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7/31/2019 Dyadic Green's Function
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Dyadic Greens FunctionDyadic Greens Function
Textbook: Sec. 7.1, 7.2, 7.3, 7.5, 2.10
National Taiwan University
Outline
3.0 Introduction
3.1 Greens Functions (1-4)
3.2 Greens Functions for Hn0 Modes in a
Rectangular Guide (2-4)
3.3 Dyadics (3-5)
3.4 Dyadic Greens Functions (4-3)
3.5 Infinity-Space Dyadic Greens Functions (5-5)
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Excitation of waveguides
Antenna (probe or loop) excitation:
Aperture coupling:
Wire/Aperture antenna in waveguide:
Transitions
Coaxial-to-waveguide transition
3.0 Introduction
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3.1 Greens Function
1
Basic Concept: ( ) ( )
: given linear differential operator (DE+BC)
( ): given source distribution;
( ): the field to be determined
key: the inverse integral operator with kernel ( ,
r S r
S r
r
G r
L
L
L
1
1 1
) :
( , ) ' the solution:
( ) ( ) ( ) ( , ) ( ) '
( ) ( , ) ( ) ' [ ( , )] ( ) ' ( );
( , ) ( ) where ( , ) the Green's
r
G r r dv
r r S r G r r S r dv
r G r r S r dv G r r S r dv S r S
G r r r r G r r
L
L L L
L L L
L function ofL
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Properties of Greens Functions -11Math: Green's function ( , ) is the kernel of inverse operator .G r r
L
Phy: ( , ) is the solution or field measured at field point due to a
unit point source ( ) at source point .
G r r r
r r r
Solution ( ) ( , ) ( ) superposition of point-source fields.r G r r S r dv
If * self-adjoint operator
( , ) ( , ) :math: symmetric
phy: reciprocity
G r r G r r
L L
1 1 2 2
1 2 1 2
1 2 1 2
1 2 2 1
pf: ( , ) ( ), ( , ) ( )
= *
( , ), ( , ) ( , ), ( , )
( ) ( , ) ( , ) ( )
( , ) ( , )
G r r r r G r r r r
G r r G r r G r r G r r
r r G r r dv G r r r r dv
G r r G r r
L L
L L
L L
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Properties of Greens Functions -2
DE: ( , ) ( ) ,( , ) :
BC: ( , ) 0,
=linear differential operator
= linear differential boundary operator
( ):
G r r r r r G r r
G r r r
r
L
B
L
B
DE: ( ) ( , ) ( ) ' ,
BC: ( , ) 0 ,
Same homogeneous boundary conditions
pf: ( ) ( , ) ( ) ' [ ( , )] ( ) '
r G r r S r dv r
G r r r
r G r r S r dv G r r S r dv
B
L L L
0
( ) ( ) ( ) ,
( ) ( , ) ( ) ' [ ( , )] ( ) ' 0,
r r S r dv S r r
r G r r S r dv G r r S r dv r
B B B
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2 2 2 2
2 22 2 2
2 2
,
( ) ,
Uniform unit line current: ( ') ( ')
, 0
0 (field) 0
( ) ( ) (
y
y x z
y y y
Jy y
x z
AE j A H A
j
k A J k
J yJ y x x z z
A yA A A
k A k A J x
') ( ')
xA
x z z
A
0
y
y
A
x
0zA
0
2 2 2
2 2
0
,
DE: ( ) ( ') ( ')
BC: ( 0) 0 and ( ) 0
RC: ( ) outgoing
y y y y
y
y y
y
z
x z
E j A j A y E y E j A
k E j x x z z
E x E x a
E z
Greens Fx. for Hn0 Modes in Rectangular Guide
x
y
z0
a
x
0 zz
y
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Derivation to Greens Functions
2 2 2
2 2DE: ( ) ( , ; ', ') ( ') ( ') (A)
BC: ( 0) 0 and ( ) 0 (B)
RC: ( ) outgoing (C)
y
x zk G x z x z x x z z
G x G x a
G z
A G
, .y yE j A j G
0
1
mod
(B) ( , , ', ') ( ; ', ') sin
n
n
n
H e
n xG x z x z g z x z
a
2
2 2 22 2 2
2 2 2(A) ( ) sin { [( ) ]} sin ( ') ( ')
n
n n
n n
n x d n n xk g k g x x z z
x z a dz a a
0
sina n x
dxa
22
2
2 'DE: ( ) ( sin ) ( ')
BC: ( ) outgoing
m m
m
d m xg z z
dz a a
g z
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Derivation to Greens Functions -22 2
2
2
DE: ( ) 0
Source cond.: ( ' ) ( ' )
( ' ) ( ' ) sin
RC: ( ) outgoing
m m
m m
m m
m
d
dz
dg dg m xa adz dz
g
g z g z
z z
g z
' '
2
' '
at ' ' ~ , '' ~ '
at '
2 '( " ) ( sin ) ( ')
m m m
m
z z
m m m
z z
g C z z g g
g C z z
m xg g dz z z dz
a a
( ')
( ')
'
; '
; '
2 'sin '( ' ) '( ' ) ( ) 2
1 ' 'sin ; sin
m
m
m
z z
m z z
m m m m m
z z
m
m m
Ae z zg
Ae z z
m xg z g z A A A
a am x e m x
A ga a a a
'
2 2
1
'( , , ', ') sin sin ; ;
m z z
n cn cn
n m
e m x m x nG x z x z k k k
a a a a
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Physical Intepretation
0
0
a) modes are excited, sinn y
n
n xH E G
a
2 2
1
1
;b) propagation constant: ;
;
, 1, min mod
, 2, higher-order evanescent modes
n cn
n cn cn
n cn
c
c
a
a
j k k nk k k
k k a
k k n do ant propagating e
k k n
2 2
1 ' 1c) modal coefficient sin ,
( )na
n x
a nk
'2
ax
JJ
'4
ax
d) excitation ofHno
modes: e) excitation of Hno
modes:
'2
by
J
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3.3 Dyadic Basic Concept
1 2 3
1 2 3
Cartesian coordinate systems: ( , , ) ( , , )
Basis vectors: ( , , ) ( , , ): Constant orthonormal (ON) vectors
Differential operator: , ( , , )
Vector differentia
j
j
x x x x y z
i i i x y z
Dx x y z
3
1 2 3
11 2 3
l (or del) operator, :
j j j j
j
x y z i i i i D i Dx y z x x x
1 1 2 2 3 3
331 2
1 1 2 3
Scalar: ( ) Vector: ( )
; = component of in
x y z
j j j j j j
j
r
A A r A x A y A z A i A i A i
ii iA i A i A i
A A A
A
Matrix representation
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Tensor of Second Order
3 3
1 1
( )
jk j k
jk j k
j k
T T r T i i
T i i
dentity dyadic :
1 0 0
0 1 0 identity matrix;
0 0 1
1,Kronecker delta
0,
jk j k
jk
I
I i i
j k
j k
Notation:
Scalar:
Vector: , , , , ,
Dyadic: , , ,
Triadic: , , ,
A A A A
T T T
T T T
A
31 2
11 1 1 12 1 2 13 1 3
11311 12
21 2 1 22 2 2 23 2 3matrix
221 22 23representaion
31 3 1 32 3 2 33 3 331 32 33 3
ii iT i i T i i T i i
iTT TT i i T i i T i i
iT T TT i i T i i T i i
T T T i
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Algebraic Operations
1311 12 1 1
21 22 23 2 2
31 32 3 333
( ) ( ) vector
kl
jk j k l l j k l jk l j jk k j j
jk k j
T A T i i A i i i i T A i T A i B B
TT T A B
T A B T T T A B
T T A BT
( ) ( )kl
jk j k l l j k l jk l j jk k j jI A i i A i i i i A i A i A A
A I A
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Differential Invariants
vector differential (or del) operator
scalar, =vector, =dyadic function
j j jjx
i D i
A T
vector; ( ) ( ) dyadic
( ) ( ) scalar
jk
j j j j k k j k j k
j j k k j k j k j j
i D A i D A i i i D A
A i D A i i i D A D A
2
2
( ) vector
( ) triadic
( ) vector
( ) dyadic
j j k k j j
j j kl k l j k l j kl
j j kl k l j k l j kl l j kl
j j kl k l j k l j kl j l ijk j kl
A A i D i D A D D A
T i D T i i i i i D T
T i D T i i i i i D T i D T
T i D T i i i i i D T i i D T
T T
( ) dyadicj j k k j k j k j ji D i D T i i D D T D D T
1, (i, j, k)=(1, 2, 3), (2, 3, 1), (3, 1, 2)
-1, (i, j, k)=(1, 3, 2), (2, 1, 3), (3, 2, 1)
0, (i, j, k)=otherwise
ijk
31 2
1 2 3
1 2 3
( ) ( ) ( ) vectorj j k k j k j k i ijk j k
ii i
A i D A i i i D A i D A D D D
A A A
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Identities
0
( )
0
i i j j kl k l i j k l i j kl i j k l i j kl
j i k l j i kl i j k l i j kli j S S S
T i D i D T i i i i i i D D T i i i i D D T
T i i i i D D T i i i i D D T
2
( ) ( )
[( ) ( ) ] ( )( )
i i j j kl k l i j k l i j kl
i k j l i j k l i j kl j j i i k l kl i i j j k l kl
T i D i D T i i i i i i D D T
i i i i i i i i D D T i D i D i i T i D i D i i T
T T T T
( ) ( ) ( ) [( ) ]
( ) ( ) ( ) ( ) ( )
j j k l kl j k l j kl j k l j kl j kl
j j k l kl j j k l kl
T i D i i T i i i D T i i i D T D T
i D i i T i D i i T T T
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3.4 Scalar Greens Functions
scalar field ( ) ~ scalar source ( ); Linear problem:
DE: ( ) ( )
Linear operator equation: BC (homogeneous) for bound space
RC for space
r S r S
r S r
L
1
1
The inverse operator with kernel ( , ) such that
( ) ( , ) ( ) '
( ) ( , ) ( ) ' ( , ) ( ) ' ( ),
G r r
r S G r r S r dv
r G r r S r dv G r r S r dv S r S
L
L
L L L
1
DE: ( , ) ( );
same BC (homogeneous) & RC
( , ) : kernel of , Green's function of
G r r r r
G r r
L L
L
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Dyadic Greens Functions
Vector field ( ) ~ vector source ( ); Linear problem:
DE: ( ) ( )
Linear operator equation: BC (homogeneous) for bound space
RC for space
F r S r F S
F r S r
L
1
1
The inverse operator with kernel ( , ) such that
( ) ( , ) ( ) '
( ) ( , ) ( ) ' ( , ) ( ) ' ( ),
G r r
F r S G r r S r dv
F r G r r S r dv G r r S r dv S r S
L
L
L L L
1
DE: ( , ) ( );
same BC (homogeneous) & RC
( , ) ( , ) : kernel of , Dyadic Green's function of jk j k
G r r I r r
G r r G r r i i
L L
L
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Properties of Dyadic Greens Functions
( , ) ( , ) ( ) ( )
each comp. ( , ) : ( , ) ( )
( , ) ( ), 1, 2, 3; ( , ) 0,
jk j k j k jk
jk jk jk
ii ij
G r r G r r i i I r r i i r r
G r r G r r r r
G r r r r i G r r i j
L L
L
L L
( ) ( , ) ( ) ' ' ' '
( ) ( , ) ( ) ' ( , ) ( ) ''
For ( ) ( ) , k-directed unit point source a
jk j k l l j k l jk l j jk k j j
j jk k jk k
k jk
F r G r r S r dv G i i S i dv i i i G S dv i G S dv i F
F r G r r S r dv G r r S r dv
S r r r
t
( ) ( , ) ( ) '' ( , ) : j-comp. field at .j jk jk
r
F r G r r r r dv G r r r
the j-comp. field at due to aPhysical interpretation: ( , ) .
k-directed unit point source atjk
rG r r
r
field source
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-Space Greens Dyadic for A2 2 2 2
DE: ( ) ( , ) ( );( , )
RC: ( ) outgoing 4
jk r rG k g r r r r k eg r r
G r r r
L
2 2
1
Dyadic Green's fx. ( , ) for : vector potential ( ) vector source ( )
DE: ( ) ( ) ( ) ( )( ) ( ) ( , ) (
RC: ( ) outgoing
A
A
G r r A A r J r
A r k A r J rA r J r G r r J
A r
LL ) '
( ) ( , ) ( ) ' ( ),A
r dv
A r G r r J r dv J r J
L L
( ) [ ( , ) ] ( ) ' ( , ) ( ) 'A r g r r I J r dv g r r J r dv
2 2DE.: ( , ) ( ) ( , ) ( )
RC.: ( ) outgoing
[ ( , )] ( , ) ( ) ( , ) is a solution
uniqueness theorem solution
A A
A
G r r k G r r I r r
G r
Ig r r I g r r I r r Ig r r
G
L
L L
( , ) ( , )A
r r I g r r
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-Space Greens Dyadic for E
2
Vector field , ( ) vector source ( )
( )
E H E J r
E j HE j H j j E J k E j J
H j E J
2
1DE: ( ) ( ) ( ) ( ) ( ) ( ) ( , ) ( ) 'RC: ( ) outgoing
( ) ( , ) ( ) ' ( ),
E r k E r j J rE r J r j G r r J r dv
E r
E r j G r r J r dv j J r
LL
L L J
2
0
0
2
To find
[ ( )]
( )
1( )
G
G k G I r r
I I
G r rk
2DE.: ( , ) ( - ) ( , ) ( )
RC.: ( ) outgoingE
G r r k G r r I r r
G r
L
2
2 2 ( )
G G G
G G k G I r r
2 2
2( ) ( , ) ( ) ( )k G r r I r r
k
r
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-Space Greens Dyadic for E2 2
2
2 2 2 2
2 2
DE: ( ) ( , ) ( ) ( ),
RC: ( ) outgoing
consider ( )[ ( ) ( , )] ( )( ) ( , )
k G r r I r r k
G r
k I g r r I k g r r k k
2
2
2
( ) ( )
by uniqueness theorem, the solution:
( , ) ( ) ( , ) : -Space Green's dyadic for
( ) [( ) ( , )] ( ) '
I r rk
G r r I g r r E k
E r j I g r r J r dvk
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Reciprocity for Greens Dyadic
Two completely independent (or unrelated) problems with
same freq. and environment, in linear isotropic media (,).
, ( , ), ( , ))
source field
ReciprocityProb. A: ( , ) ( , )
, ,Prob. B: ( , ) ( , )
a b a a b b a b a
a a a a
a b b ab b b b
f s E H J M E J H
J M E H
f s f sJ M E H
bV
M dv
Given Green's dyadic , : ( ) , ( ) ' .
( ) ( ): ( ) , ( ) ' Reciprocity:
( ) ( ): ( ) , ( )
a a a a
b b b b
G r r E r G r r J r dv
E r J r E r G r r J r dv
E r J r E r G r r J r dv
( ) ( ) ( ) ( ) '
( ) , ( ) ' ( ) , ( ) ' , ,
b a a b
b a a b
J r E r dv J r E r dv
J r G r r J r dvdv J r G r r J r dvdv G r r G r r
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Problems