2.3        FBG properties

2.3.2       FBG types

a)   Homogeneous FBG

Homogeneous gratings are characterized by a rectangular function for the modulation amplitude envelope Dn_ac and the index offset Dn_dc with L kept constant (Fig. 2-6 left). The strong index step at the input and output of the grating induces important reflections bands, called side-lobes, outside the main Bragg peak. This effect can be understood by considering the grating edge as a Fabry-Perot structure.

Fig. 2-6 FBG index profile : homogeneous (left), apodized (right)

The spectral reflectivity r(l) = |r(l)|×exp(i×f(l)) of such a grating has been calculated by the T-Matrix method that will be extensively explained in section 3.1.4. The results are presented in Fig. 2-7. The parameters used for the simulations are : an effective refractive index of 1.45, a maximal refractive index modulation of 2×10^-4, a grating period corresponding to a Bragg condition of l_b = 1300 nm and a grating length of 5 mm (a fringe visibility of 1 is assumed). The homogeneous FBG is calculated in one layer. We observe in Fig. 2-7 that in the reflection amplitude |r(l)|, the side-lobes are equally placed at both sides of the main Bragg peak resonance. The maximal reflectivity is 97 % and the first side-lobes show a reflectivity of 22 %. The delay time df(l)/dw (where w is the angular frequency) tends asymptotically to a value of 24 ps for |l-l_b|>>0. For strong gratings, part of the light is coupled in the cladding modes, inducing excess losses. This effect cannot be studied with reflection spectra but looking at the transmission light.

Fig. 2-7 FBG reflection intensity in linear scale (top), in dB (middle) and time delay (bottom) for an homogeneous FBG (solid line), an apodized FBG (dashed line) and a period chirped FBG (dashed-dotted line)

The amplitude and power reflection coefficients r(n) and R(n) = |r(n)|², respectively, are given by [2-16, 2-17]

where g² = |q|² - d², d = b - p/L, |q| = k = h×p×Dn_ac/l, Arg(q) = p/2, b = 2pn_eff/l and n_eff = n₀ + Dn_dc.

The maximal reflectivity R_max is given by

The grating bandwidth Dl_BW, defined as the wavelength range between the first zeros apart from the Bragg peak is given by

Depending on the grating parameters, the Bragg reflector can operate as a narrow-band or a broadband filter or mirror.

b)   Apodized FBG

A variation along the fiber in the envelope of the refractive index modulation amplitude, Dn_ac, is called apodization. The period L and the DC refractive index function Dn_dc are considered constant. Since the apodization can prevent any discontinuities in the Dn_ac profile (Fig. 2-6 right), the Fabry-Perot effect observed for rectangular gratings is greatly reduced. This can be observed in Fig. 2-7 where the side-lobes are suppressed by about 30 dB. The simulation has been performed by considering a 100-layered grating and a Hann apodization function. The other parameters are the same used for the homogeneous grating. The reflectivity at the resonance is reduced to 68 % due to the refractive index envelope. The delay time range is reduced by a factor 4.5.