1. Introduction

By focusing the laser beam on the resist surface and controlling the energy point by point using a computer, the laser direct writing lithography (LDWL) can realize surface relief structures with different depths. The technique is widely used in the fabrication of various micro-optical devices, such as linear grating, diffractive optical element, micro-optical waveguide, and micro-lens array [1–6]. However, in the actual fabrication process, the focused spot of the laser beam is not infinitesimal, but with a size of micrometer, which limits the resolution of LDWL. Moreover, the spatial distribution of light intensity of the spot is not uniform, meaning that the exposure of each point will affect the surrounding points, which is the so-called proximity effect [7,8]. In addition, the obtained surface pattern is also influenced by the exposure dose, the type of photoresist, the developing time, the development process, and the stability of the motorized stage used in the laser direct writing system [9,10,11]. These factors could lead to a large deviation between the surface pattern of the final fabricated and that of the desired.

In order to eliminate these deviations and achieve high-precision fabrication of micro-optical devices, the initial exposure intensity distribution data imported to the laser direct writing system needs to be compensated and modified. Some researchers have focused on optimizing the manufacturing process of LDWL by establishing the exposure model that can describe the relationship between various experimental parameters and the actual structure. For example, J. R. Salgueiro et al. presented a model to describe the widths of the lines written on a photoresist as a function of various experimental parameters. The model can be used in the calibrating and fast tuning the laser direct writing system [12]. T. Onanuga et al. introduced a simulation flow that models light diffraction, exposure, polymerization, and development processes in LDWL and the model can be applied in the formulation of proximity ion algorithm [13]. N. Xie et al. proposed a comprehensive model based on the exposure intensity distribution of the laser beam and the critical dimension (CD) bias which can help to achieve dimensional accuracy in the exposure process [14]. However, these methods are relatively complicated because various factors should be considered, and mainly apply to improve and optimize specific simple structures. For three-dimensional microstructures such as aspheric micro-lens array, these methods generally need repeated modification and verification, which require highly experienced designers and are time-consuming.

In recent years, with the development and progress of computational science and hardware, artificial intelligence and especially deep learning have been widely used in many fields of scientific research and engineering technology [15–18]. Unlike traditional machine learning, deep learning uses multi-layers neural networks to analyze data, extract features, and make decisions, so that it can use existing data to learn and solve complex problems. In the field of lithography, deep learning has also gained some applications. For instance, Moojoon Shin et al. applied the neural network to the hotspot detection of lithography. By adjusting various parameters of the network model and analyzing the influence of each parameter on the performance of the model, a set of convolutional neural networks with excellent performance were obtained, and the accuracy of hot spot detection was improved [19,20]. To improve the accuracy of lithography, Xu Ma et al. developed a model-driven convolution neural network in the inverse lithography techniques, which can effectively improve the computational efficiency and further improve the imaging performance of coherent lithography systems [21]. Y. Watanabe et al. proposed a new compact resist model based on convolution neural network to reduce CD prediction errors and improve lithography simulation accuracy [22]. The above-mentioned works mainly focus on the fabrication of binary optical elements and simple line structures, while the processing of three-dimensional relief micro-optics elements requires higher precision in controlling the exposure dose. These methods are not applicable, and related work has not been reported.

In this paper, a method based on U-Net convolutional neural network for fabrication error modification of LDWT of three-dimensional microstructure is proposed. Traditionally, U-Net convolutional neural network is mainly used to establish the correlation of different graphical features, such as image segmentation [23]. Taking full advantage of the local feature recognition and nonlinear characteristics of the convolution neural network, we use the U-Net convolutional neural network to describe the complex process of lithography. The structural data actually fabricated by the LDWL is used as the input of the network model, and the intensity distribution data imported to the laser direct writing system is used as the output of the network model. By building and training the U-Net convolutional neural network, the mapping relationship between the input and the output is established and thereby the process model of the LDWL is developed for the modification of the exposure error. This can be used in the optimizing of the manufacturing. Compared with the traditional construction of complex physical models or multiple test modifications, this method is much simpler and more efficient, and has more general applicability.

2. Principle and method

2.1. Description of LDWL processing deviation

As the sketch of the laser direct writing system shown in Fig. 1, the laser beam is focused by an objective lens and then scanned through the photoresist to generate the desired patterns. However, due to the size of the laser beam spot is finite and its intensity distribution is not uniform, the exposure dose on the photoresist of each pixel will be affected by the exposure of its surrounding pixels, which is called the optical proximity effects. As shown in the Fig. 1, the desired exposure dose of pixel B is marked as E. In the actual fabricate process, the exposure dose of the surrounding pixel A and pixel C would affect the value of E, making the pattern of pixel A deviates from the designed pattern. Similarly, the exposure dose of a specific pixel will not only be affected by the exposure dose of itself, but also that of its surrounding pixels.

 

Fig. 1. Sketch of the laser direct writing system and the optical proximity effect.

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In addition, according to the classical exposure model proposed by Frederick H. Dill [9], the exposure process of photoresist is a complex nonlinear photochemical reaction. When the light radiates on the photoresist, chemical reactions occur within the photoresist, which will further cause a change in the light distribution inside the photoresist, and that in turn affects the chemical reaction. Coupled with the optical proximity effects mentioned above, there is a large deviation between the designed pattern and actual fabricated structure. Especially when the desired microstructure becomes more complicated, this deviation would be more serious accordingly.

2.2. Building of the U-Net convolutional neural network

We define the intensity distribution data imported to the laser direct writing system as ${D_{design}}$, which is a two-dimensional matrix. The value of each pixel represents the intensity coefficient of the focused spot radiated on the corresponding pixel. At the same time, we define the measurement data of the actual fabricated structure on the photoresist after exposure and development as ${D_{measure}}$, which is also a two-dimensional matrix. In this article, the measurements can be carried out by the atomic force microscope (AFM). The value of each pixel represents the surface sag of the corresponding pixel. For this process, the following relationship is satisfied.

In this paper, we construct a U-Net convolutional neural network to solve ${F^{ - 1}}$. The U-Net architecture stems from the so-called fully convolutional network. By introducing a number of feature channels in the expansive path, which allow the network to propagate context information to higher resolution layers, the output and the input can have the same resolution. The overall framework is shown in Fig. 2.

 

Fig. 2. The brief architecture of the U-Net neural network structure. The meaning of all types of signs is marked on the bottom of the picture.

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2.2.1. Data collection and processing

Based on the maximum surface sag of the desired microstructure, we empirically determine the laser power of the system, the photoresist and the development time. Then the surface data of the desired microstructure is processed into the intensity distribution data ${D_{design}}$ that can be recognized by the laser direct write system. A 128-level data processing is carried out because the system is generally capable of 128 levels of laser intensity [24]. Then the ${D_{design}}$ is imported to the laser direct write system for exposure. After development and measurement, the surface sag of the actual structure can be obtained. In order to compare with ${D_{design}}$, we also processed it into 128 level as ${D_{measure}}$. Based on these procedures, a set of ${D_{measure}}$ and ${D_{design}}$ can be used as the input and output of the neural network, respectively. As shown in the upper part of Fig. 2.

In order to increase the diversity and the quantity of the data to improve the accuracy of the neural network, we also transformed the ${D_{design}}$ into $D_{design}^{\prime}$ using the Eq. (4).

2.2.2. U-Net construction

Firstly, we define the training dataset and test dataset. We normalize the ${D_{measure}}$ and ${D_{design}}$ to be the input tensor and target tensor of the neural network respectively. 4% of the data are randomly selected as test dataset and the rest as training dataset.

Then, as shown in the bottom half of Fig. 2, we build a U-Net convolutional neural network to establish the mapping relationship between inputs and targets. The entire network consists of a contracting path and an expansive path. The contracting path consists of repeated application of convolution with activation function and an operation of batch normalization, and each followed a max pooling operation. The convolution operation is used to emphasize features of the input data through convolution kernels. The output matrix $u^{\prime}({i,j} )$ in each convolution operation with activation function is given by Eq. (5).

Each layer in the expansive path consists of transposed convolution, a concatenation with the correspondingly cropped feature map from the contracting path, and the convolution operation followed by Leaky ReLu and batch normalization. To ensure the size of the final output tensor has the same size of the input, a convolution operation is implemented at the end of the network.

2.2.3. Training and prediction

Finally, the input tensors and target tensors are fed into the U-Net convolutional neural network we built. The model is trained on the training data with Adam as the optimizer and MSE as the loss function [25], and the model is tested on the test data to prevent overfitting. After training, we take the desired microstructure ${D_{desire}}$ as an input of the model, and then make predictions through the model to get the ${D_{modify}}$, which is our modified intensity distribution data based on the desired microstructure.

3. Experiment and discussion

We now apply the proposed method to the fabrication of concave micro-lens array which is mainly used in the field of laser beam shaping. The micro-lens has an aperture of 24 µm×24 µm and its expression is shown in Eq. (7).

 

Fig. 3. (a) The parabolic shape surface of the desired concave micro-lens; (b) The image of ${D_{design}}$; (c) The AFM measurement of the fabricated structure using the ${D_{design}}$ (d) The image of ${D_{measure}}$; (e) The center cross-sectional diagrams along the y-axis direction of ${D_{design}}$ and $D_{design}^{\prime}$ with different values of A.

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We use AZ4562 photoresist, and the Heidelberg's DWL66FS laser direct writing system, which is capable of 128 levels of laser intensity. The step size of the focused laser spot is set to be 0.25 µm, meaning that the size of each pixel is 0.25 µm. Therefore, we discretize the lens surface into a two-dimensional matrix with a size of 96×96 and the values of the matrix elements are integers between 0 and 127. The matrix is just the ${D_{design}}$ and is shown in Fig. 3(b). The pixel with the maximum value of 127 represents the place where the exposure dose is the largest, corresponding to the place where the surface sag is the smallest, and vice versa.

The ${D_{design}}$ is imported to DWL66FS for exposure, and the power of the laser is set to be 30 milliwatt. Then the samples are finally developed for 120sec in the diluted AZ 400 K developer, and then measured by AFM. The 3D structure of the fabricated micro-lens is shown in Fig. 3(c). In order to compare with the ${D_{design}}$, we discretize the measurement date into a two-dimensional matrix with a size of 96×96 and the values of the matrix elements are integers between 0 and 127. The matrix is just the ${D_{measure}}$ and is shown in Fig. 3(b). The pixel with the maximum value of 127 represents the lowest point of the fabricated concave micro-lens; vice versa. It can be seen from Fig. 3(b) and Fig. 3(d) that there is indeed a relatively large deviation between the actual fabricated and desired surface shape.

Here, to increase the diversity and the quantity of data, we define the transformation matrix $\omega ({i,j} )$ as Eq. (8).

The $D_{design}^{\prime}$ can be obtained by the following Eq. (9).

Keep the parameters ${A_l}$ and $\; {A_o}$ the same as before, and import the $D_{design}^{\prime}$ into the DWL66FS. After exposure, development and measurement, the $D_{measure}^{\prime}$ is obtained. To further improve the quantity and diversity of the data, all of the data include ${D_{design}}$, $D_{design}^{\prime}$, ${D_{measure}}$ and $D_{measure}^{\prime}$ are rotated and mirrored by rotating and mirroring the matrices respectively. We have an original dataset of 16 matrix pairs and 96 pairs after rotated and mirrored, which are divided into 70 pairs for training, 22 pairs for validation, and 4 pairs for blind testing.

The U-Net convolutional neural network is implemented in Python 3.7.9 based on the TensorFlow version 2.1.0 platform on a computer with an Intel Xeon CPU E5-2678 V3, 64 GB main memory, and an NVIDIA GeForce RTX 2080 Ti GPU. The Adam optimizer with a learning rate of 0.005 is adopted to optimize the weights and biases. It takes about 4 min with 2000 epochs. After training, we take the ${D_{design}}$ as the ${D_{desire}}$ and import it to the model to predict the ${D_{modify}}$. As shown in Fig. 4(a) and Fig. 3(b), the ${D_{modify}}$ is quite different from the original ${D_{design}}$. We use the $\; {D_{modify}}$ instead of the original ${D_{design}}$ for the fabricating by LDWL with the same parameters ${A_l}$ and ${A_o}$. The results are shown in Fig. 4.

 

Fig. 4. The results of the prediction and the measurement. (a) The image of ${D_{modify}}$ predicted by the trained U-Net convolutional neural network; (b) The AFM measurement of the structure fabricated by LDWL using the ${D_{modify}}$; (c) The image of the ${D_{measure}}$ of the modified structure; (d)(e)(f)(g) The cross-sectional diagrams along the line S1, S2, S3, S4 respectively. The line S1, S2, S3, S4 are marked in (c); (h) The graph of the loss function during training.

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Compared to the microstructure fabricated using the original ${D_{design}}$, the surface shape of modified structure is significantly improved. As the cross-sectional diagrams in four lines shown in Figs. 4(d)–4(e), the modified structure is basically consistent with the desired structure. Although there is a little difference in the y-axis direction, the improvement is obvious compared with the unmodified structure. In order to evaluate the three-dimensional microstructure quantitatively, we also calculate the mean squared error (MSE) and the peak signal-to-noise ratio (PSNR) of the unmodified and modified structure compared to the desire structure [26]. The MSE measures the average of the squares of the errors, and the smaller its value, the smaller differences between the actual structure and the desire structure. The PSNR is an image quality metrics, and used to measure the similarity of the actual structure with the desire structure here. A higher PSNR value provides a higher similarity, while a small value implies high differences between the structures. The results show that the MSE is decreased from 100 to 36, while PSNR is increased from 22 dB to 26.5 dB, which show a significant improvement in the performance.

Furthermore, we introduce a different concave micro-lens to verify whether the proposed method is suitable for other types of three-dimensional microstructures. The micro-lens structure has an aperture of 24µm×24 µm and has an expression shown in Eq. (7) with $\mathrm{\alpha} = 0.0168$ and $\mathrm{\beta} ={-} 0.00474$. As shown in Fig. 5(a), the lens has a saddle shape surface. We perform the same data collection and processing operations. By combining the saddle data with the previous parabolic data, we have an original dataset of 32 matrix pairs and 192 pairs after rotated and mirrored, which are divided into 140 pairs for training, 44 pairs for validation, and 8 pairs for blind testing. After training, we also perform a series of operations of prediction, fabrication, and measurement. The results are shown in Fig. 5. The MSE and PSNR of the unmodified and modified structure are calculated and shown in Table 1.

 

Fig. 5. (a) The saddle shape surface of the desired concave micro-lens; (b) The AFM measurement of the structure fabricated by LDWL using the original ${D_{design}}$; (c) The AFM measurement of the structure fabricated by LDWL using the ${D_{modify}}$; (d) The image of the ${D_{measure}}$ of the modified structure; (e)(f)(g)(h) The cross-sectional diagrams along the line S1, S2, S3, S4 respectively. The line S1, S2, S3, S4 are marked in (d); (i) The graph of the loss function during training.

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Table 1. The MSE and PSNR of the unmodified and modified micro-lens.

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Figure 5(b) shows a large deviation of the actual structure fabricated using the original ${D_{design}}$, compared to the desire surface shape shown in Fig. 5(a). Meanwhile, the structure fabricated using the ${D_{modify}}$ is improved significantly, especially at the aperture edge of the micro-lens, as shown in Fig. 5(c). The cross-sectional diagrams also show that the modified structure match better with the desired structure. As shown in the Table 1, the MSE is decreased from 151 to 50, while PSNR is increased from 20 dB to 25 dB.

The new trained U-Net convolutional neural network is also applied to modify the previous parabolic concave micro-lens and the results are shown in Fig. 6. The MSE and PSNR of the unmodified and modified structure are calculated and shown in Table 1.

 

Fig. 6. (a) The AFM measurement of the structure fabricated by LDWL using the ${D_{modify}}$; (b) The image of the ${D_{measure}}$ of the modified structure; (c)(d)(e)(f) The cross-sectional diagrams along the line S1, S2, S3, S4 respectively. The line S1, S2, S3, S4 are marked in (b).

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It can be seen from Fig. 5, Fig. 6 and Table 1 that the modified structure has a significant improvement. That means the method proposed has an effective modification both on the parabolic and the saddle microstructure. Actually, the method proposed here is also applicable to other types of three-dimensional microstructures.

It should be point out that the MES and PSNR of the modified parabolic structure is 22 and 29 dB respectively, which is much better than that of the previous modified parabolic structure with 36 and 26.5 dB. The same phenomenon could be found through the Figs. 6(c)–6(f), compared to the Figs. 4(d)–4(g). The reason is that with the import of the concave micro-lens data, the quantity and diversity of the training dataset have been increased, and the prediction accuracy of the model has also been improved. With the fabrication of other types structures in the future, the quantity and diversity of training dataset can be further increased, and the prediction accuracy of the network will be better, signifying there still has rooms for the improvement of our proposed method.

We also built the optical system for comparing the performances of the fabricated devices, as shown in Fig. 7(a). In the experiment, a He-Ne laser with the wavelength 632.8 nm is used as the light source. A diffusion film manufactured by BrightView technology company is used to increase the beam size and its central uniformity. After passing through the diffuser, the divergence angle of collimated beam is ± 2.5 degree and the stray light is filtered out by an aperture with 2.0 mm diameter. The laser beam passes through the micro-lens array and reaches at the observation screen placed at the position of 100 mm behind the micro-lens array. A camera with lens is used to capture the beam spot. Numerical simulations using ray tracing method are also performed. The experimental and simulation results are shown in Figs. 7(b)–7(g). As can be seen from Figs. 7(b), 7(e), the simulated beam spots of the saddle and parabolic micro-lens array are both rectangular. However, the results are quite different between the simulation and the experiment without the application of the modification method, which are obvious in the Figs. 7(c), 7(f). As a comparison, the beam spots shaped by the modified micro-lens array show a good agreement with the simulation results both for the saddle and parabolic micro-lens array, which are shown in Figs. 7(d), 7(g). This demonstrate that our method for the modification of the micro-lens structure is effective.

 

Fig. 7. (a) The schematic diagram of the optical setup for the beam shaping experiment; (b) The simulated beam spot of the of saddle microstructure; (c) (d) The experiment results of the unmodified and modified saddle microstructure, respectively; (e) The simulated beam spot of the of parabolic microstructure; (f) (g) The experiment results of the unmodified and modified parabolic microstructure, respectively;

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4. Conclusion

In this study, a modification method for LDWL of three-dimensional microstructures based on a U-Net convolution neural network is proposed. By using the intensity data of laser direct writing and the surface profile data of the actual fabricated microstructure as training dataset, the U-Net convolution neural network is trained, and the inverse model of the whole LDWL process is thus obtained. Through this model, a modified intensity distribution data is predicted, and so as to complete the modification of the LDWL. The experiments on parabolic and saddle micro-lens show the effectiveness of the method in the high-precision fabrication using LDWL. The method proposed here is also applicable to other types of three-dimensional microstructures. Compared with other methods, this method is simple, efficient and effective, and provides a new way for high-precision laser direct writing relief surface fabrication.