Fig. 3: Visualising the topology of the focal volume.
The optical texture is reconstructed from the polarisation and phase measurements via the VFFR of the optical beam. The measured volume is coloured following the Poincaré sphere and reveals the topological structure of the Hopf fibration (a). Two C lines, the black loop and the threading straight white line, organise the texture into nested tori. Each toroidal surface represents points characterised by the same ellipticity. The colours wind nontrivially around each torus, and a few polarisation filaments making up these tori are shown in the insets: in (b), the lighter surface (\({S}_{3}=0.398\)) is made of lines characterised by RH elliptic polarisation; in (c), the L surface (\({S}_{3}=0\)) is made of lines along which the polarisation state is linear23,26; in (d), the darker surface (\({S}_{3}=-0.775\)) is made of lines characterised by LH elliptic polarisation. In each inset, the cyan and red filaments, corresponding to \(\beta =0,\pi\) are shown to form a Hopf link. Every pair of filaments in the texture link in this way, consistent with the Hopf fibration. The 3D rendering of this experimental skyrmionic hopfion is in the Supplementary Video.
