To evaluate the quality of microlithography result, massive aerial images are often generated for careful inspection using applications such as OPC LCC (Lithography Compaliance Check). The number of the pixels used in a 2D aerial image is in the order of O(n * n), where n is the image resolution, which means the runtime scales in a n~2 fashion. However, most of the quality indexes such as CDs or EPE (Edge Placement Error) can be readily observed using contours only and the number of pixels in a specific contour is around O(n) in general. Therefore, there is a huge waste (at least O(n)) of both computation time and memory in most microlithography aerial image simulation tools. The question is: "how to compute an image contour without explicitly generate images?". In this paper, we show that it is indeed feasible to know the image contour with an explicit image formation. The concept is to represent the image in an implicit way. In our algorithm, we utilize hierarchical region-wise function such as 2D polynomials to fit the aerial image kernels instead of using a bitmap type fit. Therefore, any LUT (Look-up-table) operation can be transformed into a polynomial look up and mathematical operations. Since there are only additive and subtractive operations during aerial image generation, we only need to apply same operations to the polynomial coefficients. Once the LUT operation is done, we have analytical forms of the light intensity in each region. To know the contour at each region, we only need to solve a local 2D polynomial equation. With these simple forms, the equation solution can be performed with great efficiency. Furthermore, due to the hierarchical structure, we can reduce the search time from 0(m~*m) to O(m logm), where m is the number of regions. Further speed up is achieved this way. Finally, our algorithm also does not suffer from increased feature count caused by the feature size reduction between process nodes because once the regional polynomial representation is complete, no future LUT search is necessary. Therefore, the runtime of our algorithm is significantly faster than the traditional image LUT method. The initial experiment demonstrates over hundreds of times speed up over the traditional LUT methods.
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