Electrodynamics on a grid: The finite-difference time-domain method applied to optics and cloaking
Yee’s arrangement of the field components in a cubic lattice. Each field component is surrounded by the components necessary to calculate the curl terms in Eqs. (1) and (2). For instance, has in the direction and in the direction. The Yee lattice can be seen as a mesh of interlinked loops of magnetic and electric fields. Note that the plane, for integer , represents the arrangement of field components appropriate for a two-dimensional TM system and the plane represents the arrangement of components for a two-dimensional TE system.
The arrangement of field components on a two-dimensional grid for the TM mode. is out of the page. The dotted lines define a boundary for the calculation of the divergence.
Simple optical phenomena arising from Maxwell’s equations. The plots show components of the TM waves in the grid with the scale at the right. Gray represents zero field and the incident wave has an amplitude of 1.0. A grid of is used for each simulation. (a) Simulation of an incident Gaussian pulse reflecting off a perfectly conducting boundary, which is on the left side. (b) Diffraction of a sinusoidal wave through a perfectly conducting double slit. Absorbing boundary conditions are used on the sides in this simulation. The tick marks on the left side mark the positions of maximum intensity. (c) Refraction of a sinusoidal wave through a glass converging lens.
Spatial diagrams of the tensor elements , , and of the permeability that are necessary for bending a TM wavefront around a central region in two dimensions. The numbers indicate the value of the property on the axis at the inner and outer boundary when the ratio of the boundary diameters is 2. The diagram includes the values on the diagonal axis. The solid line is a contour for a value of 1.0 and the dashed line is for . The values of the permittivity vary from 0 at the inner to at the outer boundary and are symmetric. These properties are calculated for a plane wave inbound from the left.
Comparison of the scattering of an incident sinusoidal wave by (a) a perfectly conducting cylinder of radius and (b) the same cylinder with a concentric cloak of radius . The incident wave is to the right (the direction). The shading scale for the electric field is the same as in Fig. 3. The cloaked cylinder produces some reflection due to the incident wave coming in contact with the cloaked object because of the numerical limits on , , and when is near .
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