Page 108 - Fiber Optic Communications Fund

Optical Fiber Transmission 89

If C < 0, at the fiber input, the leading edge is up-shifted in frequency whereas the trailing edge is
down-shifted. The frequency components corresponding to the leading edge travel faster than those
corresponding to the trailing edge in an anomalous dispersion fiber. Therefore, as shown by the arrows in
Fig. 2.42(a), the leading edge moves to the left (earlier time) and the trailing edge moves to the right, which
leads to pulse broadening. The frequency chirp at the fiber input in this case has the same sign (as that due to
dispersion as given by Eq. (2.167)) and, therefore, these two chirps add up, leading to enhanced broadening
as seen in Fig. 2.40 (C =−4) compared with the case of an unchirped pulse.

Exercises
2.1 A step-index fiber has a cutoff wavelength = 900 nm, and NA = 0.22. (a) Calculate the core radius.
(b) What could be the maximum allowable core radius to make this fiber single-moded at 500 nm?
(Ans: (a) 3.44 μm; (b) 2.29 μm.)
2.2 Consider a small fiber section of length ΔL as shown in Fig. 2.43. Let F(ΔL)= P(ΔL)∕P .Next,
in
consider a cascade of identical fiber sections as shown in Fig. 2.44. Let M be the total number of fiber
sections. When M → ∞ (or ΔL → 0), show that
P out
F tot = = exp (−L), (2.271)
P in
where =−dF∕d(ΔL).

Δ
Δ
Δ

Figure 2.43 An infinitesimal fiber section.

in out

Δ

Figure 2.44 A fiber of length L with M sections of length ΔL.

Hint: F =[F(ΔL)] L∕ΔL , expand F(ΔL) in a Taylor series with F(0)= 1 and use
tot
M
e = Lim (1 + 1∕M) . (2.272)
M→∞