Nonlinear Fiber Optics - 4 ed. Agrawal

44 Chapter 2. Pulse Propagation in Fibers
In effect, the nonlinearity is assumed to be lumped at the midplane of each segment
(dashed lines in Figure 2.3).
In practice, the split-step Fourier method can be made to run faster by noting that
the application of Eq. (2.4.8) over M successive steps results in the following expression:
A(L,T ) ≈ e − 1 2 h ˆD
( M∏
e h ˆD e h ˆN
m=1
)
e 1 2 h ˆD A(0,T ). (2.4.10)
where L = Mh is the total fiber length and the integral in Eq. (2.4.9) was approximated
with h ˆN. Thus, except for the first and last dispersive steps, all intermediate steps can
be carried over the whole segment length h. This feature reduces the required number
of FFTs roughly by a factor of 2 and speeds up the numerical code by the same factor.
Note also that a different algorithm is obtained if we use Eq. (2.4.7) with â = h ˆN and
ˆb = h ˆD. In that case, Eq. (2.4.10) is replaced with
)
A(L,T ) ≈ e − 1 2 h ˆN
( M∏
e h ˆN e h ˆD
m=1
e 1 2 h ˆN A(0,T ). (2.4.11)
Both of these algorithms provide the same accuracy and are easy to implement in practice
(see Appendix B). Higher-order versions of the split-step Fourier method can be
used to improve the computational efficiency [70]. The use of an adaptive step size
along z can also help in reducing the computational time for certain problems [71].
The split-step Fourier method has been applied to a wide variety of optical problems
including wave propagation in atmosphere [76], graded-index fibers [77], semiconductor
lasers [78], unstable resonators [79], and waveguide couplers [80]. It is referred to
as the beam-propagation method when applied to the propagation of CW optical beams
in nonlinear media when dispersion is replaced by diffraction [77]–[81].
For the specific case of pulse propagation in optical fibers, the split-step Fourier
method was first applied in 1973 [35]. Since then, this method has been used extensively
for studying various nonlinear effects in optical fibers [82]–[90], mainly because
of its fast execution compared with most finite-difference schemes [46]. Although the
method is relatively straightforward to implement, it requires that step sizes in z and T
be selected carefully to maintain the required accuracy [71]. The optimum choice of
step sizes depends on the complexity of the problem, and a few guidelines are available
[91]–[94].
The use of FFT imposes periodic boundary conditions whenever the split-step
Fourier method is employed. This is acceptable in practice if the temporal window
used for simulations is made much wider than the pulse width. Typically, window size
is chosen to be 10 to 20 times the pulse width. In some problems, a part of the pulse
energy may spread so rapidly that it may be difficult to prevent it from hitting the window
boundary. This can lead to numerical instabilities as the energy reaching one edge
of the window automatically reenters from the other edge. It is common to use an “absorbing
window” in which the radiation reaching window edges is artificially absorbed
even though such an implementation does not preserve the pulse energy. In general, the
split-step Fourier method is a powerful tool provided care is taken to ensure that it is
used properly. Several generalizations of this method have been developed that retain