A variation of a Luneberg lens simulated at 15GHz with XFdtd^®, formed by two concentric dielectric spheres.

A variation of a Luneberg lens is simulated at 15 GHz with XFdtd.The lens is formed by two concentric dielectric spheres with relative permittivities of 1.7 and 1.4 for the inner and outer spheres. The inner sphere has a radius of 125 mm; the radius of the outer sphere is 250 mm. A 600x600x10 mm dielectric sheet with a relative permittivity of 4 is placed 100 mm behind the sphere to represent the surface of a plastic box enclosing the lens.A vertically polarized 15 GHz plane wave is incident on the opposite side of the sphere from the dielectric plate.

This geometry measures approximately 47 wavelengths (for permittivity of 4) on each side for a total volume of over 100,000 cubic wavelengths. It is not possible to obtain a full wave solution for such a large volume including dielectric by any other calculation method.Yet an approximate calculation that does not consider the diffraction effects, such as optical ray tracing, will not provide accurate results. Due to the enormous size of the geometry, image planes are used by XFdtd in two dimensions to reduce the calculation space to one-quarter of the original size. A perfect electric conductor boundary is placed at Z=1 to image the electric fields while a perfect magnetic conductor boundary is placed at X=1 for the magnetic fields. This geometry is shown in the XFdtd interface illustrated in Figure 1 (xy plane) and Figure 2 (xz plane). The cell size is chosen as 1.5 mm which corresponds to approximately 10 cells per wavelength in the highest dielectric. The total space is 231x468x231 cells.

This calculation was run for 4000 time steps using the XFdtd Multiprocessor Module on a dual-processor 700 MHz Pentium III computer running Red Hat Linux v6.1.The execution time was just under 19 hours and approximately 730 MB of memory were used.

Figure 3 is an animation (~395KB) which displays the transient electric field propagation through the sphere.The magnitude of the electric fields are shown on a dB scale.In the initial frame the sphere geometry is drawn but for subsequent images the geometry has been turned off for better viewing of the fields. The focusing effect is clearly seen. Figure 4 and Figure 5 display the steady-state electric field magnitudes in the Z=1 and X=2 planes. Figure 6 shows the steady state electric field magnitude in the predicted focal plane.

This example demonstrates the power of the image plane features of XFdtd and the large calculations which can be performed using XFdtd on current computer equipment.