The input beam is split into two by the beam splitter: 50% of the light is reflected toward mirror M₁, while the remaining 50% is transmitted toward mirror M₂. After being reflected from M₁ and M₂, half of the light from each mirror is redirected by the beam splitter toward the viewing screen as shown in Figure 1 and Figure 2 .

Thus, the original light beam splits, and the resulting beams are recombined. Since the beams originate from the same source, their phases are highly correlated. When a focusing lens is placed between the laser source and the beam-splitter, the light ray spreads out after the focal point, producing an interference pattern of dark and bright rings on the viewing screen. For theoretical analysis, we first define the electromagnetic field of the input laser as E, with its strength denoted as E₀.

After passing through the beam splitter, the incident electromagnetic field E₀ is divided into two components: E₁ = rE₀, the reflected part, and E₂ = tE₀, the transmitted part, where r and t are the reflection and transmission coefficients, respectively. The output field E_out is then the superposition of these two components as they recombine after reflecting from the mirrors. The resulting interference pattern depends on the phase difference between E₁ and E₂, which leads to constructive or destructive interference on the viewing screen.

Mathematically, this can be expressed as:

$E_{\text{out}}=r{E}_1+t{E}_2$ (1)

$r^2+t^2=1$

If we introduce the phase difference $\Delta \phi ={\phi }_2-{\phi }_1$ between the two beams due to vibrations or other disturbances, the intensity I_out of the output will be affected by the interference of the two beams.

The general equation for the intensity of the output is:

$I_{\text{out}}={\left|E_{\text{out}}\right|}^2$ (2)

Since the electric field components are now influenced by the phase difference Δφ, the expression for the combined electric field becomes:

$E_{\text{out}}=E_{\text{1}}+E_{\text{2}}{\text{e}}^{i\Delta \varphi }$ (3)

Thus, the intensity becomes:

$I_{\text{out}}={\left|E_{\text{1}}+E_{\text{2}}{\text{e}}^{i\Delta \varphi }\right|}^2$ (4)

Expanding this:

$I_{\text{out}}={\left|E_{\text{1}}\right|}^2+{\left|E_{\text{2}}\right|}^2+2\left|E_{\text{1}}\right|\left|E_{\text{2}}\right|\cos \left(\Delta \varphi \right)$ (5)

Assuming E₁ = rE₀ and E₂ = tE₀, where r and t are the reflection and transmission coefficients mentioned before, the intensity simplifies to:

$I_{\text{out}}={\left|E_{\text{0}}\right|}^2\left(r^2+t^2+2rt\cos \left(\Delta \varphi \right)\right)$ (6)

This equation shows how the output intensity depends on the phase difference Δφ, which is affected by any vibrations introduced into the system. The constructive or destructive interference leads to variations in the intensity depending on Δφ.

If a small signal δ(t), this experimental test, is input by handwriting on the optical desk, the final output signal I_{out_signal} will be:

$I_{\text{out\_signal}}=I_{\text{0}}{\cos }^2\left(\frac{\Delta {\varphi }_0+\Delta \left(t\right)}{2}\right)$ (7)

where Δ(t) represents the phase vibration generated by the small signal δ(t), and Δϕ₀ is the initial phase difference.

Vibration mode data is collected using a camera recording at 1080p resolution and 60 frames per second. A Michelson interferometer with a 532 nm laser beam is aligned parallel to the optical desk. The laser beam is expanded to form a larger parallel beam, and a plane mirror is placed perpendicular to the beam. A 50/50 beam splitter cube is positioned between the plane mirror and the light source. The interference pattern is generated by combining the two light beams reflected off the plane mirror as they pass through the beam splitter cube, the complete setup is shown in Table 1 .

To generate vibration modes, lowercase English letters (a-z) were handwritten on the optical desk, as shown in Figure 3 . Each writing instance was recorded as a video, with each video serving as a data sample for training and testing. Additionally, during intervals between handwritten characters, a pattern with no writing behavior was recorded and labeled as “other” or “_”. This occurs during the idle time between each letter. Fifty samples were captured for each letter, all written by the same author.

This results in a total of 50 × 26 = 1300 samples for the letters “a-z” and an additional 1300 samples for the “_” label. To prevent overfitting, we selected 50 “other” samples from the 1300 available for training and evaluation. The data was shuffled and split, with 20% reserved for testing, 16% for validation, and the remaining 64% for training.

In this experiment, the vibration signals are a type of time series data, where the focus is on identifying patterns over time and monitoring continuous steps throughout the entire sequence. Therefore, certain well-known machine learning models that primarily focus on image classification and recognition, such as Support Vector Machines (SVM) and K-Nearest Neighbors (KNN), as well as deep learning models like Autoencoder + CNN and Autoencoder + Transformer, are not considered in this paper. Instead, we will focus on models such as LSTM (Long Short-Term Memory) with Attention, ViViT (Video Vision Transformer), and LRCN (Long-term Recurrent Convolutional Network), which will be described in detail in the following sections.

The LRCN (Long-term Recurrent Convolutional Networks), as proposed by Donahue et al. [20] , is a model designed for processing 2D video data by combining the strengths of convolutional neural networks (CNNs) and recurrent neural networks (RNNs). It efficiently processes interferogram frames, with the 2D CNN serving as a feature extractor that captures spatial patterns and relevant information. This network employs convolutional layers with batch normalization to stabilize training, Leaky ReLU activation to enhance feature learning, and max-pooling to downsample the feature maps. Additionally, dropout is applied during training to prevent overfitting.

The spatial features extracted by the CNN are then passed through a flattening layer, converting the 2D representation into a 1D format suitable for the LSTM layer, which manages the temporal patterns in the data. Additionally, batch normalization is applied to assist with optimization, further enhancing the model’s structure and performance, as described in [21] .

The unique strength of the LRCN model lies in its ability to capture both spatial and temporal dependencies in the interference patterns. This hybrid approach, which combines CNNs for spatial feature extraction and LSTMs for sequential analysis, makes it particularly well-suited for analyzing video data. The model excels at classifying vibration modes by effectively capturing both spatial and temporal patterns, resulting in competitive performance on the interferogram classification task.

The ViViT model, developed by Arnab et al. [22] , is a state-of-the-art architecture designed for sequence classification tasks using the Transformer’s self-attention mechanism. In this study, the ViViT model is applied to 2D video data from interferograms. It leverages multiple attention heads and layers, allowing the model to capture long-range dependencies and interactions within the data. By using self-attention, the ViViT model dynamically focuses on different regions of the interferogram frames, emphasizing the most salient features that are crucial for accurate classification.

The key strength of the ViViT model lies in its ability to capture complex spatial relationships across entire interferogram sequences. By incorporating attention mechanisms, the model outperforms traditional CNN-based methods in capturing global contextual information, which is essential for accurately classifying vibration modes with intricate spatial patterns. Additionally, the inclusion of label smoothing during training enhances model generalization, resulting in improved accuracy and robustness in the interferogram classification task.

On the other hand, the LRCN model, designed for 1D video data, takes a different approach to handling the concentric circular nature of the diffraction pattern. Instead of processing the entire pattern, the model extracts a single diameter from the interferogram. This simplification preserves essential spatial information while reducing the input data’s complexity. The 1D CNN then captures spatial features through layers of 1D convolutions, max-pooling, and dropout, effectively analyzing the one-dimensional time-series patterns of the interferogram.

The main advantage of the LRCN model is its efficiency in handling 1D video data, providing a compact representation while retaining key spatial information. Although it achieves moderate accuracy on the interferogram classification task, the model may face challenges when dealing with more complex spatial patterns due to the simplified input representation.

The LSTM-Attention model [23] , on the other hand, is designed to capture temporal dependencies and subtle variations within the 1D time series patterns of the interferogram. Equipped with 128 LSTM units, this model excels at learning long-term dependencies, which is crucial for accurately identifying vibration modes generated by handwritten behavior. The attention mechanism further enhances the model by dynamically focusing on relevant parts of the input sequence, improving its ability to recognize intricate temporal patterns.

Self-attention allows the LSTM-Attention model to dynamically focus on specific parts of the time series patterns, enhancing classification accuracy. This attention mechanism enables the model to adaptively assign weights to different time steps, enhancing its ability to capture and prioritize relevant features critical for accurate classification. Among the models tested, the LSTM-Attention model demonstrates superior performance in classifying vibration modes, achieving higher accuracy. The interpretability analysis of the model further reveals insights into its decision-making process, providing a clearer understanding of how it identifies crucial temporal features for precise classification.

Each of the three models—LSTM-Attention for 1D, LRCN for 1D/2D, and ViViT for 2D—was trained on preprocessed data using TensorFlow and Keras. During training, categorical crossentropy was used as the loss function, while Adam and RMSProp served as optimization algorithms.

After training, the models were evaluated on a separate testing set. Metrics such as accuracy were calculated to compare the models’ effectiveness. A confusion matrix was also generated to visually assess the classification performance of each model.

Comprehensive analysis of the models’ results revealed each model’s strengths and weaknesses in classifying the vibration modes generated by handwritten behavior. Additionally, hyperparameter tuning and optimization were applied to improve classification accuracy on the interferogram data.

The findings from this study help identify the most suitable deep learning architecture for classifying vibration modes generated by handwritten movements on an optical desk. The research also contributes valuable insights into applying deep learning for material science and vibration pattern recognition.

We begin with a 1920 × 1080 RGB video recorded at 60 fps. The video is then converted to grayscale. After that, we crop the video to focus on the concentric circle pattern produced by the interferometer. The cropped video is resized to 175 × 175 pixels and normalize the pixel values by dividing each pixel by 255. This normalization ensures that the data values remain fall the range of 0 to 1, represented as float32. An example of this preprocessing steps is shown in Figure 4 .

The above-mentioned preprocessing steps were used while training the LRCN 2D model and ViViT. However, since the pattern consists of concentric circles, we realized we could simplify the input data by extracting a single line of pixels from each video frame, passing through the center of the pattern. This extracted line effectively represents the pattern’s information. A period of one data sample is shown in Figure 5 , where each line corresponds to a single frame of the video.

We can summarize our preprocessing steps as follows, and the workflow is illustrated in Figure 6 .

1) Convert 1920 × 1080 @ 60 fps RGB video data to 1920 × 1080 @ 60 fps grayscale.

2) Crop the video to match the concentric circle (700 × 700).

3) Resize it to 175 × 175 using the cv2 default resize algorithm (inter-linear interpolation).

4) Normalize the pixel values to be between 0 and 1 by dividing by 255.

5) Extract the horizontal centerline intensity of the image if using LRCN 1D or LSTM-Attention.

The data labeling process utilizes one-hot encoding for 27 classes, which includes 26 English lowercase letters and one additional class for patterns generated during inactivity on the optical desk. Each class contains 50 data points. To prevent overfitting, label smoothing [24] with a factor of 0.1 was applied.

Initially, we trained our model using 2D video data. Since the LRCN model is known for its ability to handle this type of data, we applied it to our dataset. The architecture of the model is illustrated in Figure 7 , and the detailed workflow is depicted in Figure 8 .

After applying the LRCN model, we also implemented the ViViT model for comparison. Given ViViT’s strength in handling video data through its transformer-based architecture, we explored its performance on our dataset. This allowed us to evaluate the differences in classification accuracy and model efficiency between the two approaches.

There are four variations of the ViViT model proposed by Arnab et al. [22] . For simplicity, we focus on evaluating the Spatio-temporal attention variation. The ViViT workflow of our study is shown in Figure 9 . The first step involves performing Tubelet Embedding using a conv3D layer. Since conv3D requires a fixed input length, we limit the data length to 175 frames. If the data contains fewer than 175 frames, it is zero-padded, while longer data is cropped.

To improve model performance through data simplification, we applied a feature extraction method that leverages the concentric nature of the diffraction pattern. By extracting only one diameter of the pattern, we preserve the necessary information while reducing complexity.

After completing the feature extraction, we applied the LRCN 1D model, which is simpler than the LRCN 2D model previously used. The architecture of the LRCN 1D model is shown in Figure 10 .

The LRCN 1D model shares the same architecture as the LRCN 2D model, with

the key difference being the use of 1D CNN layers for feature extraction instead of 2D CNN layers. In addition, we also explored LSTM-Attention models, which were adapted from the framework proposed by Wang et al. [23] . The architecture of the LSTM-Attention model is depicted in Figure 11 , with the detailed workflow outlined in Figure 12 . This model utilizes global average pooling, originally proposed by Lin et al. [25] , which functions similarly to a fully connected layer but with fewer parameters, thereby reducing the likelihood of overfitting. Additional research related to the comparative analysis of speaker identification performance using deep learning can be referenced [26] , which provides a strong foundation for the use of multiple classifiers in complex classification tasks.