Fig. 1: Concept.

a, An optical skyrmion of arbitrary order n passing through specially designed passive structured matter can have the effect of addition or subtraction by an arbitrary integer, n↦n ± k (only n↦n + k is shown here). Note that the same medium can be used to perform both addition and subtraction. Such structured matter can be realized in many different ways including SLM cascades, metasurfaces, inkjet printing of liquid crystal structures, gradient-index systems, direct laser writing of birefringent structures in silica and more. This figure also depicts the Stokes fields of a standard Néel-type skyrmion passing through such a medium. Throughout this paper, colour is used to represent the azimuthal angle on the Poincaré sphere (PS), and saturation to represent height (similar to ref. ¹³). b, Stokes fields of skyrmions passing through different adders of the first, second and third orders, and their respective skyrmion numbers. The linear retarder array described in the main text is used as an example, with spatially varying material properties illustrated using cylinders, where the local-axis orientation determines the shape and colour, and local retardance determines the height. Half-wave plates (HWPs) that control addition and subtraction are also shown (see the main text for details). c, Adders of the first, second and third orders using highly disordered materials, demonstrating the robustness of our proposed adder to imperfections of the medium. Here we consider perturbations to the linear retarder arrays that respect the conditions proposed in the main text, resulting in ellipticity of the axes and changes in retardance. Note that the chosen disorder here is merely an example that abstractly represents an arbitrary level of distortion.