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 [3–5], beam shaping [6–9], particle accelerating and trapping [10–14], laser materials processing [15–17], superresolution techniques [18–20], and optical metrology [21–23]. Recent studies have also demonstrated potential applications of CV beams in optical communication [24–26], quantum information processing [27–29], spin and orbital angular momentum effects [30–34], inverse energy flux [35,36], plasmonic nanostructures [37,38], and fiber and integrated optics [39–42].

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 [43–45], 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 [50–52], special phase plates [53,54], anisotropic crystals [55–58], diffractive optical elements [59,60], subwavelength gratings [61–63], and metasurfaces [64–67]. 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 [68–71], Mach–Zehnder interferometers [72,73], and other interferometers [74–78]. 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 [79–82] and split-screen method by a single SLM [83–88]. 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 P≤2 and 2P−1 for P≥3, 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.

The VHP Jones matrix Jm,σ with fast axis at the direction θ can be expressed as Jm,σ=[cos2θsin2θsin2θ−cos2θ]=[cos(mψ+2σ)sin(mψ+2σ)sin(mψ+2σ)−cos(mψ+2σ)],(2)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: m2Φ−Φ=ϕ⇒Φ=2ϕm−2.(3)

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.

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//=[cosψsinψsinψ−cosψ]·[10]=[cosψsinψ],(4)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],(5)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⊥=[sinψ−cosψ]=[cos(ψ−π/2)sin(ψ−π/2)]=eiπ·EAP,(6)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)].(7)

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, E−π/41 can be obtained by inputting a horizontally polarized beam for the VHP(1, −π/8) or VHP(1, 3π/8) as the following: $$\begin{gathered} E−π/41=J1,−π/8·E//=[cos(ψ−π/4)sin(ψ−π/4)] \\ =eiπ·[cos(ψ+3π/4)sin(ψ+3π/4)]=eiπ·J1,3π/8·E//. \end{gathered}$$(8)

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(m−2)]. Obviously, the larger the order m (for m≥3) 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: $$\begin{gathered} E−αm=Jm,0·Eα=[cos(mψ)sin(mψ)sin(mψ)−cos(mψ)]·[cosαsinα] \\ =[cos(mψ−α)sin(mψ−α)]. \end{gathered}$$(9)

This means that each polarization direction is rotated by −α anti-clockwise on the basis of the radial polarization. The superscript m in E−αm represents polarization order number P=m of the CV beams. So, the ERP,EAP,Eπ/4m,E−π/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=[100−1]·[cosψsinψ]=[cos(−ψ)sin(−ψ)].(10)

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 $$\begin{gathered} EAAP=J0·EAP′=Jπ/4·ERP=[0110]·[cosψsinψ] \\ =[sinψcosψ]=[cos(−ψ+π/2)sin(−ψ+π/2)]. \end{gathered}$$(11)

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 high-order CV beams will be obtained when a linearly polarized beam passes through the VHP with m≥2.

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[1−icos2θ−isin2θ−isin2θ1+icos2θ].(12)

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[1−i]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: $$\begin{gathered} Jm,σ·EL=22[cos2θ+isin2θsin2θ−icos2θ]=22[ei2θei(2θ−π/2)] \\ =22[1−i]·ei(mψ+2σ)=ER·ei(mψ+2σ)=EARCV, \end{gathered}$$(13)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: $$\begin{gathered} Jm,σ·ER=22[cos2θ−isin2θsin2θ+icos2θ]=22[ei(−2θ)ei(−2θ+π/2)] \\ =22[1i]·e−i(mψ+2σ)=EL·e−i(mψ+2σ)=ECLCV, \end{gathered}$$(14)which is the clockwise vortex beam with left-handed circular polarization, i.e., clockwise left-handed circularly polarized vortex (CLCV) beam.

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: $$\begin{gathered} Jπ/4(λ/4)·EARCV=[0−i]·ei(mϕ+2σ)=E⊥·ei(mψ+2σ−π/2)=EAVLV, \\ J−π/4(λ/4)·EARCV=[10]·ei(mϕ+2σ)=E//·ei(mψ+2σ)=EAHLV. \end{gathered}$$(15)

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: $$\begin{gathered} Jπ/4(λ/4)·ECLCV=[10]·e−i(mϕ+2σ)=E//·e−i(mψ+2σ)=ECHLV, \\ J−π/4(λ/4)·ECLCV=[0i]·e−i(mϕ+2σ)=E⊥·e−i(mψ+2σ−π/2)=ECVLV. \end{gathered}$$(16)

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: $$\begin{gathered} Jm2,σ2·EAHLV=[cos(m2ψ+2σ2)sin(m2ψ+2σ2)]·ei(m1ψ+2σ1)=ERm2Vm1, \\ Jm2,σ2·EAVLV=[sin(m2ψ+2σ2)−cos(m2ψ+2σ2)]·ei(m1ψ+2σ1−π/2)=EAm2Vm1. \end{gathered}$$(17)

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: $$\begin{gathered} Jm2,σ2·ECHLV=[cos(m2ψ+2σ2)sin(m2ψ+2σ2)]·e−i(m1ψ+2σ1)=ERm2Vm1′, \\ Jm2,σ2·ECVLV=[sin(m2ψ+2σ2)−cos(m2ψ+2σ2)]·e−i(m1ψ+2σ1−π/2)=EAm2Vm1′. \end{gathered}$$(18)

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 $$\begin{gathered} J2·J1=Jm2,σ2·Jm1,σ1=[cosα2sinα2sinα2−cosα2]·[cosα1sinα1sinα1−cosα1] \\ =[cos[(m2−m1)ψ+2(σ2−σ1)]−sin[(m2−m1)ψ+2(σ2−σ1)]sin[(m2−m1)ψ+2(σ2−σ1)]cos[(m2−m1)ψ+2(σ2−σ1)]], \end{gathered}$$(19)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 (m2−m1)ψ+2(σ2−σ1) 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 $$\begin{gathered} (J2·J1)·J0=[cos(α2−α1)−sin(α2−α1)sin(α2−α1)cos(α2−α1)]·[cos0sin0sin0−cos0] \\ =[cos(α2−α1+0)sin(α2−α1+0)sin(α2−α1+0)−cos(α2−α1+0)]. \end{gathered}$$(20)

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(α3−α2+α1)sin(α3−α2+α1)sin(α3−α2+α1)−cos(α3−α2+α1)].(21)

It can be found from the expression that the cascaded device forms the new VHP with m=m3−m2+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=m5−m4+m3−m2+m1. By analogy, more types of VHPs can be generated.

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 [91–94] and S-wave plate [95–97]. 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].

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+I−π/4=IR+IL=I0S1=Ix−Iy=S0−2Iy=2Ix−S0S2=Iπ/4−I−π/4=S0−2I−π/4=2Iπ/4−S0S3=IR−IL=S0−2IL=2IR−S0,(22)where I0 is the total transverse intensity of the beam. Ix, Iy, Iπ/4, and I−π/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,(23)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 specific tilt offsets are measured by pixel positions p1−p6 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=(p3−p1+p6−p4)/(2N), in which N is the fringe number between p1 (or p4) and p3 (or p6). The fringe offset d is obtained by d=(p4−p1+p5−p2+p6−p3)/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…).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 |R−1| for l<0) is close to the topological charge l, which could be expressed as follows: {l=Ri/R1,l>0l=Ri/|R−1|,l<0.(24)

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.

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 P≥3. 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.

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.

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 P≤2 and 2P−1 for P≥3, 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.

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