This example demonstrates the ability of XFdtd to simulate dielectric materials with off-diagonal terms in the permittivity tensor. The case in point here is the computation of bistatic radar cross section from an anisotropic sphere excited by a plane wave.
The sphere is chosen to have a size ka = 0.5 where k is the wave number given by 2π/λ, where λ is the free space wavelength. For this simulation, the frequency is chosen to be 300 MHz giving a wavelength of 1 m. In turn, the radius of the sphere will be 79.58 mm. The permittivity of the material under consideration for this example has a tensor in the frequency domain given by:
At 300 MHz, the terms (5 - 0.1j) convert to a permittivity of 5 and a conductivity of 0.00167 S/m. Similarly the conductivity for the off-diagonal terms (+/- j) convert to a conductivity of +/- 0.01669 S/m. A view of this material entered into the XF7 material editor is shown in Figure 1. This material is slightly lossy due to the imaginary terms on the diagonal. The off-diagonal imaginary terms have opposite signs and therefore do not add loss to the material, but they do make it gyrotropic.
A sphere is created in XF7 of the appropriate size and meshed in a 5mm grid. An incident plane wave with a frequency of 300 MHz is created to be incident from the -X direction (theta = 90 degrees, phi = 180 degrees) with the electric field Z polarized (theta direction). With this excitation, if the material were diagonally isotropic there would be no Y-polarized electric field.
For comparison, an initial calculation is made for an isotropic sphere with a permittivity of 5. The bistatic scattering patterns in the E and H planes are computed for the isotropic sphere and are shown in Figures 2 and 3. For the E-plane pattern, the angular variation is in theta around the XZ plane of the sphere and the wave is incident from 270 degrees. The H-plane pattern has variation in phi around the XY plane with the wave incident from the 180 degree direction. The co-polarized plots are visible in the figures but the cross-polarized signal is so small that it does not appear on the plots. This is as expected for the isotropic sphere.
When the anisotropic sphere is substituted for the isotropic one, the resulting output is different. As can be seen in Figures 4 and 5, the bistatic scattering pattern shows a cross-polarized component in addition to the co-polarized signal. The cross polarized pattern is a result of the anisotropic nature of the material.
In the near-field, the Y-polarized electric field is visible for the anisotropic sphere. In Figure 6, the steady state field magnitude is displayed for the Y component of the field in the E-plane around the anisotropic sphere.