Nonlinear Fiber Optics - 4 ed. Agrawal

448 Chapter 11. Highly Nonlinear Fibers
Figure 11.20: (a) Cross section of a microstructured fiber made with tellurite glass; (b) transmission
view observed with a microscope. (After Ref. [105]; c○2003 OSA.)
nature, bismuth-oxide fibers are being used for a variety of applications, including
supercontinuum generation [108], a topic we turn to in the next chapter.
Problems
11.1 The refractive index of an optical fiber is specified as n = n 0 + n 2 I, where n 0 =
1.45, n 2 = 2.7×10 −20 m 2 /W and I is the optical intensity. If it is written in terms
of the electric field as n = n 0 + n E 2 |E|2 , find the magnitude of n E 2 .
11.2 Light from two CW lasers differing in their wavelengths by a small amount Δλ
is transmitted through a fiber of length L. Derive an expression for the optical
spectrum at the fiber output after including the effects of SPM-induced phase
shift. Assume equal input powers for the two lasers and neglect fiber loss and
dispersion for simplicity.
11.3 Explain how the optical spectrum obtained in the preceding problem can be used
to estimate the nonlinear parameter γ. Also, derive the expression for the power
ratio given in Eq. (11.1.5).
11.4 Discuss how the nonlinear phenomenon of XPM can be used to deduce γ for an
optical fiber. Why is it important to depolarize the pump light when a pump–
probe configuration is employed?
11.5 The measured values of γ are found to be smaller by about 20% when pulses
shorter than 1 ns are employed. Explain the physical process responsible for this
reduction. What would happen if femtosecond pulses are employed for measuring
γ?
11.6 Explain why large values of γ cannot be realized by reducing the core diameter
of a fiber designed with silica cladding to below 1 μm? Using A eff = πw 2 with
w = a/ √ lnV , prove that the maximum value of γ occurs when the V = √ e,
where V parameter of the fiber is defined as in Eq. (11.2.1).
11.7 Solve the eigenvalue equation (2.2.8) given in Section 2.2.1 for a tapered fiber
with air cladding using n 1 = 1.45, n 2 = 1, and a = 1 μm and plot β(ω) in the