Theory And Method For Fast Simulation Of Optical Image In Lithography Based On The Method Of Seperation Of Variables

With the development of integrated circuit and semiconductor technology, the feature sizes on the photomask are below the illumination wavelength. Therefore, optical proximity effect (OPE) has been brought in optical lithography, resulting in difference between the output mask patterns on the wafer with the target mask patterns. In order to eliminate this effect, the optical proximity correction (OPC) technique is widely utilized. Generally, an OPC process involves forward modeling known as lithography simulation and an inverse procedure. Since the forward model is repeated many times in the inverse optimization algorithm, fast and accurate optical image simulation is highly desirable as one of the most critical components in the forward modeling simulations. With the continuous decrease of lithography node, feature size of integrated circuit is sensitive to the process variations. For low numerical aperture (NA) systems, process variations, such as defocus, dose and aberrations, have a great impact on the imaging results, which can be simulated by scalar imaging theory. While for high-NA systems, these process variations can be simulated by vectorial theory. In order to keep high accuracy, vectorial imaging simulation should take more variations into consideration, such as the incidence angles of thick mask model and radial position parameter in resist model.Although a few simplified methods of optical imaging in Hopkins’theory or Abbe’s theory achieve significant speedup. However, thes simplified methods are usually obtained under the nominal (best) process condition. For a different process condition, a simulator may have to repeat the costly eigen-decomposition and mask-kernel convolutions. In this dissertation, we focus on investigating the physical properties of optical imaging in lithography, thus introduce the method of separation of variables in Mathematical Physics as the fundmental theory to deal with a wide range of process variations. The imaging formula can be rearranged by two parts, one part with only one variation, while the remaining part independent with the variation. Accordingly, we propose a rigorous methodology from first principles to speed up variational image simulations. For image simulations under a variety of process variations, we have done some research in both scalar theory and vectorial theory, mainly including:A general methodology called convolution-variation separation (CVS) is proposed to speed up scalar image simulations for process variations without sacrificing accuracy. This method firstly filters the mutual intensity (MI) of illumination source by a low pass filter, and then eigen-analyzes the filtered MI, finally decomposes the point spread function (PSF) into a series based on the method of separation of variables. By utilizing the eigenfunctions of the filtered MI and the series of PSF, we can derive the CVS method from first principles as a form of series expression, which consists of a set of predetermined basis functions weighted by a set of expansion coefficients. The basis functions are independent of the process variations and thus may be computed and stored in advance, while the expansion coefficients depend only on the process variations. When the variations change, the image can be obtained quickly by summing up the product of the predetermined basis functions and the changed coefficients.A generalized method to efficiently represent the incident-angle-dependent mask transmittance function (MTF) of a thick mask is proposed. This method establishes an explicit expression between incidence angles and MTF by expanding the MTF into a series based on Taylor expansion. The fast algorithm is subsequently developed. And a MTF-based vectorial imaging process is proposed, which is easy to implement and yields superior performance under the condition of the mulitiple incident angles of partially coherent illumination source.Based on the method of vectorial imaging process under the partially coherent illumination source, this dissertation proposes a novel theory and method to meet the demand of fast and accurately calculating the three-dimensional image in a stratified resist medium, using the polarized light formulations of reflection from and transmission through a stratified medium. This model is derived as a form of series expression, which can calculate image efficiently in different depth of the reisit.In order to verify the proposed scalar and vectorial image calculation methods, simulations for a variety of different process variations are performed by using the proposed methods against commerical software, which have confirmed that propoded methods yield a superior quality of image with an accuracy of10-3and superior performance of speed. Therefore, the proposed methods provide a novel theory and practical means for OPC and orther resolution enhancement technologies (RETs) in optical lithography.