Advanced model and fast algorithm for aerial image computation with well controlled accuracy

Abstract

Modeling of an aerial image in projection lithography was considered as an important part of R&D for a long time. Its role evolved from an "illustrative" to "predictive" method. In last years it became one of the basic components for optimization of both layouts (PSM, OPC, DFM) [1][2][3][4][5] and optical systems (scanner NA, shape and size of illuminator etc) [6][7][8] . Mathematical models and algorithms for the simulation of aerial images were developed based on either Hopkins's or Abbe's or coherent decomposition approaches. In this paper, we describe an analytical advancement of Abbe's method. We present new algorithms for faster simulation of aerial images for integrated circuit patterns. Our approach can be efficiently used for repetitive simulations of varied optical systems (no need for time-consuming re-generation of kernel functions or TCC); the accuracy of simulation is well-controlled and does not depend on the problem of "square grid over circular NA". THEORETICAL APPROACHIn Abbe approach, a partially coherent image (the light intensity distribution in the image plane) is described as the integral of coherent intensity over the source (illuminator) with respect to the source brightness (radiance):( 1 )Here B is the brightness of the source point, s s k ϑ λ π sin 2 = is the absolute value of projection of the wave vector on the mask plane and θ s is the angle between the wave vector and the optical axis. Coherent intensity is the square of the absolute value of the (complex) electromagnetic field, ( ) ( ) 2 , , r k F r k I s s = .Field F can be expressed as:Here r m are the coordinates of a point on the mask plane scaled to the image coordinates r, ϑ λ π sin 2 = k is the absolute value of projection of the wave vector on the lens plane, T is the mask (complex) transparency, L is the function of the lens that includes the obliquity factor [9,10] , lens aberrations etc., NA is the lens numerical aperture, n is the refraction index of the media on the resist/substrate side of the lens. It is preferable to use dimensionless variables.Below all distances will be normalized to the wavelength (λ), and ϑ κ sin n = will be used instead of k. Constant factors (combined in C) will not be considered because they will be cancelled after the image intensity normalization. Thus, intensity is computed as six-fold (3 x 2D) quadrature. However, it is possible to reduce the computational task by taking specifics of the lithography tasks and environment into account [11,12] . Analytical solutions for some of the quadratures from (2) dramatically simplify the computations and/or increase the accuracy of the results. Bearing in mind that the final result is intensity which is the square of the absolute value of the field, we can multiply F by any complex unity without affecting the result: