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Waves in dispersive media
Bound electron model of dielectric constant/index of refraction
The behavior of electrons when driven by an electromagnetic radiation can be understood
using the well-known equation of forced oscillator, in terms of the mass, oscillation
frequency, and damping coefficient:
E
m
e x dt
dx
dt
d x
2 2 0
2
σ ω (1)
where x is the displacement vector,
1 / 2
ω 0 = ( κ / m ) and σ is a damping constant.
The polarization density of the medium is the sum of the dipole moments of N-atoms per unit
volume so that P = Nex. Eq (1) becomes
P E
dt
dP
dt
d P 0 0
2 0
2 2 0
2
where
2 0 0
2
χ 0 = e N / m ε ω is the susceptibility.
For a monochromatic electric field of frequency ω,
j E 0 0 E
2 0
2 0
2
ν ν ν ν
ν χ ν χ − + ∆
j
2 2 0
2 0 ( ) 0
where the bandwidth π
σ ν 2
∆ =. The susceptibility is a complex number. The meaning of
damping?
The real and imaginary parts are
2 2 2 2 0
2 2 0
2 0 0
'
( ) ( )
ν ν ν ν
ν ν ν χ χ − + ∆
2 2 2 2 0
2 0 0
''
(ν ν ) (ν ν )
ν ν ν χ χ − + ∆
If the atoms are placed in a medium of
index of refraction n 0 ,
∆ν
ν
ν
χχχχ ’
- χχχχ ’’
0
'
0 2
n
n n
χ ν ν = + (5)
''
0 0
χ ν
πν α ν
n c
The imaginary part of the susceptibility can lead to gain or absorption loss.
Pulse propagation in dispersive medium
Consider a plane-wave pulse U(z,t) propagating in the z-direction, the propagation of the
pulse may be analyzed by treating the traveling of the individual frequency components.
U ( z , t )= A ( z , t )exp( j ( 2 πν 0 t − β 0 z ) (7)
where β 0 = β(ν 0 ) is the central wavenumber and A is the complex envelop of the pulse which
is slow-varying. This is a wave packet of central frequency ν0. The propagation can be
treated by considering the frequency components of the wave at the initial point z=0.
∞
−∞
A ( 0 , t ) = a ( 0 , f )exp( j 2 π ft ) df (8)
and amplitude for the frequency f is given by
∞
−∞
a ( 0 , f )= A ( 0 , t )exp(− j 2 π ft ) df
Here it is assumed that f is the frequency deviation from the central frequency and f<< ν 0.
The “frozen” wave at z=0 in (7) then is
∞
−∞
U ( 0 , t )= a ( 0 , f )exp( j 2 π ft )exp[+ j 2 πν 0 t ] df (9)
By expanding β ( ν ) surrounding ν 0 , the wave then travel to z according to
j Dzf j z df V
z a f j f t j f
zdf j z d
d z j f d
d a f exp j f t jf
U zt a f j ft j f z j df
g
( 0 , )exp( 2 [( )])exp( 2 )exp( )exp( )
) exp( ) 2
( 0 , ) [ 2 ( )]exp( )exp(
( ,) ( 0 , )exp( 2 )exp[ ( ) ] exp[ 2 ]
0
2 0
2 0
2 2 0
0 0
= − + ×
∞
−∞
∞
−∞
∞
−∞
where the group velocity
ν
β
π
d
d
Vg
= and dispersion coefficient 2
2
ν
β
π d
d D =. (11)
Case I Dispersion free medium
For D=0, (10) becomes
Maintaining pulse shape in a dispersive medium by frequency chirping
In Eq. (12), if the phase of the original Gaussian pulse is phase modulated during the pulse by
a factor exp( )
2
+ j π Dzf to cancel the broadening effect, the pulse duration may be maintained
while propagating in a dispersive mediu--optical soliton
Problem 5.5-
Problem 5.6-
Pulse broadening in optical fibers 5.6-
