26 CHAPTER 2. OPTICAL FIBERS
such as the pulse shape, it is clear intuitively that∆Tshould be less than the allocated
bit slot (TB= 1 /B). Thus, an order-of-magnitude estimate of the bit rate is obtained
from the conditionB∆T<1. By using Eq. (2.1.5) we obtain
BL<n 2
n^21c
∆. (2.1.6)
This condition provides a rough estimate of a fundamental limitation of step-index
fibers. As an illustration, consider an unclad glass fiber withn 1 = 1 .5 andn 2 =1.
The bit rate–distance product of such a fiber is limited to quite small values since
BL< 0 .4 (Mb/s)-km. Considerable improvement occurs for cladded fibers with a small
index step. Most fibers for communication applications are designed with∆< 0 .01.
As an example,BL<100 (Mb/s)-km for∆= 2 × 10 −^3. Such fibers can communicate
data at a bit rate of 10 Mb/s over distances up to 10 km and may be suitable for some
local-area networks.
Two remarks are in order concerning the validity of Eq. (2.1.6). First, it is obtained
by considering only rays that pass through the fiber axis after each total internal re-
flection. Such rays are calledmeridional rays. In general, the fiber also supportsskew
rays, which travel at angles oblique to the fiber axis. Skew rays scatter out of the core at
bends and irregularities and are not expected to contribute significantly to Eq. (2.1.6).
Second, even the oblique meridional rays suffer higher losses than paraxial meridional
rays because of scattering. Equation (2.1.6) provides a conservative estimate since all
rays are treated equally. The effect of intermodal dispersion can be considerably re-
duced by using graded-index fibers, which are discussed in the next subsection. It can
be eliminated entirely by using the single-mode fibers discussed in Section 2.2.
2.1.2 Graded-Index Fibers
The refractive index of the core in graded-index fibers is not constant but decreases
gradually from its maximum valuen 1 at the core center to its minimum valuen 2 at
the core–cladding interface. Most graded-index fibers are designed to have a nearly
quadratic decrease and are analyzed by usingα-profile, given by
n(ρ)={
n 1 [ 1 −∆(ρ/a)α]; ρ<a,
n 1 ( 1 −∆)=n 2 ; ρ≥a,(2.1.7)
whereais the core radius. The parameterαdetermines the index profile. A step-index
profile is approached in the limit of largeα.Aparabolic-index fibercorresponds to
α=2.
It is easy to understand qualitatively why intermodal or multipath dispersion is re-
duced for graded-index fibers. Figure 2.3 shows schematically paths for three different
rays. Similar to the case of step-index fibers, the path is longer for more oblique rays.
However, the ray velocity changes along the path because of variations in the refractive
index. More specifically, the ray propagating along the fiber axis takes the shortest path
but travels most slowly as the index is largest along this path. Oblique rays have a large
part of their path in a medium of lower refractive index, where they travel faster. It is
therefore possible for all rays to arrive together at the fiber output by a suitable choice
of the refractive-index profile.