Nonlinear Fiber Optics - 4 ed. Agrawal

Chapter 2
Pulse Propagation in Fibers
For an understanding of the nonlinear phenomena in optical fibers, it is necessary to
consider the theory of electromagnetic wave propagation in dispersive nonlinear media.
The objective of this chapter is to obtain a basic equation that governs propagation
of optical pulses in single-mode fibers. Section 2.1 introduces Maxwell’s equations
and important concepts such as the linear and nonlinear parts of the induced polarization
and the frequency-dependent dielectric constant. The concept of fiber modes
is introduced in Section 2.2 where the single-mode condition is also discussed. Section
2.3 considers the theory of pulse propagation in nonlinear dispersive media in the
slowly varying envelope approximation with the assumption that the spectral width of
the pulse is much smaller than the frequency of the incident radiation. The numerical
methods used to solve the resulting propagation equation are discussed in Section 2.4.
2.1 Maxwell’s Equations
Like all electromagnetic phenomena, the propagation of optical fields in fibers is governed
by Maxwell’s equations. In the International System of Units (see Appendix A),
these equations take the form [1]
∇ × E = − ∂B
∂t , (2.1.1)
∇ × H = J + ∂D
∂t , (2.1.2)
∇ · D = ρ f , (2.1.3)
∇ · B = 0, (2.1.4)
where E and H are electric and magnetic field vectors, respectively, and D and B are
corresponding electric and magnetic flux densities. The current density vector J and
the charge density ρ f represent the sources for the electromagnetic field. In the absence
of free charges in a medium such as optical fibers, J = 0 and ρ f = 0.
The flux densities D and B arise in response to the electric and magnetic fields E
and H propagating inside the medium and are related to them through the constitutive
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