Nonlinear Fiber Optics - 4 ed. Agrawal

4.3. Semianalytic Techniques 103
In the case of a chirped Gaussian pulse, the field U(z,T ) at any distance z has the
form
U(z,T )=a p exp[− 1 2 (1 + iC p)(T /T p ) 2 + iφ p ], (4.3.6)
where all four pulse parameters, a p , C p , T p , and φ p , are functions of z. The phase φ p
does not appear in Eqs. (4.3.4) and (4.3.5). Even though φ p changes with z, it does not
affect other pulse parameters and can be ignored. The peak amplitude a p is related to
the energy as E p = √ πa 2 pT p . Since E p does not change with z, we can replace it with
its initial value E 0 = √ πT 0 . The width parameter T p is related to the RMS width σ p of
the pulse as T p = √ 2σ p . Using Eq. (4.3.6) and performing integrals in Eqs. (4.3.4) and
(4.3.5), the width T p and chirp C p are found to change with z as
dT p
dz = β 2C p
,
T p
(4.3.7)
dC p
dz =(1 +C2 p) β 2
Tp
2 + γP 0 e −αz T
√ 0
.
2Tp
(4.3.8)
This set of two first-order differential equations can be used to find how the nonlinear
effects modify the width and chirp of the pulse.
Considerable physical insight can be gained from Eqs. (4.3.7) and (4.3.8). The
SPM phenomenon does not affect the pulse width directly as the nonlinear parameter
γ appears only in the chirp equation (4.3.8). The two terms on the right side of this
equation originate from dispersive and nonlinear effects, respectively. They have the
same sign for normal GVD (β 2 > 0). Since SPM-induced chirp in this case adds to
the GVD-induced chirp, we expect SPM to increase the rate of pulse broadening. In
contrast, when GVD is anomalous (β 2 < 0), the two terms on the right side of Eq.
(4.3.8) have opposite signs, and the pulse broadening should be reduced in the presence
of SPM because of smaller values of C p in Eq. (4.3.7). In fact, this equation can be
integrated to obtain the following general relation between pulse width and chirp:
∫ z
Tp 2 (z)=T0 2 + 2 β 2 (z)C p (z)dz. (4.3.9)
0
The equation shows explicitly that the pulse compresses whenever the quantity β 2 C p <
0, a result obtained earlier in Section 3.2.
4.3.2 Variational Method
The variational method is well known from classical mechanics and is used in many
different contexts [81]–[83]. It was applied as early as 1983 to the problem of pulse
propagation inside optical fibers [35]. Mathematically, it makes use of the Lagrangian
L defined as
∫ ∞
L = L d (q,q ∗ )dT, (4.3.10)
−∞
where the Lagrangian density L d is a function of the generalized coordinate q(z) and
q ∗ (z), both of which evolve with z. Minimization of the “action” functional, S =