Chung-Ang University Field & Wave Electromagnetics 8-9 Normal Incidence at Multiple Dielectric Interfaces Transmitted wave ErEr HrHr a nr Reflected wave

Chung-Ang University Field & Wave Electromagnetics

8-9 Normal Incidence at Multiple Dielectric Interfaces

2H

2na

2E

2H

2E

2na

3H

3E

3na

Transmittedwave

Er

Hr

anr

Reflectedwave

Hi

Ei

ani

Incidentwave

Medium 3(ε3 , μ3)

z=0 z=d

Medium 2(ε2 , μ2)

Medium 1(ε1 , μ1)

x

Antireflection coating for eye glasses.

Radome (for the protection of radars of ships)

1 1

1 1

1 0 0

1 0 01

( )

1( )

j z j zx i r

j z j zy i r

E = a E e E e

H = a E e E e

2 2

2 2

2 2 2

2 2 22

( )

1( )

j z j zx

j z j zy

E = a E e E e

H = a E e E e

3

3

3 3

3 33

1

j zx

j zy

E = a E e

H = a E e

0 2 2 3, , ,rE E E E

Unknowns :

Boundary conditions at z=0 , z=d

1 2

1 2

0, (0) (0)

(0) (0)

at z E E

H H

2 3

2 3

0, ( ) ( )

( ) ( )

at z E d E d

H d H d

The procedure is straightforward and purely algebraic.


1 1

1 1

11 1

1

( )( )

( )

i z i zx

i z i zy

E z e eZ z

H z e e

1 1

1 1

1

11

( )

( )

i z i zx i r x io

i z i zioy i r y

E (z)= E (z)+E (z)= a E e e

EH (z)= H (z)+H (z)= a e e

η: intrinsic impedance

8-9.1 Wave Impedance of the total field

Wave impedance

(includes the effect of medium 2)

1( )Z z In an unbounded medium , for +z direction wave

1( )Z z In an unbounded medium, for -z direction wave

z

z=0

Medium 1 Medium 2

O

1 1

1 1

1 2 11 1

1 2 1

( )( ) ( )

( )

j l j lx

j l j ly

E l e eZ l

H l e e

2 1 1 1

11 1 2 1

cos sin

cos sin

l j l

l j l

z l

1( )Z l

1 2 1 1) : ( ) no reflectioni Z l

2) perfect conductor, 0 ( ( ) 0, ( ) 0 )V I

ii E Hm m

1 1 ( ) tanZ l j l

which is the same as the input impedance of a transmission line of length l

that has a characteristic impedance and terminates in a short circuit.1


8-9.2 Impedance Transformation with Multiple Dielectrics

3 2 2 22 2

2 2 3 2

cos sin(0)

cos sin

d j dZ

d j d

We can transform to an arbitrary value 3 3( )Z z

of by choosing and

2 (0)Z 2 .d

when , There is no reflection.

2 1(0)Z 0 2 1 2 1

0 2 1 2 1

(0)

(0)r

i

E Z

E Z

2 (0)Z 0z z d

1 2 3 3( )Z z

l d

1 2 2' (0)Z

1 1

1 1

11 1

1

( )( )

( )

j l j lx

j l j ly

E l e eZ l

H l e e

2 1 1 1

11 1 2 1

cos sin

cos sin

l j l

l j l

3 2 2 22

2 2 3 2

cos sin( )

cos sin

d j dZ d

d j d

Ex. 8-12 p405


Incident

wave

x

z

z=0

Medium 1

(ε1,μ1)

Medium 2

(ε2,μ2)

ani

anr ant

A

A’ B

O’

O

Reflected

waveRefracted

wave

θt

θi

θr

8-10 Oblique Incidence at a plane Dielectric Boundary ' 'OA AO

'time timeOB AO

'cos( ) 'cos( )2 2r iOO OO i r

2 2

'sin t

p p

OOOB

u u

1

'cos( )2 i

p

OO

u

n : refractive index

Snell’s Law of refraction -The angle of reflection is equal to the angle of incident)

2 11 1 1 21 1 2

1 2 2 12 2 2 2

1

sin

sin

o

p o r rt

i p o r r o

u n

u n


8.10-1 Total Reflection

1 2when , 1

2

sin sint i

n

n

1

2

When sin 1 sin2 c

n

n

1

2

1n

n

(Total reflection)1 1

2

sinc

n

n

1 2

1

sin

1

2

sin sin 1t i

2cos 1 sint t 21

2

sin 1ij

Medium 1

ε1

Medium 2

ε2

xa

i c What happen in Medium 2 when ? z

x

z=0

Oθi

θt

Critical Angle

21 12

2 2

sin sin 1i ij x j z

e

sin cosnt x t z ta a a 2 2 ( sin cos )nt t tj a R j x ze e

��

21 12 2

2 2

sin 1 sini iz j x

e e

Evanescent wave along z

Propagation in x direction

tightly bound to the surface⇒ surface

wavez (no power transmission)If medium 2 is not thick enough,

there is transmitted wave even though TIR condition is met!


8-10.2 Perpendicular Polarization

1 ( sin cos )( , ) i ij x zyi ioE x z a E e

1 ( sin cos )

1

( , ) ( i ij x zioi x zi i

EH x z a cos +a sin )e

��

1 ( sin cos )( , ) r rj x zyr roE x z a E e

1 ( sin cos )

1

( , ) ( i ij x zror x zr r


��

2 ( sin cos )( , ) t tj x zyt toE x z a E e

2 ( sin cos )

2

( , ) ( i ij x ztot x zi i


��? , ?ro toE E ��

For given incident wave,

Findz

x

z=0

Oθi

θr

iH��

iE��

•


( , ) ( , ) ( , )iy ry tyE x o E x o E x o

( , ) ( , ) ( , )ix rx txH x o H x o H x o ��

We know that ,i r 1

2

sin,

sint

i

n

n

io ro toE E E

1 2

1( )cos costo

io ro i t

EE E

? , ?ro toE E ��

8-10.2 Perpendicular Polarization (2)

z

x

z=0

Oθi

θr

iH��

iE��

•


2 1

2 1

cos cos

cos cosro i t

io i t

E

E

Fresnel’s Equation

2

2 1

2 cos

cos costo i

io i t

E

E

1

Assume that we found the solutions, ,ro toE E��

8-10.2 Perpendicular Polarization (3)


8-10.3 Parallel Polarization

1 ( sin cos )( , ) ( i ij x zi x zio i iE x z E a cos a sin )e

��

1 ( sin cos )

1

( , ) i ij x zioi y

EH x z a e

��

1 ( sin cos )( , ) ( r rj x zr x zro r iE x z E a cos a sin )e

��

1 ( sin cos )

1

( , ) t tj x zror y

EH x z a e

��

2 ( sin cos )( , ) ( t tj x zt x zto t tE x z E a cos a sin )e

��

2 ( sin cos )

2

( , ) t tj x ztot y

EH x z a e

��

For given incident wave,

Find

? , ?ro toE E ��

x

z=0

Oθi

iH��

iE��

•

rH��

rE��

θt

x


Boundary conditions 1tan 2tan ,E E��

1tan 2tan ,H H��

Z=0

( ) cos cosio ro i to tE E E

1 2

1 1( )io ro toE E E

2 1

2 1

cos cos

cos cosro t i

io t i

E

E

2

2 1

2 cos

cos costo i

io t i

E

E

cos1 ( )

cost

i

8-10.3 Parallel Polarization (2)


2 1

2 1

cos cos

cos cost i

t i

0 2 1cos cos 0t iwhen

22 21

22

cos 1 sin 1 sint t i

n

n

(assume μ1= μ2)

2 22 1 1

2 2 1

1 sin 1 sini i

find such i

Snell’s law

(No reflection)



- (1)2 21

22 2 1 1

1 1 1sin sini i

22 1 (1)

2 2 2 2 21 2 1 2 2sin sini i

2 2 22 1 2 1 2( ) ( )sin i

1

2

sin 1/(1 )i



1 1 12 2

1 11

2

1sin tan tan

1i B

n

n

2 1

2 1

cos cos

cos cosi t

i t

1 2

2 2 12

1

2

1

sin

1

1 2 1 2

1

2

1( , , sin )

1

If

Brewster Angle


What about perpendicular polarization?

However this is the rare situation in nature!


0 ,iIf , 1

2iIf

1

2

B

i

2

2


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Chung-Ang University Field & Wave Electromagnetics 8-9 Normal Incidence at Multiple Dielectric Interfaces Transmitted wave ErEr HrHr a nr Reflected wave