2
2.1
2.2
2.3
3.4
Classical propagation
Propagation of light in a dense optical medium
The dipole oscillator model
Dispersion
Optical anisotropy: birefringence
Chapter 2 Classical propagation
E ( z, t )  E0ei ( k z t ) ,
k  (n  i) / c
 E0 e z ei ( nz / c t ) .
Two propagation parameters:
n, 
Model:
Light:
electromagnetic wave
Atom and molecule: classical dipole oscillator
n(),  ()
2.1 Propagation of light in a dense optical medium
Three types of oscillators:
1. bound electron (atomic) oscillator
2. vibrational oscillator;
3. free electron oscillators
2.1.1 Atomic oscillators
1
1
1


,
 m0 mN
KS
,


 
p  q (r  r )
0 
p (t )  ex (t )
2.1 Propagation of light in a dense optical medium
If   0, non-resonant, transparent
2.1.1 Atomic oscillators
If  = 0,
The oscillators follow the driving wave,
but with a phase lag. The phase lag
accumulates through the medium and
retards the propagation of the wave
front, leading to smaller velocity than
in free space (v =c / n).
-- the origin of n
resonant absorption
(Beer’s law)
h  = E2 - E1
re-radiated photon – luminesce
radiationless transition
2.1.2 Vibrational oscillators
0 
KS
 1012  1013 Hz

Infrared spectral region
In a crystalline solid form the condensation of polar
molecules, these oscillations are associated with
lattice vibrations (phonons).
2.1.3 Free electron oscillators
Free electrons, Ks = 0, 0 = 0
Classical model of a polar molecule
(an ionic optical medium)
Drude-Lorentz model
2.2 The dipole oscillator model
2.2.1 The Lorentz oscillator
d 2x
m0 2  m0 02 x  0,
mN  m0
dt
d 2x
dx
m0 2  m0   m0 02 x  eE (t )
dt
dt
Light wave will drive oscillations at its own
Frequency:
The macroscopic polarization of medium P:
Presonant  Np   Nex
Ne2
1

E
m0 (02  2  i)
The electric displacement D:
D  0 E  P
  0 E  Pbackground  Presonant
E (t )  E0 cos( t  )
 E0e(exp( i (t  ))
 E0' e(exp( it ))
Solution;
x(t )  X 0e(exp( i (t  ' )))
 X 0e(exp( it ))
The gives:
  0 E   0 E  Presonant 0  r E
Thus
Ne 2
1
 r ()  1   
 0 m0 (02  2  i)
02  2
Ne 2
1 ()  1   
 0 m0 (02  2 ) 2  ( ) 2
Ne 2

 2 () 
 0 m0 (02  2 ) 2  ( ) 2
 m0 2 X 0 e  it  im0 X 0 e  it  m0 02 X 0 e  it  eE0 e  it
With:
X0 
 eE0 / m0
02  2  i
2.2 The dipole oscillator model
2.2.1 The Lorentz oscillator
low frequency limit:
Ne2
 r (0)   st  1   
 0 m002
high frequency:
r ()    1  
Thus
Ne2
( st    ) 
 0 m0 02
Close to resonance:
20 
4() 2   2
0
 2 ()  ( st    )
4() 2   2
1 ()     ( st    )
Frequency dependence of the real and imaginary
Parts of the complex dielectric constant of a dipole
At frequencies close to resonance. Also shown is
The real and imaginary part of the refractive index
Calculated from the dielectric constant.
1. 吸收峰位于o, 半宽= ;
2. 1的极值位于 o  ， 1出现负值;
3. 折射率在o  区间出现反常色散。
2.2 The dipole oscillator model
2.2.2
Multiple resonance
Take account of all the transitions in the medium



P  Npr   Nex


Ne 2
1
E

m0 j (02  2  i)
Ne 2
 r ()  1 
 0 m0
 (
j
1
2
2
j    i j )

 
D  0 E  P

  0  r E.
Assign a phenomenological oscillator strength
fj to each transition:
Ne2
 r ()  1 
 0 m0
where
f
j
 (
j
j
 1.
fj
2
j
 2  i j )
For each atom.
Schematic diagram of the frequency dependence of
the refractive index and absorption of a hypothetical
solid from the infrared to the x-ray spectral region.
The solid is assummed to have three resonant
frequencies with width of each absorption line has
been set to 10% of the centre frequency by
appropriate choice of the j’s.
2.2 The dipole oscillator model
2.2.3 Comparison with experimental data
1. n >>  except near the peaks of the absorption;
2. The transmission range of optical materials is
determined by the electronic absorption in UV
and the vibrational absorption in IR;
3. IR absorption is caused by the vibrational quanta
in SiO2 molecules themselves(1.4  1013 Hz
(21m) and 3.3  1013 Hz(9.1 m);
4. UV absorption is caused by interband electronic
transition(band gap of about 10 eV), threshold at
2  1013 Hz(150 nm)( ~ 108 m-1);
5. UV absorption departure from Lorentz model;
6. n actually increases with frequency in transparency region, the dispersion originates from
wings of two absorption peaks of UV and IR;
7. The phase velocity of light is greater than c in
region where n falls below unity;
8. Group velocity:
d
k dn
 (1 
)
dk
n dk
dn / dk  0,
g  c
g 
(a) Refractive index and (b) extinction coEfficient of fused silica (SiO2) glass from the
Infrared to the x-ray spectral region.
2.2 The dipole oscillator model
2.2.4 Local field correction
The actually atomic dipoles respond not
only to the external field, but also to the
field generated by all the other dipoles
Elocal  E  Eother
Eother
dipoles 
Elocal  E 
dipoles
,
P
,
3 0
P
3 0
P  N 0  a Elocal
e2
a 
 0 m0

j
P  N 0  a ( E 
fj
2
j
  2  i j
,
P
)  ( r  1) E ,
3 0
 r  1 N a

r  2
3
Clausius-Mossotti relationship
Model used to calculate the local field by
the Lorentz correction. A imaginary spherical
surface drawn around a particular atom
divides the medium into nearby dipoles and
distant dipoles. The field at the centre of the
sphere due to the nearby dipoles is sunned
exactly, while the field due to the distant
dipoles is calculated by treating the material
outside the sphere as a uniformly polarized
dielectric.
2.2 The dipole oscillator model
2.2.5 The Kramers-Kronig relationships
The discussion of the dipole oscillator shows that the
refractive index and the absorption coefficient are not
independent parameters but are related to each other.
If we invoke the law of causality (that an effect may
not precede its cause) and apply complex number
analysis, we can derive general relationships between
the real and imaginary parts of the refractive index as
follows:
  (' )
1
P
d'
  ' 

 n (' )  1
1
()   P 
d' ,



'
n()  1 
Where P indicates that the principal part of the integral
should be taken. The K-K relationships allow to calculate
n and , and vice versa.
2.2 Dispersion
This dispersion mainly originates from the interband
absorption in the UV and the vibrational absorption
in IR
Normal dispersion :
the refractive index increases
with frequency;
Anomalous dispersion:
the contrary occurs.
Refractive index of SiO2 glass in the IR, visible
And UV regions
2.2 Dispersion
• Pulse broadening
 
1
tp
Dispersion causes the very short pulse to broaden
in time as it propagates through the medium.
• group velocity dispersion (GVD)
g 
d
k dn
 (1 
),
dk
n dk
d 2
GVD 
dk 2
The Lorentz model indicates that GVD is positive
below an absorption line and negative above it.
There is a region of zero GVD around 1.3 m in
silica. So short pulses can be transmitted down
the silica fibre with negligible temporal broadening
at this wavelength.
2.2 Optical anisotropy: birefringence
The relationship of the P and E


P   0 E
 : thesusceptibility tensor.
 Px 
 11 12 13  E x 
 

 
 Py    0  21 22 23  E y 
P 
 31 32 33  E 
 z

 z 
Chos sin g x, y, z , to the principal
crystallin e axes :
0 
 11 0


   0 22 0 
 0
0 33 

Cubic:
11  22  33,
Re frative index and dielectric cons tan t tensor
for uniaxial crystal :


r  1  
isotropic;
Tetragonal, hexagonal or trigonal:
11  22  33, uniaxial;
Orthorhombic, monoclinic or triclinic:
11  22  33,
biaxial.
0
0 
1  11


r   0
1  11
0 
 0
0
1  33 

 n02

r   0
0

0
n02
0
0
0
n





2.2 Optical anisotropy: birefringence
Double refractive in a natural calcite crystal, an unpolarized incident light ray is split into two spatially
separated orthogonally polarized rays.
2.2 Optical anisotropy: birefringence
Electric field vector of ray propagating in a
uniaxial crystal with is its optic axis along the z
direction. The ray makes an angle of  with
respect to the optic axis. The polarization can be
resolved into: (a) a component along the x-axis
and (b) a component at an angle of 90o -  to
the optic axis. (a) Is o-ray and (b) is the e-ray.
no () 2  no2
no2 ne2
ne ()  2
no sin 2   ne2 cos 2 
2
Exercises:
1.
The full width at half maximum of the strongest hyperfine component of the sodium D2
line at 589.0 nm is 100 MHz. A beam of light passes through a gas of sodium with an
atom density of 11017 m-3. Calculate: (i) The peak absorption coefficient due to this
absorption line. (ii ) The frequency at which the resonant contribution to the refractive
index is at a maximum. (iii) The peak value of the resonant contribution to the refractive
index.
( i); 1.7*103m-1; ii) 50 MHz below the line center; iii) 3.95 * 10-5)
2.
A damped oscillator with mass, natural frequency 0, and damping constant  is being
driven by a force of amplitude F0 and frequency . The equation of motion for the
displacement x of the oscillator is:
d 2x
dx
m 2  m  m02 x  F0 cos t.
dt
dt
What is the phase of x relative to the phase of the driving force?
(-tan-1[/(02-2)])
3. Show that the absorption coefficient of a Lorentz oscilator at the line centre does not
depend on the value of 0.