Broadband generation of perfect Poincaré beams via dielectric spin-multiplexed metasurface

The term Poincaré beam, which describes the space-variant polarization of a light beam carrying spin angular momentum (SAM) and orbital angular momentum (OAM), plays an important role in various optical applications. Since the radius of a Poincaré beam conventionally depends on the topological charge number, it is difficult to generate a stable and high-quality Poincaré beam by two optical vortices with different topological charge numbers, as the Poincaré beam formed in this way collapses upon propagation. Here, based on an all-dielectric metasurface platform, we experimentally demonstrate broadband generation of a generalized perfect Poincaré beam (PPB), whose radius is independent of the topological charge number. By utilizing a phase-only modulation approach, a single-layer spin-multiplexed metasurface is shown to achieve all the states of PPBs on the hybrid-order Poincaré Sphere for visible light. Furthermore, as a proof-of-concept demonstration, a metasurface encoding multidimensional SAM and OAM states in the parallel channels of elliptical and circular PPBs is implemented for optical information encryption. We envision that this work will provide a compact and efficient platform for generation of PPBs for visible light, and may promote their applications in optical communications, information encryption, optical data storage and quantum information sciences.

(2) where f is the focal length of lens. By substituting eq. 1 into eq. 2, the complex field amplitude of the POV in the focal plane is obtained as: where ! is an l-th order modified Bessel function of the first kind, = @( ) ) + ( ) ) , = actan( / ), 7 = " / , is a scale factor and controls the ellipticity of POV, (x, y) are Cartesian coordinates in the rear focal plane of the lens, 7 = " / , = 2 ⁄ is the wave vector, f is the focal length of the lens, " can be controlled by the numerical aperture of the axicon NA, and 7 = 2γ / & is waist of Gaussian beam at the rear focal plane of the lens. For small 7 at the focus of the lens and large 7 , ! 0

Supplementary Note 2. Phase distributions of the metasurface for generation of PPBs
As shown in the main text, the PPBs are composed of orthogonal circularly polarized POVs, and the POVs can be generated by the Fourier transformation of a higher-order Bessel-Gaussian beam. In the experiments, a Gaussian beam can become POV after successively passing through a spiral phase plate, axicon and Fourier transformation lens. A metasurface can implement this multielement multifunctional process for either RCP or LCP incident light by satisfying the superposition of the phase distributions of a spiral phase plate <=(">! ( , ) , an axicon >?(@AB ( , ) and a Fourier transformation lens !CB< ( , ) which are expressed as: where (x, y) is the geometric coordinate of the metasurface and = E or B for RCP or LCP light respectively. Thus the phase profile ECF> ( , ) = G ( , ) or ) ( , ) encoded respectively on the metasurface for RCP or LCP light is described as: Supplementary Note 3. Derivation of the Jones matrix and its eigenvalues and 5 eigenvectors: An arbitrary polarized light beam normally incident onto the metasurface can be decomposed into RCP and LCP spin eigenstates with corresponding two-dimensional (2D) Jones vectors given by: |LCP⟩ = 3 1 6 and |RCP⟩ = 3 1 − 6. In order to generate two completely different POVs represented by two poles of the HyOPS, the metasurface is required to provide two independent spatial phase profiles G ( , ) and ) ( , ) corresponding to |LCP⟩ and |RCP⟩, where and eigenvectors as: . Thus, the Jones matrix can be rewritten in terms of its 6 eigenvectors and eigenvalue as: Since the matrix ( , ) operates in the linear polarization basis, the diagonal matrix Λ determines the phase shifts

Supplementary Note 4. Calculation of polarization conversion efficiency of the unit cells.
For an arbitrary polarized light | (B ⟩ incident on the metasurface unit cell, the transmitted light | ATF ⟩ can be expressed as 4 : and O . For an incident light with circular polarization (|LCP⟩ or |RCP⟩), the polarization conversion efficiency 6 ) or H ) can be obtained according to the eq. 14 as:  Fig. 6) of π and thus the geometric phase is wavelength-insensitive.

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As for the propagation phase, it is accumulated through light wave propagating in the nanopillar behaving as a truncated waveguide, which can be expressed as: where In order to see the phase shifts between design wavelength and other wavelengths more clearly, the cross sections of these in-plane phase profiles are extracted from Supplementary   Fig. 9 along x-axis depicted by dashed white lines and unwrapped, as shown in Supplementary   Fig. 10. The simulated results also agree well with the theoretical results. 11

Supplementary Note 6. Stokes polarimetry of PPBs and spherical wave interferometry
The polarization state of PPBs on the HyOPS is mapped by representing the Stokes parameters in the spherical Cartesian coordinates. Stokes polarimetry is implemented by measuring a series of intensity distributions to obtain the values of Stokes parameters expressed as 5 : ) . Thus, the spatial distribution of polarization orientation angle Θ for the PPB described by any point on the HyOPS can be obtained as: In order to analyze the polarization distribution of the PPBs in experiment, a linear polarizer and a quarter waveplate are inserted at the front of the camera to measure these four intensities distributions ( 4 , S_ , ^4 and 6 ). In addition, in order to accurately distinguish state A and state F in Figure 1a, Stokes polarimetry and spherical wave interferometry should be performed on the two beams, respectively. First, a measurement of four intensities ( 4 , S_ , ^4 and 6 ) of the two beams after transmission through a quarter waveplate and a linear polarizer must be performed. According to these intensities ( Supplementary Fig. 13 Fig. 14. According to the number of lobes, one can determine the topological charges of RCP and LCP perfect vortex to be 5 and 10, respectively. Therefore, using the measurement sequence outlined above, one can distinguish state A from state F and accurately determine the polarization distribution and topological charges of the two beams. As for the measurement of other states on the HyOPS, Stokes polarimetry only needs to be implemented for these beams. 13

Supplementary Note 7. Intensity calculation for PPBs with superpositions of POVs
According to the Eq. 4, the magnitude of a POV with topological charge l is expressed as: As shown in the main text, the PPB is the superposition of two orthogonal circular polarization POVs. For an arbitrary polarization state of light incident on the metasurface device, the output light is a superposition of the form given by eq. 1. The intensity of the output PPB corresponding to point (α, β) on the HyOPS is calculated by the following equation: