Extended Data Fig. 10: Quantitative comparison of experimental and theoretical phase-shift images.

a, Experimental electron optical phase-shift image recorded using off-axis electron holography at T = 180 K in an external magnetic field of 200 mT. b, Theoretical phase-shift image calculated for the equilibrium configuration in the micromagnetic model, as shown in d. The grey levels used to display the magnitude of the phase shift, φ, are identical for the experimental and theoretical images. c, Line profiles of the phase shift extracted along the paths marked in a and b. The experimental and theoretical profiles indicate that the skyrmion inside the hopfion ring induces an electron optical phase shift of approximately 3.5 radians, whereas the hopfion ring induces an electron optical phase shift of approximately 1.0 radians. d, Magnetic texture obtained from micromagnetic simulations for an external magnetic field of 200 mT. The magnetization field is represented by the isosurface m_z = 0.