A Plane Wave Model for Direct Simulation of Reflection and Transmission by Discretely Inhomogeneous Plane Parallel Media

Dr Daniel Mackowski (Auburn)

A formulation is developed for numerically solving the frequency domain Maxwell's equations in plane parallel layers of inhomogeneous media.   As has been done in recent direct simulation models using the discrete dipole approximation (DDA), the plane parallel layer is modeled as an infinite square lattice of W  x W x H unit cells, with W being a sample width of the layer and H the layer thickness.    As opposed to the 3D volume integral/DDA formulation, the derivation begins with a Fourier expansion of the electric field amplitude in the lateral plane, and leads to a coupled system of 1D ordinary differential equations in the depth direction of the layer.   A 1-D dyadic Green's function is derived for this system and used to construct a set of coupled 1-D integral equations for the field expansion coefficients.  The resulting mathematical formulation is considerably simpler and more compact than that derived, for the same system, using the DDA applied to the periodic plane lattice.    Furthermore, the fundamental property variable appearing in the formulation is the Fourier transformed complex permittivity distribution in the unit cell, and the method obviates any need to define or calculate a dipole polarizability.   Although designed primarily for random media calculations, the method is also capable of predicting the single scattering properties of individual  particles; comparisons are presented with demonstrate that the method can accurately reproduce the polarimetric scattering properties of single and multiple spheres.  The derivation of the dyadic Green's function allows for an analytical preconditioning of the equations, and it is shown that this can result in significantly accelerated solution times when applied to densely-packed systems of particles.    Calculation results demonstrate that the method, when applied to inhomogeneous media, can predict coherent backscattering and polarization opposition effects.