## Abstract

Imaging-based measurement methods of polarization aberration (PA) are indispensable in hyper-numerical aperture projection optics for advanced lithography. However, the current methods are derived from the Kirchhoff model and ignore the 3D mask effect of the test mask, which will impact the measurement accuracy. In this paper, a novel imaging-based measurement method of PA is proposed based on a rigorous imaging model to improve the measurement accuracy. Through the quantitative description of the 3D mask effect, a rigorous imaging-based measurement model of PA is established. A synchronous orientation measurement method is designed to effectively reduce the cost of establishing the overdetermined equations. A deep neural network is used to retrieve the PA accurately. Simulations show that the proposed method effectively eliminates the impact of the 3D mask effect of test mask on PA measurement, and the measurement error is reduced by 72% compared with the measurement method based on the Kirchhoff model.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Polarization aberration (PA) characterizes the changes in amplitude, phase, and polarization of light [1–4]. With the continuous improvement of integrated circuit technology nodes, the impact of PA on imaging deterioration is increasingly prominent in hyper-numerical aperture projection optics (PO) [5,6]. Many resolution enhancement techniques consider and compensate PA of the hyper-numerical aperture PO to improve image fidelity and process stability [7–9]. As a prerequisite for PA compensation, high-precision in situ PA measurement methods have become very important in advanced lithography.

Imaging-based techniques have been widely used for the in situ measurement of aberration [10–12] and PA [13–17] due to the advantages of lower cost and easier implementation in lithography tool. Liu established aberration measurement methods based on a scalar imaging model [10,11], which retrieved aberrations up to the 37th Zernike coefficients through the scalar aerial image and the linear or non-linear sensitivity matrix. Dong established an aberration measurement method based on a vector imaging model and improved the accuracy of aberration retrieval [12]. Then, based on the vector modeling model, some studies have established linear [13–16] and non-linear [17] imaging-based PA measurement techniques, and measured 6 × 37 orientation-Zernike coefficients, 6 × 37 Pauli-Zernike coefficients, and 8 × 37 pseudo-Zernike coefficients of PA, respectively. (non-linear method has higher accuracy). It can be seen that the imaging model is a critical part of imaging-based techniques, which improves the accuracy of aberration measurement and extends the measurement technology from aberration measurement to PA measurement. However, the vector imaging models adopted by these measurement methods all treat the test mask as a thin mask (Kirchhoff model) and ignore the 3D mask effect, which refers to the polarization phenomenon of the 3D structure of mask [18,19]. Many studies have shown that it can cause PA and degrade the imaging performance [20–24]. This PA will be doped with the PA of PO and may affect the measurement accuracy. Therefore, quantitative analysis and consideration of the PA caused by the 3D mask effect of test mask are required to further improve the measurement accuracy of the PA of PO.

Rigorous imaging models were proposed several years ago [25,26], which obtain the mask near-field spectrum through rigorous electromagnetic field simulation, so the 3D mask effect can be considered quantitatively. However, due to its extremely computationally expensive and non-analytical, it is difficult to directly use in lithography techniques such as imaging prediction, inverse lithography, and imaging-based measurement. Therefore, until recently, some innovative methods have proposed fast rigorous imaging models and inverse lithography based on rigorous imaging models through machine learning, derivative-free optimization, particle swarm optimization and other technologies [27–30].

In this paper, a novel imaging-based measurement method of PA is proposed based on a rigorous imaging model to improve the measurement accuracy. Through the quantitative description of the 3D mask effect as the form of polarization aberration ${{\mathbf J}_{\mathbf M}}$, a semi-analytical rigorous imaging model is obtained, and a rigorous analytical relationship between the aerial image spectrum and the PA is established based on it. According to the characteristics of the 3D mask effect, a synchronous orientation measurement method is designed, which effectively reduces the cost of building the ${{\mathbf J}_{\mathbf M}}$ library. A Deep neural network (DNN) is used to retrieve the PA accurately. Simulation and analysis show that the 3D mask effect of test mask will lead to a relatively large PA. The proposed method effectively eliminates the impact of the 3D mask effect of test mask on PA measuring, and the measurement error is reduced by 72% compared with the measurement method based on the Kirchhoff model.

## 2. Rigorous imaging model and 3D mask effect

The imaging model is a critical part for imaging-based measurement methods, which largely determines the accuracy of measurement method. In this section, we will introduce the rigorous imaging model and the description of the 3D mask effect.

#### 2.1 Rigorous imaging model

According to Abbe imaging theory, the rigorous imaging model can be expressed as follows [31,32]:

*S*is the effective source. ${\mathbf V}^{\prime}$ can be expressed as: where $A$, ${\mathbf V}$, and

*U*are the correction factor, the transfer matrix in the exit pupil of PO, and the low pass filter, respectively. Their specific forms are given in Appendix A. ${\mathbf J}$ is the PA expressed in form of Jones pupil, and we choose pseudo-Zernike basis [33] to expand it. Pseudo-Zernike basis can be described as: and ${\mathbf J}$ can be expressed as:

#### 2.2 Description of the 3D mask effect

The 3D mask effect refers to the polarization phenomenon of the 3D structure of mask, and this polarization phenomenon can cause focus shift, placement error, and critical dimension (CD) error [18,19]. In our earlier work [24], we proposed a method to quantify the 3D mask effect and describe it as the form of polarization aberration ${{\mathbf J}_{\mathbf M}}$:

## 3. Rigorous imaging-based measurement model of PA

Since the traditional rigorous imaging model is non-analytic and not conducive to the establishment of PA measurement model, we develop a semi-analytic rigorous imaging model. Import Eq. (5) into Eq. (1), the rigorous imaging model can be re-expressed as follows:

Figure 1 shows the principle of this imaging model and the traditional rigorous imaging model, where ① is the traditional rigorous imaging model, and ② is the semi-analytic rigorous imaging model. In this semi-analytic rigorous imaging model, the analytical far-field spectrum is beneficial to simplify the imaging model and establish the PA measurement model. Moreover, as long as the corresponding ${{\mathbf J}_{\mathbf M}}$ is imported, the rigorous imaging model can be completely analytic. Therefore, the semi-analytic rigorous imaging model lay a foundation for establishing a rigorous imaging-based measurement model of PA.

After obtaining the semi-analytic rigorous imaging model, a rigorous imaging-based measurement model of PA is proposed based on it. The one-dimensional dense line binary mask is selected as the test mask to discretize the rigorous imaging model. Its far-field spectrum under the Kirchhoff model is:

*t*and

*t*′ are the cutoff frequency, which can be expressed as:

## 4. Measurement method

In this section, the synchronous orientation measurement method is proposed to reduce the cost of establishing the overdetermined equations, and a DNN is designed to solve the overdetermined equations in reverse with high precision. They are given separately in the following two subsections.

#### 4.1 Synchronous orientation measurement method

According to the proposed PA measurement model, overdetermined equations can be established by measuring multiple aerial images, and PA can be obtained by solving the overdetermined equations in reverse. For each imaging in the establishment of the overdetermined equations, the corresponding ${{\mathbf J}_{\mathbf M}}$ needs to be imported, which requires a high cost to build a ${{\mathbf J}_{\mathbf M}}$ library. Therefore, in order to reduce the cost of establishing overdetermined equations, the synchronous orientation measurement method is proposed.

The 3D mask effect is related to the effective source and illumination polarization, as well as the thickness, pitch and orientation of the mask. Therefore, when the orientation of the mask, the effective source, and the polarization are changed synchronously, the relative position of the illumination and the mask does not change. They only change an orientation relative to the PO, so the changed ${{\mathbf J}_{{\mathbf M} + \theta }}$ is simply rotated relative to the previous ${{\mathbf J}_{{\mathbf M}0}}$, as shown in Fig. 2.

According to this principle, the specific steps of this method are as follows:

- (1) Select
*m*test masks with different pitches $({p_1},{p_2}\ldots {p_m})$, and for each mask, import its corresponding ${{\mathbf J}_{\mathbf M}}$ under the preset effective source and polarization to perform imaging: - (2) Select
*n*orientations $({\theta _1},{\theta _2},{\theta _3}\ldots {\theta _n})$, and synchronously set the orientation of the test mask, preset polarization, and preset effective source to these angles for imaging. The required ${{\mathbf J}_{{\mathbf M},\theta }}$ can be obtained by a simple rotation transformation of ${{\mathbf J}_{\mathbf M}}$: - (3) Bring these ${{\mathbf J}_{{\mathbf M},\theta }}$ into the measurement model, and considering m different pitches and $l$-order aerial image spectrum, a total of
*m × n × l*equations can be obtained:

The preset effective source and polarization can be rotationally invariant (such as Coherent Illumination, Conventional Illumination, Annular Illumination, TE polarization, TM polarization, etc.), and do not need to be changed during multiple imaging. It can be seen that according to the synchronous orientation measurement method, a ${{\mathbf J}_{\mathbf M}}$ can be multiplexed n times, and the rigorous nonlinear overdetermined equations of PA and aerial image spectrum can be established at low cost.

#### 4.2 Deep neural network algorithm

Neural network has shown tremendous ability in dealing with nonlinear problems, and has been widely used in wave-front sensing [36,37], aberration correction [38,39], multivariate regression [40], and other fields. Therefore, it is adopted to solve the nonlinear overdetermined equations to achieve high-precision retrieval of PA. Equation (18) can be expressed as the following nonlinear problem:

where $\{{\tilde{I}} \}$ is the measured multiple sets spectrum information of aerial images, $f({} )$ is the nonlinear function of $\{{\tilde{I}} \}$ and ${\mathbf P}$. The solution of this nonlinear problem is to obtain a function $g({} )$ that satisfies the following conditions:That is, the $g({} )$ function can obtain $\hat{{\mathbf P}}$ through multiple sets of spectrum information $\{{\tilde{I}} \}$, and the Euler distance between $\hat{{\mathbf P}}$ and the ${\mathbf P}$ is the smallest. A neural network is designed and trained to fit the function $g({} )$. In order to achieve high-precision fitting, a DNN framework is selected. Similarly, multiple hidden layers, nonlinear activation functions, Batch Normalization, and Drop are set to improve the accuracy of the fitting. In addition, hybrid samples are established in multiple ranges around the PA design value to train the neural network and improve its robustness. Finally, this algorithm fits the function $g({} )$ with high precision, and achieves high-precision retrieval of PA.

## 5. Simulation

This section contains two simulations. The first simulation analyzes the PA caused by the 3D mask effect of test masks with different pitches and patterns to evaluate its impact on PA measuring. The second simulation verifies the proposed method in measuring the PA coefficients up to 37th order, and compares it with the previous method based on the Kirchhoff model [17].

#### 5.1 PA of the 3D mask effect of test mask

According to Eq. (5), we analyze the polarization aberration ${{\mathbf J}_{\mathbf M}}$ of the 3D mask effect with different test mask pitches and patterns at different source points. In this simulation, two patterns are selected, as shown in Fig. 3. According to the commonly used pitch range of the test mask in the PA measurement method [13–17], the pitch range of the test mask in this analysis is set to 360 nm-1440 nm (corresponding to the image side line width 45 nm-180 nm), and some simulation parameters are given in Table 1. The calculations in the simulations are implemented in MATLAB, and the rigorous mask near-field spectrum is obtained by the FDTD algorithm of the commercial software PROLITH.

The ${{\mathbf J}_{\mathbf M}}$ pupils of pattern 1 and pattern 2 with a pitch of 360 nm at the source point (0, 0) are shown in Fig. 4. It can be seen that the values of $J{m_{xx}}$ and $J{m_{yy}}$ are in the order of 10^{−1}λ, and most of them are negative, which are quite different from their ideal value 1 + 0iλ (Ideally, each entry of ${{\mathbf J}_{\mathbf M}}$ is a 2 × 2 identity matrix and has no effect on the polarization of incident light). The $J{m_{xy}}$ and $J{m_{yx}}$ are in the order of 10^{−3} ∼ 10^{−2}λ, which are close to their ideal value 0 + 0iλ.

Similarly, the ${{\mathbf J}_{\mathbf M}}$ of the pattern 1 and pattern 2 with pitches from 360 nm–1440 nm at the source points (0, 0) and (0.76, 0) are also calculated (0.76 and 0 are the source plane coordinates ${f_s}$ and ${g_s}$, respectively), and the mean value of the ${{\mathbf J}_{\mathbf M}}$ pupil is selected to evaluate the average of ${{\mathbf J}_{\mathbf M}}$ in the entire pupil (it also includes the positive or negative of ${{\mathbf J}_{\mathbf M}}$), as shown in Fig. 5. In addition, the energy of the peaks (spectral points) of the spectrum is much higher than other points, which makes a greater contribution to imaging. Therefore, the values of ${{\mathbf J}_{\mathbf M}}$ at the spectral points also have greater effect on imaging, so they are also selected separately, as shown in Figs. 6 and 7 (0-order and 1-order spectrum points).

It can be seen from Figs. 5 to 7 that in the pitch range of 360 nm–1440 nm, the mean values of $J{m_{xx}}$ and $J{m_{yy}}$ are in the range of -1.2λ to -0.2λ, and the values of $J{m_{xx}}$ and $J{m_{yy}}$ at the spectral points are in the range of -1λ to -0.2λ, which are quite different from their ideal values. The mean values and the spectral points values of $J{m_{xy}}$ and $J{m_{yx}}$ are very small, close to their ideal values. Therefore, the 3D mask effect will lead to a relatively large PA, mainly $J{m_{xx}}$ and $J{m_{yy}}$, which may mainly affect the measurement accuracy of ${J_{xx}}$ and ${J_{yy}}$ in the imaging-based measurement technology.

#### 5.2 Comparison of PA measurement accuracy

Next, we simulate the proposed method to measure PA coefficients up to 37th order, and compare it with the previous method based on the Kirchhoff model. The simulation process is shown in Fig. 8, which is divided into two parts: imaging and retrieval.

In the imaging part, a rigorous imaging model is adopted, and a random PA is added to the PA design value as the PA input value (true value), where random PA indicates the deviation of the PA true value from the PA design value due to the processing, installation, and use of PO. Then import the preset test mask and illumination, and use the synchronous orientation measurement method to obtain multiple sets of aerial image information. In the retrieval part, Fourier transform is performed on the aerial images to extract their spectrum, and then these spectrum are brought into the proposed method (rigorous imaging-based measurement method of PA) and the previous method based on Kirchhoff model to retrieve the PA. Finally, by comparing the PA input value and the PA retrieved value of the two methods, the measurement accuracy of the two methods is analyzed. Some parameters in the simulation are shown in Table 2. The PA design value in the simulation is obtained by ray tracing the PO designed by our laboratory using CODE V [6], as shown in Fig. 9. Five hundred sets of PA input values are set for simulation by adding random PA coefficients in the range of ±0.02λ to the PA design value. The calculations in the simulations are implemented in MATLAB and Python, and the rigorous mask near-field spectrum is obtained by the FDTD algorithm of the commercial software PROLITH.

For DNN, considering accuracy and training time, the framework of 5 layers (4 hidden layers, number of neurons in each layer: 3000∼600) is excellent. The activation function is Tanh, and the Drop is in the range of 0.2 to 0.3. The hybrid samples are as follows: the ±0.03λ range of the design value accounts for 10%, the ±0.024λ range accounts for 20%, the ±0.02λ range accounts for 50%, and the ±0.0006λ range accounts for 20%. A learning algorithm called *Adam* is used for optimizing the network with an initial learning ratio of 0.003 [41].

According to the above parameters, the measurement accuracy of the proposed PA measurement method (rigorous imaging-based measurement method) and the previous method (based on Kirchhoff model) are compared. 500 sets of PA measurement results are obtained, and the median is taken for comparison. The PA coefficients errors of the proposed method and the previous method are shown in Figs. 10–13, and the Jones pupil errors of them are shown in Fig. 14.

For the previous method, the measurement errors of *b* and *b*′ are in the order of 10^{−4}λ, but the measurement errors of *a* and *a*′ are in the order of 10^{−3}λ. This is because the 3D mask effect mainly causes $J{m_{xx}}$ and $J{m_{yy}}$, it has a relatively large impact on the measurement accuracy of *a* and *a*′, and has a relatively small impact on the measurement accuracy of *b* and *b*′. For the proposed method, the measurement errors of the PA coefficients are all in the order of 10^{−4}λ. Compared with the previous method, the errors of the real and imaginary parts of *a*, and *a*′ are reduced by 89%, 54%, 92%, and 58%, respectively. The errors of the real and imaginary parts of *b* and *b*′ are reduced by 44%, 26%, 21%, and 36%, respectively. It can be seen that the proposed method effectively eliminates the impact of the 3D mask effect on the measurement results, and greatly improves the measurement accuracy, especially for *a* and *a*′.

The comparison of Jones pupil between the proposed method and the previous method gives a similar conclusion. It can be seen from Fig. 14 that the ${J_{xx}}$ and ${J_{yy}}$ errors of the proposed method are significantly smaller than those of the previous method. In addition, the root mean square errors (RMSE) of Jones pupil are also compared, as shown in Table 3. It can be seen that the RMSEs of each pupil measured by the proposed method are in the order of 10^{−4}λ. Compared with the previous method, the RMSEs of the 8 pupils are greatly reduced, with an average reduction of 72%. It shows that the proposed method greatly improves the measurement accuracy, especially for ${J_{xx}}$ and ${J_{yy}}$.

In addition, the impact of metrology errors on the proposed method is also analyzed. According to some studies, the lateral and axial metrology errors of the aerial image will affect the accuracy of the spectrum extraction, and they are assumed to be 1 nm and 5 nm respectively [10,12]. These metrology errors are added to the imaging model and the 500 sets of PA input values are measured again using the proposed method. The RMSEs of the Jones pupil measured with metrology errors are calculated and compared with those without metrology errors, as shown in Table 4. It can be seen that with metrology errors, the RMSEs of proposed method is increased by 2% on average, and the overall accuracy is still on the order of 10^{−4}λ. Therefore, the proposed method performs well in the presence of metrology errors.

In general, the 3D mask effect of the test mask will cause a relatively large PA (especially for the $J{m_{xx}}$ and $J{m_{yy}}$), which impacts the measurement accuracy of PA of PO. The proposed method applies the rigorous imaging model to the measurement model, which effectively eliminates the impact of the 3D mask effect on PA measuring, and greatly improves the PA measurement accuracy.

## 6. Conclusion

In this paper, a rigorous imaging-based measurement method of PA is proposed to improve the measurement accuracy. Using the quantitative description of the 3D mask effect ${{\mathbf J}_{\mathbf M}}$, the semi-analytic rigorous imaging model is obtained. Then the rigorous imaging-based measurement model of PA is established, and the rigorous nonlinear relationship between PA and aerial image spectrum orders is obtained. According to the characteristics of the 3D mask effect, a synchronous orientation measurement method is proposed to reduce the cost of building the ${{\mathbf J}_{\mathbf M}}$ library. A DNN is used to solve the overdetermined equations to achieve high-precision retrieval of PA. Simulation results show that the 3D mask effect of the test mask will cause a relatively large PA (especially for $J{m_{xx}}$ and $J{m_{yy}}$), which impacts the measurement accuracy of the PA of PO. The proposed method effectively eliminates the impact of the 3D mask effect of the test mask on the PA measurement, and compared with the measurement method based on the Kirchhoff model, the measurement error is reduced by 72%.

## Appendix A: Specific forms of A, V, and U

In Eq. (2), the expression of ${\mathbf V}^{\prime}$ is given, and the specific forms of $A$, ${\mathbf V}$, and *U* are as follows. *A* is the correction factor which can be expressed as:

$U$ is the low pass filter which can be expressed as:

## Appendix B: Specific forms of ${\mathbf C}_{11}^1$, ${\mathbf C}_{12}^1$, ${\mathbf C}_{21}^1$, ${\mathbf C}_{22}^1$, ${\mathbf C}_{31}^1$, ${\mathbf C}_{32}^1$, and ${\mathbf C}_{11}^2$, ${\mathbf C}_{12}^2$, ${\mathbf C}_{21}^2$, ${\mathbf C}_{22}^2$, ${\mathbf C}_{31}^2$, ${\mathbf C}_{32}^2$

In Eq. (13), we give the relationship between PA coefficients and areal image. The specific forms of ${\mathbf C}_{11}^1$, ${\mathbf C}_{12}^1$, ${\mathbf C}_{21}^1$, ${\mathbf C}_{22}^1$, ${\mathbf C}_{31}^1$, ${\mathbf C}_{32}^1$, and ${\mathbf C}_{11}^2$, ${\mathbf C}_{12}^2$, ${\mathbf C}_{21}^2$, ${\mathbf C}_{22}^2$, ${\mathbf C}_{31}^2$, ${\mathbf C}_{32}^2$ are:

## Funding

Major Scientific Instrument Development Project of National Natural Science Foundation of China (11627808); National Natural Science Foundation of China (61435005); National Natural Science Foundation of China (61975013); National Science and Technology Major Project (2017ZX02101006-001).

## Acknowledgments

We gratefully acknowledge KLA-Tencor Corporation for providing academic use of PROLITH.

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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