Dyadic Green's Function

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


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


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


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


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


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


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


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


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


-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


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

Documents

Dyadic Green's Function