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Full Field FEM Solutions To Maxwell's Equations
Finite Element ApproachIn the finite element approach to electromagnetic radiation scattering, a combined form of Maxwell’s equations is solved for electric field throughout a computational volume V which has been discretized into finite volume elements. In the absence of free charges, the time-domain point form of electric field derived from Maxwell’s equations is: In Equation 1, E is electric field amplitude, e is the permittivity of the medium, s is the conductivity of the medium, and m is the permeability of the medium. The inner dot product of this governing differential equation is formed with the arbitrary vector weighting function G, which is a function of both time and position, and integrated over each volume element in the model: Electric field values are defined at the nodes of the finite element mesh, and fields are interpolated within each volume element using a linear shape function. In the finite element implementation used in the present research, the weighting function is chosen to have the same spatial variation as the shape function (the Galerkin formulation), yielding a symmetric system of linear equations for the unknown electric field. The temporal second derivative appearing in Equation 1 and Equation 2 is approximated using an explicit scheme based upon the central difference approximation: Equation 3 where Dt is the time step used in the computation. The time-domain finite element software used in the present research has been applied by other investigators to a number of systems of practical interest, including the determination of light scattering properties of features on silicon wafers and evaluation of the performance of integrated optical devices and waveguides.,, The finite element models in the present study are constructed in a Cartesian coordinate systems by specifying edge dimensions, the finite element mesh density, particle position(s) and shape(s), and the optical properties of the constituent materials at the wavelength of interest. Typical models contain 1-2 million elements. Each individual computation employs a single wavelength of electromagnetic radiation which propagates in the model as a polarized plane wave with a specified Poynting vector. The electromagnetic radiation is allowed to propagate in the finite element model in the time domain, with electric field amplitude and phase computed at each node in the finite element model per time step. A sufficient number of time steps are employed in a computation to ensure that steady state is achieved; steady state is confirmed by reviewing time histories of electric field amplitude at selected nodes in the model and verifying convergence. The near-field scattered electric field amplitudes are extrapolated to far field by positioning a Kirchoff surface located a few elements inside the walls of the finite element model, on which equivalent electric and magnetic currents are computed from the tangential components of scattered electric and magnetic field. From these equivalent currents the scattered electric field at near field is extrapolated onto the surface of a far-field sphere having a radius which is much greater than the wavelength of light using the equivalence theorem. Extrapolation eliminates evanescent electric field components which are present at near field so that only propagating electromagnetic waves contribute to the far-field result. The simulation of diffuse illumination is accomplished by superimposing the far-field scattered intensities computed in a series of individual computations in which the illumination directions have been varied step-wise over the full range of possible orientations. The platform used in these computations is a Cray C-94 supercomputer. Once extrapolation of scattered electric field and intensities has been accomplished, it is possible to extract macroscopically observable physical quantities such as the scattering cross section Csca of the microstructure being modeled, and the angular distribution of scattered light. The scattering cross section is defined in Equation 4 and is expressed in units of area: It is useful to normalize the scattering cross section by the volume of the scattering material in the finite element model. The scattering coefficient S is defined in Equation 5 and is expressed in units of inverse length. The scattering coefficient provides information about the strength of light scattering by a particular microstructure on a per-volume basis. It provides no direct information, however, about the efficiency of a microstructure in deflecting light from the incident direction. In a paint film, for example, a combination of strong scattering per volume and strong deflection of light is desired, since it is this combination that imparts hiding power to a paint film. The asymmetry parameter g (Equation 6) is the average cosine of the scattering angle associated with a scattering microstructure, weighted by the scattered intensity as a function of angle: The asymmetry parameter g is unitless and varies between - 1 (for perfect backwards scattering) and 1 (for perfect forward scattering). Both the asymmetry parameter and the scattering coefficient play prominent roles in multiple scattering theory, where the term defined here as the angle-weighted scattering coefficient s (Equation 7) expresses the combined ability of a scattering feature to strongly deflect incident electromagnetic radiation off the incident direction: The term s, like the scattering coefficient S, is expressed in units of inverse length. It can be considered a definitive figure of merit in the context of the ability of a scattering feature in a paint film to contribute to hiding power. Throughout this study, scattering results are expressed in terms of both the scattering coefficient S and the angle-weighted scattering coefficient s.
Figure 8. Percent error in scattering coefficient S for finite element calculations compared to Mie theory for the case of a 197nm diameter, n=2.74 sphere in n=1.514 resin; the light wavelength is 560 nm.
Figure. Near field solutions for the scattering of 7 spheres.
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Comment: (c) 2003 Roger H. French ,
frenchrh@lrsm.upenn.edu |
