Theory And Method For Lens Aberration Measurement In Optical Lithographic Tools Based On Analytical Sensitivity Functions

As the limit of optical lithography is pushed and feature densities continue to increase, lens aberration has become one of the most important factors to evaluate the imaging quality of lithographic tools. In order to meet the requirement of optical path tolerances on the order of several nanometers over extremely large aperture of current projection lens, the higher-order coefficients of Zernike polynomials are becoming increasingly important for monitoring lens performance on a regular basis. Moreover, in the current optical lithography processes for semiconductor manufacturing, OAI (Off-Axis Illumination) technique has been widely used for the need of stringent CD (Critical Dimension) control. Therefore, it is highly desired for manufactures of lithographic tools to develop in-situ techniques to accurately measure aberrations up to the 37th order, suitable for off-axis illumination sources or even free-form illumination sources. In this dissertation, we focus on investigating the physical properties of aberrated intensities induced by different kinds of lens aberrations, thus propose an analytical sensitivity functions based theory and method for in-situ measurement of Zernike aberrations up to the 37th order, under arbitrarily shaped illumination source settings.Firstly, a parametric analytical source model for overall representation of the physical distribution properties of off-axis illumination sources is proposed. A Sigmoid function is adopted as a kernel function to construct the analytical model for the multiple mainstream off-axis sources. Corrected parametric terms are subsequently presented for characterizing different physical distortions and deviations of real illumination sources. The corrected parametric terms can be decomposed into Fourier series which have the special physical meaning of respectively indicating different distortion types including shift of the center, tilt, ellipticity, etc.Secondly, the theory of aberrated intensities induced by lens aberrations is investigated, and simulations of aberrated intensities are performed. A cross triple correlation (CTC) based quadratic aberration model is proposed for in-depth analyzing aberrated intensities induced by multiple Zernike aberrations. By decomposition of the transmission cross coefficient (TCC) into CTCs, the Zernike aberration-induced intensities in the quadratic aberration model, including linearly aberrated and quadratically aberrated terms, can be quickly calculated and clearly separated from each other. This model has the advantage of efficiently and accurately characterizing aberrated intensities, and reveals the physical essential of influences of different Zernike orders on aerial image intensity distributions.Thirdly, this dissertation proposes a theory and method under the conventional circular source for accurately in-situ measuring lens aberrations up to the 37th Zernike coefficient. Using a TIS (Transmission Image Sensor), this method requires the acquisition and analysis of aerial image intensities of binary gratings simply designed with different pitches and orientations. By in-depth investigating the through-focus image intensities in the frequency domain based on Hopkins theory of optical imaging, the analytical linear relationships between intensity spectra and Zernike coefficients are established, and the aberration sensitivity functions suitable for the conventional circular source are further derived. The aberration sensitivity matrices are subsequently constructed as a compact expression by odd and even sensitivity functions, which can be easily obtained in advance by numerical calculation instead of by lithographic simulators, and then used to determine the Zernike coefficients of odd aberration and even aberration respectively.Finally, based on the method for aberration measurement under the conventional circular source, this dissertation proposes a novel theory and method for fast and accurately in-situ measuring lens aberrations up to the 37th order, using generalized formulations of odd and even aberration sensitivities suitable for arbitrarily shaped illumination sources. The generalized formulations are determined by independent variables of the Zernike order, pupil position and source distribution. With a set of Zernike orders, the generalized formulations can be treated as a set of analytical kernels which succeed in constructing a sensitivity function space. The analytical kernels take the advantage of realizing a linear and analytical relationship between the Zernike coefficients to be measured and the measurable physical signals, and reveal the physical essence of the proposed measurement method by mapping the measurable signals into the sensitivity function space where the weights of corresponding kernel components indicate the Zernike coefficients to be measured.A variety of mainstream illumination sources with spatially variable intensity distributions are input into the PROLITH for the simulation work of the proposed aberration measurement method, which have validated the theoretical derivation and confirms that such a method yields a superior quality of wavefront estimate with an accuracy of Zernike coefficients on the order of 0.1mλs and an accuracy of wavefronts on the order of mλs. The proposed method has the advantages of simple to implement, low cost and time efficient, and thus provides a novel theory and practical means for optimizing controlling and monitoring projection systems in lithographic tools.