Deep diffractive optical neural networks for detecting Skyrmionic topologies of light

TL;DR

The study tackles the challenge of detecting Skyrmionic topologies in light, where topological invariants are not orthogonal. It introduces a dual-channel deep diffractive optical neural network (D$^2$NN) that sorts OAM components in each polarization and reads out the Skyrmion number $N = p \, \Delta\\ell$, enabling deterministic, all-optical topology detection. With two 5-layer channels and a Fourier-basis phase-screen parameterization, the system achieves high readout fidelity (≈93% visibility) and substantial channel efficiency (>75%), validated over 81 input states spanning $N \\in [-7,7]$, and demonstrates practical utility by image transmission with a 14-level topology alphabet and 100% reconstruction. The detector shows strong robustness to isotropic noise and environmental disturbances, suggesting real-world applicability and potential extension to quantum-topology sensing and communications.

Abstract

Optical Skyrmions are topological forms of structured light with the potential of an infinite encoding alphabet that is immune to disturbance. This attractive prospect is hindered by the lack of any topological detector, a challenging problem due to the non-orthogonal nature of the topological invariant (N). Here we demonstrate the first deterministic detector for Skyrmionic topologies of light using a deep diffractive optical neural network. Our network uses two independent processing channels of 5 diffractive layers each to map incoming topologies to spatially separated Gaussian channels from which N can be detected. We overcome the complexity of the training by using a spatial mode basis rather than pixels, reducing the training variables by x1000 compared to current methods. We use the detector on an input set of 81 input topologies, showing high accuracy even in the presence of significant levels of noise. Finally, to show the practical utility of the device, we transmit and receive an image encoded in a 14-level topological alphabet with no discernible cross-talk. Our work offers a new paradigm for the emergent field of diffractive optical networks and can easily be extended to other forms of optical topologies, setting a clear pathway for their deployment in real-world applications.

Figures (9)

• Figure 1: Concept of a deterministic detector for optical Skyrmions. (a) Vector beam carrying skyrmion topology with orbital-angular-momentum (OAM) of $\ell_{1,2}$ and polarisation states, $H/V$. (b) Each vector beam defines a spatially varying Stokes spin field $\mathbf{S}(\mathbf{r})$ in the transverse plane, which is mapped onto the unit sphere via inverse stereographic projection, establishing the topological mapping $\mathcal{R}^2 \rightarrow \mathcal{S}^2$. The Skyrmion number $N =p \Delta\ell$ depends only on the polarity $p=\text{sgn}(|\ell_1|-|\ell_2|)$ and the vorticity $\Delta\ell=\ell_2-\ell_1$. (c) Multiple vector beams can therefore belong to the same topological class, making the mapping from vector modes to Skyrmion classes surjective. For example, the beams $(\ell_1,\ell_2)=(4,2)$ and $(1,3)$ both yield $N=-2$ despite being orthogonal. (d) Conversely, distinct Skyrmion classes can contain overlapping or non-overlapping vector modes. (e) A diffractive neural network that resolves the OAM for each polarisation component in a single shot so that we can leverage the OAM/polarisation information to determine (f) the polarity $p$, vorticity $\Delta \ell$, and thus deterministic classification of $N$.
• Figure 2: Sorting modes with modes. (a) A set of input OAM eigenmodes $\ket{\Psi_j}\equiv\ket{\ell_j}$ is injected into the system, with the goal of mapping each OAM value to a unique spatial location. (b) The sorting operation is implemented using a diffractive neural network that realises an approximate unitary transformation $\hat{T}$ composed of cascaded free-space propagation ($P$) and phase-modulation layers ($D_m$). (c) The target transformation maps each input OAM mode to a distinct displaced Gaussian output mode $\ket{\Phi_j}$, forming spatially separated detection bins. (d) Instead of pixel-wise phase modulation, each diffractive layer is constructed from a weighted superposition of Fourier modes. The trainable weights $\mu_{m,n}$ are optimised using simultaneous perturbation stochastic approximation (SPSA) to minimise a cost function that penalises incorrect mode-to-bin mapping. This architecture enables efficient, all-optical sorting of OAM modes for both polarisation components in the subsequent Skyrmion detection scheme.
• Figure 3: Training and validation of the D$2$NN using OAM. (a) Examples of the simulated input OAM modes and their corresponding target mode is shown alongside the experimentally measured intensity obtained by mapping the input using channel 1 of our D$^2$NN, which will correspond to the horizontally polarised component of our input Skyrmion field. (b) A crosstalk matrix constructed from the normalised inner products between each input mode and all target modes, quantifying the detection probability for each input. (c) The behaviour of the visibility and cost function at each epoch during the phase screen training process, displaying the convergence of the system to its optimum solution. (d) Similar measurement examples are shown for channel 2, corresponding to the vertically polarised component of our input Skyrmion field. (e) Using these projective measurements a crosstalk matrix is again compiled from the normalised visibilities to visualise the probability distribution of the detected modes. (f) The behaviour of the various performance variables at each epoch is again shown for channel 2, where the solved coefficients from channel 1 served as the initial ansatz for the training procedure.
• Figure 4: Topology detection. (a) To confirm the quality of the states entering the detector, the polarisation structure of the input Skyrmion fields are reconstructed using a full set of Stokes measurements. (b) The individual $\ket{H}$ and $\ket{V}$ components of the various Skyrmion fields are measured in their respective channel regions, allowing the extraction of their constituent topological charges $\ell_{1,2}$, and the determination of the topological number $N$. (c) Measured topological numbers $N\in[-7,7]$ for $81$ Skyrmion states formed using all possible combinations of $\ell_{1,2}\in[-4,4]$. Multiple input states $\ket{\Psi_j^{\text{in}}}$ within the same class are measured with the same $N$, illustrating a many-to-one (surjective) mapping. (d) A crosstalk matrix representing the detection probability for each topological class, constructed from the measurement of all $81$ Skyrmion states.
• Figure 5: Image reconstruction using Skyrmion topology. The reconstructed image compiled using single-shot measurements taking by our topology detector. The image that is transmitted to the detector is shown as an inset and is constructed using a 14-levelled encoding scheme, utilising various Skyrmion topologies (N) as the information carriers.