Optical Bit Error Rate

Chapter 2.13.1 - Modal Dispersion

2.13.1   Modal Dispersion

An optical signal launched into fiber may be viewed as a bundle of nonparallel rays
transmitted within a small cone. Because the rays are not parallel, they travel along
different paths in the fiber in a zigzag manner, as electromagnetic waves do in
waveguides; each zigzag path represents a different mode (Figure 2.12). For simpli-


Figure 2.12. Modal dispersion causes a narrow pulse to widen.


fication of this description, we assume ideal monochromaticity. Thus, ray A travels
in a straight path along the core of the fiber (one mode), whereas ray B travels at an
angle, bouncing off the cladding (another mode). Thus, as rays at different modes
travel different distances, they arrive at the receiver at different times. As a result,
the initial narrow pulse spreads out. This is known as modal dispersion and it is
more pronounced in multimode fibers.

The min–max travel time of two rays is expressed as

 τmin = (Ln1)/v

and

 τmax = (Ln1)/v cos Θ

for Θ = Θcrit (total reflection), and from Snell’s law,

 cos Θcrit = n1/n2

The difference in travel time (assuming total reflection) is

 Δτ = τmax –  τmin = [(Ln1)/v]Δn/n1

Hence, the maximum bit rate, Rb, is calculated from

 Rb < 1/(4Δτ)=(1/4)(v/Ln1)(n1n)

Now, consider the case of two connected multimode fibers. Clearly, the connection
point of the two fibers represents a perturbation or a discontinuity in the optical
path. When light rays reach the end of the first fiber, they are launched into the second.
However, since there are many modes, that is, different rays travel in different
angles, each ray enters the second fiber at a different angle and thus is refracted dif-
ferently, changing from one mode to another mode. This is known as mode mixing
and it occurs in interconnected multimode fibers only.

Mode mixing affects the actual transmitted bandwidth (BWact) over the multimode
fiber length (L), defining an effective bandwidth. To distinguish the effect of
mode mixing on bandwidth, an empirical scaling factor, γ, has been devised, defining
the effective bandwidth as

 BWeff = BWact/Lγ

where γ is between 0.7 and 1.0

The difference in travel time (assuming total reflection) is improved if a graded-index
fiber is used, with a graded-index refraction profile n(r). In this case, the fiber
dispersion is improved if this condition holds:

 Rb 2vg/(ngLΔn2)

where ng is the group refractive index, Δn is the maximum relative index between
core and cladding, and vg is the group velocity in the medium.

 

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