Nonlinear Fiber Optics - 4 ed. Agrawal

42 Chapter 2. Pulse Propagation in Fibers
where ˆD is a differential operator that accounts for dispersion and losses within a linear
medium and ˆN is a nonlinear operator that governs the effect of fiber nonlinearities on
pulse propagation. These operators are given by
ˆD = − iβ 2
2
ˆN = iγ
∂ 2
∂T 2 + β 3
6
(
|A| 2 + i
ω 0
1
∂ 3
∂T 3 − α 2 , (2.4.2)
∂
∂|A| 2 )
A ∂T (|A|2 A) − T R . (2.4.3)
∂T
In general, dispersion and nonlinearity act together along the length of the fiber. The
split-step Fourier method obtains an approximate solution by assuming that in propagating
the optical field over a small distance h, the dispersive and nonlinear effects
can be assumed to act independently. More specifically, propagation from z to z + h
is carried out in two steps. In the first step, the nonlinearity acts alone, and ˆD = 0
in Eq. (2.4.1). In the second step, dispersion acts alone, and ˆN = 0 in Eq. (2.4.1).
Mathematically,
A(z + h,T ) ≈ exp(h ˆD)exp(h ˆN)A(z,T ). (2.4.4)
The exponential operator exp(h ˆD) can be evaluated in the Fourier domain using the
prescription
exp(h ˆD)B(z,T )=FT −1 exp[h ˆD(−iω)]F T B(z,T ), (2.4.5)
where F T denotes the Fourier-transform operation, ˆD(−iω) is obtained from Eq. (2.4.2)
by replacing the operator ∂/∂T by −iω, and ω is the frequency in the Fourier domain.
As ˆD(iω) is just a number in the Fourier space, the evaluation of Eq. (2.4.5) is straightforward.
The use of the FFT algorithm [73] makes numerical evaluation of Eq. (2.4.5)
relatively fast. It is for this reason that the split-step Fourier method can be faster by
up to two orders of magnitude compared with most finite-difference schemes [52].
To estimate the accuracy of the split-step Fourier method, we note that a formally
exact solution of Eq. (2.4.1) is given by
A(z + h,T )=exp[h( ˆD + ˆN)]A(z,T ), (2.4.6)
if ˆN is assumed to be z independent. At this point, it is useful to recall the Baker–
Hausdorff formula [74] for two noncommuting operators â and ˆb,
(
exp(â)exp(ˆb)=exp â + ˆb + 1 2 [â, ˆb]+ 1
)
12 [â − ˆb,[â, ˆb]] + ··· , (2.4.7)
where [â, ˆb]=âˆb − ˆbâ. A comparison of Eqs. (2.4.4) and (2.4.6) shows that the splitstep
Fourier method ignores the noncommutating nature of the operators ˆD and ˆN. By
using Eq. (2.4.7) with â = h ˆD and ˆb = h ˆN, the dominant error term is found to result
from the commutator 1 2 h2 [ ˆD, ˆN]. Thus, the split-step Fourier method is accurate to
second order in the step size h.
The accuracy of the split-step Fourier method can be improved by adopting a different
procedure to propagate the optical pulse over one segment from z to z + h. In
this procedure Eq. (2.4.4) is replaced by
) ( ∫ h z+h
)exp( )
A(z + h,T ) ≈ exp(
2 ˆD
h
exp ˆN(z ′ )dz ′ 2 ˆD A(z,T ). (2.4.8)
z