l = +1 exceeds that of the mode l = −5 by about 80% and that of the mode l = +7 by about 76%: the disclination exhibits a monstar pattern with two parabolic sectors and one hyperbolic sector, for q = +1/2, and a monstar pattern with six parabolic sectors wider than in the case n 1 = 2/3. • n 1 = 1/6, the squared-amplitude of the OAM mode l = +1 exceeds that of the mode l = −5 by slightly less than 50% and that of the mode l = +7 by only 33%: the disclination exhibits a monstar pattern with four parabolic sectors and one hyperbolic sector, for q = +1/2, and a monstar pattern with six parabolic sectors wider than in the case n 1 = 3/10. Importantly, the modes at play, in this case, are more than three.
In all three cases, the modes l = −5 and l = +1 are in phase with the mode l = 0, while the mode l = +7 is in phase opposition. In these examples, there are four active modes at play, three of which-the modes l = −5, l = +1 and l = +7are simultaneously in the right-circular polarization state and the fourth-the mode l = 0-is in the left-circular polarization state. The degree of asymmetry increases as the three modes introduced through the FFH mode tend to be comparable.

Modal analysis of optical beams bearing monstar disclinations: experimental findings
In order to test the theoretical predictions reported in the previous section, a modal decomposition with digital holograms on a phase-only spatial light modulator (SLM) has been performed on the optical fields described by equation (3), and generated through a suitably fabricated SVAP [2,9]. Immediately beyond the SVAP-i.e. in the near-field observation plane-the optical field is the product of the TEM 00 fundamental Gaussian mode and the azimuthal phase factor e i2Ψ(ϕ) . Therefore, the OAM modal analysis, in the near-field, can be carried out neglecting the radial dependence, which is cancelled out upon integration. Along free-space propagation, in the paraxial approximation, the OAM mode-amplitude distribution is conserved, but the intermodal phases change and exhibit a radial dependence that is missing in the near-field. Besides, due to diffraction, the rotational symmetry of the FFH modes encapsulated in their phase modulation affects also the intensity profile of the generated beam, so that light power appears equally partitioned among the m equally spaced sectors of the phase profile [9], giving birth to m-fold rotationally invariant dark hollow beams. The relative squared amplitude of an OAM mode l can be quantitatively measured by selecting the on-axis far-field intensity of the light beam fraction diffracted by the phase hologram producing the mode e -ilϕ . This optical scheme is equivalent to the projection |c l | 2 = |⟨e i2Ψ(ϕ) | e ilϕ ⟩| 2 of the input beam on the target mode e ilϕ . The intermodal phase arg c l − arg c 0 , with respect to the fundamental mode l = 0, can be measured by adopting an analogous projection scheme in which the hologram for the mode l is replaced with two distinct holograms producing the superpositions 1 + e −ilϕ and 1 + i e −ilϕ [27,28]. In this case the onaxis far-field intensity of the light beam fraction diffracted by these holograms is equivalent to the following projections: from which the intermodal phase can be extracted: t being an integer number. The experimental setup adopted to implement the described measurements is sketched in figure 2. A linearly polarized TEM 00 Gaussian beam from a Helium-Neon laser (λ = 632.8 nm and maximum power P max = 5 mW) was  [20]. The peak's height represents the relative squared amplitude of the corresponding OAM component: (a) theoretical prediction and (d) experimental findings for the SVAP Q 1 with n 1 = 2/3; (b) theoretical prediction and (e) experimental findings for the SVAP Q 2 with n 1 = 3/10; (c) theoretical prediction and (f) experimental findings for the SVAP Q 3 with n 1 = 1/6. expanded by a telescope including the converging lenses L 1 and L 2 with focal lengths of f 1 = 5 cm and f 2 = 20 cm, respectively, to magnify the beam by about a factor four. A quarter-waveplate (QWP 1 ) is used to make the laser beam polarization state left-circular and an SVAP (Q i ) designed for generating the required FFH mode is electrically tuned to the retardation δ = π for maximum conversion efficiency. For the sake of simplicity, we identify the SVAPs relative to the indices n 1 = 2/3, 3/10, 1/6 as Q 1 , Q 2 and Q 3 , respectively. The beam emerges from the inserted SVAP rightcircularly polarized in the required FFH mode. A second quarter-waveplate QWP 2 and a half-waveplate HWP are inserted to make the beam polarization linear again and parallel to the optic axis of an SLM, used to display the above-mentioned probe holograms. The optical near-field immediately beyond the SVAP is imaged on the SLM through a 4 f optical system including the converging lenses L 3 and L 4 having equal focal lengths f 3 = f 4 = 15 cm. A final converging lens L 5 , with focal length f 5 = 30 cm provides the Fourier-transform of the wavefront immediately beyond the SLM. A CCD camera located in the focal plane of L 5 and distant 2f 5 from the SLM is used to collect intensity patterns.
The measured spectra are shown in figures 3(d) and (f) next to their theoretical counterparts. The results are in good agreement with the theoretical predictions. In figure 3(f), however, the overall method sensitivity is too low to enable the measurements of the relative intensities of the modes l = −17, l = 13, l = 19. In table 1, the measured values of the relative intensities of the OAM modes are reported with their corresponding uncertainties.
The far-field intensity distributions of the FFH modes here considered are shown in figure 4. They have been obtained for a left-circulary polarized input beam passing through the fabricated SVAPs used as half-waveplates. As increasing the parameter 1 n1 , the intensity patterns come to be narrower, as a consequence of the increasing azimuthal modulation amplitude.
The uncertainties on the squared amplitudes |c l | 2 and on the phases arg c l − arg c 0 of the OAM spectral coefficients reported in tables 1 and 2, respectively, have been estimated as follows. As mentioned above, all expectation values of |c l | 2 are determined as the on-axis values of the far-field intensity after the holographic filter. The uncertainty on the unnormalized value of each coefficient is primarily ascribed to the uncertainty of the actual beam axis position in the Fourier plane and to the corresponding alignment of the central singularity of the filtering hologram with respect to the beam axis itself. We estimated first a maximal circular area for localizing the beam axis position on the image detector; the uncertainty in the un-normalized coefficients was then obtained from the range of intensities measured within that circular area. This method provides, in our opinion, a good estimate of the maximal possible error (or uncertainty interval). Next, the coefficients |c l | 2 were normalized with respect to the sum over all the l's for which the measurement procedure had returned nonvanishing values (setting to zero those values which are smaller than the corresponding uncertainty). In the cases analyzed in the manuscript, the nonvanishing coefficients were obtained for l = −5, +1, +7. The un-normalized coefficient |c −11 | 2 was lower than its experimental uncertainty and was therefore excluded from the subsequent analysis. Its value is reported in figure 3(f) just to provide a qualitative indication of the barely detectable signal for l = −11. The same method of the circular area is then adopted to estimate the uncertainty on the I cos l and I sin l appearing in equations (10) and (11). Standard uncertainty-propagation techniques are then finally applied to obtain the uncertainty on arg c l − arg c 0 from equation (11).

Far-field polarization disclination patterns
Adopting the same polarization imaging system reported in [2,20], and shown in figure 5, we also reconstructed the polarization disclination patterns in the far-field zone in all three cases. The experimental findings (figures 6, 7 and 8(c) and (d)) are in satisfactory agreement with the theoretical predictions (figures 3(a), (b), 6 and 7). It is quite clear that these disclinations tend to lose their asymmetric signature, despite the fact that their azimuthal mode composition remains unchanged. We ascribe this behavior to the stronger radial divergence of higher-order azimuthal modes, which reduces their contribution to the pattern structure close to the singularity. Moreover, the intermodal phases change along propagation, a fact which may also contribute to altering the pattern structure. Experimental setup for spatially-resolved polarimetric measurements. A telescope including the lenses L 1 and L 2 expands the light beam from a He-Ne laser source (λ = 632.8 nm, maximum power output Po = 5 mW, beam waist w 0 = 0.5 mm). The linear polarizer P 1 and the QWP 1 are used to make the input polarization circular. The latter is replaced with a half-waveplate HWP 1 when the input polarization is linear horizontal or vertical. Half-waveplate HWP 4 is used for swapping left-/right-handed circular polarized output states, when needed. The SVAP, the lenses L 3 and L 4 , and the CCD camera are positioned in the 4 f configuration for near field measurements. A single lens is adopted for far field measurements. The half-waveplate HWP 3 , the quarter-waveplate QWP 2 and the linear polarizer P 2 are used for measuring the local Stokes parameters over the wavefront. Reproduced from [2]. © The Author(s). Published by IOP Publishing Ltd. CC BY 4.0. and experimental ((c), (d)) far-field polarization patterns related to Q 1 for I C = −1/2 and I C = +1/2 respectively. In the images, brightness represents total intensity, while the locally calculated or experimentally reconstructed polarization state is represented by the small ellipses.

Conclusions
We have investigated the OAM spectrum of modulated Poincaré beams bearing monstar disclinations, both theoretically and experimentally. Consistently with [3], we have come to the conclusion that such class of asymmetric disclinations, in general, is not necessarily a blend of lemon and star disclinations, as early conjectured for |I C | = 1/2-the monstar was thought to occur for the same index as the lemon and the same number of radial lines as the star [21][22][23][24]. Indeed, we have shown that this is the case neither for |I C | = 1/2: modulus and sign of the topological charge can be maintained fixed as increasing the number of radial lines enclosing parabolic sectors. Importantly, there is no simple recipe prescribing which OAM spectral distribution gives birth to a specific monstar disclination. For this reason we believe that the geometric approach described in [20] is more inclusive and versatile, besides being easier to implement, thus opening further investigation prospects.
As a final remark, we notice that monstar disclinations tend to lose their asymmetric character in the far field. Indeed, in this zone, radial diffraction evolves the spatial modes and the monstar disclination lines that preserve their asymmetric nature tend to glide away from the beam axis. Here, the transverse intensity achieves its maximum and the disclinations tend to become symmetric. The pattern always tends to asymptotically acquire a lemon-or star-like character according to the sign of I C .

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.