Use of the G-TFSF and AFP Methods to Study Scattering from 3D Defects in Edges or Wedges

1. Use of the G-TFSF and AFP Methods to Study Scattering from 3D Defects in Edges or Wedges Zhen Chen, John Schneider School of EECS FDTD Group Abstract/Introduction Comparison of G-TFSF and G-AFP-TFSF The generalized total-field/scattered-field (G-TFSF) method was developed to study the fields scattered from infinite scatterers such as wedges [Anantha and Taflove, IEEE Trans. Antennas Propagat., 50(10):1337-1349, 2002]. The G-TFSF method provides a way of illuminating these scatterers with a plane. In the G-TFSF method, portions of the TFSF boundary are embedded in the perfectly match layer (PML) that terminates the grid. To account for the effect the PML has on the incident field, in the original G-TFSF formulation, the equations used to correct the fields along the TFSF boundary employed scalar correction factors. These scalar factors do not account for the fact that the PML is dispersive. Thus, where the TFSF boundary passes through the PML there is leakage of spurious fields in excess of what it would be if one were to account for the dispersion within the PML. Independent of the G-TFSF method, in the past decade a new type of TFSF technique, known as the analytic field propagation (AFP) method, was been developed [e.g., Schneider and Abdijalilov, IEEE Trans. Antennas Propagat., 54(9):2531-2542, 2006]. In the AFP method the incident field is calculated analytically but in such a way as to perfectly mimic the numerical dispersion inherent in the finite-difference time-domain (FDTD) grid. Because the AFP method uses, as an intermediate step, a spectral description of the incident field, it can be coupled with the G-TFSF method to account for the dispersive nature of the PML. In this way the spurious leaked fields can be greatly reduced. In this work we demonstrate the use of a hybrid G-TFSF/AFP technique, that we identify as the G-AFP-TFSF method, that provides a high fidelity approach to study not just scattering from wedges but also scattering from 3D “defects” in wedges or edges. Nodes adjacent to a TFSF boundary must have the incident field either added or subtract as part of the updating process. The original G-TFSF method accounts for the presence of the PML by scaling the incident field by a scalar constant based on the attenuation caused by the PML. This would be the correct approach if the PML attenuation were independent of frequency (which, from Fig. 2(a), it nearly is) and it did not impart a phase shift. But, from Fig. 2(b), the PML clearly does impart a phase shift which is nearly linear as a function of frequency. Figure 3 shows the field at an observation point in the scattered-field region of two simulations where no scatterer was present and hence the scattered field should be zero. One simulation used the original G-TFSF method while the other used the G-AFP-TFSF method. A pulsed plane wave with unit peak amplitude was used. The G-AFP-TFSF method has about 35 dB less leaked field. Figure 3. Spurious leaked field measured using original G-TFSF method (magnitude) and G-AFP-TFSF method (magnitude and phase). [1] Calibration A crucial step in implementing of the G-AFP-TFSF method is a calibration that measures the effect the PML has on the amplitude and phase of the spectral components of the incident field. Two auxiliary simulations are performed: one in which nodes are embedded in the PML and one in which the nodes are outside the PML. A “standard” TFSF boundary can be used in these simulations and, other than the displacement of the PML, all other simulation parameters are held fixed, i.e., incident angle, incident spectrum, displacement between nodes and TFSF boundary, etc. Figure 1 depicts these two separate simulations in which the time-domain fields at the measured nodes are recorded. By transforming the time-domain fields obtain from the two auxiliary simulations to the frequency domain, one can obtain the the transfer function that describes the effect the PML has on each node. Figure 2 shows the magnitude and phase as a function of frequency for nodes at three different depths. [1] Schneider and Chen,IEEE Trans. AntennasPropagat., acceptedforpublication 2011. TMz Illumination of A Half-Plane in 3D Figure 4 depicts a 2D simulation in which a TMzplane wave illuminates a PEC half-plane. The incident field consists of both the incoming plane wave and the reflected plane wave—both are introduced over the TFSF boundary. The AFP method, which assumes the existence of a PEC plane, is used to introduce this field Note that the TFSF boundary is only two-sided. One segment of the TFSF boundary (shown in red) passes through the PML on the right side of the grid. The nodes adjacent to the TFSF boundary that are within the PML use the incident field as described by the G-AFP-TFSF method. This 2D simulation is subsequently used as an auxiliary simulation to provide the incident field over the TFSF boundary in a 3D simulation. Figure 4. TMzsimulation in which a plane wave illuminates a PEC half-plane. Figure 1. Auxiliary calibration simulations. (a) Measured nodes inside PML. (b) Measured nodes outside PML. [1] Figure 5 depicts a 3D grid together with the 2D grid that is used to provide the incident field over the TFSF boundary (the 2D grid appears the 3d grid). The total-field (TF) region in the 3D grid is the cuboid bound on top and bottom by the yellow rectangles. The yellow lines in Fig. 4 correspond to locations of the vertical faces of the TFSF boundary in 3D. Because the incident field used over the 3D TFSF boundary comes from an appropriately constructed FDTD simulation (as opposed to some analytic calculation), it inherently and exactly incorporates all the numerical dispersion of the FDTD grid. Thus there is never any leakage of fields from the 3D TFSF boundary that exceeds machine precision. However, this is not to say the 3D TFSF implementation is a perfect realization ofplane-wave illumination. There are some slight imperfectionsin the 2D simulation, such as reflections from the PML, that necessarily deviate from true plane-wave illumination. The implementation here requires that the propagation direction be orthogonal to one of the grid axes (in Fig. 5, propagation is orthogonal to the z-direction.) (a) (b) Figure 5.An auxiliary TMzgrid is shown above a 3D grid. The incident field for the 3D TFSF boundary is taken from the auxiliary 2D grid. The PEC half-plane has a notch in it in the TF region Figure 2. (a) Magnitude and (b) phase of transfer function measured at three different depths within the PML. [1]

2. Figure 6. Snapshots of theEzfield taken from the auxiliaryTMzgrid when the PEC plane spans the computational domain. Slight scattering is visible where the TFSF boundary passes into the PML, but otherwise no scattered field is present in thescattered-field (SF)region. Figure 7. Snapshots of the Ezfield taken from the auxiliary TMzgrid when the PEC is a half-plane. Since the AFP method assume an infinite full-plane, the field scattered from the edge enters the SF region. Figure 8. Snapshots of Ezover a constant-z plane running through the center of the 3D grid when the notch is not present. No scattered fields appear in the SF region. Figure9. Snapshots of Ezover a constant-z plane running through the center of the 3D grid when the notch is present. Results Figure 9shows the same field as Fig. 8 except the notch is now present. The fields scattered from this notch do propagate into the SF region. Note that Figs. 7 and9correspond to the fields associated with the grids depicted in Fig. 5. Figure 6 shows a colormap of the Ezfield taken from the 2D simulation in which the PEC plane spans the entire computational domain. These snapshots were taken at 300, 400, 500, and 600 time-steps into the simulation. The incoming and reflected wavefrontsare visible in the TF region. Since the incident field in the AFP method assume the existence of an infinite plane, there are (almost) no scattered fields present in thescattered-field (SF)region. In these color maps the fields are visible over four decades and hence the absence of a visible field indicates the field at that location is four (or more) orders of magnitude smaller than the peak of the incident field (this is usual for judging spurious leaked fields). Figure 7 shows the same field as Fig. 6 except now the PEC is a half-plane. The fields scattered from the edge are visible in the SF region. The geometry of this simulation is the one depicted in Fig. 4 and the top portion of Fig. 5. Figure 8shows snapshots of the Ezfield taken from a 3D simulation. Fields are recorded over a constant-z planethat bisects the 3D grid. InFig. 8 there is PEC half-plane but no notch is present. Thus, there are no scattered fields present in the SF region since the incident field in the 3D simulation (which is take from the 2D simulation) includes the scattering from the edge. Conclusions By embedding the TFSF boundary in a PML, the GTFSF method permits the efficient modeling of scattering from infinite scattererssuch as wedges. We have extended the method by merging it with the AFP technique. This hybrid G-AFP-TFSF method, which accounts for the dispersion of the PML, greatly reduces the spurious scattering associated with the passage of the TFSF boundary into the PML. Furthermore the AFP method directly accounts for the reflected (and transmitted) fields associated with infinite scatterers. Thus in 2D the TFSF boundary merely has to be two-sided instead of four. We demonstrated how the G-AFP-TFSF technique can be used to study scattering in a 3D problem in which there is a “defect” in an edge. The technique can be used for dielectric as well as PEC wedges. Both TMzand TEzpolarization are allowed although the incident field is confined to having a wave vector that lies in the xy-plane. Contact information: John Schneider Email: schneidj@eecs.wsu.edu School of Electrical Engineering and Computer Science, Washington State University