Nonlinear Fiber Optics - 4 ed. Agrawal

102 Chapter 4. Self-Phase Modulation
locked Yb-doped fiber lasers can also emit parabolic-shape pulses under suitable conditions
[70]. The self-similar nature of parabolic pulses is helpful for generating highenergy
pulses from lasers and amplifiers based on Yb-doped fibers [73]–[76].
4.3 Semianalytic Techniques
The results of the preceding section are based on the numerical solutions of the NLS
equation with the split-step Fourier method of Section 2.4.1. Although a numerical
solution is necessary for accuracy, considerable physical insight is gained if the NLS
equation can be solved approximately in a semianalytic fashion. In this section we employ
two such techniques for solving the NLS equation (2.3.45). Using A = √ P 0 e −αz U,
this equation can be written in the form
4.3.1 Moment Method
i ∂U
∂z − β 2 ∂ 2 A
2 ∂T 2 + γP 0e −αz |A| 2 A = 0. (4.3.1)
The moment method was used as early as 1971 in the context of nonlinear optics [77].
It can be used to solve Eq. (4.3.1) approximately, provided one can assume that the
pulse maintains a specific shape as it propagates down a fiber even though its amplitude,
width, and chirp change in a continuous fashion [78]–[80]. This assumption may
hold in some limiting cases. For example, it was seen in Section 3.2 that a Gaussian
pulse maintains its shape in a linear dispersive medium even though its amplitude,
width, and chirp change during propagation. If the nonlinear effects are relatively weak
(L NL ≫ L D ), a Gaussian shape may remain approximately valid. Similarly, it was seen
in Section 4.1 that a pulse maintains its shape even when nonlinear effects are strong
provided dispersive effect are negligible (L NL ≪ L D ). It will be seen in Chapter 5 in the
context of solitons that, even when L NL and L D are comparable, a pulse may maintain
its shape under certain conditions.
The basic idea behind the moment method is to treat the optical pulse like a particle
whose energy E p , RMS width σ p , and chirp C p are related to U(z,T ) as
E p =
C p =
∫ ∞
−∞
|U| 2 dT,
i ∫ ∞
T
E p −∞
σp 2 = 1 ∫ ∞
T 2 |U| 2 dT, (4.3.2)
E p
(
U ∗ ∂U ∂U
∗
−U
∂T ∂T
−∞
)
dT. (4.3.3)
As the pulse propagates inside the fiber, these three moments change. To find how they
evolve with z, we differentiate Eqs. (4.3.2) and (4.3.3) with respect to z and use Eq.
(4.3.1). After some algebra, we find that dE p /dz = 0butσp 2 and C p satisfy
dσp
2
dz = β ∫ ∞
(
2
T 2 Im U ∗ ∂ 2 )
U
E p −∞ ∂T 2 dT, (4.3.4)
dC p
dz = 2β ∫ ∞
2
∂U
2 E p
∣ ∂T ∣ dT + γP 0 e −αz 1 ∫ ∞
|U| 4 dT. (4.3.5)
E p
−∞
−∞