Page 177 - Fiber Bragg Gratings

154 Chapter 4 Theory of Fiber Bragg Gratings

The reason why the reflection peak is at a longer wavelength than
the Bragg wavelength is because the average refractive mode index Arc
continuously increases with a positive refractive index modulation.
For nonidentical modes, the integral in Eq. (4.3.5) has to be integrated
numerically. However, the integral is simply a weighting factor, 0 < 77 < 1,
dependent on the mode and refractive index profiles. It is for this reason
that 77 has been introduced in Eq. (4.6.3), normalized to unity for identical
modes.
There are several definitions of bandwidth. However, the most easily
identifiable one is bandwidth between the first minima on either side of
the main reflection peak (with reference to Fig. 4.11). This may be calcu-
lated by equating the argument oL in Eq. (4.3.11) to TT,

Therefore,

which, after rearranging, becomes

From Eq. (4.3.12), assuming K dc = 0 and d<$dz = 0 (no chirp in the
grating), we get

so that the detuning from the peak to the first zero is

For identical modes, /u, = v, using Eq. (4.3.4) we get,

where the bandwidth from the peak to the first zero is A A. Combining
Eqs. (4.16.12) and (4.6.13), and noting that the bandwidth between the