Skip to main content
U.S. flag

An official website of the United States government

Official websites use .gov
A .gov website belongs to an official government organization in the United States.

Secure .gov websites use HTTPS
A lock ( ) or https:// means you’ve safely connected to the .gov website. Share sensitive information only on official, secure websites.

Scattering by Dielectric Strips on a Dielectric Layer of Finite Thickness



Egon Marx


Accurate simulation of optical images of lines and trenches placed on semiconductors are of great interest to industry, especially in the overlay process.  Similarly, optical images of photomasks in the transmission mode are also of interest.  The basic scattering problem is solved exactly by reducing it to a system of singular integral equations for boundary functions defined on the interfaces between homogeneous regions.  These equations are equivalent to Maxwell's equations [1].  To allow for images in the transmission mode, we generalize previous codes [2] to place the features in a layer of finite thickness on top of a substrate.  This substrate can be free space to allow for transmission images.  Images obtained from our codes have been compared to ones obtained from a code in which the system is approximated by a grating [3] and to measured images.Homogeneous fields defined in the absence of the scatterer are subtracted from the total fields to obtain scattered fields that vanish at infinity.  There are two homogeneous fields in the layer, one propagating upwards and one propagating downwards, which have to be matched to the incident, reflected, and refracted fields.  The number of equations is minimized using the single-integral-equation method for dielectric scatterers.  We take advantage of the nature of the plane interface to eliminate one of the variables without having to resort to matrix inversion.  Kernels have singularities that are no worse than 1/R.  These equations are then solved numerically using, for instance, the point-matching method.  Strips and trenches have sharp edges, where boundary functions diverge.  We have explored these divergences numerically for a finite dielectric wedge, and we have established that the use of hypersingular equations results in boundary functions that remain constant near the edge.  One of the kernels then has a singularity that is of the 1/R2 type, requiring special handling not only of the self-patch contributions but also of the neighboring-patch ones [4], a requirement that makes these equations difficult to solve until we have a better grasp of the singular terms.We derive the singular integral equations for the scattering problem and the integrals that allow us to compute the fields on a plane above and below the layer once the equations are solved.  These fields allow us to determine fields on any other plane above the strip or below the layer.  They contain evanescent waves, induction fields, and radiation fields.  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 what Fourier components of the fields contribute to the image.  The scattered fields are computed at a relatively large distance over the lines and substrate to eliminate the evanescent and induction fields.  The image is then computed in the focal plane of the microscope, defined by some best-focus algorithm.  Images obtained using electric or magnetic fields differ, possibly due to induction fields and  the addition of the reflected or refracted field.  We show some examples of simulated images obtained in this manner.REFERENCES1. E. Marx, Single-Integral-Equation Method for Scattering by Dielectric Cylinders, PIERS 2001 Proceedings, Osaka, Japan, 537, 2001.2. E. Marx, Computed Images of a Dielectric Strip on a Substrate, PIERS 2003 Proceedings, Hawaii, U.S.A., 2003.3. M. Davidson, Metrology, Inspection, and Process Control XIII, Proceedings of the SPIE, Vol. 3677, 866-875, 1999.4. E. Marx, Numerical Experiments in Scattering by a Dielectric Wedge, PIERS 2003 Proceedings, Hawaii, U.S.A., 2003.
Proceedings Title
PIERS Meeting
Conference Dates
January 1, 2004
Conference Location
Unknown, USA
Conference Title
PIERS 2004: Progress in Electromagnetics Research Symposium


electromagnetic scattering, integral equations, microscope image simulation


Marx, E. (2004), Scattering by Dielectric Strips on a Dielectric Layer of Finite Thickness, PIERS Meeting, Unknown, USA (Accessed May 26, 2024)


If you have any questions about this publication or are having problems accessing it, please contact

Created January 1, 2004, Updated June 2, 2021