Electromagnetic Theorems 2

UCF Fields from Surface Currents (1)sJ sMand in homogeneous space

r'rR

FAAE

1)(1jj

AFFH

1)(1jj

sJAA 22 k

sMFF 22 k

')'()( dsRG )(rJrA s

')'()( dsRG )(rMrF s

|| r'r RHomogeneous space

Green’s functionR

eGGRGjkR

4,,

r)(r')r'(r)(

UCF Fields from Surface Currents (2)

]')'([1

]}')'([{')'(

dsRG

dsRGjdsRGj

)(rM

)(rJ)(rJE

s

ss

Note: operate on r rather than r’

)(rJ)(rJ ss RGRG )'(])'([

Use math formula AAA )(

)(rMrM)()(rM

s

ss

RGRGRG

)'(

)'(])'([

Use math formula AAA )(

UCF Fields from Surface Currents (3)

')'(

]}'])'([1')'({

dsRG

dsRGj

dsRGj

)(rM

)(rJ)(rJE

s

ss

Since

kZ ,

Likewise (or from Duality Theorem)

')'(

]}'])'([1')'({

dsRG

dsRGjk

dsRGjkY

)(rJ

)(rM)(rMH

s

ss

')'(

]}'])'([1')'({

dsRG

dsRGjk

dsRGjkZ

)(rM

)(rJ)(rJE

s

ss

UCF Kirchhoff-Huygens Formula (1)

HE j

EH j

aaa j MHE aaa j JEH

Region I

Region II

Homogeneous space, source free µ, ϵ

n

Auxiliary problem:

homogeneous space

µ, ϵaJ

aΜ

s

Assume fields in Region II (free space) are E and H

Assume fields from Ja and Ma in free space are Ea and Ha

UCF Kirchhoff-Huygens Formula (2)

EEHEHHEH

JEEEHEMHHHEH

aa

aa

aaa

aaa

jj

jj

aaaa

aaaaaa

MHJEHEHE

MHJEEHHEHEEH

)(Integrate both sides on the whole volume of Region 2, we have

dvds aa

SS

aa )()()(

MHJEnHEHE

Note: 0...

dsS

dv

ds

aa

S

aa

)(

)()(

MHJE

nHEHERegion I

Region II

nn

n

n

S

s

UCF

FAAE

1)(1jja

AFFH

1)(1jja

ak JAA 22

ak MFF 22

')'()( dvRGa )(rJrA

ReGGRG

jkR

4,,

r)(r')r'(r)(

|| r'r R

o

aJaΜ

Rr'

r

,

Kirchhoff-Huygens Formula (3)

')'()( dvRGa )(rMrF

UCF

If ( ) ( '), ( ) 0 can be either or or a au u x y z J r a r r M r a a a a

( ') ( ' ') ( 0 )au u J r a r r a

( ) , ' uGA r (r r )a 0)( rF1( ) , ' ( , ' )

1( ) [ , ' ]

au u

au

j G j G

G

E r ( r r ) a ( r r ) a

H r ( r r ) a

2

( ) ( , ' )

( ) [ , ' ]

au

au

k Gj

G

E r ( r r ) a

H r ( r r ) a2' '( ') ( , ' )

( ') ' [ , ' ]

au

au

k Gj

G

E r ( r r ) a

H r ( r r ) a

Kirchhoff-Huygens Formula (4)

)'(')''()'(

)0(')'()'(

xfdxxxxf

fdxxxf

UCF

From dvdsn aa

SS

aa )()ˆ()(

MHJEHEHE

( ) ( ') , ( ) 0 a au J r a r r M r

( ') [ ( ) ( ) ( ) ( )] ( )a au

S

ds a E r E r H r E r H r n

2

( ) [ ( ') ( ') ( ') ( ')] ( ) '

[ ( ')] ( ') ' [ ( ')] ( ')] '

' '[ ( ')] [ ( , ' )] ' [ ( ')] [ ( , ' )] '

[ ( ')] ' [ , ' ]

a au

S

a a

S S

u uS s

uS

ds

ds ds

k G ds G dsj j

G d

a E r E r H r E r H r n

n H r E r n E r H r

n H r (r r )a n H r (r r )a

n E r (r r )a

's

Note: ' [ , ' ] ' , ' ' , 'u u uG G G (r r )a (r r ) a a (r r )

Kirchhoff-Huygens Formula (5)

UCF

SinceR

eGGRGjkR

4,,

r)(r')r'(r)( || r'r R

RRR

dRdGG R

,

RR

RGRdRdGG

R' ,''

Likewise ' ' ( ) ( )u uG R G R a a

[ ( ')] ' [ , ' ] [ ( ')] [ , ' ] { [ ( ')]} {[ ( ')] }

u u

u

u

G GG

G

n E r (r r )a n E r a (r r )a n E r

a n E r {[ ( ')] }u G a n E r

3

1

3

1

ˆ[ ( ')] [ ] [ ( ')] ( ) {[ ( ')] }

ˆ ({[ ( ')] } ) ([ ( ')] )

u ii i

i ui i

G R G RG R xu x u

x G R G Ru x

( ) ( )n H r ( )a n H r n H r

n H r ( ) a n H r ( )

z

y

x

x

xx

a

aa

3

2

1

ˆ

ˆˆ

Kirchhoff-Huygens Formula (6)

UCF

2

( ) [ ( ') ] [( , ' ) ] '

1 [ ( ') ] , ' ' [ ( ') ] '

u uS

u us S

G d sj

G d s G d sj

a E r a n H r (r r )

a n H r (r r ) a n E r

1( ) { [ ( ')] , ' ' [ ( ')] , ' '}

[ ( ')] 'S s

S

Z jk G ds G dsjk

Gds

E r n H r (r r ) n H r (r r )

n E r

')]'([

}'))]'(([1')]'([{)(

dsRG

dsRGjk

dsRGjkZ

S

sS

)(rEn

)(rHn)(rHnrE

Likewise, if we let (or from duality), we have( ) 0, ( ) ( ') a au J r M r a r r

')]'([

}'))]'(([1')]'([{)(

dsRG

dsRGjk

dsRGjkY

S

sS

)(rHn

)(rEn)(rEnrH

Kirchhoff-Huygens Formula (7)

UCFKirchhoff-Huygens Formula (8)

')]'([

}'))]'(([1')]'([{)(

dsRG

dsRGjk

dsRGjkZ

S

sS

)(rEn

)(rHn)(rHnrE

')]'([

}'))]'(([1')]'([{)(

dsRG

dsRGjk

dsRGjkY

S

sS

)(rHn

)(rEn)(rEnrH

')'(

]}'])'([1')'({

dsRG

dsRGjk

dsRGjkY

)(rJ

)(rM)(rMH

s

ss

')'(

]}'])'([1')'({

dsRG

dsRGjk

dsRGjkZ

)(rM

)(rJ)(rJE

s

ss

compare

with

EnM s HnJs

UCF

Surface Equivalence PrincipleCase I

aJ

aΜRegion I Region II

00 ,

Case I

s

n

00 ,

Impressed source J and M and source scatters are in free space.

Region I includes all sources and inhomogeneous objects.Region II is considered to be homogeneous.

The fields in region II can be uniquely determined by the following Kirchhoff-Huygens formulas:

(1) ')]'([

}'))]'(([1')]'([{0

00

dsRG

dsRGjk

dsRGjkZ

S

sS

)(rEn

)(rHn)(rHnE

(2) ')]'([

}'))]'(([1')]'([{0

00

dsRG

dsRGjk

dsRGjkY

S

sS

)(rHn

)(rEn)(rEnH

UCF Case IINow if we let

(4) EnMs

(3) HnJs

Zero fields

Region I

00 ,

nsJ

sM 0,0, HE

Region II

Case II- Solution in Region II is equivalent to solution in Region II

in case I

And designate Region I as E=0, H=0, the boundary condition of this Case II has:

(3) as same the0)-( sJHn II

(4) as same the 0)-( sMEn II

This means the solution of Region II in Case II is equivalent as the solution of Region II in Case I.

UCF Example 1 – Case I

00 ,

y

x

z0k

Consider a uniform plane wave propagating in free space. The Case I of this example is:

(5) 0zjkx AeE

(6) 00

zjky AeYH

Region I as z < 0, Region II as z > 0For Case I, at z = 0

| 0 AE zx

| 00 AYH zy

E

H

UCF Example 1 – Case IIIntroduce

(8) yxxz AE aaaM s

(7) 0 xyyz AYH aaaJ s

00 ,

x

z

zasJ

x

0z

sM

0,0,

HE

Region I Region II

Assume in Region II 0zjkII

x CeE

00

zjkIIy CeYH

From boundary condition:

(9) 00 sMaa )-|E( zII

xxz

(10) 00 sJaa )-|H( zII

yyz

(12) --

(11) C

00 xx

yy

AYCY

A

aa

aa

ACThe fields in Region II of Case II are the same as (5)&(6)

UCF Case IIIBoth Region I and Region II are free space

Region I

00 , nsJ

sM

Region II

Case III- In the free space, the fields generated in region II is the same as the fields of region II in case I and

case II

00 ,

UCF Case III of Example 1

00 ,

sJ

x sM

Case II of Example I

00 ,

Region I Region II

In order to investigate Case III of Example 1, we look at example 2 and 3.

UCF Example 2

00 ,

Region I Region II

00 ,

yyM aMs

z

Without loss of generality and for simplicity, this source will generate a plane wave propagating in +z direction in Region II and a plane wave propagating in –z direction in Region I

For region II

(14)

(13) 0

0

0 yzjkII

xzjkII

GeY

Ge

aH

aE

For region I

(16) '

(15) '0

0

0 yzjkI

xzjkI

eGY

eG

aH

aE

At z =0

(18) 0)(

(17) )(

III

z

IIIz

-

-

HHa

MEEa s

(20)2

(19) 2

M

G' -M

G yy

For region II

2

2)21(

0

0

0 yzjkyII

xzjkyII

eM

Y

e-M

aH

aE

For region I

2

2)22(

0

0

0 yzjkyI

xzjkyI

eM

Y

eM

aH

aE

UCF Example 3

xxJ aJs

Region I Region II Assume the general solution the same as (13)-(16)

At z =0

(24) )(

(23) 0)(

sIII

z

IIIz

-

-

JHHa

EEa

J

ZYJ

GG xx

22' 0

0

For region II

2

2)25(

0

00

yzjkxII

xzjkxII

eJ

eJZ

aH

aE

For region I

2

2)26(

0

00

yzjkxI

xzjkxI

eJ

eJZ

aH

aE

UCF Example 4xxJ aJs yyM aM s If both are at z =0,

the fields will be superposition of Example 2 & 3

For region II

)(2

)22

(

)(21)

22(

)27(00

00

00

0

00

yzjk

xyyzjkxyII

xzjk

xyxzjkxyII

eJZMYeJMY

eJZMeJZM

aaH

aaE

For region I

)(2

)22

(

)(21)

22(

)28(00

00

00

0

00

yzjk

xyyzjkxyI

xzjk

xyxzjkxyI

eJZMYeJMY

eJZMeJZM

aaH

aaE

UCF Case III of Example 1xAY aJs 0 yAaM s In this case This is AM y AYJ x 0

in example 4. From (27) and (28), we have

For region II

)(2

)(21

)29(00

00

0000

00

yzjk

yzjkII

xzjk

xzjkII

AeYeAYZAY

AeeAYZA

aaH

aaE

For region I

0

0)30(

I

I

H

E

(29) Is the same as (5) & (6). This shows the Case III of Region II is the same as Case I’s & II’s Region II

(30) Is not the same as (5) & (6).

UCF Case IV

PEC

nsM

Region II

Case IV - Replace Region I by PEC only on the surface, Region II of Case IV is equivalent

to Region II of Cases I, II and III.

sM

UCF Case V

PMC

nsJ

Region II

Case V - Replace Region I by PMC, only on the surface, Region II of Case V is equivalent (or the

same as )to Region II of Cases I, II, III and IV.

sJ

UCF Example 5 (1)

00 ,

xPEC

Region II

za

yAaMs

Region I

00 ,

x

Region II

za

yAaMs 22

Region I

x

00 , Equivalent

This has been solved in example 2. Substitute My =-2A into (21)&(22)

yzjkII

xzjkII AeYAe aHaE 00

0

the same as (5) & (6).

Case IV of Example 1

UCF Example 5 (2)

xPEC yAaMs xyAaMs

xZA aJ s0

xsM2

It is noticed from example 2&3 that if the solution of Example 2&3 are the same for Region II. But there is a sign flip in Region I.

Js is the induced current on the boundary. It’s contribution is equivalent of another Ms for Region II.

xy JZM 0

x

UCF Case VI - IXRegion II

Case VI

J

Μ Region I

s

n

00 , 'n

'sJ'sM

00 , 0,0, HE Region II

J

Μ Region Is

n

00 , 'n

'sJ'sM

Case VII

J

Μ Region I

s00,

'n

'sM

PEC

J

Μ Region I

s00,

'n

'sJPMCCase VIII

Case IXss

ss

MEnMJHnJ

nn

''''

'

UCF Example 6 (1)

00 ,

Region I Region I’

00 ,

yyM aMs

z

Sources:A uniform plane wave propagating in +z directionMS

eEYH

eEE

yz-jkinc

xz-jkinc

a

a0

0

00

0)31(

For region I

2

2)32(

00

00

000

0

yz-jk

yzjkyI

xz-jk

xzjkyI

eEYeM

Y

eEeM

aaH

aaE

For region I’

2

2)33(

00

00

000'

0'

yz-jk

yzjkyI

xz-jk

xzjkyI

eEYeM

Y

eEeM

aaH

aaE

E

H k0

UCF Example 6 (2)

At we have4

z

)2

()4

(

)2

()4

()34(

00'

0'

yyI

xyI

EM

jY

EM

j

aH

aE

Now we divide the whole space at the equivalent source are: 4

z

(35) )

2()

4()('

)2

()4

()('

0

00

yyI

zs

xyI

zs

EM

j

EM

jY

aEaM

aHaJ

zan '

UCF Example 6 - Case VI (1)

00 , Region I Region I’

00 ,

yyM aMs

z

'sJ

4

z0z

'sM

0,0

Region II

The incident fields are still present

General solutions for total fields in Region I and I’ (including UPW) are:

)(

)()36(

00

00

000

0

yz-jk

yzjkI

xz-jk

xzjkI

eEYPeYz

eEPez

aaH

aaE

)()(

)()()37(

000

000

000'

0'

yzjk

yzjkzjkI

xzjk

xzjkzjkI

eEYCeBeYz

eECeBez

aaH

aaE

Must use total fields since Region II is not assumed to be free space here so thatboundary conditionsrequire total fields.

UCF Example 6 - Case VI (2)At z = 0

0)(

)('

'

II

z

sII

z

HHa

MEEa)38(

0

PCB

MPCB y

4

zAt )39( ')0(

')0('

'

sI

z

sI

z

JHa

MEa

2

2

00

00

EM

CEB

EM

CEB

y

y

UCF

2

02

y

y

MP

C

MB

The same as original problem’s Region I and I’

2

2

00

00

000

0

yz-jk

yzjkyI

xz-jk

xzjkyI

eEYeM

Y

eEeM

aaH

aaE

2

2

00

00

000'

0'

yz-jk

yzjkyI

xz-jk

xzjkyI

eEYeM

Y

eEeM

aaH

aaE

Example 6 - Case VI (3)

UCF Example 6 - Case VII (1)

00 , Region I Region I’

00 , sM

z

'sJ

4

z0z

'sM

Region II

Incident wave

)(

)(0

0

0 yzjksI

xzjksI

AeYz

Aez

aH

aE

)()(

)()(00

00

0'

'

yzjkzjksI

xzjkzjksI

CeBeYz

CeBez

aH

aE

)(

)(0

0

0 yzjksII

xzjksII

DeYz

Dez

aH

aE

00 ,

General solution of Region I and I’ from (excluding UPW) are:sss JMM ',' ,

Since the whole space is free space.Superposition will work.

UCF

At z = 0 0)(

)('

'

sIsI

z

ssIsI

z

-

-

HHa

MEEa

0

ACB

MACB y

4

zAt

')(

')('

'

ssIIsI

z

ssIIsI

z

-

-

JHHa

MEEa

0

)2

( 0

C

EM

DB y

0,0,2

,2

EDCM

BM

A yy

2

)(

2

)(

0

0

0 yzjkysI

xzjkysI

eM

Yz

eM

z

aH

aE

2

)(

2

)(

0

0

0'

'

yzjkysI

xzjkysI

eM

Yz

eM

z

aH

aE

)(

)(0

0

00

0

yzjksII

xzjksII

eEYz

eEz

aH

aE

The same as (32)&(33)

2

2

00

00

000

0

yz-jk

yzjkyincsII

xz-jk

xzjkyincsII

eEYeM

Y

eEeM

aaHHH

aaEEE

0)(

0)(

z

zII

II

H

E

Not the same as (33)

Example 6 – Case VII (2)

Adding UPW

2

2

00

00

000'

0'

yz-jk

yzjkyI

xz-jk

xzjkyI

eEYeM

Y

eEeM

aaH

aaE

UCF Two More ExamplesIn the above cases, Region I includes all sources and inhomogeneous objects andRegion II is free space. It is noted that Region I and II can be either closed oropen regions. Source can exist in both regions.

Region I

Region II is homogeneous with

different

1J 1M

n'n

Region II

2J 2M

incEincH Region II

Region I

'n

n1J 1M

incEincH

11 , 11 ,

22,

22 ,

22,

UCF Equivalence for Region II

Region I

nRegion II

2J 2M

00

HE

sJ

sM

Region II

Region I

sM

sJ

n 0,0, HE

Region I

nRegion II

2J 2M

sJ

sM

22 , Region IIRegion I

sM

sJ

n

22 ,

Case II

Case III

22 ,

22 ,

22,

22,

UCFEquivalence for Region II

Region II

Region I

PMC

sJ

n

nRegion II

2J 2M

sM

Region I, PEC

Region II

Region I

PEC sMn

Region I

PMC

nRegion II

2J 2M

sJ

22 ,

Case IV

Case V

Only Js

22 ,

22 ,

22,

Only Ms

UCF Equivalence for Region I

Region IIRegion I

J M

incEincH

Region I

1J 1M

'nincEincH

0,0, HE

Region II

'sJ 'sM

ss

ss

MEn'MJHnJ

'

''

Region IIRegion I

J M

incEincH

0,0, HE

Region I

1J 1MincEincH

'sJ 'sMA

B

Only when object A and B are not present, we can use free space

green’s function

A

B

'sJ

'sM

'sJ

'sM

'n

'n

'n

Region II

Case VII

Case VI

11 ,

11 ,

11 ,

11,

11 ,

11 ,

UCF Equivalence for Region I

Region IIRegion I

J M

incEincH

Region I

1J 1MincEincH'sM

Region II

PEC

Region I

J M

incEincH

Region I

1J 1MincEincH

'sJA

B

'sM

'sJ

PEC 'n

'n

'n

'n

Case IX

Case VIII

11 ,

11 ,

11 ,

11 ,

PMC

UCF Some Application Examples (1)

Region iRegion 0,

Zero fields, µ, ϵ

incEincH

n

HnJ s

sJ

PEC

incEincH

Region iRegion 0,

Zero fields, µ, ϵ

incEincH

n

EnM s

sM

PMC

incEincH

Region i

Region i

Region 0,

Region 0,

Region i equivalent

Region i equivalent

Only Js on the boundary. Fields from Js are scattered fields

Only Ms on the boundary. Fields from Ms are scattered fields

,

, ,

,

UCF Some Application Examples (2)

Region aRegion d,

Zero fields

incEincH

nsJ

incEincH

sM

Region a

dielectric

Region d,

Region a equivalent

Region d equivalentsJ

incEincH

sMZero fields

Fields from Js and Ms are scattered fields

'n

, ,

,dd ,

dd , dd ,