Page 454 - Fiber Bragg Gratings

9.3 Phase and temporal response of Bragg gratings 431

shown in Fig. 9.23. As has been seen in Chapter 3 on fabrication of gratings
with a phase mask, the incident light must have a wavelength less than
the period of the grating in the fiber in order to have a first-order diffrac-
tion. Referring to Fig. 9.23, which defines the angles, and from phase
matching conditions, we find [31]

where n eff is the effective index of the fiber at the probe wavelength, so
that the probe wavelength must be greater than the Bragg wavelength
of the grating by a factor of the effective index of the fiber. The input light
is reflected at the incident angle, 6 t. In the weak scattering limit, the
cross-section a- t (as fraction of the incident peak-power density) is given by
the following expression, assuming that the grating has a pure sinusoidal
period [31], and the focused spot size w t in the core is much greater than
the core radius a:

Here Arc is the local refractive index modulation amplitude, k is the
wave vector at the incident wavelength A probe, and y core is the angle be-
tween the reflected and incident beams and ignores reflection losses.
Owing to the geometry of the scattering region, y core is polarization sensitive
and s-polarized light; the reflected power is maximum at y core = 77/2.
Typical parameters used for the experiment are a beam waist of 5
/mm, using a 10-mm focal length focused ~37 /mi in front of the surface
of the fiber to give a spot size of approximately 10 /um at the core (after
focusing from the core-cladding surface) with an incident angle of 45.3°.
The equivalent internal angle 6 core = 29.4°. The input power needs to be
high for a good signal-to-noise ratio, and was reported in the experiments
to be 5 mW. The resolution in this arrangement is limited to the spot size
of 10 /tni.
The fiber grating is scanned in front of the fixed laser beam, so that
the data may be recorded as a function of position along the grating. Good
correlation between the measurement and the simulated transmission
spectrum of the grating has been reported [31,35]. The side-diffraction
profile of a Gaussian apodized grating is shown in Fig. 9.24. From this
profile, it is simple to simulate the grating transfer function to establish
a correlation.