This example demonstrates the ability of XFdtd^® 6 to include anisotropic dielectric materials with off-diagonal terms in the permittivity tensor.

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  Figure 1
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  Figure 2
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  Figure 3
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  Figure 4
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  Figure 5
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  Figure 6
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  Figure 7
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  Figure 8
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  Figure 9

The permittivity tensor for the example material is given in the frequency domain as

|	7	0	0	|
|	0	(5 - 0.1j)	-j	|
|	0	j	(5 - 0.1j)	|

It is desired to calculate the bistatic scattering pattern for a sphere of this material of size ka = 0.5, where k is the wave number given by 2pi / lambda where lambda is the free space wavelength. To implement this calculation we pick the frequency to be 300 MHz giving a wavelength of 1m. This determines the radius of the sphere to be 0.5/2pi or 0.07958 m. At a frequency of 300 MHz the XFdtd Material Parameter Calculator converts a complex permittivity of (5 – 0.1j ) to a permittivity of 5 and a conductivity of 0.00167. Similarly the conductivity for the off-diagonal terms is +/- 0.01669. The material is entered into XFdtd with the Edit Electrical Material menu as shown in Figure 1.

The 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 do make it gyrotropic. After the material parameters are entered, a sphere of radius 7.958 cm is generated in and meshed using 5mm cells. Excitation is a 300 MHz plane wave incident from the –X direction (theta =90, phi=180) with the electric field Z (theta) polarized. 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 permittivity 5. The bistatic scattering patterns in the E and H planes are shown in Figures 2 and 3. For the E plane pattern theta varies while for the H plane pattern phi varies, so that for the E plane pattern plot the incident field comes from the 270 degree direction but for the H plane pattern plot it is incident from the 180 degree direction. The cross-polarized scattering cross section sigma is not exactly zero due to numerical approximations, but is too low in amplitude to appear on these plots. Next consider the corresponding results for the anisotropic sphere in Figures 4 and 5. The cross-polarized scattering cross section sigma, while weaker than the co-polarized, is definitely present in the results.

The cross-polarized Y component of the electric field is easily observable in near zone field displays. Figures 6, 7, and 8 show transient near field snapshots of the X, Y, and Z polarized electric field components respectively in the E plane of the sphere. Figure 9 shows the steady state Ey field in the E plane for the anisotropic sphere.