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Unformatted text preview: 72 CHAPTER 2. OPTICAL FIBERS rectangular array which is placed inside a polyethylene tube. The mechanical strength
is provided by using steel rods in the two outermost polyethylene jackets. The outer
diameter of such ﬁber cables is about l—l.5 cm. Connectors are needed to use optical ﬁbers in an actual communication system.
They can be divided into two categories. A permanent joint between two ﬁbers is
known as a ﬁber splice, and a detachable connection between them is realized by using
a ﬁber connector. Connectors are used to link ﬁber cable with the transmitter (or the
receiver), while splices are used to join ﬁber segments (usually 5—10 km long). The
main issue in the use of splices and connectors is related to the loss. Some power is
always lost, as the two ﬁber ends are never perfectly aligned in practice. Splice losses
below 0.1 dB are routinely realized by using the technique of fusion splicing .
Connector losses are generally larger. State-of—the—art connectors provide an average
loss of about 0.3 dB . The technology behind the design of splices and connectors
is quite sophisticated. For details, the reader is referred to Ref. , a book devoted
entirely to this issue. Problems 2.1 A multimode ﬁber with a 50-;tm core diameter is designed to limit the inter-
modal dispersion to 10 ns/km. What is the numerical aperture of this ﬁber?
What is the limiting bit rate for transmission over 10 km at 0.88 [.tm? Use 1.45
for the refractive index of the cladding. 2.2 Use the ray equation in the paraxial approximation [Eq. (21.8)] to prove that
intermodal dispersion is zero for a graded-index ﬁber with a quadratic index
proﬁle. 2.3 Use Maxwell’s equations to express the ﬁeld components E p, E¢, Hp, and H1, in
terms of EZ and Hz and obtain Eqs. (2.2.29)—(2.2.32). 2.4 Derive the eigenvalue equation (2.2.33) by matching the boundary conditions at
the core—cladding interface of a step—index ﬁber. 2.5 A single-mode ﬁber has an index step 111 — n2 2 0.005. Calculate the core radius
if the ﬁber has a cutoff wavelength of 1 ,um. Estimate the spot size (FWHM) of
the ﬁber mode and the fraction of the mode power inside the core when this ﬁber
is used at 1.3 ,um. Use 111 = 1.45 mm. 2.6 A 1.55—um unchirped Gaussian pulse of 100-ps width (FWHM) is launched into
' a single—mode ﬁber. Calculate its FWHM after 50 km if the ﬁber has a dispersion
of 16 ps/(km-nm). Neglect the source spectral width. 2.7 Derive an expression for the conﬁnement factor F of single-mode ﬁbers deﬁned
as the fraction of the total mode power contained inside the core. Use the Gaus-
sian approximation for the fundamental ﬁber mode. Estimate 1" for V : 2. 2.8 A single-mode ﬁber is measured to have 22(d2n / £122) : 0.02 at 0.8 #111. Cal-
culate the dispersion parameters B2 and D. PROBLEMS 73 —m0de expressions for the minimum width and the ﬁber
length at which the minimum occurs. 2.10 Estimate the limiting bit rate for a 60—km ‘um wavelengths assuming transform—limited, 50—ps (FWHM) input pulses. As— sume that [32 = 0 and —20 psZ/km and ﬁ3 = 0.1 ps3/km and O at 1.3- and 1.55-um
wavelengths, respectively. Also assume that Va, << 1. single—mode ﬁber link at 1.3- and 1.55- se of a single~mode semiconductor laser for which ate is limited by B(l[33|L)1/3 < 0.324. What is the
= 100 km if m = 0.1 pS3/km? Va, << 1 and show that the bit r
limiting bit rate for L ystem operating at 5 Gb/s is using Gaus-
M) chirped such that C = —6. What is the
ength? How much will it change if the pulses phase shift induced by SPM b
neglect ﬁber losses. 2.19 Calculate the power launched into a 40-km-long single-mode ﬁber for which
the SPM—induced nonlinear phase shift becomes 180°. Assume l = 1.55 pm,
A63 = 40 ,umz, a = 0.2 dB/km, and r72 = 2.6 >< 10*20 m2M. 74 CHAPTER 2. OPTICAL FIBERS 2.20 Find the maximum frequency shift occurring because of the SPM—induced chirp
imposed on a Gaussian pulse of 20-ps width (FWHM) and 5—mW peak power af—
ter it has propagated 100 km. Use the ﬁber parameters of the preceding problem but assume ()6 = 0. References  J. Tyndall, Proc. Roy. Inst. 1, 446 (1854).  J. L. Baird, British Patent 285,738 (1927).
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