Nonlinear Fiber Optics - 4 ed. Agrawal

2.3. Pulse-Propagation Equation 31
Figure 2.1: Variation of mode-width parameter w with V obtained by fitting the fundamental
fiber mode to a Gaussian distribution. Traces on the right show the quality of fit for V = 2.4.
(After Ref. [8]; c○1978 OSA.)
where the width parameter w is determined by curve fitting or by following a variational
procedure. Figure 2.1 shows the dependence of w/a on the fiber parameter V defined
by Eq. (2.2.10). The comparison of the actual field distribution with the fitted Gaussian
is also shown for a specific value V = 2.4. The quality of fit is generally quite good [8],
particularly for V values in the neighborhood of 2. Figure 2.1 shows that w ≈ a for V =
2, indicating that the core radius provides a good estimate of w for telecommunication
fibers for which V ≈ 2. However, w can be significantly larger than a for V < 1.8. An
analytic approximation for w that is accurate to within 1% for 1.2 < V < 2.4 isgiven
by [8]
w/a ≈ 0.65 + 1.619V −3/2 + 2.879V −6 . (2.2.15)
This expression is of considerable practical value as it expresses the mode width in
terms of a single fiber parameter V .
2.3 Pulse-Propagation Equation
The study of most nonlinear effects in optical fibers involves the use of short pulses
with widths ranging from ∼10 ns to 10 fs. When such optical pulses propagate inside
a fiber, both dispersive and nonlinear effects influence their shapes and spectra. In
this section we derive a basic equation that governs propagation of optical pulses in
nonlinear dispersive fibers. The starting point is the wave equation (2.1.7). By using
Eqs. (2.1.8) and (2.1.17), it can be written in the form
∇ 2 E − 1 ∂ 2 E
c 2 ∂t 2 = μ ∂ 2 P L
0
∂t 2 + μ ∂ 2 P NL
0
∂t 2 , (2.3.1)
where the linear and nonlinear parts of the induced polarization are related to the electric
field E(r,t) through Eqs. (2.1.9) and (2.1.10), respectively.