Presentation at the Days on Diffraction International Conference, May 27 – 31, 2013, St. Petersburg, Russia
Solution of 3D scattering problems from 2D ones in short waves
L. I. Goray
The paper reports on development of a boundary integral equation technique of characterization of scattering by rough three-dimensional (3D) surfaces at a short wavelength of λ. The effect of roughness on the mirror scattering intensity can be rigorously taken into account with the model in which an uneven surface is represented by a grating with a large period of di in different perpendicular planes i, which includes an appropriate number of random asperities with a correlation length of ξi. The code analyzes the complex structures which, while being multilayer gratings from a mathematical viewpoint, are actually rough surfaces for di>>ξi. If ξi~λ and the number of orders is large, the continuous angular distribution of the energy reflected from randomly rough boundaries can be described by a discrete distribution η(#) in order # of a grating [1]. A study of the scattering intensity starts with obtaining statistical realizations of profile boundaries of the structure to be analyzed, after which one calculates the intensity for each realization, to end with the intensity averaged out over all realizations. By selecting large enough samples, one comes eventually to properly averaged properties of the rough surface; however, this approach does not involve approximations, including averaging by the Monte Carlo method. The more general case of bi-periodic gratings (or 3D surfaces) may be considered in a similar way or by expressing the solution of the 3D Helmholtz equation through solutions of the 2D equation described below, an approach which may be resorted to in some cases [2]. General equivalent rules for determination of the efficiencies of reflected orders of bi-periodic gratings from those calculated for one-periodic gratings can be found, for example, in [3]. The general approach used is based on expansion of the efficiency of a bi-grating with profile boundaries symmetric relative to the horizontal plane in a Taylor series in powers of a boundary profile depth h, with the principal terms of the series retained in the h<d case. Then the efficiencies e0,m+ and e0,n+ of the orders numbered (0,m) and (n,0) propagating in the upper (+) medium for arbitrary linear polarization of light can be defined through the leading (quadratic in h) terms of the expansion as
where en(m),1(2)+ are the values of the efficiencies of the corresponding mutually perpendicular one-periodic gratings calculated with the position of the polarization vector left unchanged, and R is the Fresnel reflection coefficient of the grating material. For non-deterministic surface functions some modification of the general approach is required. As follows from a comparison with the results of rigorous calculations performed in [3] and by the present author, the approximate relations (1) give a high-accuracy solution for cosθihi<<di and λ<di, where θi is an incidence angle. In the cases where one minus real part of the refractive index and imaginary part of the material are small, h can be large enough.
References
[1] Goray L. I., 2010, Application of the rigorous method to x-ray and neutron beam scattering on rough surfaces, J. Appl. Phys., Vol. 108, pp. 033516-1-10.
[2] Goray L.I., 2011, Solution of the inverse problem of diffraction from low-dimensional periodically arranged nanocrystals, Proc. SPIE, Vol. 8083, 80830L-1-12.
[3] Petit, R., ed., 1980, Electromagnetic Theory of Gratings (Springer, Berlin).
© 2013 `Days on Diffraction‘, POMI.
The lecture is published at IEEE Proceedings. Click here to download the full text.