International Journal of Optics
Volume 2020 (2020), Article ID 2812803, 33 pages
https://doi.org/10.1155/2020/2812803

Review Article

Phase Singularities to Polarization Singularities

Ruchi, P. Senthilkumaran, and Sushanta Kumar Pal

Department of Physics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India

Correspondence should be addressed to P. Senthilkumaran; psenthilk@yahoo.com

Received 9 March 2020; Revised 10 May 2020; Accepted 22 May 2020; Published 14 July 2020

Academic Editor: E. Bernabeu

Copyright © 2020 Ruchi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Polarization singularities are superpositions of orbital angular momentum (OAM) states in orthogonal circular polarization basis. The intrinsic OAM of light beams arises due to the helical wavefronts of phase singularities. In phase singularities, circulating phase gradients and, in polarization singularities, circulating Stokes phase gradients are present. At the phase and polarization singularities, undefined quantities are the phase and Stokes phase, respectively. Conversion of circulating phase gradient into circulating Stokes phase gradient reveals the connection between phase (scalar) and polarization (vector) singularities. We demonstrate this by theoretically and experimentally generating polarization singularities using phase singularities. Furthermore, the relation between scalar fields and Stokes fields and the singularities in each of them is discussed. This paper is written as a tutorial-cum-review-type article keeping in mind the beginners and researchers in other areas, yet many of the concepts are given novel explanations by adopting different approaches from the available literature on this subject.

1. Introduction

A singular point is characterized by an undefined physical parameter surrounded by a region of high gradient [1–3]. In electromagnetic fields, at a phase singularity, the phase and, at a polarization singularity, the polarization parameter azimuth is undefined. Singular optics is a new area which studies the singularity that occurs in any of the parameters that define optical fields. Azimuth refers to the angle the major axis of an ellipse of an elliptical polarization state makes with respect to a reference direction (say axis) [4–7]. The phase of the Stokes field indicates the azimuth [8–10]. At the immediate neighborhood of a phase singularity, all phase values ranging from 0 to are present [11, 12], where is the topological charge. Similarly, in the neighbourhood of a polarization singularity, the Stokes phase has values ranging from 0 to [8, 9], where is the Stokes polarization singularity index. The phase gradient in a phase singularity and azimuth gradient in a polarization singularity circulate around the respective singularities. The phase and azimuth contours emanate from a phase and a polarization singularity, respectively.

While there are plethora of research articles available in literature on phase singularity [11, 13–30], the literature on polarization singularity is limited in number. Due to additional parameters associated with polarized light fields, the subject of polarization singularity has more complexity. In this article, some of the basics of polarization singularity and related useful tools in its understanding are presented. The paper is organized as follows: the article starts with a brief note on phase singularity in Section 2, followed by polarization singularity in Section 3; the subject of phase control to azimuth ( Stokes phase) control and generation method for polarization singularities is presented in Section 4; polarization parameters and Stokes phases, useful in the study of polarization singularities, are presented in Sections 5 and 6, respectively; factors deciding the ellipticity and azimuth in different orthogonal decomposition schemes are also presented in Section 5; Stokes parameters and Poincaré sphere are also covered in Section 5 for completeness of the subject; Stokes phase distribution is explained in Section 6; converting phase distribution into Stokes phase distribution is presented in Section 7; we experimentally demonstrate the generation of polarization singularities such as bright C-points (lemons and stars), dark C-points, polarization flowers, and spider webs using Mach–Zehnder-type interferometer in Section 8; recently introduced topological spheres to represent ellipse and vector field singularities are presented in Section 9; in the last section, literature survey on phase and polarization singularities is presented that explains the current status of the field. This paper is written as a tutorial-cum-review-type article keeping in mind the beginners and researchers in other areas. Care is taken to avoid complications by adopting to simple explanations, yet many of the concepts are given novel explanations by adopting different approaches from the available literature on this subject.

2. Phase Singularity

The complex field of a phase singular beam is of the form given as follows:

where is the amplitude distribution. The phase singularity is also known as a phase vortex or a scalar vortex or an optical vortex in literature. The index in equation (1) is called topological charge of the vortex and is defined as

The wave fronts have helical shape [11, 14, 31, 32]. The topological charge can take positive and negative integer values [16, 32] depending on the handedness of the helical wavefront. The phase distribution is given by azimuthally varying function , where is the polar angle. The transverse phase gradient [18, 33–35] for this vortex is given by . This phase gradient is mainly circulating, and near the vortex core, its magnitude is high [11, 32, 36] and it also has some radial component [37–39]. The phase contours of different phase values terminate at the vortex (singular point), resulting in phase ambiguity, and therefore, the amplitude is zero at the vortex core. Figure 1 shows phase distributions, phase gradients, phase contours, and wavefronts of the scalar vortex beams with different topological charges. The amplitude distribution in a phase singular beam can be that of Laguerre–Gaussian, or of the form , and so on [40]. Some possible intensity distributions for a vortex beam of topological charge are depicted in Figure 2. Note that, in all types of intensity variations, at the vortex core (), the amplitude is zero and is surrounded by a doughnut-type intensity distribution. At the singular point, the zero of real part as well as the zero of imaginary part of the wave function crosses each other.

Figure 1: Phase distributions, phase gradients, and wavefronts of phase singular beams with different topological charges. Phase contours are shown below each wavefront.

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Figure 2: Possible intensity distributions for an optical vortex beam of topological charge . Note that amplitude is zero at the vortex core () and is surrounded by a doughnut-type intensity distribution. (a) LG variation; (b) r variation; (c) tanh variation.

Phase singularities have many applications [22, 41–53]. However, they are not useful in some situations [54–56]. Use of a fork grating [32] or a spiral phase plate [57] is common among many generation methods [58–69] reported in literature. Many reports exist in the arrays of phase singularities [19, 70–72]. Generation of phase singular beams has also been reported in nonlinear media [73–77]. Among the various detection methods of phase singular beams, interferometric [78, 79] and diffractive [80, 81] methods are commonly used due to their simplicity.

3. Polarization Singularity

Beams with slowly and spatially varying polarization distributions have attracted interest in recent years [8, 9, 82–89]. Polarization singularities occur in inhomogeneously polarized light fields that have spatially varying polarization distribution. Fields in which the distribution of state of polarization (SOP) is predominantly elliptical are called ellipse fields, and one such field is shown in Figure 3(a). On the contrary, fields in which the predominant SOP distribution is that of linearly polarized states are called vector fields. A vector field distribution is shown in Figure 3(b). Polarization singularities of ellipse fields are called C-points and that of vector fields are called V-points. In random fields, elliptical and linear polarization states can occur in different regions of the same field.

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Figure 3: Inhomogeneous polarization distributions: (a) an ellipse field distribution; (b) a vector field distribution.

In an inhomogenously polarized ellipse field, an isolated polarization singular point is termed as C-point at which the SOP is circular. At circular polarization, the orientation of the major axis of polarization ellipse is undefined, and hence, it is a singularity in the azimuth distribution. In vector fields, the V-point singularity is an intensity null point at which the polarization azimuth is undefined. Basically, in both types of singularities, in the immediate neighborhood of the singularity, as one goes around the singular point in a positively oriented closed path (anticlockwise), the azimuths undergo rotation in a clockwise or anticlockwise sense. This rotation of the azimuths of neighbouring SOPs around a C-point and V-point singularity is depicted in Figures 4(a) and 4(b), respectively. The amount of rotation the azimuth undergoes around a polarization singularity can be found by evaluating the integral , where is the azimuth of the polarization ellipse. In case of linear polarization, the handedness is undefined. Linear polarization that normally occurs in ellipse fields along points on a curve is referred as a L-line. It segregates the regions of right and left handedness in an ellipse field distribution.

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Figure 4: Rotation of the azimuths of the surrounding SOP states around a singular point: (a) a left-handed positive index C-point with ; (b) a negative index V-point with . The pink arrow denotes the sense of traversal of a closed path around the singularity, i.e., anticlockwise. The green arrow depicts the sense of rotation of the major axes of ellipses (plane of polarization of linearly polarized state in case of a V-point) around the singularity.

3.1. C-Points

For C-point singularities, C-point index is defined by

For C-points, the attributes such as index, bright, dark, left-handed, and right-handed are decided by their OAM superpositions in the orthogonal circular basis states [90–94]. Generic C-points are lemon (), monstar (), and star (). The SOP distribution of a lemon and a star is shown in Figures 5(a) and 5(b), respectively. In these figures, filled ellipses in red color are right-handed (RH) and those drawn in blue color are left-handed (LH). C-points can be further classified based on handedness . Thus, a C-point with a given C-point index can be right-handed or left-handed [91, 95–97]. This means, for example, in Figure 5(a), SOP distribution for a RH lemon is shown and a lemon which is LH is also possible. C-points can occur at any value of intensity. For example, C-points in Figures 5(a), 5(b), and 5(d) with respective indices , , and are bright, whereas C-point () in Figure 5(c) occurs at intermediate value of intensity. Two examples of dark C-points with index and are shown in Figures 5(e) and 5(f), respectively. In Figures 5(a)–5(f), the RH- and LH-handed regions are separated by a closed curve (L-line) on which SOPs are all linear. Anisotropic C-point is a monstar [98–101], and it has the index value . The C-point singularities are referred in recent literature as hybrid-order Poincaré sphere beams.

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Figure 5: Polarization distributions containing (a–f) C-points and (g–i) V-points. Indices of C-points and V-points are mentioned above each figure. (a) and (b) are generic C-points and are commonly known as lemon and star, respectively. (a–d) Bright C-points; (e-f) dark C-points. Note that, in (d) and (e), a C-point with the same index can be dark or bright. V-points always occur at zero intensity. (g) A generic V-point known as radially polarized; (h) a polarization flower; (i) a spider web.

3.2. V-Points

V-point singularities are the polarization singularities occurring in linearly polarized light fields. They are characterized by Poincaré–Hopf index defined as

The handedness of V-points is undefined as they are made of linear states. V-points always occur at intensity null [102]. In Figures 5(g)–5(i), V-points with three different Poincaré–Hopf indices are shown. Even though V-points are devoid of handedness, during dissociation, they disintegrate into equal number of left- and right-handed C-points [103]. When diffracted through fork grating [104], a V-point segregates into C-points of one handedness in positive diffraction orders and of opposite handedness in negative diffraction orders. Helicity conservation has also been observed in diffraction scattering [105]. In recent literature, the V-point polarization singularities are referred as higher order Poincaré sphere beams. Spirally polarized beams are superposition states of radially and azimuthally polarized beams. The critical points in spirally polarized beams are characterized using streamline morphologies equivalent to the stability theory of autonomous systems of ordinary differential equations [106].

4. Phase Gradient to Azimuth Gradient

A comparison of equation (2) with equations (3) and (4) reveals the similarity in the form of line integrals. The term in equation (2) is replaced by in equations (3) and (4). To have a polarization singularity, conversion of phase variation into azimuth variation is required. In the next section, we make a comparative study of the conventional linear decomposition of polarization state with circular decomposition. It is shown that, in circular decomposition, the phase difference between the right and left circular polarization components of a given SOP can be varied to achieve the azimuth variation. In superpositions, the phase difference between two fields which are in same SOP leads to intensity modulation, whereas the phase difference between orthogonal polarization basis states modulates the SOP of the light. This phase difference between orthogonal states is termed as the Stokes phase. Stokes phase and its importance are explained in subsequent sections. The phase singularities in the Stokes phase distributions are called Stokes singularities. This entails a revisit to the basics of polarization, which is presented in the next section.

5. Polarization

Polarization of light [4–7] refers to the study of temporal variation in electric field vector of light. We consider here two-dimensional paraxial fields, which means that the orientations of polarization ellipses in three-dimensional space are safely projected on to a two-dimensional plane. The amplitude and phase of the component oscillations of a given SOP during orthogonal decomposition are crucial. In this section, a brief note on the polarization parameters of interest to us is given.

5.1. Azimuth and Ellipticity

Even though there are various parameters available to describe the SOP of light, we first introduce the following two parameters that are crucial in the understanding of polarization singularities. These two parameters are ellipticity and azimuth . For a fully polarized light, ellipticity is defined as , where and are the major and minor axis of the polarization ellipse, positive sign is for right-handed ellipse, and the negative sign is for the left-handed ellipse. It varies between and . The azimuth is the orientation angle of the major axis of the ellipse with respect to a reference direction (usually axis). It varies between and . In Figure 6, polarization ellipses with different azimuths and/or ellipticities are depicted. SOPs with constant ellipticity but different azimuths are shown in first row in Figure 6. In the second row, polarization ellipses with constant azimuth but different ellipticities are shown. The polarization ellipses with varying both azimuth and ellipticity are depicted in the third row of Figure 6.

Figure 6: Polarization ellipses with different azimuths and/or ellipticities.

5.2. Decomposition

The azimuth and ellipticity of a given SOP can be given in terms of component polarization states. We present here two types of decompositions, namely, linear and circular decompositions.

5.2.1. Linear Decomposition

Any SOP of light can be decomposed into two linearly polarized orthogonal states. In other words, any SOP can be represented as a superposition of two linearly polarized states. Consider the following superposition

where are the component amplitudes of the simple harmonic oscillations (for a coherent monochromatic light) occurring in and planes and are the corresponding component phases. Circularly polarized light occurs when and . Linear polarization occurs when irrespective of the values of and . Elliptical polarization occurs when linear and circular polarization conditions are not met. Linear decomposition is explained in most common text books on polarization [4–7]. In linear decomposition, the azimuth is decided by the ratio of component amplitudes, and ellipticity is decided by the difference between the component phases. To have a nonzero ellipticity, the condition on phase difference between the two component oscillations is . But there is also another way of decomposition, namely, circular decomposition, which is very useful in the study of polarization singularities and is given below.

5.2.2. Circular Decomposition

Light in any SOP can be decomposed into two circularly polarized orthogonal states [9, 91, 92, 107]. In this case, the two component oscillations are clockwise (left) and counterclockwise (right) rotating circularly polarized light with amplitudes and , respectively. The superposition state here is given by

where and are the phases of the component oscillations. In this decomposition, the linear states occur when the component oscillations have the same amplitude, i.e., , and elliptical states occur when the component oscillations are such that . The orientation of the plane of polarization or the orientation of the major axis of the ellipse is decided by the phase difference between two circular components .

Between the two types of decompositions, in achieving the azimuth and ellipticity controls, the conditions on component amplitudes and phases are reversed. Figure 7 summarizes the variation in azimuth and ellipticity in linear and circular decompositions. Note that, in linear decomposition, as one moves from left to right, change in amplitude changes the azimuth of polarization state, but ellipticity remains constant. Similarly, as one moves from top to bottom, change in phase difference between two orthogonal linear states leads to ellipticity change and azimuth remains constant. On the contrary, in circular decomposition, as one moves from left to right (top to bottom), change in amplitude (phase) between two orthogonal circular polarization components produces change in ellipticity (azimuth).

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Figure 7: Role of amplitude and phase difference variation in azimuth and ellipticity in (a) linear and (b) circular basis.

5.3. Stokes Parameters

The state of polarization of light can be described using Stokes parameters [4–7]. Using intensity measurements, these parameters can be found as

where , , , , , and are the component intensities when a given SOP is decomposed into linear states oriented along , and circular states: left circularly polarized (LCP) and right circularly polarized (RCP) states, respectively. For a fully polarized light, . For a linearly polarized light, the Stokes parameter and ; for circularly polarized light, both and and . Unlike homogeneous polarization distribution, for inhomogeneous polarization distribution, all the Stokes parameters are functions of position coordinates. This means the Stokes parameters to 3, are functions of two variables.

5.4. Geometric Representation of Polarization

Any general SOP of light can be represented by a point on the surface of a Poincaré sphere. Poincaré sphere is a unit radius sphere constructed using three normalized Stokes parameters (, , and ) as three coordinate axes. The poles on the sphere represent orthogonal circular polarization states, and points on the equator represent linear polarization states. All other points in Northern and Southern hemisphere correspond to right- and left-handed elliptical polarization states, respectively. A Poincaré sphere with various SOPs is shown in Figure 8(a). The parameters azimuth () and ellipticity () of a polarization ellipse are related to Stokes parameters as follows:

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Figure 8: (a) A Poincaré sphere: geometric representation of homogeneous polarization distribution; (b) polarization variation along azimuth and ellipticity contours.

The SOPs on a particular latitude maintain constant ellipticity, whereas SOPs on a longitude maintain constant azimuth. This is depicted in Figure 8(b).

Stokes parameters when defined in terms of the orthogonal polarization field components take the form, as mentioned in Table 1. The subscripts correspond to in linear decomposition, in linear diagonal decomposition, and in circular decomposition. The Stokes parameter is always the sum of orthogonal component intensities in respective decomposition, whereas the expressions of Stokes parameters (, and ) change cyclically in three different field decompositions.

Table 1: Stokes parameters: cyclic rotation of Stokes parameters, , in three different decompositions.

Decomposition
Linear
Linear diagonal
Circular

The subscripts “p” and “q” in linear basis represent “x” and “y”; in diagonal basis, they represent “+45°” and “−45°”; in circular basis, they represent “R” and “L,” respectively.

5.5. Helicity and Spin

Helicity and spin are different [108], and treating them as synonymous to each other is a common mistake committed by many. Photons are bosons, integer spin (spin = ) particles, with spin angular momentum (SAM) of for right circular polarized light and for left circular polarized light. In the circular basis decomposition (equation (6)), we have seen that any polarization state of light is shown as a superposition of right and left circular polarization components (superposition of positive and negative spin states). The component amplitudes decide the helicity (or handedness) of the superposition state. If the right circular polarization component is larger than the left circular polarization component , then the resulting elliptical polarization state is said to be right-handed ellipse state and so on. Traditionally, right-handed (left-handed) polarizations are considered as positive (negative) helicity (handedness). For example, a left elliptically polarized light has negative helicity (handedness) but has both positive and negative spin components. Therefore, spin and helicity are different parameters. For linear polarization, both the right and left circular polarization components are equal , and therefore, there is no handedness associated with the linear states.

6. Stokes Phase

Using the Stokes parameters, Stokes fields can be constructed. These fields are mathematical constructions. For example, using Stokes parameters and , a complex field, namely, field, can be constructed [8, 9, 109, 110]. The Stokes phase and is equal to , as given in equation (8). Hence, the phase vortices of complex Stokes field are the polarization singularities. Therefore, constructing a Stokes field from the measured Stokes parameters is helpful in identifying the polarization singularities. In Figure 9(a), in the polarization distribution, presence of two polarization singularities with opposite index can be identified as two phase singularities in the Stokes phase distributions. A V-point singularity and its Stokes phase distribution is shown in Figure 9(b).

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Figure 9: Polarization distribution and Stokes phase of (a) dipole field and (b) a V-point field. (c) Limitations in Stokes phase: the Stokes phase has same spatial distribution for a V-point of index , right-handed and left-handed bright C-point of index , and a dark C-point of index , and thus, the Stokes phase is degenerate.

However, there are certain limitations in using the Stokes phase. The Stokes phase distribution does not distinguish the right- and left-handed C-point singularities. It does not distinguish between dark and bright C-point singularities. Also, it does not distinguish between integer charged C-points and V-point singularities. As an example, four different polarization distributions with the same Stokes phase distribution are illustrated in Figure 9(c). The Stokes index is connected to IC index and Poincaré Hopf index η by the relation . Similar to Stokes phase degeneracy, intensity degeneracies have been observed in interference and diffraction of polarization singularities [111].

6.1. Stokes Phase and Scalar Fields

Phase distribution of Stokes field is found to be related to phase difference between RCP and LCP components in circular basis. From Table 1, the corresponding expressions for and in circular basis are

Using equation (9), we construct Stokes field as

Likewise, using expression of S₂ and S₃ from Table 1, the phase distribution of Stokes field is found to be related to phase difference between and components in linear basis as

Similarly using Table 1, phase distribution of Stokes field is found to be related to phase difference between and components in linear diagonal basis as

It is surprising to see by using component intensities in linear states, the phase difference between circular basis states can be obtained. That is by using S₁ and S₂ the phase difference between left and right circular basis states can be obtained. Similarly, by using S₂ and S₃ the phase difference between vertical and horizontal component states ϕ₂₃ can be obtained. Note neither S₂ nor S₃ need intensity measurements in horizontal and vertical states. Similar arguements hold good for ϕ₃₁.

7. Phase Distribution and Phase Difference Distribution

The phase distribution of a scalar field is that of the complex field, whereas the Stokes phase distribution is that of phase difference distribution. Phase distribution of a scalar field corresponds to wavefront structure, which is seen in homogeneous polarization distribution. In the interference of two scalar beams both in the same SOP, the resulting intensity variation depends on the phase difference between the interfering beams. These intensity modulations are referred as intensity fringe pattern in text book. But if the two interfering scalar fields are such that one is in one SOP and other in the orthogonal SOP to the first one, then the phase difference between these beams does not modulate the intensity but changes the SOP in the resultant field. In other words, one can realize polarization fringes instead of intensity fringes by changing the phase difference between orthogonal polarization components of any field. Interestingly, this phase difference distribution is found to be linked to phase distribution of a complex Stokes field. Referring to equations (10)–(12), the Stokes phases are related to phase differences and are reproduced as , , and . Surprisingly, the Stokes phases which are phase differences between orthogonal polarization states can be obtained from the Stokes parameters which are pure intensity measurements. Of the three Stokes phases, the Stokes phase is related to the azimuth of the SOP, and hence, it plays an important role in the polarization singularities as they have circulating . This can be seen from equations (3) and (4). Polarization singularities are Stokes phase vortices. Presence of a vortex in Stokes phase indicates that there may be vortices in and Stokes phases also [110, 112].

8. Experimental Realization of Polarization Vortices from Phase Vortices

In this section, we demonstrate experimental generation of polarization singularities such as bright C-points (lemons and stars), dark C-points, polarization flowers, and spider webs. The interferometer presented in Figure 10 is universal in the sense that it can be used for generating any type of phase as well as polarization singularities.

8.1. Experimental Setup

Collimated He-Ne laser light illuminates a polarizer at . The light coming from the polarizer is equally split into two arms by a polarizing beam splitter (PBS). By inserting spiral phase plates (SPPs) of different charges and in the two arms of the Mach–Zehnder-type configuration, it is possible to realize bright C-points, dark C-points, and V-points at the output. In the two arms of the interferometer, SPPs transform the orthogonal linearly polarized plane waves ( and ) into and polarized vortex beams of charges and , respectively. Note that the beams coming from the two arms are homogeneously polarized, and each contain a phase vortex. Therefore, by blocking one of the arms of the interferometer, this setup can be used for the generation of phase singularities. Since these beams are homogeneously polarized, we can call them as scalar vortices. The beam splitter (BS) combines these two scalar vortices to form polarization singularities. For superposition to be in circular basis, a quarter wave plate (QWP) at is used after the BS. The Stokes camera (SALSA: Full Stokes Polarization Imaging camera, Bossa Nova, USA) is used to record the experimental Stokes parameters, and these parameters are used to plot the corresponding polarization distributions. The different combinations of and lead to generation of bright C-points, dark C-points, and V-points. These are illustrated in Figure 11 and elucidated in following sections.

Figure 10: Experimental generation of Stokes phases and polarization singularities using scalar vortices. P, polarizer; (P)BS, (polarizing) beam splitter; and , mirrors; SPP, spiral phase plate; QWP, quarter wave plate; L, lens; stokes camera.

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Figure 11: Experimentally recorded polarization distributions containing (a–f) C-points and (g–i) V-points. Indices of C-points and V-points are mentioned above each figure. (a–d) Bright C-points; (e-f) dark C-points. V-points always occur at intensity null. (g) A generic V-point; (h) a polarization flower; (i) a spider web.

8.2. Case 1: Bright C-Points or

In interference, bright C-points are generated when one of the superposing beams is a nonvortex beam, i.e., the beam has a plane wavefront. Figures 11(a)–11(d) depict bright C-points with () combinations as (), (), (), and (), where and are topological charges of scalar vortices in right circular polarization (RCP) and left circular polarization (LCP) states, respectively. The index of the C-point is mentioned in each figure and is given as . Note that, in each case, the handedness of the C-point is decided by the handedness of the nonvortex beam. In the superposition, the amplitude corresponding to the vortex core is zero, whereas the amplitude corresponding to the nonvortex beam is nonzero at the same point. Hence, the resultant state is circular, and its handedness is that of nonvortex beam. At every other neighbourhood points, since the amplitudes of the two beams are unequal in the circular basis superposition, elliptical states result. The phase difference between the two beams (Stokes phase) is that of helical phase, and this leads to rotation of azimuth around the C-point. The C-point in Figure 11(a) is right-handed (RH), whereas they are left-handed (LH) in Figures 11(b)–11(d). C-points can occur at any value of intensity. For example, in Figures 11(a), 11(b), and 11(d), C-points occur at intensity maxima, whereas C-point in Figure 11(c) occurs at intermediate value of intensity. Note for the C-point generation, both the interfering beams must have dissimilar amplitude distribution, and the phase difference between them must be helical.

8.3. Case 2: Dark C-Points

When both the beams in the interferometer contain vortices such that , dark C-points are produced. Note this combination satisfies the amplitude and phase difference condition for C-point generation. The only difference here is that since both the beams have dark vortex cores, the resulting C-point is a dark C-point. Two examples of dark C-points with indices , for the and combination (), and , for the and combination (−1, 4), are shown in Figures 11(e) and 11(f), respectively. Note that, in case of a dark C-point, the handedness is decided by the circular polarization component that has a lower magnitude of topological charge. As the magnitude of vortex in LCP (RCP) component has a lower value, the dark C-point in Figure 11(e) (Figure 11(f)) is LH (RH).

8.4. Case 3: V-Points

To generate a V-point, same intensity variation should exist in two orthogonal circular polarization components. This can be achieved by introducing SPPs in two arms of the interferometer such that . This combination of vortices generates V-points with index . Some experimentally generated V-points with index , , and are shown in Figures 11(g)–11(i), respectively. The respective (, ) combination for each case is (, 1), (, 2), and (2, −2). As both the circular components contain a vortex, V-points always occur at intensity minima, as depicted in Figures 11(g)–11(i). Figure 11(h) is an example of a polarization flower as it has two petals, whereas Figure 11(i) is an example of a spider web.

9. Superpositions

We have seen that the ellipse field and vector field singularities can be expressed as the superposition of beams in orthogonal spin and orbital angular momentum states:

where and are right and left circular unit basis vectors, respectively. and are constants, and are vortex charges in each beams, and is the constant phase shift. In the above equation, the superposition is between two phase vortex beams of unequal charges, and they are in orthogonal polarization states. As far as spatial modes are concerned, two vortex beams with unequal charges themselves are orthogonal to each other. By considering different combinations of vortex charges in right and left circular polarization states, different polarization singularity distributions can be constructed and that state can be represented as a point on either one of the spheres, namely, higher order Poincaré sphere (HOPS) [113–117] or hybrid order Poincaré sphere (HyOPS) [118–121]. Figure 12 depicts construction of various spheres by considering two orthogonal OAM states in same SOP, as basis states to form modal sphere, and then two orthogonal OAM states in two orthogonal SAM states, as basis states to form other types of spheres.

Figure 12: Construction of various spheres. (a) Fundamental Poincaré sphere: orthogonal spin angular momentum (SAM) states at the poles. (b) Modal sphere: orthogonal orbital angular momentum (OAM) states at the poles. (c) Higher order Poincaré sphere: orthogonal SAM and OAM states at the poles with equal magnitude of OAM states. (d) Hybrid order Poincaré sphere: orthogonal SAM and OAM states at the poles with different OAM states. Note that in each sphere, the eigenstates are different.

For every pair of vortex state in right and left circular polarization states, a HOPS or HyOPS can be constructed. With infinite number of orthogonal vortex states available, it is therefore possible to realize infinite number of these types of spheres and hence infinite number of polarization singularity distributions. In each sphere, every point represents a polarization singularity distribution, and all the points on the sphere can be realized by changing the values of and in that superposition. All the possible polarization singularity distributions in a given sphere have the same polarization singularity index. Because of the way these spheres are constructed, the beams represented by points on each of these spheres have different topological features. Beams represented by points on HOPS are constant ellipticity fields, and beams represented by points on HyOPS are Poincaré beams [112, 122–129]. So far introduced, V-points lie on the surface of HOPS (equatorial points), whereas C-points lie on the surface of HyOPS (equatorial points). It is interesting to note that superposition of vortex beams in orthogonal linear polarization states will also produce Poincaré beams [112]. Elliptically polarized base states were also tried for vector field generation [130]. In the reverse conversion, it is also possible to realize homogeneous polarization distributions as superposition of polarization singularity distributions [131]. We have seen that HOPS beams and HyOPS beams are superpositions. These beams are also called spin-orbit beams or beams in nonseparable states of spin and orbital angular momenta. A separable state is a state that can be written as a single product of spin state and orbital state similar to the way separable functions are defined, for example, . Beams having homogeneous polarization across the beam cross section such that the spin part and orbital part can be separately determined are said to be in separable state, like in scalar vortex beams. But polarization singularities are in nonseparable states. Each point on the equator of a HyOPS represents a C-point, whereas each point on the equator of a HOPS is a V-point polarization singularity. Beams represented by these spheres are called HyOPS beams and HOPS beams. Nonpolar and nonequatorial points on HOPS represent a vector vortex beam (VVB) with constant, nonzero ellipticity, and varying azimuth. For the generation of HyOPS beams, use of hologram [132], sectorial phase plate [133], and electrically driven devices [134] have been reported. These HyOPS beams can be considered as Poincaré beams, and there is lot of interest in them [122–129, 135, 136]. By anisotropic polarization modulation [137], Poincaré beams can be generated. Other beams such as Mathieu–Poincaré beams [138], entangled vector vortex beams [139], and Poincaré–Bessel beams [140] are also subject of interest. These elliptic field singularities are also generated in photoelastic stressed medium [141, 142] and studied. Tight focusing of full Poincaré beams [143] and the forces exerted by these beams on submicron particles were also studied [144].

10. Literature Survey of Phase and Polarization Singularities

10.1. Phase Singularities

In early seventies, the idea of phase singularity in electromagnetic waves was first introduced by Nye and Berry [11] while studying radio echoes from the bottom of Antarctic ice sheets. Like the crystal defects, optical wavefronts also exhibit phase defects. These defects are found in large numbers in random fields [15, 68, 145–149]. Presence of point phase defects in laser modes [150] is reported in 1983. There are also some early studies in the 80s [151, 152]. Vortices appear as solutions of wave equations in cylindrical coordinates. Many authors have studied their propagation characteristics [39, 153–156], with aberrations such as astigmatism [157, 158] and coma [159]. Role of Gouy phase during propagation was studied in [160]. Propagation through obstructions also came under study [161–164]. The concept of optical vortices has also been applied in phase retrieval algorithms [55, 165–167].

In scalar optical fields, the properties of phase singular beams came under study [31, 36]. The topological properties of vortices are studied in detail [37–39, 168–173]. The studies on critical points such as maxima, minima, saddles, and singular points in vortex rich optical fields came under thorough study [174–176]. There are also sign rules that describe the distribution of these critical points [12, 17, 174–177]. Vortex trajectories with the help of topological manifolds, leading to formation of knots, links, and loops, were studied [170, 171, 178–184]. Other types of phase defects such as edge and mixed-type phase defects [32, 34], anisotropic vortices [185], and perfect [76, 186–194] and fractional vortices came under study [195–204]. Some of the earliest optical elements capable of producing phase singularities were reported [58, 59, 61]. But the widely cited papers on the use of spiral zone plate for vortex generation were reported [60, 205] in 1992.

At the same time, reports on orbital angular momentum [206] carried by the helical waves appeared. Research articles on orbital angular momentum of light described [206–215] the central role played by the phase singularities. Other than LG beams, beams with helical wavefronts such as Bessel beams [216], Mathieu beams [217], and Ince–Gaussian beams [218] were also found to carry orbital angular momentum. Helico-conical beams also are known to contain OAM states [219, 220] since they are the product of helical and conical waves [221]. These beams have self-healing property [222, 223]. Vortex preserving statistical optical beams [224] were also reported. The analogy between paraxial optics and quantum mechanics was used [225, 226] to explain the OAM in light beams. Divergence of vortex beams was studied [227–231] to understand the propagation characteristics [39, 40, 160, 185, 232]. In [233], the transformation of vortex beams from fractional fork holograms due to Gouy phase was demonstrated.

The energy flows in a vortex beam have circulating components about the vortex core. The transverse flow has two contributions coming from spin and orbital angular momentum. The virtual spin part [234] is due to polarization of light. The energy flow in an optical field can be visualized by Bekshaev–Bliokh–Soskin method [235–239]. Helmholtz–Hodge decomposition method has also been applied to demonstrate internal energy flows in scalar optical fields [35, 240]. An analogy between azimuthons (found in nonlinear media) and rotating transverse energy flow structures in paraxial beams is demonstrated in [241]. Patterns of energy flows in dipole vortex beams were studied using a knife-edge test [242]. Effect of astigmatism [243] and coma [244] on the transverse energy distributions was also investigated. Mechanical action of the spin part of the energy flow was reported in [245, 246]. The skew angle of Poynting vector in a helically phased beam was measured in [247]. The handedness and azimuthal energy flow in optical vortex beams is also reported [248]. The radial [38, 169, 249] components of the propagation vector in a vortex beam came under scrutiny of few groups. Several other theoretical [25, 26, 250–254] and experimental [255–260] investigations exist for Poynting singularities in transverse energy flow.

10.1.1. Generation

Conversion between Hermite–Gaussian modes to Laguerre–Gaussian modes using cylindrical lenses is one of the early methods [261, 262] of vortex generation. Optical vortices can be generated in diffraction orders [263, 264] of specially designed diffraction grating. Use of chiral fiber grating is also reported for vortex generation [265]. Intracavity generation by introducing a spot defect in one of the resonator mirrors is also demonstrated [266–268]. Mirrors bent in the shape of a ramp [213, 269, 270] and digital micromirror device [192] can also be used for phase singularity generation. There are also reports [271–279] on the generation of vortices in optical fibers. Vortex generation in high power laser is possible with the use of fused silica fibers [280]. Use of micropatterned optical fiber tip has also been reported for the generation of optical vortices [281].

For the vortex generation use of diffractive lens [62] and newly designed diffractive optical elements (DOE) [282–284], mode converters [261, 262], spiral phase plate [57, 285], and a new element called q-plate [286–288] have been reported. Other methods of vortex generation include use of nonspiral phase plates [63], plexi glass [64], wedge plates [156, 289, 290], stack of wedge plates [291], spatial light modulators [187, 292], anisotropic media [293], spatial filtering [65], laser etched mirrors [66], micro-electromechanical systems [294], adaptive mirrors [67, 295, 296], and laser with large Fresnel number [23]. Use of photo polymerization and micromachining has also been introduced [297, 298] for SPP generation. It is also possible to generate vortices using nonspiral phase plate [63], where half of the beam cross section passes through a glass plate which is curved and adjustable phase plates [64] in which the amount of twist given to the plate can yield higher charges. Generation of vortices is also possible by controlling polarization of light [299] and by engineering astigmatism [300]. Spin to orbital momentum conversion methods [301, 302] were also reported for vortex generation. Spin to orbital angular momentum conversion is also possible by focusing, imaging, and scattering [303]. Tunable vortex generation method was also reported [304].

Spiral zone plates can be used [13, 60, 305] for vortex generation. Other than wedge plates, local tilts introduced in parts of the wavefront [19, 20] can be used for the generation of array of vortices of same charge. Sagnac interferometer can be used for vortex generation [306]. In the array form also, these vortices were generated [307]. There are many interference-based methods reported [71, 308–313] for vortex array generation in which multiple beams with non-coplanar propagation vectors are made to interfere. Vortex array generation by spiral Dammann zone plates [314] and spiral square zone plate [315] was also reported.

10.1.2. Detection

Formation of fork fringes by interference was suggested for vortex detection [78, 316], and this was the first reported and widely used method. On the contrary, fork gratings can be used for vortex generation also [317]. For phase singularity detection, however, initially, there were very few methods reported [318]. Later, many diffraction-based methods were reported [80, 81, 263, 319–323]. Some interference-based methods for vortex detection include phase shifting [190, 324], modified Mach–Zehnder [325, 326], and Fizeau interferometer [327]. Another interference method, based on lateral shear interferometry, was reported [33, 79] in 2008. Being a self-referencing method, this is one of the simplest methods available for vortex detection. A correlation-based detection technique was also employed for vortex detection [328]. The effect of aberrations, on vortices [158, 329–334], was also studied for vortex detection. Other methods include the use of Shack–Hartmann sensor [335]. Some other vortex sorters are presented in [336] and [337]. It has been shown that OAM content can be determined by using a cylindrical lens pair [338]. Diffraction patterns produced by phase singular beams are different, and they can be used for the detection of vortices in a beam. Normally, a higher topological charge vortex is unstable [339] and disintegrates into unit charged vortices under perturbation. Diffraction by single [80, 340], double [38, 319, 341], and multiple slits [320] was studied and used for vortex detection. Multiple slits correspond to grating, and special gratings were also designed [263, 321, 342] for the diffraction study. Apertures of different shapes ranging from triangular [81, 343], circular [344, 345], diamond-shaped [161], hexagonal [346], regular polygon [347], and annular [151, 155, 348, 349] were also used in diffraction experiments. Additional methods for vortex charge determination based on using twisted phase element [350], hyperbolic grating [351], axicon [352], spiral spectra [353], single point detector [354], and Talbot effect [355] were also reported in literature.

10.1.3. Applications

Singular beams found many applications. They are useful in optical meteorology for wavefront tilt measurement [356, 357], wavefront reconstruction [41, 358], phase unwrapping [165–167], vortex sign [359, 360], and vortex charge determination [361, 362] by optical vortex interferometer. Optical vortices are also used for collimation testing [42] and in spiral interferometry [44, 363, 364] in which the peak and valley can be detected unambiguously. Problems arising during misalignment of vortices is discussed in [365]. By using elements having vortex transmittance function, radial Hilbert transform mask isotropic edge enhancement [46, 61, 366] is possible. Modification to this radial Hilbert mask leads to selective edge-enhancing capabilities [202, 366–370] in the spatial filtering systems. The spiral phase filter introduced in microscopy [364, 371, 372] leads to phase contrast imaging. Optical vortices can also be used to perform high precision astronomy and tip/tilt correction [373] and used in coronagraphs [48, 374]. Annular intensity pattern of STED beam can be achieved using a vortex phase plate [375–380]. Introduction of vortex in diffracting field can offer better phase retrieval [45] algorithms with better capabilities. Studies on vortices lead to the construction of speckle-free reconstruction of phase randomized holograms [54, 56, 381] and diffractive optical elements. Application of singular optics arising due to the OAM carried by vortex beams [382–391], include trapping [392–399] and rotation [210, 400–407] of microscopic particles. These OAM states being orthogonal can be used as transmitting channels [22, 52, 53, 408–410] in free space as well as in fibers for communication. Vortex states were also tried for underwater communication [411] and free space communication [412]. There are some weak measurements [413, 414] suggested with singularities. Vortices also have healing properties [415, 416] so that restoration is possible.

Shaping the focal structure in optical systems has been an active area of research for a long time. Focal shaping is possible by pupil function engineering by modifying amplitude, phase, and polarization distribution of the wave that is focused. In high numerical aperture (NA) systems, the polarization distribution of the beam also plays a vital role. Focusing of singular beam leads to generation of a doughnut structure in the focal plane of a lens in contrast to the well-known Airy pattern. Doughnut intensity structure is useful in several applications in fields such as microscopy, optical trapping, lithography, and astronomy. It has been observed that even with optics considered well-corrected, the intensity distribution of a singular beam gets distorted in the presence of small amount of azimuthally dependent aberrations, in comparison to that of the nonsingular beam [417]. Structural modifications in the focused structure of the singular beam have been carried out in [418]. Singular beams focused by aberrated system disturb the doughnut pattern significantly. The focal plane intensity distribution under the influence of spherical aberration [419–422], astigmatism [329, 330, 423, 424] and coma [425–428] are also studied. Astigmatism was used to invert the sign of the topological charge of a vortex [429].

10.2. Polarization Singularities

Polarization singularities occur naturally in daylight sky [18, 430] and has been subject to various atmospheric studies [431–435]. Polarization patterns consisting of polarization singularities in cosmic sky have also been investigated [436, 437]. Topological singularities have also been observed in disclinations in liquid crystals [438, 439], fingerprints [440, 441], and umbilic points in the curvature of surfaces [442, 443].

In optical fields, polarization singularities may occur where state of polarization varies with position. There is a large interest in paraxial fields with slowly varying polarization distributions in recent years [82–84, 444–446]. Singularities in the polarization state of partially coherent wavefields are also gaining interest [447]. Ellipse fields have spatial distribution of elliptical SOPs, whereas vector fields have spatially varying linear SOPs. In polarization singularities, the direction of polarization azimuth is a crucial parameter [109]. Similar to phase in optical phase vortex, all possible values of polarization azimuth occur at the polarization vortex point. In an inhomogeneous polarization distribution of ellipse fields, C-points are points of circular polarization state, whereas L-lines are linear polarization states at which polarization azimuth and handedness are, respectively, undefined. The neighborhood of C- point has SOP distribution consisting of polarization ellipses with their azimuths oriented in clockwise or anticlockwise directions. In a spatially varying distribution of linear states, polarization singularities are V-points. The sense and number of rotation(s) of the azimuth in one complete closed path around the singularity decides the polarity and index of the polarization vortex, respectively. The polarization singularities form optical Mobius strips in three-dimensional fields [448, 449]. In three dimensions, circular polarization occurs along C lines and linear along L surfaces [448, 450–453]. Poincaré vortices [454] are another type of Stokes singularity at which the SOP is linear. There are reports on refraction [455] and reflection [456] of C-lines. During propagation, the SOP distribution in the polarization singular beams are found to rotate due to Gouy phase [457, 458]. The pattern can also undergo rotation by acquiring the Pancharatnam phase while travelling through appropriate optical elements [97]. Hamiltonian based on the Maxwell–Schrödinger equation has been used for the analysis of Pancharatnam–Berry phase of VVBs [459].

Similar to phase vortices, polarization singularities (which essentially are phase vortices in Stokes phase) in a distribution are also governed by sign rule. According to the sign principle, adjacent vortices along a stokes phase contour must alternate the index sign [10, 460–462]. Fields laced with only C-points [110, 463] and of purely V-points but with opposite indices [102] have been reported. Interference field distributions interlaced with mixtures of C-points and V-points are also there [93, 109, 464]. Sign conservation is followed during diffraction also. It has been observed that a V-point with index disintegrates into number of C-points, each with index , and same polarity as that of a V-point. Of these generated C-points during diffraction, half the number of C-points are right-handed and half are left-handed. This means that handedness conservation is also observed during diffraction.

10.2.1. Generation

One of the methods to generate polarization singularities is by using a universal interferometer presented in Section 8. Usage of a spiral phase is unavoidable in all the interferometric setups. These methods are highly sensitive and demand precise alignment of the cores of the overlapping vortices [465, 466]. Other efficient methods to generate radially polarized beams are by an image rotating resonator [467] or with a double interferometer [468, 469]. Polarization singularity generation by using a Twyman–Green interferometer [470] and a Wollaston prism [471] have also been reported. Vector beams embedded with polarization singularities can also be realized by a noninterferometric technique [472].

Cylindrical vector beams can be generated by introducing polarization-selective mirror inside a laser resonator cavity [473–475]. In this technique, the resonator is made polarization selective by a binary dielectric diffraction grating which is etched at the backsurface of the mirror substrate. Use of calcite crystal [476], windows [477], axicon [478], polarization selective grating mirror [479], image rotating mirror arrangement [467], polarization selective GIRO (giant reflection to zero order) mirror [473], conical Brewster prism [480], polarization-based beam displacer [481], an undoped c-cut crystal [482], conical prism [483] inside a resonator have been reported for the generation of cylindrical vector beams. Generally, these methods produce positive index V-point singularities. To produce their negative counterparts, a HWP can be inserted outside the cavity. Intracavity methods can be employed for generation of only V-points and not for C-points. Generation of radially polarized beams can also be achieved by conical diffraction [484], spatially varying subwavelength grating structures [485–487] and diffractive optical elements [488].

Commercially available spatial light modulators (SLMs) can also be used to generate polarization singularities. As SLMs respond to only one linear polarization, this fact can be used to tailor the phase of the beam [107, 489–494]. Simultaneous generation of multiple vector beams on a single SLM is reported in [495]. Similar to SLMs, recently developed digital micromirror devices (DMDs) are being used for producing structured light fields [496, 497].

Another method of generation of vector vortex beams utilizes segmented spatially varying waveplates [498, 499] or subwavelength grating structures [485, 486, 500, 501] or metasurfaces [502–505] that work on diffraction phenomenon. Generation of VVB with a single plasmonic metasurface is depicted in [506]. S-waveplates are also used for generation of vector beams. S-waveplates are segmented half waveplates, with each segment having different orientations of crystal’s optic axis. Depending on the input plane of polarization of linearly polarized light, radial or azimuthal or superposition states of radial and azimuthal can be realized. Similar to S-waveplates, liquid crystal-based q-plates are also used to generate and manipulate vector beams [69, 507–511]. The q plates can be combined with HWPs to obtain higher order polarization singularities [512]. Use of q-plate as a coupler has been demonstrated recently [513]. There is also increased research activity in q-plate fabrication methods [286, 507, 510, 514–520].

Generation of vector vortex beams using polarization gratings has also been reported recently [521]. Vector vortex beams were also generated in optical fibers [522–524]. Generation methods also include metamaterial-based Pancharatnam–Berry phase elements [525–528]. There are also other methods reported for VVB generation [130, 529–532]. The topology of VVB was also discussed [258, 533]. Generation of vector beams using a double-wedge depolarizer [534], ring resonator [535], caustic-based approach [536], and parametric oscillator [537] was also demonstrated. Generation of broadband vector beams with tunable phase and polarization has been demonstrated in [538]. Achromatic VVBs can be produced from a glass cone [539]. Fractional polarization vortices by Sagnac interferometer [540] and radially polarized fractional vortex beam [541] were also generated and studied. A single integrated on chip device has been proposed that allows switchable radially and azimuthally polarized VVB [542].

Polarization singularities can also be found in random fields. In structured distributions, they appear in lattice form. There are several reports on lattices of only C-points [110, 463], only V-points [102], and C-points interlaced with V-points [93, 109, 464]. A spatially varying lattice of C-points and V-points has also been generated [543]. All these engineered lattice fields obey sign rule and are generated by interference of multiple plane waves.

10.2.2. Detection

In inhomogeneous polarization distributions, Stokes parameters are spatially varying and can be measured by adopting standard Stokes polarimetry technique. Polarization singularities in an optical field can be identified by measurement of these Stokes parameters, as they are phase vortices of Stokes field [110, 112, 497]. The quality of the vector beams can be measured by vector quality factor (VQF) tool [544]. Another deterministic detection mechanism for vector vortex states utilizes classical and weak coherent states [545].

Detection of C-points in a field distribution is based on the identification of closed L-lines (s-contours) in multiple recorded interferograms [546]. The information about the presence of fork fringes in the interferograms can be used to locate and track path taken by L-lines. L-lines separate regions of right and left handedness in a polarization distributions and enclose a C-point singularity. Detection of V-points is relatively easier than C-points. For a V-point singularity with index , a polarizer produces lobes of intensity pattern. Another method of detection of V-points is based on diffraction. One of the first experiments on diffraction of V-points through triangular apertures of two types was reported recently [547]. In this method, both diffraction and polarization transformations were used to uniquely determine all states of V-points [548]. Diffraction of VVB through diamond-shaped aperture [549] and circular aperture [550, 551] has been reported recently. In diffracted near fields, exceptional polarizations structures have been found [552].

10.2.3. Applications

Polarization distribution of the beam becomes important in shaping the point spread function (PSF) in high NA focusing [553–557], and manipulation of the PSF by polarization distribution is referred to as polarization engineering [558]. Radially and azimuthally polarized light can be used to realize smallest focal point beyond scalar optics limit [554, 555, 559, 560]. These beams can also be used for particle acceleration [561, 562], trapping of metallic Rayleigh particles [563], optical manipulation [564], and optical signal processing [370, 565, 566]. Both point and edge phase dislocations are present in radial and azimuthal polarized beams [86] depending on which polarization component (either linear or circular) is extracted. Depolarization effects of laser beam propagation in turbulent atmosphere was studied [567, 568]. Propagation of radial/azimuthally polarized beams through turbulence also caught the attention of researchers [569–571]. Synthesized vortex beams are also subject of interest in the turbulent atmosphere [572]. Irradiance [573] and scintillation [574, 575] of radial/azimuthally polarized beams and Poincaré beams [576] propagating through atmosphere were examined. Use of C-point beams in turbulence-resistant robust beam generation has been proposed [577–580]. Reports on the use of polarization singularities for enhancing chiral light matter interaction are also there [88, 581–583]. Data-carrying fiber vector eigenmode multiplexing has also been reported [584] in communication [585, 586]. The nonseparability of VVB can also be used to encode information for optical communication [587]. Switching between phase and polarization singularity using metasurfaces [588] is also reported. Vector beams under the effect of perturbations form pair of fundamental and stable singularities that may be useful in weak field measurements [589, 590]. Splitting of C-points can be used as a tool for weak measurement of elliptical dipole moments [414]. Entanglement studies are also reported [591]. Characterization and manipulation of these vector vortex beams is a subject of study for laser matter interaction experiments [592]. Deep learning algorithms are also applied for turbulence aberration correction for VVBs [593].

The research area of polarization vortices (vector vortex beams or Poincaré beams) is relatively a new field compared to its scalar counterpart, i.e., optical phase vortices. Polarization vortices are still an emerging and active area of research with few review articles [18, 29, 594–596] published. There is also a review article [597], highlighting the applications of vector beams. Few review articles are also there in the area of structured light [598–600]. They are at a much higher and more eclectic level and hence may not be suitable for beginners. This article is believed to bridge the gap between the researchers at two extremes of the spectrum.

11. Conclusion

In conclusion, we have presented the method of generating polarization singularities from phase singularities from first principles. The azimuthal phase variation of the scalar vortices is converted into Stokes phase variation in a Mach–Zehnder-type interferometer. The interferometer presented here is universal in the sense that it can be used for generating both phase as well as polarization singularities. This interferometer can produce azimuth vortices with different attributes such as dark, bright, left-handed, right-handed, ellipse, and vector field singularities. We have presented new and intuitive explanations for Stokes phases and the connection between phase and polarization singularities. Experimental and simulations are included. In the first part, tutorial on the subject of polarization singularities as a natural extension of phase singularities is presented. Towards the end, a survey of activities on these two areas is presented.

Data Availability

The experimental data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The authors acknowledge the financial support from CSIR, India, through the research grant 03(1430)/18/EMR-II.

References

J. W. Brown and R. V. Churchill, Complex Variables and Applications, McGraw-Hill, New York, NY, USA, 1996.

G. Gbur, Mathematical Methods for Optical Sciences, Cambridge University Press, New York, NY, USA, 2011.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, Elsevier, Amsterdam, Netherlands, 2005.

D. H. Goldstein, Polarized Light, CRC Press, Boca Raton, FL, USA, 2011.

E. Hecht, Optics, Addison-Wesley, Reading, MA, USA, 4th edition, 1974.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, Cambridge University Press, Cambridge, UK, 7th edition, 1999.

R. Azzam and N. Bashara, Ellipsometry and Polarized Light, 1977.

I. Freund, “Polarization singularity indices in Gaussian laser beams,” Optics Communications, vol. 201, no. 4–6, pp. 251–270, 2002.

M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Optics Communications, vol. 213, no. 4–6, pp. 201–221, 2002.

I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, “Stokes singularity relations,” Optics Letters, vol. 27, no. 7, pp. 545–547, 2002.

J. Nye and M. Berry, “Dislocations in wave trains,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 336, no. 1605, pp. 165–190, 1974.

I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Physical Review A, vol. 50, no. 6, pp. 5164–5172, 1994.

P. Senthilkumaran, J. Masajada, and S. Sato, “Interferometry with vortices,” International Journal of Optics, vol. 2012, Article ID 517591, 2012.

N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, and B. Y. Zel’dovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” Journal of the Optical Society of America, vol. 73, no. 5, pp. 525–529, 1983.

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Optics Communications, vol. 101, no. 3-4, pp. 247–264, 1993.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Physical Review A, vol. 56, no. 5, pp. 4064–4075, 1997.

F. S. Roux, “Optical vortex density limitation,” Optics Communications, vol. 223, no. 1–3, pp. 31–37, 2003.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Chapter 5 singular optics: optical vortices and polarization singularities,” Progress in Optics, vol. 53, no. 8, pp. 293–363, 2009.

S. Vyas and P. Senthilkumaran, “Two dimensional vortex lattices from pure wavefront tilts,” Optics Communications, vol. 283, no. 14, pp. 2767–2771, 2010.

S. Vyas and P. Senthilkumaran, “Vortices from wavefront tilts,” Optics and Lasers in Engineering, vol. 48, no. 9, pp. 834–840, 2010.

A. Khoroshun, O. Chernykh, H. Tatarchenko et al., “Chain of optical vortices synthesized by a Gaussian beam and the double-phase-ramp converter,” OSA Continuum, vol. 2, no. 2, p. 320, 2019.

J. Wang, “Advances in communications using optical vortices,” Photonics Research, vol. 4, no. 5, pp. B14–B28, 2016.

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Optics Communications, vol. 73, no. 5, pp. 403–408, 1989.

M. V. Berry, M. R. Dennis, and M. Soskin, “Quantum cores of optical phase singularities,” Journal of Optics A: Pure and Applied Optics, vol. 6, no. 5, pp. S178–S180, 2004.

M. V. Berry, “Optical currents,” Journal of Optics A: Pure and Applied Optics, vol. 11, no. 9, 2009.

M. V. Berry and M. R. Dennis, “Stream function for optical energy flow,” Journal of Optics, vol. 13, no. 6, pp. 3–6, 2011.

P. Senthilkumaran, S. Sato, and J. Masajada, “Singular optics,” International Journal of Optics, 2012.

A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” in Progress in Optics, E. Wolf, Ed., vol. 47, pp. 291–391, 2005.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” in Progress in Optics, E. Wolf, Ed., vol. 42, pp. 219–276, 2001.

M. Soskin, S. V. Boriskina, Y. Chong, M. R. Dennis, and A. Desyatnikov, “Singular optics and topological photonics,” Journal of Optics, vol. 19, no. 1, 2017.

V. Bazhenov, M. Vasnetsov, and M. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Letters, vol. 52, no. 8, pp. 429–431, 1990.

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Optics Communications, vol. 103, no. 5-6, pp. 422–428, 1993.

D. P. Ghai, P. Senthilkumaran, and R. S. Sirohi, “Shearograms of an optical phase singularity,” Optics Communications, vol. 281, no. 6, pp. 1315–1322, 2008.

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Optics Communications, vol. 119, no. 5-6, pp. 604–612, 1995.

M. Bahl and P. Senthilkumaran, “Helmholtz Hodge decomposition of scalar optical fields,” Journal of the Optical Society of America A, vol. 29, no. 11, p. 2421, 2012.

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” Journal of Modern Optics, vol. 39, no. 5, pp. 985–990, 1992.

M. V. Berry, “Phase vortex spirals,” Journal of Physics A: Mathematical and General, vol. 38, no. 45, pp. L745–l751, 2005.

P. Senthilkumaran and M. Bahl, “Young's experiment with waves near zeros,” Optics Express, vol. 23, no. 9, pp. 10968–10973, 2015.

P. Lochab, P. Senthilkumaran, and K. Khare, “Near-core structure of a propagating optical vortex,” Journal of the Optical Society of America A, vol. 33, no. 12, pp. 2485–2490, 2016.

F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre-Gaussian beams,” Optics Communications, vol. 250, no. 4-6, pp. 218–230, 2005.

J. Masajada, A. Popiołek-Masajada, and D. M. Wieliczka, “The interferometric system using optical vortices as phase markers,” Optics Communications, vol. 207, no. 1–6, pp. 85–93, 2002.

P. Senthilkumaran, “Optical phase singularities in detection of laser beam collimation,” Applied Optics, vol. 42, no. 31, pp. 6314–6320, 2003.

K. V. Sriram, M. P. Kothiyal, and R. S. Sirohi, “Self-referencing collimation testing techniques,” Optical Engineering, vol. 32, no. 1, pp. 94–101, 1993.

S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral interferometry,” Optics Letters, vol. 30, no. 15, pp. 1953–1955, 2005.

M. K. Sharma, C. Gaur, P. Senthilkumaran, and K. Khare, “Phase imaging using spiral-phase diversity,” Applied Optics, vol. 54, no. 13, pp. 3979–3985, 2015.

K. Crabtree, J. A. Davis, and I. Moreno, “Optical processing with vortex-producing lenses,” Applied Optics, vol. 43, no. 6, pp. 1360–1366, 2004.

P. Bouchal and Z. Bouchal, “Selective edge enhancement in three-dimensional vortex imaging with incoherent light,” Optics Letters, vol. 37, no. 14, pp. 2949–2951, 2012.

G. Foo, D. M. Palacios, and G. A. Swartzlander, “Optical vortex coronagraph,” Optics Letters, vol. 30, no. 24, pp. 3308–3310, 2005.

P. Bianchini, C. Peres, M. Oneto, S. Galiani, G. Vicidomini, and A. Diaspro, “STED nanoscopy: a glimpse into the future,” Cell and Tissue Research, vol. 360, no. 1, pp. 143–150, 2015.

A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Optics Letters, vol. 11, no. 5, pp. 288–290, 1986.

P. Senthilkumaran, Singularities in Physics and Engineering, IOP Publishing, Bristol, UK, 2018.

T. Lei, M. Zhang, Y. Li et al., “Massive individual orbital angular momentum channels for multiplexing enabled by Dammann gratings,” Light: Science & Applications, vol. 4, no. 3, Article ID e257, 2015.

C. Perumangatt, N. Lal, A. Anwar, S. Gangi Reddy, and R. P. Singh, “Quantum information with even and odd states of orbital angular momentum of light,” Physics Letters A, vol. 381, no. 22, pp. 1858–1865, 2017.

P. Senthilkumaran and F. Wyrowski, “Phase synthesis in wave-optical engineering: mapping- and diffuser-type approaches,” Journal of Modern Optics, vol. 49, no. 11, pp. 1831–1850, 2002.

D. C. Ghiglia and M. D. Pritt, Two Dimensional Phase Unwrapping: Theory, Algorithms and Software, Wiley, New York, NY, USA, 1998.

P. Senthilkumaran, F. Wyrowski, and H. Schimmel, “Vortex Stagnation problem in iterative Fourier transform algorithms,” Optics and Lasers in Engineering, vol. 43, no. 1, pp. 43–56, 2005.

M. W. Beijersbergen, R. P. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical wavefront laser beams produced with a spiral phase plate,” Optics Communications, vol. 112, no. 5-6, pp. 321–327, 1994.

P. Szwaykowski and K. Patorski, “Moire fringes by evolute gratings,” Applied Optics, vol. 28, no. 21, pp. 4679–4681, 1989.

C.-W. Chang and D.-C. Su, “Collimation method that uses spiral gratings and Talbot interferometry,” Optics Letters, vol. 16, no. 22, pp. 1783-1784, 1991.

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Optics Letters, vol. 17, no. 3, pp. 221–223, 1992.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” Journal of Modern Optics, vol. 39, no. 5, pp. 1147–1154, 1992.

F. S. Roux, “Diffractive lens with a null in the center of its focal point,” Applied Optics, vol. 32, no. 22, pp. 4191-4192, 1993.

G.-H. Kim, J.-H. Jeon, K.-H. Ko, H.-J. Moon, J.-H. Lee, and J.-S. Chang, “Optical vortices produced with a nonspiral phase plate,” Applied Optics, vol. 36, no. 33, pp. 8614–8621, 1997.

C. Rotschild, S. Zommer, S. Moed, O. Hershcovitz, and S. G. Lipson, “Adjustable spiral phase plate,” Applied Optics, vol. 43, no. 12, pp. 2397–2399, 2004.

C.-S. Guo, Y. Zhang, Y.-J. Han, J.-P. Ding, and H.-T. Wang, “Generation of optical vortices with arbitrary shape and array via helical phase spatial filtering,” Optics Communications, vol. 259, no. 2, pp. 449–454, 2006.

J. Strohaber, T. D. Scarborough, and C. J. G. J. Uiterwaal, “Ultrashort intense-field optical vortices produced with laser-etched mirrors,” Applied Optics, vol. 46, no. 36, pp. 8583–8590, 2007.

D. P. Ghai, P. Senthilkumaran, and R. S. Sirohi, “Adaptive helical mirror for generation of optical phase singularity,” Applied Optics, vol. 47, no. 10, pp. 1378–1383, 2008.

W. Wang, T. Yokozeki, R. Ishijima, M. Takeda, and S. G. Hanson, “Optical vortex metrology based on the core structures of phase singularities in Laguerre-Gauss transform of a speckle pattern,” Optics Express, vol. 14, no. 22, pp. 10195–10206, 2006.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin to orbital angular momentum conversion in inhomogeneous anisotropic media,” Physical Review Letters, vol. 96, no. 16, Article ID 163905, 2006.

A. Dreischuh, S. Chervenkov, D. Neshev, G. G. Paulus, and H. Walther, “Generation of lattice structures of optical vortices,” Journal of the Optical Society of America B, vol. 19, no. 3, pp. 550–556, 2002.

J. Xavier, S. Vyas, P. Senthilkumaran, C. Denz, and J. Joseph, “Sculptured 3D twister superlattices embedded with tunable vortex spirals,” Optics Letters, vol. 36, no. 17, pp. 3512–3514, 2011.

P. Upadhaya and D. Prakash, “Computational analysis of higher order optical vortex,” Optik, vol. 127, no. 2, pp. 722–727, 2016.

X. Chen, S. Chen, and S. Wang, “Existence of vortices in nonlinear optics,” Journal of Mathematical Physics, vol. 59, no. 10, 2018.

Q. H. Quang, T. T. Doan, T. D. Quoc, and T. N. Manh, “Nonlinear optical tweezers for longitudinal control of dielectric particles,” Optics Communications, vol. 421, pp. 94–98, 2018.

Y. Chen and J. Atai, “Dynamics of optical-vortex solitons in perturbed nonlinear media,” Journal of the Optical Society of America B, vol. 11, no. 10, pp. 2000–2003, 1994.

N. A. Chaitanya, M. V. Jabir, and G. K. Samanta, “Efficient nonlinear generation of high power, high order, ultrafast perfect vortices in green,” Optics Letters, vol. 41, pp. 1348–1351, 2016.

D. Rozas and G. A. Swartzlander Jr., “Observed rotational enhancement of nonlinear optical vortices,” Optics Letters, vol. 25, no. 2, pp. 126–128, 2000.

A. G. White, C. P. Smith, N. R. Heckenberg et al., “Interferometric measurements of phase singularities in the output of A visible laser,” Journal of Modern Optics, vol. 38, no. 12, pp. 2531–2541, 1991.

D. P. Ghai, S. Vyas, P. Senthilkumaran, and R. S. Sirohi, “Detection of phase singularity using a lateral shear interferometer,” Optics and Lasers in Engineering, vol. 46, no. 6, pp. 419–423, 2008.

D. P. Ghai, P. Senthilkumaran, and R. S. Sirohi, “Single-slit diffraction of an optical beam with phase singularity,” Optics and Lasers in Engineering, vol. 47, no. 1, pp. 123–126, 2009.

J. Hickmann, E. Fonseca, W. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Physical Review Letters, vol. 105, no. 5, p. 53904, 2010.

J. V. Hajnal and J. F. Nye, “Singularities in the transverse fields of electromagnetic waves. I. Theory,” Proceedings of the Royal Society of London A: Mathematical and Physical Sciences, vol. 414, no. 1847, pp. 433–446, 1987.

J. V. Hajnal and J. F. Nye, “Singularities in the transverse fields of electromagnetic waves. II. Observations on the electric field,” Proceedings of the Royal Society of London A: Mathematical and Physical Sciences, vol. 414, no. 1847, pp. 447–468, 1987.

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, vol. 457, no. 2005, pp. 141–155, 2001.

I. Freund, M. S. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial optical fields,” Optics Communications, vol. 208, no. 4-6, pp. 223–253, 2002.

M. Verma, S. K. Pal, S. Joshi, P. Senthilkumaran, J. Joseph, and H. C. Kandpal, “Singularities in cylindrical vector beams,” Journal of Modern Optics, vol. 62, no. 13, pp. 1068–1075, 2015.

A. Aadhi, P. Vaity, P. Chithrabhanu, S. G. Reddy, S. Prabakar, and R. P. Singh, “Non-coaxial superposition of vector vortex beams,” Applied Optics, vol. 55, no. 5, p. 1107, 2016.

C. T. Samlan and N. K. Viswanathan, “Chiral dynamics of exceptional points in weakly absorbing biaxial crystal,” Optics Letters, vol. 43, no. 15, pp. 3538–3541, 2018.

S. Vyas, Y. Kozawa, and S. Sato, “Polarization singularities in superposition of vector beams,” Optics Express, vol. 21, no. 7, pp. 8972–8986, 2013.

S. K. Pal and P. Senthilkumaran, “Cultivation of lemon fields,” Optics Express, vol. 24, no. 24, pp. 28008–28013, 2016.

B. S. B. Ram, Ruchi, and P. Senthilkumaran, “Angular momentum switching and orthogonal field construction of C-points,” Optics Letters, vol. 43, no. 9, pp. 2157–2160, 2018.

S. K. Pal, Ruchi, and P. Senthilkumaran, “Polarization singularity index sign inversion by a half-wave plate,” Applied Optics, vol. 56, no. 22, pp. 6181–6189, 2017.

S. K. Pal and P. Senthilkumaran, “Lattice of C points at intensity nulls,” Optics Letters, vol. 43, no. 6, pp. 1259–1262, 2018.

S. K. Pal and P. Senthilkumaran, “Hexagonal vector field of polarization singularities with a gradient basis structure,” Optics Letters, vol. 44, no. 8, pp. 2093–2096, 2019.

I. Freund, “Second harmonic generation of optical ellipse fields,” Optics Communications, vol. 213, no. 1–3, pp. 129–149, 2002.

I. Freund, “Second harmonic generation of polarization singularities,” Optics Letters, vol. 27, no. 18, p. 1640, 2007.

Ruchi, B. S. Bhargava Ram, and P. Senthilkumaran, “Hopping induced inversions and Pancharatnam excursions of C-points,” Optics Letters, vol. 42, no. 20, pp. 4159–4162, 2017.

V. Kumar, G. M. Philip, and N. K. Viswanathan, “Formation and morphological transformation of polarization singularities: hunting the monstar,” Journal of Optics, vol. 15, no. 4, 2013.

M. V. Vasnetsov, M. S. Soskin, V. A. Pas’ko, and V. I. Vasil’ev, “A Monstar portrait in the interior,” Journal of Optics, vol. 18, no. 3, 2016.

E. J. Galvez and B. Khajavi, “Monstar disclinations in the polarization of singular optical beams,” Journal of the Optical Society of America A, vol. 34, no. 4, p. 568, 2017.

B. A. Cvarch, B. Khajavi, J. A. Jones, B. Piccirillo, L. Marrucci, and E. J. Galvez, “Monstar polarization singularities with elliptically-symmetric q-plates,” Optics Express, vol. 25, no. 13, p. 14935, 2017.

Ruchi, S. K. Pal, and P. Senthilkumaran, “Generation of V-point polarization singularity lattices,” Optics Express, vol. 25, no. 16, pp. 19326–19331, 2017.

S. N. Khan, S. Deepa, and P. Senthilkumaran, “Helicity conservation in V-point diffraction,” Optics Letters, vol. 44, no. 16, pp. 3913–3916, 2019.

S. Deepa, B. S. Bhargava Ram, and P. Senthilkumaran, “Helicity dependent diffraction by angular momentum transfer,” Scientific Reports, vol. 9, no. 1, Article ID 12491, 2019.

F. J. Gilman, J. Pumplin, A. Schwimmer, and L. Stodolsky, “Helicity conservation in diffraction scattering,” Physics Letters B, vol. 31, no. 6, pp. 387–390, 1970.

B. Perez-Garcia, R. I. Hernández-Aranda, C. López-Mariscal, and J. C. Gutiérrez-Vega, “Morphological transformation of generalized spirally polarized beams by anisotropic media and its experimental characterization,” Optics Express, vol. 27, no. 23, pp. 33412–33426, 2019.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New Journal of Physics, vol. 9, no. 3, p. 78, 2007.

R. P. Cameron, S. M. Barnett, and A. M. Yao, “Optical helicity of interfering waves,” Journal of Modern Optics, vol. 61, no. 1, pp. 25–31, 2014.

S. K. Pal, Ruchi, and P. Senthilkumaran, “C-point and V-point singularity lattice formation and index sign conversion methods,” Optics Communications, vol. 393, pp. 156–168, 2017.

S. K. Pal and P. Senthilkumaran, “Synthesis of Stokes vortices,” Optics Letters, vol. 44, no. 1, pp. 130–133, 2019.

Ruchi and P. Senthilkumaran, “Polarization Singularities and Intensity Degeneracies,” Frontiers in Physics, vol. 8, p. 140, 2020.

G. Arora, Ruchi, and P. Senthilkumaran, “Full Poincaré beam with all the Stokes vortices,” Optics Letters, vol. 44, no. 22, pp. 5638–5641, 2019.

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher order Poincaré sphere, stokes parameters, and the angular momentum of light,” Physical Review Letters, vol. 107, no. 5, pp. 1–4, 2011.

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam-Berry phase and the angular momentum of light,” Physical Review Letters, vol. 108, no. 19, pp. 1–4, 2012.

Y. Liu, X. Ling, X. Yi, X. Zhou, H. Luo, and S. Wen, “Realization of polarization evolution on higher order Poincaré sphere with metasurface,” Applied Physics Letters, vol. 104, no. 19, 2014.

D. Naidoo, F. S. Roux, A. Dudley et al., “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nature Photonics, vol. 10, no. 5, pp. 327–332, 2016.

C. Chen, Y. Zhang, L. Ma et al., “Flexible generation of higher-order Poincaré beams with high efficiency by manipulating the two eigenstates of polarized optical vortices,” Optics Express, vol. 28, no. 7, pp. 10618–10632, 2020.

X. Yi, Y. Liu, X. Ling et al., “Hybrid order Poincaré sphere,” Physical Review A, vol. 91, no. 2, pp. 14–19, 2015.

X. Ling, X. Yi, Z. Dai, Y. Wang, and L. Chen, “Characterization and manipulation of full Poincaré beams on the hybrid Poincaré sphere,” Journal of the Optical Society of America B, vol. 33, no. 11, p. 2172, 2016.

Z. Liu, Y. Liu, Y. Ke et al., “Generation of arbitrary vector vortex beams on hybrid-order Poincaré sphere,” Photonics Research, vol. 5, no. 1, pp. 15–21, 2017.

S. Chen, X. Zhou, Y. Liu, X. Ling, H. Luo, and S. Wen, “Generation of arbitrary cylindrical vector beams on the higher order Poincaré sphere,” Optics Letters, vol. 39, no. 18, p. 5274, 2014.

A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Optics Express, vol. 18, no. 10, p. 10777, 2010.

E. J. Galvez, S. Khadka, W. H. Schubert, and S. Nomoto, “Poincaré-beam patterns produced by nonseparable superpositions of Laguerre-Gauss and polarization modes of light,” Applied Optics, vol. 51, no. 15, p. 2925, 2012.

B. Khajavi and E. J. Galvez, “Preparation of Poincaré beams with a same-path polarization/spatial-mode interferometer,” Optical Engineering, vol. 54, no. 11, p. 111305, 2015.

J. A. Jones, A. J. D’Addario, B. L. Rojec, G. Milione, and E. J. Galvez, “The Poincaré-sphere approach to polarization: formalism and new labs with Poincaré beams,” American Journal of Physics, vol. 84, no. 11, pp. 822–835, 2016.

C. Alpmann, C. Schlickriede, E. Otte, and C. Denz, “Dynamic modulation of Poincaré beams,” Scientific Reports, vol. 7, no. 1, pp. 1–9, 2017.

J. C. G. de Sande, M. Santarsiero, and G. Piquero, “Full Poincaré beams obtained by means of uniaxial crystals,” in Proceedings of the Third International Conference on Applications of Optics and Photonics, p. 140, Faro, Portugal, September 2017.

J. Wang, L. Wang, and Y. Xin, “Generation of Full Poincaré beams on arbitrary order Poincaré sphere,” Current Optics and Photonics, vol. 1, no. 6, pp. 631–636, 2017.

Z. Gu, D. Yin, F. Gu et al., “Generation of concentric perfect Poincaré beams,” Scientific Reports, vol. 9, no. 1, pp. 1–9, 2019.

D. Xu, B. Gu, G. Rui, Q. Zhan, and Y. Cui, “Generation of arbitrary vector fields based on a pair of orthogonal elliptically polarized base vectors,” Optics Express, vol. 24, no. 4, pp. 4177–4186, 2016.

Ruchi, S. K. Pal, and P. Senthilkumaran, “Basis construction using generic orthogonal C-points,” Journal of Optics, vol. 21, no. 8, Article ID 085603, 2019.

S. Fu, Y. Zhai, T. Wang, C. Yin, and C. Gao, “Tailoring arbitrary hybrid Poincaré beams through a single hologram,” Applied Physics Letters, vol. 111, no. 21, 2017.

S. N. Khonina, A. V. Ustinov, S. A. Fomchenkov, and A. P. Porfirev, “Formation of hybrid higher order cylindrical vector beams using binary multi sector phase plates,” Scientific Reports, vol. 8, no. 1, pp. 1–11, 2018.

R. Wang, S. He, S. Chen et al., “Electrically driven generation of arbitrary vector vortex beams on the hybrid-order Poincaré sphere,” Optics Letters, vol. 43, no. 15, pp. 3570–3573, 2018.

D. Lopez-Mago, “On the overall polarisation properties of Poincaré beams,” Journal of Optics, vol. 21, no. 11, 2019.

D. Li, S. Feng, S. Nie, C. Chang, J. Ma, and C. Yuan, “Generation of arbitrary perfect Poincaré beams,” Journal of Applied Physics, vol. 125, no. 7, 2019.

S. Fu, C. Gao, T. Wang, Y. Zhai, and C. Yin, “Anisotropic polarization modulation for the production of arbitrary Poincaré beams,” Journal of the Optical Society of America B, vol. 35, no. 1, pp. 1–7, 2018.

H. Garcia-Gracia and J. C. Gutiérrez-Vega, “Polarization singularities in nondiffracting Mathieu-Poincaré beams,” Journal of Optics, vol. 18, no. 1, 2016.

V. D. Ambrosio, G. Carvacho et al., “Entangled vector vortex beams,” Physical Review A, vol. 94, Article ID 030304, 2016.

B. M. Holmes and E. J. Galvez, “Poincaré Bessel beams: structure and propagation,” Journal of Optics, vol. 21, no. 10, Article ID 104001, 2019.

A. Ariyawansa, K. Liang, and T. G. Brown, “Polarization singularities in a stress-engineered optic,” Journal of the Optical Society of America A, vol. 36, no. 3, pp. 312–319, 2019.

C. T. Samlan, D. N. Naik, and N. K. Viswanathan, “Isogyres-manifestation of spin-orbit interaction in uniaxial crystal: a closed-fringe Fourier analysis of conoscopic interference,” Scientific Reports, vol. 6, 2016.

W. Zhu, V. Shvedov, W. She, and W. Krolikowski, “Transverse spin angular momentum of tightly focused full Poincaré beams,” Optics Express, vol. 23, no. 26, pp. 34029–34041, 2015.

L.-G. Wang, “Optical forces on submicron particles induced by full Poincaré beams,” Optics Express, vol. 20, no. 19, pp. 20814–20826, 2012.

N. B. Baranova, B. Y. Zel’dovich, A. V. Mamaev, N. F. Pilipetski, and V. V. Shkunov, “Dislocations of the wavefront of a speckle inhomogeneous field,” JETP Letters, vol. 33, pp. 195–199, 1981.

K. Staliunas, A. Berzanskis, and V. Jarutis, “Vortex statistics in optical speckle fields,” Optics Communications, vol. 120, no. 1-2, pp. 23–28, 1995.

J. M. Huntley and J. R. Buckland, “Characterization of sources of 2π phase discontinuity in speckle interferograms,” Journal of the Optical Society of America A, vol. 12, no. 9, pp. 1990–1996, 1995.

W. Wang, N. Ishii, S. Hanson, Y. Miyamoto, and M. Takeda, “Phase singularities in analytic signal of white light speckle pattern with application to micro displacement measurement,” Optics Communications, vol. 248, no. 1–3, pp. 59–68, 2005.

V. P. Aksenov and O. V. Tikhomirova, “Theory of singular-phase reconstruction for an optical speckle field in the turbulent atmosphere,” Journal of the Optical Society of America A, vol. 19, no. 2, pp. 345–355, 2002.

J. M. Vaughan and D. V. Willetts, “Temporal and interference fringe analysis of TEM_01 laser modes,” Journal of the Optical Society of America, vol. 73, no. 8, pp. 1018–1021, 1983.

W. J. Condell, “Fraunhofer diffraction from a circular annular aperture with helical phase factor,” Journal of the Optical Society of America A, vol. 2, no. 2, pp. 206–208, 1985.

K. W. Nicholls and J. F. Nye, “Three-beam model for studying dislocations in wave pulses,” Journal of Physics A: Mathematical and General, vol. 20, no. 14, pp. 4673–4696, 1987.

G. S. Mcdonald, K. S. Syed, and W. J. Firth, “Optical vortices in beam propagation through a self defocusing medium,” Optics Communications, vol. 94, pp. 469–476, 1996.

D. Rozas, C. T. Law, and G. A. Swartzlander Jr., “Propagation dynamics of optical vortices,” Journal of the Optical Society of America B, vol. 14, no. 11, pp. 3054–3065, 1997.

Z. Mei, D. Zhao, and J. Gu, “Propagation of elegant Laguerre-Gaussian beams through an annular apertured paraxial ABCD optical system,” Optics Communications, vol. 240, no. 4–6, pp. 337–343, 2004.

X. C. Yuan, B. P. S. Ahluwalia, H. L. Chen, J. Bu, and J. Lin, “Generation of high quality optical vortex beams in free space propagation by micro-fabricated wedge with filtering technique,” Applied Physics Letters, vol. 91, pp. 511031–511033, 2008.

A. Wada, T. Ohtani, Y. Miyamoto, and M. Takeda, “Propagation analysis of the Laguerre-Gaussian beam with astigmatism,” Journal of the Optical Society of America A, vol. 22, no. 12, pp. 2746–2755, 2005.

S. G. Reddy, S. Prabhakar, A. Aadhi, J. Banerji, and R. P. Singh, “Propagation of an arbitrary vortex pair through an astigmatic optical system and determination of its topological charge,” Journal of the Optical Society of America A, vol. 31, no. 6, pp. 1295–1302, 2014.

A. Wada, H. Ohminato, T. Yonemura, Y. Miyamoto, and M. Takeda, “Effect of comatic aberration on the propagation characteristics of Laguerre--Gaussian beams,” Optical Review, vol. 12, no. 6, pp. 451–455, 2005.

S. M. Baumann, D. M. Kalb, L. H. MacMillan, and E. J. Galvez, “Propagation dynamics of optical vortices due to Gouy phase,” Optics Express, vol. 17, no. 12, p. 9818, 2009.

Y. Liu, S. Sun, J. Pu, and B. Lü, “Propagation of an optical vortex beam through a diamond-shaped aperture,” Optics & Laser Technology, vol. 45, pp. 473–479, 2013.

Z. Li, M. Zhang, G. Liang, X. Li, X. Chen, and C. Cheng, “Generation of high-order optical vortices with asymmetrical pinhole plates under plane wave illumination,” Optics Express, vol. 21, no. 13, pp. 15755–15764, 2013.

Z. Li and C. Cheng, “Generation of second-order vortex arrays with six-pinhole interferometers under plane wave illumination,” Applied Optics, vol. 53, no. 8, pp. 1629–1635, 2014.

A. Mobashery, M. Hajimahmoodzadeh, and H. R. Fallah, “Detection and characterization of an optical vortex by the branch point potential method: analytical and simulation results,” Applied Optics, vol. 54, no. 15, pp. 4732–4739, 2015.

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Applied Optics, vol. 21, no. 15, pp. 2758–69, 1982.

F. Wyrowski and O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” Journal of the Optical Society of America A, vol. 5, no. 7, pp. 1058–1065, 1988.

J. R. Fienup and C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” Journal of the Optical Society of America A, vol. 3, no. 11, pp. 1897–1907, 1986.

M. W. Kowarz, “Homogeneous and evanescent contributions in scalar near-field diffraction,” Applied Optics, vol. 34, no. 17, pp. 3055–3063, 1995.

M. V. Berry, “Waves near zeros,” in Proceedings of the Conference on Coherence and Quantum Optics, Optical Society of America, Rochester, NY, USA, 2007.

I. Freund, “Critical foliations,” Optics Letters, vol. 26, no. 8, pp. 545–547, 2001.

I. Freund, “Optical vortex trajectories,” Optics Communications, vol. 181, no. 1–3, pp. 19–33, 2000.

J. H. Hannay, “Vortex reconnection rate, and loop birth rate, for a random wavefield,” Journal of Physics A, vol. 50, no. 16, 2017.

M. R. Dennis, Y. S. Kivshar, M. S. Soskin, and G. A. Swartzlander Jr., “Singular optics: more ado about nothing,” Journal of Optics A: Pure and Applied Optics, vol. 11, no. 9, Article ID 090201, 2009.

I. Swartzlander Jr., “Saddles, singularities, and extrema in random phase fields,” Physical Review E, vol. 52, no. 3, pp. 2348–2360, 1995.

I. Freund, “Saddle point wave fields,” Optics Communications, vol. 163, no. 4–6, pp. 230–242, 1999.

I. Freund, “Critical point explosions in two-dimensional wave fields,” Optics Communications, vol. 159, no. 1–3, pp. 99–117, 1999.

N. Shvartsman and I. Freund, “Vortices in random wave fields: Nearest Neighbor Anticorrelations,” Physical Review Letters, vol. 72, no. 7, pp. 1008–1011, 1994.

M. V. Berry and M. R. Dennis, “Phase singularities in isotropic random waves,” Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, vol. 456, no. 2001, pp. 2059–2079, 2000.

I. Freund, “Critical foliations and Berry’s paradox,” Optics & Photonics News, vol. 56, 2001.

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Knotted threads of darkness,” Nature, vol. 432, no. 7014, p. 165, 2004.

M. Wilson, “Holograms tie optical vortices in knots,” Physics Today, vol. 63, no. 3, pp. 18–20, 2010.

M. R. Dennis, R. P. King, B. Jack, K. O’Holleran, and M. J. Padgett, “Isolated optical vortex knots,” Nature Physics, vol. 6, no. 2, pp. 118–121, 2010.

J. Leach, M. Dennis, J. Courtial, and M. Padgett, “Vortex knots in light,” New Journal of Physics, vol. 7, no. 55, pp. 1–11, 2005.

M. Dennis, “Topological configurations of optical phase singularities,” Topologica, vol. 2, no. 007, pp. 1–10, 2009.

G.-H. Kim, H. J. Lee, J.-U. Kim, and H. Suk, “Propagation dynamics of optical vortices with anisotropic phase profiles,” Journal of the Optical Society of America B, vol. 20, no. 2, pp. 351–359, 2003.

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Optimal phase element for generating a perfect optical vortex,” Journal of the Optical Society of America A, vol. 33, no. 12, pp. 2376–2384, 2016.

A. S. Ostrovsky, C. Rickenstorff-Parrao, and V. Arrizón, “Generation of the “perfect” optical vortex using a liquid-crystal spatial light modulator,” Optics Letters, vol. 38, no. 4, pp. 534–536, 2013.

M. K. Karahroudi, B. Parmoon, M. Qasemi, A. Mobashery, and H. Saghafifar, “Generation of perfect optical vortices using a Bessel-Gaussian beam diffracted by curved fork grating,” Applied Optics, vol. 56, no. 21, pp. 5817–5823, 2017.

M. Jabir, N. Apurv Chaitanya, A. Aadhi, and G. Sharma, “Generation of perfect vortex of variable size and its effects in angular spectrum of the down converted photons,” Scientific Reports, vol. 6, Article ID 21877, pp. 1–8, 2016.

H. Ma, X. Li, Y. Tai et al., “In situ measurement of the topological charge of a perfect vortex using the phase shift method,” Optics Letters, vol. 42, no. 1, pp. 135–138, 2017.

J. García-García, C. Rickenstorff-Parrao, R. Ramos-García, V. Arrizón, and A. S. Ostrovsky, “Simple technique for generating the perfect optical vortex,” Optics Letters, vol. 39, no. 18, pp. 5305–5308, 2014.

Y. Chen, Z.-X. Fang, Y.-X. Ren, L. Gong, and R.-D. Lu, “Generation and characterization of a perfect vortex beam with a large topological charge through a digital micromirror device,” Applied Optics, vol. 54, no. 27, pp. 8030–8035, 2015.

D. Deng, Y. Li, Y. Han et al., “Perfect vortex in three-dimensional multifocal array,” Optics Express, vol. 24, no. 25, p. 28270, 2016.

S. Fu, T. Wang, and C. Gao, “Perfect optical vortex array with controllable diffraction order and topological charge,” Journal of the Optical Society of America A, vol. 33, no. 9, pp. 1836–1842, 2016.

M. R. Dennis, “On the Burgers vector of a wave dislocation,” Journal of Optics A: Pure and Applied Optics, vol. 11, no. 9, p. 094002, 2009.

S. S. R. Oemrawsingh, E. R. Eliel, J. P. Woerdman, E. J. K. Verstegen, J. G. Kloosterboer, and G. W. t. Hooft, “Half-integral spiral phase plates for optical wavelengths,” Journal of Optics A: Pure and Applied Optics, vol. 6, no. 5, pp. S288–S290, 2004.

I. V. Baistiy, V. A. Pasko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A Pure Appl. Opt, vol. 6, pp. S160–S166, 2004.

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” Journal of Optics A: Pure and Applied Optics, vol. 6, no. 2, pp. 259–268, 2004.

W. M. Lee, X.-C. Yuan, and K. Dholakia, “Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step,” Optics Communications, vol. 239, no. 1-3, pp. 129–135, 2004.

S. Vyas, R. Kumar Singh, and P. Senthilkumaran, “Fractional vortex lens,” Optics & Laser Technology, vol. 42, no. 6, pp. 878–882, 2010.

B. K. Singh, D. S. Mehta, and P. Senthilkumaran, “Visualization of internal energy flows in optical fields carrying a pair of fractional vortices,” Journal of Modern Optics, vol. 60, no. 13, pp. 1027–1036, 2013.

M. K. Sharma, J. Joseph, and P. Senthilkumaran, “Fractional vortex dipole phase filter,” Applied Physics B, vol. 117, no. 1, pp. 325–332, 2014.

S. Vyas, R. Singh, D. Ghai, and P. Senthilkumaran, “Fresnel lens with embedded vortices,” International Journal of Optics, vol. 2012, Article ID 249695, p. 7, 2012.

S. Maji and M. M. Brundavanam, “Topological transformation of fractional optical vortex beams using computer generated holograms,” Journal of Optics, vol. 20, no. 4, 2018.

N. R. Heckenberg, R. MCDuff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, “Laser Beams with Phase Singularities,” Optical and Quantum Electronics, vol. 24, no. 9, pp. S951–S962, 1992.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Physical Review A, vol. 45, no. 11, pp. 8185–8189, 1992.

L. Allen, “Orbital angular momentum - a personal memoir,” Phil.Trans.R.Soc.A, vol. 375, no. 11, pp. 8185–8189, 2017.

L. Allen, J. Courtial, and M. J. Padgett, “Matrix formulation for the propagation of light beams with orbital and spin angular momenta,” Physical Review E, vol. 60, no. 6, pp. 7497–7503, 1999.

M. J. Padgett, “Orbital angular momentum 25 years on [Invited],” Optics Express, vol. 25, no. 10, p. 11265, 2017.

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: An optical spanner,” Optics Letters, vol. 22, no. 1, pp. 52–54, 1997.

J. P. Torres and L. Torner, Twisted Photons: Applications of Light with Orbital Angular Momentum, John Weily & Sons Inc, 2011.

A. E. Willner, H. Huang, Y. Yan et al., “Optical communications using orbital angular momentum beams,” Advances in Optics and Photonics, vol. 7, no. 1, pp. 66–106, 2015.

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Advances in Optics and Photonics, vol. 3, no. 2, p. 161, 2011.

K. Y. Bliokh, A. Y. Bekshaev, A. G. Kofman, and F. Nori, “Photon trajectories, anomalous velocities and weak measurements: a classical interpretation,” New Journal of Physics, vol. 15, no. 7, p. 073022, 2013.

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nature Photonics, vol. 9, no. 12, pp. 796–808, 2015.

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemporary Physics, vol. 46, no. 1, pp. 15–28, 2005.

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Optics Letters, vol. 25, no. 20, pp. 1493–1495, 2000.

M. A. Bandres and J. C. Gutiérrez-Vega, “Ince-Gaussian beams,” Optics Letters, vol. 25, no. 2, pp. 1493–1495, 2000.

E. Engay, A. Bañas, A.-I. Bunea, S. D. Separa, and J. Glückstad, “Interferometric detection of OAM-carrying Helico-conical beams,” Optics Communications, vol. 433, pp. 247–251, 2019.

N. P. Hermosa and C. O. Manaois, “Phase structure of helico-conical optical beams,” Optics Communications, vol. 271, no. 1, pp. 178–183, 2007.

C. A. Alonzo, P. J. Rodrigo, and J. Glückstad, “Helico-conical optical beams: a product of helical and conical phase fronts,” Optics Express, vol. 13, no. 5, pp. 1749–1760, 2005.

B. K. Singh, D. S. Mehta, and P. Senthilkumaran, “Conical light sword optical beam and its healing property,” Optics Letters, vol. 39, no. 7, pp. 2064–2067, 2014.

N. Hermosa, C. Rosales-Guzmán, and J. P. Torres, “Helico-conical optical beams self-heal,” Optics Letters, vol. 38, no. 3, pp. 383–385, 2013.

Z. Xu, X. Li, X. Liu, S. A. Ponomarenko, Y. Cai, and C. Liang, “Vortex preserving statistical optical beams,” Optics Express, vol. 28, no. 6, pp. 8475–8483, 2020.

S. J. Van Enk and G. Nienhuis, “Eigenfunction description of laser beams and orbital angular momentum of light,” Optics Communications, vol. 94, no. 1-3, pp. 147–158, 1992.

G. Nienhuis and L. Allen, “Paraxial wave optics and harmonic oscillators,” Physical Review A, vol. 48, no. 1, pp. 656–665, 1993.

R. Philips and L. Andrews, “Spot size and divergence for Laguerre-Gaussian beams of any order,” Appl. Opt, vol. 22, pp. 643-644, 1983.

M. J. Padgett, F. M. Miatto, M. P. J. Lavery, A. Zeilinger, and R. W. Boyd, “Divergence of an orbital-angular-momentum-carrying beam upon propagation,” New Journal of Physics, vol. 17, no. 2, p. 023011, 2015.

E. H. K. Stelzer and S. Grill, “The uncertainty principle applied to estimate focal spot dimensions,” Optics Communications, vol. 173, no. 1-6, pp. 51–56, 2000.

M. Padgett, “On the focussing of light, as limited by the uncertainty principle,” Journal of Modern Optics, vol. 55, no. 18, pp. 3083–3089, 2008.

S. G. Reddy, C. Permangatt, S. Prabhakar, A. Anwar, J. Banerji, and R. P. Singh, “Divergence of optical vortex beams,” Applied Optics, vol. 54, no. 22, pp. 6690–6693, 2015.

G. Indebetouw, “Optical vortices and their propagation,” Journal of Modern Optics, vol. 40, no. 1, pp. 73–87, 1993.

S. Maji, A. Mandal, and M. M. Brundavanam, “Gouy phase-assisted topological transformation of vortex beams from fractional fork holograms,” Optics Letters, vol. 44, no. 9, pp. 2286–2289, 2019.

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Dual electromagnetism: helicity, spin, momentum and angular momentum,” New Journal of Physics, vol. 15, no. 3, p. 033026, 2013.

A. Y. Bekshaev and M. S. Soskin, “Transverse energy flows in vectorial fields of paraxial beams with singularities,” Optics Communications, vol. 271, no. 2, pp. 332–348, 2007.

A. Bekshaev, K. Y. Bliokh, and M. Soskin, “Optical currents Related content Internal flows and energy circulation in light beams,” vol. 11, pp. 1–50, 2009.

A. Bekshaev, K. Y. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” Journal of Optics, vol. 13, no. 5, 2011.

A. Y. Bekshaev, O. V. Angelsky, S. V. Sviridova, and C. Y. Zenkova, “Mechanical Action of Inhomogeneously Polarized Optical Fields and Detection of the Internal Energy Flows,” Advances in Optical Technologies, vol. 2011, no. 2, pp. 1–11, 2011.

A. Y. Bekshaev, “Internal energy flows and instantaneous field of a monochromatic paraxial light beam,” Applied Optics, vol. 51, no. 10, p. C13, 2012.

F. Petronetto, A. Paiva, M. Lage, G. Tavares, H. Lopes, and T. Lewiner, “Meshless Helmholtz-Hodge decomposition,” IEEE Transactions on Visualization and Computer Graphics, vol. 16, no. 2, pp. 338–349, 2010.

A. Bekshaev and M. Soskin, “Rotational transformations and transverse energy flow in paraxial light beams: linear azimuthons,” Optics Letters, vol. 31, no. 14, p. 2199, 2006.

B. K. Singh, M. Bahl, D. S. Mehta, and P. Senthilkumaran, “Study of internal energy flows in dipole vortex beams by knife edge test,” Optics Communications, vol. 293, pp. 15–21, 2013.

M. Bahl and P. Senthilkumaran, “Focal plane internal energy flows of singular beams in astigmatically aberrated low numerical aperture systems,” Journal of the Optical Society of America A, vol. 31, no. 9, pp. 2046–2054, 2014.

M. Bahl, B. K. Singh, R. K. Singh, and P. Senthilkumaran, “Internal energy flows of coma-affected singular beams in low-numerical-aperture systems,” Journal of the Optical Society of America A, vol. 32, no. 4, p. 514, 2015.

O. V. Angelsky, A. Y. Bekshaev, P. P. Maksimyak et al., “Circular motion of particles suspended in a Gaussian beam with circular polarization validates the spin part of the internal energy flow,” Optics Express, vol. 20, no. 10, p. 11351, 2012.

O. V. Angelsky, A. Y. Bekshaev, P. P. Maksimyak, A. P. Maksimyak, S. G. Hanson, and C. Y. Zenkova, “Orbital rotation without orbital angular momentum: mechanical action of the spin part of the internal energy flow in light beams,” Optics Express, vol. 20, no. 4, pp. 3563–3571, 2012.

J. Leach, S. Keen, M. J. Padgett, C. Saunter, and G. D. Love, “Direct measurement of the skew angle of the Poynting vector in a helically phased beam,” Optics Express, vol. 14, no. 25, p. 11919, 2006.

J. Arlt, “Handedness and azimuthal energy flow of optical vortex beams,” Journal of Modern Optics, vol. 50, no. 10, pp. 1573–1580, 2003.

M. V. Berry, Faster than Fourier in Quantum coherence and Reality, World Scientific, Singapore, 1994.

I. Mokhun, R. Khrobatin, A. Mokhun, and J. Viktorovskaya, “The behavior of the Poynting vector in the area of elementary polarization singularities,” Applied Optics, vol. 37, no. 3, pp. 261–277, 2007.

I. Mokhun, A. Mokhun, and J. Viktorovskaya, “Singularities of Poynting vector and the structure of optical fields,” Ukrainian Journal of Physical Optics, vol. 7, no. 3, pp. 129–141, 2006.

C. Y. Zenkova, M. P. Gorsky, P. P. Maksimyak, and A. P. Maksimyak, “Optical currents in vector fields,” Applied Optics, vol. 50, no. 8, p. 1105, 2011.

H.-I. Lee and J. Mok, “Orbital and spin parts of energy currents for electromagnetic waves through spatially inhomogeneous media,” Journal of Modern Optics, vol. 65, no. 9, pp. 1053–1062, 2018.

A. V. Novitsky and L. M. Barkovsky, “Poynting singularities in optical dynamic systems,” Physical Review A, vol. 79, no. 3, p. 033821, 2009.

I. Mokhun, A. Arkhelyuk, Y. Galushko, Y. Kharitonovtta, and J. Viktorovskaya, “Experimental analysis of the Poynting vector characteristics,” Applied Optics, vol. 51, no. 10, p. C158, 2012.

V. Kumar and N. K. Viswanathan, “Topological structures in the Poynting vector field: an experimental realization,” Optics Letters, vol. 38, no. 19, p. 3886, 2013.

E. Otte, E. Asché, and C. Denz, “Shaping optical spin flow topologies by the translation of tailored orbital phase flow,” Journal of Optics, vol. 21, no. 6, p. 064001, 2019.

V. Kumar and N. K. Viswanathan, “Topological structures in vector-vortex beam fields,” Journal of the Optical Society of America B, vol. 31, no. 6, p. A40, 2014.

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Radial dependence of the angular momentum density of a paraxial optical vortex,” Physical Review A, vol. 97, no. 5, pp. 1–6, 2018.

V. V. Kotlyar, A. G. Nalimov, and S. S. Stafeev, “Exploiting the circular polarization of light to obtain a spiral energy flow at the subwavelength focus,” Journal of the Optical Society of America B, vol. 36, no. 10, pp. 2850–2855, 2019.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Optics Communications, vol. 96, no. 1-3, pp. 123–132, 1993.

M. J. Padgett and L. Allen, “Orbital angular momentum exchange in cylindrical-lens mode converters,” Journal of Optics B: Quantum and Semiclassical Optics, vol. 4, no. 2, pp. S17–S19, 2002.

S. Fu, T. Wang, S. Zhang, and C. Gao, “Integrating 5 × 5 Dammann gratings to detect orbital angular momentum states of beams with the range of −24 to +24,” Applied Optics, vol. 55, no. 7, pp. 1514–1517, 2016.

J. Yu, C. Zhou, W. Jia et al., “Three-dimensional Dammann vortex array with tunable topological charge,” Applied Optics, vol. 51, no. 13, pp. 2485–2490, 2012.

Y. Zhang, Z. Bai, C. Fu et al., “Polarization-independent orbital angular momentum generator based on a chiral fiber grating,” Optics Letters, vol. 44, no. 1, p. 61, 2018.

A. Ito, Y. Kozawa, and S. Sato, “Generation of hollow scalar and vector beams using a spot-defect mirror,” Journal of the Optical Society of America A, vol. 27, no. 9, pp. 2072–2077, 2010.

S. Vyas, Y. Kozawa, and S. Sato, “Generation of a vector doughnut beam from an internal mirror He-Ne laser,” Optics Letters, vol. 39, no. 7, pp. 2080–2082, 2014.

K. Kano, Y. Kozawa, and S. Sato, “Generation of a Purely Single Transverse Mode Vortex Beam from a He-Ne Laser Cavity with a Spot Defect Mirror,” International Journal of Optics, vol. 2012, p. 359141, 2012.

A. Khoroshun, A. Chernykh, J. Kirichenko, O. Ryazantsev, and A. Bekshaev, “Singular skeleton of a Laguerre-Gaussian beam transformed by the double-phase-ramp converter,” Applied Optics, vol. 56, no. 12, pp. 3428–3434, 2017.

A. N. Khoroshun, A. V. Chernykh, A. N. Tsimbaluk et al., “Properties of an axial optical vortex generated with the use of a Gaussian beam and two ramps,” Journal of Nanoscience and Nanotechnology, vol. 16, no. 2, pp. 2105–2107, 2016.

D. McGloin, N. B. Simpson, and M. J. Padgett, “Transfer of orbital angular momentum from a stressed fiber-optic waveguide to a light beam,” Applied Optics, vol. 37, no. 3, pp. 469–472, 1998.

P. Z. Dashti, F. Alhassen, and H. P. Lee, “Observation of Orbital Angular Momentum Transfer between Acoustic and Optical Vortices in Optical Fiber,” Physical Review Letters, vol. 96, p. 43604, 2006.

C. Won, S. H. Yoo, K. Oh, U.-C. Paek, and W. Jhe, “Near-field diffraction by a hollow-core optical fiber,” Optics Communications, vol. 161, no. 1-3, pp. 25–30, 1999.

R. Kumar, D. Singh Mehta, A. Sachdeva, A. Garg, P. Senthilkumaran, and C. Shakher, “Generation and detection of optical vortices using all fiber-optic system,” Optics Communications, vol. 281, no. 13, pp. 3414–3420, 2008.

M.-L. Hu, C.-Y. Wang, Y.-J. Song et al., “A hollow beam from a holey fiber,” Optics Express, vol. 14, no. 9, pp. 4128–4134, 2006.

P. Vayalamkuzhi, S. Bhattacharya, U. Eigenthaler et al., “Direct patterning of vortex generators on a fiber tip using a focused ion beam,” Optics Letters, vol. 41, no. 10, pp. 2133–2136, 2016.

L. Yan, P. Kristensen, and S. Ramachandran, “Vortex fibers for STED microscopy,” APL Photonics, vol. 4, no. 2, 2019.

W. Zhang, L. Huang, K. Wei et al., “High-order optical vortex generation in a few-mode fiber via cascaded acoustically driven vector mode conversion,” Optics Letters, vol. 41, no. 21, pp. 5082–5085, 2016.

K. Weber, F. Hütt, S. Thiele, T. Gissibl, A. Herkommer, and H. Giessen, “Single mode fiber based delivery of OAM light by 3D direct laser writing,” Optics Express, vol. 25, no. 17, pp. 19672–19679, 2017.

S. Pidishety, S. Pachava, P. Gregg, S. Ramachandran, G. Brambilla, and B. Srinivasan, “Orbital angular momentum beam excitation using an all-fiber weakly fused mode selective coupler,” Optics Letters, vol. 42, no. 21, pp. 4347–4350, 2017.

R. S. Rodrigues Ribeiro, P. Dahal, A. Guerreiro, P. Jorge, and J. Viegas, “Optical fibers as beam shapers: from Gaussian beams to optical vortices,” Optics Letters, vol. 41, no. 10, pp. 2137–2140, 2016.

J. Zhou, Y. Liu, Y. Ke, H. Luo, and S. Wen, “Generation of Airy vortex and Airy vector beams based on the modulation of dynamic and geometric phases,” Optics Letters, vol. 40, no. 13, pp. 3193–3196, 2015.

J. Lin, X.-C. Yuan, S. H. Tao, and R. E. Burge, “Collinear superposition of multiple helical beams generated by a single azimuthally modulated phase-only element,” Optics Letters, vol. 30, no. 24, pp. 3266–3268, 2005.

A. V. Carpentier, H. Michinel, J. R. Salgueiro, and D. Olivieri, “Making optical vortices with computer-generated holograms,” American Journal of Physics, vol. 76, no. 10, pp. 916–921, 2008.

R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Efficient formation of pure helical laser beams,” Optics Communications, vol. 182, no. 1-3, pp. 205–208, 2000.

L. Marrucci, E. Karimi, S. Slussarenko et al., “Spin-to-Orbital Optical Angular Momentum Conversion in Liquid Crystal “q-Plates”: Classical and Quantum Applications,” Molecular Crystals and Liquid Crystals, vol. 561, no. 1, pp. 48–56, 2012.

M. Gecevicius, M. Ivanov, M. Beresna et al., “Toward the generation of broadband optical vortices: extending the spectral range of a q-plate by polarization-selective filtering,” Journal of the Optical Society of America B, vol. 35, no. 1, pp. 190–196, 2018.

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Applied Physics Letters, vol. 94, no. 23, p. 231124, 2009.

X.-C. Yuan, B. P. S. Ahluwalia, S. H. Tao et al., “Wavelength-scalable micro-fabricated wedge for generation of optical vortex beam in optical manipulation,” Applied Physics B, vol. 86, no. 2, pp. 209–213, 2007.

J. Lin, X.-C. Yuan, J. Bu, B. P. Ahluwalia, Y. Y. Sun, and R. E. Burge, “Selective generation of high-order optical vortices from a single phase wedge,” Optics Letters, vol. 32, no. 20, pp. 2927–2929, 2007.

Y. Izdebskaya, V. Shvedov, and A. Volyar, “Generation of higher-order optical vortices by a dielectric wedge,” Optics Letters, vol. 30, no. 18, pp. 2472–2474, 2005.

A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Advances in Optics and Photonics, vol. 8, no. 2, p. 200, 2016.

A. Ciattoni, G. Cincotti, and C. Palma, “Circularly polarized beams and vortex generation in uniaxial media,” Journal of the Optical Society of America A, vol. 20, no. 1, pp. 163–171, 2003.

G. Zhou and F. S. Chau, “Helical wave front laser beam generated with a micro electro mechanical systems, MEMS) based device,” IEEE Phot. Tech. Lett., vol. 18, pp. 292–294, 2005.

O. Bokyo, T. A. P. Mercere, C. Valentin, and P. Balcou, “Adaptive shaping of a focused intense laser beam into a doughnut mode,” Optics Communications, vol. 246, pp. 131–140, 2005.

D. P. Ghai, “Generation of optical vortices with an adaptive helical mirror,” Applied Optics, vol. 50, no. 10, pp. 1374–1381, 2011.

S. S. R. Oemrawsingh, J. A. W. van Houwelingen, E. R. Eliel et al., “Production and characterization of spiral phase plates for optical wavelengths,” Applied Optics, vol. 43, no. 3, pp. 688–694, 2004.

K. Sueda, G. Miyaji, and M. Nakatsuka, “Laguerre-Gaussian beam generated with a multilevel spiral phase plate for high intensity laser pulses,” Optics Express, vol. 12, pp. 3584–3589, 2004.

S. Yan, B. Yao, and M. Lei, ““Comment on”optical Orbital Angular Momentum from the Curl of Polarization,” Physical Review Letters, vol. 106, no. 18, pp. 1–4, 2011.

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Accelerating Beams,” Vortex Laser Beams, vol. 26, no. 1, pp. 141–158, 2018.

J. Zhu, Y. Chen, Y. Zhang, X. Cai, and S. Yu, “Spin and orbital angular momentum and their conversion in cylindrical vector vortices,” Optics Letters, vol. 39, no. 15, p. 4435, 2014.

L. Marrucci, E. Karimi, S. Slussarenko et al., “Spin to orbital conversion of the angular momentum of light and its classical and quantum applications,” Journal of Optics, vol. 13, no. 6, 2011.

K. Y. Bliokh, E. A. Ostrovskaya, M. A. Alonso, O. G. Rodríguez-Herrera, D. Lara, and C. Dainty, “Spin to orbital angular momentum conversion in focusing, scattering, and imaging systems,” Optics Express, vol. 19, no. 27, pp. 26–132, 2011.

V. Sharma, G. K. Samanta, S. Chaitanya Kumar, R. P. Singh, and M. Ebrahim-Zadeh, “Tunable ultraviolet vortex source based on a continuous-wave optical parametric oscillator,” Optics Letters, vol. 44, no. 19, pp. 4694–4697, 2019.

M. K. Sharma, R. K. Singh, J. Joseph, and P. Senthilkumaran, “Fourier spectrum analysis of spiral zone plates,” Optics Communications, vol. 304, pp. 43–48, 2013.

D. N. Naik and N. K. Viswanathan, “Generation of singular optical beams from fundamental Gaussian beam using Sagnac interferometer,” Journal of Optics, vol. 18, no. 9, p. 095601, 2016.

T. D. Huang and T. H. Lu, “Controlling an optical vortex array from a vortex phase plate, mode converter, and spatial light modulator,” Optics Letters, vol. 44, no. 16, p. 3917, 2019.

J. Masajada, “The interferometry based on regular lattice of optical vortices,” Applied Optics, vol. 37, no. 1-2, pp. 167–185, 2007.

S. Vyas and P. Senthilkumaran, “Interferometric optical vortex array generator,” Applied Optics, vol. 46, no. 15, pp. 2893–2898, 2007.

S. Vyas and P. Senthilkumaran, “Vortex array generation by interference of spherical waves,” Applied Optics, vol. 46, no. 32, pp. 7862–7867, 2007.

J. Xavier, S. Vyas, P. Senthilkumaran, and J. Joseph, “Tailored complex 3D vortex lattice structures by perturbed multiples of three-plane waves,” Applied Optics, vol. 51, no. 12, pp. 1872–1878, 2012.

M. Boguslawski, P. Rose, and C. Denz, “Increasing the structural variety of discrete nondiffracting wave fields,” Physical Review A, vol. 84, Article ID 013832, 2011.

J. Xavier, R. Dasgupta, S. Ahlawat, J. Joseph, and P. Kumar Gupta, “Three dimensional optical twisters-driven helically stacked multi-layered microrotors,” Applied Physics Letters, vol. 100, no. 12, p. 121101, 2012.

J. Yu, C. Zhou, W. Jia et al., “Generation of dipole vortex array using spiral Dammann zone plates,” Applied Optics, vol. 51, no. 28, pp. 6799–6804, 2012.

A. Sabatyan, S. M. Taheri Balanoji, and S. M. Taheri Balanoji, “Square array of optical vortices generated by multiregion spiral square zone plate,” Journal of the Optical Society of America A, vol. 33, no. 9, pp. 1793–1797, 2016.

J. M. Vaughan and D. V. Willetts, “Interference properties of a light beam having a helical wave surface,” Optics Communications, vol. 30, no. 3, pp. 263–267, 1979.

A. Bekshaev, O. Orlinska, and M. Vasnetsov, “Optical vortex generation with a “fork” hologram under conditions of high-angle diffraction,” Optics Communications, vol. 283, no. 10, pp. 2006–2016, 2010.

M. Padgett, J. Arlt, N. Simpson, and L. Allen, “An experiment to observe the intensity and phase structure of Laguerre-Gaussian laser modes,” American Journal of Physics, vol. 64, no. 1, pp. 77–82, 1996.

H. I. Sztul and R. R. Alfano, “Double-slit interference with Laguerre-Gaussian beams,” Optics Letters, vol. 31, no. 7, pp. 999–1001, 2006.

I. Moreno, J. A. Davis, B. M. L. Pascoguin, M. J. Mitry, and D. M. Cottrell, “Vortex sensing diffraction gratings,” Optics Letters, vol. 34, no. 19, pp. 2927–2929, 2009.

N. Zhang, X. C. Yuan, and R. E. Burge, “Extending the detection range of optical vortices by Dammann vortex gratings,” Optics Letters, vol. 35, no. 20, pp. 3495–3497, 2010.

L. E. E. de Araujo and M. E. Anderson, “Measuring vortex charge with a triangular aperture,” Optics Letters, vol. 36, no. 6, pp. 787–789, 2011.

A. Bekshaev, L. Mikhaylovskaya, S. Patil, V. Kumar, and R. P. Singh, “Optical-vortex diagnostics via Fraunhofer slit diffraction with controllable wavefront curvature,” Journal of the Optical Society of America A, vol. 37, no. 5, pp. 780–786, 2020.

H. Gao, Y. Han, Y. Li, D. Zhu, M. Sun, and S. Yu, “Topological charge measurement of concentric OAM states using the phase-shift method,” Journal of the Optical Society of America A, vol. 35, no. 1, pp. A40–A44, 2018.

P. Kumar and N. K. Nishchal, “Modified Mach-Zehnder interferometer for determining the high-order topological charge of Laguerre-Gaussian vortex beams,” Journal of the Optical Society of America A, vol. 36, no. 8, pp. 1447–1455, 2019.

P. Kumar and N. K. Nishchal, “Self-referenced spiral interferogram using modified lateral shearing Mach-Zehnder interferometer,” Applied Optics, vol. 58, no. 25, pp. 6827–6833, 2019.

S. Cui, B. Xu, S. Luo et al., “Determining topological charge based on an improved Fizeau interferometer,” Optics Express, vol. 27, no. 9, pp. 12774–12779, 2019.

S. Pachava, A. Dixit, and B. Srinivasan, “Modal decomposition of Laguerre Gaussian beams with different radial orders using optical correlation technique,” Optics Express, vol. 27, no. 9, pp. 13182–13193, 2019.

R. K. Singh, P. Senthilkumaran, and K. Singh, “The effect of astigmatism on the diffraction of a vortex carrying beam with a Gaussian background,” Journal of Optics A: Pure and Applied Optics, vol. 9, no. 5, pp. 543–554, 2007.

R. K. Singh, P. Senthilkumaran, and K. Singh, “Influence of astigmatism and defocusing on the focusing of a singular beam,” Optics Communications, vol. 270, no. 2, pp. 128–138, 2007.

A. Y. Bekshaev and A. I. Karamoch, “Astigmatic telescopic transformation of a high-order optical vortex,” Optics Communications, vol. 281, no. 23, pp. 5687–5696, 2008.

A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focusing by an astigmatic lens,” Optics Communications, vol. 241, no. 4-6, pp. 237–247, 2004.

P. Vaity, J. Banerji, and R. P. Singh, “Measuring the topological charge of an optical vortex by using a tilted convex lens,” Physics Letters A, vol. 377, no. 15, pp. 1154–1156, 2013.

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Astigmatic transforms of an optical vortex for measurement of its topological charge,” Applied Optics, vol. 56, no. 14, pp. 4095–4104, 2017.

M. Chen, F. S. Roux, and J. C. Olivier, “Detection of phase singularities with a Shack-Hartmann wavefront sensor,” Journal of the Optical Society of America A, vol. 24, no. 7, pp. 1994–2002, 2007.

G. F. Walsh, “Pancharatnam-Berry optical element sorter of full angular momentum eigenstate,” Optics Express, vol. 24, no. 6, p. 6689, 2016.

S. Fu, S. Zhang, T. Wang, and C. Gao, “Measurement of orbital angular momentum spectra of multiplexing optical vortices,” Optics Express, vol. 24, no. 6, pp. 6240–6248, 2016.

J. Lu, C. Cao, Z. Zhu, and B. Gu, “Flexible measurement of high-order optical orbital angular momentum with a variable cylindrical lens pair,” Applied Physics Letters, vol. 116, no. 20, Article ID 201105, 2020.

F. Ricci, W. Löffler, and M. P. van Exter, “Instability of higher-order optical vortices analyzed with a multi-pinhole interferometer,” Optics Express, vol. 20, no. 20, pp. 22961–22975, 2012.

Q. S. Ferreira, A. J. Jesus-Silva, E. J. S. Fonseca, and J. M. Hickmann, “Fraunhofer diffraction of light with orbital angular momentum by a slit,” Optics Letters, vol. 36, no. 16, pp. 3106–3108, 2011.

C. Acevedo, Y. T. Moreno, and Á. Guzmán, “Far-field diffraction pattern of an optical light beam with orbital angular momentum through of a rectangular and pentagonal aperture,” Optik, vol. 164, pp. 479–487, 2018.

K. Dai, C. Gao, L. Zhong, Q. Na, and Q. Wang, “Measuring OAM states of light beams with gradually-changing-period gratings,” Optics Letters, vol. 40, no. 4, pp. 562–565, 2015.

M. Bahl and P. Senthilkumaran, “Energy circulations in singular beams diffracted through an isosceles right triangular aperture,” Physical Review A, vol. 92, Article ID 013831, pp. 1–6, 2015.

A. Ambuj, R. Vyas, and S. Singh, “Diffraction of orbital angular momentum carrying optical beams by a circular aperture,” Optics Letters, vol. 39, no. 19, p. 5475, 2014.

Y. Taira and S. Zhang, “Split in phase singularities of an optical vortex by off-axis diffraction through a simple circular aperture,” Optics Letters, vol. 42, no. 7, pp. 1373–1376, 2017.

Y. Liu and J. Pu, “Measuring the orbital angular momentum of elliptical vortex beams by using a slit hexagon aperture,” Optics Communications, vol. 284, no. 10-11, pp. 2424–2429, 2011.

A. Ambuj, R. Vyas, and S. Singh, “Diffraction of Laguerre-Gauss vortex beams by regular polygons,” in Proceedings of the Frontiers in Optics 2014, Optical Society of America, Tucson, AZ, USA, 2014.

L. Yongxin, T. Hua, P. Jixiong, and L. Baida, “Detecting the topological charge of vortex beams using an annular triangle aperture,” Optics & Laser Technology, vol. 43, no. 7, pp. 1233–1236, 2011.

S. Zheng and J. Wang, “Measuring Orbital Angular Momentum (OAM) states of vortex beams with annular gratings,” Scientific Reports, vol. 7, Article ID 40781, 2017.

D. Shen and D. Zhao, “Measuring the topological charge of optical vortices with a twisting phase,” Optics Letters, vol. 44, no. 9, pp. 2334–2337, 2019.

Z. Liu, S. Gao, W. Xiao et al., “Measuring high-order optical orbital angular momentum with a hyperbolic gradually changing period pure-phase grating,” Optics Letters, vol. 43, no. 13, p. 3076, 2018.

Y. Han and G. Zhao, “Measuring the topological charge of optical vortices with an axicon,” Optics Letters, vol. 36, no. 11, pp. 2017–2019, 2011.

B. Ni, L. Guo, C. Yue, and Z. Tang, “A novel measuring method for arbitrary optical vortex by three spiral spectra,” Physics Letters A, vol. 381, no. 8, pp. 817–820, 2017.

S. Li, P. Zhao, X. Feng et al., “Measuring the orbital angular momentum spectrum with a single point detector,” Optics Letters, vol. 43, no. 19, pp. 4607–4610, 2018.

P. Panthong, S. Srisuphaphon, S. Chiangga, and S. Deachapunya, “High-contrast optical vortex detection using the Talbot effect,” Applied Optics, vol. 57, no. 7, pp. 1657–1661, 2018.

A. Popiolek-Masajada, P. Kurzynowski, W. A. Wozniak, and M. Borwinska, “Measurements of the small wave tilt using the optical vortex interferometer with the Wollaston compensator,” Applied Optics, vol. 46, no. 33, pp. 8039–8044, 2007.

M. Borwinska, A. Popiolek-Masajada, and B. Dubik, “Reconstruction of a plane wave’s tilt and orientation using an optical vortex interferometer,” Optical Engineering, vol. 46, no. 7, 2007.

W. Fraczek and J. Mroczka, “Optical vortices as phase markers to wave front deformation measurement,” Metrology and Measurement Systems, vol. 15, no. 4, pp. 433–440, 2008.

P. Kurzynowski, M. Borwinska, and J. Masajada, “Optical vortex sign determination using self interference methods,” Applied Optics, vol. 40, no. 1, pp. 165–175, 2010.

I. Mokhun and Y. Galushko, “Detection of vortex sign for scalar speckle fields,” Ukrainian Journal of Physical Optics, vol. 9, no. 4, pp. 246–255, 2008.

E. Fraczek, W. Fraczek, and J. Masajada, “The new method of topological charge determination of optical vortices in the interference field of the optical vortex interferometer,” Optik, vol. 117, no. 9, pp. 423–425, 2006.

E. Fraczek, W. Fraczek, and J. Mroczka, “Experimental method for topological charge determination of optical vortices in a regular net,” Optical Engineering, vol. 44, no. 2, 2005.

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Spiral interferogram analysis,” Journal of the Optical Society of America A, vol. 23, no. 6, pp. 1400–1408, 2006.

S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral phase contrast imaging in microscopy,” Optics Express, vol. 13, no. 3, pp. 689–694, 2005.

A. Y. Bekshaev and S. V. Sviridova, “Effects of misalignments in the optical vortex transformation performed by holograms with embedded phase singularity,” Optics Communications, vol. 283, no. 24, pp. 4866–4876, 2010.

J. A. Davis, D. E. McNamara, D. M. Cottrell, and J. Campos, “Image processing with the radial Hilbert transform: theory and experiments,” Optics Letters, vol. 25, no. 2, pp. 99–101, 2000.

M. K. Sharma, J. Joseph, and P. Senthilkumaran, “Selective edge enhancement using anisotropic vortex filter,” Applied Optics, vol. 50, no. 27, pp. 5279–5286, 2011.

M. K. Sharma, J. Joseph, and P. Senthilkumaran, “Selective edge enhancement using shifted anisotropic vortex filter,” Journal of Optics, vol. 42, no. 1, pp. 1–7, 2013.

A. W. Lohmann, E. Tepichín, and J. G. Ramírez, “Optical implementation of the fractional Hilbert transform for two-dimensional objects,” Applied Optics, vol. 36, no. 26, pp. 6620–6626, 1997.

G. Situ, G. Pedrini, and W. Osten, “Spiral phase filtering and orientation-selective edge detection/enhancement,” Journal of the Optical Society of America A, vol. 26, no. 8, pp. 1788–1797, 2009.

G. Situ, M. Warber, G. Pedrini, and W. Osten, “Phase contrast enhancement in microscopy using spiral phase filtering,” Optics Communications, vol. 283, no. 7, pp. 1273–1277, 2010.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Upgrading a microscope with a spiral phase plate,” Journal of Microscopy, vol. 230, no. 1, pp. 134–142, 2008.

G. Anzolin, F. Tamburini, A. Bianchini, G. Umbriaco, and C. Barbieri, “Optical vortices with starlight,” Astronomy & Astrophysics, vol. 488, no. 3, pp. 1159–1165, 2008.

G. A. Swartzlander, “Peering into darkness with a vortex spatial filter,” Optics Letters, vol. 26, no. 8, pp. 497–499, 2001.

J. Keller, A. Schönle, and S. W. Hell, “Efficient fluorescence inhibition patterns for RESOLFT microscopy,” Optics Express, vol. 15, no. 6, pp. 3361–3371, 2007.

P. Török and P. R. T. Munro, “The use of Gauss-Laguerre vector beams in STED microscopy,” Optics Express, vol. 12, no. 15, pp. 3605–3617, 2004.

K. Willig, J. Keller, M. Bossi, and S. Hell, “STED microscopy resolves nanoparticle assemblies,” New Journal of Physics, vol. 8, no. 106, pp. 1–8, 2006.

N. Bokor, Y. Iketaki, T. Watanabe, K. Daigoku, N. Davidson, and M. Fujii, “On polarization effects in fluorescence depletion microscopy,” Optics Communications, vol. 272, no. 1, pp. 263–268, 2007.

S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Optics Letters, vol. 19, no. 11, pp. 780–782, 1994.

L. Yan, P. Gregg, E. Karimi et al., “Q-plate enabled spectrally diverse orbital-angular-momentum conversion for stimulated emission depletion microscopy,” Optica, vol. 2, no. 10, pp. 900–903, 2015.

T. Aoki, T. Sotomaru, T. Ozawa, T. Komiyama, Y. Miyamoto, and M. Takeda, “Two-dimensional phase unwrapping by direct elimination of rotational vector fields from phase gradients obtained by heterodyne techniques,” Optical Review, vol. 5, no. 6, pp. 374–379, 1998.

L. Allen, M. J. Padgett, and M. Babiker, “IV The orbital angular momentum of light,” Progress in Optics, Elsevier, Amsterdam, Netherlands, 1999.

L. Allen, S. M. Barnett, and M. J. Padgett, Optical angular momentum, Institute of Physics Publishing, Bristol and Philadelphia, 2003.

M. Padgett, J. Courtial, and L. Allen, “Light’s orbital angular momentum,” Physics Today, vol. 57, no. 5, pp. 35–40, 2004.

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Physical Review, vol. 50, no. 2, pp. 115–125, 1936.

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Physical Review Letters, vol. 87, no. 2, Article ID 23902, 2001.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Physical Review Letters, vol. 88, no. 5, Article ID 53601, 2002.

A. Y. Bekshaev, “Mechanical properties of the light wave with phase singularity,” Proceedings of SPIE, vol. 3994, pp. 131–139, 1999.

A. Y. Bekshaev, “Manifestation of mechanical properties of light waves in vortex beam optical systems,” Optics and Spectroscopy, vol. 88, no. 6, pp. 904–910, 2000.

S. H. Tao, X. C. Yuan, J. Lin et al., “Residue orbital angular momentum in interferenced double vortex beams with unequal topological charge,” Optics Express, vol. 14, pp. 353–541, 2006.

M. Zürch, C. Kern, P. Hansinger, A. Dreischuh, and C. Spielmann, “Strong-field Physics with singular light beams,” Nature Physics, vol. 8, no. 10, pp. 743–746, 2012.

K. T. Gahagan and G. A. Swartzlander, “Optical vortex trapping of particles,” Optics Letters, vol. 21, no. 11, pp. 827–829, 1996.

N. R. Heckenberg, T. A. Nieminen, M. E. J. Friese, and H. Rubinsztein-Dunlop, “Trapping microscopic particles with singular beams,” Proc. SPIE., vol. 3487, no. 46, 1998.

T. M. Grzegorczyk and J. A. Kong, “Analytical prediction of stable optical trapping in optical vortices created by three TE or TM plane waves,” Optics Express, vol. 15, no. 13, pp. 8010–8018, 2007.

K. T. Gahagan and G. A. Swartzlander Jr., “Trapping of low-index microparticles in an optical vortex,” Journal of the Optical Society of America B, vol. 15, no. 2, pp. 524–534, 1998.

K. T. Gahagan and G. A. Swartzlander Jr., “Simultaneous trapping of low-index and high-index microparticles observed with an optical-vortex trap,” Journal of the Optical Society of America B, vol. 16, no. 4, pp. 533–537, 1999.

J. Liesener, M. Reicherter, T. Haist, and H. J. Tiziani, “Multi-functional optical tweezers using computer-generated holograms,” Optics Communications, vol. 185, no. 1-3, pp. 77–82, 2000.

D. S. Bradshaw and D. L. Andrews, “Interactions between spherical nanoparticles optically trapped in Laguerre-Gaussian modes,” Optics Letters, vol. 30, no. 22, pp. 3039–3041, 2005.

W. M. Lee and X.-C. Yuan, “Observation of three-dimensional optical stacking of microparticles using a single Laguerre-Gaussian beam,” Applied Physics Letters, vol. 83, no. 25, pp. 5124–5126, 2003.

M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Physical Review Letters, vol. 75, pp. 826–829, 1995.

N. B. Simpson, L. Allen, and M. J. Padgett, “Optical tweezers and optical spanners with Laguerre-Gaussian modes,” Journal of Modern Optics, vol. 43, no. 12, pp. 2485–2491, 1996.

A. T. O’Neil and M. J. Padgett, “Three dimensional optical confinement of micron sized metal particles and the decoupling of the spin and orbital angular momentum within an optical spanner,” Optics Communications, vol. 185, pp. 139–143, 2000.

W. M. Lee, B. P. S. Ahluwalia, X.-C. Yuan, W. C. Cheong, and K. Dholakia, “Optical steering of high and low index microparticles by manipulating an off-axis optical vortex,” Journal of Optics A: Pure and Applied Optics, vol. 7, no. 1, pp. 1–6, 2005.

P. Galajda and P. Ormos, “Complex micromachines produced and driven by light,” Applied Physics Letters, vol. 78, no. 2, pp. 249–251, 2001.

K. Ladavac and D. G. Grier, “Microoptomechanical pumps assembled and driven by holographic optical vortex arrays,” Optics Express, vol. 12, no. 6, pp. 1144–1149, 2004.

M. E. J. Friese, H. Rubinsztein-Dunlop, J. Gold, P. Hagberg, and D. Hanstorp, “Optically driven micromachine elements,” Applied Physics Letters, vol. 78, no. 4, pp. 547–549, 2001.

A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “An optical vortex as a rotating body: mechanical features of a singular light beam,” Journal of Optics A: Pure and Applied Optics, vol. 6, no. 5, pp. S170–S174, 2004.

N. Bozinovic, Y. Yue, Y. Ren et al., “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science, vol. 340, no. 6140, pp. 1545–1548, 2013.

Y. Yan, G. Xie, M. P.J. Lavery et al., “High capacity milli meterwave communications with orbital angular momentum multiplexing,” Nature Communications, vol. 5, no. 1, 2014.

G. Gibson, J. Courtial, M. J. Padgett et al., “Free-space information transfer using light beams carrying orbital angular momentum,” Optics Express, vol. 12, no. 22, pp. 5448–5456, 2004.

S. Avramov-Zamurovic, A. T. Watnik, J. R. Lindle, and K. P. Judd, “Designing laser beams carrying OAM for a high-performance underwater communication system,” Journal of the Optical Society of America A, vol. 37, no. 5, pp. 876–887, 2020.

Y. Yue, S. Dolinar, M. Tur, and A. Willner, “Terabit free space data transmission employing orbital angular momentum multiplexing,” Nature Photonics, vol. 6, pp. 488–496, 2012.

G. Puentes, N. Hermoss, and J. Torres, “Weak measurements with orbital angular momentum pointer states,” Physical Review Letters, vol. 109, no. 4, Article ID 040401, 2012.

S. Nechayev, M. Neugebauer, M. Vorndran, G. Leuchs, and P. Banzer, “Weak measurement of elliptical dipole moments by C-point splitting,” Physical Review Letters, vol. 121, no. 24, p. 243903, 2018.

M. V. Vasnetsov, I. G. Marienko, and M. S. Soskin, “Self-reconstruction of an optical vortex,” Journal of Experimental and Theoretical Physics Letters, vol. 71, no. 4, pp. 130–133, 2000.

V. Gorshkov, A. Kononenko, and M. Soskin, “Diffraction and self restoration of a severely screened vortex beam,” Proceedings of SPIE, vol. 4403, pp. 127–137, 2001.

B. R. Boruah and M. A. A. Neil, “Susceptibility to and correction of azimuthal aberrations in singular light beams,” Optics Express, vol. 14, no. 22, pp. 10377–10385, 2006.

F. K. Fatemi and M. Bashkansky, “Focusing properties of high charge number vortex laser beams,” Applied Optics, vol. 46, no. 30, pp. 7573–7578, 2007.

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of primary spherical aberration on high-numerical-aperture focusing of a Laguerre-Gaussian beam,” Journal of the Optical Society of America A, vol. 25, no. 6, pp. 1307–1318, 2008.

R. Singh, P. Senthilkumaran, and K. Singh, “Focusing of a vortex carrying beam with Gaussian background by a lens in the presence of spherical aberration and defocusing,” Optics and Lasers in Engineering, vol. 45, pp. 773–782, 2007.

R. K. Singh, P. Senthilkumaran, and K. Singh, “Focusing of a singular beam in the presence of spherical aberration and defocusing,” Optik, vol. 119, no. 10, pp. 459–464, 2008.

R. K. Singh, P. Senthilkumaran, and K. Singh, “Tight focusing of linearly and circularly polarized vortex beams; effect of third-order spherical aberration,” Optics and Lasers in Engineering, vol. 47, no. 7-8, pp. 831–841, 2009.

R. K. Singh, P. Senthilkumaran, and K. Singh, “Tight focusing of vortex beams in presence of primary astigmatism,” Journal of the Optical Society of America A, vol. 26, no. 3, pp. 576–588, 2009.

R. K. Singh, P. Senthilkumaran, and K. Singh, “Focusing of linearly-, and circularly polarized Gaussian background vortex beams by a high numerical aperture system afflicted with third-order astigmatism,” Optics Communications, vol. 281, no. 24, pp. 5939–5948, 2008.

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of primary coma on the focusing of a Laguerre-Gaussian beam by a high numerical aperture system; vectorial diffraction theory,” Journal of Optics A: Pure and Applied Optics, vol. 10, no. 7, Article ID 075008, 2008.

R. K. Singh, P. Senthilkumaran, and K. Singh, “Focusing of a vortex carrying beam with Gaussian background by an apertured system in presence of coma,” Optics Communications, vol. 281, no. 5, pp. 923–934, 2008.

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of coma on the focusing of an apertured singular beam,” Optics and Lasers in Engineering, vol. 45, pp. 488–494, 2007.

R. K. Singh, P. Senthilkumaran, and K. Singh, “Structure of a tightly focused vortex beam in the presence of primary coma,” Optics Communications, vol. 282, no. 8, pp. 1501–1510, 2009.

F. S. Roux, “Topological charge inversion in polynomial astigmatic Gaussian beams,” Optics Communications, vol. 281, no. 17, pp. 4205–4210, 2008.

M. V. Berry, M. R. Dennis, and R. L. Lee, “Polarization singularities in the clear sky,” New Journal of Physics, vol. 6, p. 162, 2004.

K. L. Coulson, Polarization and intensity of light in the atmosphere, Studies in geophysical optics and remote sensing, A. Deepak Pub., Hampton, VA, USA, 1988.

J. Gál, G. Horváth, V. B. Meyer-Rochow, and R. Wehner, “Polarization patterns of the summer sky and its neutral points measured by full-sky imaging polarimetry in Finnish Lapland north of the Arctic Circle,” Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, vol. 457, no. 2010, pp. 1385–1399, 2001.

G. Horváth, J. Gál, I. Pomozi, and R. Wehner, “Polarization portrait of the Arago point: video-polarimetric Imaging of the neutral points of skylight polarization,” Naturwissenschaften, vol. 85, no. 7, pp. 333–339, 1998.

R. L. Lee, “Digital imaging of clear-sky polarization,” Applied Optics, vol. 37, no. 9, pp. 1465–1476, 1998.

R. Hegedüs, S. Åkesson, R. Wehner, and G. Horváth, “Could Vikings have navigated under foggy and cloudy conditions by skylight polarization? On the atmospheric optical prerequisites of polarimetric viking navigation under foggy and cloudy skies,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 463, no. 2080, pp. 1081–1095, 2007.

A. D. Dolgov, A. G. Doroshkevich, D. I. Novikov, and I. D. Novikov, “Geometry and Statistics of Cosmic Microwave Polarization,” International Journal of Modern Physics D, vol. 8, no. 2, pp. 189–212, 1999.

D. Huterer and T. Vachaspati, “Distribution of singularities in the cosmic microwave background polarization,” Physical Review D, vol. 72, no. 4, Article ID 043004, 2005.

F. C. Frank, “I. Liquid crystals. On the theory of liquid crystals,” Discussions of the Faraday Society, vol. 25, pp. 19–28, 1958.

Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, “The fine structure of singular beams in crystals: colours and polarization,” Journal of Optics A: Pure and Applied Optics, vol. 6, no. 5, pp. S217–S228, 2004.

M. Kücken and A. C. Newell, “A model for fingerprint formation,” Europhysics Letters (EPL), vol. 68, no. 1, pp. 141–146, 2004.

R. Penrose, “The topology of ridge systems,” Annals of Human Genetics, vol. 42, no. 4, pp. 435–444, 1979.

M. V. Berry and J. H. Hannay, “Umbilic points on Gaussian random surfaces,” Journal of Physics A: Mathematical and General, vol. 10, no. 11, pp. 1809–1821, 1977.

I. R. Porteous, Geometric Differentiation: For the Intelligence of Curves and Surfaces, Cambridge University Press, Cambridge, UK, 2001.

J. F. Nye, “Lines of Circular Polarization in Electromagnetic Wave Fields,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 389, no. 1797, pp. 279–290, 1983.

M. V. Berry, “The electric and magnetic polarization singularities of paraxial waves,” Journal of Optics A: Pure and Applied Optics, vol. 6, no. 5, pp. 475–481, 2004.

I. Freund, “Hidden order in optical ellipse fields: I. Ordinary ellipses,” Optics Communications, vol. 256, no. 4–6, pp. 220–241, 2005.

W. S. Raburn and G. Gbur, “Singularities of Partially Polarized Vortex Beams,” Frontiers in Physics, vol. 8, p. 168, 2020.

I. Freund, “Optical Möbius strips in three-dimensional ellipse fields: I. Lines of circular polarization,” Optics Communications, vol. 283, no. 1, pp. 1–15, 2010.

I. Freund, “Polarization Mobius strips on elliptical paths in three dimensional optical fields,” Optics Letters, 2020, in press.

I. Freund, “Multitwist optical Möbius strips,” Optics Letters, vol. 35, no. 2, pp. 148–150, 2010.

R. W. Schoonover and T. D. Visser, “Polarization singularities of focused, radially polarized fields,” Optics Express, vol. 14, no. 12, p. 5733, 2006.

K. S. Grigoriev, P. S. Ryzhikov, E. B. Cherepetskaya, and V. A. Makarov, “Structure of polarization singularities of a light beam at triple frequency generated in isotropic medium by singularly polarized beam,” Optics Express, vol. 25, no. 21, Article ID 25416, 2017.

K. Y. Bliokh, M. A. Alonso, and M. R. Dennis, “Geometric phases in 2D and 3D polarized fields: geometrical, dynamical, and topological aspects,” Reports on Progress in Physics, vol. 82, no. 12, Article ID 122401, 2019.

I. Freund, “Poincaré vortices,” Optics Letters, vol. 26, no. 24, pp. 1996–1998, 2001.

J. H. Hannay and J. F. Nye, “Refraction of C-line vortices,” Journal of Optics, vol. 15, no. 1, 2013.

W. Loffler, J. F. Nye, and J. H. Hannay, “An experiment on the shifts of reflected C-lines,” Journal of Optics, vol. 16, no. 8, 2014.

M. M. Sánchez-López, J. A. Davis, I. Moreno, A. Cofré, and D. M. Cottrell, “Gouy phase effects on propagation of pure and hybrid vector beams,” Optics Express, vol. 27, no. 3, p. 2374, 2019.

Y. Zhang, X. Guo, L. Han et al., “Gouy phase induced polarization transition of focused vector vortex beams,” Optics Express, vol. 25, no. 21, pp. 25725–25733, 2017.

M. Suzuki, K. Yamane, K. Oka, Y. Toda, and R. Morita, “Analysis of the Pancharatnam-Berry phase of vector vortex states using the Hamiltonian based on the Maxwell-Schrödinger equation,” Physical Review A, vol. 94, no. 4, 2016.

I. Freund, “Observer-dependent sign inversions of polarization singularities,” Optics Letters, vol. 39, no. 20, p. 5873, 2014.

A. I. Mokhun, M. S. Soskin, and I. Freund, “Elliptic critical points: C-points, a-lines, and the sign rule,” Optics Letters, vol. 27, no. 12, pp. 995–997, 2002.

M. S. Soskin, V. Denisenko, and I. Freund, “Optical polarization singularities and elliptic stationary points,” Optics Letters, vol. 28, no. 16, pp. 1475–1477, 2003.

S. K. Pal and P. Senthilkumaran, “Phase engineering methods in polarization singularity lattice generation,” OSA Continuum, vol. 1, no. 1, p. 193, 2018.

S. K. Pal and P. Senthilkumaran, “Generation of orthogonal lattice fields,” Journal of the Optical Society of America A, vol. 36, no. 5, p. 853, 2019.

W. A. Wozniak, P. Kurzynowski, and A. Popiolek-Masajada, “Polarization vortices as a superposition of orthogonal phase vortices,” Optics Communications, vol. 441, pp. 155–159, 2019.

R. Kalita, M. Gaffar, and B. R. Boruah, “The generation of arbitrary vector beams using a division of a wavefront based setup,” Journal of Optics, vol. 18, no. 7, 2016.

D. J. Armstrong, M. C. Phillips, and A. V. Smith, “Generation of radially polarized beams with an image-rotating resonator,” Applied Optics, vol. 42, no. 18, pp. 3550–3554, 2003.

S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Applied Optics, vol. 29, no. 15, p. 2234, 1990.

S. C. Tidwell, G. H. Kim, and W. D. Kimura, “Efficient radially polarized laser beam generation with a double interferometer,” Applied Optics, vol. 32, no. 27, pp. 5222–5229, 1993.

S. Fu, C. Gao, Y. Shi, K. Dai, L. Zhong, and S. Zhang, “Generating polarization vortices by using helical beams and a Twyman Green interferometer,” Optics Letters, vol. 40, no. 8, p. 1775, 2015.

J. Xin, C. Gao, C. Li, and Z. Wang, “Generation of polarization vortices with a Wollaston prism and an interferometric arrangement,” Applied Optics, vol. 51, no. 29, pp. 7094–7097, 2012.

P. Kumar, S. K. Pal, N. Nishchal, and P. Senthilkumaran, “Non-interferometric technique to realize vector beams embedded with polarization singularities,” Journal of the Optical Society of America, vol. 37, pp. 1043–1052, 2020, in press.

T. Moser, J. Balmer, D. Delbeke, P. Muys, S. Verstuyft, and R. Baets, “Intracavity generation of radially polarized CO₂ laser beams based on a simple binary dielectric diffraction grating,” Applied Optics, vol. 4, no. 33, pp. 8517–8522, 2006.

Y. Mushiake, K. Matsumura, and N. Nakajima, “Generation of radially polarized optical beam mode by laser oscillation,” Proceedings of the IEEE, vol. 60, no. 9, pp. 1107–1109, 1972.

S. Quabis, R. Dorn, and G. Leuchs, “Generation of a radially polarized doughnut mode of high quality,” Applied Physics B, vol. 81, no. 5, pp. 597–600, 2005.

D. Pohl, “Operation of a Ruby laser in the purely transverse electric mode TE01,” Applied Physics Letters, vol. 20, no. 7, pp. 266-267, 1972.

A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre-Gaussian laser beams,” Journal of the Optical Society of America A, vol. 15, no. 10, pp. 2705–2711, 1998.

K.-C. Chang, T. Lin, and M.-D. Wei, “Generation of azimuthally and radially polarized off-axis beams with an intracavity large-apex-angle axicon,” Optics Express, vol. 21, no. 13, pp. 16035–16042, 2013.

T. Moser, H. Glur, V. Romano et al., “Polarization-selective grating mirrors used in the generation of radial polarization,” Applied Physics B, vol. 80, no. 6, pp. 707–713, 2005.

Y. Kozawa and S. Sato, “Generation of a radially polarized laser beam by use of a conical Brewster prism,” Optics Letters, vol. 30, no. 22, pp. 3063–3065, 2005.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Applied Physics Letters, vol. 77, no. 21, pp. 3322–3324, 2000.

Y. Kozawa, K. Yonezawa, and S. Sato, “Radially polarized laser beam from a Nd:YAG laser cavity with a c-cut YVO4 Crystal,” Applied Physics B, vol. 88, no. 1, pp. 43–46, 2007.

Y. Kozawa, T. Yoneyama, and S. Sato, “Direct generation of cylindrical vector beam from Nd:YAG laser cavity,” in Proceedings of the 2005 Pacific Rim Conference on Lasers & Electro-Optics, pp. 809-810, Tokyo, Japan, 2005.

C. F. Phelan, J. F. Donegan, and J. G. Lunney, “Generation of a radially polarized light beam using internal conical diffraction,” Optics Express, vol. 19, no. 22, pp. 21793–21802, 2011.

A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Formation of linearly polarized light with axial symmetry by use of space-variant subwavelength gratings,” Optics Letters, vol. 28, no. 7, pp. 510–512, 2003.

G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Formation of helical beams by use of Pancharatnam-Berry phase optical elements,” Optics Letters, vol. 27, no. 21, pp. 1875–1877, 2002.

Z. Yang, D.-F. Kuang, and F. Cheng, “Vector vortex beam generation with dolphin-shaped cell meta-surface,” Optics Express, vol. 25, no. 19, pp. 22780–22788, 2017.

A. V. Nesterov, V. G. Niziev, and V. P. Yakunin, “Generation of high-power radially polarized beam,” Journal of Physics D: Applied Physics, vol. 32, no. 22, pp. 2871–2875, 1999.

E. J. Galvez and X. Cheng, Space Variant Polarization States of Photons, vol. 1, 2014.

E. J. Galvez, B. L. Rojec, K. Beach, and X. Cheng, “C-point Singularities in Poincaré Beams,” Coherence Quantum Opt. X, 2014.

E. J. Galvez and S. Khadka, “Poincaré modes of light,” Proceedings of SPIE, vol. 8274, Article ID 82740, 2012.

E. J. Galvez, B. L. Rojec, and K. Beach, Isolated Polarization Singularities in Optical Beams, vol. 1, 2014.

F. Fan, T. Du, A. K. Srivastava, W. Lu, V. Chigrinov, and H. S. Kwok, “Axially symmetric polarization converter made of patterned liquid crystal quarter wave plate,” Optics Express, vol. 20, no. 21, pp. 23036–23043, 2012.

M.-Q. Cai, Z.-X. Wang, J. Liang et al., “High-efficiency and flexible generation of vector vortex optical fields by a reflective phase-only spatial light modulator,” Applied Optics, vol. 56, no. 22, p. 6175, 2017.

C. Rosales-Guzmán, N. Bhebhe, and A. Forbes, “Simultaneous generation of multiple vector beams on a single SLM,” Optics Express, vol. 25, no. 21, p. 25697, 2017.

L. Gong, Y. Ren, W. Liu et al., “Generation of cylindrically polarized vector vortex beams with digital micromirror device,” Journal of Applied Physics, vol. 116, no. 18, 2014.

A. Manthalkar, I. Nape, N. T. Bordbar et al., “All-digital Stokes polarimetry with a digital micromirror device,” Optics Letters, vol. 45, no. 8, pp. 2319–2322, 2020.

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarized beams,” Optics Communications, vol. 281, no. 4, pp. 732–738, 2008.

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Efficient extracavity generation of radially and azimuthally polarized beams,” Optics Letters, vol. 32, no. 11, pp. 1468–1470, 2007.

Y. Gorodetski, G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Space-variant polarization manipulation for far-field polarimetry by use of subwavelength dielectric gratings,” Optics Letters, vol. 30, no. 17, p. 2245, 2005.

Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Space variant Pancharatnam-Berry phase optical elements with computer generated subwavelength gratings,” Optics Letters, vol. 27, no. 13, p. 1141, 2007.

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nature Materials, vol. 13, no. 2, pp. 139–150, 2014.

R. C. Devlin, A. Ambrosio, N. A. Rubin, J. P. B. Mueller, and F. Capasso, “Arbitrary spin-to-orbital angular momentum conversion of light,” Science, vol. 358, no. 6365, pp. 896–901, 2017.

X. Yi, X. Ling, Z. Zhang et al., “Generation of cylindrical vector vortex beams by two cascaded metasurfaces,” Optics Express, vol. 22, no. 14, pp. 17207–17215, 2014.

N. A. Rubin, A. Zaidi, M. Juhl et al., “Polarization state generation and measurement with a single metasurface,” Optics Express, vol. 26, no. 17, p. 21455, 2018.

F. Yue, D. Wen, J. Xin, B. D. Gerardot, J. Li, and X. Chen, “Vector vortex beam generation with a single plasmonic metasurface,” ACS Photonics, vol. 3, no. 9, pp. 1558–1563, 2016.

L. Marrucci, “The q-plate and its future,” Journal of Nanophotonics, vol. 7, no. 1, Article ID 078598, 2013.

F. Cardano, E. Karimi, S. Slussarenko, L. Marrucci, C. de Lisio, and E. Santamato, “Polarization pattern of vector vortex beams generated by q-plates with different topological charges,” Applied Optics, vol. 51, no. 10, p. C1, 2012.

S. Slussarenko, A. Murauski, T. Du, V. Chigrinov, L. Marrucci, and E. Santamato, “Tunable liquid crystal q-plates with arbitrary topological charge,” Optics Express, vol. 19, no. 5, pp. 4085–4090, 2011.

M. Vergara and C. Iemmi, “Generalized -plates and alternative kinds of vector and vortex beams,” Physical Review A, vol. 100, no. 5, pp. 1–9, 2019.

S. Liu, S. Qi, Y. Zhang et al., “Highly efficient generation of arbitrary vector beams with tunable polarization, phase, and amplitude,” Photonics Research, vol. 6, no. 4, p. 228, 2018.

S. Delaney, M. M. Sánchez-López, I. Moreno, and J. A. Davis, “Arithmetic with q-plates,” Applied Optics, vol. 56, no. 3, pp. 596–600, 2017.

S. Bansal, S. Kumar Pal, and P. Senthilkumaran, “Use of -plate as a coupler,” Applied Optics, vol. 59, pp. 4933–4938, 2020, in press.

C.-H. Yang, Y.-D. Chen, S.-T. Wu, and A. Y. G. Fuh, “Independent Manipulation of Topological Charges and Polarization Patterns of Optical Vortices,” Scientific Reports, vol. 6, pp. 1–12, 2016.

B. Wei, S. Qi, S. Liu et al., “Auto-transition of vortex- to vector-airy beams via liquid crystal q-airy-plates,” Optics Express, vol. 27, no. 13, pp. 18848–18857, 2019.

B. Radhakrishna, G. Kadiri, and G. Raghavan, “Wavelength adaptable effective -plates with passively tunable retardance,” Scientific Reports, vol. 9, no. 1, Article ID 11911, 2019.

I. Moreno, M. M. Sanchez-Lopez, K. Badham, J. A. Davis, and D. M. Cottrell, “Generation of integer and fractional vector beams with q-plates encoded onto a spatial light modulator,” Optics Letters, vol. 41, no. 6, pp. 1305–1308, 2016.

D. Marco, M. d. M. Sánchez-López, P. García-Martínez, and I. Moreno, “Using birefringence colors to evaluate a tunable liquid-crystal q-plate,” Journal of the Optical Society of America B, vol. 36, no. 5, pp. D34–D41, 2019.

D. Li, S. Feng, S. Nie, J. Ma, and C. Yuan, “Scalar and vectorial vortex filtering based on geometric phase modulation with a -plate,” Journal of Optics, vol. 21, no. 6, 2019.

A. Rubano, F. Cardano, B. Piccirillo, and L. Marrucci, “Q-plate technology: a progress review,” Journal of the Optical Society of America B, vol. 36, no. 5, pp. D70–D87, 2019.

M. Sakamoto, Y. Nakamoto, K. Kawai et al., “Polarization grating fabricated by recording a vector hologram between two orthogonally polarized vector vortex beams,” Journal of the Optical Society of America B, vol. 34, no. 2, p. 263, 2017.

N. K. Viswanathan and V. V. G. Inavalli, “Generation of optical vector beams using a two-mode fiber,” Optics Letters, vol. 34, no. 8, pp. 1189–1191, 2009.

C. N. Alexeyev, N. A. Boklag, and M. A. Yavorsky, “Higher order modes of coupled optical fibres,” Journal of Optics, vol. 12, no. 11, 2010.

S. Pidishety, B. Srinivasan, and G. Brambilla, “All-fiber fused coupler for stable generation of radially and azimuthally polarized beams,” IEEE Photonics Technology Letters, vol. 29, no. 1, pp. 31–34, 2017.

S. Liu, L. Han, P. Li, Y. Zhang, H. Cheng, and J. Zhao, “A method for simultaneously measuring polarization and phase of arbitrarily polarized beams based on Pancharatnam-Berry phase,” Applied Physics Letters, vol. 110, no. 17, 2017.

L. Marrucci, C. Manzo, and D. Paparo, “Pancharatnam-Berry phase optical elements for wave front shaping in the visible domain: switchable helical mode generation,” Applied Physics Letters, vol. 88, no. 22, pp. 1–4, 2006.

A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Polychromatic vectorial vortex formed by geometric phase elements,” Optics Letters, vol. 32, no. 7, pp. 847–849, 2007.

Y. Liu, Z. Liu, J. Zhou et al., “Measurements of Pancharatnam-Berry phase in mode transformations on hybrid-order Poincaré sphere,” Optics Letters, vol. 42, no. 17, p. 3447, 2017.

S. Mamani, E. Bendau, J. Secor, S. Ashrafi, J. J. Tu, and R. R. Alfano, “Hybrid generation and analysis of vector vortex beams,” Applied Optics, vol. 56, no. 8, pp. 2171–2175, 2017.

E. Otte, C. Alpmann, and C. Denz, “Higher order polarization singularitites in tailored vector beams,” Journal of Optics, vol. 18, no. 7, pp. 1–7, 2016.

W. Zhu and W. She, “Electrically tunable generation of inhomogeneously polarized light beam,” Optics Communications, vol. 311, pp. 212–215, 2014.

D. Wu, Y. Li, W. Jia et al., “Generation of arbitrary vector vortex beams based on the dual-modulation method,” Applied Optics, vol. 58, no. 6, p. 1508, 2019.

H.-T. Wang, X.-l. Wang, Y. Li, J. Chen, C.-S. Guo, and J. Ding, “A new type of vector fields with hybrid states of polarization,” Optics Express, vol. 18, no. 10, p. 10786, 2010.

C. T. Samlan and N. K. Viswanathan, “Generation of vector beams using a double-wedge depolarizer: Non-quantum entanglement,” Optics and Lasers in Engineering, vol. 82, pp. 135–140, 2016.

S. A. Schulz, T. Machula, E. Karimi, and R. W. Boyd, “Integrated multi vector vortex beam generator,” Optics Express, vol. 21, no. 13, pp. 16130–16141, 2013.

R. P. Chen, Z. Chen, K. H. Chew et al., “Structured caustic vector vortex optical field: Manipulating optical angular momentum flux and polarization rotation,” Scientific Reports, vol. 5, pp. 1–8, 2015.

V. Sharma, S. C. Kumar, A. Aadhi, H. Ye, G. K. Samanta, and M. Ebrahim-Zadeh, “Tunable vector vortex beam optical parametric oscillator,” Scientific Reports, vol. 9, no. 1, pp. 1–7, 2019.

K. J. Mitchell, N. Radwell, S. Franke-Arnold, M. J. Padgett, and D. B. Phillips, “Polarisation structuring of broadband light,” Optics Express, vol. 25, no. 21, Article ID 25079, 2017.

N. Radwell, R. D. Hawley, J. B. Götte, and S. Franke-Arnold, “Achromatic vector vortex beams from a glass cone,” Nature Communications, vol. 7, 2016.

P. H. Jones, M. Rashid, M. Makita, and O. M. Maragò, “Sagnac interferometer method for synthesis of fractional polarization vortices,” Optics Letters, vol. 34, no. 17, pp. 2560–2562, 2009.

J. Zeng, C. Liang, H. Wang et al., “Partially coherent radially polarized fractional vortex beam,” Optics Express, vol. 28, no. 8, pp. 11493–11513, 2020.

Z. Shao, J. Zhu, Y. Zhang, Y. Chen, and S. Yu, “On-chip switchable radially and azimuthally polarized vortex beam generation,” Optics Letters, vol. 43, no. 6, p. 1263, 2018.

G. Arora, S. K. Pal, and P. Senthilkumaran, “Spatially varying lattice of C points,” OSA Continuum, vol. 2, no. 2, p. 416, 2019.

B. Ndagano, H. Sroor, M. McLaren, C. Rosales-Guzmán, and A. Forbes, “Beam quality measure for vector beams,” Optics Letters, vol. 41, no. 15, pp. 3407–3410, 2016.

B. Ndagano, I. Nape, B. Perez-Garcia et al., “A deterministic detector for vector vortex states,” Scientific Reports, vol. 7, no. 1, pp. 1–8, 2017.

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Physical Review E, vol. 65, Article ID 036602, pp. 1–5, 2002.

B. S. B. Ram, A. Sharma, and P. Senthilkumaran, “Diffraction of V-point singularities through triangular apertures,” Optics Express, vol. 25, no. 9, pp. 10270–10275, 2017.

B. S. B. Ram, A. Sharma, and P. Senthilkumaran, “Probing the degenerate states of V-point singularities,” Optics Letters, vol. 42, no. 18, pp. 3570–3573, 2017.

S. N. Khan, S. Deepa, G. Arora, and P. Senthilkumaran, “Perturbation-induced morphological transformations in vector-field singularities,” Journal of the Optical Society of America B, vol. 37, no. 6, pp. 1577–1586, 2020.

X. Cui, C. Wang, and X. Jia, “Nonparaxial propagation of vector vortex beams diffracted by a circular aperture,” Journal of the Optical Society of America A, vol. 36, no. 1, p. 115, 2018.

X. Zhao and X. Jia, “Vectorial structure of arbitrary vector vortex beams diffracted by a circular aperture in the far field,” Laser Physics, vol. 28, no. 1, Article ID 015004, 2018.

R. Yu, Y. Xin, Q. Zhao, Y. Shao, and Y. Chen, “Exceptional polarization structures near the C-lines in diffracted near fields,” Journal of the Optical Society of America A, vol. 32, no. 8, p. 1468, 2015.

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Optics Express, vol. 7, no. 2, pp. 77–87, 2000.

Q. Zhan and J. Leger, “Focus shaping using cylindrical vector beams,” Optics Express, vol. 10, no. 7, pp. 324–331, 2002.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Physical Review Letters, vol. 91, no. 23, 2003.

C. J. R. Sheppard and A. Choudhury, “Annular pupils, radial polarization, and superresolution,” Applied Optics, vol. 43, no. 22, pp. 4322–4327, 2004.

W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Optics Communications, vol. 265, no. 2, pp. 411–417, 2006.

B. Hao, J. Burch, and J. Leger, “Smallest flattop focus by polarization engineering,” Applied Optics, vol. 47, no. 16, pp. 2931–2940, 2008.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Optics Communications, vol. 179, no. 1–6, pp. 1–7, 2000.

L. E. Helseth, “Smallest focal hole,” Optics Communications, vol. 257, no. 1, pp. 1–8, 2006.

Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Optics Express, vol. 18, no. 10, pp. 10828–10833, 2010.

Y. I. Salamin, “Acceleration in vacuum of bare nuclei by tightly focused radially polarized laser light,” Optics Letters, vol. 32, no. 23, pp. 3462–3464, 2007.

Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Optics Express, vol. 12, no. 15, pp. 3377–3382, 2004.

P. Shi, L. Du, and X. Yuan, “Structured spin angular momentum in highly focused cylindrical vector vortex beams for optical manipulation,” Optics Express, vol. 26, no. 18, p. 23449, 2018.

B. S. Bhargava Ram, P. Senthilkumaran, and A. Sharma, “Polarization-based spatial filtering for directional and nondirectional edge enhancement using an S-waveplate,” Applied Optics, vol. 56, no. 11, pp. 3171–3178, 2017.

B. S. B. Ram and P. Senthilkumaran, “Edge enhancement by negative Poincaré-Hopf index filters,” Optics Letters, vol. 43, no. 8, pp. 1830–1833, 2018.

E. Collett and R. Alferness, “Depolarization of a laser beam in a turbulent medium,” Journal of the Optical Society of America, vol. 62, no. 4, pp. 529–533, 1972.

M. A. Cox, C. Rosales-Guzmán, M. P. J. Lavery, D. J. Versfeld, and A. Forbes, “On the resilience of scalar and vector vortex modes in turbulence,” Optics Express, vol. 24, no. 16, pp. 18105–18113, 2016.

G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” Journal of the Optical Society of America A, vol. 25, no. 1, pp. 225–230, 2008.

W. Cheng, J. W. Haus, and Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Optics Express, vol. 17, no. 20, p. 17829, 2009.

R. Tao, X. Wang, L. Si, P. Zhou, and Z. Liu, “Propagation of focused vector laser beams in turbulent atmosphere,” Optics & Laser Technology, vol. 54, pp. 62–67, 2013.

V. P. Aksenov, V. V. Dudorov, V. V. Kolosov, and M. E. Levitsky, “Synthesized vortex beams in the turbulent atmosphere,” Frontiers in Physics, vol. 8, no. 143, 2020.

Y. Cai, Q. Lin, H. T. Eyyuboglu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Optics Express, vol. 16, no. 11, pp. 7665–7673, 2008.

Y. Gu and G. Gbur, “Reduction of turbulence-induced scintillation by nonuniformly polarized beam arrays,” Optics Letters, vol. 37, no. 9, pp. 1553–1555, 2012.

Y. Gu, O. Korotkova, and G. Gbur, “Scintillation of nonuniformly polarized beams in atmospheric turbulence,” Optics Letters, vol. 34, no. 15, p. 2261, 2009.

C. Wei, D. Wu, C. Liang, F. Wang, and Y. Cai, “Experimental verification of significant reduction of turbulence-induced scintillation in a full Poincaré beam,” Optics Express, vol. 23, no. 19, pp. 24331–24341, 2015.

P. Lochab, P. Senthilkumaran, and K. Khare, “Robust laser beam engineering using polarization and angular momentum diversity,” Optics Express, vol. 25, no. 15, pp. 17524–17529, 2017.

P. Lochab, P. Senthilkumaran, and K. Khare, “Designer vector beams maintaining a robust intensity profile on propagation through turbulence,” Physical Review A, vol. 98, no. 2, pp. 1–9, 2018.

P. Lochab, P. Senthilkumaran, and K. Khare, “Investigating polarization singular beams for robust propagation through a random medium,” in Proceedings of the Imaging and Applied Optics 2018 (3D, AO, AIO, COSI, DH, IS, LACSEA, LS&C, MATH, pcAOP), Orlando, FL, USA, 2018.

P. Lochab, P. Senthilkumaran, and K. Khare, “Propagation of converging polarization singular beams through atmospheric turbulence,” Applied Optics, vol. 58, no. 23, p. 6335, 2019.

F. Araoka, T. Verbiest, K. Clays, and A. Persoons, “Interactions of twisted light with chiral molecules: An experimental investigation,” Physical Review A, vol. 71, no. 5, pp. 1–3, 2005.

K. A. Forbes and D. L. Andrews, “Optical orbital angular momentum: twisted light and chirality,” Optics Letters, vol. 43, no. 3, p. 435, 2018.

C. T. Samlan, R. R. Suna, D. N. Naik, and N. K. Viswanathan, “Spin orbit beams for optical chirality measurement,” Applied Physics Letters, vol. 112, no. 3, pp. 1–7, 2018.

J. Liu, S. M. Li, L. Zhu et al., “Direct fiber vector eigenmode multiplexing transmission seeded by integrated optical vortex emitters,” Light: Science & Applications, vol. 7, no. 3, pp. 1–6, 2018.

S. Fu, T. Wang, Z. Zhang, Y. Zhai, and C. Gao, “Selective acquisition of multiple states on hybrid Poincaré sphere,” Applied Physics Letters, vol. 110, no. 19, 2017.

B. Ndagano, I. Nape, M. A. Cox, C. Rosales-Guzman, and A. Forbes, “Creation and detection of vector vortex modes for classical and quantum communication,” Journal of Lightwave Technology, vol. 36, no. 2, pp. 292–301, 2018.

G. Milione, T. A. Nguyen, J. Leach, D. A. Nolan, and R. R. Alfano, “Using the nonseparability of vector beams to encode information for optical communication,” Optics Letters, vol. 40, no. 21, p. 4887, 2015.

Y. He, Y. Li, J. Liu et al., “Switchable phase and polarization singular beams generation using dielectric metasurfaces,” Scientific Reports, vol. 7, no. 1, pp. 1–10, 2017.

A. D’Errico, M. Maffei, B. Piccirillo, C. De Lisio, F. Cardano, and L. Marrucci, “Topological features of vector vortex beams perturbed with uniformly polarized light,” Scientific Reports, vol. 7, pp. 1–11, 2017.

H. Kobayashi, K. Nonaka, and Y. Shikano, “Stereographical visualization of a polarization state using weak measurements with an optical vortex beam,” Physical Review A, vol. 89, no. 5, 2014.

E. Karimi, J. Leach, S. Slussarenko et al., “Spin orbit hybrid entanglement of photons and quantum contextuality,” Physical Review A, vol. 82, Article ID 022115, 2010.

E. Allahyari, J. J. Nivas, F. Cardano et al., “Simple method for the characterization of intense Laguerre-Gauss vector vortex beams,” Applied Physics Letters, vol. 112, no. 21, Article ID 211103, 2018.

Y. Zhai, S. Fu, J. Zhang, X. Liu, H. Zhou, and C. Gao, “Turbulence aberration correction for vector vortex beams using deep neural networks on experimental data,” Optics Express, vol. 28, no. 5, pp. 7515–7527, 2020.

S. Pachava, R. Dharmavarapu, A. Vijayakumar et al., “Generation and decomposition of scalar and vector modes carrying orbital angular momentum: a review,” Optical Engineering, vol. 59, no. 4, Article ID 041205, 2019.

G. Piquero, R. Martínez-Herrero, J. C. G. de Sande, and M. Santarsiero, “Synthesis and characterization of non-uniformly totally polarized light beams: tutorial,” Journal of the Optical Society of America A, vol. 37, no. 4, pp. 591–605, 2020.

Y. Shen, X. Wang, Z. Xie et al., “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light: Science & Applications, vol. 8, no. 1, 2019.

C. Rosales-Guzmán, B. Ndagano, and A. Forbes, “A review of complex vector light fields and their applications,” Journal of Optics, vol. 20, no. 12, 2018.

A. Forbes, “Structured light: tailored for purpose,” Optics & Photonics News, vol. 31, no. 6, pp. 1–60, 2020.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry et al., “Roadmap on structured light,” Journal of Optics, vol. 19, no. 1, 2017.

O. V. Angelsky, A. Y. Bekshaev, S. G. Hanson, C. Y. Zenkova, I. I. Mokhun, and Z. Jun, “Structured light: ideas and concepts,” Frontiers in Physics, vol. 8, p. 114, 2020.