J. Eur. Opt. Society-Rapid Publ. 22, 27( 2026) 277
Rayleigh-Sommerfield diffraction formula [ 14 ], with Fourier-based convolutions, applying band-limiting and zeropadding to maintain sampling resolution and avoid aliasing.
The detector plane was partitioned into ten output regions arranged in three rows( 3-4-3), corresponding to digit classes( 0-9). The first, second, and third rows represent digits { 0-2 }, { 3-6 }, and { 7-9 }, respectively. Optical intensity was evaluated within these regions at the output plane, and the predicted digit was assigned to the region exhibiting the highest intensity. Classification was therefore performed using class-specific detectors associated with the target categories. During training, the MCM-algorithm iteratively adjusts the binary configuration of the magnetic domain pattern to minimize the classification error, evaluated through the cross-entropy-loss function comparing the predicted and target intensity distributions: L ¼� XC i¼1
Q i ðxÞlog ðP i ðx; vÞÞ
where C is the number of output classes, Q i ðxÞ is the ground-truth one-hot label vector( the true class probability distribution), and P i ðx; vÞ 2 ½ 0; 1Š is the predicted class probability output for class i, under configuration v. Through repeated stochastic updates and forward propagation calculations, the network converges toward a configuration that optimally maps the input to the desired output.
2.2.1 Network architecture
Calculations were performed using a single-layer MO-D 2 NN model shown in Figure 2, comprisingN trainable neurons in a100 100 neuron( 1 lm width). Although the proposed training scheme is not restricted to a single hidden layer and has been numerically validated extensive simulations beyond the single-layer configuration, involving 2, 3, 4, 5, 10 and 15 diffractive layers, within the same training conditions, the MCM training jointly optimizes all layers simultaneously rather than sequentially optimizing each layer. However, deeper systems experimentally require significantly higher alignment precision and fabrication control. Therefore, for clarity and consistency with the online experimental study [ 12 ], we report computations for the singlelayer configuration, which provide the theoretical foundation of our proposed approach. Diffractive layer was made from a perpendicular anisotropic magnetic material, with two out-of-plane magnetization domains rᵢ = ± 1( upor down), encoding phase shifts of 0 or p(+ 1 or �1), and updated via MCM. The network was driven by linearly polarized light at a wavelength of 532 nm. This wavelength was selected due to the strong magneto-optical response of bismuth-gallium substituted yttrium iron garnet( BIG), which served as the hidden layer in the network. BIG exhibits pronounced Faraday rotation in the visible spectral range, peaking near the green region around( 532 nm) [ 12 ]. As reported in our previous study [ 15 ], the magneto-optical performance can be evaluated using the figure of merit( FOM), defined as the ratio of Faraday rotation relative to optical absorption, which quantifies the efficiency of polarization rotation to optical loss. At longer wavelength such as 633 nm, the Faraday rotation decreases due to the reduced
Fig. 1. Algorithmic Flowchart of the employed MCM process.
MO response, optical absorption is also lower, resulting in higher transmittance and a comparable FOM. Consequently, the choice of 532 nm provides an optimal balance between maximizing Faraday rotation and minimizing optical loss, consistent with the intrinsic optical and MO properties of BIG films. The separations between the input-to-hidden layer( d 1 = 3.0 mm) and between the hidden-to-output layer( d 2 = 0.5 mm) were selected following an optimization of the inter-layer spacing, which identified 3.0 mm and 0.5 mm as the configurations yielding the highest accuracy. The refractive indices were set to n = 1forfree space and n = 2 for the substrate. Faraday rotation angle h F and the ellipticity g F were assumed to be 3.3 ° and 0 °, respectively, to maximize modulation and ensure optimal interaction with the magneto-optical layer. This configuration provides strong Faraday rotation while maintaining high optical transparency and stable diffraction patterns. To maintain consistency, the same polarization parameters were applied in both simulations and experiment. A polarizer oriented at 90 ° was placed between the last layer and detector, so that it was perpendicular to the polarization of the incident light.
2.2.2 Datasets and task
Classification was performed on the MNIST handwritten digits dataset, with 5,000 training and 10,000 testing grayscale images of size 28 28. Each image was rescaled and