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Abstract
A practical direct-view scheme for generating arbitrary high-order cylindrical vector (HCV) beams by cascading vortex half-wave plates (VHPs) is presented. The combination of odd number 2n-1 VHPs for n≥1 can realize (m2n-1-m2n-2+…+m1)-order CV beams, in which m is the order number of VHP and the corresponding subscript 2n-1 represents the arrangement number of VHPs, and the cascading of even number 2n ones can obtain (m2n-m2n-1+…+m2-m1)-order CV beams. All 1-12 order CV beams, including the high-order anti-vortex CV (ACV) beams, are generated only by selectively cascading the VHPs with m=1, 3 and 8. The polarization properties of the generated HCV beams are investigated by measuring the corresponding Stokes parameters. It is experimentally demonstrated that arbitrary HCV beams are effectively achieved by the proposed method. The order numbers of CV beams can be greatly expanded by cascading limited types of VHPs.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The cylindrical vector (CV) beams have attracted widespread attention due to a doughnut intensity distributions and cylindrically symmetric polarization properties [1,2]. For example, the radially polarized (RP) and azimuthally polarized (AP) beam as the typical representatives of CV beams, have been found important applications in the fields such as tight focusing [3,4], beam shaping [5,6], particle accelerating and trapping [7–9], laser materials processing [10,11], super-resolution techniques [12,13], optical communication [14,15], quantum information process [16,17], spin and orbital angular momentum effects [18–20], reverse flow [21,22], laser structuring [23,24] and fiber and integrated optics [25–28], and so on.
Various methods and schemes have been proposed to generate CV beams, including intra-cavity techniques such as the use of crystal birefringence [29,30], Brewster angle characteristics [31,32], cavity configuration design [33] and geometric phase control [34], and extra-cavity conversion devices such as spiral phase plates or segmented spiral varying retarders [35–37], anisotropic crystals [38,39], diffractive optical elements [40,41], subwavelength gratings [42,43], metasurfaces [44–47], Sagnac and Mach–Zehnder interferometers [48–51], and most used spatial light modulator (SLM) [52–56].
The intra-cavity methods have the advantages of high quality and energy conversion efficiency but lack of flexibility due to limited space. The extra-cavity conversion methods are relatively more flexible, but there are more or less problems of stability, complexity, efficiency, cost performance, and so on. For example, the interferometric methods often need high stability and precise control, and the SLM methods have low conversion efficiency due to the diffraction effect and are not cost-effective. Besides, relatively less researches are reported on the generation [51,57–59] of HCV beams, but which have highly significant applications [60–62]. Although the HCV beam can be realized by a directly designed converter with a determined order [63], when large numbers of CV beams with different orders are needed, such as in optical communication systems, the manufacturing cost will be significantly increased. Therefore, it is highly desirable to generate HCV beams with adjustable orders. Although a similar idea has been mentioned in the Ref. [64], it is composed of low-order converters such as first-order and second-order ones. As it is said, the conversion efficiency of low-order converter combination is lower. In addition, too many half-wave plates (HPs) are needed to connect the converters for realizing higher-order CV beams, which increases the design cost and reduces the flexibility. Moreover, the generation mechanism lacks a more general description and explanation. For example, it is only a special case for the expression “Mm=Mm-nHMn; M-m=HMmH”, where Mm and H represent the Jones matrix of a m-order CV beam converter and a HP, respectively.
In this paper, a practical and concise direct-view scheme is proposed to generate arbitrary high-order cylindrical vector (HCV) beams by cascading vortex half-wave plates (VHPs). The VHP is a special HP with a consistent delay π but the direction of the fast axis changing continuously around the center, which has high transmittance just like a HP thus with high energy conversion. The function of VHP is similar to the general q-plate [65–67] and S-waveplate [68,69], which is a kind of metasurfaces. Generally, the q-plate is a thin (nematic) liquid crystal film sandwiched between two coated flat glasses, which requires additional electric or temperature control. And the S-waveplate is nano subwavelength periodic structure usually written inside a glass plate by femtosecond laser, behaving as a uniaxial crystal whose optical axis is parallel and perpendicular to the subwavelength grooves. The q-plate generally is an active device needing additional electric or temperature control, which increases costs and is not easy to operate and use because of control lines. And the S-waveplate involves micro fabrication, which suffers from high cost of precise design and machining accuracy, especially difficult for high order ones. These devices are characterized by singular optical axis distributions with topological charge q, generally known as q-plates [70,71].
A general mechanism is deduced in detail. The (m2n-1-m2n-2+…+m1)-order CV beams can be obtained by cascading odd number 2n-1 VHPs for n≥1, where m is the VHP order number and the corresponding subscript 2n-1denotes the arrangement number of VHPs. And correspondingly, even number 2n cascaded VHPs can realize (m2n-m2n-1+…+m2-m1)-order CV beams. The 1-12 order CV beams, including partial anti-vortex CV (ACV) beams, are formed by selectively cascading the VHPs with m=1, 3 and 8. The polarization properties of the HCV beams are investigated in detail, and the polarization azimuth are obtained by measuring the corresponding Stokes parameters. It is experimentally demonstrated that arbitrary HCV beams are effectively realized by the presented scheme. On the basis of the existing experimental setup, up to 36-order CV beam can be realized if there are another three VHPs with m=8. The order numbers of CV beams can be greatly expanded by cascading limited types of VHPs, which is contribute to reduce costs and improve utilization.
2. Principles
2.1 Vortex half-wave plate
The vortex half-wave plate (VHP) is a kind of retarder with a constant retardance π, but its fast axis rotates continuously around the center. The distribution of the fast axis can be described as the following:
In which, φ is the VHP azimuth angle; θ is the VHP fast axis direction at a certain azimuth angle; m is the order number, which is an integer; and φ0 is the fast axis direction when φ=0. The VHP with m and φ0 will be expresses as VHP(m, φ0) for short in the following. Especially when m = 0, the VHP becomes an ordinary half-wave plate (HP) with the fast axis orientated at φ0, also called φ0 HP, i.e. VHP(0, φ0). The VHP Jones matrix ${J_{m,{\varphi _\textrm{0}}}}$ with fast axis at the direction θ can be expressed as,
The VHP(m, φ0) can be converted to the VHP(m, 0) (except for m = 2) by rotating an angle Φ. The conversion process of 1-order and 4-order VHPs are depicted in Fig. 1. The arrows represent the fast axis distributions of VHPs, and the mark “0” means the fast axis orientates to azimuth direction. For the VHP(1, φ0), the rotation angle Φ=−2φ0 and for VHP(4, φ0), the rotation angle Φ=φ0. The rotation angle Φ for any VHP(m, φ0) converting to the VHP(m, 0) can be expressed as follows:
Fig. 1. The conversion process of 1-order and 4-order VHPs. (a1)-(b2) represent VHP(1, φ0), VHP(1, 0), VHP(4, φ0) and VHP(4, 0), respectively. Arrows represent the fast axis distributions.
When Φ is a negative value, it represents a clockwise rotation, and a positive value devotes a counter-clockwise rotation.
2.2 Generation of arbitrary cylindrical vector beams
When a linearly polarized (LP) beam with the polarization direction orientated at α according to the horizontal direction (x axis), expresses as the α LP beam for short, passes through the VHP(m, φ0), arbitrary CV beam can be obtained with the Jones vector expressed as following,
This means that each polarization direction is rotated by 2φ0-α counter-clockwise on the basis of the m-order radial polarization. Thus, changing φ0 by rotating VHP or incident polarization direction α can realize multi-type CV beams. According to Eq. (3), 2△φ0, the change value of 2φ0, could be achieved by rotating the VHP Φ=2△φ0/(m-2), which needs higher precision control for high-order m and cannot realize multi type for m=2. So, fixing the φ0 and changing α by rotating a half-wave plate (HP) is a better solution. For the convenience of research, φ0 = 0 is selected. And the Jones vector of obtained CV beam can be described as $E_{ - \alpha }^m = {J_{m,0}}\cdot {E_\alpha } = {\left[ {\begin{array}{cc} {\cos (m\varphi - \alpha )}&{\sin (m\varphi - \alpha )} \end{array}} \right]^T}$. The superscript m in $E_{ - \alpha }^m$ represents polarization order number P = m and the subscript -α denotes the polarization rotation angle around the m-order radial polarization direction. It means m-order radially polarized (RP) beam when α=0, and for α=π/2 or α=-π/2, it is m-order azimuthally polarized (AP) beam.
2.3 Cascading of VHPs
For the VHP that has been manufactured, the order number m is determined. Only a few CV beams of fixed order can be obtained by using limited types of VHPs alone. However arbitrary high-order CV beams can be generated by cascading two or more VHPs. The combined Jones matrix of two VHPs with φ0 = 0 can be expressed as:
That is, each polarization direction is rotated by α counter-clockwise on the basis of the (m2-m1)-order radial polarization. The (m2-m1)-order RP beam can be generated when the horizontally LP (HLP) beam passes through the combined device. And (m2-m1)-order AP beam can be obtained when passed through by the vertically LP (VLP) beam. When m2-m1<0, the generated CV beam is called |m2-m1|-order anti-vortex CV (ACV) beam [72], such as the anti-vortex RP (ARP) beam and anti-vortex AP (ARP) for m2-m1=−1. Especially m1-order ACV beam is obtained when m2=0. Or they can be collectively called (m2-m1)-order CV beam, in which (m2-m1) < 0 refers to ACV beam.
The combined Jones matrix of three VHPs with φ0 = 0 can be expressed as:
It can be found from the expression that the cascaded device forms the new VHP with the order number M3=m3-r2=m3-m2+m1. That is to say three cascaded VHPs form the new VHP(M3, 0). Especially when one of them is a 0° HP, i.e., VHP (0,0), and the other two are VHPs (m1,0) and VHP (m2,0), changing their front and back positions can generate many other kinds of CV beams such as (m2±m1)-order and (m1-m2)-order CV beams.
Correspondingly, four cascaded VHPs produce new rotation function with the Jones matrix ${R_4} = {J_4}\cdot J_3^{\prime}$ and rotation order number r4=m4-M3=m4-m3+m2-m1. And five cascaded VHPs form a new VHP with the Jones matrix $J_5^{\prime} = {J_5}\cdot {R_4}$ and order number M5=m5-r4=m5-m4+m3-m2+m1.
By analogy, the cascading of even number 2n VHPs for n≥1 can generate r2n-order CV beam with r2n=m2n-m2n-1+…+m2-m1, and the combination of odd number 2n-1 VHPs can form the new VHP with M2n-1=m2n-1-m2n-2+…+m1. Therefore, arbitrary high-order CV beams can be realized by directly cascading limited VHPs without the connection of other devices.
3. Experimental setup
The direct-view experimental setup for generation of arbitrary high-order cylindrical vector (HCV) beams by cascading Vortex Half-wave Plate (VHP) is shown in Fig. 2. The incident beam from a linearly polarized He-Ne laser is expanded six times by two well aligned lenses. The combination of half-wave plate 1 (HP1) and Gran Taylor prism (GTP) can both modulate the incident beam intensity and ensure the incident beam is horizontally linearly polarized. And the HP2 is used to change the linearly polarized direction for generating more types of CV beams, such as the RP beam and AP beam. The VHP, composed of single or cascaded VHP in the black dashed line, is modulated to φ0 = 0 and can convert the linearly polarized beam into arbitrary HCV beams. The intensity distribution of the generated beam is captured by a 12-bit CCD camera with the resolution of 1024×1024 pixels and pixel pitch 5.5×5.5 µm.
Fig. 2. Experimental setup for the direct-view generation of arbitrary HCV beams. Lenses 1 and 2 constitute a beam expander; HP: half-wave plate; GTP: Gran Taylor prism; VHP: vortex half-wave plate.
The VHP in the experimental setup is made of a liquid crystal polymer, similar to nematic liquid crystals, which requires the polymer molecules to be aligned. To accomplish this, an alignment layer is created by coating a substrate in photo-alignment material and exposing it to polarized laser light. A 20 to 30 nm layer of photo-alignment material is deposited on a glass substrate by spin coating. The coated substrate is then placed in the photo-alignment system for alignment, where it is rotated at a certain speed while being exposed to a linearly polarized line of light. The molecules in the coating align with the polarization axis of the incident line of light. The center of rotation for the substrate is positioned on the light line to minimize center misalignment.
In order to get the spatial polarization distributions of generated beams, Stokes parameters S0, S1, and S2 are measured by placing a polarizer in the red dashed line before the CCD in the following equation:
4. Results and discussion
In our experiment, the cascading of VHPs with m=1, 3 and 8 are used to generate 1-12 order CV beams, including the corresponding anti-vortex CV (ACV) beams. Figure 3 shows the generated RP, 3-order AP (3-AP for short) and 8-order RP (8-RP for short) beam by a single VHP(m, 0) with m=1, 3 and 8 respectively. For the RP and 8-RP beam, the incident beam is horizontally linearly polarized (HLP) with the fast axis of HP2 orientated at 0° direction (or the HP2 removed). And the incident beam is vertically linearly polarized (VLP) with the fast axis of HP2 orientated at −45° direction for the 3-AP beam. More types of CV beams can be obtained when rotating the HP2. Figures 3(a1)-(c1) are the whole intensity distributions of RP, 3-AP and 8-RP beams respectively, i.e. I0. And Figs. 3(a2)-(c4) are the intensity distributions after passing through a linear polarizer with the polarization direction orientated at 0°, 45° and 90° respectively, i.e. Ix, Iπ/4 and Iy. All intensity distributions of I0, Ix, Iπ/4 and Iy are normalized for mutual comparison. The corresponding polarization azimuth (PA) distributions are depicted in Figs. 3(a5)-(c5).
Fig. 3. Intensity and polarization azimuth distributions of generated RP, 3-AP and 8-RP beams. (a)-(c): RP, 3-AP and 8-RP beam; 1-5: I0, Ix, Iπ/4, Iy and PA. Two-way arrowheads indicate the transmission direction of the polarizer P.
Polarization rotation direction (PRD) marked by dark arrowheads in a white circle and spatial polarization direction (SPD) marked by one-way arrowheads are displayed in Fig. 3(a5) and Fig. 3(b5) to illustrate the polarization information in detail. The PRDs of RP and 3-AP beams are both counterclockwise, i.e., the SPD rotates counterclockwise along counterclockwise direction. The horizontal and vertical SPD are marked by gray and yellow one-way arrowheads respectively. And the 45° (or 225°) and −45° (or 135°) SPD are marked by black and white one-way arrowheads. The color of 45° and 225° PA in the color bar is the same, so is the −45° and 135° PA. As shown in Fig. 3(b5), the SPD of 3-AP beam is vertical in the horizontal direction. It is obvious that the SPD of 3-AP beam is rotated by 90° on the basis of 3-RP beam. The experimental results show that the high-order CV beams with high quality are well generated.
Other HCV beams can be obtained when the cascaded VHPs with m=0, 1, 3 and 8 are placed in the black dashed line in the experimental setup. The VHP (0, 0) is actually a half-wave plate (HP) with fast axis orientated at 0° direction, i.e., 0° HP. The cascading of two VHPs with m = m1 and m2 can generate (m2-m1)-order CV beams. The 2-RP beam shown in Fig. 4. (a1)-(a5) is obtained when the incident HLP beam passes through the VHP(1, 0) and VHP(3, 0) in sequence. Especially ACV beams can be obtained when m2=0. The 3-order anti-vortex RP (3-ARP or -3-RP for short) beam, depicted in Figs. 4(b1)-(b5), is generated when the incident HLP beam passes through the VHP(3, 0) and 0° HP in sequence. The PRD and partial SPD are both displayed in Figs. 4(a5) and Fig. 4(b5). It is obvious that the PRDs of RP and ARP beams are opposite. The PRD of 3-ARP beam is clockwise, i.e., the SPD rotates clockwise along counterclockwise direction. When the incident HLP beam passes through the VHP(3, 0) and VHP(8, 0) in sequence, the 5-RP beam is realized shown in Figs. 4(c1)-(c5).
Fig. 4. Intensity and polarization azimuth distributions of generated 2-RP, 3-ARP and 5-RP beams. (a)-(c): 2-RP, 3-ARP and 5-RP beam; 1-5: I0, Ix, Iπ/4, Iy and PA. Two-way arrowheads represent the transmission direction of the polarizer P.
The odd number 2n+1 cascaded VHPs can realize (m2n+1-m2n+…+m1)-order CV beams. The 6-RP beam is obtained when the incident HLP beam passes through the VHP(1, 0), VHP(3, 0) and VHP(8, 0) in sequence. Especially when m2n=0, the relative maximum-order CV beams can be realized. The 11-RP beam is generated when the HLP beam passes through the VHP(3, 0), 0° HP and VHP(8, 0) in sequence. And when the HLP beam passes through the VHP(1, 0), 0° HP, VHP(3, 0), 0° HP and VHP(8, 0) in sequence, 12-RP beam is obtained. The generated 6-RP, 11-RP and 12-RP beams are shown in Fig. 5. Figures 5(a1)-(c3) are the Stokes parameters S1, S2 and S0 distributions of 6-RP, 11-RP and 12-RP beams respectively. And Figs. 5(a4)-(c4) show the corresponding PA distributions.
Fig. 5. Stokes parameters and polarization azimuth of generated 6-order, 11-order and 12-order RP beams. (a)-(c): 6-RP, 11-RP and 12-RP beam; 1-4: S1, S2, S0 and PA.
Also, other-order CV beams, such as 4-RP, 7-RP, 9-RP and 10-RP, are generated shown in Fig. 6. They are realized by arranging the cascaded VHPs in the expression (3-0 + 1), (8-1), (8-0 + 1) and (8-1 + 3) respectively.
Fig. 6. Intensity and polarization distributions of generated 4-RP, 7-RP, 9-RP and 10-RP beams. 1-2: I0 and PA; (a)-(d): 4-RP, 7-RP, 9-RP and 10-RP beam.
It is easy to find from Figs. 3–6 that the higher the order is, the larger the central hollow area of the generated HCV beam is. And the polarization direction for even-order RP beams in both horizontal and vertical directions is horizontally polarized. While for odd-order RP beams, the polarization direction in vertical direction is vertically polarized. Based on the existing experimental setup, all 1-20 order CV beams can be generated when there is another VHP (8, 0), including both positive and negative orders. And if there are another three ones, up to 36-order CV beams can be realized. There may be a little difference between the theoretical and experimental results due to the manufacturing precision of VHP, the arrangement accuracy of VHPs and the imaging quality of CCD. Higher-order CV beams can be obtained when cascading other types or more fixed types of HVPs.
5. Conclusion
In summary, we have demonstrated a practical direct-view scheme to generate arbitrary high-order cylindrical vector (HCV) beams by cascading vortex half-wave plates (VHPs). The principle and application of VHP are introduced in detail. The odd number 2n-1 cascaded VHPs for n≥1 can realize (m2n-1-m2n-2+…+m1)-order CV beams, in which m is the order number and the corresponding subscript 2n-1represents the arrangement number of VHPs, and (m2n-m2n-1+…+m2-m1)-order CV beams can be obtained by cascading even number 2n VHPs. 1-12 order CV beams, including the high-order anti-vortex CV (ACV) beams, are generated only by selectively cascading the VHP(1, 0), VHP(3, 0) and VHP(8, 0). The polarization properties of the HCV beams are investigated in detail, and the polarization azimuth are obtained by measuring the corresponding Stokes parameters. The polarization direction in vertical direction is horizontally polarized for even-order RP beams, while vertically polarized for odd-order RP beams. On the basis of the existing experimental scheme, all 1-20 order CV beams can be generated when there is another VHP (8, 0), and up to 36-order CV beam can be realized if there are another three ones. The order numbers of CV beams can be greatly expanded by cascading other types or more fixed types of VHPs, which is contribute to reduce costs and improve utilization.
Funding
Natural Science Foundation of Hunan Province (2019JJ40341); Natural Science Foundation of Anhui Province (1908085MA26); National Magnetic Confinement Fusion Program of China (2018YFE0310400); National Natural Science Foundation of China (11704411, 11872070, 11975022, 12005002, 12072366, 62005207, 62005317).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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