High Frequency Techniques in Electromagnetics Ayhan Altıntaş Bilkent University, Dept. of Electrical Engineering, Ankara, Turkey E-mail: [email protected].

High Frequency Techniques in Electromagnetics

High Frequency Techniques in Electromagnetics

Ayhan Altıntaş

Bilkent University, Dept. of Electrical Engineering, Ankara, Turkey

E-mail: [email protected]

Ayhan Altıntaş

Bilkent University, Dept. of Electrical Engineering, Ankara, Turkey

E-mail: [email protected]

OutlineOutline

Ray-based Techniques Geometrical Optics (GO) Geometrical Theory of Diffraction (GTD-UTD)

Integral-based Techniques Physical Optics (PO) Physical Theory of Diffraction (PTD) Equivalent Edge Currents (EEC)

Scattering ProblemScattering Problem

J J: induced surface current

Escat()

Einc()

PEC Scatterer

E = Einc() + Escat()

Total Field Radiated by J

Determine E or Escat !

Geometrical OpticsGeometrical Optics

factorphase

sj

factorSpread

ess

EsE

))((

)0()(21

21

PROPERTIES•Abides power conservation in the ray tubes•Phase factor is introduced along rays (local plane waves)•Polarization is preserved in ray-fixed coordinates•Can be derived from Maxwell’s Equations

DIFFICULTY•Not valid in caustics

s12

Astigmatic Ray Tube

0 sLine Caustics

are two caustic distances21,

Geometrical OpticsGeometrical Optics

Properties:Conceptually simple Localized scatteringRequires only tracing of incident and reflected raysPinpoints flash points

Shadow

LitEEE

refincGO

;0

;

Reflected rays

Incident raysShadow RegionScatterer

Shadow boundary

Shadow boundary

Disadvantages: Requires finding of reflection point

on the surface Predicts null field in shadow

regions Predicts discontinuous field along

shadow boundaries

Geometrical OpticsGeometrical Optics

Geometrical Optics for reflection

Source

Image

s

Qr

n̂

Wavefront

cos)(

2

'

11

ro Qas

S’

Caustic distance for reflected rays

)( ro Qa Radius of curvature of the surface at Qr

sjincref es

REE

Note that in 2-D there is only one caustic distance

Geometrical Optics Example – A stripGeometrical Optics Example – A strip

Half Plane Fields Half Plane Fields

Geometrical Theory of Diffraction (GTD)Geometrical Theory of Diffraction (GTD)

Incident ray

Q1Q2

Diffractedrays

Surface diffraction

'

)2( n

Diffracted raysIncident ray

Edge diffraction

s

dQ

Observation direction

Shadow boundary

Shadow boundary

•Ray Theory•Solves some of GO difficulties

GTD CalculationGTD Calculation

GTD Formulation:GTD Formulation:

dGOGTD EEE

sjhsd

incd esADQEE )()( ,

fieldDiffractedE d :

factorspreadsA :)(

factorphasee sj :

.:, coeffndiffractioDyadicD hs

Properties: Conceptionally simple Local phenomena Tracing of diffracted rays Pinpoints flash points Predicts non-zero field in shadow

regions A higher order approximation than

GO in terms of frequency Uniform versions yield smooth and

continuous fields at and around shadow boundaries (transition regions)

Disadvantages: Requires searching for diffraction

points on the edge Requires finding of attachment and

launching points and geodesics on the surface

Fails at caustics where many diffracted rays merge

Properties: Conceptionally simple Local phenomena Tracing of diffracted rays Pinpoints flash points Predicts non-zero field in shadow

regions A higher order approximation than

GO in terms of frequency Uniform versions yield smooth and

continuous fields at and around shadow boundaries (transition regions)

Disadvantages: Requires searching for diffraction

points on the edge Requires finding of attachment and

launching points and geodesics on the surface

Fails at caustics where many diffracted rays merge

e

Edge

s´

o´

sKeller cone

Plane of Diffraction

Eio´

Ei´

Ed

Edo o

Incident ray

Diffracted ray

3-D Edge Diffraction3-D Edge Diffraction

Keller Cone becomes a disk in 2-D problems

Edge Diffraction CoefficientsEdge Diffraction Coefficients

sj

sA

hsdincd e

ssDQEE

)(

,)(

)(

sj

E

inc

inc

D

h

s

E

d

d

esAE

E

D

D

E

E

inc

o

hsd

o

)(

0

0

'

'

,

Note there is only one caustic distanceWhere is the other one?

Keller’s Diffraction Coefficients (GTD)Keller’s Diffraction Coefficients (GTD)

Keller´s edge diffraction coefficients

nnnno

jk

hskn

neD '

coscos

1'

coscos

14

,sin22

)/sin(

nnnno

jk

hskn

neD '

coscos

1'

coscos

14

,sin22

)/sin(

'Not valid when Non-uniform

Numerical Result – GTDNumerical Result – GTD

Numerical Result - UTDNumerical Result - UTD

•In the Uniform Geometrical Theory of Diffraction (UTD) Ds,h contain Fresnel integrals to make them valid in transition regions. (Invented at Ohio State University by Kouyoumjian and Pathak

•Uniform Asypmtotic Theory(UAT) is similar to UTD but uses Keller diffraction and modifies reflected field, not very suitable for numerical work.(Invented at U.of Illinois)

•In the Uniform Geometrical Theory of Diffraction (UTD) Ds,h contain Fresnel integrals to make them valid in transition regions. (Invented at Ohio State University by Kouyoumjian and Pathak

•Uniform Asypmtotic Theory(UAT) is similar to UTD but uses Keller diffraction and modifies reflected field, not very suitable for numerical work.(Invented at U.of Illinois)

GTD-UTD Example – A DiskGTD-UTD Example – A Disk

Backscattering from a square plateBackscattering from a square plate

z

y

x

a

a

einc

hinc

Diffracted Ray Caustics

Diffracted Ray

Caustics

Flat Plate ModelingFlat Plate Modeling

•Scattered field for RCS has many Caustics•Ray based techniques fail at caustics

Physical Optics approximationPhysical Optics approximation

regionShadow

regionLitHnJ

IntegralRadiationdSGJE

EEE

incPO

S

POPO

POinc

;0

;ˆ2

;'

is the GO based surface current.POJ

Properties:•Simple•No need to search for flash points•Stationary phase evaluation of the radiation integral yields reflected field, so PO asymptotically reduces to GO•Stays bounded in the caustics•Suited well for the RCS of targets build up with flat polygonal plates

Disadvantages:•Surface integral required•Reciprocity is not satisfied•Not accurate away from specular reflection

Physical Theory of DiffractionPhysical Theory of Diffraction

fwPO E

S

fw

E

S

POinc

fwPO

dSGJdSGJEE

JJJ

''

We do not know J fw ! How do we calculate the second integral?

Use High frequency asymptotic approximation to E !

fwPO JJJ

Incident Plane Wave

Half plane

Physical Theory of DiffractionPhysical Theory of Diffraction

fw

S

POinc

sjPO

hshsdincPO

ddfw

E

fwPOd

E

POs

inc

EdSGJEE

esADDQEEEE

EEEEEdGO

'

:PTD

)()()( ,,

pointEnd

phaseStationary

analysis asymptoticApply

Note that singularities of and cancel so is valid in transition regionshsD ,

PO

hsD ,fwE

PTD Equivalent Edge Currents (EEC)PTD Equivalent Edge Currents (EEC)

PTD - EEC Derived from the integration of fringe wave currents on a half plane. Then use asymptotic methods to convert the 2-D surface integral into a 1-D line integral.

dteMtsYItssr

ejkZE

dSeJssr

ejkZE

stjktf

C

fjkr

fw

syzjk

s

fwjkr

fw

ˆˆ

ˆ)ˆˆ(

]ˆˆˆˆˆ[4

'ˆˆ4

fm

oo

if

fem

o

if

eo

if

Dk

tHZjM

Dk

tHjD

kZ

tEjI

sinsin

ˆ2

sin

ˆ2

sin

ˆ2

'

'2'2

Surface Integral:

Line Integral:

PTD CoefficientsPTD Coefficients

fm

fem

fe DDD ,, coefficients depending on angles

of the geometry

Various approaches exist to determine these coefficients, most useful ones are by Mitzner (ILDC) and Ando.

RCS of a Flat PlateRCS of a Flat Plate

z

y

x

a

einc

hinc

Disk Example – RevisitedDisk Example – Revisited

Disk - Cross Polar RadiationDisk - Cross Polar Radiation

HF work of A. AltintasHF work of A. Altintas

HF Work of A. AltintasHF Work of A. Altintas

HF Work of A.AltintasHF Work of A.Altintas

HF Work of A.AltintasHF Work of A.Altintas

HF Work of A.AltintasHF Work of A.Altintas

End of the ShowEnd of the Show

EndEnd

GTD Equivalent Edge Currents (EEC)GTD Equivalent Edge Currents (EEC)

Advantages:Finite fields at or around caustics.Field prediction even when there is no GO/GTD ray reaching the observation (corner diffraction).Spatial variations of the incident field are inherently included.

Problems:Not valid in the transition regions of shadow boundaries.Derived heuristically.

GTD - EEC Replaces the edge with non-uniform electric and magnetic line sources.

)(8

sin

)ˆ()(

)(8

sin

)ˆ()(

4

4

hj

o

im

sj

o

ie

Dek

tHI

DekZ

tEI