Nonlinear Fiber Optics - 4 ed. Agrawal

104 Chapter 4. Self-Phase Modulation
∫ L dz, requires that Ld satisfy the Euler–Lagrange equation
( )
∂ ∂Ld
+ ∂ ( ) ∂Ld
− ∂L d
= 0, (4.3.11)
∂T ∂q t ∂z ∂q z ∂q
where q t and q z denote the derivative of q with respect to T and z, respectively.
The variational method makes use of the fact that the NLS equation (4.3.1) can be
derived from the Lagrangian density
L d = i (
U ∗ ∂U )
∂U
∗
−U + β 2
∂U
2
2 ∂z ∂z 2 ∣ ∂T ∣ + 1 2 γP 0e −αz |U| 4 , (4.3.12)
with U ∗ acting as the generalized coordinate q in Eq. (4.3.11). If we assume that the
pulse shape is known in advance in terms of a few parameters, the time integration
in Eq. (4.3.10) can be performed analytically to obtain the Lagrangian L in terms of
these pulse parameters. In the case of a chirped Gaussian pulse of the form given in
Eq. (4.3.6), we obtain
L = β 2E p
4Tp
2 (1 +C 2 p)+ γe−αz Ep
2 √ + E (
p dCp
8πTp 4 dz − 2C p
T p
)
dT p dφ p
− E p
dz dz , (4.3.13)
where E p = √ πa 2 pT p is the pulse energy.
The final step is to minimize the action S = ∫ L (z)dz with respect to the four
pulse parameters. This step results in the reduced Euler–Lagrange equation
d
dz
( ) ∂L
− ∂L
∂q z ∂q
= 0, (4.3.14)
where q z = dq/dz and q represents one of the pulse parameters. If we use q = φ p in
Eq. (4.3.14), we obtain dE p /dz = 0. This equation indicates that the energy E p remains
constant, as expected. Using q = E p in Eq. (4.3.14), we obtain the following equation
for the phase φ p :
dφ p
dz = β 2
2T 2 p
+ 5γe−αz E p
4 √ 2πT p
. (4.3.15)
We can follow the same procedure to obtain equations for T p and C p . In fact, using
q = C p and T p in Eq. (4.3.14), we find that pulse width and chirp satisfy the same two
equations, namely Eqs. (4.3.7) and (4.3.8), obtained earlier with the moment method.
Thus, the two approximate methods lead to identical results in the case of the NLS
equation.
4.3.3 Specific Analytic Solutions
As a simple application of the moment or variational method, consider first the case of
a low-energy pulse propagating in a constant-dispersion fiber with negligible nonlinear
effects. Recalling that (1 +C 2 p)/T 2 p is related to the spectral width of the pulse that
does not change in a linear medium, we can replace this quantity with its initial value