Fig. 1. Illustration of a resonator with eightfold rotation symmetries and eightfold reflection symmetries. These symmetries compose the D8 group. In (a), the resonator (in orange) is assumed to be made of metal. The resonator in (a) is abstracted as eight points to demonstrate the symmetries [see the orange points in (b)]. In (b), three symmetries operations are given as examples: C83, is a rotation by 3π/8 with respect to the z axis (see the purple arrow); and σv and σd are reflections with respect to the planes made by the z axis and the green line, and by the z axis and the red line, respectively.

Fig. 2. Illustration of the magnitude and the phase distributions of the z component of the eigen electric fields. The colors in the figure are coded from blue to red to denote the strength and the phase variations. Unless otherwise stated, the same color coding will be applied to the rest of this work.

Fig. 3. Illustration of the intensity and the polarization distributions of the in-plane components of the eigen electric fields. In the figure, |E∥|2=Ex2+Ey2, and “Pol. Mor.” is an abbreviation of “polarization morphology.” To describe the polarization morphologies, three symbols, i.e., the black lines, the blue hollow ellipses, and the red solid ellipses, are introduced to represent the linear, the right-handed, and the left-handed polarization states, respectively. The topological charges carried by the polarization vortices are marked at the centers of the polarization distributions.

Fig. 4. Illustration of the devices with the one- and the two-port feeding networks. (a), (b), and (c) show the top, the bottom, and the side views of the device with the one-port feeding network. (d) and (e) show the side views of the device with the two-port feeding network. (f) shows the details of the two-port feeding network. In (a)–(c), the adopted parameters are: ri=12  mm, ro=24  mm, rd=52  mm, rf1=26  mm, rf2=28  mm, dr=4  mm. The width of the microstrip line is 1.1 mm to ensure that the input impedance of the microstrip line is 50 Ω. In (c) and (e), two additional vias marked by “via 3” and “via 4” are to connect the ground of the SMA connector with the fourth layer, i.e., the ground of the microstrip lines.

Fig. 5. Illustration of the simulation results for the one-port case in Figs. 4(a)–4(c). The reflection coefficient S11 of the MPR is shown in (a). In (a), the dips in the reflection coefficient are marked by m1, m2, m3, M4, and M0 (from low frequencies to high frequencies), respectively. In (b), the real parts of the Ez component at the five dips are demonstrated. The fields are taken at the z=10  mm cut plane. In the inset of (b), a dashed black circle delineates the profile of the MPR; and a gray arrow points at the position of the via 0 in Fig. 4(c). In (b), the real part of the electric field is normalized to the range between −1 and +1.

Fig. 6. Projections of the electric fields at the five dips [in Fig. 5(a)] along the dimensions of the irreps of the D8 group. (a)–(e) correspond to the dips m1, m2, m3, M4, and M0, respectively. In each subplot, the projections along the dimensions of the irreps (i.e., the vertical axis) are plotted against the dimensions of the irreps (i.e., the horizontal axis). The insets illustrate the phase distributions of the Ez components of the projected fields.

Fig. 7. Illustration of the (a) and (c) intensity and the (b) and (d) polarization distributions of the in-plane components of the excited electric fields at the dips M4 and M0. The topological charges are marked at the center of (b) and (d).

Fig. 8. Illustration of the out-of-plane and the in-plane components, i.e., Ez and E∥ of the excited fields in the two-port feeding designs in Figs. 4(d)–4(f). The fields are taken on a cut plane at z=20  mm. (a) and (b), (c) and (d), (e) and (f) show the Ez and the E∥ belonging to the two dimensions indexed by j=1,7 of the E1 irrep, to the two dimensions indexed by j=2,6 of the E2 irrep, and to the two dimensions indexed by j=3,5 of the E3 irrep, respectively. For the case of j=1 and j=7, the lengths of the delay lines Δd are chosen as −27.5  mm and 27.5 mm, so that −90° and 90° phase delays between two ports are achieved at the targeted frequency 1.72 GHz (in the final simulation, 1.73 GHz). For the cases of j=2 and j=6, the lengths of the delay lines are chosen as −20.8  mm and 20.8 mm, so that −90° and 90° phase delays between two ports are achieved at the targeted frequency 2.28 GHz (in the final simulation, 2.30 GHz). For the case of j=3 and j=5, the lengths of the delay lines Δd are chosen as −18.7  mm and 18.7 mm, so that −90° and 90° phase delays between two ports are achieved at the targeted frequency 2.54 GHz (in the final simulation, 2.56 GHz).

Fig. 9. Illustration of the magnitude distribution and the phase distribution of the Stokes field S12 (corresponding to Fig. 3 in the main text).

Fig. 10. Illustration of the normalized Stokes parameters S3/S0 (corresponding to Fig. 3 in the main text).

Fig. 11. Illustration of the magnitude and phase distributions of the EL and the ER components of the in-plane eigen electric fields for all the irreps of the D8 group.

Fig. 12. Illustration of the magnitude and phase distributions of the Ez component of the eigen electric fields corresponding to the irreps of the D7 group.

Fig. 13. Illustration of the intensity and the polarization distributions of the in-plane components of the eigen electric fields corresponding to the irreps of the D7 group.

Fig. 14. Illustration of the magnitude and the phase distributions of two circular components of the in-plane eigen electric field corresponding to the second dimension (row) of the E3 irrep (marked by j=4) of the D7 group. It can be observed that the ER and the EL exhibit two scalar vortex modes with topological charge of −3 and 2, respectively.

Fig. 15. Illustration of the magnitude and the phase distributions of the Stokes field S12, and the intensity and the polarization distributions of the in-plane eigen electric field corresponding to the second dimension (row) of the E3 irrep (marked by j=4) of the D7 group. It can be observed that the Stokes field S12 exhibits a scalar vortex with a topological charge of −5, and thus the in-plane eigen electric field carries a polarization singularity with a topological charge of −5/2.

Fig. 16. Projections of the electric field radiated by the electric dipole oscillating at 1.72 GHz along the dimensions of the irreps of the D8 group. The projections along the dimensions of the irreps (i.e., the vertical axis) are plotted against the dimensions of the irreps (i.e., the horizontal axis). The insets illustrate the phase distributions of the Ez components of the projected fields. Notably, here for the sake of compactness, we only demonstrate the projected fields at the first resonant point (i.e., 1.72 GHz). For other resonant frequency points (e.g., 2.28, 2.54, 2.63, and 3.78 GHz), similar behaviors can be observed as well.

Fig. 17. Illustration of a spatial configuration of a 16-port feeding system to excite the A2 irrep.

Fig. 18. Illustration of the spatial configuration of a 16-port feeding system to excite the eigen electric field belonging to the B2 irrep.

Fig. 19. Illustration of the simulated reflection coefficient S11 and the excited electric field. (a) shows the reflection coefficient. The magnitude and the phase of the out-of-plane component, i.e., Ez, are plotted in (b) and (c), while the intensity and the polarization distributions of the in-plane components, i.e., E||, are plotted in (d) and (e).

Fig. 20. Illustration of the simulated magnitude and phase distributions of the EL and the ER components of the in-plane electric fields. The figure includes eight columns. The columns marked by A1(j=0) and B1(j=4) correspond to Fig. 7 in the main text. The rest of the columns correspond to Fig. 8 in the main text. The figure includes two subplots. In each subplot, there are two panels. The upper panel and the lower panel correspond to the EL and the ER components of the in-plane eigen electric fields, respectively. The first row and the second row of the upper (lower) panel are the magnitude and the phase distributions of the EL (ER) component of the in-plane electric fields. All the phase distributions are zoomed-in plots in the squares encircled by white lines in the B1(j=4) column.

Fig. 21. Illustration of the magnitude [abs(S12)] and the phase [angle(S12)] distributions of the Stokes field S12. Similar to Fig. 20, the figure consists of eight columns. The columns marked by A1(j=0) and B1(j=4) correspond to Fig. 7 in the main text. The rest of the columns correspond to Fig. 8 in the main text. Further, in the column marked by E1(j=7), a white dashed circle and two solid circles mark three loop paths. And, in the same column, a black square denotes a square region where the intensity and the polarization distributions in Fig. 23 are plotted. Additionally, in the columns marked by E1(j=1), E2(j=2), and E2(j=6), the arrows point at the C-points whose topological charge is 1/2. Lastly, all the phase distributions are plotted in the square encircled by the white line in the column marked by B1(j=4).

Fig. 22. Illustration of the normalized Stokes parameters S3/S0 corresponding to Fig. 8 in the main text. Two white arrows point at the locations where the S3/S0 equals 1.

Fig. 23. Illustration of the E|| in the black square in Fig. 21. (a) Intensity; (b) polarization distribution. In (b), the black lines denote the streamlines of the local polarization states, from which it can be observed that there are two C-points with a topological charge of 1/2, as pointed at by the green arrows, and one (quasi-)L line.

Fig. 24. Reflection coefficients S11 of different two-port feeding networks. The reflection coefficients S11 in (a)–(f) correspond to (a)–(f) in Fig. 8 of the main text. The design parameters of the two-port feeding networks can be found in the caption of Fig. 8 of the main text.

Fig. 25. Illustration of the out-of-plane and the in-plane components of the electric field radiated by the resonator in Fig. 4 of the main text. Here, the electric fields are plotted at 1.73 GHz on two cut planes. (a) z=0.5λ and (b) z=λ (where λ is the vacuum wavelength corresponding to 1.73 GHz), and the electric fields belong to the E1(j=7) irrep. In (a) and (b), the intensities of the Ez are normalized by the maximal intensities of the E||, respectively. “Pol. Mor.” is the abbreviation of “polarization morphology.”