Page 338 - Fiber Bragg Gratings
P. 338
7.1 General characteristics of chirped gratings 315
Transmission in a linearly dispersive system broadens the pulse, but
does not change its shape, and is simply related to the input pulse width
by [37]
which is the well-known formula for linear pulse broadening, remember-
ing that the dispersion length is a function of the spectral width of the
pulse and the linear group velocity dispersion (GVD) of the fiber, /3 2, as
Combining Eqs. (7.1.6) and (7.1.7) leads to
Since the effect of dispersion in a linear system is additive, we may
now modify the GVD parameter /3 2 by including the dispersion of the
grating for the bandwidth of the pulse so that the pulse broadening is
where /3 g is the GVD of the grating of length L g. We note that the dispersion
D f of the fiber is related to the GVD by [37]
A similar expression relates the fiber grating dispersion D g to the
grating GVD, but with the sign depending on the sign of the grating chirp.
We now find that the pulse broadening for the Gaussian pulse is
In Eq. (7.1.11) we have used the relationship between a transform-
limited pulse width and its spectrum (&yT 0 = 1).
