Implementation of Gaussian Beam Implementation of Gaussian Beam Sources in FDTD for Scattering Problems Sources in FDTD for Scattering Problems

Electromagnetic Communication Electromagnetic Communication LaboratoryLaboratory

The Pennsylvania State UniversityThe Pennsylvania State University

University Park, PA 16802 University Park, PA 16802

Emails: Emails: lcma@ieee.orglcma@ieee.org, , rajmittra@ieee.orgrajmittra@ieee.org

Lai-Ching Ma & Raj Mittra

OutlineOutline

Motivation Implementation and characteristics of

the new Gaussian beam sourceParametric study of the new source

implementationNumerical examplesConclusions

MotivationMotivationTo implement a focused source To implement a focused source

distribution, namely a Gaussian beam, for distribution, namely a Gaussian beam, for scattering problems in FDTD.scattering problems in FDTD.– To eliminate the edge effectsTo eliminate the edge effects– To investigate the local behavior of the To investigate the local behavior of the

scattering phenomenonscattering phenomenon

SF

TF

TF/SF Interface

k

PML

PML

PML

PML

PML Some applications Some applications include scattering by: rough include scattering by: rough surfaces; photonic crystals.surfaces; photonic crystals.

Commonly used FDTD sources in scattering problemsCommonly used FDTD sources in scattering problems Total-field/Scattered Field Total-field/Scattered Field

FormulationFormulation

Incident fields are needed on the Incident fields are needed on the TF/SF interface only.TF/SF interface only.

The scatterers must be totally The scatterers must be totally enclosed by the TF/SF interface.enclosed by the TF/SF interface.

Scattered Field FormulationScattered Field Formulation

Incident fields are needed over the Incident fields are needed over the entire volume of the scatterer.entire volume of the scatterer.

Computation of incident fields is Computation of incident fields is difficult when the scatterer is difficult when the scatterer is comprised of frequency dependent comprised of frequency dependent materials.materials.

SF

TF

TF/SF interface

SF

Implementation of Gaussian beam sourcesImplementation of Gaussian beam sourcesBased on the TF/SF formulation for plane wave

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.

Einc = Einco * exp( -2/w2 )

Einco = plane wave amplitude, w = beam width,

= distance from the beam axis

Incident field amplitude on TF/SF

interfaceSF

TF

TF/SF interface

scattererAll 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

PML

PML

PML

PML

Characteristics of the new sourceCharacteristics of the new sourceNo 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.

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.

SF

TF

TF/SF Interface

k

PML

PML

PML

PML

PML

w

Parametric study for the implementation of Parametric study for the implementation of Gaussian beam sourcesGaussian 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

Observation: How close is the field distribution at the TF/SF interface to the incident field distribution ?

Frequency

Test 1: Varying the TF/SF Test 1: Varying the TF/SF interface area interface area Normal incidence, Ex-pol.Beam width w = 90 mm

Frequency = 10 GHzBeam width is 3 at 10GHz

TF/SF interface (LxL)Case 1: L = 2w = 180 mmCase 2: L = 3w = 270 mmCase 3: L = 4w = 360 mm

Case 1 edge: -9 dB

Case 2 edge: -20 dB

Case 3 edge: -35 dB

Incident |Ex| set at the TF/SF interface

Case 1

Case 3: L = 4 wCase 2: L = 3 w

FDTD Computed |Ex| at the TF/SF interface at 10 GHz

Case 1: L = 2 w

L L

Incident direction

No distortions for all cases.

Comparison of field distribution on TF/SF interface with Comparison of field distribution on TF/SF interface with incident field for normal and oblique incidence at 10 GHzincident field for normal and oblique incidence at 10 GHz

Inc. from (=0o,=0o)*E /Ex-polarized

Inc. from (=30o,=180o)*E-polarized

Inc. from (=30o,=270o)*E-polarized

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

A B

*Incident angles are defined by phase progression on the TF/SF for plane wave )A B

30o

A B

30oIncident direction*

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.

Test 2: Varying the frequency / Test 2: Varying the frequency / electric size of beam width electric size of beam width

Normal incidence, Ex-pol.Beam width w = 90 mm

TF/SF size L = 3w = 270 mm

Frequency/Electric size of beam widthf = 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

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 = 3.0

FDTD Computed |Ex| at the TF/SF interfaceIncident |Ex| set at the TF/SF interface

(same at all freq.)

L = 3wL

Distortion

Incident direction

Comparison of field distribution on TF/SF interface with Comparison of field distribution on TF/SF interface with incident field for normal and oblique incidenceincident field for normal and oblique incidence

Cut A

Cut B

Cut A

Cut B

Cut A

Cut B

Inc. from (=0o,=0o)*E /Ex-polarized

Inc. from (=30o,=180o)*E-polarized

Inc. from (=30o,=270o)*E-polarized

A B

*Incident angles are defined by phase progression on the TF/SF for plane wave A B

30o

A B

30oIncident direction*

- Distortions for w = 0.5 in all cases.

- Slight distortion for w = 1.0 at oblique incidence (E-polarized)

Example 1: Scattering by a Example 1: Scattering by a homogeneous dielectric slab at homogeneous dielectric slab at

oblique incidenceoblique incidence

Vertical field distribution |EY| at 15 GHz

Dielectric slab r=4L=W=80mm, D=19mm

(D=1.9 at 15 GHz in dielectric)TF/SF interface

Incident waves comes from (=150o,=0o)*, E-polarizedGaussian beam width = 11 mm = 0.55 at 15 GHz

SF

Free space

Free space

Transmitted beam

Reflected beam Incident direction

TF

SF

Slab

The incident beam is not seen in the figure because it is in the scattered field region.

TF

SF

TF/SF Interface k

PML

PML

PML

PMLPML

*Incident angles are defined by the phase progression on the TF/SF for plane wave

Example 2:Example 2:Scattering by EBG Array at Normal / Oblique incidenceScattering by EBG Array at Normal / Oblique incidence

Array settings:Ele. Separation: 2.25 mm x 5 mm x 4 mmEle. Separation in : 0.1125 x 0.25 x 0.20Total number: 38 x 17 x 6 = 3876Total number falls within beam width = 34 ( X:10, Y:5)

85.5 mm = 4.3 85 mm = 4.3

24 mm = 1.2

22 mm = 1.1

FDTD Computational domainPhysical size: 85.5 mm x 85 mm x 67 mm

Cell number: 680 x 684 x 536 = 2.5 x 108 cells

= wavelength at 15.0 GHz

Guassian beam

Geometry of one element

Oblique incidenceNormal incidence

TF/SF Interface

k

PML

PML

PML

PMLPML

k

PMLPML

TF

SF

Comparisons of Transmission/Reflection Coefficients with infinite arrayComparisons of Transmission/Reflection Coefficients with infinite array Periodic boundary conditions(PBC)/Plane wave Periodic boundary conditions(PBC)/Plane wave

VS Finite Array/Gaussian Beam for Normal IncidenceVS Finite Array/Gaussian Beam for 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.

k

E = (0,Ey,0)H = (Hx,0,0)

Normal incidence

k

PML

PMLPML

TF

PML

SF

TF/SF interface

Comparisons of Transmission/Reflection Coefficients Comparisons of Transmission/Reflection Coefficients with infinite arraywith infinite array

Periodic boundary conditions(PBC)/Plane wave Periodic boundary conditions(PBC)/Plane wave VS Finite Array/Gaussian Beam for TE and TM VS Finite Array/Gaussian Beam for TE and TM

Oblique Incidence (30Oblique Incidence (30oo))

TEzk

E = (0,Ey,0)H = (Hx,0,Hz)

=30oTMz k

E = (0,Ey,Ez)H = (Hx,0,0)

=30o

TFSF

k

PML

PML

PML PML

Normal IncidenceNormal Incidence-- Field Distribution at 15.0 GHz-- Field Distribution at 15.0 GHz

P3: XZ Plane P4: YZ Plane

P1: XY Plane(just above the array) P2: XY Plane (~1 from

array)

Array: 6 layers

P1

P2

P3 P4

Transmission/Reflection

Incident direction

TF/SF interface just below the array

P4P3

P1

P2

Array

Transverse Field Distribution at 15.0 GHzTransverse Field Distribution at 15.0 GHz

P1 (just above the array) P2 (~1 from array)

Incident direction

Vertical Field Distribution Vertical Field Distribution at 15.0 GHzat 15.0 GHz

Free space Dielectric Slab EBG Array

Free space Dielectric Slab EBG Array

P4: YZ Plane

P4: YZ Plane in Transmission region

P1

P2

P4

Array

Incident direction

Incident direction

ConclusionsConclusions

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.

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.

TF/SF Interface

SF

TF

TF/SF Interfacek

PML

PML

PML

PML

PML

w