PHY 712 Electrodynamics
9-9:50 AM Olin 103
Plan for Lecture 16:
Read Chapter 7
1. Plane polarized electromagnetic waves
2. Reflectance and transmittance of electromagnetic
waves – extension to anisotropy and complexity
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PHY 712 Spring 2017 -- Lecture 16
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PHY 712 Spring 2017 -- Lecture 16
2
For linear isotropic media and no sources: D   E; B   H
Coulomb's law:
E  0
E
Ampere-Maxwell's law:   B  
0
t
B
Faraday's law:
E 
0
t
No magnetic monopoles:   B  0
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PHY 712 Spring 2017 -- Lecture 16
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Analysis of Maxwell’s equations without sources -- continued:
Coulomb' s law :
E  0
Ampere - Maxwell' s law :   B  
E
0
t
B
0
t
No magnetic monopoles :   B  0
E 
   E 

2
     B  


B



t 
t

Faraday's law :
E 
 2B
  B   2  0
t
2
B 
   B 

2
   E 


E


t 
t

2

E
  2 E   2  0
t
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Analysis of Maxwell’s equations without sources -- continued:
Both E and B fields are solutions to a wave equation:
1
 B 2
v
1
2
 E 2
v
2
B
0
2
t
2
E
0
2
t
2
 0 0 c
where v  c
 2

n
2
2
2
Plane wave solutions to wave equation :

Br, t    B 0 e
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ik r it


Er, t    E 0 e
PHY 712 Spring 2017 -- Lecture 16
ik r it
5

Analysis of Maxwell’s equations without sources -- continued:
Plane wave solutions to wave equation :

Br, t    B 0 e
ik r it
    n 
k   

v  c 
2
2

2

Er, t    E 0 e
ik r it


where n 
 0 0
Note: , , n, k can all be complex; for the moment we will
assume that they are all real (no dissipation).
Note that E 0 and B 0 are not independent;
B
from Faraday's law :   E 
0
t
k  E 0 nkˆ  E 0
 B0 


c
also note : kˆ  E 0  0 and kˆ  B 0  0
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Analysis of Maxwell’s equations without sources -- continued:
Summary of plane electromagnetic waves :
 nkˆ  E 0 ik r it 

Br, t   
e

c


    n 
k   

v  c 
2

Er, t    E 0 e ik r it


where n 
and kˆ  E 0  0
 0 0
2
2
Poynting vector and energy density:
S
u
1 
2
ˆ

k
E0 kˆ
2 c
2 
n E0
avg
avg
2
1
2
  E0
2
E0
B0
k
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Reflection and refraction of plane electromagnetic waves at a
plane interface between dielectrics (assumed to be lossless)
’ ’
k’
q

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ki
kR
i
R
PHY 712 Spring 2017 -- Lecture 16
8
Reflection and refraction -- continued
In medium  '  ':

E' r, t    E'0 e
’ ’
q
k’
  ki i R kR

i c n 'kˆ 'r  ct


n' ˆ
B' r, t   k 'E' r, t    '  'kˆ 'Er, t 
c
In medium  :

Ei r, t    E 0i e

i c nkˆ i r  ct


nˆ
B i r, t   k i  Ei r, t    kˆ i  Ei r, t 
c
i c nkˆ R r  ct 
E R r, t    E 0 R e

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
nˆ
B R r, t   k R  E R r, t    kˆ R  E R r, t 
c
PHY 712 Spring 2017 -- Lecture 16
9
Reflection and refraction -- continued
’ ’
ẑ
q
k’
  ki i R kR
x̂
Snell' s law - - matching phase factors at boundary plane :
r  xxˆ  yyˆ  0zˆ
kˆ 'r  x sin q
kˆ i  r  x sin i  kˆ R  r  i  R
n' kˆ 'r  nkˆ  r  n' x sin q  nx sin i
i
Snell' s law : n' sin q  n sin i
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Reflection and refraction -- continued
’ ’
ẑ
q
k’
  ki i R kR
x̂
Continuity equations at boundary with no sources :
D  0
B  0
D
B
H 
0
E 
0
t
t
Matching field amplitudes at boundary plane :
D  zˆ , B  zˆ continuous
H  zˆ , E  zˆ continuous
PHY 712 Spring 2017 -- Lecture 16
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11
Reflection and refraction -- continued
Matching field amplitudes at boundary plane :
D  zˆ continuous :
ẑ
’ ’
q
k’
  ki i R kR
x̂
 E 0i  E 0 R   zˆ   ' E'0 zˆ
B  zˆ continuous :
n kˆ i  E 0i  kˆ R  E 0 R  zˆ 
n' kˆ 'E' zˆ


0i
E  zˆ continuous :
E0i  E0 R  zˆ  E'0 zˆ
H  zˆ continuous :


n ˆ
n' ˆ
ˆ
k i  E 0i  k R  E 0 R  zˆ  k 'E'0i zˆ

'
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Reflection and refraction -- continued
’ ’
ẑ
s-polarization – E field “polarized”
perpendicular to plane of incidence
q
E  zˆ continuous :
E0i  E0 R  zˆ  E'0 zˆ
k’
  ki i R kR
x̂
H  zˆ continuous :

E0 R
E0 i

n ˆ
n' ˆ
ˆ
k i  E 0i  k R  E 0 R  zˆ  k 'E'0i zˆ

'

n cos i  n' cos q
E '0
2n cos i
'




E
0
i
n cos i  n' cos q
n cos i  n' cos q
'
'
Note that : n' cos q  n'2 n 2 sin 2 i
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Reflection and refraction -- continued
p-polarization – E field “polarized”
ẑ
parallel to plane of incidence
’ ’
q
k’
  ki i R kR
x̂
D  zˆ continuous :
 E 0i  E 0 R   zˆ   ' E'0 zˆ
H  zˆ continuous :


n ˆ
n' ˆ
ˆ
ˆ
k i  E 0i  k R  E 0 R  z  k 'E'0i zˆ

'

n' cos i  n cos q
E0 R  '


E0 i
n' cos i  n cos q
'
E '0
2n cos i

E0i  n' cos i  n cos q
'
Note that : n' cos q  n'2 n 2 sin 2 i
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Reflection and refraction -- continued
’ ’
ẑ
q
k’
x̂
  ki i R kR
Reflectance, transmittance :
S R  zˆ E0 R
R

S i  zˆ
E0 i
Note that
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2
2
E '0 n'  cos q
S'zˆ
T

S i  zˆ E0i n  ' cos i
R  T 1
PHY 712 Spring 2017 -- Lecture 16
15
For s-polarization

n cos i  n' cos q
E0 R
'

E0i n cos i   n' cos q
'
E '0
2n cos i

E0i n cos i   n' cos q
'
Note that : n' cos q  n'2 n 2 sin 2 i
For p-polarization

n' cos i  n cos q
E0 R  '


E0 i
n' cos i  n cos q
'
E '0
2n cos i

E0i  n' cos i  n cos q
'
Note that : n' cos q  n'2 n 2 sin 2 i
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Special case: normal incidence (i=0, q=0)

n'n
E0 R  '
E '0
2n




E0 i
E
0
i
n' n
n' n
'
'
Reflectance, transmittance :
2
E0 R
R
E0 i

n'n
'


n' n
'
2
2
2
E '0 n ' 
2n
n' 
T

E0i n  '  n' n n  '
'
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Multilayer dielectrics
n1
ki
kR
(Problem #7.2)
n3
n2
ka
kb
kt
d
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Extension of analysis to anisotropic media --
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Consider the problem of determining the reflectance from
an anisotropic medium with isotropic permeability 0 and
anisotropic permittivity 0 k where:
 k xx

κ  0
 0

0
k yy
0
0 

0 
k zz 
By assumption, the wave vector in the medium is
confined to the x-y plane and will be denoted by
kt 

c
(nx xˆ  n y yˆ ), where nx and n y are to be determined.
The electric field inside the medium is given by:
E  ( Ex xˆ  E y yˆ  Ez zˆ )e
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
i ( nx x  n y y ) it
c
PHY 712 Spring 2017 -- Lecture 16
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20
Inside the anisotropic medium, Maxwell’s equations are:
H  0
 κ E  0
  E  i0 H  0
  H  iò0κ  E  0
After some algebra, the equation for E is:
 k xx  n y2
  Ex 
nx n y
0

 
2
k yy  nx
0
 nx n y
  E y   0.
2
2 

 0
0
k

(
n

n
)
E
zz
x
y  z 

From E, H can be determined from
H
E (n xˆ  n yˆ )  ( E n

c
1
z
y
x
y
x
 Ex n y )zˆ  e

i ( nx x  n y y ) it
c
0
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.
The fields for the incident and reflected waves are the
same as for the isotropic case.
ki 
kR 

c

c
(sin ixˆ  cos iyˆ ),
(sin ixˆ  cos iyˆ ).
Note that, consistent with Snell’s law: nx  sin i
Continuity conditions at the y=0 plane must be applied for
the following fields:
H ( x, 0, z , t ), Ex ( x, 0, z , t ), E z ( x, 0, z , t ), and Dy ( x, 0, z , t ).
There will be two different solutions, depending of the
polarization of the incident field.
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Solution for s-polarization
Ex  E y  0
E  Ez zˆ e
 n y2  k zz  nx2

i ( nx x  n y y ) it
c
H
E (n xˆ  n yˆ ) e

c
1
z
y

i ( nx x  n y y ) it
c
x
0
Ez must be determined from the continuity conditions:
E0  E0  Ez
( E0  E0 ) cos i  E z n y
( E0  E0 ) sin i  E z nx
E0 cos i  n y

.
E0 cos i  n y
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Solution for p-polarization
Ez  0
k xx
 n 
(k yy  nx2 ).
k yy
2
y

k xx nx
E  Ex  xˆ 

k yy n y

 i  ( nx x  ny y ) it
yˆ  e c
.


Ex k xx i c ( nx x  ny y ) it
H
zˆ e
.
0 c n y
Ex must be determined from the continuity conditions:

0
( E0  E ) cos i  Ex

0
( E0  E ) 
k xx
ny
Ex

0
( E0  E ) sin i 
k xx nx
ny
Ex .
E0 k xx cos i  n y

.
E0 k xx cos i  n y
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Extension of analysis to complex dielectric functions
For simplicity assume that    0
Suppose the dielectric function is complex :

2
 nR  inI     i
0
   R  i I
1/ 2
1/ 2
     
     


nR  
nI  




2
2




i c nkˆ r  ct 
i c n R kˆ r  ct   c n I kˆ r
Er, t    E 0 e
  E0e
e
2

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2
 
PHY 712 Spring 2017 -- Lecture 16
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
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