Nonlinear Fiber Optics - 4 ed. Agrawal

52 Chapter 3. Group-Velocity Dispersion
two length scales, known as the dispersion length L D and the nonlinear length L NL [1]–
[3]. Depending on the relative magnitudes of L D , L NL , and the fiber length L, pulses
can evolve quite differently.
Let us introduce a time scale normalized to the input pulse width T 0 as
τ = T T 0
= t − z/v g
T 0
. (3.1.2)
At the same time, we introduce a normalized amplitude U as
A(z,τ)= √ P 0 exp(−αz/2)U(z,τ), (3.1.3)
where P 0 is the peak power of the incident pulse. The exponential factor in Eq. (3.1.3)
accounts for fiber losses. By using Eqs. (3.1.1)–(3.1.3), U(z,τ) is found to satisfy
i ∂U
∂z = sgn(β 2) ∂ 2 U
2L D ∂τ 2 − exp(−αz) |U| 2 U, (3.1.4)
L NL
where sgn(β 2 )=±1 depending on the sign of the GVD parameter β 2 and
L D = T 0
2
|β 2 | , L NL = 1 . (3.1.5)
γP 0
The dispersion length L D and the nonlinear length L NL provide the length scales over
which dispersive or nonlinear effects become important for pulse evolution. Depending
on the relative magnitudes of L, L D , and L NL , the propagation behavior can be classified
in the following four categories.
When fiber length L is such that L ≪ L NL and L ≪ L D , neither dispersive nor
nonlinear effects play a significant role during pulse propagation. This can be seen by
noting that both terms on the right-hand side of Eq. (3.1.4) can be neglected in that case.
(It is assumed that the pulse has a smooth temporal profile so that ∂ 2 U/∂τ 2 ∼ 1.) As
a result, U(z,τ)=U(0,τ), i.e., the pulse maintains its shape during propagation. The
fiber plays a passive role in this regime and acts as a mere transporter of optical pulses
(except for reducing the pulse energy because of fiber losses). This regime is useful for
optical communication systems. For L ∼ 50 km, L D and L NL should be larger than 500
km for distortion-free transmission. One can estimate T 0 and P 0 from Eq. (3.1.5) for
given values of the fiber parameters β 2 and γ. Atλ = 1.55 μm, |β 2 |≈20 ps 2 /km, and
γ ≈ 2W −1 km −1 for standard telecommunication fibers. The use of these values in Eq.
(3.1.5) shows that the dispersive and nonlinear effects are negligible for L < 50 km if
T 0 > 100 ps and P 0 < 1 mW. However, L D and L NL become smaller as pulses become
shorter and more intense. For example, L D and L NL are ∼0.1 km for T 0 ∼ 1psand
P 0 ∼ 1 W. For such optical pulses, both the dispersive and nonlinear effects need to be
included if fiber length exceeds 10 meters.
When the fiber length is such that L ≪ L NL but L ∼ L D , the last term in Eq. (3.1.4)
is negligible compared to the other two. The pulse evolution is then governed by GVD,
and the nonlinear effects play a relatively minor role. The effect of GVD on propagation
of optical pulses is discussed in this chapter. The dispersion-dominant regime is