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Computed Images of Dielectric Strips on a Substrate



Egon Marx


Accurate computed optical images of lines and trenches placed on semiconductors are of great interest to the manufacturers of computer components, especially in the overlay process for different layers. These images do not accurately reflect the form of these lines and trenches when their dimensions are comparable to the wavelength of the light used for illumination. Different kinds of computer codes are capable of computing the fields scattered by these elements, and some of them were compared in particular cases. The codes used in the work described here are based on an exact solution of singular integral equations equivalent to Maxwell's equations. There are, of course, computation errors and modeling errors, e.g., from the assumed infinite length of features. To compute an image formed in an optical microscope, the incident light is approximated by a number of plane waves with different directions of incidence and polarization that cover the illumination aperture. The microscope has a collection aperture that determines the amount of the scattered light that goes to form the image. The scattered fields are computed at a relatively large distance over the lines and substrate so that it is mostly in the far field. The light is then transformed back onto the top of the lines, the substrate, or some other height as desired so as to be in the focal plane of the microscope. The contributions of the different incident plane waves are added incoherently to form the computed optical image. Even though the integral equations are exact for infinite lines and trenches, a number of approximations are involved in the actual numerical computations. The include the subdivision of the cross sections of the interface into patches, the distribution of the wave vectors of the incident plane waves, and the approximate computation of Fourier transforms. Of particular concern is the divergence of the unknown boundary functions near sharp edges, which suggests that points should be concentrated near the edges. Alternatively, hypersingular integral equations could be used, but they have problems of their own. Simulation is also useful to determine the effects of deviations of the shape of the features from their design shapes or their separation. Another factor of interest is the absorption and phase shift of the waves by conducting strips such as found on photomasks. Eventually the method will be extended to finite thick substrates so that images formed by the transmitted fields of photomasks can be determined. Images will be shown for individual and multiple lines and trenches of different sizes and configurations. The effects of the choice of parameters and the distribution of points near the edges in the simulation will also be presented.
Proceedings Title
PIERS Meeting
Conference Dates
January 1, 2003
Conference Location
Unknown, USA
Conference Title
PIERS 2003: Progress in Electromagnetics Research Symposium


divergent boundary functions, electromagnetic scattering, numerical methods, photomasks, singular integral equations, two-dimensional scattering problems


Marx, E. (2003), Computed Images of Dielectric Strips on a Substrate, PIERS Meeting, Unknown, USA (Accessed May 30, 2023)
Created January 1, 2003, Updated June 2, 2021