16 March 2009 Abbe-PCA (Abbe-Hopkins): microlithography aerial image analytical compact kernel generation based on principle component analysis
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Abstract
In the year of 1873, Professor E. Abbe, cooperating with Carl Zeiss Inc, summarized his discovery of the microscope imaging principle which states that the final image is the superposition result of all the diffracted images entering at different angles oblique to the pupil. This discovery forms the foundation analytical methods to analyze optics resolving power. Later, in 1951, the Hopkin's equations, derived by Professor H. H Hopkins, clarified the correlation relationship in the image from both spatial and frequency domain. Based on Hopkin's theory, many microlithography aerial image simulation tools have been developed. In this paper, we claimed that by combing Abbe's theory with the Principle Component Analysis (PCA) method, which is specific to the covariance eigen-decomposition method rather than the SVD (Singular Value Decomposition) method, we can achieve an extremely efficient computational algorithm to generate the essential kernels for aerial image simulation. The major reason for this speed up is from our discovery that the covariance matrix of Abbe's kernels, which is in the dimension of number of discretized condenser sources, can be easily constructed analytically as well as decomposed to a basis set. As a result, an analytical form of compact decorrelated Abbe's kernels can be obtained even without explicit formation of the kernel images. Furthermore, the asymptotic eigenvalues and eigenvectors of the covariance matrix can also be obtained without much computational effort. This discovery also creates a new way to analyze the relationship between sources and final images which can be easily utilized to optimize source shape for lithography process development. Several imaging phenomenon have been readily explained by this method. Extensive experimental results demonstrate that Abbe-PCA is 10X- 40X faster than the state-of-the-art algorithm Abbe-SVD.
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Meng-Fong Tsai, Shi-Jei Chang, Charlie Chung Ping Chen, Lawerence S. Melvin, "Abbe-PCA (Abbe-Hopkins): microlithography aerial image analytical compact kernel generation based on principle component analysis", Proc. SPIE 7274, Optical Microlithography XXII, 72742D (16 March 2009); doi: 10.1117/12.814419; https://doi.org/10.1117/12.814419
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