Elsevier

Optics Communications

Volume 329, 15 October 2014, Pages 63-68
Optics Communications

Aberration measurement based on principal component analysis of aerial images of optimized marks

https://doi.org/10.1016/j.optcom.2014.05.003Get rights and content

Abstract

We propose an aberration measurement technique based on principal component analysis of aerial images of optimized marks (AMAI-OM). Zernike aberrations are retrieved using a linear relationship between the aerial image and Zernike coefficients. The linear relationship is composed of the principal components (PCs) and regression matrix. A centering process is introduced to compensate position offsets of the measured aerial image. A new test mark is designed in order to improve the centering accuracy and theoretical accuracy of aberration measurement together. The new test marks are composed of three spaces with different widths, and their parameters are optimized by using an accuracy evaluation function. The offsets of the measured aerial image are compensated in the centering process and the adjusted PC coefficients are obtained. Then the Zernike coefficients are calculated according to these PC coefficients using a least square method. The simulations using the lithography simulators PROLITH and Dr.LiTHO validate the accuracy of our method. Compared with the previous aberration measurement technique based on principal component analysis of aerial image (AMAI-PCA), the measurement accuracy of Zernike aberrations under the real measurement condition of the aerial image is improved by about 50%.

Introduction

The aberration measurement techniques based on aerial images [1] have been studied extensively [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14]. The transmission image sensor using multiple illumination settings (TAMIS) technique [2] and the techniques based on TAMIS [3], [4], [5] were used widely to measure low order Zernike coefficients. In order to measure high order aberrations, the Z37 AIS proposed technique was proposed by Nikon corporation and then developed by Liu et al. [6], [7]. Recently, Duan et al. have proposed aberration measurement by use of aerial images (AMAI) based on the principal component analysis (PCA) technique, named AMAI-PCA [8], [9], [10]. AMAI-PCA just adopts a single illumination. Isolated spaces oriented along 0° and 90° directions are used as the test marks. Zernike aberrations are retrieved from the aerial image intensities of the test marks. AMAI-PCA contains two processes: the modeling process and the aberration retrieval process. In the modeling process, a statistic method is first utilized to obtain one group of Zernike coefficients [9]. Then these Zernike coefficients are inputted into a lithography simulator to generate aerial images for modeling. After the principal component analysis (PCA) of the aerial images, a group of principal components (PCs) and the corresponding PC coefficients are obtained. The relationship between Zernike coefficients and PC coefficients is built by a regression matrix (RM), which is obtained by the multiple linear regression analysis method. In the aberration retrieval process, PC and RM are used to retrieve Zernike aberrations from a measured aerial image. AMAI-PCA has the merit of high speed, low cost, and high accuracy, and it has been verified by various lithography experiments [10]. Dongbo Xu et al. [11] have adopted multiple illuminations to improve the accuracy of AMAI-PCA. Yan et al. [12] and Yang et al. [13] have improved the accuracy of AMAI-PCA further based on an optimized source and an optimized process. Yang et al. have proposed a denoising method to improve the repeatability of AMAI-PCA [14]. However, all these improvements didn׳t consider position errors of the measured aerial image. Because of the position errors, AMAI-PCA can only be used to measure Z7 to Z9 and Z14 to Z16 in a real lithography tool.

In order to decrease the impacts of position offsets of the measured aerial image, a centering process is introduced. A test mark composed of three spaces with different widths is designed to improve the centering accuracy and aberration measurement theoretical accuracy together. The parameters of the test mark are optimized based on an accuracy evaluation function, and an optimized mark (OM) is obtained. Utilizing the principal components of the aerial image of the optimized test mark, the offsets of the measured aerial image are compensated in the centering process and the adjusted principal component coefficients are obtained. Then Zernike coefficients are retrieved using a least square method. We named the proposed method as AMAI-OM for short. The measurement accuracy of the proposed method is validated by the lithography simulators PROLITH [16] and Dr.LiTHO [17] and compared with that of AMAI-PCA.

Section snippets

Lithography imaging model

The through-focus aerial image can be calculated according to Abbe method as follows [18]:I(x,y,z)=+J(f,g)|++O(f,g)H0(f+f,g+g)×exp{ikW(f+f,g+g)}exp{i2πkzpz}×exp{i2π(xf+yg)}dfdg|2dfdg,where x and y are normalized coordinates in the wafer plane, f and g are normalized coordinates in the pupil plane, z is the defocus, J(f, g) is the effective source distribution, O is the spectrum of the test mark (a test mark is a mask pattern for the aberration measurement), H0

Compensation of aerial image offset

The accuracy of AMAI-PCA for the real aerial image in a lithography tool decreases a lot compared to the theoretical accuracy [10]. Position offsets and noises of aerial images are the chief reasons for the accuracy reduction. Fig. 1 shows the schematic diagram of the position and sampling points of a real aerial image.

In Fig. 1, the red box represents the nominal position of the aerial image, while, the blue box represents the real position. There are always offsets between the nominal and

Simulations

The theoretical aberration measurement accuracy was verified by the PROLITH [16] and Dr.LiTHO [17] simulations. The maximum difference of the aerial images generated by the two simulators is on the order of 1e−4, so the difference nearly has no effect on the measurement accuracy of the Zernike coefficients. An annular illumination source with σout=0.96, σin=0.58 was used in both AMAI-PCA and AMAI-OM. NA was set to 0.75. In the lateral direction, the aerial image was measured between −900 nm and

Conclusions

In conclusion, an aberration measurement technique based on principal component analysis of aerial images with optimized marks (AMAI-OM) has been proposed. The characteristics of the real aerial image have been analyzed. The position offsets of the real aerial image are the chief reasons of the accuracy reduction. A centering process is introduced to compensate the position offsets of the aerial image. The final measurement accuracy of Zernike coefficients is affected by the theoretical

Acknowledgement

The authors would like to thank Dr. Erdmann for his help in aerial image simulation, Dr. Bourov for his help in formula derivation, and the support of the Chinese National Engineering Research Center for Lithographic Equipment. This work was supported by grants from the National Natural Science Foundation of China (Nos. 61275207 and 61205102).

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