Nonlinear Fiber Optics - 4 ed. Agrawal

20 Chapter 1. Introduction
Chapters 11 and 12 focus on highly nonlinear fibers developed during the 1990s.
Various techniques used to measure the nonlinear parameter are described first in Chapter
11. The remainder of this chapter then focuses on four types of highly nonlinear
fibers and their properties. The focus of Chapter 12 is on the novel nonlinear effects
that have become possible with the advent of such fibers. After discussing intrapulse
Raman scattering and FWM, the chapter focuses on supercontinuum generation. The
last section is devoted to second- and third-harmonic generation in optical fibers.
Problems
1.1 Calculate the propagation distance over which the injected optical power is reduced
by a factor of two for three fibers with losses of 0.2, 20, and 2000 dB/km.
Also calculate the attenuation constant α (in cm −1 ) for the three fibers.
1.2 A single-mode fiber is measured to have λ 2 (d 2 n/dλ 2 )=0.02 at 0.8 μm. Calculate
the dispersion parameters β 2 and D.
1.3 Calculate the numerical values of β 2 (in ps 2 /km) and D [in ps/(km-nm)] at 1.5 μm
when the modal index varies with wavelength as n(λ)=1.45 − s(λ − 1.3 μm) 3 ,
where s = 0.003 μm −3 .
1.4 For silica fiber doped with 7.9% of germania, parameters appearing in the Sellmeier
equation have the following values [69]: B 1 = 0.7136824, B 2 = 0.4254807,
B 3 = 0.8964226, λ 1 = 0.0617167 μm, λ 2 = 0.1270814 μm, and λ 3 = 9.896161
μm. Plot n, n g , and β 2 in the wavelength range of 0.5 to 1.6 μm and comment
on changes from values shown in Figures 1.4 and 1.5.
1.5 Using the parameter values given in the preceding problem, calculate the values
of third- and fourth-order dispersion parameters (β 3 and β 4 ) at the zerodispersion
wavelength λ D of the fiber. Calculate β 2 and D when the input wavelength
exceeds λ D by 10 nm.
1.6 A 1-km-long single-mode fiber with the zero-dispersion wavelength at 1.4 μm
is measured to have D = 10 ps/(km-nm) at 1.55 μm. Two pulses from Nd:YAG
lasers operating at 1.06 and 1.32 μm are launched simultaneously into the fiber.
Calculate the delay in the arrival time of the two pulses at the fiber output assuming
that β 2 varies linearly with wavelength over the range 1.0–1.6 μm.
1.7 Define the dispersion parameters D and β 2 and derive a relation between the two.
Prove that D = −(λ/c)(d 2 n/dλ 2 ).
1.8 Explain the concepts of birefringence and beat length. Why does an optical fiber
exhibit some residual birefringence that varies randomly along its length?
1.9 What is meant by polarization-mode dispersion in the context of optical fibers?
What happens to optical pulses when a fiber exhibits randomly varying birefringence
along its length?
1.10 Sketch a design for a polarization-maintaining fiber. Under what conditions do
such fibers maintain polarization? What happens to the state of polarization
when input light is polarized linearly at an angle of 10 ◦ from the slow axis of the
fiber?