Appendix C
General Formula for Pulse
Broadening
The discussion of pulse broadening in Section 2.4 assumes the Gaussian-shape pulses
and includes dispersive effects only up to the third order. In this appendix, a general
formula is derived that can be used for pulses of arbitrary shape. Moreover, it makes
no assumption about the dispersive properties of the fiber and can be used to include
dispersion to any order. The basic idea behind the derivation consists of the observation
that the pulse spectrum does not change in a linear dispersive medium irrespective of
what happens to the pulse shape. It is thus better to calculate the changes in the pulse
width in the spectral domain.
For pulses of arbitrary shapes, a measure of the pulse width is provided by the
quantityσ^2 =〈t^2 〉−〈t〉^2 , where the first and second moments are calculated using the
pulse shape as indicated in Eq. (2.4.21). These moments can also be defined in terms
of the pulse spectrum as
〈t〉=∫∞−∞t|A(z,t)|^2 dt≡−i
2 π∫∞−∞A ̃∗(z,ω)A ̃ω(z,ω)dω, (C.1)〈t^2 〉=∫∞−∞t^2 |A(z,t)|^2 dt≡1
2 π∫∞−∞|A ̃ω(z,ω)|^2 dω, (C.2)whereA ̃(z,ω)is the Fourier transform ofA(z,t)and the subscriptωdenotes partial
derivative with respect toω. For simplicity of discussion, we normalizeAandA ̃such
that ∫∞
−∞|A(z,t)|^2 dt=1
2 π∫∞−∞|A ̃(z,ω)|^2 dω= 1. (C.3)As discussed in Section 2.4, when nonlinear effects are negligible, different spectral
components propagate inside the fiber according to the simple relation
A ̃(z,ω)=A ̃( 0 ,ω)exp(iβz)=[S(ω)eiθ]exp(iβz), (C.4)whereS(ω)represents the spectrum of the input pulse andθ(ω)accounts for the ef-
fects of input chirp. As seen in Eq. (2.4.13), the spectrum of chirped pulses acquires