The Pancharatnam–Berry (PB) phase, arising from the polarization conversion, has attracted rapidly growing interests in the fundamental physics and SoP modulations.^[63–67] Here, we generalize such phenomena and conclude a corresponding modulation model, of which the PB phase generation is schematically illustrated in Fig. 1.^[60] As shown, for an incident light field with homogeneous SoP, it can be depicted as E_in = E₀[aexp(–φ₀/2)e_R+bexp(φ₀/2)e_L], where E₀ is the amplitude, φ₀ is a constant phase, a and b are the normalized amplitudes with a² + b² = 1, e_R,L denote the right-hand (RH) and left-hand (LH) circular polarizations, respectively.^[33,68] Supposing that the two spin states are independently controllable in the polarization conversion process with identical transmission rate and light path, thus the SoP of the output light field is dependent on the phases appending on the two spin components, namely, the PB phases, which are generally described by the solid angles corresponding to the polarization conversion trajectory denoted on the Poincar´e sphere, as shown in Fig. 1(b).

Fig. 1.

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	Fig. 1. (color online) (a) Schematic illustration of PB phases arising from polarization conversion and corresponding evolution dynamics of two spin components under the modulation effects of PB phases. (b) PB phases corresponding to the point denoted by red spot in inserts of A and B. R and L denote the RH and LH circular polarizations, respectively.[60]

In general, the polarization conversion shown in Fig. 1(a) can be described by an inhomogeneous half wave plate, which has spatially variant optical axis denoted as φ(x,y) but identical transmission rate t(x,y) = 1. Considering that the LH spin component is input into the system, the corresponding output component can be expressed as^[60] 1 where M is the Jones matrix of the inhomogeneous wave plate. Equation (1) clearly indicates that the LH spin component obtains an additional phase denoted as ψ_L = 2φ(x,y), this special phase arising from polarization conversion is the very PB phase. Moreover, it is notable that this component flips the SoP into a RH circular one. Likewise, the RH spin component obtains an additional phase denoted as ψ_R = –2φ(x,y) and transforms into a LH circular polarization. Supposing , the output field immediately from the optical system can then be consequently described as 2 Equation (2) indicates that the output field transfers local SoP along an angle (ψ_R - ψ_L)/2 + φ₀, but keeps its intensity profile at the immediate output plane. Importantly, the appending phases ψ_R and ψ_L, i.e., the PB phases, make it possible to control the evolution of the two spin components by engineering phase structures, resulting in intriguing phenomena such as spin-dependent separation and spin-dependent guiding.^[69–72] Take the typical selection of PB phase that has a linear function as an example, e.g., ψ_{L, R} = ± k_xx. For this case, two spin states would obtain two mutually conjugate titled phases, resulting in the transverse separation with each other upon propagation. Such a macroscopic spin-dependent dynamic of light was firstly implemented by Hasman using spatially variant subwavelength grating.^[73] Remarkably, Zhang et al. observed extremely enhanced spin-dependent transverse separation by using meatsurface based on PB phase theory.^[74] Ling et al. reported a giant spin-dependent separation by using dielectric metasurface fabricated by the femtosecond laser self-assembly nanostructure.^[75] Analogous to the dynamic behavior of spin electrons in solid-state system, i.e., spin Hall effect, such kind of transverse separation, referring to two opposite spin states of light, is also named as SHEL.^[76–78]

In addition, combining the linear phase with spiral phase, we realized the manipulation of SAM and OAM separation simultaneously. For instance, figure 2 displays the transverse separation of two constituent components with opposite SAMs and OAMs.^[60] Figure 2(a) depicts the intensity and polarization distributions at the output plane of polarization conversion system. For such a special case, the PB phases are designed as ψ_{L, R} = ± (k_xx + ϕ). As a result, the resultant field presents a hollow core, and horizontally variant SoP. Figures 2(b)–2(d) display the experimentally measured two constituent components, SoP distributions, and evolution process, respectively. Clearly, two components carrying opposite OAMs occur spin-dependent separation upon propagation, under the modulation of linear PB phase.

Fig. 2.

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	Fig. 2. (color online) Spin-dependent separation induced by PB phases upon propagation. (a) Initial field with polarization direction marked by red arrowheads. (b) Horizontally and vertically polarized components of output beam. (c) The distribution of output beam after propagation (top), and the corresponding S3 distribution (bottom). (d) Side view of the beam propagation.[60]

Owing to the geometric flexibility, the PB phase was further designed to realize fetching modulation, e.g., longitudinal spin-dependent separation.^[60] Two conjugate spherical phases that enable the shift of focal length are introduced into PB phases, for which the convergence and divergence phases denoting as ψ_{L, R} = ± (αr²) are encoded into a square phase of exp(iαr²). For this case, two spin components respectively focus into the focal planes f(α₀ ± α), resulting in the longitudinal spin-dependent splitting. According to this principle, Hasman et al. created PB phase lens element composed by subwavelength grating with radially variant orientation.^[79,80] When illuminating this PB phase element, longitudinal multi focal spots with variant SoP appear. Furthermore, the PB phase-based multi-foci lens has also been realized by using segmented metasurface with azimuthally variant phase modulation effect.^[81] Interestingly, such multi-focal polarization-dependent lens generates focal field with longitudinally variant spot number and SoP that depend on the degree of polarization of incident beam.

Beyond that, combining this longitudinal spin separation with the transverse one, it is possible to achieve arbitrary spin-dependent separation in 3D space.^[60] The PB phase in this case can be expressed as a polynomial like ψ_R = k_xx + k_yy + αr². The first two terms and the third term are the phase factors of plane and spherical waves, respectively, providing three degrees of freedom along x, y, z axes to control the spin-dependent separation in 3D space. The positions of focal points (x_R, y_R, z_R) and (x_L, y_L, z_L) are determined by the parameters k_x, k_y, α, following x_{R L} = k_xz_{R L}/k, y_{R L} = k_yz_{R L}/k, and z_{R L} = f(α₀ ± α ). Figure 3 shows an experimentally measured 3D spin-dependent separation. Obviously, this spin-dependent separation increasing along z direction actually performs an equivalently longitudinal SoP modulation. As above presentation, this modulation principle has been successively applied to various functional elements for the implementations such as geometric-phase lens and prism, spin-dependent hologram, and so on.

Fig. 3.

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	Fig. 3. (color online) 3D spin-dependent separation. (a) Horizontally and vertically polarized components. (b) Distributions of S3 at z = 25 mm and 75 mm. (c) Side view of intensity evolution in the x–z plane.[60]

In addition to the PB phase modulation strategy, some modulations affecting the wave front of vector beams have also been reported to achieve spin-dependent separation. Similarly, these methods are both based on the principle that the vector beam is the coaxial superposition of two spin components carrying opposite spiral phases. For a cylindrical vector beam, two opposite spin components generally have equivalent phase topological charges, i.e., |l_R| = |l_L|, exhibiting identical propagation dynamics.^[68–84] As a whole, the SoP in the cross-section remains invariant during propagation. However, once the vector beam has an additional phase, e.g., the vector vortex beam, of which two spin components have different topological charges, i.e., l_R ≠ l_L, these distinct phases will significantly influence the evolutions of the two spin components due to the strong dependence of cross-sectional intensity profile and Gouy phase on the topological charge.^[82,85–87] That is, the two spin components that compose the vector vortex beam do not keep identical intensity profiles and dynamic phases during propagation. Meanwhile, considering the rotational symmetry, the two spin components consequently occur radial spin-dependent separation. As a whole, the vector vortex beam simultaneously presents radially and longitudinally variant SoP.

We note that, such a z-dependent variation has the maximum effect at far-field, i.e., z → ∞. In experiment, the Fourier transform with a lens is a common method to achieve such a special condition. We demonstrated such kind of spin-dependent separation and 3D SoP variation by detecting the SoP distribution at the Fourier plane of vector vortex beams.^[88] Figure 4 shows the experimentally measured SoP transition of focused vector vortex beams with different polarization orders and phase topological charges. The corresponding theory and experiment results show that, the SoP transition is related to the parity of smaller modulus between polarization order m and topological charge l. Moreover, the focal fields of vector vortex beams show radially variant polarization distributions resulting from the unequal intensity proportion of two spin components, namely, radial spin-dependent separation. Remarkably, this radial spin-dependent separation has been successfully used to enrich the optical manipulation of micro-particles^[89] and construct focal fields with typically spatial properties.^[90,91]

Fig. 4.

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	Fig. 4. (color online) SoP transition of vector vortex beams with different polarization orders and topological charges. The polarization order and topological charge are denoted as m and l, respectively.[88]

The caustic phases,^[92] which produce significant evolution dynamics such as arbitrary curve trajectory and transverse acceleration,^[93–100] have recently been introduced into vector beams to manipulate SoP, SAM and OAM fluxes in 3D space, producing physical interests such as spin–orbit interaction and polarization singularities conversion. Figure 5 shows the autofocusing of radially polarized Airy beams without and with a single charged spiral phase.^[101] From the experimental results, it is clear that, under the modulation of spiral phase, the radially polarized autofocusing Airy beam occurs abrupt polarization transition, associated with spin–orbit interaction. Likewise, this impressive phenomenon has been revealed according to mode superposition theory. However, it is noteworthy that, when the attached topological charge and the polarization order of the Airy beam are equal in number, the local SoP undergoes an abrupt transition from linear to circular polarization at the focal point, exhibiting an intuitional result that the associated OAM partially coverts into the spin of photons.

Fig. 5.

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	Fig. 5. (color online) Autofocusing of radially polarized Airy beams without (top) and with (bottom) a single charged vortex phase. (a) and (c) Intensity patterns at pupil and focal planes, respectively. (b) Side view of the beam propagation from numerical simulation. (d) Measured beam polarizations at focal plane.[101]

We have further encoded complex caustic phases onto the vector vortex beams. Figure 6 shows the spin-dependent separation of vector abruptly autofocusing beam encoding with cosine-azimuthal variant phase denoted as Φ = cos(nφ).^[102] Obviously, the spin-dependent separation induced by this typical caustic phase presents strong dependence on the parity of factor n. This relationship was approximately mapped by employing the local spatial frequency, which establishes a simple mapping relationship between the focal field intensity and the pertinent phase distribution of the input field.^[99] According to this theoretical principle, focal fields with desirable intensity, SoP, and phase structures were realized by consciously managing the cosine-azimuthal phase. Actually, this mapping theory supplies a commonly inverse approach to engineer the caustic phase, for the steering of spin–orbit interaction and optical micro manipulation.^[103,104]

Fig. 6.

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	Fig. 6. (color online) Cosine-azimuthal phase modulated spin-dependent separation of vector autofocusing Airy beams with polarization order of m = 2. The top, middle, and below rows correspond to the intensity patterns of total, RH, and LH circularly polarized components, respectively.[102]

Moreover, it is possible to steer the spin-dependent separation by modulating the amplitude profile of vector beams in the cross-section. Owing to the rotational symmetry of cylindrical vector beams, various amplitude modulations aiming to break this property have been proposed, resulting in conceptual and application interests.^{[82–84,105,106]} Remarkably, the fan-shaped masks that have sectorial photic region, attaching onto a vortex beam, produce transverse energy flow to gradually transfer the energy to the opaque region upon beam propagation.^[105] Figure 7 schematically illustrates the transverse energy flow induced spin-dependent separation of cylindrical vector beam, under the amplitude modulation of a fan-shaped mask.^[106] As shown, two spin components carrying opposite spiral phases take opposite angular rotations, under the effect of transverse energy flow arising from the symmetry broken of OAM.

Fig. 7.

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	Fig. 7. (color online) Schematic illustration of spin-dependent separation of azimuthally polarized (AP) beam induced by amplitude modulation.[106]

This diffraction dynamic can be semi-quantitatively described by the angular diffraction theory.^[105] That is, after the fan-shaped aperture, the transmitted field can be described as a series of OAM spectra, with distinct complex amplitudes and topological charges. Then the total diffraction field is the superposition of these diffractive OAM spectra, which can be described by the Fresnel diffraction theory. According to this, we proposed a theoretical model for explicitly explaining the diffractive dynamics of azimuthally broken vector vortex beams.^[105] This model reveals that the spin-dependent azimuthal separation is closely related to the polarization order of the modulated vector beam and the diffraction distance. The results shown in Fig. 8 are the dependence of rotation angle of two spin components on the polarization order and diffractive distance.^[82] For the vector beams carrying zero OAM, two spin components rotate equivalent angles along opposite azimuthal directions. While for the vector vortex beams carrying nonzero OAM, the additional OAM enables the manipulation of spin-dependent separation, namely, the angle width between two spin components.^[107]

Fig. 8.

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	Fig. 8. (color online) Rotation angle |Δφ| versus the (a) polarization order |l| and (b) diffractive distance of fan-shaped vector beams. The diffractive distance corresponding to (a) is z = 25 cm, and the angular widths of photic regions are π/2.[82]

From the results shown in Fig. 8(b), it can be concluded that the rotation angle has the maximum value of π/2 at far-field. This means that the spin-dependent azimuthal separation reaches its maximum value of π. This phenomenon has been demonstrated in the focal field of azimuthally broken vector vortex beams. According to this principle, some rotationally symmetric amplitude and phase masks, as well as vector beams with multiple singularities, have been proposed to redistribute SAM, OAM, and energy flow in the focal plane.^{[41,108–110]} Interestingly, this modulation method and controllable spin-dependent separation have been demonstrated with the application possibility in optical information process.^[111]

To construct a vector beam with longitudinally variant SoP, we started from the SoP description method.^[119] Figure 9 depicts the common Poincaré sphere description method of SoP. Each point on the Poincaré sphere, denoting by angular position (2φ, 2χ), corresponds to one SoP that consists of arbitrary two bases with orthogonal SoPs, that is, two points that are centrally symmetric with respective to the sphere center.^[120] For instance, two crossing points of prime meridian and equator that correspond to the horizontally and vertically polarized states are a pair of bases. Accordingly, each point on the spherical shell can be decomposed into these two bases with variant amplitude ratio or phase difference.

Fig. 9.

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	Fig. 9. (color online) Poincaré sphere description method of SoP.

For simplicity, we coaxially superposed two constituent beams with opposite linear polarizations, i.e., the horizontally and vertically linear polarizations, and supposed these two constituent beams having Bessel functional intensity profiles in the cross-section. It is well known that, Bessel beams are spatially structured beams with the property of non-diffraction and self-healing, typically maintaining their SoP during propagation.^[121–128] Based on this, we further assumed that the total field has a Bessel profile and longitudinally varying SoP, the electric field E can thus be expressed as^[112] 3 where E_{H V}(z) are normalized on-axis field distributions (axial envelopes) corresponding to the horizontally and vertically polarized bases, respectively, and δ_{H V}(z) are phase retardations independent of dynamic phases. The SoP depending on the Jones vector can be intuitively characterized by the ellipse angle χ and orientation angle ψ, which are given by 4 Equation (4) clearly provides two schemes to steer the longitudinal variation of SoP, that is, engineering two components with z-dependent axial envelopes and phase difference, i.e, ensure E_H(z)/E_V(z) ≠ const. or δ_H(z) – δ_V(z) ≠ const.

As is well known, any azimuthally independent field can be divided into a serial of Bessel spectra. This means that the constituent Bessel beams described in Eq. (3), with varying axial envelops denoting as E_{H V}(z), can be constructed from the superposition of Bessel functional spatial spectra.^[129] Accordingly, the spatial spectra can be created based on the inverse Fourier transform expressed as^[112] 5 The corresponding modulation mechanism is schematically illustrated in Fig. 10. As shown, a linearly polarized input beam is split into horizontally and vertically polarized beams, then two constituent beams after independently spatial spectrum modulation are transformed into two quasi-Bessel beams with pre-shaped axial envelops.^[112] Note that, here we engineered two constituent beams having complementary intensity profiles, which enables the total field to present longitudinally variant SoP and uniform intensity profile. Clearly, the axial variation of intensity ratio results in the longitudinal variation of SoP. With this method, we constructed zeroth order quasi-Bessel beams with uniform axial intensity but longitudinally varying SoP. Moreover, we demonstrated the possibility and flexibility of manipulating the trajectory, speed, and period of SoP transformation by changing the basis states, axial envelopes, and initial phases.

Fig. 10.

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	Fig. 10. (color online) Schematic illustration of reshaping the axial intensity envelops of quasi-Bessel beams.[112]

There is another implementation method for functional intensity envelope, the Frozen wave (FW),^[130–132] which consists of a suitable superposition of co-propagating Bessel beams with equal frequency but different transverse and longitudinal wavenumbers, having the advantages of static envelope and arbitrary adjustable longitudinal intensity pattern. Furthermore, the FWs have been successively designed to construct 3D vector beam with higher polarization order. The electric field of FW composed by 2N + 1 Bessel beams of order l can be given as^[131] 6 where A_j is the weighting factor for Bessel spectrum, determined by the distribution of desired light field. Figure 11 schematically shows the construction principle based on the FWs with sinusoidal functional axial envelopes and opposite spin states.^[117] Suppose that two FWs have a constant phase difference φ₀ = –π/2 and independent OAMs denoting as mℏ and nℏ. Therefore, the composed field varies its inhomogeneous SoP along the red trajectory on the hybrid order Poincaré sphere,^[133–135] as shown in Fig. 11(b).

Fig. 11.

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	Fig. 11. (color online) (a) Schematic diagram of polarization oscillating vector beam based on Frozen waves. (b) SoP oscillating trajectory on the hybrid order Poincaré sphere,.[117]

Combining above schemes, a new class of 3D vector beams with longitudinally oscillating SoP within centimeters spatial interval have been proposed via the superposition of two co-propagating optical FWs with pre-shaped axial envelopes and transverse phase structures. The results shown in Fig. 12 are the experimentally measured SoP and Stokes parameter S₃ distributions at four transverse planes with equivalent interval. The initial conditions are φ₀ = –π/2 and m = –n = 1. In such a special case, the oscillation trajectory corresponds to the red curves shown in Fig. 11(b), but on a first order Poincaré sphere, on which the diagonal and antidiagonal points denote the first order hybrid states, as shown in Figs. 12(a) and 12(b), respectively. Considering the static property of FWs, we defined these two states as nodes, while the pole states corresponding to homogeneous circular polarization as antinodes. As the experimental results shown, the composed field longitudinally oscillates its transverse SoP between nodes and antinodes. Consequently, we defined such 3D vector beams as polarization oscillating beams.

Fig. 12.

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	Fig. 12. (color online) Experimentally measured SoP distributions at two adjacent (a) nodes and (b) antinodes. (c)–(f) Left, polarization orientation (background) and polarization ellipticity distributions; right, S3 distributions. Black line and ellipse depict the linear and ellipse polarizations, respectively.[117]

Furthermore, we created other kinds of polarization oscillating beams with fundamental and higher order SoPs to demonstrate the feasibility of our approach. Likewise, Corato–Zanarella et al. reported similar 3D vector beams constructed from FWs with another pre-shaped axial envelops. It is obvious that this method has the ability to construct 3D vector beams with uniform and other functional axial intensity, which are particularly useful for many applications such as optical manipulation, light guiding of atoms, polarization-sensitive sensing, and so on.^[118]

Besides the amplitude modulation, some attention has been devoted to the modulation of phase along propagation direction. Several modulation mechanisms have been proposed to construct 3D vector beams. Remarkably, Moreno et al. proposed 3D vector beams by engineering axicon phase profiles for two orthogonally polarized constituent beams.^[113,114] The construction principle is schematically shown in Fig. 13. The modulation phases attached onto two constituent beams are Φ_1,2(r, ϕ, z = 0) = ± (lϕ + πr/d), with a radial period of d. Meanwhile, such two constituent beams are linearly focused into lth order Bessel beams with longitudinally varying phases Φ_1,2(r, ϕ, z) = ± (lϕ + πz/Λ), after an axicon or equivalent concial phase, as shown in Fig. 13(a).^[115] For such a specific case, considering the local SoP at one point in the cross-section, it periodically varies along the propagation direction, as schematically shown in Fig. 13(b). As a whole, the vector beam presents 3D variant SoP. Employing this transverse-to-longitudinal structuring strategy, we created zeroth and higher order vector Bessel beams with 3D variant SoP. Figures 13(c)–13(h) display the experimental results of a zeroth order vector Bessel beam.

Fig. 13.

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Fig. 13. (color online) (a) Construction principle of 3D vector beams based on longitudinally variant phases. (b) Local polarization orientation in the cross-section versus propagating distance. (c) Intensity profile in the y–z plane of zeroth order vector Bessel beam. (d)–(h) Intensity distributions in the cross-section after a polarizer at different propagation distances shown in (c). The arrows in (c)–(h) denote the polarization orientation of the polarizer.[115]

Furthermore, we explored the self-healing of these vector Bessel beams. Figure 14 depicts the measured reconstruction of zeroth order vector Bessel beams after a circle obstacle.^[115] The results show that, similar to the self-healing of their intensity profiles, the spatial property of SoP, especially the SoP z-dependence can self-heal. This self-healing capability is important in practical applications of light–matter interaction at micro scalar.

Fig. 14.

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	Fig. 14. (color online) Measured reconstruction of zeroth order vector Bessel beams after a circle obstacle. (a) Intensity distribution immediately after the obstacle. (b) Intensity distribution in the y–z plane after the obstacle. (c1)–(c5) The reconstructed beams at planes z1 to z5 denoted in (b), respectively.[115]

It is noteworthy that, the above described 3D vector beams composed by co-propagating constituent beams with pre-shaped axial envelops or phase structures have controllable periods, e.g., the polarization oscillating beams. However, two counter-propagating circularly polarized beams with different amplitude ratios and phase structures have also been proposed to construct 3D vector beams, e.g., vector standing waves with periodically oscillating SoPs.^[136] This special field is also named as superspiral light,^[137] because its oscillating period of SoP is shorter than the wave length. This continuous change of SoP along the axial direction produces polarization gradient,^[138] exhibiting application potential in optical lattices for sub-Doppler cooling of atoms.^[139,140] Remarkably, cylindrically symmetric ‘Sisyphus’ and ‘corkscrew’ types polarization gradients have been constructed based on the superposition of counter-propagating vector Laguerre–Gauss (LG) beams with different types of SoPs,^[141–143] such as the radially and azimuthally polarized LG beams.

Most recently, the 3D vector beams, configured from the counter-propagating cylindrical vector beams, have been proved the entanglement dynamics in free space under unitary conditions.^[144] Due to the non-separability of cylindrical vector vortex fields,^[145,146] the longitudinally oscillating SoP structure produces oscillating degree of local entanglement during propagation in free space, as a result of spin–orbit interaction. Figure 15 shows the schematic representation of the 3D vector field with a z-dependent degree of entanglement. Such a 3D vector field is equivalently composed by two counter-propagating fields with radially and azimuthally polarized SoPs. As shown in top three rows, the z-dependent dynamic phases produce the longitudinal variation of transverse SoP structures, that is, the SoP oscillates between fully vector and fully scalar modes. Notably, during propagation, the total angular momentum is constant, i.e., J_z = 0, indicating the periodic coupling between SAM and OAM. As a result, the degree of entanglement oscillates periodically along the propagation direction. This classical entanglement dynamic arising from the 3D SoP property gives a new insight of spin–orbit coupling of light and offers a new tool to enrich the application of structured fields.

Fig. 15.

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	Fig. 15. (color online) Schematic representation of a 3D vector field with z-dependent degree of entanglement.[144]

Different from the additional phase strategy, the Gouy phase,^[147] a typical part of dynamic phase that nonlinearly increases with beam propagating, has also been reported to construct 3D vector beams,^[148–150] because of its close dependence on the mode orders,^[151] i.e., the radial and azimuthal orders denoted as p and l, respectively. For instance, Cardano et al. generated Poincaré beams that rotate transverse SoP distribution with beam propagating by coaxially overlapping the LG_0,0 and LG_0,l modes. Utilizing a spiral wave plate, the incident light field with LG_0,0 mode is partly transformed into the LG_0,l mode carrying a vortex phase. Then it axially overlaps with the idle part of LG_0,0 mode, resulting in Poincaré beam with lemon, star, and spiral polarizations that depend on the topological charge of the LG_0,l mode. In this process, an extra phase difference emerges with the expression of ΔΦ = |l| \tan⁻¹(z/z₀), called Gouy phase shift. This z-dependent phase difference gives rise to the rotation of polarization upon beam propagation. Figure 16 shows the variations of transverse SoP of Poincaré beams with lemon, star, and spiral SoP structures.^[149]

Fig. 16.

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	Fig. 16. (color online) Variations of transverse SoP of Poincaré beams with (a) lemon, (b) star, and (c) spiral SoP structures, respectively.[149]

Since the cylindrically polarized beams present intriguing tightly focusing property, the research referring to tightly focusing of light fields is one of the most attractive topics in optics. As is well known, the tightly focused field of radially polarized beam presents a much smaller focal spot, supplying a desirable protocol for super-resolution imaging and lithography. Importantly, such a focal field has an extensively enhanced longitudinal component, for which conceptual interests such as transverse SAM^[152–154] and Möbius trip^[155] have been reported. According to such two typical features of tightly focused vector beams, considerable tightly focused fields with specific intensity, polarization, phase structure, as well as energy flow distribution, by means of amplitude and phase modulations in the pupil plane,^{[13,156–159]} have been proposed toward promising applications such as super-resolution imaging, optical trapping, and machining.

Obviously, constructing focal field with 3D SoP structure, even geometrical angular momentum in the space near focal plane, would supply more vector beams and further enrich the light–matter interaction.^[160–162] For instance, the enhanced longitudinal component has application interests in light–matter interaction, which benefits polarization information encryption with ultra-security and optical manipulation of micro-particles (Fig. 17).^[162]

Fig. 17.

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	Fig. 17. (color online) 3D alignment determination of single gold nanorods in two-photon fluorescence scanning images. (a) Schematic illustration of an arbitrary 3D polarization beam interacting with randomly aligned gold nanorods. (b)–(e) Schematic 3D alignment of gold nanorods and their associated calculated fluorescence rate images.[162]

Controlling the focal field with arbitrary 3D SoP in the vicinity of focus has attracted extensive researches recently. For instance, Abouraddy et al. proposed an approach to optical microscopy that enables full control over the 3D SoP at the focal spot by controlling the azimuthal harmonic content of the input field.^[163] Figure 18 shows the resultant 3D vector focal field. As shown, the projected SoP is elliptical in the x–y plane, circular in the y–z plane, and linear in the x–z plane. By combining the electric dipole radiation and a vector diffraction method, Chen et al. proposed an analytical approach for full control over the 3D SoP and field distributions near the tight focus.^[164] Moreover, Zhu et al. reported focal fields with tunable 3D SoPs in 4Pi focusing system by uniformly modulating the amplitude or polarization of one half of the input beam.^[165] Utilizing this focusing scheme, tightly focused fields with special 3D SoP and SAM distributions have been proposed.

Fig. 18.

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	Fig. 18. (color online) (a) An example of 3D SoP and its projections on three orthogonal planes. (b) The corresponding amplitude and phase distributions of incident field.[163]

The promising applications of 3D vector focal fields motive researchers to find a generalized engineering scheme.^[166] Subsequently, inverse design methods, namely, reverse engineering approaches, for complete shaping of the focal field with prescribed distribution of intensity, phase, and SoP, have been proposed.^[167,168] Most recently, Ding et al. further presented a method of shaping 3D vector focal field with controllable SoP variation along arbitrary curves in 3Ds.^[169] Two curved laser beams with orthogonal polarizations and pre-designed intensity and phase structures were coaxially superposed to produce a 3D vector focal field. Figure 19 gives the experimental results of the generated vector Archimedean spiral (left) and the hybrid vector beams (right), of which the linear polarization state varies continuously along the curve. This efficient and noniterative method exhibits application potentials in realms related to optical data storage, microscopy, material processing, and optical tweezer.

Fig. 19.

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	Fig. 19. (color online) Experiment results of the generated vector Archimedean spiral, where the linear polarization state varies continuously along the curve.[169]