Nonlinear Fiber Optics - 4 ed. Agrawal

2.4. Numerical Methods 45
the basic idea behind the split-step technique but employ an expansion other than the
Fourier series; examples include splines and wavelets [72].
2.4.2 Finite-Difference Methods
Although the split-step Fourier method is commonly used for analyzing nonlinear effects
in optical fibers, its use becomes quite time-consuming when the NLS equation
is solved for simulating the performance of wavelength-division-multiplexed (WDM)
lightwave systems. In such systems, the temporal resolution should be a small fraction
of the entire bandwidth of the WDM signal. For a 100-channel system, the bandwidth
approaches 10 THz, requiring a temporal resolution of ∼10 fs. At the same
time, the temporal window should be typically 1 to 10 ns wide, resulting in more than
10 5 mesh points in time domain. Even though each FFT operation is relatively fast, a
large number of FFT operations on a large-size array leads to an overall computation
time measured in hours (even days) on a state-of-the art computer. For this reason,
finite-difference methods continue to attract attention.
Several finite-difference schemes have been used to solve the NLS equation (see
Refs. [52] and [61]). Some of the common ones among them are the Crank–Nicholson
scheme and its variants, the hopscotch scheme and its variants, and the leap-frog
method. A careful comparison of several finite-difference schemes with the split-step
Fourier method shows that the latter is efficient only when the field amplitude varies
slowly with time [61]. However, it is difficult to recommend a specific finite-difference
scheme because the speed and accuracy depend to some extent on the number and form
of the nonlinear terms included in the generalized NLS equation. A linearized Crank–
Nicolson scheme can be faster by more than a factor of five under certain conditions.
Another situation in which finite-difference schemes are useful corresponds to propagation
of ultrashort optical pulses whose width is so short that the pulse contains only
a few optical cycles. The slowly varying envelope approximation does not always hold
for such short pulses. In recent years attempts have been made to relax this approximation,
and several new techniques have been proposed [95]–[101]. Some of them
require the use of a finite-difference method in place of the split-step Fourier method.
Finite-difference techniques for solving the paraxial wave equation have developed in
parallel with the split-step Fourier method and are sometimes the method of choice.
They can be extended beyond the validity of the paraxial approximation by using techniques
such as the Lanczos orthogonalization [96] and the Padé approximation [101].
Other extensions include algorithms that can handle bidirectional beam propagation
[100]. Most of these techniques have been developed in the context of beam propagation
in planar waveguides, but they can be readily adopted for pulse propagation in
optical fibers.
There are several limitations inherent in the use of the NLS equation for pulse
propagation in optical fibers. The slowly varying envelope approximation has already
been mentioned. Another one is related to the total neglect of backward propagating
waves. If the fiber has a built-in index grating, a part of the pulse energy will be
reflected backward because of Bragg diffraction. Such problems require simultaneous
consideration of forward and backward propagating waves. The other major limitation
is related to the neglect of the vector nature of the electromagnetic field. In essence,