Perturbation-resilient integer arithmetic using optical skyrmions

TL;DR

This work addresses the challenge of robust, high-bandwidth photonic computing in noisy, analog regimes by encoding information in optical skyrmions, which carry a discrete topological number. It introduces a boundary-driven design principle for passive structured matter that implements perturbation-resilient integer arithmetic: the skyrmion number changes by an integer $k$ according to the boundary state, with the fundamental relation $\deg \mathcal{S}' = \deg \mathcal{S} \pm k$. The authors develop two classes of devices: conventional skyrmion adders and generalized skyrmion adders, the latter delivering multi-dimensional topological charges via a tuple like $(n_1,n_2,\dots)$ and offering enhanced robustness to boundary and material perturbations. Experimental demonstrations employ gradient-index retarders and cascaded spatial light modulators, achieving addition and subtraction operations with significant tolerance to disorder and paving the way for scalable, energy-efficient digital photonic computing based on topological quantities. The generalized framework further suggests high information density through multi-component topological charges, with potential extensions to multiplication and division, frames a route toward high-TOPS photonic processors.

Abstract

The decline of Moore's law coupled with the rise of artificial intelligence has recently motivated research into photonic computing as a high-bandwidth, low-power strategy to accelerate digital electronics. However, many modern-day photonic computing strategies are analog, making them susceptible to noise and intrinsically difficult to scale. Optical skyrmions offer a route to overcoming these limitations through digitization in the form of a discrete topological number that can be assigned to the analog optical field. Apart from an intrinsic robustness against perturbations, optical skyrmions represent a new medium that has yet to be fully exploited for photonic computing, namely spatially varying polarization. Here, we propose and experimentally demonstrate a method for performing perturbation-resilient integer arithmetic with optical skyrmions and passive optical components. To the best of our knowledge, this is the first time such discrete mathematical operations have been directly achieved using optical skyrmions without external energy input.

Figures (4)

• Figure 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, $n \mapsto n\pm k$ (only $n \mapsto n+k$ 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 spatial light modulator (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, color is used to represent azimuthal angle on the Poincaré sphere (PS), and saturation to represent height (similar to shen_optical_2023). b, Stokes fields of skyrmions passing through different adders of first, second, and third order, 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 color, and local retardance determines the height. Half-wave plates (HWPs) that control addition and subtraction are also shown (see main text for details). c, Adders of first, second, and third order 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.
• Figure 2: Adder modules and experimental results.a, Order $k$ adder modules can be constructed using a linear retarder array placed before a half-wave plate while order $k$ subtractor modules can be constructed using a linear retarder array placed after a half-wave plate. The addition of the half-wave plates resolves incompatibilities in polarization states on the boundary as explained in the main text, allowing for the different modules to be cascaded indefinitely. Notice also that order $k$ adders and subtractors can be realized using the same hardware, with one direction performing addition and the other performing subtraction. Lastly, while gradient index systems and a 3-SLM cascade are used to implement the adder in our work, it is worth emphasizing that an adder can be implemented in many different ways, provided its properties at the boundary are properly constrained. b, A subset of the measured Stokes fields of optical skyrmions passing through adders of order 2 realized using gradient index systems. The operations $1\pm 2 \pm 2$ and $3 \pm 2 \pm 2$ are shown with detailed implementation presented in Supplementary Note 1 and the full dataset presented in Supplementary Fig. 2. c, A subset of the measured Stokes fields of optical skyrmions passing through adders of order 1 and 3 realized using a 3 SLM cascade, and where disorder is introduced by a random pixel-wise noise to the voltage levels of the SLMs. Since only a single adder is used in this experiment, no half-wave plate is included. Three levels of disorder are shown as indicated by the color of the star, increasing from left to right. Details of the implementation can be found in Supplementary Note 1, and a complete dataset is presented in Supplementary Fig. 3.
• Figure 3: Generalized skyrmion adders.a, Concept of a generalized skyrmion photo-adder, which is a passive component that converts a skyrmion of degree $n$ into a generalized skyrmion of degree $(n+k_1, n+k_2, n+k_3, \ldots)$. Note that the function of the adder is robust to perturbations in both the input field and material parameters, with this robustness extending even to situations where perturbations occur at the boundary. b, Given a polarization field, a single generalized skyrmion number can be defined for each connected component of the Poincaré sphere carved out by the image of the boundary curve. A field with (left) one component and (right) three components, along with the corresponding images of their boundary curves on the Poincaré sphere are shown. A stereographically projected version of the boundary curve is also shown. Note that for a given boundary condition, any continuous extension of the boundary to the entire domain will have the same number of generalized skyrmion numbers. c, A generalized skyrmion adder works by manipulating the boundary to create new connected components. For each newly created component, the original skyrmion number is increased once for each time the boundary curve encircles the component, accounting for orientation. The figure depicts an example of a $(n) \mapsto (n+1,n-1,n)$ adder, with input field $n=2$ and where the Stokes fields and stereographically projected boundary curves are shown. Finally, the skyrmion number and generalized skyrmion numbers of the two fields are provided.
• Figure 4: Experimental results (generalized skyrmion adders).a, (Left) The different regions of the Poincaré sphere on which the function of our adder is stable. The top region corresponds to addition, the bottom to subtraction and the middle to both addition and subtraction simultaneously. (Right) A subset of the measured output Stokes fields and computed generalized skyrmion numbers is shown for different levels of disorder (increasing from left to right) and different incident SoPs selected to demonstrate addition, subtraction, and simultaneous addition and subtraction. A stereographic projection of the boundary curve is also shown, with the change in skyrmion number of each region labeled. Note the color of these labels indicate the generalized skyrmion number corresponding to that region. A complete dataset including details of the incident SoPs and measured Mueller matrices is presented in Supplementary Fig. 4 . Technical details such as the formation of small loops due to disorder are also discussed. Lastly, details of the implementation and the experimental assembly can be found in Supplementary Note 1 and Supplementary Fig. 1, respectively. b, The skyrmion number and generalized skyrmion numbers at each level of disorder for different incident SoPs. Note that the generalized skyrmion number is topologically stable even though the usual skyrmion number is not.