Download MASTER THESIS

Intramodal or Chromatic Dispersion. Since the optical propagating signal is limited in time its energy is
spread over a frequency band. Therefore, the energy into each propagating mode suffers additional
differential delays for every frequency component over the signal’s band. This is known as intramodal
dispersion. The propagation delay differences between the different spectral components of the
transmitted signal cause intramodal dispersion.
In this PFC we will be only considering monomode optical fibres and therefore we will be dealing
exclusively with chromatic dispersion.
In order to present the chromatic dispersion effect, we introduce the expression of a pulse which travels
along the fibre and analyze its components.
V (ω , z ) = V (ω,0)e − jβ (ω ) z
(1)
V (ω,0) refers to the pulse value at the input of the fibre. The term e − jβ (ω )z reflects the phase constant of
the fundamental propagating mode, which depends on frequency (the minus sign indicates propagation
along the ‘z’ axis positive direction).
The expression above relates β and ω, and takes into account the chromatic dispersion as the main limiting
effect in fibre propagation. In this case other phenomena (such as losses, nonlinear effects and so on) are
considered of lower order. This approximation is relevant to several practical applications and allows to
simplify the study of dispersion.
We now concentrate on the phase constant represented by e − jβ (ω ) z . In an ideal case, the phase constant
suffers a linear dependency on frequency, so all spectral components experience the same delay and, at
reception, there will be the same signal without distortion but only delayed. In a real case, as it happens in
a dispersive channel, the dispersion relation is not linear; this effect leads to different arrival times of
different frequency components. The signal recovered in reception will differ from the transmitted signal.
An exact solution can be difficult to treat analytically, so in view of the fact that the phase constant slowly
varies in the signal frequency bandwidth, it is possible to use a Taylor approximation to express β(ω) in a
valid way:
β (ω ) = β (ω0 ) + (ω − ω0 )
(ω − ω0 ) 2 ∂ 2 β
(ω − ω0 ) 3 ∂ 3 β
∂β
|ω =ω0 +
|
+
|ω =ω0 +...
ω =ω0
∂ω
2
∂ω 2
6
∂ω 3
= β 0 + ∆ωβ1 +
(2)
∆ω
∆ω
β2 +
β 3 + ...
2
6
2
3
where the third and higher terms can be neglected if we suppose that ∆ω = ω − ω 0 << ω0 . In this way,
the (2) expression can be rewritten as:
16