Page 98 - Fiber Optic Communications Fund

Optical Fiber Transmission 79

In the case of multi-mode fibers, modes propagate at different speeds and arrive at different times. Similarly,
in a single-mode fiber, x-(y-)polarization component propagates at the speed of 1∕ (1∕ ) and, therefore,
1x
1y
the time delay ΔT between two polarization components at the output of the fiber of length L is
ΔT = L| − |. (2.228)
1x
1y
The above equation is valid when there is no coupling between x- and y-polarization components. However, for
standard telecommunication fibers, there is a random coupling between these components due to perturbations
such as stress and micro-bending. The fiber vector transfer function given by Eq. (2.227) does not take into
account the random coupling between x- and y-polarization components. In general, the fiber vector transfer
function can be written as [38]
[ ]
H (Ω, L) H (Ω, L)
xx
xy
H(Ω, L)= . (2.229)
H (Ω, L) H (Ω, L)
yx
yy
The transfer functions H (Ω, L) and H (Ω, L) represent the random coupling between x- and y-polarization
xy yx
components. Because of the random nature of the coupling, it is hard to characterize these functions. Nev-
ertheless, these functions change over a time scale that is longer than the symbol period and, therefore, it is
possible to estimate H(Ω, L) and compensate for it using digital signal processing (see Chapter 11) in coherent
communication systems.

2.7.6 Spot Size
The transverse extent of the field distribution of the fundamental mode plays an important role in determining
splice loss between fibers, bending loss, fiber dispersion, and the threshold power required to have significant
nonlinear effects (discussed in Chapter 10). The root mean square (r.m.s.) spot size or Petermann-1 spot size
is defined as [39–40]
[ ∞ ] 1∕2
2 ∫ Φ (r)r dr
3
2
= 0 , (2.230)
p1 ∞
∫ Φ (r)rdr
2
0
where Φ(r) is the transverse field distribution of the fundamental mode, which is radially symmetric. When
a fiber mode has a large transverse extent the spot size is large, leading to enhancement of bending losses.
On the contrary, large spot size is desirable to reduce the effect of fiber nonlinearity on optical pulses and,
thereby, transmission performance can be improved. This is because, for the given launch power, the power
per unit cross-sectional area (= optical intensity) is larger when the spot size is smaller and the nonlinear
change in refractive index is directly proportional to the optical intensity. Typically, as the spot size increases,
the dispersion slope increases too. Therefore, the refractive index profile n(r) of a fiber should be optimized so
that (i) it has a single mode and has low loss at the desired wavelength range and (ii) the spot size is sufficiently
large for the transmission performance to not be impaired by nonlinear effects and, yet, be small enough so
that the dispersion slope and bending losses are not enhanced.

2.8 Dispersion-Compensating Fibers (DCFs)
For long-haul and/or high-bit-rate optical communication systems, the pulse broadening due to intramodal
dispersion leads to intersymbol interference, which degrades transmission performance. The pulse broadening
can be compensated using a DCF, as shown in Fig. 2.37. Using Eq. (2.107), the transfer functions of the