Download 5 Waves in dispersive media

5 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:
d 2x
dx
e
+σ
+ ω 02 x = E
2
dt
m
dt
(1)
where x is the displacement vector, ω 0 = (κ / m)1 / 2 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
d 2P
dP
+σ
+ ω 02 P = ω 02ε 0 χ 0 E
2
dt
dt
where χ 0 = e 2 N / mε 0ω 02 is the susceptibility.
(2)
For a monochromatic electric field of frequency ω,
(−ω 2 + jωσ + ω 02 ) E = ω 02ε 0 χ 0 E
χ (ν ) = χ 0
ν 02
ν 02 − ν 2 + jν∆ν
where the bandwidth ∆ν =
damping?
σ
. The susceptibility is a complex number. The meaning of
2π
The real and imaginary parts are
ν 02 (ν 02 − ν 2 )
χ ' = χ0 2
(3)
(ν 0 − ν 2 ) 2 + (ν∆ν ) 2
χ '' = − χ 0
ν 02ν ∆ν
(4)
(ν 02 − ν 2 ) 2 + (ν∆ν ) 2
-χ’’
∆ν
ν
χ’
ν
If the atoms are placed in a medium of
index of refraction n0 ,
n(ν ) = n0 +
χ ' (ν )
(5)
2n 0
2πν
χ '' (ν )
n0 c 0
α (ν ) = −
6)
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 (2πν 0 t − β 0 z )
(7)
where β 0 = β(ν0 ) 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 2πft )df
(8)
and amplitude for the frequency f is given by
∞
a(0, f ) =
−∞
A(0, t ) exp(− j 2πft )df
Here it is assumed that f is the frequency deviation from the central frequency and f<<ν0.
The “frozen” wave at z=0 in (7) then is
U (0, t ) =
∞
−∞
a(0, f ) exp( j 2πft ) exp[+ j 2πν 0 t ]df
(9)
By expanding β(ν) surrounding ν0 , the wave then travel to z according to
U ( z, t ) =
=
∞
−∞
∞
a(0, f ) exp( j 2πft ) exp[− jβ (ν 0 + f ) z ] × exp[ j 2πν 0 ]df
a(0, f )exp[ j 2π (ν 0 + f )t ] exp(− jf
−∞
=
∞
−∞
1
dβ
d 2β
z ) exp(− j f 2
z )df exp(− jβ 0 z ) (10)
dν
2
dν 2
a(0, f ) exp( j 2π [(ν 0 + f )t ]) exp(− j 2πf
where the group velocity V g =
Case I Dispersion free medium
For D=0, (10) becomes
z
) exp(− jπDzf 2 ) exp(− jβ 0 z )df
Vg
2π
1 d 2β
and dispersion coefficient D =
.
dβ
2π dν 2
dν
(11)
U ( z, t ) =
∞
=
−∞
a (0, f ) exp( j 2π [(ν 0 + f )t ]) exp(− j 2πf
∞
= [ a (0, f ) exp[ j 2πf (t −
−∞
z
νg
z
) exp(− jβ 0 z )df
Vg
)df ] exp( j 2πν 0 t − jβ 0 z )
The leads to an envelop function A(t −
z
, z ) which is centered at ν g t to without changing
vg
shape.
Assuming that the initial pulse shape A(0,t) has a Gaussian profile of width τ 0:
t2
A(0, t ) = A0 exp(−π 2 )
τ0
Substituting a (0, f ) = exp[−π ( fτ 0 ) 2 ] into (10)
U ( z, t ) =
∞
−∞
exp[−π ( fτ 0 ) 2 ] exp( j 2π [(ν 0 + f )t ]) exp(− j 2πf
∞
= { exp[( j 2πf (t −
−∞
z
νg
z
) exp(− jπDzf 2 ) exp(− jβ 0 z )df
Vg
)] exp[−π ( f τ ' ) 2 ]df } × exp( j 2πν 0 − jβ 0 z )
(12)
where τ ' 2 = τ 0 + jDz . After the integration, the electric field has the following form
2
U ( z , t ) ∝ exp(−π
where τ 2 = τ 02 +
(t − z / v g 2 ) 2
τ
2
)
D2 z2
π 2τ 02
The integration of Eq (12) is simply Eq (8), shifted to a new location z = ν g t and with a
Gaussian shape of width τ , which is a complex number.
(13)
(14)
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(+ jπDzf 2 ) to cancel the broadening effect, the pulse duration may be maintained
while propagating in a dispersive mediu--optical soliton
Problem 5.5-2
Problem 5.6-1
Pulse broadening in optical fibers 5.6-3