Cancellation of the topological charge in the inverse Faraday effect: why an integrating detector cannot see the winding number

Abstract

Reliable optical readout is a bottleneck for magnetic textures, and the inverse Faraday effect looks like an obvious route for transferring the orbital angular momentum of light into a magnetisation. I show that in its simplest form this route is closed, and closed exactly rather than approximately. For a vortex with spatially uniform circular polarisation, the out-of-plane component of the optical spin density equals sigma |u(r)|^2. The phase factor exp(i l theta) cancels between the two transverse components, so this quantity carries no trace of the topological charge l, neither its sign nor its magnitude. The statement holds at every beam waist down to the diffraction limit: the difference between l = +1 and l = -1 is zero in S_z, whereas in the in-plane components it arises only through the longitudinal field and reaches merely 2.9e-2 even at a waist of 0.5 lambda. A receiver with out-of-plane magnetisation driven by such a field therefore cannot register the winding number, for reasons of symmetry. The cancellation is lifted as soon as the polarisation itself is spatially structured. In a microcavity the transverse-longitudinal splitting does this without further effort. Computed for a two-component condensate, the result has the following structure: the generated partner population and the integrated S_z are exactly equal for +l and -l; the entire difference lives in the radial profile. A detector averaging over the area therefore stays at chance level, while a spatially resolved one achieves full separation. At tenfold noise the topological fidelity of a resolved readout is 0.943 for |l| = 2 and 0.571 for |l| = 1, whereas the integrating detector stays between 0.503 and 0.519 at every noise level. A prescription for the experiment follows: use charge +-2 rather than +-1, because the generated partner then carries winding zero and sits exactly on the axis, where the seed has a node. The contrast of the radial profiles grows from 0.433 to 1.543. This work contains no measurement. All quantities are dimensionless, and the relative TE-TM mass splitting, on which the effect depends quadratically, has not been measured for perovskite cavities. 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.A companion record carries the same structure one stage further, into the magnetic texture: "What an optical readout can see of a topological texture: a classification by symmetry with quantified noise limits", DOI 10.5281/zenodo.22308692.