Extended Data Fig. 9: Micromagnetic simulations for hopfion rings, with modulations penetrating the whole thickness, and a comparison of the contrast with experimental measurements.

a, Three initial states. b,c, Corresponding magnetic textures after energy minimization. In b, a cross-cut of the simulated state is shown; in c, the same magnetic texture is visualized using isosurfaces m_z = 0. The calculations for all three cases were conducted for an external magnetic field of 243 mT, with a very small tilt angle not exceeding 2°. For details, see Mumax Script II in the Supplementary information. d, Corresponding cross-cuts for the median plane and theoretical phase images calculated for each of the three states. e, Corresponding theoretical Lorentz TEM images, assuming a defocus distance of 400 μm, as in the experimental set-up. f, Experimental Lorentz TEM images recorded in an external magnetic field of 243 mT and at a defocus distance of 400 μm. The experimental images match the theoretical images well for States 1a and 1b with a compact hopfion ring. In State 1a, the vortex is centred on the hopfion ring, whereas in State 1b this is not the case. Note the bumps on the hopfion ring in State 1b visible in the experimental and theoretical Lorentz images. For State 2, which exhibits magnetic modulations extending along the thickness, the theoretical Lorentz TEM images show two distinct features: a strongly contrasting bright ring in the over-focus image and an additional bright ring in the under-focus image. No high-intensity bright ring in the over-focus image or second bright ring in the under-focus image are observed in the experimental images.