EM scattering 3 new - Sharifee.sharif.edu/~emscattering_ms/Lecture 3B.pdf · Scattering from layered media 23 Scattering matrix of the interface Let us consider a generalized version

Scattering from layered media 23

Scattering matrix of the interface

� Let us consider a generalized version of the problem: when

waves are incident on the interface from both media

0 0,��1 0,��

� �0,exp zA jk z� � �0,exp zB jk z

� �1,exp zC jk z�� 1,exp zD jk z

� TE scattering:

� � � �0, 0,0 exp expy z zz E A jk z B jk z� � � � ��

� � � �1, 1,0 exp expy z zz E C jk z D jk z� � � � ��

0z �



� Matching the solutions results in the scattering matrix of the

interface: matrix relating the amplitudes of the reflected waves

to those of the incident waves at the interface

TE TE

TE TE

R TB A

T RC D

��

1,

1, 0,

21 1 z

TE TE TEz z

kT R R

k k� ��

�

TES

1TE TET R� �

0 0,��1 0,��



0z �

TE TER R� � �



� For the TM case:

TM TM

TM TM

R TB A

T RC D

��

0 1,

1 0, 0 1,

21 1 z

TM TM TMz z

kT R R

k k� ��

��

� �

� � � �1, 1,0 exp expy z zz H A jk z B jk z� � � � ��

� � � �2, 2,0 exp expy z zz H C jk z D jk z� � � � ��

TMS

1TM TMT R� �

0 0,��1 0,��



0z �

TM TMR R� � �



� If the interface is not located at z = 0, but at z = z0 we can still

use the same result, but should use the correct amplitudes at

the interface

� ��

� ��

0, 0 0, 0

1, 0 1, 0

exp exp

exp exp

z z

z z

jk z B jk z AR T

T Rjk z C jk z D

� � � ��

0 0,��1 0,��



0z z�


Scattering by a dielectric slab

� Let us go one step further and consider the scattering by a

dielectric slab of finite thickness

rk

t i�k k

0 0,��

1 0,��

2 0,��

d

� We again distinguish the TE and

TM cases

� The tangential wave vector is

again the same in all layers

� Solution can be obtained by

matching the fields at the two

interface planesik

i� i�


TE Scattering by a dielectric slab

� Solution for the electric field inside each layer for reduced fields

0 0,��

1 0,��

2 0,��



� �2,exp zE jk z�

0z �

z d�



� Instead of matching fields use the interface scattering matrices

10 10

10 10

10

1TE TE

TE TE

B AR Rz

C DR R

� ��

0, 1,10

0, 1,

z zTE

z z

k kR

k k

��

�

� ��

� �21 211, 1,

21 212,

exp exp1

1exp 0

z zTE TE

TE TEz

jk d D jk d CR Rz d

R Rjk d E

� � � ��

1, 2,21

1, 2,

z zTE

z z

k kR

k k

��

�



� Resulting overall reflection and transmission coefficients

� ��

10 211,

10 211,

exp 2

1 exp 2TE TE z

TE TE z

R R jk dBR

A R R jk d

� ��

� �

� ��

10 211,

10 211,

1 1 exp 2

1 exp 2TE TE z

TE TE z

R R jk dET

A R R jk d

� � ��

� �

� Similar results for TM scattering, just replace

1 0, 0 1,10 10

1 0, 0 1,

z zTE TM

z z

k kR R

k k

��

��

� �2 1, 1 2,21 21

2 1, 1 2,

z zTE TM

z z

k kR R

k k

��

��

� �


Scattering by a dielectric slab in vacuum

� A more practical case: a dielectric slab in vacuum

ik

1 0d�� d

20, 1,10

20, 1,

cos sin

cos sin

z z i d iTE

z z i d i

k kR

k k

� �

� �

� � ��

� � �

�

�

1, 0,21 10

1, 0,

z zTE TE

z z

k kR R

k k

��

�

0�

0�

210

2

cos sin

cos sin

d i d iTM

d i d i

R� �

� �

� ��

� �

� �

� �

21 10TM TMR R� � rkik

i� i�



� Overall reflection coefficient for TE scattering:

� ��

� ��

101,

2101,

2 20, 1,

2 20 , 1, 0 , 1, 1,

1 exp 2

1 exp 2

2 cot

TE z

TE

TE z

z z

z z z z z

R jk dR

R jk d

k k

k k jk k k d

� ��

��

� �

� �� 0

1

cos 2 2 ( ) cos cot ( )d

TEd i i i i

Rj k d� � � � � �

��

� ��

�

2 20, 0 1, 0cos , sin , ( ) sinz i z d i i d ik k k k� � � � ��



� Overall reflection coefficient for TM scattering:

� ��

� �

101,

2101,

2 2 2 21 0, 0 1,

2 2 2 21 0, 0 1, 0 1 0 , 1, 1,

1 exp 2

1 exp 2

2 cot

TM z

TM

TM z

z z

z z z z z

R jk dR

R jk d

k k

k k j k k k d

� ��

��

� �

� �

� � � �

� �2 2 2

2 2 20

cos sin

cos sin 2 ( ) cos cot ( )d i i d

TMd i d i d i i i

Rj k d

� ��

� ��

� � ��

� � �

2 20, 0 1, 0cos , sin , ( ) sinz i z d i i d ik k k k� � � � ��



� Reflection coefficient of the TE and TM mode becomes zero if

� �1,1 exp 2 =0 zjk d� � �2

21,

0

, 1,2, sinz i d

nk d n n

k d

��

� ��

� ��

� Phenomenon caused by (almost) standing waves in slab due to

multiple reflections, similar to Fabry-Perrot, resonant tunneling

� Can occur when the slab is thicker than half propagation wave

length inside the slab

0d dk d k d ��



� In addition, we have the usual Brewster angle for TM scattering

where reflection becomes zero

� Normal incidence � same TE and TM values apart from

a minus sign (why?)

2 2 2 2cos sin 0 sin1

dd i i d i

d

� � ��

� ��

0i� �

� �� 0

1

1 2 cot

dTE TM

d d d

R Rj k d

��

� �

�

� � �

� Almost horizontal incidence / 2i� ��

1TER � � 1TMR � �


Reflection from a dielectric slab in vacuum

� Plots for a slab with a relative dielectric constant of 4

(deg)i�

2TER

2TMR

2dk d �

(deg)i�

2TER

2TMR

6.5dk d �


Reflection from a dielectric slab in vacuum

� Note also that if we keep the angle constant and change the

frequency then an oscillatory behavior is observed due to the

standing wave phenomenon

0k d

2TER

2TMR

0i� �

0k d

45 degi� �


Scattering from a general layered medium

� So far we considered a number of specific cases, now we turn

to the general case

1 0,��

�

0,N ��

0 0,��

1 0,N ��

� �0 0,exp zA jk z� � �0 0,exp zB jk z


� �,expN N zA jk z� � �,expN N zB jk z

� �1 1,expN N zA jk z� ��

xk



� Our aim is to find the overall reflection and, perhaps,

transmission coefficients

0

0

BR

A� 1

0

NAT

A��

� Like for a single slab, we would like to use the expressions for

interface scattering

� But, the scattering matrix is not the right quantity for analyzing

a stack of multiple layers

� Combining scattering matrices of different layers is not easy



� Let us again return to a single interface between 2 media

� ��

� ��

, ,

1, 1,

exp exp

exp exp

p z p p z p

p z p p z p

B jk z A jk zR T

T RC jk z D jk z� �

� � � ��

� Instead of relating

amplitudes of scattered

waves to those of incident

waves, we relate waves in

layer i+1 to those in layer i

S

0,p ��1 0,p ��

� �,exp p zA jk z� � �,exp p zB jk z

� �1,exp p zC jk z�� 1,exp p zD jk z�

pz z�



� Let us again return to a single interface between 2 media

� ��

� ��

1, ,21 12 11 22 22

11121, ,

exp exp11exp exp

p z p p z p

p z p p z p

C jk z A jk zS S S S S

SSD jk z B jk z

�

�

� � � ��

M

11

11

R

RR

��

M

� In terms of the reflection

coefficient

Transfer matrix

0,p ��1 0,p ��

� �,exp p zA jk z� � �,exp p zB jk z

� �1,exp p zC jk z�� 1,exp p zD jk z�

pz z�



� Note again, that the matrix relates the wave amplitudes

� Now consider the full problem

1�

�

N�

0�

1N��




� �1 1,expN N zA jk z� �� 1 1,expN N zB jk z� �

0z

1z

1Nz �

Nz



� We have

� ��

� ��

1, 1 ,1,

1, 1 ,

exp exp

exp exp

p z p p p z p pp p

p z p p p z p p

jk z A jk z A

jk z B jk z B

� � �

� �

� � � ��

M

� � � �1 1,1, ,

1

p pp pp z p p z p

p p

A Ak z k z

B B� �

��

� � � ��

� � � �Q M Q

� � � ��

exp 0

0 exp

j

j

��

��

� � �� Q

1,1,

1, 1,

11

1 1

p pp p

p p p p

R

R R

��

� �

� ��

M



� ��

� ��

1, 0, 0

1, 0, 0

1 0

1 0

exp exp

exp exp

N z N z

N z N z

N

N

jk z jk z

jk z jk z

A A

B B

�

�

�

�

� ��

� � � �� M

1�

�

N�

0�

1N ��




� �1 1,expN N zA jk z� �� 1 1,expN N zB jk z� �

0z

1z

1Nz �

Nz

1p p pd z z��

,

,

0

0

p z p

p z pp

jk d

jk d

e

e

��

Q

1, 3,2 2,1 1,02 1

N NN

�� M M Q M Q M Q M�



� ��

� ��

1, 0, 0

1, 0, 0

1 011 12

21 221 0

exp exp

exp exp

N z N z

N z N z

N

N

jk z jk z

jk z jk z

A AM M

M MB B

�

�

�

�

� ��

� � � ��

� We have found the overall transfer matrix of the structure

� The overall reflection and transmission coefficient:

� ��

0, 0

0, 0

1,

0, 0

0 211

220

1 12 2111

220

exp0

exp

exp

exp

z

z

N z N

z

N

N

jk z

jk z

jk z

jk z

B MB R

MA

A M MT M

MA�

�

�

�

�

�

� � � � �

� � �



� Example: single dielectric slab in vacuum

0z �

z d�

2,1 1,01� � �M M Q M

0�

d�

0�

101,0

10 10

11

1 1

R

R R

� ��

M

102,1

10 10

11

1 1

R

R R

� ��

M

1,1

1,

exp( ) 0

0 exp( )z

z

jk d

jk d

��

Q � ��

101,

2101,

1 exp 2

1 exp 2

z

z

R jk dR

R jk d

� ��



� This method involves the multiplication of a lot of matrices if

there are many layers

� There exist another technique which yields a simpler, recurrent

equation for the reflection coefficient

� Consider two adjacent layers, p and p+1

1p��

�

p� � �,expp p zA jk z� � �,expp p zB jk z

� �1 1,expp p zA jk z� �� 1 1,expp p zB jk z� �

pz

1pz �



� Imagine that somehow we know the ratio of the forward and

backward moving waves in the (p+1)-th layer at

1p��

�

p� � �,expp p zA jk z� � �,expp p zB jk z

� �1 1,expp p zA jk z� �� 1 1,expp p zB jk z� �

pz

1pz �

� ��

1 1, 1

1

1 1, 1

exp

exp

p p z p

p

p p z p

B jk zr

A jk z

� � ��

� � �

��

Reflection coefficient at 1pz �

� What can be said of this ratio in the p-th at ?

1pz �

pz



� Use the transfer matrix (or scattering matrix)

� ��

� ��

1, ,

1, ,

1, 1,1 11 12

1, 1,21 221

exp exp

exp exp

p z p p z p

p z p p z p

p p p pp p

p p p pp p

jk z jk z

jk z jk z

A AM M

M MB B

�

�

� ��

� ��

� ��

� � � ��

� ��

,

,

exp

exp

p p z p

p

p p z p

B jk zr

A jk z�

�

� ��

1, 1,11 1 21 1, 1

1, 1,22 1, 1 12 1

exp 2

exp 2

p p p pp p z p

p p p p pp z p p

M r M jk dr

M jk d M r

� ��

� ��

��

�

� One finds the recurrent relationship



� In terms of interface reflection parameters:

� ��

1,1 1, 1

1,1, 1 1

exp 2

exp 2

p pp p z p

p p pp z p p

r R jk dr

jk d R r

��

��

��

�

� Remember that for TE and TM scattering:

, 1,1,

, 1,

p z p zp pTE

p z p z

k kR

k k��

�

��

�1 , 1,1,

1 , 1,

p p z p p zp pTM

p p z p p z

k kR

k k� ��

� �

��

�

� �

� �

� But, what about the ‘initial condition’ of this recurrent relation?



� There is no reflection in the topmost layer!

� Our initial condition is

1 0Nr � �

1�

�

N�

0�

1N ��

0r

1r

Nr

� The recurrent relation can then

be calculated all the way down

to the overall reflection

coefficient

0R r�

��

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EM scattering 3 new - Sharifee.sharif.edu/~emscattering_ms/Lecture 3B.pdf · Scattering from layered media 23 Scattering matrix of the interface Let us consider a generalized version