AI-based analysis algorithm incorporating nanoscale structural variations and measurement-angle misalignment in spectroscopic ellipsometry

2.1 Modeling structural variations and measurement-angle misalignment

MMSE characterizes periodic nanostructures by measuring the polarization changes of light reflected from fabricated samples. These changes are represented by a single MM, which captures the averaged optical response within the illuminated area [14], [38]. In practice, however, fabricated nanostructures often deviate from ideal periodic designs, exhibiting nanometer-scale variations in shape and dimension across periods due to inherent process fluctuations [31], [32]. Such structural variations induce subtle differences in the optical response of each structure, thereby influencing the measured MM.

In this work, we consider a one-dimensional SiO₂ grating, defined by the structural parameters and measurement conditions illustrated in Figure 1. To investigate the effect of structural variations on the MM, we analyzed several representative cases, as shown in Figure 2(a). Specifically, we examined a structure with geometrical variations composed of four different linewidths (35, 37, 39, and 41 nm) arranged sequentially, mimicking a non-uniform specimen measured by SE. For this structure, the MM was first calculated for a grating with the averaged value p _average = 38 nm, using RCWA simulations. Motivated by the fact that SE measurements reflect the average optical response from multiple structures, we then implemented an MM averaging approach, in which the RCWA-simulated MMs of individual periodic structures (35, 37, 39, and 41 nm) were averaged to produce a representative MM. Figure 2(b) compares the M ₂₃ element of the MM obtained from the two approaches. Although both correspond to the average value of 38 nm, noticeable differences are observed, demonstrating that assuming a perfectly periodic structure with an averaged value is insufficient. Accounting for structural variations is therefore crucial for precise nanostructure analysis.

Figure 1:

Schematic of the 1D grating structure showing the definitions of the structural parameters and the incident beam.

Figure 2:

Overall comparison of the M ₂₃ element for representative non-uniform structures with geometrical variations: (a) schematic of three representative structures; (b) results obtained using the averaged structural parameter (p _average) and averaged MM approaches; and (c) comparison between the averaged MM and the direct MM calculated using FDTD simulations for non-uniform structures with geometrical variations.

To further validate the MM averaging approach, the same non-uniform specimen was modeled using finite-difference time-domain (FDTD) simulation, which is significantly more computationally demanding. Figure 2(c) shows that the M ₂₃ element obtained from the MM averaging approach closely matches the FDTD result. Across 40 different non-uniform specimens, the MM results from the two methods were compared using the mean squared error (MSE), and the averaged MSE of 3.12 × 10⁻⁵ confirms the high level of agreement (see Supplementary Section 1). These results demonstrate that the proposed MM averaging approach can effectively analyze measured MMs from non-uniform structures with geometrical variations.

Building on this concept, we defined the parameter space of structural variations and generated representative discrete combinations using an efficient sampling strategy. In parallel, a neural-network-based MM generator was developed to rapidly produce MMs for large numbers of structures, enabling efficient large-scale analysis while accurately capturing the effects of structural variations.

2.1.2 Random symmetric sampling

In this study, we propose an efficient high-dimensional sampling strategy termed Random Symmetric Sampling. Within the parameter domain Ω, a local displacement vector p _Δ is generated relative to p _c , where each component corresponding to a structural parameter is randomly sampled from the interval [0, p _r /2], as illustrated in Figure 3. Since p _Δ is four-dimensional, independently assigning positive or negative signs to its components generates 2⁴ = 16 symmetric points. By generating n such local displacement vectors and applying this symmetric expansion, a total of N = 16n sampling points is obtained. These points are then used as inputs for the MM-generation process.

Figure 3:

Schematic illustration of the data-generation process accounting for structural variations.

To ensure uniform coverage across the parameter space, the number of sampling points per unit hypervolume was fixed. Specifically, the sampling density was set to 96 points per (1 nm)⁴, calculated based on the total parameter-space volume and applied uniformly within each p-region (see Supplementary Section 2 for details). Consequently, the total number of samples scales proportionally with the four-dimensional parameter-space volume, maintaining representativeness across all ranges. This approach ensures that the sampling density remains consistent, rather than allocating an equal number of points to both small (e.g., 1 nm) and large (e.g., 10 nm) variation ranges. The sampling procedure within a defined parameter region is illustrated in Figure 3. For intuitive understanding, a simplified two-dimensional parameter space is shown under a specific measurement-angle. In this example, local displacements are generated, symmetrically extended, and uniformly distributed within the sampling region. Random symmetric sampling thus combines local randomness with symmetric extension, providing efficient coverage of the parameter space while substantially reducing computational cost compared with exhaustive grid sampling.

2.2 M _avg generation through the development of MM generation neural networks (MMgen)

In Section 2.1, sampling points representing different parameter combinations were generated within the defined parameter space. Obtaining the corresponding MM for each point typically requires RCWA or other electromagnetic simulations. However, performing thousands of such simulations is computationally intensive, creating a bottleneck for real-time implementation and large-scale training data generation.

As an alternative, Jiang et al. [4] proposed a neural network–based model that learns the relationship between structural parameters and reflection ellipsometry measurements (Ψ, Δ), replacing RCWA simulations. Building on this approach, we introduce a high-speed, neural network–based MM-generation framework, termed MMgen, which directly generates the MM corresponding to both structural and measurement-angles parameters. MMgen consists of two sequential neural networks: the first generates the MM with reference to structural parameters, and the second corrects the MM for specific measurement-angle conditions. The architecture is illustrated in Figure 4(a).

Figure 4:

Overall framework of the enhanced two-step analyzing algorithm for structural parameter prediction accounting for specimen-induced structural variations and measurement-angle misalignment: (a) Part 1: M _avg data-generation framework using MMgen neural networks; (b) Part 2: Training framework for the two-step analyzing algorithm using M _avg.

The first neural network generates the MM corresponding to a given structural parameter. A reference library, precomputed using RCWA simulations at 5 nm intervals, is used to locate the closest lattice point and its corresponding MM. The network input includes the reference parameter vector (p _ref), the difference (p _sample − p _ref), and the reference MM, MM (p _ref, A _ref), obtained from the RCWA library. The output is the MM, MM (p _sample, A _ref), corresponding to the actual structural parameter p _sample evaluated at fixed reference measurement-angles A _ref (θ = 70°, ϕ = 45°). The second neural network refines this result by correcting for the actual measurement-angle conditions. It takes as input the MM generated by the first network, MM (p _sample, A _ref), together with p _sample and the experimental angles A _exp. The output is the corrected MM, MM (p _sample, A _exp), corresponding to the full parameter set (p _sample and A _exp). Using MMgen, MMs for arbitrary combinations of p _sample and A _exp can be generated with high speed while maintaining RCWA-level accuracy. This efficiency is further enhanced through GPU-based parallelization, ensuring that computational cost does not scale linearly with the number of samples. As a result, MMgen is highly effective for computing M _avg across numerous samples reflecting structural variations. The MMs generated by MMgen for each sampling point are averaged according to the structural distribution:

where M (p) denotes the predicted MM for a given structural parameter p, and f (p) is the probability density function describing the structural distribution. In this study, we assume that all structures occur with equal probability, i.e., f (p) = const. Under this assumption, the average MM can be simplified to:

where N is the number of sample points accounting for structural variations. The resulting M _avg provides a representative MM that approximates the optical response of structures with geometrical variations within the illuminated region. Based on this framework, parameter combinations (p, A) are sampled within the extended space (Ω × A) (Section 2.1). MMgen generates the MM for a large number of structures at each sampling point, which are then averaged to obtain M _avg. This integrated procedure ensures that the entire pipeline – from MM-generation at sampling points to M _avg computation – proceeds as a streamlined process, enabling rapid and accurate generation of extensive datasets. The resulting M _avg values provide essential input for subsequent algorithm development and structural parameter analysis.

2.3 Enhanced two-step analyzing algorithm accounting for structural variations and measurement-angles misalignment

Based on the training dataset generated in the preceding steps, we developed an enhanced two-step analyzing algorithm that simultaneously predicts structural parameters and measurement-angle conditions while accounting for fabrication-induced structural variations. This framework extends our previously proposed two-step algorithm [21] and can flexibly adjust the output layer to match the number of parameters to be predicted. The networks are trained in a supervised manner using simulated MMs as inputs and the corresponding structural parameters and measurement angles as labels, enabling accurate parameter inference from MM data. In this study, the architecture is explicitly extended to incorporate both structural variations and measurement-angle uncertainty.

The overall framework consists of two main parts, as illustrated in Figure 4.

Part 1: M _avg data-generation framework using the MMgen neural networks

In Part 1 (Figure 4(a)), random regions are selected within the structural parameter space (Ω × A), and multiple sampling points are generated within each region to represent structural variations. For each sampling point, the MMgen neural networks generate the corresponding MM. Precomputed reference data are incorporated to enhance generation accuracy, while GPU-based parallel mini-batch processing enables efficient computation of large numbers of MMs. By averaging the MMs of all structures in a specific region, the average Mueller matrix M _avg is obtained, providing an optical response that realistically reflects fabrication-induced structural variations for a given measurement-angles.

Part 2: Training framework of the two-step analyzing algorithm using M _avg

In Part 2 (Figure 4(b)), the enhanced two-step algorithm is trained using the M _avg datasets generated in Part 1. The training dataset is produced by defining the parameter-space domain via p _c and p _r (see Figure 3), sampling the space, and generating the corresponding MMs. The algorithm takes a single M _avg as input and predicts the center parameter vector p _exp[p _c ] of the structural distribution along with the measurement-angles A _exp.

In the first step, as shown in the upper part of Figure 4(b), the node-matching algorithm takes the input M _avg and identifies the closest node point. Specifically, it compares the center point of the experimental parameter space, denoted as p _exp[p _c ], with the predefined node points distributed at 5-nm intervals across the parameter space. The node p_n with the shortest Euclidean distance to p _exp[p _c ] is selected. The center parameter vector of this node p _n , along with its associated reference angles, serves as the zeroth-order reference for subsequent refinement.

In the second step, illustrated in the lower part of Figure 4(b), the input M _avg is combined with the MMs of the surrounding −1 and +1 order nodes. The node p _n predicted by the first network is defined as the zeroth-order node (p _n , A _n ), where A _n = A _ref corresponds to the reference measurement angles (e.g., p _r = 5 nm, θ = 70°, ϕ = 45°). Using this reference, the +1 node (p _n+1, A _n+1) is obtained by incrementing p _c by +5 nm and adopting the maximum range of p _r and measurement angles (e.g., p _r = 10 nm, θ = 72°, ϕ = 49°). Similarly, the −1 node (p _n−1, A _n−1) is defined by decrementing p _c by −5 nm and adopting the minimum range of p _r and angles (e.g., p _r = 0 nm, θ = 68°, ϕ = 41°). The MMs corresponding to these ±1-order nodes are retrieved from the pre-generated M _avg-based library, constructed using the MM-generation framework described in Section 2.2. Together with the measured MM, these three MMs are used as inputs to the second-step neural network, which is trained to predict the relative displacement vector (p _relative, A _relative). This vector represents the offset of the zeroth-order node (p _n , A _n ). The final predicted parameters are reconstructed by adding this relative vector to the reference node, i.e., (p _exp, A _exp) = (p _n , A _n ) + (p _relative, A _relative).

In this manner, the two-step analyzing algorithm, trained using the M _avg dataset, can predict structural parameters directly from measured MM data obtained via spectroscopic ellipsometry. This approach enables accurate estimation of nanostructure parameters even in the presence of fabrication-induced structural variations and measurement-angle misalignment.

3.1 Generation of M _avg using MMgen neural networks

The M _avg data-generation framework consists of two sequential MMgen neural networks designed for efficient M _avg computation. The first network generates the MM for a given p _sample using three inputs: (i) the reference parameter vector p _ref (4 × 1), (ii) the reference MM, MM (p _ref, A _ref) (320 × 15 × 1) obtained from the RCWA-based library, and (iii) the difference (p _sample − p _ref) (4 × 1). The target output is MM (p _sample, A _ref) (320 × 15 × 1), based on the RCWA dataset. The second MMgen network refines this output to generate MM (p _sample, A _exp) (320 × 15 × 1) using three inputs: (i) the MM output of the first network (320 × 15 × 1), (ii) p _sample(4 × 1), and (iii) A _exp (2 × 1) as inputs. This network accounts for variable measurement-angles, with further details of the dataset provided in Supplementary Section 3. Both MMgen networks share a Swin U-Net backbone, which is widely recognized for its efficient learning capabilities in high-resolution image segmentation tasks. The hyperparameters were set as follows: C = 48, W = 20, L = 14, and h = 6 [21], [41].

After training, the two networks were connected sequentially and evaluated on 3,510 test samples from the RCWA dataset with variable A _exp, achieving an average MSE of 9.99 × 10⁻⁸. As shown in the histogram of Figure 5, most test samples exhibit MSE values below 9 × 10⁻⁸. Figure 6 presents an example case with p _sample = (80.03, 38.61, 9.06, and −7.13 nm) and A _exp = (69.98°, 43.80°), yielding an MSE of 1.1 × 10⁻⁷, demonstrating high accuracy across the full spectral range and all MM elements. For M _avg generation, the two networks were executed in batch mode on an NVIDIA A40 GPU (48 GB). With a total model size of ∼82 MB, a batch size of 512 provided the optimal balance between computation time (∼7 s for 10,000 cases), GPU memory usage (∼42 GB), and data transfer latency. Table 1 summarizes inference times and GPU memory usage for different batch sizes.

Figure 5:

Histogram of MSE values between the MMs generated by the MMgen neural networks and those calculated by RCWA for 3,510 test datasets.

Figure 6:

Comparison of MM spectra predicted by the MMgen neural networks with those obtained from RCWA simulations.

These results confirm that MMgen enables efficient and accurate M _avg generation across diverse structural and measurement conditions.

3.2 Two-step analyzing algorithm trained using an RCWA dataset

The two-step analyzing algorithm was initially trained using only the RCWA dataset, employing supervised learning to predict the four structural parameters. The neural network inputs and targets were defined as follows:

Step 1: The input was the MM corresponding to the perfect periodic structure p _exp, and the target was the node p _n (4 × 1) with the smallest Euclidean distance from p _exp.

Step 2: The inputs included the MMs corresponding to the neighboring p _n+1 and p _n−1 nodes (extracted from the RCWA-based library), along with the original MM. The target was the relative displacement vector, p _relative = p _exp − p _n (4 × 1).

3.3 Two-step analyzing algorithm trained using a dataset accounting for structural variations and measurement-angle misalignment

To improve prediction accuracy, the two-step analyzing algorithm was trained using a combined dataset comprising the RCWA dataset with variable A _exp and the M _avg dataset, yielding a total of 70,204 samples. Supervised learning was employed to predict four structural parameters and two measurement angles.

Step 1: The input was the M _avg corresponding to (p _exp[p _c , p _r ], A _exp), and the target was the node p _n (4 × 1) with the smallest Euclidean distance from (p _exp[p _c ]).

Step 2: The inputs included MMs for the +1 order (maximum range/angle) and −1 order (minimum range/angle), together with the original M _avg. The target was the relative displacement vector (6 × 1), (p _relative, A _relative) = (p _exp[p _c ]−p _n , A _exp −A _n ), with A _n = (70°, 45°).

Both algorithms were validated using the M _avg test dataset by evaluating their mean absolute errors (MAEs). As summarized in Table 2, the algorithm trained using the RCWA dataset showed relatively high MAEs for structural parameters, whereas the algorithm trained using the M _avg dataset achieved consistently low MAEs for both structural parameters and measurement angles. Prediction–target distributions for the algorithm trained on the M _avg dataset are presented in Figure 7(a) and (b), demonstrating that training with the M _avg dataset effectively enables the algorithm to account for structural variations of the specimen and measurement-angle misalignment.

Figure 7:

Evaluation of structural parameter (p) and measurement-angle (A) prediction performance on the test datasets: (a) p predictions using the algorithm trained on the M _avg dataset. (b) A predictions using the algorithm trained using the M _avg dataset.

3.4 Experimental evaluation with the fabricated 1D grating

To evaluate the performance of the proposed algorithm on real fabricated structures, a one-dimensional SiO₂ grating with a designed period of 76 nm was fabricated on a silicon substrate, as shown in Figure 8. MMSE measurements (Woollam RC2, J.A. Woollam Co., Nebraska) were conducted at an incident angle of 70° and an azimuthal angle of 45° across 50 points within the patterned region. SEM imaging revealed noticeable linewidth variation across the sample, with representative values of 44.36 nm and 45.85 nm. Mean values and 95 % confidence intervals for each structural parameter were extracted from SEM images obtained at 20 different locations.

Figure 8:

Cross-sectional SEM image of the fabricated 1D grating nanostructure.

Table 3 summarizes the nominal design parameters, SEM-measured values, and average predictions from the two-step algorithms trained using the RCWA dataset and the M _avg dataset, respectively. Detailed experimental results, including additional parameter configurations, are provided in Supplementary Section 4. The most significant discrepancy occurred in the delta-width parameter. The algorithm trained using the RCWA dataset substantially overestimated this value (16.15 nm), whereas the algorithm trained using the M _avg dataset produced a prediction of 5.39 nm, which closely matched the SEM-measured value of 5.7 ± 1.08 nm.

Overall, the RCWA-trained model showed noticeable deviations from SEM measurements, while the M _avg -trained model – explicitly incorporating fabrication-induced structural variations – exhibited strong agreement with all SEM-derived parameters. Notably, the delta-width prediction error was reduced to only 0.1 nm.

The M _avg -dataset trained algorithm also estimated the measurement angles with high fidelity, predicting an incident angle of 70.08° and an azimuthal angle of 47.7°, both consistent with the experimental setup. In summary, experimental validation using MMSE measurements on a real fabricated specimen demonstrates that the algorithm trained with the M _avg dataset achieves the highest overall accuracy. By integrating structural variations and measurement-angle uncertainty into the training process, the algorithm achieved an MAE below 0.4 nm across the four structural parameters and delivered superior performance in predicting delta width. These results highlight its strong capability for precise metrology of fabrication-induced nanoscale structural variations.