Modulation of orbital angular momentum on the propagation dynamics of light fields

Abstract

Optical vortices carrying orbital angular momentum (OAM) have attracted extensive attention in recent decades because of their interesting applications in optical trapping, optical machining, optical communication, quantum information, and optical microscopy. Intriguing effects induced by OAMs, such as angular momentum conversion, spin Hall effect of light (SHEL), and spin–orbital interaction, have also gained increasing interest. In this article, we provide an overview of the modulations of OAMs on the propagation dynamics of scalar and vector fields in free space. First, we introduce the evolution of canonical and noncanonical optical vortices and analyze the modulations by means of local spatial frequency. Second, we review the Pancharatnam–Berry (PB) phases arising from spin–orbital interaction and reveal the control of beam evolution referring to novel behavior such as spin-dependent splitting and polarization singularity conversion. Finally, we discuss the propagation and focusing properties of azimuthally broken vector vortex beams.

Keywords

orbital angular momentum / polarization / spin angular momentum / Pancharatnam–Berry (PB) phase / angular diffraction

1 Introduction

Scientists have realized that light fields carry linear and angular momenta after the pioneering work of Poynting [1]. The angular momentum carried by a light field can be divided into the spin and orbital parts [2,3], which are conventionally associated with a circular polarization state and vortex phase, respectively. The spin angular momentum (SAM) is an intrinsic property of photons, similar to the SAM of electrons; it has quantitated values of±ħ and depends on the circular polarization state of the light field. The orbital angular momentum (OAM), however, can be further divided into intrinsic and extrinsic OAMs [2]. In 1992, Allen et al. recognized that light fields with spiral phases carry intrinsic OAMs [4]. This intrinsic OAM carried by each photon can be many times greater than the SAM [2,4]. Immediately after the detection reported by Woerdman [5–7], several studies in this new area of light physics began. Meanwhile, light beams carrying intrinsic OAMs have found practical applications in optical trapping, optical machining, optical communication, quantum information, optical microscopy, and so on [8–10]. Extrinsic OAMs arise from the cross product of the total linear momentum transported by the beam and position of its axis relative to the origin of coordinates [2]. Therefore, the total angular momentum of a light field can be presented in the form of three summands.

Simultaneously, particular interest was paid to the interaction of angular momenta, which produce nontrivial evolution of light fields [11–16]. The interaction between SAMs and extrinsic OAMs leads to the transverse deflection of circularly polarized beams when propagating in inhomogeneous media, including the polarization-dependent Imbert–Fedorov effect of a light beam reflected or refracted at the interface between two media and splitting of rays in inhomogeneous anisotropic media [13,17–20]. The interaction between intrinsic and extrinsic OAMs leads to significantly enhanced transverse deflection with respect to the values for the topological charge of the vortex phase [17,21,22], similar to the Magnus effect of light [23]. In this article, we review the modulations of OAMs on the propagation dynamics of light fields, considering both scalar and vector fields. In Section 2, we introduce the basic theoretical description of OAM for generic light fields. In Section 3, we describe the evolutions of canonical and noncanonical optical vortices and emphatically analyze the modulation based on local spatial frequency. In Section 4, we present the Pancharatnam–Berry (PB) phases that stem from spin–orbital interaction and discuss the novel propagation dynamics induced by spatially variant PB phases. Finally, we give an overview on the propagation and focusing properties of azimuthally broken vector vortex beams.

2 OAM of light fields

First, we consider a monochromatic plane wave propagating along the z-axis in free space, of which the electric and magnetic components are characterized as E and B, respectively. Considering the time-harmonic feature of a light filed, the time-averaged linear momentum density can be expressed as

(1)$\mathit{p}={\displaystyle \frac{{\epsilon }_{\mathrm{0}}}{\mathrm{2}}}\mathrm{Re}\left({\mathit{E}}^{\ast }\times \mathit{B}\right).$

A uniformly polarized paraxial light field has vector potential denoted as A = u(x,y,z)exp[−i(wt−kz)](e_x + ise_y), where u(x,y,z) is the normalized complex amplitude, w is the frequency, k = w/c is the wavenumber, e_x and e_y are the unit vectors, s = i(ab* -a*b) is the polarization ellipticity, and a and b are the complex amplitude of two orthogonal polarizations with |a|² + |b|² = 1. The time-averaged linear momentum density thus can be rewritten in the cylindrical coordinate system as

(2)$\mathit{p}={\displaystyle \frac{\mathrm{i}{\epsilon }_{\mathrm{0}}\omega }{\mathrm{2}}}\left(u^{\ast }\nabla u-u\nabla u^{\ast }\right)+{\epsilon }_{\mathrm{0}}\omega k{\left|u\right|}^{\mathrm{2}}{\mathit{e}}_z+\frac{{\epsilon }_{\mathrm{0}}\omega \sigma }{\mathrm{2}}{\displaystyle \frac{\partial {\left|u\right|}^{\mathrm{2}}}{\partial r}}{\mathit{e}}_{\phi },$

where e_z and e_φ are the longitudinal and azimuthal unit vectors in the cylindrical coordinate system, respectively. Thus, the total angular momentum can be obtained by integrating the cross product of linear momentum density with the radius vector r = (r,0,z)—i.e., J = ε₀(r× p)dr [24]. From Eq. (2), we can find that, when s = 0—that is, for the linearly polarized plane waves—there cannot be any component of angular momentum, whereas for the circularly polarized plane waves—i.e., s ≠ 0—there are pure SAMs with respective to the spin state s.

Nevertheless, Allen et al. noted that, for the Laguerre–Gaussian (LG) mode with complex amplitude [6,25]

${\mathrm{LG}}_{l,p}\left(r,\phi \right)=\sqrt{\frac{\mathrm{2}p!}{\mathrm{\pi }\left(p+\left|l\right|\right)!}}\frac{\mathrm{1}}{w\left(z\right)}{\left({\displaystyle \frac{\sqrt{\mathrm{2}}r}{w\left(z\right)}}\right)}^{\left|l\right|}$

$\times \exp \left(-{\displaystyle \frac{r^{\mathrm{2}}}{w{\left(z\right)}^{\mathrm{2}}}}\right)L_p^{\left|l\right|}\left[{\displaystyle \frac{\mathrm{2}{r}^{\mathrm{2}}}{w{\left(z\right)}^{\mathrm{2}}}}\right]$

(3)$\times \exp \left(\mathrm{i}l\phi \right)\exp \left[-\mathrm{i}k{\displaystyle \frac{r^{\mathrm{2}}z}{\mathrm{2}\left(z_R^{\mathrm{2}}+z^{\mathrm{2}}\right)}}\right]\exp \left[-\mathrm{i}\mathrm{\Phi }\right],$

where $L_p^{\left|l\right|}$ is a normalized Laguerre polynomial with l and p denoting the azimuthal and radial variation of phase fronts, z_R = k$w_0^2$/2 is the Rayleigh range, w(z) = w₀$\sqrt{\mathrm{1}+z^{\mathrm{2}}/z_R^{\mathrm{2}}}$ is the beam waist, F = (2p + |l| + 1)tan⁻¹(z/z_R) corresponds to the Gouy phase, and (r,f,z) are the cylindrical coordinates. The total angular momentum density can be described as [26]

(4)${\mathit{j}}_z=\left({\displaystyle \frac{l}{\omega }}{\left|u\right|}^{\mathrm{2}}+{\displaystyle \frac{\sigma r}{\mathrm{2}\omega }}{\displaystyle \frac{\partial {\left|u\right|}^{\mathrm{2}}}{\partial r}}\right){\mathit{e}}_z.$

Integrating this equation, we obtain the total angular momentum of (l + s)ћ per photon. Obviously, the total angular momentum is divided into the orbital and spin parts with respect to the topological charge of the vortex phase and spin state, respectively.

For the generic fields, the linear momentum density can be deduced from the vector potential and can be given by [27]

(5)$\mathit{p}={\displaystyle \frac{{\epsilon }_{\mathrm{0}}}{\mathrm{2}\omega }}\left\{\mathrm{Im}\left[{\mathit{E}}^{\ast }\cdot (\nabla )\mathit{E}\right]+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\nabla \times \mathrm{Im}\left[{\mathit{E}}^{\ast }\times \mathit{E}\right]\right\}.$

These two terms in the right part of the above equation correspond to the orbital and spin parts of linear momentum. Particularly, the orbital part of linear momentum density is closely related to the local wave vector, depending on the phase gradient perpendicular to the wave vector. In consequence, the OAM caused by the gradient of the wave front is rewritten as

(6)${\mathit{j}}_o=\mathit{r}\times {\mathit{p}}_o={\displaystyle \frac{{\epsilon }_{\mathrm{0}}}{\mathrm{2}\omega }}\mathrm{Im}\left[{\mathit{E}}^{\ast }\cdot \left(\mathit{r}\times \nabla \right)\mathit{E}\right].$

Integrating over the whole transverse field, one can obtain the OAM per photon.

The simplest example of a light field carrying OAMs is that with a canonical vortex phase in the transverse plane of exp(ilf). Similar to classical mechanics, the OAM of a photon can exert forces and torques on both macroscopic and quantum objects [28–31]. It has been transferred from light fields to microsized particles and to atoms and Bose–Einstein condensates [2,8,32]. Moreover, the modes carrying distinct l are mutually orthogonal. Relying on this property, the OAMs have been regarded as another degree of freedom in optical and quantum communications [33–37]. Specifically, the OAM state is theoretically infinite, which makes the beams carrying OAMs appropriate candidates for developing the transmission capacity [38–41]. Furthermore, the OAM-based beams have promoted many other developments from optical microscopy to light processing and entanglement and quantum information [42–46]. However, these prospective applications are based on the investigation about the fundamental diffractive properties of beams carrying OAMs [47]. In other words, exploiting other OAM beams can further develop the practical applications [48–50].

3 Diffraction of fields carrying OAMs

3.1 Canonical vortices

As the scalar solution of Helmholtz equation in the cylindrical coordinate system, the LG eigenmodes were initially demonstrated carrying OAMs [4,51], and then recognized as the canonical vortices carrying OAMs. These beams have cylindrical symmetries of complex amplitude with an on-axis intensity null and vortex phase. Importantly, as described in Eq. (3), the LG beams are characterized by the parameters l and p. l is the azimuthal order giving an OAM of lћ per photon—i.e., the so-called topological charge of vortex phase; p is the radial index denoting the ring number of intensity distribution. Figures 1(a)–1(d) show the intensity and phase distributions of four canonical vortices.

For LG beams, the OAMs are irrespective to the radial distribution of the beam. Hence, most attention was devoted to the LG beams with the zeroth radial index—namely, the LG_0,l beams with a single-ringed profile. The evolution of LG_0,l beams, as well as the focusing properties, were investigated immediately after the reported work of Allen, including on- and off-axis beams with different topological charges [52–54] and beams having single and multiple singularities [55–57]. The remarkable diffraction property of LG_0,l beams is their divergence; that is, the LG_0,l beams undergo obvious diffraction and rapid expansion of the hollow core with respect to the modulus of the topological charge—i.e., |l|. Moreover, the beam divergence angle increases with increasing OAM. Figure 1(e) displays the divergence of LG_0,l beams generated from the transformation of a cylindrical lens (blue) and the forked diffraction grating (red), respectively. w₀ and r_rms correspond to the waist of Gaussian background and square root of the radial variance for the intensity distribution, respectively [58].

Fig.1 (a)−(d) Intensity and phase (insets) distributions of canonical vortices; (e) the divergence of vortex beams keeping the Gaussian waist or rrms constant. Two kinds of vortex beams are generated by using forked diffraction grating or cylindrical lens, respectively [58]

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Recently, the LG beams with large radial index were reported with quasi-nondiffraction within the same propagation region as Bessel beams [59–62]. For the LG beams described in Eq. (2) with a large radial index, the amplitude terms can be approximately equivalent to the elegant Bessel function as [60]

(7)$\exp \left(-{\displaystyle \frac{r^{\mathrm{2}}}{w{\left(z\right)}^{\mathrm{2}}}}\right){\left({\displaystyle \frac{\sqrt{\mathrm{2}}r}{w\left(z\right)}}\right)}^{\left|l\right|}{L}_p^{\left|l\right|}\left({\displaystyle \frac{\mathrm{2}{r}^{\mathrm{2}}}{w{\left(z\right)}^{\mathrm{2}}}}\right)\approx {\displaystyle \frac{\mathrm{\Gamma }\left(p+\left|l\right|+\mathrm{1}\right)}{p!N^{\left|l\right|/\mathrm{2}}}}{J}_l\left(\mathrm{2}\sqrt{\mathrm{2}N}{\displaystyle \frac{r}{w\left(z\right)}}\right),$

where N = p + (|l| + 1)/2, G(·) is the gamma function, and J_l(·) is the first type of Bessel function with order l. Figure 2 depicts the evolution behavior of a zeroth-order Bessel–Gauss beam as well as LG beams with l = 1, p = 10 and l = 5, p = 10. The Gauss background has the same waist of w₀ = 1 mm. It is clear that the LG beams with a high radial index maintain the intensity profile in a large propagation distance; namely, such LG beams have a nondiffractive property. This property supports the LG beams with a high radial index as an alternative representation of BG beams, and vice versa [59].

It is also worth noting that the radial variation of the wave front has potential in the application of optical communication; the LG beams with nonzero radial indices could help enlarge the communication capacity. Moreover, the nondiffractive property in a certain short transmission distance is useful for receiving higher signal power, especially when the receiver has a very limited aperture size [63].

Fig.2 Evolutions of (a) zeroth-order Bessel beam and LG beams with parameters of (b) l = 0, p = 10 and (c) l = 2, p = 10, respectively

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3.2 Noncanonical vortices

In addition to the canonical vortices having a helical wave front, noncanonical vortices that carry variable OAMs, such as nonsymmetric vortex beams [64,65], Helico-Conical light beams [66–68], fractional vortex beams [69–71], airy vortex beams, and so on, have been proposed with impressive propagation behaviors and practical applications [72–74]. For example, optical vortices with an azimuthally nonuniform phase gradient, including single and multiple singularities, have been successfully created to manipulate the three-dimensional intensity and OAM distributions, especially for engineering the distributions in the vicinity of the focal plane [75–77]. Based on this, optical tweezers that drive colloidal microspheres around quite complicated trajectories have been experimentally implemented [76–79]. Figure 3 shows an optical tweezer based on a noncanonical vortex possessing a triangle phase structure with l = 34 [78]. Moreover, the beams having Helico-Conical phase structures, of which both phase and amplitude express a helical profile as the beam propagation, have demonstrated their capacity to maintain high photon concentration even at higher values of topological charge [68].

Fig.3 (a) Phase structure of a triangle vortex with l = 34; (b) beam shape at the focal plane; (c) experimental results of optical tweezer [78]

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The novel propagation dynamics of noncanonical vortices can be expected and explained by the local spatial frequencies [67,79]. For a generic light beam having a field described as E(x,y) = E₀(x,y)exp[iy(x,y)], where E₀ and y are the transverse amplitude and phase distributions, respectively, the local spatial frequencies are given as [67,79]

(8)$f_y={\displaystyle \frac{\mathrm{1}}{\mathrm{2}\mathrm{\pi }}}{\displaystyle \frac{\partial }{\partial y}}\psi \left(x,y\right).$$f_x={\displaystyle \frac{\mathrm{1}}{\mathrm{2}\mathrm{\pi }}}{\displaystyle \frac{\partial }{\partial x}}\psi \left(x,y\right),$

According to Eq. (8), the frequency mapping to the far-field can be plotted, which provides an intuitive mapping for the intensity distribution. Here, we take the vortex beam having an azimuthally nonuniform phase gradient as example, of which the phase is described as y(r,f) =2πl(f/2π)ⁿ, where n is an integer. The corresponding local spatial frequencies are thus given as [79]

(9)$f_y={\displaystyle \frac{nl{\phi }^{n-\mathrm{1}}\cos \phi }{{\left(\mathrm{2}\mathrm{\pi }\right)}^{n}r}}.$$f_x={\displaystyle \frac{nl{\phi }^{n-\mathrm{1}}\sin \phi }{{\left(\mathrm{2}\mathrm{\pi }\right)}^{n}r}},$

According to Eq. (9), the mapping (f_x, f_y) to the far-field can be plotted. Figure 4(a) shows the calculated local spatial frequency mapping for the case of n = 3, l = 7, and r = 1. Obviously, the curve represents an Archimedes spiral [79].

In this principle, we attach this noncanonical vortex phase onto the autofocusing Airy beams (AABs) [79], which maintain cylindrical intensity profiles as beam propagation, to modulate their evolution and focusing properties. Figure 4(b) shows the experimentally obtained intensity distribution in the focal plane. It is clear that the experimental result is coincident with the corresponding frequency map; that is, AAB yields to the propagation trajectory depicted in Fig. 4(a).

Fig.4 (a) Local spatial frequencies mapping to the focal field; (b) focal filed intensity of AAB with a power-exponent-phase vortex [79]

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Figure 5 shows the simulation results including the side view of the AAB propagation at the y-z plane and the intensity distributions at different propagation distances near the focal points. AAB has parameters of l = 8 and n = 2. Compared with the smooth focusing behavior of AABs carrying classical OAMs [80–83], the light energy gradually converges, and the intensity presents slight oscillation when close to the focal point, as shown in Figs. 5(b)–5(e). This remarkable spiral focusing demonstrates that arbitrary manipulation on the propagation dynamic of light fields can be realized by engineering the locally spatial frequencies.

Fig.5 (a) Side view of propagation dynamic (in y-z plane) of AAB with l = 8 and n = 2; (b)−(e) intensity distributions at different propagation distances of z = 19.52, 20.50, 21.48 and 22.45 cm, respectively [79]

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Further, we introduce variant OAMs into vector AABs to realize the combined manipulation on intensity and polarization distributions in the focal plane of vector fields [84]. Based on the local spatial frequency shown in Eq. (8), we realized intriguing abrupt spin-dependent splitting of vector AABs by encoding cosine-azimuthal variant vortex phases. For a vector AAB with a cosine-azimuthal variant phase, the electric field can be described by the superposition of two spin components as follows:

$\mathit{E}\left(r,\phi \right)=E_{\mathrm{0}}\left(r\right)\left[\cos \left(m\phi \right){\mathit{e}}_x+\sin \left(m\phi \right){\mathit{e}}_y\right]\exp \left(\mathrm{i}\cos n\phi \right)$

$=E_{\mathrm{0}}\left(r\right)\{\exp \left[-\mathrm{i}\left(m\phi -\cos n\phi \right){\mathit{e}}_L\right]$

(10)$+\exp \left[\mathrm{i}\left(m\phi +\cos n\phi \right){\mathit{e}}_R\right]\},$

where E₀(r) = Ai[(r₀−r)/w]exp[α(r₀−r)/w], Ai(·) denotes the Airy function, r₀ is the radius of the primary ring, w is a scaling factor, a is the exponential decay factor, m is the pol-arization order, and e_L = [1, + i]^T/$\sqrt{\mathrm{2}}$ and e_R = [1, −i]^T/$\sqrt{\mathrm{2}}$ are the unit vectors of the left- and right-handed circular polarizations, respectively. According to Eq. (10), two spin components have distinct phase structures—i.e., y_R,L =±(mf±cosnf), respectively. According to Eq. (8), the local spatial frequencies in the cylindrical coordinate system are given as

(11a)$\mathbf{L}:\{\begin{array}{l}{f}_{\rho }={\displaystyle \frac{\mathrm{1}}{\mathrm{2}\mathrm{\pi }r}}\left|m+n\left(\sin n\phi \cos {\displaystyle \frac{n\mathrm{\pi }}{\mathrm{2}}}+\cos n\phi \sin {\displaystyle \frac{n\mathrm{\pi }}{\mathrm{2}}}\right)\right|,\hfill \\ \\ \phi =\varphi -{\displaystyle \frac{\mathrm{\pi }}{\mathrm{2}}},\hfill \end{array}$

(11a)$\mathbf{R}:\{\begin{array}{l}{f}_{\rho }={\displaystyle \frac{\mathrm{1}}{\mathrm{2}\mathrm{\pi }r}}\left|m-n\left(\sin n\phi \cos {\displaystyle \frac{n\mathrm{\pi }}{\mathrm{2}}}-\cos n\phi \sin {\displaystyle \frac{n\mathrm{\pi }}{\mathrm{2}}}\right)\right|,\hfill \\ \\ \phi =\varphi +{\displaystyle \frac{\mathrm{\pi }}{\mathrm{2}}}.\hfill \end{array}$

Considering the parity of phase parameter n, we divided the relationship between local spatial frequencies of two spin states into two cases. For an odd n, the local spatial frequencies have an identical closed-form expression, indicating two spin components presenting the same profile. When n is even, owing to the complementarity of trigonometric functions, the mapping corresponding to local spatial frequencies of two spin components are spaced from each other along the azimuthal direction. Figure 6 displays the intensity distributions of vector AAB with polarization order m = 2 and different phase parameter n in the focal plane. In agreement with the theoretical expectation, the vector AABs exhibit distinct focusing behavior with respect to the parity of parameter n. It is noteworthy that when n is odd, although the local spatial frequencies of two spin components have identical spatial distributions, opposite transverse energy flux arising from the phase terms exp(±imf) leads to nonuniform intensity distributions. As a result, two spin components would not overlap with each other completely, leading to a transverse spin-dependent splitting.

Fig.6 Simulation intensity distributions of vector AAB with polarization order m = 2 and different phase parameters n in the focal plane. I0, IL and IR correspond to the total intensity, left- and right-handed spin components, respectively. The insets represent the phase profiles. The dashed curves denote the local spatial frequencies distributions [84]

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As discussed above, according to the close mapping relationship between local spatial frequency and the geometrical shape of the focal field intensity distribution, we can consciously control the focusing of two spin components and further manipulate the transverse spin-dependent splitting and phase distributions of the arbitrary vector field by engineering phases corresponding to two spin components [85].

4 Modulation of PB phases on the vector fields

4.1 PB phases

So far, the generation method of vector fields can be divided into two categories: 1) interference superposition of light fields with different polarizations [86–88]; 2) polarization transformation from spatial modulator, such as inhomogeneous wave plate or polarizer, liquid crystal plate, subwavelength grating and metasurface [89–95]. The first one can generate vector fields with an arbitrary complex polarization distribution, whereas the latter is convenient and efficient. These two methods both arise from the phase modulation of two spin states—i.e., the spin–orbital interaction [96,97]. Here, we take the latter method as an example. We suppose that the wave plate has a spatial variant optical axis denoted as ϕ(x,y), with transmission rate t(x,y) = 1. Thus, the Jones matrix of the inhomogeneous wave plate can be expressed as

(12)$M=\left[\begin{array}{cc}\hfill \cos \mathrm{2}\varphi \hfill & \hfill \sin \mathrm{2}\varphi \hfill \\ \hfill \sin \mathrm{2}\varphi \hfill & \hfill -\cos \mathrm{2}\varphi \hfill \end{array}\right].$

For a left-handed circularly polarized field (or right-handed circularly polarized field) with E_in = E₀e_L (E_in = E₀e_R), after passing through this inhomogeneous wave plate, the transmitted field has complex amplitude and polarization denoted as E_out = M∙E_in = E₀exp(i2ϕ)e_R [E_out = E₀exp(−i2ϕ)e_L for the right-handed polarized one]. Therefore, two spin states have distinct responses; that is, two spin states acquire additional geometric phases denoted as y_L = 2ϕ(x,y) and y_R = -2ϕ(x,y), respectively. These special geometric phases arising from polarization transformation are also called the PB phase [98].

Figure 7 schematically shows the PB phase generation during polarization transformation; the inset illustrates the polarization transformation process on the Poincaré sphere. A focused Gaussian beam with linear polarization A passes through an inhomogeneous optical system (or wave plate), which imposes PB phases y_L and y_R onto two spin components, changing its polarization state to B [99]. Supposing the incident beam has the field distribution E = E₀[aexp(−if₀)e_L + bexp(if₀)e_R]/$\sqrt{\mathrm{2}}$, the output thus can be described as

(13)$\mathit{E}=E_{\mathrm{0}}\left\{a\exp \left[\mathrm{i}\left({\psi }_{\mathrm{L}}-{\phi }_{\mathrm{0}}\right)e_{\mathrm{R}}\right]+b\exp \left[\mathrm{i}\left({\psi }_{\mathrm{R}}+{\phi }_{\mathrm{0}}\right){\mathit{e}}_{\mathrm{L}}\right]\right\}/\sqrt{\mathrm{2}}.$

Equation (13) indicates that by utilizing the spatially inhomogeneous PB phases, one can conveniently reshape the wave front without changing the intensity profile immediately after the polarization transformation optical system. Based on this result, various optical elements with architectures including patterned subwavelength grating, liquid crystal, and metasurface have been broadly used to generate vector beams.

In addition to the standard Poincaré sphere shown in Fig. 7, other kinds of Poincaré spheres have been proposed to represent the inhomogeneous polarization states [100–103], such as the higher-order Poincaré spheres composed of two opposite spin states with opposite spiral phases [100] and the hybrid-order Poincaré sphere composed of two opposite spin states with different spiral phases [102]. For these cases, the resulting PB phases of two spin components are closely related with the topological charges and spin assigned to the sphere [100,102].

Fig.7 Schematic illustration of PB phases generation in the process of polarization transformation. Insert: schematic illustration of polarization transformation on the Poincaré sphere [99]

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4.2 Polarization singularity conversion

The PB phase, in turn, supports an approach to manipulate the propagation dynamics of two spin states because of the corresponding relationship between the phase gradient structures in position space and wave vector in momentum space. Utilizing tailored PB phases, some intriguing manipulations of spin states in three dimensions, involving spin-orbital interaction, spin Hall effect of light (SHEL), and light-guiding confinement, have been implemented [104–109].

The most common PB phases are the canonical vortex phases—i.e., 2ϕ =±lf—which transform an incident Gaussian beam with linear polarization into a cylindrical vector beam [86,88,105]. In such cases, two spin components carrying opposite OAMs exhibit identical diffraction, as mentioned in Section 3.1. As a result, the vector beam maintains its polarization state with a slowly varying intensity profile upon propagation. Nevertheless, when l_R≠l_L, two spin components exhibit obvious divergence during beam propagation [110,111]. Owing to the cylindrical symmetry, the divergence leads to spin-dependent separation along the radial direction. Considering the total field output from the optical system (or wave plate), one can see that it is a cylindrical vector vortex beam characterized by parameters m = (l_L−l_R)/2 and l = (l_L + l_R)/2. Here, m and l are the polarization order and topological charge of the additional vortex phase, respectively. On the other hand, as a whole, under the modulation of an additional OAM, the vector fields exhibit spin-dependent splitting upon propagation. Therefore, this phenomenon is also called the radial SHEL. Shu et al. realized such radial spin-dependent splitting by using a dielectric metasurface with a radially varying optical axis [112].

For the vector vortex field with larger OAM, more obvious divergence occurs, resulting in more serious polarization distortion [111]. In other words, according to the dependence of the amplitude profile on the topological charge, the vector vortex fields with the same polarization order but different OAMs usually present distinct intensity profiles. To overcome the divergence, we proposed perfect vector vortex beams, which not only have an intensity profile independent of the polarization order and additional OAM but also have a stable intensity profile and polarization at a certain propagation distance [113]. Figure 8 displays the perfect vector vortex beams generated from the superposition of two annular spin components with different vortex PB phases, which are created by the Fourier transformation of Bessel–Gauss beams with an identical transverse wave vector. Here, |H〉 denotes the state composed of two spin components with OAMs defined by subscripts. The local linear polarization orientation and phase distribution are shown in the third and fourth rows, respectively. Importantly, such vector vortex beams have been demonstrated to have the capacity of maintaining the polarization and intensity profile in a certain propagation distance [113].

Fig.8 Distributions of intensity (top row), s3 (second row), polarization orientation (third row) and phase (bottom row) of different states. |H〉 denotes the sate composed by two spin components with OAMs defined by subscripts [113]

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Furthermore, we note that the divergence of two spin components composing the vector field reaches its maximum at the far-field—i.e., z→∞. Therefore, we explored the divergence by observing the polarization state in the focal plane [114]. Figure 9 shows the numerically calculated results of the intensity and polarization distributions of the focused azimuthally polarized beam carrying vortex phases of l = 1, 2 and 3. The dotted, dashed, and solid lines in the top row depict the zero contours of s₁, s₂, and s₃, respectively; the background and short lines in the bottom row denote the ellipticity and orientation of the polarization ellipse, respectively. The results show that when an additional OAM is attached to the vector beam, the polarization in the focal plane changes dramatically. Moreover, we note that the azimuthally polarized beams focus into radially polarized ones.

Fig.9 Intensity and polarization distributions of the focal fields of the azimuthally polarized beams with vortex phases of l = 1, 2 and 3, respectively. The dotted, dashed and solid lines in the top row depict the zero contours of s1, s2 and s3, respectively; the background and short lines in the bottom row denote the ellipticity and the orientation of polarization ellipse, respectively [114]

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According to Eq. (3), one can find that at the focal plane—i.e., z→∞—vortex beams have cylindrical intensity profiles, whose size is dependent on their topological charges. In addition, it is notable that the Gouy phases of vortex beams are F = (|l| + 1)π/2, which means that two spin components composing the vector vortex beam obtain different Gouy phases with respect to their topological charges [115]. If (|l_L|−|l_R|)/2 is even, two spin components are in phase within the focal plane, and the focused field has identical polarization orientation with the incident field; if (|l_L|−|l_R|)/2 is odd, two spin components have constant phase difference of π, and the overlap area of two spin components consequently presents opposite polarization orientation. For the case shown in Fig. 9(a), the right-handed spin component has the zeroth OAM, so its corresponding focal field has an intensity profile similar to the Gaussian beam; that is, the corresponding focal field presents a solid bright spot in the center point. Accordingly, the incident vector point is transformed into a circularly polarized point. However, for the cases of Figs. 9(b) and 9(c), (|l_L|−|l_R|)/2= 1, the overlap areas are radially polarized. Meanwhile, because of the relationship between the profile size and topological charge, the spin component that has a smaller topological charge dominates the central region. As a result, the vector point is transformed into a circularly polarized point with corresponding chirality.

In addition to the abovementioned beams that slowly change their polarization in the focusing process, we also created vector AABs associated with OAM and experimentally studied the abrupt polarization transitions induced by spin–orbital interaction by engineering the PB phases corresponding to two spin components [116]. Figure 10 displays the autofocusing of radially polarized AABs without and with a single charged vortex phase. From Fig. 10, it is clear that when the polarization order and topological charge of the vortex phase are equal in number, significant polarization singularity conversion occurs, and the local polarizations undergo an abrupt transition from linear to spin states.

Fig.10 Autofocusing of radially polarized AABs without (top) and with (bottom) a single charged vortex phase. (a) and (c) depict the beam intensity patterns at input and output, respectively; (b) and (d) side view of the beam propagation from numerical simulation and measured beam polarizations at output [116]

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4.3 Spin-dependent splitting

The other typical selection of the PB phase is that with a linear function, such as 2ϕ(x,y) =±k_xx [99,117,118]. For this case, two spin states would obtain two mutually conjugate titled phases and separate transversely with each other upon propagation. This spin-dependent transverse separation was reported by Hasman in 2002 by employing spatially variant subwavelength grating [117,118]. Later, many primary works about the PB phase inducing photonic SHE were carried out. Zhang et al. observed highly enhanced spin-dependent transverse separation by using metasurface composed of metal nano-antenna arrays [108]. Ling et al. reported a giant photonic SHE by using dielectric metasurface fabricated by a femtosecond laser self-assembly nanostructure [119]. In addition to the pure phase information, the phases within which the amplitude information is encoded, such as Hermite–Gauss and Airy modes [120], even the combination of several phases [99], can be selected as PB phases. Figure 11 displays the spin-dependent splitting in the case of 2ϕ(x,y) =±(k_xx+f). The initial beam is depicted in Fig. 11(a). The side view of propagation (Fig. 11(d)) and s₃ distribution (Fig. 11(c)), as well as measured polarization, demonstrate that not only the SAM but also the OAM can be separated merely by engineering PB phases.

Fig.11 Spin-dependent separation with the bifurcation of orbital angular momentum. (a) Initial beam with polarization direction marked with red arrowheads; (b) horizontally and vertically polarized components; (c) interference pattern of output beam with a plane wave (top), and the corresponding s3 distribution (bottom); (d) side view of the beam propagation

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Of course, the PB phase can be selected as any phase expected. This means that by steering the PB phases, we can also realize longitudinal spin-dependent splitting. In geometric optics, we realize the convergence and divergence of the beam by using a square phase imparted by a lens. Therefore, we suppose that the linearly polarized incident beam has a convergent phase of exp(−ia₀r²), and the PB phases are 2ϕ(x,y) =±(ar²). For such a case, it is clear that two spin components would respectively focus into the focal planes f(a₀±a), resulting in dual-foci in the longitudinal direction—i.e., longitudinal spin-dependent splitting [106,121,122]. By relying on modern fabrication technology, metasurface and other devices with subwavelength thickness enable a PB phase lens, similar to the Fresnel lens [123–126]. Furthermore, a PB phase-based multifocus lens has also been realized by using segmented metasurface [127]. Figure 12 schematically shows the operating principle of a longitudinal three-focus lens. To realize three foci, the metasurface is divided into three annular regions with different PB phases and thereby focus the corresponding incident field to three longitudinal points.

Fig.12 (a) Schematic illustration of the longitudinal three-foci metasurface lens; each focal point is focused from segmented region with distinct PB phase response; (b) the observed light spots correspond to the three focal points. Adapted from Ref. [127]

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Additionally, by combining this longitudinal spin separation with the transverse one, we experimentally achieved a controllable spin-dependent focal shift in three-dimensional space [99]. When the PB phase is a polynomial as 2ϕ(x,y) = k_xx + k_yy + ar², two spin states hence focus to points (x_R,y_R,z_R) and (x_L,y_L,z_L), respectively. Here, x_R,L = k_xz_R,L/k, y_R,L=k_yz_R,L/k, and z_R,L = f(a₀±a). This modulation of special PB phases on the propagation dynamics of spin states reveals a series of intriguing phenomenon. More importantly, it promotes the development of spin-optics and further contributes to application; e.g., this remarkable prospect of spin-dependent separation in holographic imaging has excited new research directions [128,129].

4.4 z-dependent polarization

Accompanied by the spin-dependent separation, the light field changes its polarization upon propagation. This special light field with longitudinally variant polarization but an identical transverse intensity pattern during propagation has attracted research interest owing to its application potential in material processing, polarization meteorology, microscopy imaging, optical communication, and so on [106,121]. In 2013, Cardano et al. generated Poincaré beams that rotate the transverse polarization distribution by using spiral wave plates [130]. They transformed part of the incident light field with LG_0,0 mode and spin state into LG_0,l mode with opposite spin state. The transformed LG_0,l has a vortex phase axially overlapping with the idle LG_0,0 mode, resulting in Poincaré beams with Lemon, star, and spiral polarizations that depend on the topological charge of LG_0,l mode. As mentioned above, LG modes have a Gouy phase that is closely related to the radial index p and topological charge l. Therefore, two spin states have a longitudinally varying Gouy phase difference—i.e., DF = |l|tan⁻¹(z/z₀). This z-dependent phase difference leads to the rotation of polarization upon beam propagation.

Remarkably, Moreno et al. reported a kind of vector Bessel beam that changes transverse polarization upon propagation—i.e., vector Bessel beams having z-dependent polarization [131,132]. Combining the vortex phase with axicon phases, one can create two-dimensionally varying PB phases with different periods for two orthogonal polarizations—e.g., y_R,L(r,y) =±lf + (k_r±k₀)r. In this principle, we construct vector beams with transverse varying polarization and further generate zeroth- and higher-order Bessel beams with z-dependent polarization by employing the linear focusing of the axicon [133]. Figure 13 shows a schematic representation of z-dependent polarization distribution and transformation as well as experimentally measured intensity distributions of vertically polarized components at different planes. It is seen that the first-order vector Bessel beam longitudinally varies its local polarization upon propagation. Overall, the Bessel beam presents three-dimensionally variant polarization.

Fig.13 Schematic representation of z-dependent polarization distribution and transformation; (b)−(f) experimentally measured intensity distributions of first-order Bessel beam propagating through a vertical polarizer at planes with equal space [133]

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Furthermore, we explored the self-healing of these vector Bessel beams, including the transverse intensity profile and the three-dimensional polarization structure, by observing their transverse intensity and Stokes parameter distributions after propagating through two disparate obstacles. Figure 14 shows the measured reconstruction of second-order vector BG beams after a linear obstacle. Importantly, as the results show in Fig. 14, the three-dimensional polarization property can be reconstructed after such an obstacle. The degree of longitudinal variation in polarization and the self-healing can address many application challenges. For example, by combining it with the polarization response property of anisotropic material, these beams can be used for material processing at a special depth and to improve the axial resolution in 3D imaging. To some extent, the higher-order Bessel beams supporting quantitative OAM and SAM simultaneously, with z-dependent polarization, may induce intriguing phenomena such as SAM-OAM conversion and spin transport.

Fig.14 Measured transverse intensity and local polarization distributions of the reconstructed second-order BG beam with hybrid polarizations at propagation distances of z = 21, 23.8, 26.6, 29.5, 32.2 cm, respectively. The red and green ellipses denote the RH and LH elliptical polarizations, respectively. The linear obstacle with a diameter of about D = 70 mm is placed at plane z = 16.6 cm [133]

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5 Modulation of OAMs on the azimuthally broken fields

5.1 Vortex beams

The canonical vortex beam manifests itself as the helical phase, which generally has rotation symmetry. The transverse OAMs accordingly are null for such beams. However, as discussed in Section 3.2, the noncanonical vortex phases break this rotation symmetry and thereby produce transverse energy flow, leading to the variation of beam profile during propagation. It is natural to expect that by breaking the rotation symmetry, one can consequently introduce nonzeroth transverse OAM, then create abundant intensity structures. On the other hand, research on broken beams has demonstrated an important effect in information reconstruction [134]. Therefore, increasing attention has been devoted to the nonsymmetric optical vortices [135–140].

Remarkably, in the azimuthally broken vortex beams—namely, fan-shaped vortex beams—the transverse energy flow induced by the phase gradient gradually transfers the energy to the opaque region upon beam propagation [135]. The diffraction of fan-shaped vortex beams can be interpreted based on the angular diffraction theory [141–143]. Supposing that a fan-shaped mask has a sector photic region, the transmission function can be expressed as [144,145]

(14)$P\left(\phi \right)=\{\begin{array}{ll}t,\hfill & -\beta /\mathrm{2}\le \varphi \le \beta /\mathrm{2},\hfill \\ \mathrm{0},\hfill & \mathrm{else},\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hfill \end{array}$

where b is the angle width of the fan-shaped aperture, and t is the transmission of the photic region. Here, we set t = 1, and the incident vortex beam has complex amplitude denoted as E = exp(−r²/w²)exp(ilf), where w is the waist of the Gaussian background. According to the angular diffraction theory, the photic region is analogous to an angular grating with 2π periodic nature. The complex transmission of the angle distribution can be expressed in terms of the angular momentum distribution with a Fourier coefficient as follows [144,145]:

(15)$E\left(r,\phi \right)={\displaystyle \frac{\beta }{\mathrm{2}\mathrm{\pi }}}\exp \left(-{\displaystyle \frac{r^{\mathrm{2}}}{w^{\mathrm{2}}}}\right){\displaystyle \sum _{n=-\infty }^{+\infty }\sin c\left({\displaystyle \frac{n\beta }{\mathrm{2}}}\right)}\exp \left[\mathrm{i}\left(l+n\right)\phi \right],$

where n is an integer. From Eq. (15), one can find that after passing through the aperture, the incident beam splits into a series of OAM spectra, which are characterized by the topological charges of l + n. In consequence, the diffraction of the vortex beam can be described by the superposition of diffractive OAM spectra.

For each OAM spectrum that follows the paraxial approximation, its evolution can be described by the Fresnel diffraction as

$E_n\left(\rho ,\theta \right)={\displaystyle \frac{-\mathrm{i}\beta }{\mathrm{2}\mathrm{\pi }\lambda z}}\sin c\left({\displaystyle \frac{n\beta }{\mathrm{2}}}\right)\exp \left({\displaystyle \frac{\mathrm{i}k{\rho }^{\mathrm{2}}}{\mathrm{2}z}}\right)$

${\displaystyle \int \int \exp \left(-{\displaystyle \frac{r^{\mathrm{2}}}{w^{\mathrm{2}}}}\right)\exp \left({\displaystyle \frac{\mathrm{i}k{r}^{\mathrm{2}}}{\mathrm{2}z}}\right)}$

(16)$\times \exp \left[\mathrm{i}\left(l+n\right)\phi \right]\exp \left[-{\displaystyle \frac{\mathrm{i}kr\rho \cos \left(\theta -\phi \right)}{z}}\right]r\mathrm{d}r\mathrm{d}\phi ,$

where (r,q,z) are the cylindrical coordinates. With the help of Bessel integration, we can obtain the diffractive fields as [145]

(17)$E\left(\rho ,\theta \right)=u\left(\rho ,z\right){\mathrm{e}}^{\mathrm{i}\mathrm{\Delta }\mathrm{\Phi }}{\displaystyle \sum _{n=-\infty }^{+\infty }\sin c\left({\displaystyle \frac{n\beta }{\mathrm{2}}}\right)A_{l+n}\left(\rho ,z\right){\mathrm{e}}^{\mathrm{i}\left(l+n\right)\left(\theta -\mathrm{\pi }/\mathrm{2}\right)}},$

with

$A_{l+n}\left(\rho ,z\right)=$

(18)$\left[I_{\left(l+n-\mathrm{1}\right)/\mathrm{2}}\left(\mathit{\gamma }\right)-I_{\left(l+n+\mathrm{1}\right)/\mathrm{2}}\left(\mathit{\gamma }\right)\right]\exp \left(-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\frac{{\rho }^{\mathrm{2}}}{w{\left(z\right)}^{\mathrm{2}}}\right){\displaystyle \frac{\rho }{w_{\mathrm{0}}}},$

where DF = 3π/4−3tan⁻¹(z/z₀)/2+ zz₀(r/w₀)²/(z² + z²₀) denotes the dynamic phase, I_v(g) is the modified Bessel function of the first kind of order v, g = r²(1+ iz₀/z)/2w² _z, w_z = w₀[1+ (z/z₀)²]^1/2 is the beam waist at propagation distance z, and z₀ = kw²₀/2 is the Rayleigh length. All nonessential factors are absorbed into the envelope profile: u(r,z) = (rb/4πw₀)(z₀/z)^5/4(z₀/R_z)^3/4exp[−i(π−q₀)/2], where R_z = z[1+ (z₀/z)²] is the radius of curvature for the wave front.

According to Eqs. (17) and (18), we calculate the diffractive fields by numerically solving the above two equations. Intuitively, after propagating a distance z, the angular position of the peak intensity experiences angular rotation, which depends on l. Figure 15 presents the relationship between rotation angle and topological charge, as well as the intensity distribution of l = -4 and -20 vortex beams at z = 0 and 25 cm planes, respectively, with the parameters selected as l = 532 nm, w₀ = 1 mm [145]. In Fig. 15(a), the solid curve corresponds to the result calculated from Eqs. (17) and (18), and the blue and red squares correspond to the measured results of the two spin states. It is discernible that the fan-shaped vortex beams undergo rotation during diffraction. Importantly, the rotation angle increases with increasing topological charge. Figure 15(d) shows the intensity distributions of l =±20 vortex beams after propagating 25 cm. The red and blue lines correspond to the l = 20 and -20 order vortex beams. It is quite clear that the vortex beams having equal modulus of topological charges rotating equally along the opposite azimuthal direction. This l-dependent rotation can also be intuitively described by the transverse OAM flow, which is proportional to the azimuthal gradient of the vortex phase.

Fig.15 (a) Rotation angles |Dq| versus the topological charges; (b)−(e) Intensity distributions of fan-shaped vortex beams at z = 0 and 25 cm planes. (c) l = -4; (d) l = -20; (e) l =±20. |R〉 and |L〉 correspond to two spin states. The red and blue areas correspond to the positive and negative vortex beams, respectively

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Owing to the transverse energy flow, it is naturally seen that the rotation increases with increasing propagation distance [145]. Figure 16(a) shows the relationships between rotation angle |Dq| and propagation distance for three topological charge vortex beams, according to Eqs. (17) and (18). As expected, the rotation angle increases with increasing propagation distance. Particularly, at the far-field—i.e., z→∞—from Eq. (17) and (18), and it can be concluded that the complex amplitudes A_l+n(r,z) become real values. Meanwhile, the nth-order OAM spectrum has a constant phase retardation (l + n) (q−π/2). This means that the OAM spectra are in phase and constructively interfere at azimuthal position q + π/2; namely, the fan-shaped rotates to azimuthal position q = π/2. Figures 16(b)–16(d) display the focal intensity distributions of l = 1, 4, and 20 fan-shaped vortex beams with b = π/2, respectively. The incident fan-shaped vortex beams have the same profile as shown in Fig. 15(b).

Fig.16 (a) Rotation angles |Dq| versus propagation distance z. (b)−(d) Focal intensity distributions of l = 1, 4 and 20 fan-shaped vortex beams with b = p/2, respectively. The incident fan-shaped vortex beams have the same profile as shown in Fig. 15(b)

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The propagation of a fan-shaped scalar vortex beam in the paraxial limit has been reported, showing that the intensity pattern is rotated with beam propagation, and the direction of the rotation depends on the sign of the topological charge. This phenomenon can be attributed to the spiral phase shift of the beam. For fan-shaped Bessel–Gauss beams that have remarkable features of nondiffraction and self-healing, the spatial intensity distributions at the focal plane were well recovered [140,146,147].

5.2 Vector beams

By breaking the rotation symmetry, vector fields also produce redistributed intensity, polarization, and OAM during propagation. Figure 17 displays the measured s₃ distributions of fan-shaped pure cylindrical vector (CV) beams with b = π/2 in the planes of z = 25 cm and z→∞ (the focal plane of a lens). The red and blue areas correspond to the right- and left-handed spin components, and the dashed areas schematically show the incident beams. The polarization orders are m = 1, 2, and 4, respectively. It is clear that the two spin components represent opposite angular rotation– i.e., angular splitting—which is also regarded as the angular SHEL [148]. For a pure CV beam, l_L = −l_R = m and l = 0, the photonic SHE value is 2|Dq|. Intuitively, the spin-dependent splitting reaches its maximum at the far-field. Wang et al. reported the focal field characteristics of fan-shaped azimuthally polarized beams and revealed the focusing mechanism of such beams [138].

Fig.17 Measured s3 distributions of fan-shaped pure CV beams with b = p/2 in the planes of z = 25 cm (upper row) and z→∞ (the focal plane of a lens, below row). The red and blue areas correspond to the right- and left-handed spin components, the dashed areas schematically show the incident beams. The polarization orders are m = 1, 2 and 4, respectively

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Based on this, we introduced the rotation symmetry of a fan-shaped aperture and then investigated the focal fields of azimuthally polarized beams with multiple fold rotation symmetry [144,149]. The focusing mechanism is schematically shown in Fig. 18. Each fan-shaped lobe of the vector beam creates a pair of spin components, which rotate±π/2 at the focal plane. As a result of the interferential superposition of spin components focusing from several lobes, the focal field redistributes the polarization, intensity, and OAM. Remarkably, we found that the focal field focused from azimuthally polarized beams with odd- and even-fold symmetries are significantly different.

Fig.18 Schematic illustration of the focusing dynamics of a fan-shaped azimuthally polarized beam [149]

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Figure 19 shows the intensity and s₃ distributions in the focal fields of azimuthally polarized beams with odd- and even-fold symmetries. For the beam with even-fold rotation symmetry, it consists of even lobes. According to the results in Figs. 17 and 18, two lobes with opposite angle positions in the pupil plane create two spin components carrying opposite OAMs and SAMs at the same angle position in the focal plane. The superposition of two spin components consequently results in locally linear polarization. However, for the odd-fold beams, the spin components focused from each lobe cannot overlap at that same angle position in the focal plane, resulting in a special focal field with complex polarization and energy flow, as show in Figs. 19(a), 19(c), and 19(e).

Fig.19 Distributions of intensity and s3 in the focal field of azimuthally polarized beams with different rotation symmetries. Insets: the intensity distributions of beams in the pupil plane [149]

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5.3 Vector vortex fields

For a generic vector vortex beam having mth-order polarization and lħ OAM, these two spin components accordingly have topological charges of l_L = l + m and l_R = l−m, respectively. After the spatial modulation of the fan-shaped aperture with angle width of b, two spin components undergo angular diffraction with respective to the topological charges l_L and l_R. As shown in Fig. 15, the angular diffraction of the vortex beams is closely related to the carried OAMs. Therefore, by introducing an additional OAM into broken vector beams—namely, setting l_L≠ l_R—one can produce asymmetric rotation and then steer the angular spin-dependent splitting [145,148].

Based on this principle, we reported a new photonic SHE for a freely propagating fan-shaped vector vortex beam in a paraxial situation [145]. Figure 20 shows the asymmetric spin-dependent splitting of fan-shaped vector vortex beams with constant polarization order. The polarization order is m = 2, and the propagation distances are about 25 cm. The topological charges of the vector vortex beams are l = 2, -1, and 3, respectively. The angle width in all three cases is b = π/2. For the case of l = m = 2, the right-handed spin component does not undergo rotation because of its zero OAM, whereas the left-handed spin component rotates approximately 35°. Particularly, for the case shown in Fig. 20(c), l_R = 1 and l_L = 5, two spin components can rotate along the same angular direction. The results provide us with a direct way to manipulate the photonic SHE of a vector beam, including magnitude and orientation. Moreover, we can also change the photonic SHE by altering the shape of the vector vortex beam—namely, the size of the fan-shaped aperture [148].

Fig.20 Distributions of s3 for fan-shaped beams with angular width of p/2, after propagating 25 cm. (a) m = 2, l = 2; (b) m = 2, l = -1; (c) m = 2, l = 3 [145]

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6 Conclusions

Since Allen reported that the LG modes carry OAMs, a great variety of studies have been devoted to exploiting OAM, including the paraxial and nonparaxial propagation in homogeneous and inhomogeneous materials, as well as linear and nonlinear systems. These works have greatly developed the practical applications, evolving optical tweezers, optical and quantum communication, super-resolution microscopy and lithography, and high-precision measurements. Furthermore, the theories on optical vortices have been successfully expanded to electric beams, atom beams, and even matter wave systems. The study of OAM is still in its infancy.

Here, we have selectively focused on novel evolutions of beams carrying OAM and given an overview of the modulations of OAM on the propagation dynamics of scalar and vector beams in free space. We first introduced the evolutions of canonical and noncanonical vortex beams. Subsequently, we reviewed the Pancharatnam–Berry phases arising from spin-orbital interaction and presented novel behavior such as spin-dependent splitting, polarization singularity conversion, and z-dependent polarization. We further discussed the propagation and focusing properties of azimuthally broken vector vortex beams. New ideas and developments are emerging at a rapid pace, and it is clear that the OAM will provide new opportunities for topological photonics, spin-photonics, and spintronics.

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Acknowledgements

This work was funded by the National Natural Science Foundation of China (NSFC) (Grant Nos. 11634010, 11404262, 61675168, U1630125 and 61377035); Fundamental Research Funds for the Central Universities (No. 3102015ZY057); and Innovation Foundation for Doctor Dissertation of North-western Polytechnical University (No. CX201629).

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