POST: photonic swin transformer for automated and efficient prediction of PCSEL

2.1.1 Raw data acquisition via CWT

The epitaxial structure listed in Table 1 serves as a representative baseline for PCSEL design. While the epitaxial configuration influences parameters such as the optical Green’s function and the confinement factor of the photonic crystal (Γ_PhC) within the CWT framework, its impact on the overall device behavior is secondary to the photonic crystal design. Therefore, the proposed methodology retains broad applicability and can be readily extended to alternative epitaxial stacks without significant modification.

To evaluate the optical performance of PCSELs, we adopt the CWT to model the interaction of fundamental waves within the photonic crystal lattice. By considering four primary wave components propagating along orthogonal directions, a set of coupled partial differential equations is established to capture both diffraction feedback and radiation loss mechanisms [33].

Solving this model under appropriate boundary conditions for a finite-size square-lattice photonic crystal allows us to compute the spatial distribution of optical fields, from which key performance metrics can be derived. Among these, two indicators are particularly crucial. Surface-emitting efficiency (SE) defined as the ratio between the surface-emitting optical power and the total stimulated emission power:

where α _surface and α _total are the surface radiation loss and total radiation loss of the lasing mode, respectively. This ratio reflects how effectively the laser extracts optical energy through vertical radiation and serves as a direct metric for surface output optimization.

Quality factor (Q) quantifies the ratio of stored optical energy to energy lost per oscillation cycle, expressed as:

where a is the lattice constant. A higher Q indicates better optical confinement and lower lasing threshold.

To ensure consistency across all simulated designs, the Bragg wavelength was fixed at 980 nm by adjusting the photonic crystal lattice constant accordingly, resulting in negligible wavelength variation across the dataset. Similarly, because the structure of the epitaxial layer (Table 1) remains unchanged for all samples, the energy confinement factor does not vary significantly. Furthermore, the analysis specifically targets the fundamental Γ⁽²⁾-point band-edge mode of the photonic band structure: Variations in other bands or off-Γ⁽²⁾ modes are beyond the scope of this work. Therefore, focusing on Q and SE as prediction targets was a deliberate first step, since these quantities are well defined within the CWT framework and directly reflect device efficiency and feedback strength in the design stage.

Other important figures of merit in PCSEL design, such as device size and the accuracy of the numerical solution to differential equations, were intentionally fixed in this study to avoid introducing additional degrees of freedom that would obscure the model’s evaluation. Nevertheless, POST is inherently data-driven and architecture-agnostic. Provided that the corresponding training data are available, POST can be readily extended to predict emission wavelength shifts, confinement factors under different epitaxial stacks, device size or even full band diagrams. This extensibility ensures that the proposed framework is not limited to Q and SE, but can evolve into a comprehensive predictive tool for a broader range of PCSEL performance metrics in future work.

2.1.2 Data preprocessing

The original dataset used in this work contains 25,000 samples, which we divided into training and test sets at a 4:1 ratio. Since the actual device consists of tens of thousands of lattices arranged in a periodic square matrix, flipping or rotating a single lattice’s design pattern theoretically does not affect the final PCSEL properties. Furthermore, we observed that even translating patterns, which alter the edge structures of the periodic square matrix, has negligible impact on overall PCSEL performance. The accompanying Figure 2 demonstrates this phenomenon using a randomly generated device pattern subjected to flipping, rotation, and translation operations followed by CWT simulations.

Figure 2:

Flip-rotate-translate pattern effects simulation. (a) A randomly generated unit cell pattern of the photonic crystal. (b) The first row shows eight variants of the lattice structure in subfigure after flipping and rotation. Rows two to nine show sixty-four additional patterns generated by horizontal or vertical translation of the original structure. (c–d) Simulation results of coupled-wave theory model for the photonic crystal lattice structure shown in b after flipping, rotation, and translation. The upper and lower histograms show the distributions of simulated Q and SE, respectively.

The histograms in Figure 2 show highly consistent simulation results in all transformations, with Q variations significantly below 1 and SE variations well under 0.1 %. These results not only confirm the pattern-invariant nature of photonic crystal properties but also validate the reliability of the numerical solution component in our CWT model, particularly its convergence characteristics in the non-analytical portion of the calculations.

Physically, the invariance of Q and SE under translation arises because a rigid lateral shift of the pattern only introduces a phase change in the Fourier components of the permittivity distribution, without affecting their magnitudes. In the CWT formulation, the key coupling coefficients (κ _i,j ) that determine feedback and radiation are proportional to the magnitude of these Fourier terms and the vertical field profile. Thus, a mere translation, which shifts the phase of each Fourier coefficient ξ _i,j , leaves the magnitude of κ _i,j unchanged. Consequently, the cavity’s mode profile and thus edge losses remain virtually the same under translation, explaining why the Q and SE are insensitive to such translations.

Therefore, the dataset size can be expanded using the methods mentioned above to improve prediction accuracy. The related research is discussed in Section 2.3.

The choice of prediction targets is also investigated. SE naturally ranges between 0 and 1 with relatively uniform distribution (Figure 3). Results show that raw SE values achieve the highest prediction accuracy without preprocessing (Table 2). In contrast, the Q factor can vary dramatically from hundreds to hundreds of thousands. Taking its logarithm yields a more uniform distribution (Figure 3) and maximizes prediction accuracy (Table 2). Balanced sample distributions across all value ranges enable the neural network to better distinguish between different photonic crystal designs.

Figure 3:

Distributions of data. The histograms show the distributions of log Q and SE for all samples in the original dataset.

2.2.1 Encoding module based on Swin Transformer Block

To enhance the extraction of feature representations from the single-channel input image I, we used a Swin Transformer Block-based encoder. The resulting multi-dimensional representation Z 4 is subsequently passed through an output layer to produce the final prediction results. The overall formulation of the encoding process is expressed as:

In accordance with the requirements of our task, we adopted the architecture illustrated in Figure 4(a). The input image is first divided into non-overlapping patches, each of which is treated as an individual token. These tokens are subsequently projected to a predefined feature dimension C through a linear embedding layer, enabling the subsequent transformer layers to more effectively capture relevant features.

Figure 4:

POST network structure. Graph (a) shows the architecture of the SwinT encoder. The input single-channel image is partitioned into patches, linearly embedded, then processed through four hierarchical stages to produce the final multi-dimensional representation Z 4 . Graph (b) displays two consecutive swin transformer stacks. W-MSA and SW-MSA refer to multi-head self-attention modules with regular and shifted window configurations.

In the proposed architecture, the input tokens are initially passed through a linear embedding layer and subsequently processed by a sequence of modified self-attention modules, known as swin transformer block. These modules operate within non-overlapping local windows to capture spatially localized features while maintaining the number of tokens. We define this initial structure – comprising the linear embedding and swin transformer block – as Stage 1, which serves as the basis for subsequent hierarchical processing.

To generate more efficient hierarchical representations, SwinT incorporates a patch merging mechanism [34]. As the network deepens, the number of tokens is progressively reduced through these patch merging layers, thereby improving the model’s computational efficiency. For instance, in Stage 2, features from each group of 2 × 2 neighboring patches are concatenated into a single vector of dimension 4C, followed by a linear layer that projects it down to 2C dimensions. This operation reduces the tokens count to one-fourth of the previous stage, effectively fusing local information for subsequent processing. The resulting representation, with a resolution of H 8 × W 8 , is then processed by additional swin transformer block.

This design is extended to Stage 3 and Stage 4, where the resolutions are further reduced to H 16 × W 16 and H 32 × W 32 , respectively, resulting in progressively abstract multi-dimensional representations. The complete formulation of the process is provided in Supplementary Information.

2.2.2 Swin Transformer Block

SwinT enhances the model’s capacity for global information integration by incorporating a shifted window mechanism [34] into its architecture and constructing the swin transformer block, as illustrated in Figure 4(a). This module is a modification of the standard transformer block, in which the conventional multi-head self-attention (MSA) is replaced with localized attention mechanisms operating within regular and shifted windows – referred to as window-based MSA (W-MSA) and shifted window MSA (SW-MSA), respectively. This design facilitates cross-window information while significantly reducing computational complexity. The forward propagation of two consecutive swin transformer stacks is depicted in Figure 4(b) and the detailed algorithmic procedures are provided in Supplementary Information.

Specifically, each swin transformer block in the architecture comprises two primary components: a window-based self-attention module (either W-MSA or SW-MSA), and a two-layer multilayer perceptron (MLP) equipped with the GELU activation function. Layer normalization (LN) is applied before each submodule, while residual connections are employed following each submodule to enhance training stability in deep networks.

2.4.1 Comparison to other neuron networks

A performance comparison of multiple existing neural networks is provided in Table 4 [26], [27], [35], [36], [37]. The evaluation includes not only traditional deep learning models (e.g., FCNN, CNN, and AlexNet) but also various ViT architectures, all tested on the same dataset. The table reveals that POST achieves the highest accuracy (highest R ²) in predicting both log Q and SE. For log Q prediction, POST attains an R ² of 0.909, outperforming the second-tier models CaiT (0.883), ConViT (0.882), and LeViT (0.881). Similarly, in SE prediction, POST leads with an R ² of 0.779, surpassing CaiT (0.763). These results indicate that POST’s architectural design excels at processing smaller-scale images and more accurately captures the physical features of photonic crystal lattice structures, whereas most ViT models exhibit significant advantages only when handling images larger than 100 pixels. Nevertheless, the second-tier ViT models still surpass the milestone model AlexNet in prediction accuracy, highlighting the overall superiority of ViT architectures.

It can be observed that traditional small-scale convolutional neural networks (CNNs) exhibit slightly lower predictive performance than fully connected neural networks (FCNNs). This is attributed to the fact that small CNNs, relying on convolutional mechanisms, can only identify local correlations between pixels and their neighbors, failing to extract global information. In contrast, analyzing the optical field of photonic crystals requires consideration of the entire unit cell structure.

While POST requires greater computational resources than lightweight models, its speed of >1 × 10⁸ samples/day and superior precision make it fully capable of supporting more optimization applications.

2.4.2 Comparison between predictions and simulations

POST achieves prediction accuracies (R ²) of 0.909 for log Q and 0.779 for SE in PCSEL modeling (shown in Figure 5). It typically reaches peak accuracy within fifteen epochs, requiring only about 3 h of training time, followed by a slight overfitting trend that leads to a minor decline in test set performance.

Figure 5:

Training dynamics and predictive performance. Graphs (a) and (c) show the training curves of the POST’s predictions for log Q and SE, respectively. The blue line (Train R ²) and red line (Test R ²) indicate the goodness-of-fit of the model on the training set and test set as the training epochs progress. Graphs (b) and (d), respectively, display scatter plots of the model’s predictive performance for log Q and SE. The scatter points compare the model’s predicted values with the true values, while the black dashed line represents the ideal fit line used to evaluate prediction accuracy.

The scatter plot visually confirms that POST’s prediction accuracy for log Q is significantly higher than for SE. This discrepancy stem from the more complex partial differential numerical solving process involved in SE calculations.

2.4.3 Accuracy with limited training samples

For many simulation software tools, obtaining 25,000 raw data points remains challenging, especially when the input patterns for simulation modules have higher pixel density – simulation time costs increase quadratically. If the simulation input involves 3D structural data, the cost escalates cubically. Therefore, the learning performance of different neural networks under reduced raw data volumes (shown in Figure 6) is investigated, where the horizontal axis represents the proportion of the new training set relative to the original training set.

Figure 6:

Accuracy versus training set size. The line chart compares the prediction accuracy (measured by R ²) of seven neural networks across varying training set sizes. The horizontal axis represents the proportion of the training set used, relative to the original size of 20,000 samples (80 % of the total 25,000 samples). The reduced training set is then augmented through 4 flips, 4 rotations, and 3 translations before training. The vertical axis shows the corresponding R ² accuracy for each model.

It can be observed that POST consistently maintains the strongest predictive capability across all dataset sizes and achieves R ² accuracies of >0.8 for log Q and >0.6 for SE with only 20 % of the original training set. Additionally, we note that POST’s prediction accuracy for log Q converges with only about 60 % of the original training set, whereas its accuracy for SE may require a dataset larger than the original training set to converge. This also suggests that predicting the SE parameter is more complex than predicting Q.

Furthermore, training results with 20 % of the original training set in Figure 6 ( R SE 2 = 0.607 and R log ⁡ Q 2 = 0.813) are worse than those without translation in Table 3 ( R SE 2 = 0.654 and R log ⁡ Q 2 = 0.845). However, the former involved three additional translations in both horizontal and vertical directions, resulting in an actual training data volume that was significantly larger than the latter. This suggests that the augmented samples generated through translation carry less additional information compared to entirely new samples.

2.4.4 Fourier-space feature attribution via SHAP analysis

To further investigate whether POST has internalized the physical priors embedded in CWT, we conduct a post hoc interpretability study using SHAP (SHapley additive explanations) [38], [39]. Unlike traditional saliency-based methods that rely on image-space gradients, this analysis evaluates the relative importance of Fourier components in predicting optical properties of PCSEL structures. Specifically, we apply a 2D discrete Fourier transform (DFT) to the photonic crystal unit cell and analyze the real and imaginary parts of selected Fourier coefficients as input features.

For a 32 × 32 dielectric constant distribution representing a photonic crystal pattern, we perform a 2D Fourier transform and extract a subset of its Hermitian-symmetric coefficients according to a triangular masking rule, as illustrated in Figure 2. This results in 512 unique complex-valued coefficients, each decomposed into real and imaginary parts and concatenated to form a 1,024-dimensional input vector. Each index (m, n) corresponds to a Fourier mode ξ m , n = ξ m , n R + i ξ m , n I .

To quantify the contribution of each Fourier component to the model’s predictions, we apply the Permutation SHAP algorithm with 50 background samples and 100 evaluation samples drawn from the test dataset. The prediction function internally recovers the spatial-domain input from each perturbed Fourier vector, enabling seamless compatibility with the original POST architecture trained in the spatial domain.

Figure 7 shows the SHAP violin plots for the top 15 most influential Fourier features for both SE and log Q. The most impactful coefficients are concentrated near the center of the Fourier domain, reflecting their critical roles in determining photonic crystal performance.

Figure 7:

Distributions of SHAP values. Distributions of SHAP values for the top 15 Fourier coefficients contributing to the prediction of (a) SE and (b) log Q. Each row denotes a distinct coefficient in the Fourier domain, with the x-axis showing its corresponding SHAP value. Color shading indicates the relative magnitude of the coefficient, with red and blue representing high and low values, respectively. The violin plot contours represent the variability and concentration of each coefficient’s contribution. Positive SHAP values suggest a positive influence on the predicted outcome, while negative values imply a suppressive effect.

In particular, ξ 0,0 R corresponds to the average refractive index of the photonic crystal layer. A higher value typically indicates a lower filling factor, meaning more high-index material is present. This leads to weaker photonic crystal modulation but stronger waveguiding effect, which enhances vertical optical confinement and contributes positively to Q.

Coefficients such as ξ 1,0 R and ξ 0,1 R are directly related to vertical radiation coupling. Deviations of their values from zero increase surface emission efficiency (SE) by enhancing out-of-plane leakage. However, since this also introduces greater radiation loss, it tends to reduce Q.

In contrast, ξ − 1,1 R governs the strength of in-plane two-dimensional diffraction, contributing to lateral distributed feedback. A larger magnitude of this coefficient suggests stronger horizontal coupling, which reinforces resonant feedback and increases Q.

Although the top-ranked Fourier features differ slightly between SE and log Q, substantial overlap exists in high-impact modes. This indicates that both performance metrics are shaped by a common set of structural features, especially those affecting radiative loss, confinement, and feedback within the photonic crystal.

According to the 3D-CWT framework [40], the radiation constant α and thus the Q-factor of the lasing mode depend explicitly on three quantities: the non-Hermitian coupling coefficient μ, and the real and imaginary parts (R, I) of the effective Hermitian coupling ( κ 1 D + κ 2 D − ) e − i θ pc . Specifically,

where a is the lattice constant. The coupling coefficients κ _1D and κ _2D− are positive correlated with the Fourier coefficients ξ _1,−1 (or ξ _−1,1) and ξ _2,0 (or ξ _0,2), respectively. Therefore, the Q-factor is highly sensitive to the magnitudes of ξ _−1,1 and ξ _2,0. This theoretical insight aligns well with the SHAP analysis, which identifies ξ − 1,1 R can significantly influence Q predictions.

It should be noted that the ultimate performance of a PCSEL is governed by the combined action of Hermitian couplings and non-Hermitian coupling [40]. In this context, our SHAP analysis does not aim to replace the underlying physical derivations, but rather to evaluate the interpretability of POST. By highlighting that the most influential Fourier components align with those known to control (R, I, μ) in coupled-wave theory, SHAP provides evidence that POST has internalized meaningful physical priors, thereby enhancing trust in the model’s predictions.