2.1. GEOMETRICAL-OPTICS DESCRIPTION 27
Figure 2.3: Ray trajectories in a graded-index fiber.Geometrical optics can be used to show that a parabolic-index profile leads to
nondispersive pulse propagation within theparaxial approximation. The trajectory
of a paraxial ray is obtained by solving [22]
d^2 ρ
dz^2=
1
ndn
dρ, (2.1.8)
whereρis the radial distance of the ray from the axis. By using Eq. (2.1.7) forρ<
awithα=2, Eq. (2.1.8) reduces to an equation of harmonic oscillator and has the
general solution
ρ=ρ 0 cos(pz)+(ρ 0 ′/p)sin(pz), (2.1.9)
wherep=( 2 ∆/a^2 )^1 /^2 andρ 0 andρ 0 ′are the position and the direction of the input
ray, respectively. Equation (2.1.9) shows that all rays recover their initial positions
and directions at distancesz= 2 mπ/p, wheremis an integer (see Fig. 2.3). Such a
complete restoration of the input implies that a parabolic-index fiber does not exhibit
intermodal dispersion.
The conclusion above holds only within the paraxial and the geometrical-optics ap-
proximations, both of which must be relaxed for practical fibers. Intermodal dispersion
in graded-index fibers has been studied extensively by using wave-propagation tech-
niques [13]–[15]. The quantity∆T/L, where∆Tis the maximum multipath delay in
a fiber of lengthL, is found to vary considerably withα. Figure 2.4 shows this varia-
tion forn 1 = 1 .5 and∆= 0 .01. The minimum dispersion occurs forα= 2 ( 1 −∆)and
depends on∆as [23]
∆T/L=n 1 ∆^2 / 8 c. (2.1.10)
The limiting bit rate–distance product is obtained by using the criterion∆T< 1 /Band
is given by
BL< 8 c/n 1 ∆^2. (2.1.11)
The right scale in Fig. 2.4 shows theBLproduct as a function ofα. Graded-index fibers
with a suitably optimized index profile can communicate data at a bit rate of 100 Mb/s
over distances up to 100 km. TheBLproduct of such fibers is improved by nearly
three orders of magnitude over that of step-index fibers. Indeed, the first generation