transfer matrix method in solving em problem

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Transfer Matrix Method In Solving EM Problem. Produced by. Yaoxuan Li , Weijia Wang , Shaojie Ma. Presented by Y.X.Li. 4. Theoretical Analyzing. Introducing Transfer Matrix in Solving Laplace Equation. 1. 2. General Properties for TMM in Multi-layer Shell. 3. - PowerPoint PPT Presentation

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Page 1: Transfer Matrix Method      In Solving EM Problem
Page 2: Transfer Matrix Method      In Solving EM Problem

Introducing Transfer Matrix in Solving Laplace Equation1

General Properties for TMM in Multi-layer Shell

2

General use in EM Wave Propagating in Multi-layer3

Page 3: Transfer Matrix Method      In Solving EM Problem

Introducing Transfer Matrix in Solving Laplace Equation1

General Properties for TMM in Multi-layer Shell

2

General use in EM Wave Propagating in Multi-layer3

Page 4: Transfer Matrix Method      In Solving EM Problem

Consider a series of co-central spherical shells with εn, at the nth shell, and the radius between the nth

and n+1th level is Rn,n+1.

Page 5: Transfer Matrix Method      In Solving EM Problem

We see a simple example first.

We apply a uniform field E=E0ex , and then solve the Laplace equation in the spherical coordinate, we got solutions for the 1st order inducing field

and boundary conditions , at r = Rn-1,n

2( ) cosn

n n

BA r

r

1n n 11

n nn nr r

Page 6: Transfer Matrix Method      In Solving EM Problem

Then we would easily manifest An and Bn in terms of An-1 and Bn-1 as

1 1

11

1,

2 2 2

3 3

n n

n n nn n

n n

BA A

R

1 1

1 1, 1

1 1 2

3 3

n n

n nn n n n nB A R B

Page 7: Transfer Matrix Method      In Solving EM Problem

and further in matrix form

where

111 12

21 22 1

n n

n n

A AQ Q

Q QB B

1

22

1 2

3

n

nQ

1

21 1,

1

3

n

nn nQ R

1

121,

2 21

3

n

n

n n

QR

1

11

2

3

n

nQ

Page 8: Transfer Matrix Method      In Solving EM Problem

The matrix is called the transfer matrix for the

n-1,n th level. If at the 1st level there is A1 and B1=0( to ensure converge) and at the infinite space there is An= -E0 and Bn, multiply the transfer matrix again and again we will get

And surly we got the solution of A1 and Bn, then whichever Ak

and Bk you want could be solved by using transfer matrix.

11 121,

21 22n n

Q QQ

Q Q

0 1, 1 1, 2 2,1...

0n n n nn

E AQ Q Q

B

Page 9: Transfer Matrix Method      In Solving EM Problem

Introducing Transfer Matrix in Solving Laplace Equation1

General Properties for TMM in Multi-layer Shell

2

General use in EM Wave Propagating in Multi-layer3

Page 10: Transfer Matrix Method      In Solving EM Problem

We now start some general solution for general conditions, solutions to be

we apply the same B.C and trick in calculation

1( ) (cos )

ll l n

n n lll

BA r P

r

, 1 , 11 1, 1 , 1 1

1 1 !, 1 1 , 1 12 2

, 1 , 1

1 1

1 1( 1) ( 1)

l ln n n nl ll l

n n n nn n

l ll ln n

n n n n n n n nl ln n n n

R RR RA A

B BlR l lR lR R

Page 11: Transfer Matrix Method      In Solving EM Problem

the TMM notation will be

and elements

when l=1, the results would automatically turn to the same results in the previous story.

1, 1

1

l ln nl

n nl ln n

A AQ

B B

122

( 1)

2 1

n

n

l l

Ql

2 11

21 1,

(1 )

2 1

n

lnn n

l

Q Rl

112 2 1

1,

( 1)(1 )1

2 1

n

nln n

l

Ql R

1

11

( 1)

2 1

n

n

l l

Ql

Page 12: Transfer Matrix Method      In Solving EM Problem

Furthermore, when adding up free boundary charge, the boundary condition will turn to be

we will soon get a solution no more complex than before

where

But the additional term, called charge term, is not that neat.

1, 1 , 1

1, 1 2

, 1 , 1

1

/

2( 1)

2( 1)

l ln n n n

nln n l l

n n n n

n

R

lC

R

l

1, 1 , 1

1

l ln nl l

n n n nl ln n

A AQ C

B B

1n n 11 , 1

n nn n n nr r

0 1, 1 1, 2 2,1 , 1 , 1 , 1... ...

0

ll

n n n n n n k k k kln

E AQ Q Q Q Q C

B

Page 13: Transfer Matrix Method      In Solving EM Problem

Though tough, but physicsFor a metal layer, the potential is constant, therefore only , otherwise is 0. If in the boundary for n,n+1th layer there is a l order charge ,we would directly got

It means, the surface charge density is just like other external conditions such as E field, would only induce the same order term, as a uniform E inducing only a term.

1, 1 , 1

1 1, 1 2

1 , 1 , 1

1

/

2( 1)

2( 1)

l ln n n n

ln nl

n nl l ln n n n n

n

R

A lC

B R

l

, 1ln n

0lnA

1(cos )P

Page 14: Transfer Matrix Method      In Solving EM Problem

Introducing Transfer Matrix in Solving Laplace Equation1

General Properties for TMM in Multi-layer Shell

2

General use in EM Wave Propagating in Multi-layer3

Page 15: Transfer Matrix Method      In Solving EM Problem

Consider a multi-layer withεn in the nth layer, and position of the boundary between nth and n+1th is dn,n+1 along z direction. Assume the wave propagates along z direction, perpendicular to the layer, with the expression of field, where

Solve , we got

As we have assumed there is not any surface current, so the boundary conditions are 1xn xnH H 1yn ynE E

n nik z ik zyn n nE A e B e

0 ( ,0, )x zH H H0 (0, ,0)yE E

n nik z ik zn n

xnn

A e B eH

Z

k H E 55555555555555555555555555 55

k E H 55555555555555555555555555 55

Page 16: Transfer Matrix Method      In Solving EM Problem

Then got the solution

Simply we can change the expression into matrix

1

1

1 1

1

1 1 1

1 10 01

20 01 1

n n

n n

n nik d ik d

n n n n

ik d ik dn n n n

n n

Z Z

A Z Z Ae e

B Z Z Be eZ Z

1 1 11

1[(1 ) (1 ) ]2

n n nik d ik d ik dn nn n n

n n

Z ZB e A e B e

Z Z

1 1 11

1[(1 ) (1 ) ]2

n n nik d ik d ik dn nn n n

n n

Z ZA e A e B e

Z Z

Page 17: Transfer Matrix Method      In Solving EM Problem

and in it, we define

If there are n layer (noticing that outside the layers are air, so the 0th and n+1th layer are absolutely air terms), the final solution can be written as

1 0 00 0, , 1 1 1, 2 2 1,0 0 , 1

1 0 0

...nn n n n n n n n n n

n

A A AP M PM P M P M P QB B B

1 1

, 11 1

1 11

21 1

n n

n nn n

n n

n n

Z Z

Z ZM

Z Z

Z Z

0

0

n

n

ik d

n ik d

eP

e

Page 18: Transfer Matrix Method      In Solving EM Problem

Further if we define the starting terms as ,

the solution could be simplified to the transfer matrix

By using this method we could quickly get the answer of t and r

And from this we could use transfer method to get the propagating properties in any layers.

1

1 0n

n

A t

B

0

0

1A

B r

1

0

tQr

12 2111

22

Q Qt Q

Q 21

22

Qr

Q

Page 19: Transfer Matrix Method      In Solving EM Problem

Interestingly, from the result above, one could think that, if Q21= 0, the reflective terms would be 0, and further if Q11= 1, meaning no absorption, t=1 , the transmittance behavior would be perfect.

Specifically, we could input some data asε1=1000, ε2= -2000, d1=d2= 2mm, then we get ω= 2π*0.850 GHz, there is a perfect transmission. From a COMSOL simulation we can see the S21 is near to 1.

12 2111

22

Q Qt Q

Q 21

22

Qr

Q

Page 20: Transfer Matrix Method      In Solving EM Problem