Page 342 - Fiber Bragg Gratings

7.2 Chirped and step-chirped gratings 319

from Eq. (4.6.14), and we have assumed that each section is identical in
length 81. For most of the gratings of interest here, we assume that (K$) 2
2
<^ 7T . The phase matching condition for the section requires that

where A g is the period of the grating section. The period is nearly constant
for gratings with a small percentage chirp. Remembering that Si = L g/N,
we get

When N = 1, the bandwidth of the grating is simply the bandwidth A A'
of the unchirped grating of length L g. For the chirped grating with a
bandwidth >AA', made of sections, the bandwidth of each section can
only be greater than the bandwidth of the unchirped grating (being shorter
in length), but can equal the bandwidth of the chirped grating only if
it is the appropriate length. Applying the relationship [Eq. (7.2.3)] for
bandwidths greater than the unchirped bandwidth, A A', we simply allow
the bandwidth of each section to be identical to the bandwidth AA chirp of
the chirped grating, i.e.,

so that for a fiber Bragg grating at a wavelength of 1550 nm, N/L g =
QAkA chirp steps/(mm-nm). Finally, we arrive at the relationship between
the number of steps per unit length and the chirped bandwidth,

Here A Bragg. is the central Bragg wavelength of the chirped grating. The
simple relationships of Eqs. (7.2.4) and (7.2.5) are minimum requirements
for the step chirped grating and should approximate to a continuously
chirped grating. It may be seen immediately that there is an intuitive
feel about the conclusion — that the bandwidth of each step of the grating
should be at least as large as the chirp of the whole grating. Increasing
the number of steps, i.e., $ —> 0, approaches the continuously chirped