Abstract
1. INTRODUCTION
It is known that angular momentum of light has a spin part associated with polarization [1] and an orbital part associated with spatial distribution [2]. A cylindrical vector beam (CVB) [3–5], acting as a solution of the vectorial Helmholtz equation [6,7], combines the two parts of angular momentum. Radial polarization and azimuthal polarization are the most conspicuous CVBs. Under tight focusing [8–10], radial polarization possesses a sharper focal spot than a homogeneously polarized beam [11,12], while azimuthal polarization can be focused into a hollow spot [13]. These peculiar properties are useful for many applications, such as particle manipulation [14–18], microscopy [19–21], material processing [22–24], near-field optics [25], and nonlinear optics [26]. Recently, the degrees of freedom of CVB are extended by ray-like trajectories [27], so that CVBs also present growing potential in the area of optical encoding [28] and optical communications [29–32].
There are many methods to generate CVBs. Special intracavity resonators could directly generate CVBs from a laser when the cavity geometry is precisely controlled into a frequency-degenerate state [33–37]. Meanwhile, single-element CVB generators have been introduced to tailor Pancharatnam–Berry phases [38–41], supporting modulation of a couple of orthogonal polarization bases with conjugate phase distributions, to generate CVBs from a given basic laser mode. All of these methods need to switch in different elements to generate CVBs with different topological charges, which is inconvenient in experiments that involve multiple CVBs. For example, if there is a circumstance that requests a tunable superposition state of CVBs with arbitrary customized amplitudes on each basis, solid metasurfaces will not be satisfactory. With the same circumstance, implementations containing interferometers, where two orthogonal polarizations are modulated by spatially programmable devices individually and are then combined together [42], still work. Though ingenious devices with high robustness [43–45] are developed, their complexity of modulating two individual parts still sets them as bulky and redundant. Therefore, it is significant to create a compact passive device to generate CVBs with arbitrary topological charges.
In this paper, we provide and demonstrate a passive device to generate CVBs with arbitrary azimuthal and radial topological charges. It is composed with a couple of polarization-selective cylindrical lenses to realize the mode-dependent Gouy phase [46,47] in a single polarization. The device could remain unchanged no matter the topological charges of CVBs. It even adapts to generate superimposed CVBs by employing customized wavefront modulation in incident homogeneous polarizations. In physics, the device creates a robust connection between homogeneous polarizations and CVBs, where homogeneous polarizations can be marked on a basic Poincaré sphere (PS), and CVBs can be marked on a high-order Poincaré sphere (HOPS) [48]. Mapping relationships between states on the basic PS and the first-order HOPS are experimentally proved by Stokes parameters of five representative states. Petal-like intensities are also collected to verify the extensive potential for generating states on other HOPSs.
2. MAPPING THEORY
Here Laguerre–Gaussian (LG) modes, a complete set for presenting transverse modes, are expressed with complex amplitude:
Figure 1.(a) Basic Poincaré sphere. Selected points a1–a6 on the surface include polarization states from linear polarization to elliptical polarization. (b) First-order HOPS. Selected points b1–b6 represent six CVBs on the surface.
Both radial polarization and azimuthal polarization are located on the equator of first-order HOPS. Denoting and , Eq. (3) indicates the radial vector beam is marked with , and the azimuthal vector beam is notated with [49]. For convenience, they are abbreviated with spherical coordinates and . The corresponding states with the same coordinates and for basic PS are and polarizations, respectively. Figure 1 elucidates the connection between the basic PS and the first-order HOPS via several representative points on the sphere. Points a1–a6 selected on the surface of the basic PS as shown in Fig. 1(a) include polarization states evolving from horizontal polarization to diagonal polarization along with the equator and then turning to a general elliptic polarization along with the longitude line. Points a1–a6 correspond to points b1–b6 at the same positions of the first-order HOPS, which represent CVBs as shown in Fig. 1(b).
3. IMPLEMENTATION
Significantly, the mapping is characterized by a polarization-selective conversion where is converted to and remains unchanged. For this reason, the device needs two functions: response of polarization and inversion of index .
Inspired by the design of the -plate [38], we employ liquid crystal (LC) films coated on a pair of cylindrical lenses to realize the conversion of polarization-selective response. As shown in Fig. 2(a), a polarization-selective cylindrical lens is constructed with isotropic glass lens and inhomogeneous LC film. In fabrication, plano-convex (cylindrical) isotropic glass () provides a common dynamic phase with thickness regardless of polarization, where is chosen as the converging direction of the cylindrical lens, is the refractive index of the glass material, is the refractive index of air, is a constant thickness, and is the designed focal length. The operator of the common glass lens is expressed with , where according to Fresnel paraxial approximation. LC film is produced by four steps: (a) spin coat a photo-alignment layer onto cleaned substrate (plano-convex); (b) expose the substrate to a hologram of cylindrical wavefront implemented by a He-Cd laser beam of 325 nm wavelength; (c) spin coat LC solution onto a photo-alignment layer, which induces the alignment of LC molecules periodically under the anchoring effect; and (d) with irradiation of a mercury-xenon lamp, polymerize the LC coating under ultraviolet light. An accurate thickness with a customized focal length under 632.8 nm He–Ne laser source is obtained by multi-layer spin coating of LC solution. Thickness of LC film is mainly controlled via angular velocity of spin coating and concentration of LC solution, and then provides an extra geometric phase [50–54] by metallic distribution of fast axis angle related to the direction of the axis, written with operator , in which . The compound operator, expressed with
Figure 2.(a) Fabrication of polarization-selective cylindrical lens. (b) PGPS, which contains two symmetric polarization-selective cylindrical lenses and operates on LCP only. (c) Gouy phases accumulated by PGPS for modes with different indices
The inversion of index derives from the coefficient’s conversion of Hermite–Gaussian (HG) modes. For complex amplitude of HG mode, , and are two indices corresponding to and coordinates in the transverse plane. It can be seen that an LG mode, , can be decomposed into a set of HG modes with the same order (), written as . Real coefficient is given by [46]
In the following, a polarization-selective device is constructed operating on HG modes (), where astigmatic Gouy phase is used to supply such an extra factor associated with mode index , exactly , or written as with . corresponds to cylindrical convergences set in the direction. As shown in Fig. 2(b), two polarization-selective cylindrical lenses are set at the symmetrical positions relative to original point . The condition for first piece of lens is , causing left-handed circular polarization (LCP) to become right-handed circular polarization (RCP) accompanied with converging effect, while RCP becomes LCP without other effect. Correspondingly, the second piece of lens, which is set in symmetrical position (flipped), turns RCP back to LCP with converging effect, such that only LCP of incident light accumulates [56] between the two polarization-selective cylindrical lenses. The amount of is decided by the distance between two lenses, , and designed focal length of them, , under proper coupling conditions. Exactly, , where . The phase of wavefront performs a period, so the mode-dependent phase takes effects along the instruction of . Figure 2(c) shows amounts of for to 5 in sequence. When , special points marked with red angles perform like a phase switch between and along with the axis, showing factor is attained and index becomes successfully. means , so the device is called the phase polarizing Gouy phase shifter (-PGPS).
Figure 3 shows the experimental scheme. He-Ne laser derives a 632.8 nm Gaussian beam whose polarization is projected to horizontal by a half-wave plate (HWP) and a polarizing beam splitter (PBS). Two lenses constitute an expander to provide an almost plane wave for the spatial light modulator (SLM, Holoeye, Pluto-VIS-016). Fork-like holograms [57] are loaded on the screen of the SLM to produce LG modes. The beam splitter (BS) ensures light beam propagates in the correct path. Two lenses and the iris select the first order of diffracted light after the SLM by reducing them into an appropriate scale. The characteristics of the SLM determine that the original polarization of the selected beam is horizontal. If a pre-production of polarization is necessary, a quarter-wave plate (QWP) and an HWP will be included in the optical circuit. Then the -PGPS takes effects to generate CVBs. The following QWP, HWP, PBS, and a charge-coupled device (CCD) constitute a framework to examine Stokes parameters. By changing angles of the fast axes of QWP and HWP relative to the axis, polarizations can be reconstructed with intensities recorded by CCD. Exactly, Stokes parameters are computed via [58,59]
Figure 3.Schematic of the experimental setup for generating arbitrary CVBs.
Figure 4.Stokes parameters of CVBs. The left column displays the tailoring polarization vectors of CVBs, followed by columns of Stokes parameters
As for , higher-order HOPSs are constructed. Representatively, a state on the equator of the HOPS can be examined by casting it into horizontal polarization via transmitted port of the PBS. The transmitted intensity performs special distribution that satisfies the petal-like shape, denoted by , where subscript is the same with the index of the LG mode. Combined with Eqs. (1) and (3), it is calculated that , indicating there are pieces of intensity petals. In other words, the number of petals can be exploited to characterize the azimuthal index of the CVB, or called the order of the HOPS. Equation (1) shows that radial index is separated from the operation of , meaning is an individual parameter in the construction of the HOPS. Radial index remains unchanged, and azimuthal phase is reversed to under the effect of the proposed device, so that all LG modes are connected with non-degenerated CVB completely. In notations, LG modes and CVB modes are both defined with two topological charges and where the two indices map with the same order of each other, respectively. Expression of CVB containing is written as
Figure 5.Petal-like profiles collected by CCD when launching high-order LG modes. The first row is set for
4. CONCLUSION
We propose and demonstrate a passive device based on -PGPS to generate arbitrary CVBs with both and indices even in ultrahigh order. The device simplifies existing schemes for generation and builds a solid connection between a simple scalar field on the basic PS and sophisticated CVBs on the HOPS. Extensively, states on hybrid order Poincaré sphere (HyOPS) [60,61] can be implemented with the help of a spiral phase plate (SPP), which provides a shift of index of LG modes regardless of polarization. The device supplements a convenient operation for quantum information and communications experiments [62–64], taking effects on the polarization-selective mode index inversion. Generally, it can be extended to other polarization-selective converters besides and supports more splendid mode-dependent conversion of polarizations. The method is flexible to other techniques such as metamaterials [65–67] and metalenses [68,69], which may help to miniaturize the optical device on chips [70–72].
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