Download Dispersion in Optical Fibers

Dispersion in Optical Fibers
Henrique Salgado
December, 2007
1
Phase velocity
A monochromatic wave travelling in the z-direction may be written as
E(z, t) = E0 ej(ωt−βz)
(1)
ω is the wave frequency
β is the propagation constant
1.1
Definition of phase velocity
The velocity of an observer must maintain to travel with the field at constant
phase
ωt − βz = constant
(2)
Differentiating the previous equation and solving for dz/dt gives the phase
velocity as,
ω
dz
= vp =
dt
β
(3)
The phase velocity characterizes only the rate of phase change in space,
not the rate of power or envelope propagation, the latter is characterized by
the group velocity.
1.2
Group velocity
A modulated signal is not monochromatic. Different components travel
through the fiber with generally different phase velocities, and therefore,
accumulate different phase shifts which gives origin to phase distortion.
Group velocity is, by definition, the velocity of the envelope (amplitude
modulation of an AM signal) of the signal.
AM optical signal
Let us consider an AM optical signal given by
EAM (t, z = 0) = E0 [(1 + m cos(ω1 t)] cos(ωc t)
E: field amplitude
m: modulation index
1
(4)
ω1 : modulation frequency
ωc : light (carrier) frequency (ω1 ≪ ωc
Expanding the cosine function using the Euler formula gives
n
o 1 jωc t
m jω1 t
EAM (t, z = 0) = E0 1 +
e
+ e−jω1 t
e
+ e−jωc t
2
2
After some manipulation the previous equation can be written as
n
o
m j(ω1 +ωc )t
e
+ ej(ωc −ω1 )t
EAM (t, z = 0) = E0 · Re ejωc t +
2
(5)
(6)
This signal contains three components at frequencies, ωc , ωc −ω1 and ωc +ω1 .
Each of these components will travel at its own phase velocity and accumulate
its own phase velocity.
It is convenient to write the β(ω) as a Taylor series around the carrier
frequency
1
1 ...
β(ω) = βc + β̇(ω − ωc ) + β̈(ω − ωc )2 + β (ω − ωc )3 + . . .
2
6
∂β
β̇ = ∂ω
2 ω=ω
c
∂ β
β̈ = ∂ω
2
... ∂ 3 β ω=ωc
β = ∂ω3
(7)
ω=ωc
˙ − ωc ):
Neglecting dispersion β(ω) ≈ βc + β(ω
β(ωc ± ω1 ) = βc ± β̇ω1 = βc ± ∆β
(8)
where ∆β = β̇ω1 .
At a distance z in the fiber
o
n
m
m
EAM (t, z) = E0 · Re ejωc t−βz + ej[(ωc −ω1 )t−(βc −∆β)z] + ej[(ωc +ω1 )t−(βc +∆β)z]
2
2
= E0 cos(ωt − βc z) [1 + m cos(ω1 t − ∆βz)]
(9)
Phase shift accumulated by carrier is βc z, while the phase shift accumulated by the modulation is equal to ∆βz. The group velocity, vg is defined as
the velocity one must maintain to observe a constantt phase of the envelope.
When z = vg t, ω1 t − ∆βz must be constant:
ω1 t − ∆βz = const, ∀t
ω1 t − β̇ω1 vg t = const, ∀t
This equation has a solution if and only if const = 0 in which case
−1
1
∂β
vg = =
∂ω
β̇
1.3
(10)
(11)
Chromatic dispersion in single-mode fibers
The dominant dispersion mechanism in single-mode fibers is the so-called
chromatic dispersion: different light “colors” propagate with different velocites. Mathematically, the chromatic dispersion can be characterized either
by β̈ or by the group velocity dispersion (GVD) coefficient, usually denoted
by D.
2
Pulse spread
If τ is the propagation delay at a particular value of ω:
L
∂τ
1
∂
τ= ,
= Lβ̈
=L
vg
∂ω
∂ω vg
(12)
L is the length of the fiber and vg the group velocity generally frequency
dependent.
If a signal has a frequency spectral width of ∆ω, the difference in propagation time of parts of this signal at the opposite extremes of the spectrum
will be
∂τ ∆τ = ∆ω = |β̈| · L · ∆ω
(13)
∂ω
Thus , the pulse spread is proportional to β̈, L and the signal spectrum width
∆ω.
The pulse spread is usually specified in terms of ∆λ, the spectral width
of the signal expressed in wavelength and the Group Velocity Dispersion
coefficient, D:
∆τ = |D| · L · ∆λ
(14)
with GVD given by
1 ∂τ
1 ∂τ ∂ω
2πc
=
= − 2 β̈
L ∂λ
L ∂ω ∂λ
λ
2πc
∂ 2πc
∂ω
=− 2
=
∂λ
∂λ
λ
λ
D =
The parameter D has units of (s/m2 ) or (ps/(nm·km).
Considering that
dβ
1 dvg
d
1
β̈ =
=− 2
=
dω
dω vg
vg dω
(15)
(16)
this equation implies that at the point of zero dispersion, λD , where β̈ = 0
the group velocity, vg as a function of ω has a minimum (the derivative is
zero). For wavelengths λ < λD , β̈ > 0 and the fiber is said to exhibit normal
dispersion. In this regime (dvg /dω) < 0, which means high-frequency (∆ω >
0–blue shifted) components of an optical pulse travel slower (∆vg < 0), than
the lower frequency (red-shifted) components. By contrast, the opposite
occurs in the so-called anomalous dispersion regime in which β̈ < 0.
In digital applications chromatic dispersion gives rise to intersymbol interference, whereas in analogue applications dispersion is usually quantified
in terms of an induced power penalty. The latter is described in the next
section.
3
2
Dispersion penalty in analogue applications
To quantify the impact in analogue systems we start from the Talylor series
expansion of the propagation constant and consider the first three terms of
this series.
1
β(ω) = βc + β̇(ω − ωc ) + β̈(ω − ωc )2
2
(17)
β(ωc ) = βc
1
β(ωc ± ω1 ) = βc ± β̇ω1 + β̈ω12
2
Repeating the calculations to obtain EAM (t, z) at a distance z in the fiber,
this time including dispersion, we get
n
1
2
m
EAM (t, z) = E0 Re ejωc t−βz + ej [(ωc −ω1 )t−(βc −β̇ω1+ 2 β̈ω1 )z ]
2
m j [(ωc +ω1 )t−(βc +β̇ω1+ 21 β̈ω12 )z ] o
+
(18)
e
2
To simplify the calculation we use the complex notation and write
n
1
m
2
EAM (t, z) = E0 ej(ωc t−βz) + ej [(ωc −ω1 )t−(βc −β̇ω1 + 2 β̈ω1 )z ]
2
m −j [(ωc +ω1 )t−(βc +β̇ω1 + 1 β̈ω12 )z ]
2
+
(19)
e
2 n
io
h
m j(ω1 t−β̇ω1 z)
2
e
+ e−j(ω1 t−β̇ω1 z) e−j β̈ω1 z/2
= E0 ej(ωc t−βz) 1 +
2
h
n
io
m
j(ωc t−βz)
−j β̈ω12 z/2
1 + cos(ω1 t − β̇ω1 z)e
= E0 e
(20)
2
At the end of the fiber z = L, the photodetector responds to this signal with
a current that is proportional to square of the electric field,
∗
EAM (t, L)EAM
(t, L)
2
i h
i
h
m
m
2
= E0 1 + cos(ω1 t − β̇ω1 L)e−jθ · 1 + cos(ω1 t − β̇ω1 L)e+jθ
2
2
m2
= E02 + E02 m cos(ω1 t − β̇ω1 L) cos(θ) + E02
cos2 (ω1 t − β̇ω1 L)(21)
8
iAM (t) =
where θ = β̈ω12 L/2 is the phase shift accumulated by the sidebands over the
length L. The first term is a dc component and the third term, if expanded,
contains a dc plus a component at the frequency 2ω1 . The second term is
the term of interest at the signal frequency ω1 .
iAM (ω1 ) = E02 m cos(ω1 t − β̇ω1 L) cos(θ)
(22)
From this equation it can be verified that when θ = (2n + 1)π/2 with n =
0, 1, 2, . . . the amplitude of the received signal goes to zero and this occurs
periodically along the fiber. The first null occurs for
Dλ2 ω12 Lnull,1
π
β̈ω12 Lnull,1
=
=
2
2πc
2
2
4
(23)
Penalty (dB)
0
−5
−10
−15
0
2
4
Fiber length (km)
6
8
Figure 1: Dispersion penalty for a 60 GHz signal over a fiber with D =
17 ps · nm−1 km−1 .
That is for
Lnull,1 =
c
2Dλ2 f12
(24)
The above figure shows the dispersion penalty for a fiber with D = 17 ps ·
nm km−1 operating at λ = 1.55 µm whereas the modulating signal is f1 =
60 GHz, the first null ocurring at 1.02 km.
−1
5