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Abstract
Optical scattering measurement is one of the most commonly used methods for non-contact online measurement of film properties in industrial film manufacturing. Terahertz photons have low energy and are non-ionizing when measuring objects, so combining these two methods can enable online nondestructive testing of thin films. In the visible light band, some materials are transparent, and their thickness and material properties cannot be measured. Therefore, a method based on physical consistency modeling and machine learning is proposed in this paper, which realizes the method of obtaining high-precision thin film parameters through single-frequency terahertz wave measurement, and shows good performance. Through the experimental measurement of organic material thin films, it is proved that the proposed method is an effective terahertz online detection technology with high precision and high throughput.
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1. Introduction
Optical scattering measurement is a non-contact method for measuring the optical constants of thin films [1]. Optical constants are important for evaluating the material properties of special coatings, chip integrated circuits and other optoelectronic devices [2–4]. At the same time, optical constants, as a means to characterize the macroscopic and microscopic properties of substances, provide a reliable basis for the analysis of electronic structure [5,6], doping concentration [7] and polymer properties [8,9]. Ellipsometry [10] in optical scattering measurement is a method to calculate the optical constant of thin films by detecting the polarization state changes of incident light and reflected light. Because of its characteristics, it has been widely used in scientific research and industrial production. Therefore, the progress of optical scattering measurement technology will bring great influence and extensive benefits to the measurement field. The forward physical process of ellipsometry measurement is to measure the ratio of the complex reflection coefficients of parallel polarization p light and vertical polarization s light, and then obtain the real part amplitude ratio $\psi$ and the imaginary part phase difference $\Delta $ of the ellipsometry parameter. In practice, the ellipsometer is a process of inverse solution, which is mathematically an inverse problem, which is inherently more difficult to study than the forward problem, and such inverse problems are nonlinear regression problems without analytical solutions [11–13].
The traditional inverse problem solving method is difficult to solve this kind of optical scattering measurement problem [14,15]. Since it is a trial-and-error learning based on the experience of human experts, it is necessary to artificially provide a good initial guess to achieve the fitting convergence, so the solution of the ellipsometry problem is not unique [16,17]. So far, in addition to Cauchy's dispersion formula, there are other commonly used dispersion formulas such as Hartmann's dispersion formula, Sellmeier's dispersion formula and Hetzberger's dispersion formula to characterize the properties of materials. In order to solve specific practical problems, the rich experience of human experts is needed to make the best choice among them. In addition, the analysis of the model often needs to combine a variety of dispersion models to accurately describe the sample, resulting in too many initial parameters, and the traditional fitting method is difficult to converge to get the correct solution, resulting in the whole process has to be repeated until the correct solution is fitted. This traditional method makes the ellipsometry measurement process time-consuming and inefficient. It has been shown that by supplementing the data with auxiliary measurement information of light intensity reflection [18,19], the above mathematical fuzziness can be eliminated and a unique solution can be generated [20]. However, the introduction of additional data also brings new problems, the additional information magnifies the shortcomings of traditional methods, increases the complexity of fitting, and reduces the practicality. And in the terahertz band, there is no wide-band wave plate, which leads to the terahertz ellipsometer can only use a single wavelength for measurement, if the traditional ellipsometer method is used for measurement, the measurement accuracy of the terahertz ellipsometer will be greatly reduced.
In recent years, the field of artificial intelligence has made unprecedented progress [21], especially the use of deep learning methods to solve practical problems and improve efficiency. This efficient method, which can handle real problems in real time, solves many problems that cannot be solved by traditional methods [22–24]. This trend has also led the industry to pay attention to deep learning. The ability of neural networks to extract potential features of data that cannot be noticed by human experts and use these high-dimensional features into the calculation of results makes neural networks very effective in the field of regression. This property makes neural networks a powerful tool for analyzing problems in the field of ellipsometry measurement, and this method has been applied in the visible light band and achieved remarkable results [25–27].
In this paper, a deep learning based ellipsometry method (TEML) is proposed, which exploits the physical consistency of optical scattering, based on the physical modeling of ellipsometry, and uses neural networks to solve the problem of terahertz ellipsometry. On the basis of ellipsometry parameters, the reflection and absorption spectra based on light intensity are added to enrich the data and reduce the fuzziness of the inverse problem. Combined with TEML spectra analysis, we successfully obtained the unique solution of the measured films. The TEML method is further verified by taking the typical organic polymer materials LPDE and PBAT as examples. The method does not require the intervention of prior knowledge of human experts to achieve instantaneous and accurate calculation of the results. The TEML method shows good analytical performance, and breaks through the limitations of multi-wavelength measurement in the terahertz band and polarization wave plate without broadband response, paving the way for automatic, efficient and high-throughput optical characterization of optical thin films. This is of great significance for high-precision real-time non-destructive measurement of thin films. This will be an epoch-making step for the terahertz ellipsometer to make intelligent and fully automatic non-destructive measurement.
2. Methods and networks
2.1 Overall process
In order to solve the inverse ellipsometry problem, it is necessary to understand the forward physical process of ellipsometry, as shown in Fig. 1. The ellipsometry principle deduces the properties of a material based on the change in the polarization state of the light incident on the surface of a sample. About two centuries ago, Malus discovered that the “birefringence” behavior of light incident obliquely on a material led to a well-developed physical description of the forward process in ellipsometry. The mathematical description of this anisotropic behavior was later completed by Fresnel with a set of equations now known as Fresnel's formulas [28]. Airy then went on to propose a multi-beam interference formula to calculate the reflection r and transmission t coefficients of the thin film covering surface [29]. Further, we can calculate the reflectance R and transmittance T of two kinds of polarized light, p light and s light. Since the terahertz wave is completely reflected on the metal surface, the transmittance T of the thin film material placed on the metal substrate is 0. Therefore, for the actual terahertz ellipsometer, the reflectance R and absorptivity A can be obtained in the actual experiment according to the energy conservation Eq. (1):
Then, a set of film parameters $(n,k,d)$, where n is the refractive index, k is the extinction coefficient, and d is the film thickness, can be analytically and uniquely determined using the ellipsometry spectra $(\psi ,\Delta )$ and the supplementary spectra $(R,A)$, as follows:
In our work, TEML aims to solve the following optimization problems:
The solution of Eq. (3) is a process of iteratively updating parameters. Following this approach and powered by machine learning, we propose the following TEML approach, which is based on a deep neural network with a dataset full of ellipsometry knowledge learned from offline and online training, and can effectively solve Eq. (3). In order to bridge the gap between simulation data and real experimental data, we propose a novel network iteration method with cyclic feedback, using deep neural networks as the backbone, as shown by the gray and yellow arrows in Fig. 2. The method consists of two neural modules, the inverse module and the forward module (circled by a blue rectangle in Fig. 2), which act together in a closed loop. From the perspective of network operation flow, the method takes $(\psi ,\Delta ,R,A)$ as input, and automatically outputs a set of solutions $(n,k,d)$. After proper training, the inverse module is trained as a substitute for the functions ${F^{ - 1}}$ and ${G^{ - 1}}$ and generates candidate solutions $(n,k,d)$. The forward module is trained as a substitute for the functions F and G, automatically output a set of solutions $(\psi ,\Delta ,R,A)$ based on the candidate solutions $(n,k,d)$. Specifically, given a set of data $(\psi (0),\Delta (0),R(0),A(0))$, a set of $(n(0),k(0),d(0))$ solutions are computed through the inverse module. Then, a solution set about $(n(0),k(0),d(0))$ is formed by data degradation and data augmentation in the small neighborhood of $(n(0),k(0),d(0))$, and then the forward module is executed to simulate the calculation flow of F and G, and the ellipsometry data $(\psi ,\Delta ,R,A)$ is reconstructed by calculating the solution set $(n,k,d)$. Then, RMSE is used to calculate the error values of the forward module predicted $(\psi (1),\Delta (1),R(1),A(1))$ data and the real $(\psi (0),\Delta (0),R(0),A(0))$ data, the trainable weights in the inverse module are updated by backpropagation error values of the two modules. RMSE [30] is widely used in regression tasks and the mathematical expression of RMSE is shown in Eq. (4):
where ${x_{predict}}$ represents the predicted value of the network, ${x_{true}}$ represents the actual value, and n is the number of parameters of the same type.The updated inverse module will generate a better solution $(n(1),k(1),d(1))$. This process is repeated until the generated $(\psi (t),\Delta (t),R(t),A(t))$ of $(n(t),k(t),d(t))$ can reconstruct $(\psi (0),\Delta (0),R(0),A(0))$ well enough to consider that the network performance is optimal and the training is stopped.
In detail, before the $(\psi ,\Delta ,R,A)$ data is input into the inverse module, it needs to be split into two groups of ellipsometry data $(\psi ,\Delta )$ and supplementary spectral data $(R,A)$, multiplied by their respective weight coefficients $\gamma$, and input into the Stacked Residual U Module (SRUM) in the inverse module. The forward module also needs to configure the weight coefficient when output. The weight coefficient of each calculated group $(\psi ,\Delta ,R,A)$ is configured, and then the loss value between the calculated group and the real data is calculated and used for error transmission and automatic optimization of network gradient.
For the inverse and forward modules in the TEML framework, we use the SRUM architecture as the backbone, which can be considered a variant of U-net. In our case, both the inverse and forward modules are obviously dealing with data regression task, for which the encoder-decoder U-net and its variants are often the most efficient architecture [31].
Like the SRUM structure in forward and inverse modules, there is only a difference in the number of feature data elements in the input and output when performing different module tasks. Each SRUM consists of three identical U-nets and additional input and output layers to fit the target tensor of the input and output, as shown in Fig. 3.
Fig. 3. Stacked Residual U Module (SRUM). Except for the input and output layers where the convolution kernel size is 1, the convolution kernel size of all other convolution layers is 3.
2.2 Data degradation and data enhancement
The $(\psi ,\Delta )$ and $(R,A)$ data measured by the experiment will inevitably contain noise (such as fluctuations in the radiation intensity of the light source, ellipsometer system errors and detector noise). In order to make the neural network acquire ellipsometry knowledge, TEML is first trained offline on a large number of simulation data, which may have some deviation from the real experimental ellipsometry data. Therefore, we use data degradation and data augmentation methods to improve the robustness of the neural network. We perform a data degradation operation on the solutions $(n(t),k(t),d(t))$ generated by each iteration, first generating some random points in the neighborhood of the current solution $(n(t),k(t),d(t))$ and then calculating the corresponding pseudo-measurements using the forward module. The specific method of data degradation is as follows: 1) the input feature $(\psi ,\Delta ,R,A)$ is randomly shifted to the left or right by a scale not greater than 0.03; 2) Multiply the input feature $(\psi ,\Delta ,R,A)$ by a small random scale ${N_r}$, where g is a random number generated by a standard normal distribution and ${N_r}$ represents a noise level not greater than 10−3. Each data degradation method generates 100 sets of solutions, so a total of 300 sets of solutions plus the current solution are generated to increase the degradation space of the dataset in the neighborhood of the current solution $(n(t),k(t),d(t))$. By training the forward module on the processed solution set, the forward module can better approximate the real functions F and G in the neighborhood of the current solution. In order to clearly illustrate the effects of data degradation and data augmentation methods, especially when the training data is insufficient or does not contain the noise that needs to be processed, we conducted a series of experiments on the training set using samples with different degrees of processing, and plotted the relationship between network error RMSE and noise ${N_r}$, as shown in Fig. 4.
Fig. 4. The influence of different degree of data degradation and data augmentation methods on the accuracy of network measurement.
It can be seen that when performing data degradation, the network has strong robustness in the noise range of $[0,1 \times {10^{ - 2}}]$, indicating that the network has high measurement accuracy. As the noise value ${N_r}$ is greater than $1 \times {10^{ - 2}}$, the network error starts to increase. After data augmentation, the error is significantly reduced, indicating that the network measurement accuracy has been improved. Deeper processing leads to a more significant error reduction until it reaches a saturation point where the improvement between the augmentation of x10 and the augmentation of x20 is minimal. The data augmentation of x5 and the stronger data augmentation strategy have less impact on the error, and considering the network training efficiency, we choose the data augmentation strategy of x5 comprehensively. This shows that the proposed data processing method is an effective tool to improve the network performance. For both the inverse and forward modules, the ADAM [32] optimizer with a learning rate of ${10^{ - 3}}$ and a weight decay coefficient of ${10^{ - 5}}$ was trained on the simulation dataset. We also applied a step-decay strategy to vary the learning rate. The loss function is the root mean square error RMSE. In our experiments, the computer hardware configuration is i5 CPU and NVIDIA GTX 3090Ti GPU, and the software environment is PyTorch and Python 3.9.
2.3 Offline training and online training
During offline training, the solutions $(n(t),k(t),d(t))$ of the inverse module are processed using data processing methods. The simulation dataset consists of 1,980 pairs $(\psi ,\Delta ,R,A)$ and $(n,k,d)$, which are based on laboratory-prepared polymer materials in the theoretical range $(n,k)$ of the terahertz band and are mathematically modeled using forward physical functions of F and G combined with MATLAB, with d varying between 10 and 900 um. The step size of the material thickness is set to 10 um. The substrate is a 100 mm diameter round cake type 304 austenitic stainless steel which thickness is 10 mm, machined by computer numerical control machine tools. A high-precision optical translation platform that can be lifted and lowered freely is embedded under the substrate. Terahertz waves are fully reflected on the surface of the substrate.
Online training reads the pre-trained model obtained from offline training, and then the data set of online training is processed in the same way as the two methods of offline training, but here an additional degradation method is introduced: generating a vector with a random offset of the current solution $3 \times {10^{ - 2}}$, and then performing the same data augmentation operation. The ratio of simulation data sets assigned to offline and online training is 8:2.
2.4 Network robustness
In order to explore the robustness and accuracy of TEML method on optical film measurements, we use the network weights obtained from offline training to train the test set online, and calculate the RMSE by using the obtained $(n,k,d)$ measurements and $(n,k,d)$ theoretical values. In Fig. 5 below, we plot the RMSE of the $(n,k,d)$ values calculated by the online trained network against different noise levels ${N_r}$. It can be seen that when ${N_r}$ is less than the threshold of $1 \times {10^{ - 2}}$, TEML has high online measurement accuracy, and as the noise added to $(\psi ,\Delta ,R,A)$ is too large, $5 \times {10^{ - 2}}$, $1 \times {10^{ - 1}}$ respectively, the RMSE of $(n,k,d)$ measured by the network starts to increase, and the model measurement accuracy starts to decrease.
Fig. 5. Influence of different levels of noise ${N_r}$ on the online training accuracy of the network. The network has strong robustness and accuracy in the noise range of $[0,1 \times {10^{ - 2}}]$.
When training online, we can choose the weights $\gamma$ of the functions F and G based on their relative importance, however, there is no strong evidence in our case to favor either $(\psi ,\Delta )$ or $(R,A)$. It can be seen that this is a typical dual-objective optimization scenario [33]. If $\gamma$ is set to 1, only $(\psi ,\Delta )$ comes into play, whereas if we set $\gamma = 0$, $(R,A)$ dominates the optimization process. In this study, we chose 0.5 to balance the contributions of the two items and showed good results in our experiment (as shown in Fig. 7).
3. Experiment and analysis
As shown in Fig. 6, the terahertz ellipsometer optical measurement experimental device, the terahertz transmitter on the far right radiates terahertz waves with a frequency of 0.3 THz through the horn antenna, then passes through the beam collimating lens to form parallel light and illumines the 1/4-wave plate, and then passes through the focusing lens to form light spot with a diameter of 5 mm on the sample. The beam is scattered after interaction with the sample, collected through the collimating lens and illuminated onto the 1/2-wave plate, and finally collected through a focusing lens to the terahertz receiver horn antenna port.
Fig. 6. Experimental device diagram. The incidence and reflection angles of the ellipsometer are both 45°, and all waveplates respond at a frequency of 0.3 THz.
Fig. 7. (a) represents the actual thicknesses of LDPE films (blue dashed line) and experimentally measured thicknesses (red solid line); (b) represents the actual thicknesses of PBAT films (blue dashed line) and experimentally measured thicknesses (red solid line); (c) represents the theoretical complex refractive index of LDPE film (blue dashed line) and experimentally measured complex refractive index (red solid line). (d) represents the theoretical complex refractive index (blue dashed line) of the PBAT film and the complex refractive index (red solid line) of the experimental measurement.
We first find the angle at which the 0.3 THz linearly polarized light is changed to circularly polarized light by rotating the 1/4-wave plate by 1° and the 1/2-wave plate by 180°, and adjust the position of the 1/4-wave plate to ensure that the polarization state of the beam irradiated by the ellipsometer to the sample surface is circularly polarized. Then the angle of the 1/4-wave plate is fixed, the sample to be measured is placed on the substrate, and the 1/2-wave plate is only rotated 180° to detect the polarization change of the beam after passing through the sample.
We demonstrated TEML's performance in our experiments, measuring a range of LDPE and PBAT organic material films. The raw materials of the film are all small granular solids. By adding these small granular raw materials into the metal heating mold and melting and heating for 5 minutes at 150 degrees Celsius, the metal mold will discharge the excess liquid by extrusion during heating, so as to prepare the film with high precision. After the film solidifies, it will be cut into cuboid pieces with each side length of 50 mm. Figure 7 shows the experimental measurements of LPDE and PBAT films, respectively. The dotted blue line depicted by the dots in Fig. 7(a) and (b) represents the actual thickness data of films. We input the experimental ellipsometry data $(\psi ,\Delta )$ and supplementary spectral data $(R,A)$ into TEML, and then the network automatically calculates and outputs the predicted thickness, as shown by the red solid line depicted by the dots in Fig. 7(a) and (b). The height coincidence of the red and blue lines indicates that the thicknesses calculated by TEML has a small error with the actual thickness of the films, indicating that the measurement accuracy is high. Taking 800 um LPDE and PBAT films as an example, we measured the complex refractive index of the two materials and compared it with the theoretical complex refractive index. The results in Fig. 7(c) and (d) show the accuracy of our method, and the prediction time of each group is about 0.04s, proving that our TEML method is a single wavelength measurement method with high precision online measurement.
Our TEML method has excellent measurement capability and robustness over hundred-micron thicknesses, because the traditional fitting technique only fits the ellipsometry data $(\psi ,\Delta )$, TEML can fit both $(\psi ,\Delta )$ and $(R,A)$, here $(R,A)$ spectral data as a supplement to reduce the fitting ambiguity and select the most appropriate $(n,k,d)$ solution. The traditional ellipsometer can measure the thickness and complex refractive index of a single layer film through multi-wavelength broadband, but our method can measure the thickness and complex refractive index of a single layer film with a single wavelength, and TEML has a high measurement accuracy in a certain range smaller than the wavelength, which provides a basis for our further research on multi-wavelength measurement of multilayer films. Our method can be directly applied to all optical scattering ellipsometers without hardware replacement.
4. Summary and discussion
In conclusion, this paper presents a method based on machine learning to solve the ellipsometry optical thin film measurement problem in a highly accurate and fully automatic way. Thanks to the excellent performance of deep neural network as the backbone in solving nonlinear regression tasks and the advantages of data processing flexibility, the proposed TEML method can learn ellipsometry measurement knowledge to guide the inverse optimization process, thus avoiding the intervention of human experts, so it is more convenient to use. In addition, since the supplementary spectral data $(R,A)$ and the traditional ellipsometry data $(\psi ,\Delta )$ are analyzed simultaneously, TEML can solve the problem of $(n,k,d)$ ambiguities faced by traditional fitting techniques, thus improving the measurement accuracy.
It is worth noting that our method is based on a single wavelength to measure the thickness of a single layer film, while the traditional method uses multiple wavelengths to measure the thickness of a single layer film, and the measurement accuracy of our method is not inferior to the traditional method, which provides a theoretical and experimental basis for our further research on multi-wavelength measurement of multilayer film thickness. The machine learning driven TEML method is compatible with the existing ellipsometers, and lays a foundation for the automatic, fast and high-throughput ellipsometry measurement of terahertz thin films. This method is beneficial to real-time quality monitoring of layered structures with high precision. As a general machine learning framework for solving inverse problems without analytical solutions, the proposed TEML method can also be extended to other terahertz optical measurement techniques. In the future, we will continue to optimize the TEML method, improve the accuracy of our method and the variety of measured materials through further data processing and ablation experiments, and expand the material model, so that it can be used in a wider range of optical measurement scenarios.
Funding
National Natural Science Foundation of China; National Defense Pre-Research Foundation of China (61422160107).
Acknowledgments
K. W. conceived of the idea and Z. M. wrote the code. Z. M. performed the experimental measurements under the supervision of K. W. and Z. M. performed the main theoretical derivation and wrote the paper. All authors discussed the results and contributed to the writing of the paper.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data will be made available for research purposes upon reasonable request.
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