## Abstract

Vector beams contain complex polarization structures and they are inherently non-separable in the polarization and spatial degrees of freedom. The spatially variant polarizations of vector beams have enabled many important applications in a variety of fields ranging from classical to quantum physics. In this study, we designed and realized a setup based on Mach-Zehnder interferometer for achieving the vector beams at arbitrary points of higher-order Poincaré sphere, through manipulating two eigenstates in the Mach-Zehnder interferometer system with the combined spiral phase plate. We demonstrated the generation of different kinds of higher-order Poincaré beams, including the beams at points on a latitude or longitude of higher-order Poincaré sphere, Bell states for |*l*| = 1 and |*l*| = 2, radially polarized beams of very high order with *l* = 16, etc. Vector beams of high quality and good accuracy are experimentally achieved, and the flexibility, feasibility and high efficiency of the setup are demonstrated by the practical performance.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

As an intrinsic property of light, polarization plays an important role in various fields of optical science and engineering. Differing from the scalar beams with spatially homogeneous states of polarization (SoP), vector beams with non-separable of polarization and spatial mode have drawn great attention due to its unusual properties [1–3]. Vector beams of radial and azimuthal polarizations have been found important applications in the fields such as tightly focusing [4,5], beam shaping [6–8], optical trapping and manipulation [9–11], super-resolution techniques [12,13], optical metrology [14,15], and laser materials processing [16]. Recent studies have demonstrated that vector beams have also potential applications in optical communication [17,18] and quantum information process [19–22]. By exploiting the degrees of freedom of both spin and orbital angular momentum, the vector beams can be used in quantum key distribution (QKD), quantum teleportation, dense coding, entanglement swapping and multiple degree of freedom quantum memory.

The conventional Poincaré sphere maps complex Jones matrix to a spherical surface in Stokes parameters space, and it visualizes the spatially-invariant polarization of light. In recent years, distinctive topological geometries are proposed to describe the evolution of both polarization and phase of vector beams, such as higher-order Poincaré (HOP) sphere [23], hybrid-order Poincaré sphere (HyPS) [24], making the description of vector beams more intuitionistic and the classification more specific. HOP sphere has been proposed by Holleczek and Milione *et al.,* to describe higher-order polarization state of more general vector vortex waves [23,25], and north and south poles of the sphere represent the eigenstates of total angular momentum (TAM) that is the sum of spin (SAM) and orbital angular momentum (OAM). More specifically, a point on HOP sphere represents a vector beam that is addition of the two conjugate TAM eigenstates with optical vortices of opposite helicity carried by the two circular polarization beams of opposite handedness. The practical interference of the two eigenstates can be realized to generate higher-order cylindrical vector (CV) beams in Mach-Zehnder [26], Sagnac, or other interferometers [27–29]. Therein, the generation of vortices as one of the core procedures is prevailingly fulfilled with the spatial light modulator (SLM) [30–32], and it usually leads to significant energy loss and low power utilizing efficiency. Beside the interferometer systems, various specific optical elements have been designed to generate specific vector beams directly, such as subwavelength dielectric gratings [33], q-plates [34,35], digital micro-mirror devices [36], plasmonic metasurfaces [37,38] and spatially variable retardation plates [39]. These elements have brought great convenience in generating different vector beams, but the efficiency and flexibility are limited to some degree. Regardless of the negligible imperfections, these techniques have greatly promoted the development and application of vector beams.

As a typical non-common-path system of interference for generating HOP beams, Mach-Zehnder interferometer has the advantages of flexibility and convenience in manipulating two conjugate eigenstate vortices. Spiral phase plates (SPP) are a basic element for generating optical vortex of good quality, and it also has a high efficiency of light energy utilization. Then the setup of Mach-Zehnder interferometer combined with spiral phase plate may provide a high-efficiency method for achieving high-order Poincaré beam by the convenient and flexible manipulation of two conjugate eigenstate vortices. However, the superposed output beam from the interferometer is highly sensitive to the transverse displacements and phase shift perturbations arising from unavoidable vibrations and disturbances, leading to instability in the output beams. Liang *et al.,* [40] have concluded the lower stability of the first-order radially and azimuthally polarized vector beams produced by Mach-Zehnder interferometer, by comparing them with those produced by the common-path interferometer. Very often, the qualities of output beams from Mach-Zehnder interferometer may be degraded even by factors as the dust on the element surface and the air circulations. Apparently, to generate high-order Poincaré beams of high-stability and high-quality by utilizing Mach-Zehnder interferometer is a challenging issue.

In this paper, we generated CV beams at arbitrary point on HOP sphere using Mach-Zehnder interferometer system, with spiral phase plates as the vortex generators. Experimentally, an input plane wave of linear polarization passes through a spiral phase plate, and it is changed into a vortex beam of certain topological charge with polarization unchanged. It is divided by a polarizing beam splitter into two vortex beams with polarization perpendicular to each other. They are separately operated in the two arms of the Mach-Zehnder interferometer, for the two conjugate TAM eigenstates to be formed and to be modulated. In this process, a Dove prism is put in one of the arms to reverse the helicity of the beam vortex with topological charge becoming opposite. A quarter-wave plate is added to each arm, and with orientation of the fast axis adjusted appropriately, the beams in the two arms become a pair of the conjugate eigenstate vortices of opposite handedness and opposite topological charge. The phase shift between the two eigenstates is controlled by adjusting the phase compensation plate inserted in one of the arms, for generating vector beam corresponding to any given point on a latitude line of the HOP sphere. This adjustment is flexible, convenient and feasible, and it has a larger adjustable range. The ratio of the amplitudes of two eigenstates is tuned for achieving the vector beam at arbitrary point on a longitude line of the HOP sphere, and this is realized by changing the angle of the input linear polarization with respect to the transmission axis of the splitter. The two eigenstate vortices are recombined as one beam by the second beam splitter, and its Fourier transform is performed in transform system following the interferometer. Physically, the vortex phase at spatial point of the HOP beam after the transform is related to the angular integral on the beam from the interferometer, forming an azimuthal contribution which smooths the distortion of the wavefront and lessens the influence of the translational displacement. The recombined beam is then transformed into vector beam of cylindrical wave of high quality and robustness to perturbations. Different kinds of vector beams are practically generated, including higher-order Poincaré beams for *l* = ±3, for the four Bell states for |*l*| = 2, and radially polarized beams of *l* = 16. The use of our setup makes the generation of vector beams simple and easily feasible with high light power efficiency and lower cost. This provides more convenience for the research and the application of vector beams, and is of great significance to related fields.

## 2. Experimental setup and generation principle

In generation of high-order cylindrical vector beams on HOP sphere, two orthogonal conjugate vortex eigenstates of left- and right-handed circular polarizations with opposite helical wavefronts exp(±*ilφ*) are basically needed, and here *l* is the topological charge of the vortex. As well understood, the change of amplitudes of the two eigenstates controls the latitude value of a point representing the generated vector beam on the HOP sphere, and the change of phase shift controls the longitude value. Based on this principle, we designed the experimental setup using a Mach-Zehnder interferometer system for generating CV beams at arbitrary point on HOP sphere, realized through the flexible control of two eigenstates with arbitrary adjustment of their amplitude and relative phase, as shown in Fig. 1.

In the experiment, a vertically polarized input beam from He-Ne laser with wavelength *λ* = 632.8 nm is expanded and filtered by the spatial pinhole filter (SPF), which is placed on the front focal plane of collimating lens (L1). The beam is collimated into monochromatic planar light after passing through L1, and circular aperture (A1) is used to adjust the input beam size. The combined spiral phase plate C-SPP includes eight individual spiral phase plates (SPP) of topological charges 1 to 8, and when passing through one of them, the beam is converted into vortex beams with the corresponding topological charge. The SPP is put on a five-dimensional stage for translational and rotational adjustments, and the aperture A2 and the CCD are used to monitor the adjustment. The vortex beams pass through HWP of initial fast axis oriented at 45°, as indicated by the blue solid line in the dash-circled lower-left enlarged panel in Fig. 1, and vertically polarized vortex beam is converted to horizontal polarization, as indicated by the red solid line in the panel. If HWP rotates an angle *α* counterclockwise, with the orientation of fast axis represented by the blue dashed line, the polarization direction of outgoing beam rotates 2*α*, as indicated by the red dashed line. The polarizing beam splitter (PBS) splits vortex beams into transmission (horizontal polarization) and reflection (vertical polarization) arms. The amplitude ratio of transmitted and reflected arm is |cos2*α*/sin2*α*| and the intensity ratio is cot^{2}2*α*. Thus, the amplitude of two beams is simply adjusted. Particularly, the amplitudes of two beams are equal when *α *= 22.5°. In the transmission arm, a Dove prism is used to inverse topological charge from −*l* to + *l*, resulting in the opposite helical phase fronts *e*^{±ilφ} in the two arms. The two QWPs oriented at 135° convert horizontally or vertically polarized beams into left- or right-handed circularly polarized (LCP or RCP) beam in transmission and reflection arm, respectively. We note that the use of these two QWPs may also enable this setup to generate vector beams other than HOP beams though out of the content of the paper. In the Mach-Zehnder interferometer system, the output states of two arms can be represented by a pair of orthogonal conjugate eigenstates exp(−*ilφ*)[1 *i*]^{T} and exp(*ilφ*)[1 −*i*]^{T}, where Jones matrix [1 *i*]^{T} and [1 −*i*]^{T} denote LCP and RCP, respectively. In the reflection arm, the phase compensator that is a thin glass plate of thickness 0.138mm is inserted, and its rotation about the vertical axis passing though the optical axis, as shown by the blue dotted line on the phase compensator in Fig. 1, will alter the phase shift *δ* of two arms in the interferometer. We adjust position of mirror M1 carefully so that the phase shift of two arms is either 0 or 2*nπ* with integer *n* when beams are normally incident on the phase compensator.

The enlarged view of the compensator is shown in upper-right panel of Fig. 1. The phase shift *δ* introduced by the tilted compensator can be expressed as *δ* = *k*{*n*_{1}*t*[1−cos(*i*_{1}−*i*_{2})sec*i*_{2}]−*n*_{2}*t*(1−sec*i*_{2})}, with *k* being the wave number, *n*_{1} = 1 and *n*_{2} the refractive index of air and glass, *t* the thickness, *i*_{1} and *i*_{2} incident and refractive angle, respectively. The incident angle *i*_{1} is equal to tilted angle *β* of the compensator. The beams in two arms are recombined by the second beam splitter (BS), and the recombined beams go through round aperture (A2) of radius *R* for the outer stray light to be blocked. The light field immediately after A2 can be expressed with Jones vector:

*r*/

*R*) = 1 for

*r*≤

*R*, else circ(

*r*/

*R*) = 0. The Fourier transform of light field

*u**(*

_{l}*r*,

*φ*) is implemented by the subsequent Fourier lens (L2) with focal length

*f*

_{2}, with

*u**(*

_{l}*r*,

*φ*) on its front focal plane and the obtained CV beam in its rear focal plane. The pattern of the CV beam is imaged onto the pixels of the CCD (Princeton Instruments, ProEM-HS) with the imaging lens (L3). The component field of different polarizations can be obtained by rotating polarized analyzer (P) to different orientations. The light field

**(**

*U**ρ*,

*γ*) on Fourier plane is the Fourier transform of light field

*u**(*

_{l}*r*,

*φ*), i.e.,

**(**

*U**ρ*,

*γ*) =

*F*[

*u**(*

_{l}*r*,

*φ*)], and it is:

**(**

*U**ρ*,

*γ*) can be derived as

*ρ*,

*γ*) is polar coordinates in the Fourier plane and

*J*(

_{l}*x*) is Bessel function of the

*l*-th order and first kind. The

*x*- and

*y*-polarized component of

*P**(*

_{l}*γ*) is related to two parameters, the rotated angle

*α*of HWP and the phase shift

*δ*introduced by tilting phase compensator. From Eq. (2), the vortex phase factor exp(−

*ilγ*) in

**(**

*U**ρ*,

*γ*) is originated from the integral of vortex phase factor exp(−

*ilφ*) in

*u**(*

_{l}*r*,

*φ*) with respect to azimuthal

*φ*, and such azimuthal-to-azimuthal generation relation makes the output field

**(**

*U**ρ*,

*γ*) less sensitive to the translational displacement of the input field

*u**(*

_{l}*r*,

*φ*), and it smooths local distortion in wavefront. Thus, the introduction of the Fourier transform can improve efficiently the stability and quality of the output field. Further derivation of Eq. (3) gives the following expression,

_{1}

*F*

_{2}(

*a*,

*b*,

*c*;

*x*) is the confluent hypergeometric function [41]. The above equation can be further expressed as

**(**

*U**ρ*,

*γ*) =

*A*

_{0}(

*ρ*)

*P**(*

_{l}*γ*), with amplitude function

*A*

_{0}(

*ρ*) and Jones vector

*P**(*

_{l}*γ*) describing the amplitude and the polarization distributions, respectively. It can be seen from Eq. (5) that the amplitude

*A*

_{0}(

*ρ*) is related to the radius

*R*of aperture A2, and the radius of the first bright ring of the annular intensity pattern decreases with the increase of

*R*. The maximal intensity of first ring is further derived as where

*ρ*is the radius of first ring of light field on Fourier plane. It is clear that the maximal intensity

_{l}*I*(

_{l}*ρ*) is proportional to the first maximum of Bessel function

_{l}*J*(

_{l}*x*) of the

*l*-th order and first kind, in which the radius

*ρ*≈

_{l}*γ*

_{l}_{−1,1}

*λf*/2

*πR*, with

*γ*

_{l}_{−1,1}representing the first root of the Bessel function of (

*l*−1)-th order.

Based on the definition [3], the CV beam represented by a point at latitude and longitude (*θ*, *ϕ*) on HOP sphere can be described as

*φ*is the polar angle at real space,

*m*is the integer describing the order of helical wavefront.

*E*

_{0}(

**) is the radial spatial amplitude distribution of the beam in cylindrical coordinate system, and the Jones vector |**

*r*

*ψ**(*

_{m}*θ*,

*ϕ*)> describes the polarization states in beam cross section. cos(

*θ*/2) and sin(

*θ*/2) (0≤

*θ*≤

*π*) are in fact the weight coefficient of two eigenstates exp(−

*imφ*)[1

*i*]

^{T}and exp(

*imφ*)[1 −

*i*]

^{T}at the north and south pole on HOP sphere, respectively. The polarization states |

*ψ**(*

_{m}*θ*,

*ϕ*)> can also be expressed by higher-order Stokes parameters

*S*

_{0}

*,*

^{m}*S*

_{1}

*,*

^{m}*S*

_{2}

*, and*

^{m}*S*

_{3}

*in the Cartesian coordinates of HOP sphere, and this can be seen from the two HOP spheres of*

^{m}*m*= 3 and

*m*= −3 in Figs. 2(a) and 2(c); here the unit radius

*S*

_{0}

*= 1 is the normalized total intensity of eigenstates,*

^{m}*S*

_{1}

*and*

^{m}*S*

_{2}

*is related to the relative phase shift*

^{m}*ϕ*of the two eigenstates, and

*S*

_{3}

*denotes the difference of intensity between two eigenstates. The latitude angle*

^{m}*θ*takes value in the range of 0≤

*θ*≤

*π*and longitude angle

*ϕ*in the range of 0≤

*ϕ*≤2

*π*. Except for the poles, a state |

*ψ**(*

_{m}*θ*,

*ϕ*)> at any other points on the sphere represents a non-separable state [42], especially with maximally non-separable states on the equator. Comparing the light field obtained in the experimental setup given in Eqs. (4) and (5) to standard form of vector beam in Eq. (7), we have

Notably, Jones vector *P** _{l}*(

*γ*) in Eq. (5) is formally similar to |

*ψ**(*

_{m}*θ*,

*ϕ*)> and it follows Eq. (8a). The relation of the experimental parameters to the angles in HOP sphere is demonstrated with the above formulas, and thus the CV beams at arbitrary point (

*θ*,

*ϕ*) on HOP sphere also can be represented as (4

*α*,

*δ*). Therein, the latitude or longitude of HOP sphere, relating to the vector beams, can be experimentally implemented by controlling the HWP or the tilted angle of compensator plate, respectively. For points on the equator, the orientation of HWP is determined by

*α*=

*22.5° and*

*P**(*

_{l}*γ*) is simply given by

*l*= 0, higher-order Poincaré sphere reduces to the conventional Poincaré sphere, and

*P**(*

_{l}*γ*) is reduced to describe scalar beams with spatially invariant polarization state. While

*l*≠ 0 (

*l*is integer),

*P**(*

_{l}*γ*) represents CV beams located on equator of HOP sphere, and it depends on phase shift

*δ*and the sign of

*l*. Particularly when

*l*= +1,

*P**(*

_{l}*γ*) describes the CV beams of radial and azimuthal polarizations for

*δ*= 0 and

*δ*=

*π*, respectively. In contrast, for

*l*= −1, when

*δ*= 0 or

*δ*=

*π*,

*P**(*

_{l}*γ*) represent

*π*-radially or

*π*-azimuthally polarized states, respectively. Also, these four CV beams of the radial, azimuthal,

*π*-radial and

*π*-azimuthal polarization states are the four Bell states, as will be shown in Fig. 3.

## 3. Results of different higher-order Poincaré beams

The CV beams at points with different longitudes and latitudes on HOP sphere are experimentally generated systematically by adjusting the tilted angle of compensator plate and the orientation of HWP, with the optical vortex of topological charge *l* created by a SPP. Intuitively, we first generate the vector beams of linear polarizations on the equator (latitude value *π*/2) of HOP sphere with *l* = 3, and the vector beams of elliptical polarizations on latitude of 3*π*/4 with *l* = −3, to demonstrate the vector beam generation on certain latitude. Here we notice that the value *l* is the topological charge of vortex carried by RCP of Jones vector [1 −*i*]^{T}, namely, the sphere of *l* = 3 has the pair of conjugate eigenstates exp(−*i*3*φ*)[1 *i*]^{T} and exp(*i*3*φ*)[1 −*i*]^{T}. In contrast, the sphere of *l* = −3 has the pair of eigenstates exp(*i*3*φ*)[1 *i*]^{T} and exp(−*i*3*φ*)[1 −*i*]^{T}, and the corresponding vector beam is referred to as *π-*vector beam of order |*l*| = 3. Then we generate the four optical Bell states, including the beams of radial, azimuthal, *π-*radial, and *π-*azimuthal polarizations of order 1 and order 2, respectively, which are important in quantum physics. We also demonstrate the generation of vector beams at different latitudes on a longitude line (specifically, longitude *ϕ* = 0) with polarizations of variant ellipticity. The vector beams of orders 8 and 16 with high quality are also generated. The generations of these vector beams are depicted in the following.

#### 3.1 Generation of vector beams in equator and a latitude

For a fixed latitude angle (*θ* = *θ*_{0}), the longitude *ϕ* of a CV beam at the points of HOP sphere is equal to the phase shift *δ* between beams in the two arms, as given in Eq. (8a), and it can be achieved by rotating the phase compensator to the corresponding angle *β* as discussed in the above context. We demonstrate the generation of vector beams of different longitudes in equator (*θ* = *π*/2) on the HOP sphere of *l* = +3, and beams in latitude *θ* = 3*π*/4 on the sphere of *l* = −3 (*π*-vector beam of order 3).

In generation of the beams on the HOP sphere of *l* = +3 as shown in Fig. 2(a), the orientation of HWP is set to the angle *α* = 22.5°, and the phase compensator is rotated carefully to produce eight phase shift values of *δ _{n}* =

*ϕ*=

_{n}*nπ*/4 and

*δ*′ =

_{n}*ϕ*′ =

_{n}*π*+

*nπ*/4, with

*n*taking the integer from 0 to 3, corresponding to vector beams at eight points on the sphere. The rotation is started from

*δ*= 0 that is determined by the top two lobes in the

*x*-polarized image in the horizontal direction, as shown in Fig. 2(b) A

_{1}. For each case, the phase shift

*δ*varies from 0 to 2

*π*with the compensator rotating continuously from 0° to 9°20′ in our experiment. Owing to the small thinness of the compensator, the change of the rotation angle brings about insignificant transverse shift of the beams in two arms, and this ensures the coaxial operation in the superposition of the two beams. In Fig. 2(b) from A

_{1}to A

_{4}and A

_{1}′ to A

_{4}′, the patterns of the simulated and experimental vector beams are given respectively, at the points from A

_{1}to A

_{4}with

*δ*and A

_{n}_{1}′ to A

_{4}′ with

*δ*′ in equator (the red line) on the sphere in Fig. 2(a). Therein, the left part of each image in figures are the superposed intensity pattern of the simulated horizontally and vertically polarized components, overlaid with the polarization distribution. The right part of each image is the experimental intensity pattern of the

_{n}*x*-polarized component obtained by placing the polarized analyzer P in horizontal transmitting direction, as shown by the double-head arrow; the small insets are the corresponding simulation results.

Next, the vector beams on the HOP sphere of *l* = −3 as shown in Fig. 2(c) are generated with SPP reversely placed in the experimental setup, and latitude on the sphere is taken as *θ* = 3*π*/4, which determines the elliptical rather than linear polarization states at points of the beam cross section. The orientation of HWP is adjusted to angle *α* = *θ*/4 = 33.75° for achieving the latitude value. Similarly, the compensator is rotated to produce eight *π*-vector beams of order |*l*| = 3 at longitudes *ϕ _{n}* =

*δ*=

_{n}*nπ*/4 and

*ϕ*′ =

_{n}*δ*′ =

_{n}*π*+

*nπ*/4, corresponding to the eight points B

*to B*

_{n}*′ in Fig. 2(c), and patterns of the vector beams are given in Fig. 2(d).*

_{n}From these results in Figs. 2(b) and 2(d), we see that the experimental vector beam patterns are well consistent with simulation results, and the patterns have annular intensity distributions with dark core, demonstrating polarization and phase singularity. A vector beam pattern is composed of *N* lobes with *N *= 2|*l*|, as can be demonstrated by the six-lobe patterns both in Fig. 2(b) for *l* = +3 and in Fig. 2(d) for *l* = −3. In Fig. 2(b), the gaps between two adjacent lobes have approximately zero intensities, appearing to be radial dark lines, and this can be attributed to the linear polarization state of vector beams. In contrast, in Fig. 2(d), the minimum intensity in the gaps of vector beam pattern on latitude other than equator is not zero. The change of *δ* leads to rotation of polarization at a fixed point in real space, and this rotation can be seen by discerning the overlaid polarization state at the same point of patterns in Fig. 2(b) or in Fig. 2(d). This is reflected more intuitively in the rotation of beam patterns, we marked a lobe in each pattern with white solid arrow, and the rotation angle *τ* of pattern can be obtained from the marked lobe. Based on Eq. (9), in which the first row of Jones matrix representing *x*-component equals to 1, we know that the vector beam pattern rotates with *δ* by Δ*τ* = −*δ*/2*l*. This is readily demonstrated by the rotation in patterns in Figs. 2(b) and 2(d), where the labeled lobe rotates angle *π*/3 clockwise for *l* = 3, and counterclockwise for *l* = −3, respectively, with respect to the 2*π* change of phase shift *δ*.

Overall, these experimental results conform well with the theory in distributions of both intensity patterns and polarization states, and they demonstrate that the vector beams and the *π*-vector beams represented by a point with arbitrary longitude on certain latitude of HOP spheres can be generated by adjusting the elements in experimental setup.

#### 3.2 Generation of optical Bell states

Optical Bell states are fundamental for encoding hybrid logical qubits and increasing the channel capacity of the quantum protocol [43], and they can be prepared based on the non-separability of polarization and spatial degrees of freedom of classical light [2]. The four vector beam modes |TM>, |TE>, |HE* ^{e}*> and |HE

*> of order |*

^{o}*l*| are a set of Bell basis [44], corresponding to the vector beams and the

*π*-vector beams of linear polarizations at the points (0, 0) and (0,

*π*) on both HOP spheres of

*l*and –

*l*.

We experimentally generated optical Bell states for |*l*| = 1 and |*l*| = 2, and the simulated and the experimental results are shown in Fig. 3. To generate |TM>_{1}, namely the radial vector beam, the orientation of HWP is still set at *α* = 22.5°, the compensator is rotated to ensure *δ* = 0, and the SPP of *l* = +1 is placed in the setup. In generation of |TE>_{1}, the phase shift *δ* between two optical arms is adjusted to *π*. Then the Bell states of |HE* ^{e}*>

_{1}and |HE

*>*

^{o}_{1}are also generated by simply reversing the SPP in the setup to flip the topological charge from +1 to –1. Similarly, the four Bell states for |

*l*| = 2 are also generated with the SPP of

*l*= +2 and –2. The images of the Bell states for |

*l*| = 1 and |

*l*| = 2 are shown in four rows, respectively, in Figs. 3(a) and 3(b), where the simulated and the experimental superposed intensity patterns are given in the left two columns, respectively, with the polarization states overlaid on the simulated patterns. The other four columns are the experimental intensity patterns polarizing in the direction of

*x*-, 45°-,

*y*-, 135°-, respectively. The upper right insets are the corresponding simulated patterns, with which the experimental results are shown to be in good consistency. Again, the polarized intensity patterns have the structures of

*N*= 2|

*l*| lobes. The more detailed polarization property of the beam can be demonstrated with Stokes parameters [26]. In Figs. 3(c) and 3(d), the experimental patterns of Stokes parameters

*S*

_{1}

^{|l|},

*S*

_{2}

^{|l|}and

*S*

_{3}

^{|l|}of the four Bell states for |

*l*| = 1 and |

*l*| = 2 are shown in the four rows, respectively.

In addition, the rotation of lobe-like intensity patterns related to the analyzer P in Fig. 3 demonstrates the non-separability of polarization and spatial distribution of a beam. The rotation angle Δ*φ* of pattern also depends on the value and the sign of topological charge *l*. We define the counterclockwise rotation angle as positive Δ*φ* and Δ*φ*_{P}, with Δ*φ*_{P} describing rotation of analyzer P, and then Δ*φ* = Δ*φ*_{P}/*l* holds. This is demonstrated exemplarily by the intensity patterns for *l* = +1 in the first row in Fig. 3(a) where the patterns rotate *π* counterclockwise as the analyzer P rotates the angle *π*; for *l* = −2 in the third row in Fig. 3(b), the intensity patterns rotate *π*/2 clockwise when analyzer P anticlockwise rotates *π*.

It should be noted that the relation of Δ*φ* = Δ*φ*_{P}/*l* is also true for CV beams at other points on HOP sphere. For the more general CV beams, as can be demonstrated based on Eqs. (4) and (7) and can be seen from the overlaid polarization state distribution in Figs. 2(b) and 2(d), the polarization states are identical at points of azimuths *φ* and *φ*+(*nπ*/|*l*|) with integer *n*≤2|*l*| in the beam cross section. For instance, the polarization in panel A_{3} in Fig. 2(b) appears to be horizontal at the azimuth values *φ* = *π*/4, 7*π*/12, 11*π*/12, with the increment of Δ*φ* = *π*/3. As a summary, Table 1 lists the values of related angles and makes clear relations in them: the phase shift *δ* experimentally introduced by the tilted angle *β* of compensator determines the longitude angle *ϕ*; the rotation angle Δ*τ* of light field when the tilted angle of compensator changes from 0 to *β*; the rotational angle Δ*φ* of intensity pattern as the analyzer P rotates Δ*φ*_{P} = 2*π*.

#### 3.3 Vector beams in a longitude

Based on Eq. (8a), the generation of CV beams at points on a longitude but with different latitudes depends essentially on controlling weight of the two conjugate eigenstates. We demonstrate the generation of the beams at points (*θ*, 0) with different latitude *θ* but fixed longitude *ϕ* = 0 on HOP sphere of *l* = +3. The eigenstates in the transmission and reflection arm in Fig. 1 are again exp(−*i*3*φ*)[1 *i*]^{T} and exp(+*i*3*φ*)[1 –*i*]^{T}, respectively. Their amplitude ratio is adjusted by setting orientation of HWP at *α* = *nπ*/32 (*n* = 0, 1, 2, 3, …, 8), and the latitude *θ* = 4*α* = *nπ*/8 of generated beams on the sphere is correspondingly determined. Figure 4(a) shows seven simulated and experimental CV beams at points C* _{n}*(

*nπ*/8, 0) (

*n*= 1, 2, 3, …, 7) in the longitude line

*ϕ*= 0 with different latitudes

*θ*=

_{n}*nπ*/8, and the points are labeled on the HOP sphere in Fig. 4(b). The polarization states of the CV beams are drawn on the simulated intensity patterns in the first column, and the left and the right handedness of polarization states are distinguished in red and blue solid line. At

*α*= 0 or

*π*/4, one of beams outgoing from the PBS vanishes and the output optical field is left to be the eigenstate at the north or south pole of HOP sphere, namely, the vortex beam of LCP or RCP. The other two columns are experimental intensity patterns of

*x*- and

*y*-polarized components, and the insets are the corresponding simulated patterns.

From the patterns of *x*- and *y*-polarized components in Fig. 4(a), it can be seen that the minimum intensity values at the lobular gaps in each pattern increase with the orientation angle *α* of HWP changing from 0 to *π*/8, and decrease with *α* changing from *π*/8 to *π*/4. This change corresponds to the vector beams going from the north (with *α* = 0) to the south poles (with *α* = *π*/4). From light field in Eqs. (6)–(8), the light intensity of the first ring with radius *ρ _{l}* versus azimuthal angle

*φ*is expressed as

*l*,

*θ*and

*ϕ*are parameters of the CV beams in discussion.

*I*(

_{x}*φ*) and

*I*(

_{y}*φ*) are the intensity distribution of

*x*- and

*y*-polarized components, respectively, and their variations versus azimuth

*φ*are correlated to the cosine function, exhibiting as the form of azimuthal fringes. In each polarized component, we define the ratio of minimum

*I*

_{min}to maximum

*I*

_{max}intensity values for such azimuthal fringes as

*η*, and the ratio

_{s}*η*is readily obtained and is written as

_{s}*η*= (1–sin

_{s}*θ*)/(1+sin

*θ*) = cot

^{2}(

*θ*/2+

*π*/4) (0≤

*η*≤1), or equivalently

_{s}*η*= cot

_{s}^{2}((

*π*–

*θ*)/2+

*π*/4). This leads to identical intensity patterns of the CV beams with latitudes

*θ*and

*π*–

*θ*though the handedness of their elliptical polarization states is opposite. The handedness can be detected experimentally by using the combination of QWP and HWP. Additionally, for each polarized component, the contrast can be written as

*Γ*= (

_{s}*I*

_{max}–

*I*

_{min})/(

*I*

_{max}+

*I*

_{min}) = sin

*θ*.

In order to illustrate the accuracy of the experimentally generated CV beams, we take the experimental intensity data *I _{x}*(

*φ*) and

*I*(

_{y}*φ*) on a circle from the experimental patterns of the polarized components, and fit the functions in Eq. (10) to the data. Figure 4(d) shows the experimental data and the fit curves in the form of scattered dots and solid line of different colors respectively, corresponding to CV beams of

*x*-polarization on the points from C

_{1}to C

_{4}, and it can be seen that they are in good accordance. The ratio

*η*of experimental azimuthal fringes is extracted by the fitting, and thus the latitude of the experimentally generated CV beams is determined. The extracted values for

_{e}*η*of

_{e}*x*-polarized patterns from C

_{1}to C

_{4}are 0.448 ± 0.007, 0.166 ± 0.006, 0.045 ± 0.006 and 0.004 ± 0.006, respectively, and corresponding the values of experimental contrast

*Γ*are 0.381 ± 0.007, 0.715 ± 0.008, 0.914 ± 0.010 and 0.994 ± 0.010. To demonstrate the equivalence of

_{e}*y*- and

*x*-component in extracting the ratio

*η*, we analyze the intensity curves from the

_{e}*y*-polarized patterns from C

_{4}to C

_{7}as shown in Fig. 4(e). The ratio

*η*is 0.004 ± 0.004, 0.055 ± 0.005, 0.173 ± 0.005 and 0.485 ± 0.011, respectively, and corresponding

_{e}*Γ*is 0.992 ± 0.008, 0.896 ± 0.008, 0.706 ± 0.007 and 0.347 ± 0.009. The theoretical values of ratio

_{e}*η*are calculated as 0.447 for C

_{s}_{1}and C

_{7}, 0.172 for C

_{2}and C

_{6}, 0.040 for C

_{3}and C

_{5}, and 0 for C

_{4}, and corresponding the theoretical values of

*Γ*are 0.383, 0.707, 0.924 and 1.000, respectively. These data of the ratio

_{s}*η*(

*θ*) are shown in curves versus the latitude

*θ*in Fig. 4(c) for the CV beams on points from C

_{1}to C

_{7}of HOP sphere. The consistency in the experimental data of

*η*and

_{e}*Γ*with the theoretical values is a demonstration of the high accuracy of the vector beams generated with the setup.

_{e}#### 3.4 Vector beams of very high order and discussions

The generation of vector beams of very high order [45,46] is subject of particular interests because it provides means for manipulations of entanglement between polarization and high quanta OAM states [45] and enables applications of high-sensitivity measurements [47]. We demonstrate that by using our experimental setup, the high-dimensional vector beams with higher quality can be generated conveniently, efficiently, and practically. The radially polarized CV beams on HOP sphere of *l* = 8 and *l* = 16 are achieved for demonstration. Vector beams on HOP sphere of *l* = 8 are generated with the SPP of the corresponding topological charge, and the simulated and experimental results are shown in the upper two rows in Fig. 5. The vector beams of *l* = 16 are generated by cascading two SPP of *l* = 8, and their images are shown in lower two rows in Fig. 5. The first column is the superposed intensity patterns, and the other four columns are the intensity patterns of *x*-, 45°-, *y*-, 135°-polarized components, respectively. Interestingly, the intensity patterns of the polarized components are structured field composed of dense lobes, and they are alternately interlaced in the patterns of two perpendicular components. For the beam of *l* = 16, when the analyzer P rotates the angle of *π*/2, the rotational angle Δ*φ* of component field is as small as *π*/32, based on Δ*φ* = Δ*φ*_{P}/*l*. Additionally, from Eq. (6), the radius of first ring in an intensity pattern is dependent on the radius of filter aperture A2, and the structure of output light field can be simply adjusted by changing the radius of A2. On the whole, experimental results of the different vector beams are well consistent with theoretical predictions, and the generated vector beams can be used as the beam source for various applications due to their good quality, high efficiency and robust to perturbations.

In the practical experimental performances, the various factors may bring about errors in the generation and detection of HOP beams, except for the errors related to inherent quality of the optical elements such as the polarizers and the waveplates. The inaccurate rotations of the HWP or QWP may result in deviations of the generated vector beams from the desired latitude or from the standard HOP beams, while inaccurate rotation of the glass phase compensator may induce deviation of the beam from the set longitude. Other factors as the deviation of the C-SPP center from the optical axis will lead to the non-uniformity of the doughnut. Errors may also appear in the detection of the HOP beams with the inaccurate orientation of the analyzer. In addition, the tilting the phase compensator can cause transverse displacement Δ*x* of the vortex beam, represented by Δ*x* = *t*sin(*i*_{1}−*i*_{2})/cos*i*_{2}. For the phase shift *δ* to vary from 0 to 2*π*, corresponding to the change of tilt angle from 0° to 9°20′, the transverse displacement Δ*x* is 7.8µm, which is insignificant comparing to the radius of the vortex doughnut in the order of a few millimeters. For the maximum phase shift of 6*π* as given in Table 1, the maximum transverse displacement is 19.8µm. Seriously speaking, such displacement may cause errors in the experiment, though practically insignificant.

## 4. Conclusion

We propose a novel and efficient method to realize arbitrary CV beams on HOP sphere by use of Mach-Zehnder interferometer system. In the experimental setup, the SPP as basic optical element is employed to generate vortices, and the topological charge of vortices is reversed with Dove prism in one arm of the interferometer. The two orthogonal circularly polarized beams are created with two QWPs, respectively, and are combined with the vortices in each arm to form the two conjugated eigenstates required for the CV beams. The light intensity ratio of two eigenstates is adjusted with the HWP and PBS, and the phase shift between two eigenstates is realized by rotating the compensator plate. Through the generation of CV beams at point with different latitude and longitude on the HOP sphere, the method is shown to have the advantages of flexibility, feasibility and convenience. Besides, the high efficiency in light energy utilization and good quality of the generated CV beams enable the method to provide light sources for different applications. We expect this work to facilitate the applications of vector beams in optical field manipulations and in quantum sciences.

## Funding

National Natural Science Foundation of China (11574185, 11604183, 11904212).

## Disclosures

The authors declare no conflicts of interest.

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