chung-ang university field & wave electromagnetics 8-9 normal incidence at multiple dielectric...
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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.
Chung-Ang University Field & Wave Electromagnetics
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
Chung-Ang University Field & Wave Electromagnetics
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
Chung-Ang University Field & Wave Electromagnetics
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
Chung-Ang University Field & Wave Electromagnetics
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!
Chung-Ang University Field & Wave Electromagnetics
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
EH x z a cos +a sin )e
��������������
2 ( sin cos )( , ) t tj x zyt toE x z a E e
2 ( sin cos )
2
( , ) ( i ij x ztot x zi i
EH x z a cos +a sin )e
��������������? , ?ro toE E ����������������������������
For given incident wave,
Findz
x
z=0
Oθi
θr
iH��������������
iE��������������
•
Chung-Ang University Field & Wave Electromagnetics
( , ) ( , ) ( , )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��������������
•
Chung-Ang University Field & Wave Electromagnetics
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)
Chung-Ang University Field & Wave Electromagnetics
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
Chung-Ang University Field & Wave Electromagnetics
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)
Chung-Ang University Field & Wave Electromagnetics
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)
8-10.3 Parallel Polarization (3)
Chung-Ang University Field & Wave Electromagnetics
- (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
8-10.3 Parallel Polarization (4)
Chung-Ang University Field & Wave Electromagnetics
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
8-10.3 Parallel Polarization (5)
What about perpendicular polarization?
However this is the rare situation in nature!
Chung-Ang University Field & Wave Electromagnetics
0 ,iIf , 1
2iIf
1
2
B
i
2
2
8-10.3 Parallel Polarization (6)