What an optical readout can see of a topological texture: a classification by symmetry with quantified noise limits

Abstract

Reliable optical readout of magnetic textures is regarded as an open problem; the obstacles named are limited spatial resolution, signal-to-noise ratios degraded by thermal effects, and constraints on high-speed detection. I show that these three obstacles hit different channels with very different force, and that one of the three does not apply at all to the most important quantity. Four textures with an identical radial profile, differing only in vorticity and helicity - two Neel skyrmions of opposite helicity, a Bloch skyrmion and an antiskyrmion - are held against four observables: the out-of-plane component, to which the ordinary polar Kerr effect couples; the scalar spin chirality, to which the topological Kerr effect couples; its area integral, which a single pixel measures; and the curl of the in-plane magnetisation as a reference. The polar Kerr effect separates none of the six pairs. In thirty cases examined the topological fidelity is 0.500, even at a signal-to-noise ratio of 33. This is not a matter of sensitivity: its input is the same for all four textures. The topological Kerr effect separates the charge but not the helicity. It reaches a fidelity of 0.871 already at threefold noise and 0.999 at unit noise, while remaining at 0.500 for the helicity pairs at every level. And it is insensitive to blur: the integral of the chirality does not change to the fifth digit under convolution with a diffraction spot, because that integral is the topological charge. Resolution is irrelevant for this channel. When several objects sit inside one spot, the polar channel counts objects and the topological channel counts charge: three skyrmions and three skyrmions of mixed helicity give identical values in both channels, whereas two skyrmions with one antiskyrmion give the same polar signal as three skyrmions but the charge -1 instead of -3. A division of the three obstacles follows. The charge is cheap: a single-pixel measurement, no scanning, and a signal-to-noise ratio of one suffices for a fidelity of 0.9. The helicity is not accessible in the optical far field at all, at any resolution and any noise level. This work contains no measurement. It computes no Kerr rotations for concrete materials and therefore cannot convert the required ratios into seconds.This record contains the paper in English and German, the accompanying work log in both languages, the self-contained reproduction scripts, and the LaTeX sources. The scripts require only NumPy and check every published value against the computed one, stopping with an error on any deviation. The same structure is established one stage earlier in the chain, in the optical field itself, in a companion record: "Cancellation of the topological charge in the inverse Faraday effect: why an integrating detector cannot see the winding number", DOI 10.5281/zenodo.22308285.