Nonlinear Fiber Optics - 4 ed. Agrawal

418 Chapter 10. Four-Wave Mixing
10.2 A single CW pump beam is used to produce the signal and idler waves. Starting
from Eq. (10.1.1) derive the three nonlinear equations, similar to those in Eqs.
(10.2.1)–(10.2.4), governing this FWM process.
10.3 Solve the set of three equations obtained in Problem 10.2 assuming that the pump
beam remains undepleted. Find the parametric gain for the signal and idler waves
as a function of the pump power and the phase mismatch Δk.
10.4 Explain how self-phase modulation can be used to satisfy the phase-matching
condition for FWM to occur in a single-mode fiber. What should be the pump
power when pump and signal wavelengths are 1.50 and 1.51 μm, respectively?
Assume γ = 5W −1 /km and β 2 = −20 ps 2 /km.
10.5 FWM is observed to occur in a birefringent fiber when a 1.5-μm pump beam is
launched such that it is polarized at 40 ◦ from the slow axis. What are the wavelengths
and the directions of polarization for the spectral sidebands generated
through FWM?
10.6 Starting from Eqs. (10.2.13) and (10.2.14), derive an expression for the signal
and idler powers for a single-pump FOPA of length L. You can assume that no
idler power is launched initially.
10.7 Explain how a dual-pump FOPA can be designed to provide a nearly uniform
gain over a wide bandwidth.
10.8 How can you use FWM for wavelength conversion in WDM systems? Derive an
expression for the conversion efficiency for a dual-pump FOPA.
10.9 Derive Eqs. (10.5.3) and (10.5.4) after substituting Eq. (10.5.2) in Eq. (10.1.1).
10.10 Use Eqs. (10.5.3)–(10.5.5) to derive the set of four equations given as Eqs.
(10.5.6)–(10.5.9).
10.11 Use Eqs. (10.5.11) and (10.5.12) to prove that the FWM process is independent
of the signal polarization when the two pumps are orthogonally polarized.
Identify the terms that contribute to the signal gain when the pumps are linearly
polarized.
10.12 Prove that the solution of Eqs. (10.5.14) and (10.5.15) is given by Eq. (10.5.16)
with g(θ) given in Eq. (10.5.17).
10.13 Show that Eqs. (10.5.27) and (10.5.28) lead to the signal gain G s given in Eq.
(10.5.29).
References
[1] J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918
(1962).
[2] Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).
[3] M. Schubert and B. Wilhelmi, Nonlinear Optics and Quantum Electronics (Wiley, New
York, 1986).
[4] P. N. Butcher and D. Cotter, Elements of Nonlinear Optics (Cambridge University Press,
Cambridge, UK, 1990).