Fig. 1. Scheme for the LC-cascaded polarimeter. The exploded view exhibits the LC $q$-plate, LCPG, and $\mathit{y}$-polarizer, respectively. $a_{\text{L}}^2$ and $a_{\text{R}}^2$ depict the intensities of OVs in the $\pm 1\mathrm{st}$ diffraction orders, and $\mathit{\gamma }$ is the angle between the $\mathit{x}$ axis and the intensity minima on OV in the 0th order.

Fig. 2. (a) LC azimuthal orientations of the $q$-plate with $q=-0.5$ (up) and LCPG with $\mathrm{\Lambda }=100\mu \mathrm{m}$ (down). Blue to red indicates 0 to $\pi$. (b) POM micrographs of fabricated $q$-plate (up) and LCPG (down). The scale bar indicates $200\mu \mathrm{m}$ for all micrographs. (c) Voltage and wavelength-dependent intensity of the 0th order of the LCPG. Blue to red indicates 0 to 1. (d) Polychromatic diffraction patterns generated by the LC-cascaded polarimeter: 470 nm ($V_1=2.24\mathrm{V}$, $V_2=3.75\mathrm{V}$), 530 nm (2.01 V, 3.23 V), 590 nm (1.80 V, 2.77 V), 650 nm (1.66 V, 2.51 V), 710 nm (1.57 V, 2.29 V), 770 nm (1.50 V, 2.17 V), and 830 nm (1.41 V, 2.03 V).

Fig. 3. Different polarization states on the Poincaré sphere detected by the proposed polarimeter (green stars) and commercial polarimeter (purple points). Simulation (upper) and experimental (lower) diffraction patterns of corresponding SoP are presented. Black to white and black to green both indicate the intensity changing from 0 to 1.

Fig. 4. LC polarimeter using $q$-plate array for polarization imaging. (a) LC $q$-plate array. (b) Patterned phase retarder. (c) The diffraction patterns. (d) Calculated SoP of the phase retarder shown in panel (b). (e) Polarization image of a ruler. (f) POM image of the marked area in panel (e). (g) The diffraction patterns. (h) Calculated SoP of the region shown in panel (f).