Fig. 1

Principle of polarized structured illumination microscopy (pSIM). a A schematic setup of a typical SIM system. The excitation polarization rotates with the grating to keep the laser beams s-polarized, obtaining high-contrast interferometric stripes (PM: polarization modulation). b By stretching the fluorescent dipoles over an additional orientation dimension, we interpret them in spatio-angular hyperspace. A linearly polarized light (horizontal polarization) would excite the dipoles with different orientations in a structured manner in the angular dimension. The quantitative relationship is a cosine-squared function whose Fourier transform contains three harmonics. From this perspective, polarized excitation is intrinsically structured illumination in the angular dimension. c Under the illumination of interferometric stripes generated by the s-polarized laser beams, the sample is structurally illuminated in both the spatial and angular dimensions. Equation (1) quantitatively describes the spatial structured illumination, the angular structured illumination, and the 2D illumination pattern. d We excited uniformly distributed 20 nm fluorescent beads with polarized structured illumination and used a rotary polarizer before the sensor to directly image the illumination pattern in the x–α coordinate plane, which is consistent with the simulation results. e Fourier transform of the 2D illumination pattern in the x–α coordinate plane results in spatial harmonics (blue), angular harmonics (yellow), and cross harmonics (gray). f The Fourier transform of the experimental 2D structured illumination in d with the corresponding harmonics marked with colored circles