Page 236 - Fiber Bragg Gratings

5.2 Basic principles and methodology 213

From Eq. (5.2.6) it follows that the modulation index goes to zero for
the argument] = TT radians, so that

Equation (5.2.7) shows the obvious result that if the width of the
beam is equal to the period, then ly = v sc to wash out the refractive index
modulation. In general, the beam width W > 100 X A g, so that the velocity
of the fiber is <\.% of the scanning velocity.
The relative movement of the fiber with respect to the scanning beam
changes the Bragg wavelength of the grating, which is easily calculated
as

Combining Eqs. (5.2.7) and (5.2.8) directly leads to the chirp as a
function of the width of the UV beam,

Therefore, the maximum wavelength shift is inversely proportional
to the width of the beam. This condition is similar to the one encountered
in the MPF technique: The maximum is equivalent to the fiber moving
one period during the time it takes the UV beam to move a distance equal
to its width at the scanning velocity. Using Eq. (5.3.9) in Eq. (5.3.7),
and recalling the relationship between the grating period and the Bragg
wavelength, leads to

where

is the equivalent of "phase detuning" between the Bragg wavelength and
the grating that is being written over the width of the beam. Note here
that at constant fiber velocity (and scanning beam), the wavelength shifts;
if the fiber velocity changes during the scan, the result is a chirped grating.