2.1. GEOMETRICAL-OPTICS DESCRIPTION 25
Figure 2.2: Light confinement through total internal reflection in step-index fibers. Rays for
whichφ<φcare refracted out of the core.
One can use Eqs. (2.1.1) and (2.1.2) to find the maximum angle that the incident
ray should make with the fiber axis to remain confined inside the core. Noting that
θr=π/ 2 −φcfor such a ray and substituting it in Eq. (2.1.1), we obtain
n 0 sinθi=n 1 cosφc=(n^21 −n^22 )^1 /^2. (2.1.3)In analogy with lenses,n 0 sinθiis known as thenumerical aperture(NA) of the fiber.
It represents the light-gathering capacity of an optical fiber. Forn 1 n 2 the NA can be
approximated by
NA=n 1 ( 2 ∆)^1 /^2 , ∆=(n 1 −n 2 )/n 1 , (2.1.4)
where∆is the fractional index change at the core–cladding interface. Clearly,∆should
be made as large as possible in order to couple maximum light into the fiber. How-
ever, such fibers are not useful for the purpose of optical communications because of a
phenomenon known as multipath dispersion ormodal dispersion(the concept of fiber
modes is introduced in Section 2.2).
Multipath dispersion can be understood by referring to Fig. 2.2, where different
rays travel along paths of different lengths. As a result, these rays disperse in time at
the output end of the fiber even if they were coincident at the input end and traveled
at the same speed inside the fiber. A short pulse (called animpulse) would broaden
considerably as a result of different path lengths. One can estimate the extent of pulse
broadening simply by considering the shortest and longest ray paths. The shortest path
occurs forθi=0 and is just equal to the fiber lengthL. The longest path occurs forθi
given by Eq. (2.1.3) and has a lengthL/sinφc. By taking the velocity of propagation
v=c/n 1 , the time delay is given by
∆T=
n 1
c(
L
sinφc−L
)
=
L
cn^21
n 2∆. (2.1.5)
The time delay between the two rays taking the shortest and longest paths is a measure
of broadening experienced by an impulse launched at the fiber input.
We can relate∆Tto the information-carrying capacity of the fiber measured through
the bit rateB. Although a precise relation betweenBand∆Tdepends on many details,