Implementation of Gaussian Beam Sources in FDTD for Scattering Problems Lai-Ching Ma & Raj Mittra Electromagnetic Communication Laboratory The Pennsylvania State University University Park, PA 16802 Emails: lcma@ieee.org, rajmittra@ieee.org

Outline Motivation Implementation and characteristics of the new Gaussian beam source Parametric study of the new source implementation Numerical examples Conclusions

Motivation To implement a focused source distribution, namely a Gaussian beam, for scattering problems in FDTD. To eliminate the edge effects To investigate the local behavior of the scattering phenomenon SF TF TF/SF Interface k PML Some applications include scattering by: rough surfaces; photonic crystals.

Commonly used FDTD sources in scattering problems Total-field/Scattered Field Formulation Incident fields are needed on the TF/SF interface only. The scatterers must be totally enclosed by the TF/SF interface. Scattered Field Formulation Incident fields are needed over the entire volume of the scatterer. Computation of incident fields is difficult when the scatterer is comprised of frequency dependent materials. SF SF TF TF/SF interface

Implementation of Gaussian beam sources Based on the TF/SF formulation for plane wave Einc = Einco * exp( -2/w2 ) Einco = plane wave amplitude, w = beam width,  = distance from the beam axis The TF/SF interface is implemented on the illuminating surface, rather than on a closed box. To mimic a Gaussian beam, a Gaussian window is applied to a plane wave. Incident field amplitude on TF/SF interface SF TF TF/SF interface scatterer All six faces of computational domain are terminated by PMLs k (Direction defined by phase progression on the TF/SF interface for plane wave ) Beam width w PML

Characteristics of the new source No need to model or design a real antenna element that generates the desired source distribution. Easy to implement. The field distribution at the source location is the same as that of the desired source distribution ( unlike soft source ) while the source is transparent to the reflected fields from the scatterers (unlike hard source), provided that certain conditions are satisfied. SF TF TF/SF Interface k PML w In contrast to the situation when a closed TF/SF interface is used, the scatterer can now be allowed to touch the ABC to reduce the edge effect, and/or to model an infinitely large structure. Transmission and reflection characteristics can now be extracted easily.

Parametric study for the implementation of Gaussian beam sources The modified method truncates the TF/SF by ABC Test 1: Does any reflections come back from the absorbing boundary on the sides of interface ? We have assumed that certain incident field distributions can be sustained to propagate in free space. Test 2: Is such incident field distribution valid/physical for all frequencies ? Fix the physical beam width Test 1: Vary the TF/SF interface area Test 2: Vary the frequency => electric size of beam width Frequency Observation: How close is the field distribution at the TF/SF interface to the incident field distribution ?

Test 1: Varying the TF/SF interface area Case 1 edge: -9 dB Test 1: Varying the TF/SF interface area Case 2 edge: -20 dB Normal incidence, Ex-pol. Beam width w = 90 mm Frequency = 10 GHz Beam width is 3 at 10GHz TF/SF interface (LxL) Case 1: L = 2w = 180 mm Case 2: L = 3w = 270 mm Case 3: L = 4w = 360 mm Case 3 edge: -35 dB Case 1: L = 2 w Case 2: L = 3 w Case 3: L = 4 w Incident direction No distortions for all cases. L L FDTD Computed |Ex| at the TF/SF interface at 10 GHz Case 1 Incident |Ex| set at the TF/SF interface

Comparison of field distribution on TF/SF interface with incident field for normal and oblique incidence at 10 GHz Inc. from (=0o,=0o)* E /Ex-polarized Inc. from (=30o,=270o)* E-polarized Inc. from (=30o,=180o)* E-polarized Incident direction* A B 30o *Incident angles are defined by phase progression on the TF/SF for plane wave ) A B 30o A B Cut A |EX| on TF/SF interface at 10 GHz Cut A |EX| on TF/SF interface at 10 GHz Cut A |EX| on TF/SF interface at 10 GHz No distortions for all cases. Cut B |EX| on TF/SF interface at 10 GHz Cut B |EX| on TF/SF interface at 10 GHz Cut B |EX| on TF/SF interface at 10 GHz

Test 2: Varying the frequency / electric size of beam width f = 1.67 GHz, w = 0.5  f = 3.33 GHz, w = 1.0  f = 5.0 GHz, w = 1.5  f = 10.0 GHz, w = 3  Test 2: Varying the frequency / electric size of beam width Normal incidence, Ex-pol. Beam width w = 90 mm TF/SF size L = 3w = 270 mm FDTD Computed |Ex| at the TF/SF interface Incident |Ex| set at the TF/SF interface (same at all freq.) f = 1.67 GHz, w = 0.5  f = 3.33 GHz, w = 1.0  Distortion f = 5.0 GHz, w = 1.5  f = 10.0 GHz, w = 3.0  Incident direction L L = 3w

Comparison of field distribution on TF/SF interface with incident field for normal and oblique incidence Inc. from (=0o,=0o)* E /Ex-polarized Inc. from (=30o,=270o)* E-polarized Inc. from (=30o,=180o)* E-polarized Incident direction* A B 30o *Incident angles are defined by phase progression on the TF/SF for plane wave A B 30o A B Cut A Cut A Cut A Distortions for w = 0.5  in all cases. - Slight distortion for w = 1.0  at oblique incidence (E-polarized) Cut B Cut B Cut B

TF SF TF/SF Interface k PML Example 1: Scattering by a homogeneous dielectric slab at oblique incidence Incident waves comes from (=150o,=0o)*, E-polarized Gaussian beam width = 11 mm = 0.55  at 15 GHz Vertical field distribution |EY| at 15 GHz Transmitted beam Free space TF Dielectric slab r=4 L=W=80mm, D=19mm (D=1.9  at 15 GHz in dielectric) Slab TF/SF interface SF SF Free space Reflected beam Incident direction *Incident angles are defined by the phase progression on the TF/SF for plane wave The incident beam is not seen in the figure because it is in the scattered field region.

Example 2: Scattering by EBG Array at Normal / Oblique incidence FDTD Computational domain Physical size: 85.5 mm x 85 mm x 67 mm Cell number: 680 x 684 x 536 = 2.5 x 108 cells Normal incidence Oblique incidence TF/SF Interface k PML TF SF  = wavelength at 15.0 GHz 22 mm = 1.1  Geometry of one element 24 mm = 1.2  Guassian beam 85.5 mm = 4.3  85 mm = 4.3  Array settings: Ele. Separation: 2.25 mm x 5 mm x 4 mm Ele. Separation in : 0.1125 x 0.25 x 0.20 Total number: 38 x 17 x 6 = 3876 Total number falls within beam width = 34 ( X:10, Y:5)

Comparisons of Transmission/Reflection Coefficients with infinite array Periodic boundary conditions(PBC)/Plane wave VS Finite Array/Gaussian Beam for Normal Incidence k PML TF SF TF/SF interface k E = (0,Ey,0) H = (Hx,0,0) Normal incidence In finite-array/gaussian-beam case, the transmission/reflection coefficients are computed from the fields on the Huygen’s surfaces in the total field and scattered field region, respectively, followed by normalization using the incident power propagating in forward direction.

TF SF k PML Comparisons of Transmission/Reflection Coefficients with infinite array Periodic boundary conditions(PBC)/Plane wave VS Finite Array/Gaussian Beam for TE and TM Oblique Incidence (30o) k k TMz =30o TEz =30o E = (0,Ey,Ez) H = (Hx,0,0) E = (0,Ey,0) H = (Hx,0,Hz)

Normal Incidence -- Field Distribution at 15.0 GHz P4 P3 P2 P1 TF/SF interface just below the array P3 P4 Array: 6 layers Incident direction P1: XY Plane (just above the array) Transmission/Reflection P2: XY Plane (~1 from array) P3: XZ Plane P4: YZ Plane

Transverse Field Distribution at 15.0 GHz P2 P1 Array Incident direction P1 (just above the array) P2 (~1 from array)

Vertical Field Distribution at 15.0 GHz P1 P2 P4 Array Vertical Field Distribution at 15.0 GHz P4: YZ Plane Incident direction Free space Dielectric Slab EBG Array Incident direction P4: YZ Plane in Transmission region Free space Dielectric Slab EBG Array

TF/SF Interface SF TF k PML w Conclusions An implementation of Gaussian beam sources based on the TF/SF formulation in FDTD has been introduced. It can be used for various scattering problems that require tapered illumination, as opposed to a plane wave incident field. The present method not only preserves the desirable features of the TF/SF formulation, but also allows the scatterers to touch the absorbing boundary to reduce the edge effect. This feature enables us to model an infinitely large structure, which is not possible in the conventional TF/SF approach. The criteria for accurately constructing the Gaussian beam distribution can be on the TF/SF interface have been determined. They are: The incident field must decay to a low level at the four edges of TF/SF interface. The dimension of the Gaussian beam width should be larger than one wavelength. Two numerical examples have been presented to demonstrate the application of this new source to practical scattering problems.