Nonlinear Fiber Optics - 4 ed. Agrawal

46 Chapter 2. Pulse Propagation in Fibers
polarization effects are completely ignored. As was seen in Section 1.2.4, optical fibers
exhibit birefringence. The inclusion of the birefringence effects requires consideration
of all components of electric and magnetic field vectors.
In the case of a linear medium, the algorithms that solve Maxwell’s equations, Eqs.
(2.1.1) through (2.1.4), directly in the time domain by using finite-difference methods
have been developed for many years [102]–[104]. Such algorithms have been extended
to the case of nonlinear media [105]–[109]. In a 1992 approach [105], the delayed
nature of nonlinear response was incorporated by using Eqs. (2.3.32) and (2.3.38) together
with the functional form of the Raman response function given in Eq. (2.3.40).
This work included the dispersive effects through a single resonance frequency. In an
extension of this approach [109], dispersive effects were included through the Sellmeier
equation given in Eq. (1.2.6) with three resonance frequencies (m = 3). Conceptually,
the main difference between the finite-difference time-domain (FDTD) method and the
split-step Fourier method is that the former deals with all electromagnetic components
without eliminating the carrier frequency ω 0 , in contrast with what was done in Section
2.3 in deriving the NLS equation. For this reason, the FDTD method can be used for
pulses of arbitrary durations (as short as a single optical cycle).
The FDTD method is certainly more accurate because it solves Maxwell’s equations
directly with a minimum number of approximations. However, improvement in
accuracy is achieved only at the expense of a vast increase in the computational effort.
This can be understood by noting that the time step needed to resolve the optical carrier
is by necessity a fraction of the optical period and should often be