Method for measuring small included angle of vortex light interference system and method for measuring thickness of thin film

Abstract

This invention provides a method for measuring minute angles and thin film thickness using a vortex interferometry system. Compared to traditional conjugate vortex interferometry, which typically relies on complex optical devices and data processing algorithms to measure the angle between beams, this invention constructs a simulated vortex conjugate interferometry optical path and employs machine learning. This allows for accurate angle measurement using simpler devices and more efficient algorithms. Furthermore, the machine learning algorithm automatically learns and optimizes the model, thereby improving the accuracy and stability of the angle measurement. By learning from a large amount of training data, the machine learning model can identify and correct errors and biases that may exist in traditional methods, thus providing more accurate measurement results. This greatly simplifies experimental setup and data processing, improves experimental efficiency and reliability, and enhances accuracy and stability. This method can be used to measure thin film thickness.

Description

Technical Field

[0001] This invention belongs to the field of optical micro-measurement, specifically relating to a method for measuring the minute included angle of a vortex optical interferometer system and a method for measuring the thickness of a thin film. Background Technology

[0002] Vortex beam conjugation interferometry is a light-based interferometry technique used to measure the phase difference of optical elements. Traditional interferometry typically uses plane waves or spherical waves for interferometry, while vortex beams are beams with spin angular momentum.

[0003] Vortex light is a special type of light beam with a rotating phase structure in its optical field. This type of beam can be generated using optical elements such as vortex mirrors. In vortex conjugate interference, we use two vortex beams with opposite phase structures to interfere. These two vortex beams can be generated using spatial light modulators or other optical elements.

[0004] When two vortex beams interfere with each other, their phase difference causes a change in the interference pattern. By analyzing the changes in the interference pattern, we can infer the phase difference information of the optical element. This technique can be used to measure parameters such as the morphology, refractive index, and thickness of optical elements.

[0005] Vortex optical conjugate interferometry has advantages such as high sensitivity, high resolution, and non-contact operation. It has broad application prospects in the fields of optical component manufacturing, optical surface topography measurement, and performance evaluation of optical devices.

[0006] However, in the experimental setup for vortex interferometry, the optical paths of the two vortex beams cannot guarantee that the angle between their optical axes is 0. A non-zero angle between the optical axes will distort the interference pattern, thus affecting the final experimental results. Currently, there is no simple and highly accurate method to calculate the angle between the optical axes. Summary of the Invention

[0007] In view of this, the purpose of this invention is to provide a method for measuring the minute angle of a vortex interferometer system and a method for measuring the thickness of a thin film, which can accurately measure the optical axis angle during vortex interferometry, thereby improving the stability of the interferometer system and the efficiency of the experiment.

[0008] A method for measuring minute included angles in a vortex optical interferometer system, comprising:

[0009] Step 1: Construct the simulated optical path of the vortex light conjugate interference system in the VirtualLab simulation software;

[0010] Step 2: Set the angle between the optical axes of the two interference optical paths at intervals of 0°-0.3°, and collect the interference pattern obtained by simulating the optical path at each angle.

[0011] Feature extraction from interference patterns: First, find the center of a single vortex beam with the same topological charge. Then, locate the centroid of the petal-shaped pattern in the interference pattern. The line connecting the centroids of the two upper petals in the interference image with the center of the circle forms an angle, which is used as the input feature. The corresponding optical axis angle is used as the output result, forming a training dataset.

[0012] Step 3: Select multiple machine learning models with regression prediction capabilities and train the machine learning models based on the training dataset;

[0013] Step 4: For the vortex interferometer system under test, extract features from its output interference pattern according to the method in Step 2. Input the input features into the selected machine learning model to obtain the output optical axis angle. Take the average of the optical axis angles of all machine learning models as the angle between the two interference optical paths of the vortex interferometer system under test.

[0014] Preferably, the machine learning model includes support vector machine, BP neural network optimized by genetic algorithm, random forest, BP neural network optimized by particle swarm optimization, extreme learning machine, and Gaussian process regression prediction model.

[0015] Preferably, in the simulated optical path diagram of the constructed vortex interferometer system, two storage functions and two 2f systems constitute spatial light modulators respectively. Phase diagrams are loaded into the two spatial light modulators respectively, so that the positive and negative vortex beams are merged and interfered at the original data detector to obtain the interference result diagram; wherein, the optical axis angle is added in one of the spatial light modulators.

[0016] Preferably, the set step size is 0.002°.

[0017] A method for measuring thin film thickness based on a vortex interferometry system involves placing the thin film under test in any interference optical path, i.e., changing the deflection angle of that optical path, thereby generating an optical axis angle with another optical path; inputting the angular features extracted from the interference pattern obtained by the vortex interferometry system into a selected machine learning regression prediction model, each outputting an optical axis angle value, and averaging them to obtain the final angle; and based on the relationship between the optical axis angle and the thin film thickness, obtaining the thin film thickness through the optical axis angle.

[0018] The present invention has the following beneficial effects:

[0019] This invention provides a method for measuring minute angles and thin film thickness using a vortex interferometry system. Compared to traditional conjugate vortex interferometry, which typically relies on complex optical devices and data processing algorithms to measure the angle between beams, this invention constructs a simulated vortex conjugate interferometry optical path and employs machine learning. This allows for accurate angle measurement using simpler devices and more efficient algorithms. Furthermore, the machine learning algorithm automatically learns and optimizes the model, thereby improving the accuracy and stability of the angle measurement. By learning from a large amount of training data, the machine learning model can identify and correct errors and biases that may exist in traditional methods, thus providing more accurate measurement results. This greatly simplifies experimental setup and data processing, improves experimental efficiency and reliability, and enhances accuracy and stability. This method can be used to measure thin film thickness. Attached Figure Description

[0020] Figure 1 This is a schematic diagram of the virtuallab simulation interference of the present invention;

[0021] Figure 2 The phase diagrams are for topological charge numbers of 2 and -2, respectively.

[0022] Figure 3 is a schematic diagram of feature extraction; Figure 3(a) shows a single vortex beam with a topological charge of 2, Figure 3(b) shows the interference result of vortex beams with a topological charge of ±2, and Figure 3(c) shows the centroid localization of the interference result;

[0023] Figure 4 The selected machine learning model;

[0024] Figure 5 Comparison of Gaussian process regression predicted values ​​with actual values ​​and confidence space;

[0025] Figure 6 The relative error of the Gaussian process regression prediction test set;

[0026] Figure 7 Comparison of prediction results on the support vector machine training set;

[0027] Figure 8 For comparison of prediction results on the support vector machine test set;

[0028] Figure 9 The relative error of the support vector machine model on the test set;

[0029] Figure 10 This is the basic structure of a BP neural network;

[0030] Figure 11 A comparison of prediction results for training sets of BP neural networks based on genetic algorithm optimization and particle swarm optimization;

[0031] Figure 12 A comparison of prediction results on the test set of BP neural networks based on genetic algorithm optimization and particle swarm optimization;

[0032] Figure 13 The relative error of the test set of the BP neural network based on genetic algorithm optimization and particle swarm optimization;

[0033] Figure 14 The changes in the fitness curve of the BP neural network optimized by genetic algorithm and the changes in the iteration error of the BP neural network optimized by particle swarm optimization are shown.

[0034] Figure 15 Comparison of prediction results on the training set of random forest;

[0035] Figure 16 Comparison of prediction results on the random forest test set;

[0036] Figure 17 The relative error of the random forest model on the test set;

[0037] Figure 18 This is the basic structure of an extreme learning machine;

[0038] Figure 19 Comparison of prediction results for the training set of the Extreme Learning Machine;

[0039] Figure 20 Comparison of prediction results for the Extreme Learning Machine test set;

[0040] Figure 21 The relative error of the Extreme Learning Machine test set;

[0041] Figure 22 The mean relative error of the six model test sets. Detailed Implementation

[0042] The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0043] The method for calculating the optical axis angle of vortex conjugate interference proposed in this invention mainly includes three parts: vortex conjugate interference simulation, feature extraction, and machine learning regression prediction. Specifically, it includes:

[0044] Step 1: Simulation of vortex light conjugate interference:

[0045] The figure shows the simulated optical path diagram of the vortex light interferometer system built in the VirtualLab simulation software. Lens 1 and lens 2 together form a beam expander system with a magnification of 2. The storage function and the 2f system together form the actual spatial light modulator. Phase diagrams are loaded into the two spatial light modulators respectively, so that the positive and negative vortex beams are combined and interfered at the original data detector. The interference result diagram is then transmitted to the computer.

[0046] In the vortex light interference optical path, the laser generated by the laser is collimated and expanded by a beam expansion system composed of two ideal lenses and then reaches the beam splitter, where it is split into two beams, one vertical and one horizontal. The horizontal beam is converted into a vortex with a negative topological charge by a spatial light modulator, while the vertical beam is converted into a vortex with a positive topological charge by a spatial light modulator. The beams then interfere with the horizontal beam on the detector by a mirror.

[0047] There are two main methods for generating conjugate vortex Gaussian beams: the cylindrical lens conversion method and the Dove prism conversion method. However, since simulating the angle between two beams, an angle needs to be added to the spatial light modulator. Therefore, two spatial light modulators are added to the optical path here. This can be achieved by loading the SLM with...

[0048] The conjugate vortex light can be obtained from the phase diagram of the topological charge number conjugate shown in the figure.

[0049] Step 2: Extract features from the interference pattern:

[0050] In this invention, the angle between the optical axes of the two interference optical paths is set at intervals of 0°-0.3°, and the interference image obtained by the simulated optical path is acquired at each angle. In this embodiment, an angle is set at intervals of 0.002°, resulting in a total of 151 interference patterns.

[0051] Feature extraction from interference patterns: First, find the center of a single vortex beam with the same topological charge. Then, locate the centroid of the petal-shaped pattern after interference of the vortex beams. As shown in Figure 3(b), the centroids of the upper two petals form an angle with the center of the circle shown in Figure 3(a), as shown in Figure 3(c). This angle is used as the input feature, and the corresponding optical axis angle is used as the output result for training.

[0052] To make mathematical calculations simpler and more intuitive, and to improve calculation accuracy and reduce errors, all angles are converted to radians. Some datasets are shown in Table 1.

[0053] Table 1. Partial Datasets

[0054]

[0055] Data normalization was performed when using all models to eliminate the influence of data units and orders of magnitude.

[0056] Step 3: Use machine learning for regression prediction:

[0057] Since the dataset is small and has only one feature, some regression prediction models are not applicable. Therefore, only the six models shown in the figure that are suitable for small training sets were selected for regression prediction.

[0058] In training, shuffling the dataset for regression prediction eliminates the sequential nature of the data, preventing the model from depending on a specific order and thus improving its generalization ability. Secondly, shuffling the dataset reduces overfitting to specific samples, making the model more robust. It also reduces model bias caused by uneven data distribution, improving prediction accuracy. Furthermore, shuffling the dataset for regression prediction increases the model's global learning of the entire dataset, allowing it to better capture its features and patterns. In conclusion, shuffling the dataset effectively improves the performance and stability of regression prediction models.

[0059] Gaussian process regression: Suitable for regression prediction problems with small datasets and where uncertainty needs to be considered. Based on Bayesian theory, Gaussian process regression can make predictions based on the distribution of existing data, and the confidence interval is given as shown in the figure.

[0060] After Gaussian process regression prediction, the relative error obtained by predicting the shuffled random test set is shown in the figure. It can be seen that the maximum relative error is greater than 4%, and the prediction accuracy is poor.

[0061] Support Vector Machines (SVMs) are suitable for classification and regression problems on small datasets. SVMs perform classification or regression by finding an optimal hyperplane, exhibiting good generalization ability and robustness.

[0062] The dataset was trained using a support vector machine model, and the prediction results are shown in Figures 1 and 2. The relative error of the prediction results was calculated for the random test set, as shown in Figure 3. It can be seen that the maximum relative error can reach 15%.

[0063] Backpropagation neural networks based on genetic algorithm optimization and particle swarm optimization: suitable for regression prediction problems on small datasets, but parameter optimization is required.

[0064] Backpropagation (BP) neural networks are powerful artificial neural network models. Their basic structure, as shown in the figure, is a multi-layer feedforward neural network, typically composed of three layers of neurons, each layer containing multiple neurons that operate independently. Once the three-layer network model is successfully built, an input sample is given to the network. The sample vector is initially transmitted from the input layer neurons to the hidden layer units, processed layer by layer through intermediate layers, and then sequentially output to the output layer units. This transmission to the output layer units yields an actual sample output. This process of propagating and modifying the state layer by layer is called forward propagation. When the output sample vector does not match the expected output, an error is generated. This error needs to be propagated backward, which is also a process of propagating and modifying the connection weights of each layer. This propagation process needs to be continuously repeated until a training pattern is completed or the error reaches a minimum value, and the output result meets the expected output.

[0065] It can continuously adjust weights and biases through backpropagation to reduce prediction errors and improve the accuracy of regression predictions. When using a genetic algorithm to optimize the regression prediction problem of a BP neural network, the parameters of the BP neural network are used as the objective function of the genetic algorithm, and the prediction error is used as the fitness function to continuously update the parameters of the BP neural network. The purpose of this is to obtain better regression prediction results.

[0066] Genetic algorithms and particle swarm optimization can be used to optimize the weights and biases of a backpropagation neural network, thereby improving the network's predictive performance.

[0067] Using two optimized BP neural networks to predict the relative error on a random test set, we can see that the particle swarm optimization-based BP neural network model performs better on this test set. However, this does not mean that it will outperform the genetic algorithm-optimized BP neural network on all datasets.

[0068] The downward trend of the fitness curve can reflect the convergence of the neural network. Figure 14 The fitness value shown gradually decreases and tends to stabilize, indicating that the training process of the BP neural network based on genetic algorithm optimization and particle swarm optimization is proceeding normally and has converged to a good solution.

[0069] Random Forest: Suitable for classification and regression problems with small datasets. Random Forest is an ensemble learning method that makes predictions by building multiple decision trees and voting or averaging them, exhibiting good accuracy and robustness.

[0070] Random forest is an ensemble learning method. It trains multiple decision trees and then combines the results to obtain the final outcome. For regression problems, the prediction result of a random forest is the mean of the outputs of all decision trees.

[0071] Random forests can effectively handle missing data and imbalanced datasets, exhibiting strong robustness. Furthermore, random forests can reduce overfitting, improve the model's generalization ability, and demonstrate high accuracy and stability.

[0072] Figure 14 The changes in the fitness curve of the BP neural network optimized by genetic algorithm and the changes in the iteration error of the BP neural network optimized by particle swarm optimization are shown.

[0073] Figures 1 and 2 compare the prediction results of the random forest model on the training and test sets. The prediction results of the two figures are similar and perform well, indicating that the random forest model has good generalization ability and can make accurate predictions on new data.

[0074] The figure shows the relative error of the random forest's prediction results for a random test set. At this point, the maximum relative error for this random test set reaches 7%, and the distribution of the relative error is relatively large, indicating that there is a large difference between the model's prediction results and the true values, and the model's prediction accuracy is low.

[0075] In the test results, the random forest model may perform poorly in predicting some data, but performs well in predicting others. Therefore, we chose to retain this model and integrate it with other models at the end, using the mean of the prediction results to obtain better predictions.

[0076] Extreme Learning Machines (ELMs) are suitable for classification and regression problems on small datasets where fast training and prediction speeds are required. An ELM is a single-hidden-layer feedforward neural network that achieves rapid training and prediction by randomly initializing weights and biases.

[0077] Extreme Learning Machine (ELM) is a single hidden layer feedforward neural network algorithm (SLFN). Its model structure consists of three layers: an input layer, a hidden layer, and an output layer. Similar to artificial neural networks, the connections between each layer are made using feature mapping functions. The information from the input layer is processed by the hidden layer and then passed to the output layer. The output layer then calculates the value according to the mapping function.

[0078] Figures 1 and 2 compare the prediction results of the Extreme Learning Machine model on the training and test sets, respectively. It can be seen that the prediction results in both figures are similar and perform well, indicating that the Random Forest model has good generalization ability and can make accurate predictions on new data. Figure 3 shows the relative error on the random test set; the error is less than 2.5%, indicating that the model's training effect is good.

[0079] To reduce prediction errors, the differences in the performance of different regression prediction models when predicting the same data were investigated. Therefore, a model ensemble method was adopted, which involves averaging the prediction results of multiple models. This approach comprehensively considers the prediction results of multiple models, resulting in a more accurate prediction. This method has been proven effective in reducing prediction errors and improving prediction accuracy through multiple regression predictions.

[0080] This invention calculates the angle between the optical axes of conjugate vortex light interference by integrating multiple machine learning regression prediction models. After multiple calculations, the six models are integrated for prediction. The test results show that the error is generally maintained at around 2% within the range of 0.006°-0.3°, with only a few data points exceeding 2%, but all less than 4%. Figure 22 As shown.

[0081] This invention also provides a thin film thickness measurement method based on the aforementioned angle measurement method. For a vortex interferometry system used to determine the angle error, since the angle error is known, placing the thin film under test in any optical path with positive or negative topological charge can change the deflection angle of that optical path, thereby generating an optical axis angle with another optical path. The angle extracted from the interference pattern obtained by the vortex interferometry system is input into a selected machine learning regression prediction model, each outputting an angle value, and the average is used to obtain the final angle. Since the optical axis angle has a certain relationship with the thin film thickness, the thin film thickness (refractive index) can be obtained by calculating the optical axis angle.

[0082] In summary, the above are merely preferred embodiments of the present invention and are not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method of measuring a small included angle of a vortex light interference system, characterized by, include: Step 1: Construct the simulated optical path of the vortex light conjugate interference system in VirtualLab simulation software. In the constructed simulated optical path of the vortex light interference system, the laser generated by the laser is collimated and expanded by a beam expander system composed of two ideal lenses before reaching the beam splitter, where it is split into two beams, one vertical and one horizontal. The storage function and the 2f system constitute the actual spatial light modulator. The two storage functions and the two 2f systems constitute two spatial light modulators respectively. Phase diagrams are loaded into the two spatial light modulators respectively, so that the positive and negative vortex beams are combined and interfered at the original data detector to obtain the interference result diagram. Among them, the optical axis angle is added in one of the spatial light modulators. Step 2: Set the angle between the optical axes of the two interference optical paths at intervals of 0°-0.3°, and collect the interference pattern obtained by simulating the optical path at each angle. Feature extraction from interference patterns: First, find the center of a single vortex beam with the same topological charge. Then, locate the centroid of the petal-shaped pattern in the interference pattern. The line connecting the centroids of the two upper petals in the interference image with the center of the circle forms an angle, which is used as the input feature. The corresponding optical axis angle is used as the output result, forming a training dataset. Step 3: Select multiple machine learning models with regression prediction capabilities and train the machine learning models based on the training dataset; Step 4: For the vortex interferometer system under test, extract features from its output interference pattern according to the method in Step 2. Input the input features into the selected machine learning model to obtain the output optical axis angle. Take the average of the optical axis angles of all machine learning models as the angle between the two interference optical paths of the vortex interferometer system under test.

2. The method for measuring the minute included angle of a vortex optical interferometer system as described in claim 1, characterized in that, The machine learning models include support vector machines, BP neural networks optimized based on genetic algorithms, random forests, BP neural networks optimized based on particle swarm optimization, extreme learning machines, and Gaussian process regression prediction models.

3. The method for measuring the minute included angle of a vortex optical interferometer system as described in claim 1, characterized in that, The set step size is 0.002°.

4. A thin film thickness measurement method based on a method for measuring the minute included angle of a vortex optical interferometer system according to claim 1, 2, or 3, characterized in that, By placing the thin film under test in any interference optical path, the deflection angle of that optical path can be changed, thereby generating an optical axis angle between it and another optical path. The angle features extracted from the interference pattern obtained by the vortex optical interference system are input into the selected machine learning regression prediction model, and each outputs an optical axis angle value. The average value is then used to obtain the final angle. Based on the relationship between the optical axis angle and the film thickness, the film thickness can be obtained by using the optical axis angle.

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