Page 176 - Fiber Bragg Gratings

4.6 Properties of uniform Bragg gratings 153

Thus, a uniform period and index-modulation grating ("top hat") will
produce side lobes. However, for stronger gratings, a Fourier transform
for the fundamental component of the refractive index modulation alone
is no longer appropriate and gives rise to increasing errors.
For the uniform grating, dtydz — 0, the peak reflectivity occurs at a
wavelength at which 8= 0 [and therefore, a = K ac], and Eq. (4.3.16) leads
to

At the phase matching wavelength, the reflectivity reduces to

For identical forward- and counterpropagating modes, it is simple to
show by using the orthogonality relationship of Eq. (4.1.15) in Eqs. (4.3.5)
and (4.3.6) that

where the overlap integral 77 «* 1 for identical modes, and it therefore
follows that the peak of the Bragg reflection is at

The Bragg wavelength \ B is defined at the phase-matching point A/3
= 0 for the general case of dissimilar modes,

with the result

For identical forward- and counterpropagating modes or nearly iden-
tical mode indexes, Eq. (4.6.6) reduces to