Fig. 1. Schematic of the metasurface design principles. (a) Fundamentals for generating longitudinally tunable HOP beams with a QWP metasurface under illumination by right-circularly polarized (RCP) light. (b) Geometry of meta-atoms A and B and the transmitted light field with RCP illumination. The dimensions of the meta-atom: height H, length L, width W, and orientation angle θ. (c) Overall images of metasurface and magnified view of central region. (d) Phase profiles for different functionalities and profiles of combined propagation and geometric phases. The insets in the upper-right corners show enlarged views of the phase profiles in the central part. The patterns in the top row of the upper panel are the propagation phase profiles of the sub-metasurface SA. From left to right: helical-phase profile φph,A; hyperbolic-phase profile φlens; axicon-phase profiles φa,A, δφpa; constant propagation phase φ0,A; and phase profile φco,A of the co-polarized component of the output field of SA. The patterns in the bottom row show the geometric phase profiles of the SA. From left to right: helical-phase profile φgh,A, axicon-phase profile δφga, and phase profile φcr,A of the cross-polarized component of the output field of SA. The patterns of the top and bottom rows in the lower panel show the corresponding propagation and geometric phase profiles of SB.

Fig. 2. (a) Geometry demonstrating the generation of VBs using a metasurface. (b) Geometry for demonstrating beam deflection. The red and green double lines illustrate the parallelism of the corresponding solid light lines. (c) The propagation phases φxx, φyy, phase retardation φxx−φyy, and the amplitude Txx of selected meta-atoms.

Fig. 3. Theoretical output light fields with different phase profiles. (a) Intensity pattern of the co-polarized component in the x−z plane with a hyperbolic-phase profile but without axicon-phase profiles. (b) Intensity patterns of co- and cross-polarized components in the x−z plane with two axicon-phase profiles with periods d1=2μm and d2=70μm and without hyperbolic-phase profiles. (c) Intensity patterns of the RCP and LCP components in the x−z plane obtained based on the combination of the hyperbolic phase and two axicon-phase profiles. (d) Curves of the intensities of the RCP and LCP components on different x−y planes along the x-direction within a large focal depth.

Fig. 4. (a) Schematic of HOP sphere. The four meridians in red, green, pink, and purple correspond to the meridians of the polarization states of the HOP beams generated by samples S1−S4, respectively. (b) x-component intensity (Ix) images of the HOP beams generated by S1 to S4 and total intensity (IA) images of the HOP beams generated by S3. (c) Curves of the IRCP and ILCP of the RCP and LCP component images of VBs versus Θ generated by S1. The value of each data point was normalized using the intensities averaged over the area of the beam doughnut. (d) IRCP and ILCP images of the HOP beams generated by S1 in a unified color bar and the corresponding IA images. (e) Intensity images of RCP and LCP components of HOP beams generated by S1 in the x−z plane. (f) Ix images of the HOP beams on the equator generated by S1 to S4 and the corresponding IA images with overlaid polarization states.

Fig. 5. (a) Schematic of experimental optical path. From left to right: laser, attenuator (A), polarizer (P1), quarter-waveplate (QWP), microscope objective (MO), polarizer (P2), and s-CMOS detector. (b) Scanning electron microscopy (SEM) image and (c) SEM side-view for S1.

Fig. 6. Experimental intensity patterns of VBs generated by S1−S4. (a) Total and component intensities of HOP beams with linear polarization states. In the rows from top to bottom, the patterns show the total intensity and the x-, 45°, y-, and 135° component intensities of HOP beams on the equator generated by S1 to S4, respectively. (b) Images of HOP beams. In rows from top to bottom, the patterns show Ix images of the HOP beams generated by S1 to S4, and IA images of the HOP beams generated by S3.