Page 207 - Fiber Bragg Gratings

184 Chapter 4 Theory of Fiber Bragg Gratings

where T ps is the new phase-shift matrix for a reflection grating,

For codirectional coupling we replace the phase-shift matrix (4.8.23)
by its conjugate. The phase factor, 0/2, is any arbitrary phase and may
be as a result of a change in the n eff or a discontinuity within the grating.
The phase change could have arisen from a region in which the grating
was not present or from exposure to uniform UV radiation, thereby chang-
ing the local n eff. In either case this phase

where dn is the local phase change over a length AL. The calculation of
the reflectivity and transmissivity proceeds in the same manner as with-
out the phase step.

General conditions and restrictions for the T-matrix method
The transfer matrix method requires that certain conditions be met for
accurate simulation of grating response [10]. First, when the grating
parameters are a function of z, the minimum length of the section 81 j ^
Aj X K, where K is a suitably large number. The actual factor K is a
consequence of the slowly varying approximation. The magnitude of K
depends on chirp rate dkAJdL where AA is the total chirp bandwidth in
the grating. This value determines the upper limit to the number of
sections allowed in any simulated grating. Figure 4.33 shows the effect
on the reflectivity of reducing the number of sections from 37 periods to
one period per section of a 2-mm-long grating with a chirp of 6 nm. The
3
refractive index modulation is 2 X 10~ .
Second, care must be taken to ensure that each section j has an
integer number of grating periods in order to have a smooth transition
between sections. An abrupt change in the grating modulation is equiva-
lent to a phase jump and hence the formation of a Fabry-Perot cavity, as
has been explained in Section 3.1.13. A consequence of not maintaining
this condition over several sections is that it can lead to a deleterious
effect outside of the bandwidth of the grating, by forming a superstructure
of cascaded Fabry-Perot cavities.