Dipole radiation, antennas
The greatest intensity is
_______ to the line of
acceleration.
Index of refraction from polarization
2
2

J

E

P
2
P
 E   o o
 o
 o 2
2
t
t
t
Assumption of a linear medium
P  E
kvac 

c

2
vac
c
n   1 
v
 n2 2
A
kn 

c
vac


vac
n
(know this derivation of n from the wave
c

v   f 
equation)
n
  2 f
k
Lorentz model of spring-like oscillating dipoles to
model polarization
qe
i  k r t 
r r  r 
Eoe
me
2
o
qe
Eo
ro 
me o2   2  i
Po  Nqro if all identical
N
(Note: my notation
 N in text)
A V
Po
qe2 N
1
   

 o Eo  o me o2   2  i
Index of refraction n is now complex  n + i
qe2 N
2
1
 n  i   1      1  m   2   2  i
e o
o
Lorentz model of r spring-like oscillating dipoles
-6
2.5
x 10
magnitude
real
imaginary
2
displacement xo
1.5
1
0.5
0
-0.5
-1
-1.5
0
0.5
1
1.5
 (1016 rad/sec)
2
2.5
Lorentz model of spring-like oscillating dipoles
3.5
Phase of xo vs E
3
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
 (1016 rad/sec)
2
2.5
Effect of damping 
Meaning of complex quantities
 Im( z) 
  tan 

 Re( z) 
1
ro , Po ,  all have the same phase,
and move differently from E's phase
Amplitude and phase depend on .
A
Meaning of complex quantities
qe2 N
1
n  i  1      1 
me o o2   2  i
c
n has the same meaning as before (v and  )  n   1  
v
A 
k is complex! 
exp(ikz)  exp(i  n  i  z)
 exp(inkvac z)exp( z)
so  gives how fast the wave decays in space!
vac
n
k   n  i 

c

c
v   f 
k
n
  2 f
Complex index, dielectric constant, k
In the wave, real part n of
index determines:
And imaginary part  of index
determines:
A
Complex index, dielectric constant, k
P. Near a resonance frequency, materials with atomic
dipoles might have for example n = 2 and  = 3
When the wave has moved into the material by a
distance of one vacuum wavelength, by what factor
is the wave amplitude reduced?
A
a) exp(-2)
2 n 2

k

n
b) exp(-3)

vac
c
b) exp(-4)
c

v   f 
d) exp(-6)
n
k
P4. What is the phase change of the
wave after traveling this distance?
  2 f

  vac
n
What “spring=like“ resonances do electrons in
atoms have?
Every possible transition is a "resonant frequency"
o = energy
P2. Clear, colorless
glass has no
resonances in the
visible, but does
have them in the UV,
which is at a higher
.
Hence in clear glass,
index n _________
with as  increases
for visible light
a) increases
b) decreases
P3. If we increase the density of a gas (N), index n
a) increases for all 
b) decreases for all 
c) increases with density below resonance, decreases above
d) decreases with density below resonance, increases above
Hint: what would these
curves look like in the limit
of N  0?
Common optical glass indices vs wavelength
P4. In a glass that
absorbs green light
the index , real index
n is probably
greatest for ______
light.
a) red
b) yellow
c) green
d) blue
e) violet
P5. In a glass that absorbs green light
the index , imaginary index  is greatest for ______
(same choices)
nk
k
Organic
semiconductor for
solar cell: visible n,k
nk
k
Glass: changes in the
IR n, k due to
vibrating atoms
Review
P.
f ( x, t)  f ( x  ct)is a solution to
It is also a solution to ____:
1)
3
2)
a)
b)
c)
d)
2
1  2 f ( x, t)
f ( x, t)  2
0
2
2
x
c
t

1  3 f ( x, t)
f ( x, t)  3
0
3
3
x
c
t
4
1  4 f ( x, t)
f ( x, t)  4
0
4
4
x
c
t
eqn 1
eqn 2
both
neither