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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