2.4. DISPERSION-INDUCED LIMITATIONS 49
Figure 2.12: Variation of broadening factor with propagated distance for a chirped Gaussian
input pulse. Dashed curve corresponds to the case of an unchirped Gaussian pulse. Forβ 2 < 0
the same curves are obtained if the sign of the chirp parameterCis reversed.
Equation (2.4.17) can be generalized to include higher-order dispersion governed
byβ 3 in Eq. (2.4.15). The integral can still be performed in closed form in terms of
an Airy function [57]. However, the pulse no longer remains Gaussian on propagation
and develops a tail with an oscillatory structure. Such pulses cannot be properly char-
acterized by their FWHM. A proper measure of the pulse width is the RMS width of
the pulse defined as
σ=[
〈t^2 〉−〈t〉^2] 1 / 2
, (2.4.20)
where the angle brackets denote averaging with respect to the intensity profile, i.e.,
〈tm〉=∫∞
−∞tm|A(z,t)| (^2) dt
∫∞
−∞|A(z,t)|
(^2) dt. (2.4.21)
The broadening factor defined asσ/σ 0 , whereσ 0 is the RMS width of the input Gaus-
sian pulse (σ 0 =T 0 /
√
- can be calculated following the analysis of Appendix C and
is given by [56]
σ^2
σ 02=
(
1 +
Cβ 2 L
2 σ 02) 2
+
(
β 2 L
2 σ 02) 2
+( 1 +C^2 )^2
(
β 3 L
4√
2 σ 03) 2
, (2.4.22)
whereLis the fiber length.
The foregoing discussion assumes that the optical source used to produce the in-
put pulses is nearly monochromatic such that its spectral width satisfies∆ωL ∆ω 0
(under continuous-wave, or CW, operation), where∆ω 0 is given by Eq. (2.4.14). This