Nonlinear Fiber Optics - 4 ed. Agrawal

4.4. Higher-Order Nonlinear Effects 107
where L D , L ′ D , and L NL are the three length scales defined as
L D = T 0
2
|β 2 | , L′ D = T 0
3
|β 3 | , L NL = 1 . (4.4.2)
γP 0
The parameters s and τ R govern the effects of self-steepening and intrapulse Raman
scattering, respectively, and are defined as
s = 1
ω 0 T 0
,
τ R = T R
T 0
. (4.4.3)
Both of these effects are quite small for picosecond pulses but must be considered for
ultrashort pulses with T 0 < 1 ps.
4.4.1 Self-Steepening
Self-steepening results from the intensity dependence of the group velocity [84]–[87].
Its effects on SPM were first considered in liquid nonlinear media [2] and later extended
to optical fibers [88]–[92]. Self-steepening leads to an asymmetry in the SPMbroadened
spectra of ultrashort pulses [93]–[99].
Before solving Eq. (4.4.1) numerically, it is instructive to consider the dispersionless
case by setting β 2 = β 3 = 0. Equation (4.4.1) can be solved analytically in this
specific case if we also set τ R = 0 [86]. Defining a normalized distance as Z = z/L NL
and neglecting fiber losses (α = 0), Eq. (4.4.1) becomes
∂U
∂Z + s ∂
∂τ (|U|2 U)=i|U| 2 U. (4.4.4)
Using U = √ I exp(iφ) in Eq. (4.4.4) and separating the real and imaginary parts, we
obtain the following two equations:
∂I ∂I
+ 3sI = 0, (4.4.5)
∂Z ∂τ
∂φ ∂φ
+ sI = I. (4.4.6)
∂Z ∂τ
Since the intensity equation (4.4.5) is decoupled from the phase equation (4.4.6), it
can be solved easily using the method of characteristics. Its general solution is given
by [88]
I(Z,τ)= f (τ − 3sIZ), (4.4.7)
where we used the initial condition I(0,τ)= f (τ) and f (τ) describes the pulse shape
at z = 0. Equation (4.4.7) shows that each point τ moves along a straight line from its
initial value, and the slope of the line is intensity dependent. This feature leads to pulse
distortion. As an example, consider the case of a Gaussian pulse for which
I(0,τ) ≡ f (τ)=exp(−τ 2 ). (4.4.8)
From Eq. (4.4.7), the pulse shape at a distance Z is obtained by using
I(Z,τ)=exp[−(τ − 3sIZ) 2 ]. (4.4.9)