Polarization aberrations (PA) can be presented by Jones pupil and can also impact the imaging performance of immersion projection optics significantly. Precise PA measurement is most important for resolution enhancement technology and holistic lithography at 7nm node and below, in order to improve the pattern fidelity and processing stability. However, the current imaging-based measurement method of PA by linear approximation has not taken the coupling effect of the PA coefficients into account. This paper proposes a nonlinear measurement method of PA based on a rigorous nonlinear model to improve the measurement accuracy significantly. In this invention, the new spectrum modulation theory is developed to establish a rigorous quadratic form of PA and aerial image spectrum. A hybrid genetic algorithm is developed to solve the quadratic form inversely to obtain the PA accurately. An overall simulation validates that this method provides a superior quality of PA measurement with very high precision of .
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Polarization aberration (PA) characterizes the changes in the phase, intensity, and polarization of light after passing through the projection optics (PO) [1,2]. It can be represented by the Jones pupil, and can also be expanded into multiple forms, such as orientation Zernike polynomials , field-orientation Zernike polynomials [4,5], and pseudo-Zernike polynomials . As the critical dimensions of integrated circuits (IC) continuously shrink to 7 nm and beyond, PA cannot be neglected due to its impact to imaging performance in immersion projection optics (IPO) [7,8]. And PA needs to be measured and conducted to various resolution enhancement technologies such as source and mask optimization (SMO) [9,10], hybrid SMO (HSMO) , and source polarization mask optimization (SPMO) [12,13] to improve the pattern fidelity and processing stability. Therefore, there is a need for IPO to develop techniques and systems to accurately measure the PA.
The imaging-based measurements are widely used in aberration measurement for IPO, which have been proposed to measure the wave aberration [14–16] and PA . However, these methods established the relationship between wave aberrations and aerial image errors (e.g focus shift, lateral shift, and image placement error) based on a linear model that neglects the coupling effect of PA coefficients on imaging, thus resulting in theoretical errors. Therefore, SY Liu in  proposed an aberration measurement based on a quadratic aberration model to avoid the theoretical errors and provide a more accurate PA estimation. Similarly, it is necessary to establish a PA measurement based on the nonlinear model for a superior quality of PA estimation.
In our previous work , we established a small-scale nonlinear model and developed a method to measure the PA Zernike coefficients up to 10th, which provided higher accuracy than linear models. However, there is an incomplete analysis of the spectrum and a low utilization of spectrum information in this model. In addition, the method is used in the measurement of the PA coefficients up to 10th, so when extended it up to 37th, the solution will be more likely to fall into a local minimal value, resulting in the inability to solve the real PA.
In this paper, a nonlinear measurement of PA in IPO by spectrum analysis of aerial image is proposed. Through analysis, the spectrum modulation theory, which is the mechanism of PA impacting the imaging, is obtained to derive a quadratic form about PA coefficients and the aerial image spectrum. Based on this quadratic form, the overdetermined equations can be built by measuring multiple groups of aerial images. the PA coefficients up to the 37th order can be estimated accurately by developing a hybrid genetic algorithm and using it to solve the overdetermined equations inversely. An overall simulation is used to validate the validity and accuracy of the proposed method.
2. Spectrum modulation of PA
A critical part of the imaging-based PA measurement method is to establish the relationship between the PA coefficients and some image information. Therefore, we need to choose a kind of image information with the following characteristic: There is a simple and explicit analytical relationship between the image information and the PA, and this image information is easy to extract from the aerial image with high accuracy. For this purpose, the physical mechanism of the impact of the PA on the imaging needs to be analyzed to find the image information that satisfies these characteristics. In this Section, the vector imaging model is given first in 2.1. In 2.2, the spectrum modulation, which is the mechanism of PA's impact on imaging, is derived, and the aerial image spectrum is selected as the image information.
2.1 Vector imaging model with PA
Under the Abbe imaging principle, the rigorous vector imaging model [20,21] can be expressed in the following form:Appendix A.
2.2 Spectrum modulation
If a mask with good periodicity is used in imaging process, the Fourier spectrum will be a discrete form. Then the Eq. (1) becomes the sum of each spectrum point of , which can be expressed as:Fig. 1, we show the impact mechanism of the PA on imaging, where order is the cutoff diffraction order. It shows that , which contains the PA of PO, impacts the aerial image by directly impacting the spectrum points. Therefore, the spectrum of the aerial image contains the PA information of the PO.
If the analytical relationship between the spectrum of the aerial image and the PA of the PO is established, the PA can be measured by spectrum analysis of aerial image. Since the PA directly impacts the intensity of the imaging spectrum, the relationship between PA and the imaging spectrum can be more succinct and rigorous than the relationship between PA and some imaging errors (e.g focus shift, lateral shift, and image placement error). In addition, the spectrum analysis method extracts the period information of the image, so many random systematic errors can be eliminated by this extraction method, and their impact on accuracy is reduced to a low level.
3. The nonlinear measurement of PA
It shows that, in Eq. (3), the Fourier spectrum of the mask transmission function directly determines the analytical form of the lithography system, so selecting and constructing a reasonable mask can reduce the computational complexity and improve the measurement accuracy. In this paper, we select the one-dimensional dense line mask for the derivation of the theoretical model for the test mask. In the follow-up work, we will also select or construct other masks that have better characteristics in Fourier spectrum for this theory.
For the one-dimensional dense line masks (binary mask, alternating phase shift mask, and attenuation phase shift mask), their Fourier spectrums can be uniformly written in the following form:Eq. (4) into Eq. (3):
Traditional coherent illumination with X and Y polarization states is choose in this theory, so the effective light sources can be formulated using the delta function:Eq. (7) into Eq. (6), the discrete form of the aerial image is obtained:Appendix B. From Eq. (10) and Eq. (12), the following conclusion is obtained: The relation between PA coefficients and aerial images is quadratic and can be written as follows
In order to obtain the relation between the aerial image spectrum and the PA coefficients, a Fourier transformation is carried out to Eq. (14):
This is the quadratic form of the spectrum of the aerial image and the PA coefficients, and is its sensitivity matrix. The form of is discrete due to the periodicity of the aerial image , and the Eq. (15) shows that each matrix element of has a same form as . Therefore, each of its elements can be expressed as:Eq. (16), the sensitivity matrix of order of spectrum can be defined as a matrix composed of . Therefore, the relationship between PA and any order spectrum of aerial image can be expressed as
4. Hybrid genetic algorithm
This set of overdetermined equations is a nonlinear problem, which is easy to fall into a local minimum points when solved by some classical algorithms, such as gradient descent, nonlinear least-squares algorithm, etc. Therefore, a hybrid genetic algorithm is developed to improve the convergence speed and the calculation accuracy. This algorithm is a hybrid of genetic algorithm and classical algorithm, and draws on the advantages of both algorithms. The specific process of this algorithm is shown in Fig. 2. The initial population is set according to the initial value of PA and the error range, and the largescale genetic algorithm is used to guarantee that the final solution is the global minimum solution. The classical algorithm is used to accelerate the evolution speed of excellent individuals to improve the convergence speed and accuracy of the algorithm. The small scale genetic algorithm and optimization of optimal term are designed to reduce the number of populations to speed up the convergence of the algorithm when some individuals in the populations are very close to the target solution. Therefore, this algorithm can solve the overdetermined equations to estimate the PA quickly and accurately.
This section presents an overall simulation to verify this method in measuring the PA coefficients up to the 37th order of an arbitrary field of view in PO, and compare it to the methods based on liner approximation models. The simulation is divided into two parts, namely the imaging measurement simulation and the PA solution simulation. Figure 3 shows the process of the simulation.
In the first part, the design value of PA is obtained by the ray tracing of CODE V to an arbitrary field of view in PO designed by the laboratory, as shown in Fig. 4. Then, adding a random error to this design value represents the deviation of the true value of the PA from the design value during the production, assembly, and use of the PO. We use this as the true value of the PA of the PO in this simulation. Take it into the vector lithography imaging model and solve its aerial image. And Fourier transform it to obtain the observations (spectrum of aerial image) required by this theory. Change the mask pitch, angle and other parameters, repeat the above process to obtain a set of observations of the lens at this field of view.
In the second part, the spectrum of aerial image is input to the proposed PA measurement method to obtain the PA expansion coefficients up to the 37th order. Table 1 shows some of the simulation parameters. Figure 5 gives the comparison of the PA obtained by this method with the true values, and Fig. 6 shows the errors of measurements relative to true values.
As can be seen from the Figs. 5(a) and 5(b) and Fig. 6, the measurements obtained by this method has a high accuracy. The errors of all PA coefficients are orders of magnitude, and most of them are orders of magnitude. In contrast, the linear approximation method proposed in  established the linear relationship between Zernike coefficients of PA and the phase shift and intensity distribution of aerial image with first order approximation. By measuring the aerial images of test masks in different orientations and pitches under different illumination settings, the PA coefficients can be obtained with errors of to orders of magnitude. Therefore, compared to this linear approximation method, the accuracy of the proposed method is improved by two orders of magnitude. Convert these PA expansion coefficients to the Jones pupil, and the root mean square error (RMSE) of the measured pupil and true pupil are shown in Table 2. It can be seen that the RMSE of Jones pupils obtained by proposed method are orders of magnitude, which is also one orders of magnitude smaller than the linear approximation method. Therefore, this method provides a superior quality estimation of the PA expansion coefficients up to the 37th order.
A nonlinear measurement of PA in IPO by spectrum analysis of aerial image has been proposed to improve the quality of PA estimation. The spectrum modulation that is the mechanism of PA impacting imaging is unearthed, and the quadratic form that is the sensitivity matrix of PA coefficients and the aerial image spectrum is derived. The overdetermined equations can be built by measuring multiple groups of aerial images, and by using the hybrid genetic algorithm we developed, the PA coefficients up to the 37th order are determined by solving the overdetermined equations in reverse. An overall PA measurement simulation proves that the errors of PA coefficients are to orders of magnitude and the RMSE of Jones pupil are orders of magnitude. It shows that the method has a very high accuracy, which is one or two orders of magnitude higher than the linear approximation method.
Appendix A Specific analysis form of each item in
As presented in Section 2, the PO part of the lithography system can be described as6] to expand it. pseudo-Zernike basis can be described as
Appendix B Specific analysis form of ,,,,, and,,,,,
General Program of National Natural Science Foundation of China (No. 61675026); Major Scientific Instrument Development Project of National Natural Science Foundation of China (No. 11627808); National Science and Technology Major Project (No. 2017ZX02101006-001).
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