Abstract

A concise, efficient, and practical direct-view scheme is presented to generate arbitrary cylindrical vector (CV) beams, including CV beams, vortex beams, and cylindrical vector vortex (CVV) beams, by a vortex half-wave plate (VHP). Six kinds of first-order and other high-order CV beams, such as azimuthally polarized (AP) beams, antivortex radial polarization mode beams, and three-order AP beams, are formed by simply rotating a half-wave plate. The Stokes parameters and double-slit interference of multitype CV beams are investigated in detail. The polarization parameters, including degree of polarization, polarization azimuth, and ellipticity, are obtained, which demonstrates the efficient generation of CV beams. In addition, the double-slit interference experiment is introduced in the setup, and fringe misplacement and tilt appear for CVV beams, in which the misplacement number M is 2P+1 for P2 and 2P1 for P3, where P is the polarization order number, and the fringe tilt offset is positively related to the topological charge number l of CVV beams. In addition, new types of VHPs can be formed by cascading two or more VHPs when the types of available VHPs are limited, assisting in more flexible generation of multitype CV beams. It is experimentally demonstrated that arbitrary CV beams with high quality are effectively achieved by the proposed setup, and the double-slit interference method can be utilized to determine and analyze CV beams rapidly and concisely by practical performance, which shows the potential to be implemented as a commercial device.

© 2021 Chinese Laser Press

1. INTRODUCTION

With a cylindrically symmetric intensity profile and a variable polarization endowed with a vortex phase about the beam axis, cylindrical vector (CV) beams have attracted widespread attention [1,2]. As a typical representative of CV beams, the radially polarized (RP) and azimuthally polarized (AP) beams have been found to have important applications in the fields such as tight focusing [35], beam shaping [69], particle accelerating and trapping [1014], laser materials processing [1517], superresolution techniques [1820], and optical metrology [2123]. Recent studies have also demonstrated potential applications of CV beams in optical communication [2426], quantum information processing [2729], spin and orbital angular momentum effects [3034], inverse energy flux [35,36], plasmonic nanostructures [37,38], and fiber and integrated optics [3942].

Various techniques and schemes have been developed to generate CV beams. Generally, these methods can be categorized into two classes: intracavity (active) and external conversion (passive) [1]. The intracavity method is to generate CV beams directly in the laser by putting in special optical elements; the internal structure of the laser needs to be adjusted accordingly, including the use of crystal birefringence [4345], Brewster angle characteristics [46,47], cavity configuration design [48], and geometric phase control [49]. These methods can generate vector beams with high quality and energy conversion efficiency but lack flexibility due to limited space and complex technology to reconstruct the existing laser. The external conversion method refers to transforming spatially uniform polarized beams into CV beams through certain phase devices or decomposition and recombination outside the laser. The commonly used external devices include segmented spiral varying retarders [5052], special phase plates [53,54], anisotropic crystals [5558], diffractive optical elements [59,60], subwavelength gratings [6163], and metasurfaces [6467]. However, these conversion methods have the problems of low conversion efficiency and high design cost. Interferometric devices have also been proposed to generate CV beams, such as Sagnac interferometers [6871], Mach–Zehnder interferometers [72,73], and other interferometers [7478]. But the interferometric methods often need high stability and precise control, which reduces conversion efficiency and accuracy. Nowadays, the spatial light modulator (SLM) is mostly used to realize CV beams, including the double SLM method [7982] and split-screen method by a single SLM [8388]. However, the double SLM method is not cost-effective, and the split-screen method has the drawback of low SLM area utility. In addition, due to the diffraction effect, the SLM method has low conversion efficiency and cannot withstand high-energy laser beams because of the liquid crystal material.

Herein, we demonstrate a concise, efficient, and practical direct-view method to generate arbitrary CV beams by vortex half-wave plate (VHP) without beam reflection and deflection, including realizing CV beams, vortex beams, and CVV beams. The VHP is a special half-wave plate (HP) with a consistent delay π, but the direction of the fast axis changes continuously around the circle center. It has high transmittance just like an HP, so it has high energy utilization and can be made up to 2 inches, which can realize the conversion of high-energy beams through beam expansion and recompression. In addition, new types of VHPs can be formed by cascading two or more VHPs when the types of available VHPs are limited, which helps in more flexible generation of multitype CV beams.

In this paper, six kinds of first-order and other high-order CV beams, such as the RP beam, AP beam, and three-order AP beam, are formed by simply rotating an HP. The Stokes parameters and double-slit interference of multitype CV beams are investigated in detail. The polarization parameters, including degree of polarization (DOP), polarization azimuth, and ellipticity, are obtained, which demonstrates the efficient generation of the CV beams. Additionally, misplacement and tilt appear in the double-slit interference fringes of CVV beams, in which the misplacement number M is 2P+1 for P2 and 2P1 for P3, where P is the polarization order number, and the fringe tilt offset is positively related to the topological charge number l of the CVV beams. It is experimentally demonstrated that arbitrary CV beams with high quality are effectively achieved by the proposed setup, and the double-slit interference method can be utilized to determine and analyze CV beams rapidly and concisely through practical performance.

2. PRINCIPLES

A. VHP

The VHP is a special HP that is made of liquid crystal polymer with birefringence. It has a consistent delay π, with the direction of the fast axis changing continuously around the circle center. The distribution of the fast axis can be described as the following:

θ=m2ψ+σ,
where ψ is the VHP azimuth angle, θ is the VHP fast axis direction at a certain azimuth angle, m is the order number, and σ is the fast axis direction when ψ=0. The VHP with m and σ will be expressed as VHP(m, σ) for short in the following. The fast axis distributions of several VHPs with σ=0 and m=1, 2, 3, 4, respectively, are shown in Fig. 1.
 figure: Fig. 1.

Fig. 1. Fast axis distributions of several VHPs with σ=0. (a)–(d) are the distributions with m=1, 2, 3, 4, respectively.

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The VHP Jones matrix Jm,σ with fast axis at the direction θ can be expressed as

Jm,σ=[cos2θsin2θsin2θcos2θ]=[cos(mψ+2σ)sin(mψ+2σ)sin(mψ+2σ)cos(mψ+2σ)],
in which θ varies with the azimuth angle ψ, so that the VHP Jones matrix is spatially variable. For VHPs with the same order number m but different σ (except for m=2), they can be converted to each other by rotation. To convert the m-order VHP with σ=0 to the same order VHP with σ=ϕ, the clockwise rotation angle Φ can be expressed as follows:
m2ΦΦ=ϕΦ=2ϕm2.

When Φ is a negative value, it represents an anti-clockwise rotation, and a positive value denotes a clockwise rotation. Taking the first-order and fourth-order VHPs as examples, the VHP(1, 0) can be converted to VHP(1, π/12) by rotating 30° anti-clockwise. Similarly, the VHP(4, 0) becomes VHP(4, π/6) by rotating 30° clockwise. And the VHP(4, π/6) (or σ=5π/6) will be obtained by rotating the VHP(4, 0) by 30° anti-clockwise.

B. Generation of CV Vector Beams

We will take the VHP with m=1 to generate arbitrary first-order CV beams, for example. When the incident beam is horizontally polarized with the Jones vector E//=[10]T, for the VHP(1, 0), the Jones vector of the transmitted beam is expressed as

ERP=J1,0·E//=[cosψsinψsinψcosψ]·[10]=[cosψsinψ],
which is the RP beam. When the horizontally polarized beam passes through the VHP(1, π/4), the Jones vector of the transmitted beam is expressed as
EAP=J1,π/4·E//=[cos(ψ+π/2)sin(ψ+π/2)]=[sinψcosψ],
which is the AP beam, i.e., on the basis of radial polarization, each polarization direction is rotated 90° anti-clockwise. Or, with the vertically polarized beam input, with the Jones vector E=[01]T for the VHP(1, 0), the Jones vector of the transmitted beam is expressed as
EAP=J1,0·E=[sinψcosψ]=[cos(ψπ/2)sin(ψπ/2)]=eiπ·EAP,
which is still the AP beam, but each polarization direction is rotated 90° clockwise on the basis of radial polarization. Compared with the previous case, there is only a π-phase difference. If the horizontally polarized beam is input, for the VHP(1, π/8), the Jones vector of the transmitted beam is expressed as
Eπ/41=J1,π/8·E//=[cos(ψ+π/4)sin(ψ+π/4)].

It means that each polarization direction is rotated 45° anti-clockwise on the basis of radial polarization, which is called the first-order π/4-CV beam. The superscript 1 in Eπ/41 represents polarization order number P=1 of CV beams. Also, Eπ/41 can be obtained by inputting a horizontally polarized beam for the VHP(1, π/8) or VHP(1, 3π/8) as the following:

Eπ/41=J1,π/8·E//=[cos(ψπ/4)sin(ψπ/4)]=eiπ·[cos(ψ+3π/4)sin(ψ+3π/4)]=eiπ·J1,3π/8·E//.

We can get multitype CV beams by directly rotating the VHP with σ=0 when the horizontally or vertically polarized beam is input. But this method cannot obtain other second-order CV beams except for the second-order RP beam, and with the increase of order m, the rotation angle of VHP becomes smaller, and the accuracy will be more difficult to control. Taking the high-order AP beam (requiring ϕ=±π/4) as an example, according to Eq. (3), the rotation angle Φ of VHP with σ=0 is Φ=±π/[2(m2)]. Obviously, the larger the order m (for m3) is, the smaller the rotation angle ϕ of VHP with σ=0 becomes, and the higher rotation accuracy is needed.

So, the method of changing the polarization direction of an incident linearly polarized beam by rotating an HP to generate CV beams is proposed without rotating the VHP. And the polarization direction of the incident beam need not change with the increase of the order m for generating the same kind of other high-order CV beams. When the incident beam is a linearly polarized one, with the polarization direction orientated at α according to the horizontal direction (x axis) for the VHP(m, 0), the Jones vector of the transmitted beam is expressed as the following:

Eαm=Jm,0·Eα=[cos(mψ)sin(mψ)sin(mψ)cos(mψ)]·[cosαsinα]=[cos(mψα)sin(mψα)].

This means that each polarization direction is rotated by α anti-clockwise on the basis of the radial polarization. The superscript m in Eαm represents polarization order number P=m of the CV beams. So, the ERP,EAP,Eπ/4m,Eπ/4m can be obtained when α=0, π/2, π/4, and π/4, respectively.

In addition, when the RP beam passes through an HP with the fast axis in the horizontal (or vertical) direction, i.e., 0° HP (or 90° HP) with the Jones matrix described as J0 (or Jπ/2=eiπ·J0), the Jones vector of the transmitted beam is expressed as

EARP=J0·ERP=[1001]·[cosψsinψ]=[cos(ψ)sin(ψ)].

That is, the polarization direction at the azimuth angle ψ is ψ, which is called anti-vortex radial polarization (ARP) [89] or quasi-radial polarization (q-RP) [90] mode with the polarization order number P=1. When the AP beam passes through the 0° HP, or the RP beam passes through an HP with the fast axis at 45° direction with the Jones matrix Jπ/4, the Jones vector of the transmitted beam is expressed as

EAAP=J0·EAP=Jπ/4·ERP=[0110]·[cosψsinψ]=[sinψcosψ]=[cos(ψ+π/2)sin(ψ+π/2)].

The polarization direction is ψ+π/2 at the azimuth angle ψ, which is called the anti-vortex azimuthal polarization (AAP) or quasi-azimuth polarization (q-AP) mode. For instance, in the direction of azimuth angle ψ=0°, the polarization direction is 90°, at ψ=45°, the polarization direction is 45°, and at ψ=135°, the polarization direction is 45°. In the same azimuth direction, the polarization direction of the AAP mode is that of the ARP mode rotated 90° anti-clockwise.

The polarization distributions of all the above CV beams are shown in Fig. 2. Figures 2(a)–2(f) represent the polarization and intensity distributions of ERP,EAP,Eπ/41,Eπ/41,EARP,EAAP, respectively.

 figure: Fig. 2.

Fig. 2. Polarization distributions of various one-order CVP beams. (a)–(f) correspond to the polarization distributions of ERP,EAP,Eπ/41,Eπ/41,EARP,EAAP.

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The high-order CV beams will be obtained when a linearly polarized beam passes through the VHP with m2.

C. Generation of Vortex Beams

The Jones matrix of a quarter-wave plate (QP) with fast axis in θ direction relative to the horizontal direction (x axis) is expressed as follows:

Jθ(λ/4)=22[1icos2θisin2θisin2θ1+icos2θ].

When the horizontally polarized beam passes through the QPs with fast axis in 45° and 45° direction, respectively, right-handed and left-handed circularly polarized beams will be obtained with the corresponding Jones vectors described as ER=2/2[1i]T and EL=2/2[1i]T.

The VHP(m, σ) can transform an incident circularly polarized beam into a vortex wavefront. When the incident beam is left-handed circularly polarized, the transmitted beam is a vortex one with the topological charge number l=m, and the polarization state becomes right-handed circularly polarized. The corresponding Jones vector is as follows:

Jm,σ·EL=22[cos2θ+isin2θsin2θicos2θ]=22[ei2θei(2θπ/2)]=22[1i]·ei(mψ+2σ)=ER·ei(mψ+2σ)=EARCV,
which is an anti-clockwise vortex beam with right-handed circular polarization, i.e., anti-clockwise right-handed circularly-polarized vortex (ARCV) beam. 2σ is only an initial fixed phase and does not affect the relative phase distribution of the whole vortex beam.

When the incident beam is a right-handed circularly polarized beam, the exit beam is a vortex beam with topological charge number l=m, and the polarization state becomes left-handed circularly polarized. The corresponding Jones vector is as follows:

Jm,σ·ER=22[cos2θisin2θsin2θ+icos2θ]=22[ei(2θ)ei(2θ+π/2)]=22[1i]·ei(mψ+2σ)=EL·ei(mψ+2σ)=ECLCV,
which is the clockwise vortex beam with left-handed circular polarization, i.e., clockwise left-handed circularly polarized vortex (CLCV) beam.

D. Generation of CVV Beams

When the circularly polarized vortex beam passes through a QP with the fast axis in 45° or 45° direction, it can be converted into a linearly-polarized vortex (LV) beam. The conversion process of the Jones vector is as follows:

Jπ/4(λ/4)·EARCV=[0i]·ei(mϕ+2σ)=E·ei(mψ+2σπ/2)=EAVLV,Jπ/4(λ/4)·EARCV=[10]·ei(mϕ+2σ)=E//·ei(mψ+2σ)=EAHLV.

An anti-clockwise vertically linearly polarized vortex (AVLV) beam is obtained when the ARCV beam passes through the QP with the fast axis at 45°, and an anti-clockwise horizontally linearly polarized vortex (AHLV) beam is obtained when the ARCV beam passes through the QP with the fast axis at 45°.

Similarly, when the CLCV beam passes through the QP with the fast axis at 45°, a clockwise horizontally linearly polarized vortex (CHLV) beam is obtained. And a clockwise vertically linearly polarized vortex (CVLV) beam is obtained when the CLCV beam passes through the QP with the fast axis at 45°. The conversion process of Jones vector is as follows:

Jπ/4(λ/4)·ECLCV=[10]·ei(mϕ+2σ)=E//·ei(mψ+2σ)=ECHLV,Jπ/4(λ/4)·ECLCV=[0i]·ei(mϕ+2σ)=E·ei(mψ+2σπ/2)=ECVLV.

Combined with Section 2.B, it is easy to find that a CVV beam can be obtained when a generated LV beam passes through another VHP. The corresponding Jones vector is depicted as follows:

Jm2,σ2·EAHLV=[cos(m2ψ+2σ2)sin(m2ψ+2σ2)]·ei(m1ψ+2σ1)=ERm2Vm1,Jm2,σ2·EAVLV=[sin(m2ψ+2σ2)cos(m2ψ+2σ2)]·ei(m1ψ+2σ1π/2)=EAm2Vm1.

The m2-order radially polarized vortex (RPV) beam with topological charge number l=m1 is generated when the AHLV beam with l=m1 passes through the second m2-order VHP with σ=0. Similarly, an m2-order azimuthally polarized vortex (APV) beam with l=m1 is generated when the CVLV beam with l=m1 passes through the second m2-order VHP with σ=0. The conversion process of Jones vector is as follows:

Jm2,σ2·ECHLV=[cos(m2ψ+2σ2)sin(m2ψ+2σ2)]·ei(m1ψ+2σ1)=ERm2Vm1,Jm2,σ2·ECVLV=[sin(m2ψ+2σ2)cos(m2ψ+2σ2)]·ei(m1ψ+2σ1π/2)=EAm2Vm1.

E. Increasing Types of VHP

New types of VHPs can be formed by cascading two or more VHPs when the types of available VHPs are limited. The cascading of two and three VHPs is described, respectively, in detail. The combined Jones matrix of two VHPs can be expressed as

J2·J1=Jm2,σ2·Jm1,σ1=[cosα2sinα2sinα2cosα2]·[cosα1sinα1sinα1cosα1]=[cos[(m2m1)ψ+2(σ2σ1)]sin[(m2m1)ψ+2(σ2σ1)]sin[(m2m1)ψ+2(σ2σ1)]cos[(m2m1)ψ+2(σ2σ1)]],
in which α=mψ+2σ. It can be found from the expression that the combined Jones matrix is a rotation matrix, and the polarization direction of a linearly polarized beam through the device can rotate at an angle of (m2m1)ψ+2(σ2σ1) anti-clockwise. The cascaded device can be transformed into a VHP by the 0° HP with the Jones matrix J0, i.e., VHP(0, 0). The Jones matrix of the transformation process can be expressed as
(J2·J1)·J0=[cos(α2α1)sin(α2α1)sin(α2α1)cos(α2α1)]·[cos0sin0sin0cos0]=[cos(α2α1+0)sin(α2α1+0)sin(α2α1+0)cos(α2α1+0)].

Taking the combination of the first-order and third-order VHPs with σ=0, for example, when passing through the 0° HP, first-order and third-order VHPs successively, the combination result is equivalent to the VHP(2, 0). And when passing through the 0° HP, third-order and first-order VHPs in turn, the cascaded one corresponds to the VHP(2, 0). The negative sign indicates that the fast axis of the VHP changes continuously clockwise. In particular, when 0° HP is placed between the two VHPs, the VHP(4, 0) is formed. The combined Jones matrix of three VHPs can be expressed as

J3·(J2·J1)=[cos(α3α2+α1)sin(α3α2+α1)sin(α3α2+α1)cos(α3α2+α1)].

It can be found from the expression that the cascaded device forms the new VHP with m=m3m2+m1. Taking the σ=0 VHPs with m=2, 3, and 7, for example, the VHPs with m=±1, 2, 3, ±4, ±5, 6, 7, 8, 9, and 10 can be generated by cascading two or three of the VHPs, in which 0° HP can be placed when necessary. Similarly, the combination of five VHPs can form the new VHP with m=m5m4+m3m2+m1. By analogy, more types of VHPs can be generated.

3. EXPERIMENTAL SETUP

Figure 3 shows the direct-view experimental setup for concise and efficient generation of arbitrary CV beams by a VHP without beam reflection and deflection. The incident beam from a linearly polarized He–Ne laser is expanded six times by two well-aligned lenses. The combination of HP1 and Gran Taylor prism (GTP) can modulate the incident beam intensity and make sure the incident beam is horizontally linearly polarized, and the HP2 is used to change the linearly polarized direction. The QP1 in the green dashed line can be used to convert the linearly polarized beam into a circularly polarized beam when necessary. The HP3 in the red dashed line can be used to convert the RP or AP beam into the ARP or AAP mode polarized. And the QP2 can convert the circularly polarized beam into linearly polarized; combined with VHP2 in the blue dashed line, it helps to generate CVV beams.

 figure: Fig. 3.

Fig. 3. Direct-view experimental setup for the generation and double-slit interference of CV beams. Lenses 1 and 2 constitute a beam expander; HP, half-wave plate; GTP, Gran Taylor prism; QP, quarter-wave plate; VHP, vortex half-wave plate.

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The components in the dashed lines can be arranged or not according to the beams needed, and the HPs and QPs can be rotated accordingly. Multiple types of first-order or higher-order CV beams, such as the RP, AP, and π/4-CV beams, can be generated by rotating the HP2 when there are no components in the dashed lines. When only the HP3 in the red dashed line is added to the experimental setup, the ARP and AAP beams can be generated by rotating the HP3. When only the QP1 in the green dashed line is added to the experimental setup, the linearly polarized beam is converted into a circularly polarized beam, and after passing through the VHP1, a vortex beam with circular polarization is obtained. Meanwhile, if QP2 and VHP2 in the blue dashed line are added to the experimental setup, the CVV beam can be generated.

The intensity distribution of the generated beam is captured by the CCD camera, which is a Manta G-033B modal 8-bit product of AVT Company with the resolution of 656pixels×492pixels and a pixel pitch of 9.9μm×9.9μm. A double slit with the slit width of 99 μm and slit spacing of 1 mm in the black dashed line can be placed in front of the camera to help detect the generated CV beams.

In the experimental setup, the VHP is supplied by LBTEK (Changsha Lubang Photonics Technology Co., Ltd.), which is a liquid crystal polymer wave plate with a fast axis rotating continuously along an azimuthal coordinate, whose function resembles the general q-plate [9194] and S-wave plate [9597]. The general q-plate is essentially a thin (nematic) liquid crystal film sandwiched between two coated plane glasses, which needs extra electric or temperature control. And the S-wave plate, a kind of metasurface, is a nanosubwavelength periodic structure applied in the visible region, usually written inside a glass plate by femtosecond laser, behaving as a uniaxial crystal with the optical axes parallel and perpendicular to the subwavelength grooves. These devices are characterized by singular optic axis distributions with topological charge q and are commonly called by the joint name q-plates [98,99].

4. RESULTS AND DISCUSSION

The first-order CV beams corresponding to the Jones vector of ERP, EAP, Eπ/41, and Eπ/41 are obtained when the HP2 with the fast axis is orientated at 0°, 90°, 22.5°, and 22.5° direction, respectively, and the VHP1(1, 0) is placed in the experimental setup without components in the dashed lines. The corresponding experimental results captured by the CCD are depicted in Figs. 4(a)–4(d). The first-order ARP and AAP mode polarized beams can be obtained when the HP3 in the red dashed line with the fast axis orientated at 0° and 22.5°, respectively, passed by the RP beam is placed in the setup. The corresponding experimental results are depicted in Figs. 4(e) and 4(f). Figures 4(a1)–4(f3) are the intensity distributions after passing through a linear polarizer with the polarization direction orientated at 0°, 45°, and 45°, respectively, and the arrow heads indicate the transmission direction of the polarizers.

 figure: Fig. 4.

Fig. 4. Intensity distributions of generated first-order CV beams. (a)–(f) ERP,EAP,Eπ/41,Eπ/41,EARP,EAAP. Arrow heads represent the transmission direction of the polarizers.

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In order to get the spatial polarization distributions of generated beams, Stokes parameters S0, S1, S2, and S3 are measured by the combination of a polarizer and a QP in the following equation:

{S0=Ix+Iy=Iπ/4+Iπ/4=IR+IL=I0S1=IxIy=S02Iy=2IxS0S2=Iπ/4Iπ/4=S02Iπ/4=2Iπ/4S0S3=IRIL=S02IL=2IRS0,
where I0 is the total transverse intensity of the beam. Ix, Iy, Iπ/4, and Iπ/4 are the transmission intensity after a polarizer with polarization direction orientated at 0°, 90°, 45°, and 45°, respectively. IR means the right-handed circularly polarized component, which is the transmission intensity after a QP with the fast axis orientated at 0° (or 45°) and a polarizer with polarization direction orientated at 45° (or 90°) successively. Accordingly, IL means the left-handed circular component. It is the transmission intensity after a 0° (or 45°) QP and a 45° (or 0°) polarizer successively. According to Eq. (22), Stokes parameters can be obtained only by measuring four intensities of I0, Ix, Iπ/4, and IR.

Polarization parameters, including DOP, polarization azimuth (PA), and polarization ellipticity (PE), can be obtained by the following:

{p=S12+S22+S32S0ψ=12arctan(S2/S1)χ=12arcsinS3S12+S22+S32,
where p is the DOP, ψ is the PA, and χ is the ellipsoidal polarization angle, i.e., PE. Tan(χ) represents the ratio of short axis to long axis of elliptically polarized beam. χ=0° means the linearly polarized beam, and χ=±45° means the right-handed and left-handed circularly polarized beam, respectively.

The PA distributions of generated first-order CV beams are depicted in Fig. 5. It is easy to find that the experimental results are well consistent with the theoretical ones described in Fig. 2.

 figure: Fig. 5.

Fig. 5. Polarization azimuth distributions of generated first-order CV beams. (a)–(f) ERP,EAP,Eπ/41,Eπ/41,EARP,EAAP. Arrow heads represent the polarization direction.

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The high-order CV beams can be obtained when the VHP1 with m2 is placed in the experimental setup. The second-order and third-order RP beams are obtained for the VHP1(2, 0) and VHP1(3, 0) when the fast axis of HP2 is orientated at 0°. And the second-order and third-order AP beams are obtained for the VHP1(2, 0) and VHP1(3, 0) when the fast axis of HP2 is orientated at 45°. The corresponding measured Stokes and polarization parameters of generated second-order and third-order CV beams are depicted in Fig. 6. It is found that the DOP is basically equal to 1 in space as it is linearly polarized, which is demonstrated by nearly 0 of PE, and the PA is in good agreement with the theory. The experimental results show that the high-order CV beams with high quality are well generated.

 figure: Fig. 6.

Fig. 6. Stokes parameters and polarization parameters of generated second-order and third-order CV beams. (a), (b) Second-order and third-order RP beams; (c), (d) second-order and third-order AP beams. 1–6: S1, S2, S3, DOP, PA, PE.

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Vortex beams can be obtained when the QP1 in the green dashed line is added to the setup. CLCV beams are obtained when the fast axis of QP1 is orientated at 45°. The CLCV beams with topological charge number l=1,2,3 (i.e., first-order to third-order CLCV) are obtained for the VHP1 with m1=1,2,3, respectively. And anti-clockwise right-handed circularly-polarized vortex (ARCV) beams are obtained for the QP1 with the fast axis at 45°. The corresponding experimental results of first-order and third-order ARCV and CLCV beams are depicted in Fig. 7. In terms of intensity, vortex beams are hollow beams just like CV beams, but according to the PE, the polarization state is the same in the whole space, either right-handed or left-handed circular polarization. As shown in Figs. 7(a1)–7(d1), the PA distribution is relatively random because of circular polarization.

 figure: Fig. 7.

Fig. 7. Intensity and polarization distributions of generated first-order and third-order ARCV and CLCV beams. (a), (b) First-order and third-order ARCV beams; (c), (d) first-order and third-order CLCV beams; I0, intensity; ψ, PA; χ, PE.

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However, from the intensity and polarization information it is hard to get the topological charge number l of vortex beams, which can be measured by the double-slit interference experiment. The experimental results of double-slit interference of first-order to third-order ARCV and CLCV beams are shown in Fig. 8. It is found that the interference fringe tilt appears, and fringes for anti-clockwise vortex beams (i.e., positive topological charge number) curve to the right, while those for clockwise ones (i.e., negative topological charge number) curve to the left. And the tilt offset increases with the increase of the topological charge number l.

 figure: Fig. 8.

Fig. 8. Experimental results of double-slit interference of first-order to third-order ARCV and CLCV beams. (a)–(c) First-order to third-order ARCV beams; (d)–(f) first-order to third-order CLCV beams; p1p3, horizontal pixel positions with equal transverse fringes at y=50; p4p6, corresponding horizontal pixel positions with equal transverse fringes at y=450.

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The specific tilt offsets are measured by pixel positions p1p6 with equal transverse fringes. There are 10 fringes between p1 and p3 because of relatively little offset for l=1,2 shown in Figs. 8(a), 8(b), 8(d), and 8(e), and 8 fringes for l=3 shown in Figs. 8(c) and 8(f). To ensure the relative accuracy of measurement, p1 to p3 are selected at the same ordinate y=50. And p4 to p6, selected at the same ordinate y=450, are on the same stripe as p1 to p3, respectively. The measured values are shown in Table 1. The fringe spacing d0 is obtained by d0=(p3p1+p6p4)/(2N), in which N is the fringe number between p1 (or p4) and p3 (or p6). The fringe offset d is obtained by d=(p4p1+p5p2+p6p3)/3, and the relative offset coefficient R is defined as the ratio of d to d0, i.e., Ri=di/d0(i=2,1,0,1,2).

Tables Icon

Table 1. Measured Values of Pixel Positions and Topological Charge Number

Theoretically, the relative offset coefficient Ri should be an integer equal to the topological charge li if the double-slit spacing is close to 0 and the beam size approaches infinity. However, due to the existence of double slit spacing and limited beam size, the two-slit phase difference of the top part of the double slit is not 0, and the one of the bottom part of the double slit is less than 2lπ, in which l is the topological charge number, so the phase difference of the top and bottom parts of the double slit is less than 2lπ, i.e., the offset of interference fringes is less than l fringes. But, the ratio of Ri of vortex beam with l=i to R1 (or |R1| for l<0) is close to the topological charge l, which could be expressed as follows:

{l=Ri/R1,l>0l=Ri/|R1|,l<0.

The topological charge l can be calculated based on Eq. (24). The corresponding measurement results are listed in Table 1. It is found that the measured results are in good agreement with the theoretical values except for a little deviation, which can be improved by increasing the camera pixels and reducing the pixel size.

The double-slit interference experiment is applied to the high-order CV (HCV) beams, and the corresponding experimental results of first-order to third-order CV beams are depicted in Fig. 9. It is found that misplacements appear in the double-slit interference fringes of the HCV beams, in which the misplacement region number m in a fringe is 2P+1 for P2 and 2P1 for P3, where P is the polarization order number. And the intensity distributions of double-slit interference fringes are the same for the same polarization order but different types of CV beams because of the overall symmetrical polarization distribution. For instance, Figs. 9(b) and 9(c) show the interference fringe distributions of second-order π/4 beam and the third-order AP beam, respectively, which have the same distribution as the second-order and third-order RP beams. Besides, no tilt appears according to the red vertical line in the Fig. 9.

 figure: Fig. 9.

Fig. 9. Experimental results of double-slit interference of first-order to third-order CV beams. (a)–(c) First-order RP beam; second-order π/4 beam; third-order AP beam.

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In theory, the misplacement region number M should be equal to 2P+1 in the condition that the double-slit spacing is rather small (approximately 0). However, with the increase of the polarization number P, the polarization information of the HCV beam at the top and bottom of the double slit is masked because of non-zero double-slit spacing, and then misplacement regions reduce two parts. In this experiment, the missing appeared in the HCV beam with P3. If the double-slit spacing becomes larger, the missing may happen in the second-order CV beam. Similarly, if the spacing becomes smaller, the missing will be delayed to a higher-order CV beam. As shown in Fig. 9, although the fringe distributions of second-order and third-order CV beams both have five layers, their distribution density in the middle region is different. The higher the polarization order is, the denser the layers are.

Further, the CV vortex (CVV) beams are generated when the QP2 with the fast axis at 45° or 45° [QP2(45°) or QP2(-45°) for short in the following] and VHP2 in the blue dashed line are added to the experimental setup. According to the Section 2.D, the QP2 converts the ARCV or CLCV beam into an LV beam, and after passing through the VHP2 a CVV beam is generated. The anti-clockwise radially polarized vortex (A-RPV) beam with first-order polarization and third-order topological charge number (A-R1V3 for short) is obtained when QP1(-45°), VHP1(3,0), QP2(-45°), and VHP2(1,0) are added in the setup. Accordingly, the A-R3V1 beam is generated for QP1(-45°), VHP1(1,0), QP2(-45°), and VHP2(3,0). Also, the anti-clockwise azimuthally polarized vortex (A-APV) beam with first-order polarization and third-order topological charge number (A-A1V3 for short) is obtained when QP1(-45°), VHP1(3,0), QP2(45°), and VHP2(1,0) are added in the setup. And the A-A3V1 is generated for QP1(-45°), VHP1(1,0), QP2(45°), and VHP2(3,0). The corresponding experimental results captured by the CCD are depicted in Figs. 10(a)–10(d). Figures 10(a1)–10(d2) are the intensity distributions after passing through a 0° and 45° linear polarizer respectively. Figures 10(a3)–10(d4) are the polarization parameters of PA and PE respectively. The experimental results demonstrate that the anti-clockwise CVV (A-CVV) beams are well generated. Compared with the CV beams in Figs. 46, there is a sense of clockwise rotation in the central part, either for the intensity distribution after a polarizer or for the PA distribution. However, it is hard to get the topological charge number except the polarization information. Besides, from the PE in Figs. 10(a4)–10(d4), it is found that the middle parts of the generated CVV beams are right-handed circularly-polarized beam, which comes from the ARCV beam.

 figure: Fig. 10.

Fig. 10. Experimental results of generated A-CVV beams. (a)–(d) Intensities of A-R1V3, A-R3V1, A-A1V3, and A-A3V1 beams; 1, intensity after a 0° polarizer; 2, intensity after a 45° polarizer; 3, PA; 4, PE. Arrow heads indicate the transmission direction of the polarizers.

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Similarly, the clockwise radially polarized vortex (C-RPV) beam with first-order polarization and third-order topological charge number (C-R1V3 for short) is obtained when QP1(45°), VHP1(3,0), QP2(-45°), and VHP2(1,0) are added in the setup. Accordingly, the C-R3V1 beam is generated for QP1(45°), VHP1(1,0), QP2(-45°), and VHP2(3,0). Also, the clockwise azimuthally polarized vortex (C-APV) beam with first-order polarization and third-order topological charge number (A-A1V3 for short) is obtained when QP1(45°), VHP1(3,0), QP2(45°), and VHP2(1,0) are added in the setup. And the C-A3V1 is generated for QP1(45°), VHP1(1,0), QP2(45°), and VHP2(3,0). The corresponding Stokes and polarization parameters of generated clockwise CVV (C-CVV) beams are depicted in Fig. 11. There is a sense of anti-clockwise rotation in the central part, either for the intensity distribution after a polarizer or for the PA distribution. And, from the PE in Figs. 11(a4)–11(d4), the middle parts of the generated CVV beams are left-handed circularly-polarized beam, which comes from the CLCV beam. Also, it is hard to get the topological charge number except the polarization information.

 figure: Fig. 11.

Fig. 11. Stokes parameters and polarization parameters of generated C-CVV beams. (a)–(d) C-R1V3, C-R3V1, C-A1V3, and C-A3V1 beams; 0–5: S0, S1, S2, S3, PA, PE.

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The double-slit interference experiment is also introduced to the CVV beams to help detection, and the corresponding experimental results of anti-clockwise and clockwise CVV beams are depicted in Fig. 12. Both misplacement and tilt appear, which have the same rules as CV beams in Fig. 9 and vortex beams in Fig. 8, respectively. The misplacement region number M is related to the polarization order number P, and the fringe tilt offset is related to the topological charge number l. The fringe intensity distributions of the same P and l but different types of CVV beams are the same. And A-CVV beams curve to the right and C-CVV beams curve to the left.

 figure: Fig. 12.

Fig. 12. Experimental results of double-slit interference of anti-clockwise and clockwise CVV beams. (a)–(d) A-R1V3 beam, A-A3V1 beam, C-A1V3 beam, and C-R3V1 beam.

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The topological charge number l is calculated by measuring the tilt offset. The specific measurement process is the same as that of vortex beam. As shown in Fig. 12, the tilt offset between two red vertical lines of A-R1V3 beam is three times of that of A-A3V1 beam, and C-A1V3 beam has three times the offset of C-R3V1, i.e., l=3 for A-R1V3, and l=3 for C-R3V1 beam.

5. CONCLUSION

In summary, we have demonstrated a concise efficient and practical direct-view method to generate arbitrary CV beams by VHP, including realizing CV beams, vortex beams and CVV beams. The characteristics of VHP are analyzed in detail. New types of VHPs can be formed by cascading two or more VHPs when the types of available VHPs are limited, which helps to generate multi-type CV beams more flexibly. Six kinds of first-order and other high-order CV beams, such as the RP beam, AP beam, and third-order AP beam, are formed by simply rotating an HP. The Stokes parameters and double-slit interference of multi-type CV beams are investigated in detail. The polarization parameters, including DOP, polarization azimuth, and ellipticity, are obtained, which demonstrates the efficient generation of CV beams.

In addition, misplacement and tilt appear in the double-slit interference fringes of the CVV beams, in which the misplacement number M is 2P+1 for P2 and 2P1 for P3, where P is the polarization order number, and the fringe tilt offset is positively related to the topological charge number l of CVV beams. It is experimentally demonstrated that arbitrary CV beams with high quality are effectively achieved by the proposed setup, and the double-slit interference method can be utilized to determine and analyze CV beams rapidly and concisely by practical performance, which shows its potential to be implemented as a commercial device.

Funding

National Natural Science Foundation of China (11704411, 11872070, 11975022, 12005002, 12072366, 62005317); National Magnetic Confinement Fusion Program of China (2018YFE0310400); Natural Science Foundation of Anhui Province (1908085MA26); Natural Science Foundation of Hunan Province (2019JJ40341).

Acknowledgment

The authors acknowledge assistance from Dr. Jie Xu and Dr. Ziyang Yuan.

Disclosures

The authors declare no conflicts of interest.

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78. J. Mendoza-Hernández, M. F. Ferrer-Garcia, J. A. Rojas-Santana, and D. Lopez-Mago, “Cylindrical vector beam generator using a two-element interferometer,” Opt. Express 27, 31810–31819 (2019). [CrossRef]  

79. W. Han, Y. Yang, W. Cheng, and Q. Zhan, “Vectorial optical field generator for the creation of arbitrarily complex fields,” Opt. Express 21, 20692–20706 (2013). [CrossRef]  

80. S. Fu, C. Gao, T. Wang, S. Zhang, and Y. Zhai, “Simultaneous generation of multiple perfect polarization vortices with selective spatial states in various diffraction orders,” Opt. Lett. 41, 5454–5457 (2016). [CrossRef]  

81. S. Fu, T. Wang, and C. Gao, “Generating perfect polarization vortices through encoding liquid-crystal display devices,” Appl. Opt. 55, 6501–6505 (2016). [CrossRef]  

82. S. Fu, C. Gao, T. Wang, Y. Zhai, and C. Yin, “Anisotropic polarization modulation for the production of arbitrary Poincaré beams,” J. Opt. Soc. Am. B 35, 1–7 (2018). [CrossRef]  

83. M. Bashkansky, D. Park, and F. K. Fatemi, “Azimuthally and radially polarized light with a nematic SLM,” Opt. Express 18, 212–217 (2010). [CrossRef]  

84. J. Qi, W. Sun, J. Liao, Y. Nie, X. Wang, J. Zhang, X. Liu, H. Jia, M. Lu, S. Chen, J. Liu, J. Yang, J. Tan, and X. Li, “Generation and analysis of both in-phase and out-phase radially polarized femtosecond-pulse beam,” Opt. Eng. 45, 024201 (2013). [CrossRef]  

85. Y. Zhang, P. Li, C. Ma, S. Liu, H. Cheng, L. Han, and J. Zhao, “Efficient generation of vector beams by calibrating the phase response of a spatial light modulator,” Appl. Opt. 56, 4956–4960 (2017). [CrossRef]  

86. S. Liu, S. Qi, Y. Zhang, P. Li, D. Wu, L. Han, and J. Zhao, “Highly efficient generation of arbitrary vector beams with tunable polarization, phase, and amplitude,” Photon. Res. 6, 228–233 (2018). [CrossRef]  

87. Y. Zhou, X. Li, Y. Cai, Y. Zhang, S. Yan, M. Zhou, M. Li, and B. Yao, “Compact optical module to generate arbitrary vector vortex beams,” Appl. Opt. 59, 8932–8938 (2020). [CrossRef]  

88. S. N. Khonina, A. V. Ustinov, and A. P. Porfirev, “Vector Lissajous laser beams,” Opt. Lett. 45, 4112–4115 (2020). [CrossRef]  

89. M. Rashid, O. M. Maragò, and P. H. Jones, “Focusing of high order cylindrical vector beams,” J. Opt. A 11, 065204 (2009). [CrossRef]  

90. S. N. Khonina, “Vortex beams with high-order cylindrical polarization: features of focal distributions,” Appl. Phys. B 125, 100 (2019). [CrossRef]  

91. L. Marrucci, C. Manzo, and D. Paparo, “Pancharatnam-Berry phase optical elements for wave front shaping in the visible domain: switchable helical mode generation,” Appl. Phys. Lett. 88, 221102 (2006). [CrossRef]  

92. E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett. 94, 231124 (2009). [CrossRef]  

93. S. Slussarenko, A. Murauski, T. Du, V. Chigrinov, L. Marrucci, and E. Santamato, “Tunable liquid crystal q-plates with arbitrary topological charge,” Opt. Express 19, 4085–4090 (2011). [CrossRef]  

94. A. Rubano, F. Cardano, B. Piccirillo, and L. Marrucci, “Q-plate technology: a progress review [Invited],” J. Opt. Soc. Am. B 36, D70–D87 (2019). [CrossRef]  

95. M. Beresna, M. Gecevicius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98, 201101 (2011). [CrossRef]  

96. M. Beresna, M. Gecevičius, and P. G. Kazansky, “Polarization sensitive elements fabricated by femtosecond laser nanostructuring of glass [Invited],” Opt. Mater. Express 1, 783–795 (2011). [CrossRef]  

97. D. Wu, Y. Li, W. Jia, J. Zhou, Y. Zhao, Y. Fu, and J. Wang, “Generation of arbitrary vector vortex beams based on the dual-modulation method,” Appl. Opt. 58, 1508–1513 (2019). [CrossRef]  

98. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006). [CrossRef]  

99. W. Shu, X. Ling, X. Fu, Y. Liu, Y. Ke, and H. Luo, “Polarization evolution of vector beams generated by q-plates,” Photon. Res. 5, 64–72 (2017). [CrossRef]  

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2020 (7)

2019 (12)

S. N. Khonina, “Vortex beams with high-order cylindrical polarization: features of focal distributions,” Appl. Phys. B 125, 100 (2019).
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A. Rubano, F. Cardano, B. Piccirillo, and L. Marrucci, “Q-plate technology: a progress review [Invited],” J. Opt. Soc. Am. B 36, D70–D87 (2019).
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D. Wu, Y. Li, W. Jia, J. Zhou, Y. Zhao, Y. Fu, and J. Wang, “Generation of arbitrary vector vortex beams based on the dual-modulation method,” Appl. Opt. 58, 1508–1513 (2019).
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J. Jia, Z. Chang, H. Yang, Q. Liu, F. Wang, H. Gao, F. Li, and P. Zhang, “Mode sorter designed for (de)multiplexing vector vortex modes,” Appl. Opt. 58, 7094–7099 (2019).
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J. Mendoza-Hernández, M. F. Ferrer-Garcia, J. A. Rojas-Santana, and D. Lopez-Mago, “Cylindrical vector beam generator using a two-element interferometer,” Opt. Express 27, 31810–31819 (2019).
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C. Zhu, Q. Jiao, X. Tan, W. Wang, and Bayanheshig, “Design of a subwavelength all-metal grating for generating azimuthally polarized beams based on modified particle swarm optimization,” Appl. Opt. 58, 4052–4058 (2019).
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X. Ma, S. Zheng, Q. Chen, S. Tan, P. Zhang, Q. Lu, J. Wang, and W. Guo, “High-speed directly modulated cylindrical vector beam lasers,” ACS Photon. 6, 3261–3270 (2019).
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S. Degtyarev, D. Savelyev, S. Khonina, and N. Kazanskiy, “Metasurfaces with continuous ridges for inverse energy flux generation,” Opt. Express 27, 15129–15135 (2019).
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S. A. Syubaev, A. Y. Zhizhchenko, D. V. Pavlov, S. O. Gurbatov, E. V. Pustovalov, A. P. Porfirev, S. N. Khonina, S. A. Kulinich, J. B. B. Rayappan, S. I. Kudryashov, and A. A. Kuchmizhak, “Plasmonic nanolenses produced by cylindrical vector beam printing for sensing applications,” Sci. Rep. 9, 19750 (2019).
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U. Levy, Y. Silberberg, and N. Davidson, “Mathematics of vectorial Gaussian beams,” Adv. Opt. Photon. 11, 828–891 (2019).
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H.-F. Xu, R. Zhang, Z.-Q. Sheng, and J. Qu, “Focus shaping of partially coherent radially polarized vortex beam with tunable topological charge,” Opt. Express 27, 23959–23969 (2019).
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S. Fu, C. Guo, G. Liu, Y. Li, H. Yin, Z. Li, and Z. Chen, “Spin-orbit optical Hall effect,” Phys. Rev. Lett. 123, 243904 (2019).
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2018 (10)

P. Shi, L. Du, and X. Yuan, “Structured spin angular momentum in highly focused cylindrical vector vortex beams for optical manipulation,” Opt. Express 26, 23449–23459 (2018).
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Y. Han, L. Chen, Y.-G. Liu, Z. Wang, H. Zhang, K. Yang, and K. C. Chou, “Orbital angular momentum transition of light using a cylindrical vector beam,” Opt. Lett. 43, 2146–2149 (2018).
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M. Li, Y. Cai, S. Yan, Y. Liang, P. Zhang, and B. Yao, “Orbit-induced localized spin angular momentum in strong focusing of optical vectorial vortex beams,” Phys. Rev. A 97, 053842 (2018).
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S. N. Khonina, A. V. Ustinov, and S. A. Degtyarev, “Inverse energy flux of focused radially polarized optical beams,” Phys. Rev. A 98, 043823 (2018).
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H.-F. Xu, Y. Zhou, H.-W. Wu, H.-J. Chen, Z.-Q. Sheng, and J. Qu, “Focus shaping of the radially polarized Laguerre-Gaussian-correlated Schell-model vortex beams,” Opt. Express 26, 20076–20088 (2018).
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J. Qi, H. Zhang, B. Pan, H. Deng, J. Yang, B. Shi, H. Wang, A. Du, W. Wang, and X. Li, “A succinct method to generate multi-type HCV beams with a spatial spiral varying retardation-plate,” Europhys. Lett. 121, 54004 (2018).
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S. N. Khonina, A. V. Ustinov, S. A. Fomchenkov, and A. P. Porfirev, “Formation of hybrid higher-order cylindrical vector beams using binary multi-sector phase plates,” Sci. Rep. 8, 14320 (2018).
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S. Fu, C. Gao, T. Wang, Y. Zhai, and C. Yin, “Anisotropic polarization modulation for the production of arbitrary Poincaré beams,” J. Opt. Soc. Am. B 35, 1–7 (2018).
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X. Xu, Y. Zhou, Y.-S. Yuan, J. Wang, H.-F. Xu, and J. Qu, “Generation of cylindrical and elliptical symmetrical vector beam on the Mach–Zehnder interferometer,” AIP Adv. 8, 125007 (2018).
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S. Liu, S. Qi, Y. Zhang, P. Li, D. Wu, L. Han, and J. Zhao, “Highly efficient generation of arbitrary vector beams with tunable polarization, phase, and amplitude,” Photon. Res. 6, 228–233 (2018).
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2017 (6)

2016 (9)

P. Li, B. Wang, and X. Zhang, “High-dimensional encoding based on classical nonseparability,” Opt. Express 24, 15143–15159 (2016).
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M.-Q. Cai, P.-P. Li, D. Feng, Y. Pan, S.-X. Qian, Y. Li, C. Tu, and H.-T. Wang, “Microstructures fabricated by dynamically controlled femtosecond patterned vector optical fields,” Opt. Lett. 41, 1474–1477 (2016).
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M. Neugebauer, P. Woźniak, A. Bag, G. Leuchs, and P. Banzer, “Polarization-controlled directional scattering for nanoscopic position sensing,” Nat. Commun. 7, 11286 (2016).
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F. Yue, D. Wen, J. Xin, B. D. Gerardot, J. Li, and X. Chen, “Vector vortex beam generation with a single plasmonic metasurface,” ACS Photon. 3, 1558–1563 (2016).
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W. Shu, Y. Liu, Y. Ke, X. Ling, Z. Liu, B. Huang, H. Luo, and X. Yin, “Propagation model for vector beams generated by metasurfaces,” Opt. Express 24, 21177–21189 (2016).
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D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photonics 10, 327–332 (2016).
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S. Fu, C. Gao, T. Wang, S. Zhang, and Y. Zhai, “Simultaneous generation of multiple perfect polarization vortices with selective spatial states in various diffraction orders,” Opt. Lett. 41, 5454–5457 (2016).
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S. Fu, T. Wang, and C. Gao, “Generating perfect polarization vortices through encoding liquid-crystal display devices,” Appl. Opt. 55, 6501–6505 (2016).
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P. Li, Y. Zhang, S. Liu, C. Ma, L. Han, H. Cheng, and J. Zhao, “Generation of perfect vectorial vortex beams,” Opt. Lett. 41, 2205–2208 (2016).
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2015 (9)

Z. Chen, T. Zeng, B. Qian, and J. Ding, “Complete shaping of optical vector beams,” Opt. Express 23, 17701–17710 (2015).
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P. Yu, S. Chen, J. Li, H. Cheng, Z. Li, W. Liu, B. Xie, Z. Liu, and J. Tian, “Generation of vector beams with arbitrary spatial variation of phase and linear polarization using plasmonic metasurfaces,” Opt. Lett. 40, 3229–3232 (2015).
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G. Milione, T. A. Nguyen, J. Leach, D. A. Nolan, and R. R. Alfano, “Using the nonseparability of vector beams to encode information for optical communication,” Opt. Lett. 40, 4887–4890 (2015).
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G. Milione, M. P. J. Lavery, H. Huang, Y. Ren, G. Xie, T. A. Nguyen, E. Karimi, L. Marrucci, D. A. Nolan, R. R. Alfano, and A. E. Willner, “4 × 20  gbit/s mode division multiplexing over free space using vector modes and a q-plate mode (de)multiplexer,” Opt. Lett. 40, 1980–1983 (2015).
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Y. Zhao and J. Wang, “High-base vector beam encoding/decoding for visible-light communications,” Opt. Lett. 40, 4843–4846 (2015).
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X.-L. Wang, X.-D. Cai, Z.-E. Su, M.-C. Chen, D. Wu, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516–519 (2015).
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Y. Kozawa and S. Sato, “Numerical analysis of resolution enhancement in laser scanning microscopy using a radially polarized beam,” Opt. Express 23, 2076–2084 (2015).
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S. Berg-Johansen, F. Töppel, B. Stiller, P. Banzer, M. Ornigotti, E. Giacobino, G. Leuchs, A. Aiello, and C. Marquardt, “Classically entangled optical beams for high-speed kinematic sensing,” Optica 2, 864–868 (2015).
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S. Roy, K. Ushakova, Q. van den Berg, S. F. Pereira, and H. P. Urbach, “Radially polarized light for detection and nanolocalization of dielectric particles on a planar substrate,” Phys. Rev. Lett. 114, 103903 (2015).
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2014 (5)

S. Segawa, Y. Kozawa, and S. Sato, “Resolution enhancement of confocal microscopy by subtraction method with vector beams,” Opt. Lett. 39, 3118–3121 (2014).
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J. Zhu, Y. Chen, Y. Zhang, X. Cai, and S. Yu, “Spin and orbital angular momentum and their conversion in cylindrical vector vortices,” Opt. Lett. 39, 4435–4438 (2014).
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Y. Liu, X. Ling, X. Yi, X. Zhou, H. Luo, and S. Wen, “Realization of polarization evolution on higher-order Poincaré sphere with metasurface,” Appl. Phys. Lett. 104, 191110 (2014).
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S. Vyas, Y. Kozawa, and S. Sato, “Generation of radially polarized Bessel–Gaussian beams from c-cut Nd:YVO4 laser,” Opt. Lett. 39, 1101–1104 (2014).
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A. Yanai, M. Grajower, G. M. Lerman, M. Hentschel, H. Giessen, and U. Levy, “Near- and far-field properties of plasmonic oligomers under radially and azimuthally polarized light excitation,” ACS Nano 8, 4969–4974 (2014).
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2013 (4)

S. A. Schulz, T. Machula, E. Karimi, and R. W. Boyd, “Integrated multi vector vortex beam generator,” Opt. Express 21, 16130–16141 (2013).
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O. M. Maragò, P. H. Jones, P. G. Gucciardi, G. Volpe, and A. C. Ferrari, “Optical trapping and manipulation of nanostructures,” Nat. Nanotechnol. 8, 807–819 (2013).
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W. Han, Y. Yang, W. Cheng, and Q. Zhan, “Vectorial optical field generator for the creation of arbitrarily complex fields,” Opt. Express 21, 20692–20706 (2013).
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J. Qi, W. Sun, J. Liao, Y. Nie, X. Wang, J. Zhang, X. Liu, H. Jia, M. Lu, S. Chen, J. Liu, J. Yang, J. Tan, and X. Li, “Generation and analysis of both in-phase and out-phase radially polarized femtosecond-pulse beam,” Opt. Eng. 45, 024201 (2013).
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2012 (1)

2011 (4)

2010 (5)

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Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1–57 (2009).
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E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett. 94, 231124 (2009).
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M. Rashid, O. M. Maragò, and P. H. Jones, “Focusing of high order cylindrical vector beams,” J. Opt. A 11, 065204 (2009).
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2008 (2)

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2006 (7)

L. Marrucci, C. Manzo, and D. Paparo, “Pancharatnam-Berry phase optical elements for wave front shaping in the visible domain: switchable helical mode generation,” Appl. Phys. Lett. 88, 221102 (2006).
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L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
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W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265, 411–417 (2006).
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K. J. Moh, X.-C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett. 89, 251114 (2006).
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K. Yonezawa, Y. Kozawa, and S. Sato, “Generation of a radially polarized laser beam by use of the birefringence of a c-cut Nd:YVO4 crystal,” Opt. Lett. 31, 2151–2153 (2006).
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V. G. Niziev, R. S. Chang, and A. V. Nesterov, “Generation of inhomogeneously polarized laser beams by use of a Sagnac interferometer,” Appl. Opt. 45, 8393–8399 (2006).
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2005 (2)

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2003 (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
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2002 (2)

C. Varin and M. Piché, “Acceleration of ultra-relativistic electrons using high-intensity TM01 laser beams,” Appl. Phys. B 74, s83–s88 (2002).
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Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings,” Opt. Lett. 27, 285–287 (2002).
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Banzer, P.

M. Neugebauer, P. Woźniak, A. Bag, G. Leuchs, and P. Banzer, “Polarization-controlled directional scattering for nanoscopic position sensing,” Nat. Commun. 7, 11286 (2016).
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S. Berg-Johansen, F. Töppel, B. Stiller, P. Banzer, M. Ornigotti, E. Giacobino, G. Leuchs, A. Aiello, and C. Marquardt, “Classically entangled optical beams for high-speed kinematic sensing,” Optica 2, 864–868 (2015).
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M. Beresna, M. Gecevicius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98, 201101 (2011).
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M. Beresna, M. Gecevičius, and P. G. Kazansky, “Polarization sensitive elements fabricated by femtosecond laser nanostructuring of glass [Invited],” Opt. Mater. Express 1, 783–795 (2011).
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K. J. Moh, X.-C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett. 89, 251114 (2006).
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K. J. Moh, X.-C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett. 89, 251114 (2006).
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X.-L. Wang, X.-D. Cai, Z.-E. Su, M.-C. Chen, D. Wu, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516–519 (2015).
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X. Ma, S. Zheng, Q. Chen, S. Tan, P. Zhang, Q. Lu, J. Wang, and W. Guo, “High-speed directly modulated cylindrical vector beam lasers,” ACS Photon. 6, 3261–3270 (2019).
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P. Yu, S. Chen, J. Li, H. Cheng, Z. Li, W. Liu, B. Xie, Z. Liu, and J. Tian, “Generation of vector beams with arbitrary spatial variation of phase and linear polarization using plasmonic metasurfaces,” Opt. Lett. 40, 3229–3232 (2015).
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J. Qi, W. Sun, J. Liao, Y. Nie, X. Wang, J. Zhang, X. Liu, H. Jia, M. Lu, S. Chen, J. Liu, J. Yang, J. Tan, and X. Li, “Generation and analysis of both in-phase and out-phase radially polarized femtosecond-pulse beam,” Opt. Eng. 45, 024201 (2013).
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W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265, 411–417 (2006).
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F. Yue, D. Wen, J. Xin, B. D. Gerardot, J. Li, and X. Chen, “Vector vortex beam generation with a single plasmonic metasurface,” ACS Photon. 3, 1558–1563 (2016).
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M. Beresna, M. Gecevicius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98, 201101 (2011).
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A. Yanai, M. Grajower, G. M. Lerman, M. Hentschel, H. Giessen, and U. Levy, “Near- and far-field properties of plasmonic oligomers under radially and azimuthally polarized light excitation,” ACS Nano 8, 4969–4974 (2014).
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Guo, H.

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X. Ma, S. Zheng, Q. Chen, S. Tan, P. Zhang, Q. Lu, J. Wang, and W. Guo, “High-speed directly modulated cylindrical vector beam lasers,” ACS Photon. 6, 3261–3270 (2019).
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A. Yanai, M. Grajower, G. M. Lerman, M. Hentschel, H. Giessen, and U. Levy, “Near- and far-field properties of plasmonic oligomers under radially and azimuthally polarized light excitation,” ACS Nano 8, 4969–4974 (2014).
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Figures (12)

Fig. 1.
Fig. 1. Fast axis distributions of several VHPs with σ=0. (a)–(d) are the distributions with m=1, 2, 3, 4, respectively.
Fig. 2.
Fig. 2. Polarization distributions of various one-order CVP beams. (a)–(f) correspond to the polarization distributions of ERP,EAP,Eπ/41,Eπ/41,EARP,EAAP.
Fig. 3.
Fig. 3. Direct-view experimental setup for the generation and double-slit interference of CV beams. Lenses 1 and 2 constitute a beam expander; HP, half-wave plate; GTP, Gran Taylor prism; QP, quarter-wave plate; VHP, vortex half-wave plate.
Fig. 4.
Fig. 4. Intensity distributions of generated first-order CV beams. (a)–(f) ERP,EAP,Eπ/41,Eπ/41,EARP,EAAP. Arrow heads represent the transmission direction of the polarizers.
Fig. 5.
Fig. 5. Polarization azimuth distributions of generated first-order CV beams. (a)–(f) ERP,EAP,Eπ/41,Eπ/41,EARP,EAAP. Arrow heads represent the polarization direction.
Fig. 6.
Fig. 6. Stokes parameters and polarization parameters of generated second-order and third-order CV beams. (a), (b) Second-order and third-order RP beams; (c), (d) second-order and third-order AP beams. 1–6: S1, S2, S3, DOP, PA, PE.
Fig. 7.
Fig. 7. Intensity and polarization distributions of generated first-order and third-order ARCV and CLCV beams. (a), (b) First-order and third-order ARCV beams; (c), (d) first-order and third-order CLCV beams; I0, intensity; ψ, PA; χ, PE.
Fig. 8.
Fig. 8. Experimental results of double-slit interference of first-order to third-order ARCV and CLCV beams. (a)–(c) First-order to third-order ARCV beams; (d)–(f) first-order to third-order CLCV beams; p1p3, horizontal pixel positions with equal transverse fringes at y=50; p4p6, corresponding horizontal pixel positions with equal transverse fringes at y=450.
Fig. 9.
Fig. 9. Experimental results of double-slit interference of first-order to third-order CV beams. (a)–(c) First-order RP beam; second-order π/4 beam; third-order AP beam.
Fig. 10.
Fig. 10. Experimental results of generated A-CVV beams. (a)–(d) Intensities of A-R1V3, A-R3V1, A-A1V3, and A-A3V1 beams; 1, intensity after a 0° polarizer; 2, intensity after a 45° polarizer; 3, PA; 4, PE. Arrow heads indicate the transmission direction of the polarizers.
Fig. 11.
Fig. 11. Stokes parameters and polarization parameters of generated C-CVV beams. (a)–(d) C-R1V3, C-R3V1, C-A1V3, and C-A3V1 beams; 0–5: S0, S1, S2, S3, PA, PE.
Fig. 12.
Fig. 12. Experimental results of double-slit interference of anti-clockwise and clockwise CVV beams. (a)–(d) A-R1V3 beam, A-A3V1 beam, C-A1V3 beam, and C-R3V1 beam.

Tables (1)

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Table 1. Measured Values of Pixel Positions and Topological Charge Number

Equations (24)

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θ=m2ψ+σ,
Jm,σ=[cos2θsin2θsin2θcos2θ]=[cos(mψ+2σ)sin(mψ+2σ)sin(mψ+2σ)cos(mψ+2σ)],
m2ΦΦ=ϕΦ=2ϕm2.
ERP=J1,0·E//=[cosψsinψsinψcosψ]·[10]=[cosψsinψ],
EAP=J1,π/4·E//=[cos(ψ+π/2)sin(ψ+π/2)]=[sinψcosψ],
EAP=J1,0·E=[sinψcosψ]=[cos(ψπ/2)sin(ψπ/2)]=eiπ·EAP,
Eπ/41=J1,π/8·E//=[cos(ψ+π/4)sin(ψ+π/4)].
Eπ/41=J1,π/8·E//=[cos(ψπ/4)sin(ψπ/4)]=eiπ·[cos(ψ+3π/4)sin(ψ+3π/4)]=eiπ·J1,3π/8·E//.
Eαm=Jm,0·Eα=[cos(mψ)sin(mψ)sin(mψ)cos(mψ)]·[cosαsinα]=[cos(mψα)sin(mψα)].
EARP=J0·ERP=[1001]·[cosψsinψ]=[cos(ψ)sin(ψ)].
EAAP=J0·EAP=Jπ/4·ERP=[0110]·[cosψsinψ]=[sinψcosψ]=[cos(ψ+π/2)sin(ψ+π/2)].
Jθ(λ/4)=22[1icos2θisin2θisin2θ1+icos2θ].
Jm,σ·EL=22[cos2θ+isin2θsin2θicos2θ]=22[ei2θei(2θπ/2)]=22[1i]·ei(mψ+2σ)=ER·ei(mψ+2σ)=EARCV,
Jm,σ·ER=22[cos2θisin2θsin2θ+icos2θ]=22[ei(2θ)ei(2θ+π/2)]=22[1i]·ei(mψ+2σ)=EL·ei(mψ+2σ)=ECLCV,
Jπ/4(λ/4)·EARCV=[0i]·ei(mϕ+2σ)=E·ei(mψ+2σπ/2)=EAVLV,Jπ/4(λ/4)·EARCV=[10]·ei(mϕ+2σ)=E//·ei(mψ+2σ)=EAHLV.
Jπ/4(λ/4)·ECLCV=[10]·ei(mϕ+2σ)=E//·ei(mψ+2σ)=ECHLV,Jπ/4(λ/4)·ECLCV=[0i]·ei(mϕ+2σ)=E·ei(mψ+2σπ/2)=ECVLV.
Jm2,σ2·EAHLV=[cos(m2ψ+2σ2)sin(m2ψ+2σ2)]·ei(m1ψ+2σ1)=ERm2Vm1,Jm2,σ2·EAVLV=[sin(m2ψ+2σ2)cos(m2ψ+2σ2)]·ei(m1ψ+2σ1π/2)=EAm2Vm1.
Jm2,σ2·ECHLV=[cos(m2ψ+2σ2)sin(m2ψ+2σ2)]·ei(m1ψ+2σ1)=ERm2Vm1,Jm2,σ2·ECVLV=[sin(m2ψ+2σ2)cos(m2ψ+2σ2)]·ei(m1ψ+2σ1π/2)=EAm2Vm1.
J2·J1=Jm2,σ2·Jm1,σ1=[cosα2sinα2sinα2cosα2]·[cosα1sinα1sinα1cosα1]=[cos[(m2m1)ψ+2(σ2σ1)]sin[(m2m1)ψ+2(σ2σ1)]sin[(m2m1)ψ+2(σ2σ1)]cos[(m2m1)ψ+2(σ2σ1)]],
(J2·J1)·J0=[cos(α2α1)sin(α2α1)sin(α2α1)cos(α2α1)]·[cos0sin0sin0cos0]=[cos(α2α1+0)sin(α2α1+0)sin(α2α1+0)cos(α2α1+0)].
J3·(J2·J1)=[cos(α3α2+α1)sin(α3α2+α1)sin(α3α2+α1)cos(α3α2+α1)].
{S0=Ix+Iy=Iπ/4+Iπ/4=IR+IL=I0S1=IxIy=S02Iy=2IxS0S2=Iπ/4Iπ/4=S02Iπ/4=2Iπ/4S0S3=IRIL=S02IL=2IRS0,
{p=S12+S22+S32S0ψ=12arctan(S2/S1)χ=12arcsinS3S12+S22+S32,
{l=Ri/R1,l>0l=Ri/|R1|,l<0.