In-Situ Aberration Measurement in Optical Lithography: Fast Imaging Modeling and Aberration Retrieval

With ever  decreasing feature sizes, the impact of lens aberration has become increasingly  important for imaging quality control of projection lithographic tools. The  imaging optics configuration in lithographic tools is typically a partially  coherent system characterized by the intensity distribution of the source and  the pupil function of the projection lens. Imaging properties of such partially  coherent systems have to be described using a bilinear model, which leads to  time-consuming calculations and difficulties in comprehension, especially in  the case when wavefront aberration is involved. Therefore, we have developed  several novel methods for in situ measurement of lens aberrations up to the  37th (or even higher-order) Zernike coefficient in lithographic tools with  partially coherent illumination.

(1) Analytical  sensitivity functions for in-situ aberration measurement

By simplifying the  theoretical derivation of optical imaging under partial coherent illumination, we  propose two linear models in a compact expression with two matrices, and then  used to determine the Zernike coefficients of odd aberration and even  aberration, respectively. This technique has been extended with generalized  formulations suitable for arbitrarily shaped illumination sources by further  derivation of the analytical aberration sensitivities. The technique requires  the acquisition and analysis of optical image intensities of a set of 36 binary  gratings with different pitches and orientations. It is fully expected that  this technique will be simple to implement and will provide a useful practical  means for the in-line monitoring of imaging quality of lithographic tools under  partial coherent illumination.

(2)  Single-image method for  direct aberration retrieval

Recently, we  propose a method for retrieving small lens aberrations in partially coherent  imaging systems, which only requires measuring one single defocused image of  intensity. By deriving a linear theory of imaging systems, we obtain a  generalized formulation of aberration sensitivity in a matrix form, which  provides a set of analytic kernels that relate directly the measured intensity  distribution to the unknown Zernike coefficients. Sensitivity analysis is  performed and test patterns are optimized to ensure well-posedness of the  inverse problem.

(3)  In-situ aberration  measurement using an efficient quadratic aberration model        

In considering the  case where the aberrations are relatively large, a rapid calculation of the  quadratic or even higher order aberration model is required. We propose a fast  algorithm by      introducing a  generalized operator called cross triple correlation (CTC). By decomposition of  the transmission cross coefficient (TCC) into CTCs, the Zernike  aberration-induced intensities in the quadratic aberration model can be quickly  calculated and clearly separated from each other. Based on the proposed  algorithm, we develop a method for in-situ aberration measurement based on the  quadratic aberration model, which represents the bilinear relationship between  the optical image intensity and the Zernike coefficients. The iterative  algorithm is adopted to extract the Zernike coefficients from the through-focus  optical images of a specially designed mask pattern. The simulation work has  validated the theoretical derivation and confirms that such a method yields a  superior quality of wavefront estimates, particularly for the case when the  aberrations are relatively large.

Geometry for the trapezoidal grating conical diffraction problem and procedure of computational metrology by Mueller matrix polarimetry

Fig. 1. Characterization  of analytical kernels for measuring odd and even aberrations under a smooth  conventional illumination.

Experimental platform of the Mueller matrix polarimeter

Fig. 2. Comparison between  the input and measured aberrated wavefronts under the smooth quadrupole  illumination.

Modeling results using home-made RCWA code comparing with GSolver and for 4×4 Mueller matrix calculation

Fig. 3. Optimization  results of some object patterns for single-image method, with (a) to (d) as  four kinds of object patterns.

Scheme of nonlinear regression method for structural extraction

Fig. 4. Aberration  retrieval results. (a) Unknown aberration, (b) retrieved aberration, and (c) wavefront  error.

>Principle of global sensitivity analysis

Fig. 5. Representation  of the mathematical meaning of CTC

>Principle of global sensitivity analysis

Fig. 6. Forward modeling  and inverse problem for aberration measurement.

>Principle of global sensitivity analysis

Fig. 7. Measurement  accuracy by using the proposed quadratic model and the simplified linear model

Selected papers

  1.                 S. Xu, C. W. Zhang, H. Q. Wei, and S. Y. Liu, "A  single-image method of aberration retrieval for imaging systems under partially  coherent illumination," J. Opt. 16(7), 072001 (2014). (URL, PDF)

  2.                 S. Y. Liu, X. J. Zhou, W. Lv, S. Xu, and H. Q.  Wei, "Convolution-variation separation method for efficient modeling of optical  lithography," Opt. Lett. 38(13),  2168-2170 (2013). (URL, PDF)

  3.                 S.  Y. Liu, S. Xu, X. F. Wu, and W. Liu, "Iterative method for in situ measurement  of lens aberrations in lithographic tools using CTC-based quadratic aberration  model," Opt. Express 20(13), 14272-14283  (2012). (URL, PDF)

  4.                 S. Y. Liu, W. Liu, and T. T. Zhou, "Fast algorithm for quadratic  aberration model in optical lithography based on cross triple correlation," J. Micro/Nanolith. MEMS MOEMS 10(2),  023007 (2011). (URL, PDF)

  5.                 S. Y. Liu, W. Liu, and X. F. Wu, "Fast evaluation of aberration-induced  intensity distribution in partially coherent imaging systems by cross triple  correlation," Chin. Phys. Lett. 28(10), 104212 (2011). (URL, PDF)

  6.                 W.  Liu, S. Y. Liu, T. L. Shi, and Z. R. Tang, "Generalized formulations for aerial  image based lens aberration metrology in lithographic tools with arbitrarily  shaped illumination sources," Opt.  Express 18(19), 20096-20104 (2010). (URL, PDF)

  7.                 W. Liu,  S. Y. Liu, T. T. Zhou, and L. J. Wang, "Aerial image based technique for  measurement of lens aberrations up to 37th Zernike coefficient in lithographic  tools under partial coherent illumination," Opt.  Express 17(21), 19278-19291 (2009). (URL, PDF)