Fig. 4: Pure topological discovery within a fully continuous framework.

The parameter space (25 optical parameters) is defined by 9 beam splitter ratios (BS), 8 distances (z), and 4 wave plates (WP, with variable phase retardance η and orientation angle θ). a Discovered optical topology for STED microscopy¹⁰. The minimum value of the loss function is demonstrated in detector #3. The phase mask (PM) #2 corresponds to the radial phase pattern originally used in STED microscopy, which generates a doughnut-shaped beam. b Radial intensity profile, ∣E_x∣² + ∣E_y∣², in horizontal beam section: excitation (green), depletion (orange), and super-resolution effective STED beam (dashed blue line). Lateral position indicates lateral distance from the optical axis. The data corresponding to the original STED phase are indicated with dotted lines. The excitation and depletion beams are diffraction-limited. The effective response breaks the diffraction limit. c Discovered optical topology for Dorn, Quabis, and Leuchs (2003)⁵¹. The minimum value of the loss function is demonstrated in detector #3. Phase masks #1 and #2 correspond to the polarization converter demonstrated in Dorn, Quabis, and Leuchs (2003) and a radial phase pattern originally used in STED microscopy, respectively. Both phase patterns generate, independently, a doughnut-shaped beam. d Normalized longitudinal intensity profile, ∣E_z∣², for Dorn, Quabis, and Leuchs (2003) and the identified solution (black dotted, and green lines, respectively) and radial intensity profile, ∣E_x∣² + ∣E_y∣², of the diffraction-limited linearly polarized beam (orange dotted line). Lateral position indicates lateral distance from the optical axis. The spot size is computed as ϕ = (π/4)FWHM_x FWHM_y, where FWHM denotes Full Width Half Maximum. The discovered approach breaks the diffraction limit, demonstrating a spot size close to the reference. Also, it does not feature side lobes (indicated with a gray arrow), which can limit practical imaging techniques. Details of the initial optical setup, phase masks, and identified optical parameters can be found in Supplementary Information section Rediscovery Through Exploration of Optical Topologies.