3.3        Calculated FBG spectral and impulse responses

3.3.1        Homogeneous FBG examples

Homogeneous gratings are the simplest type of gratings that can be fabricated, and a good understanding of their spectral and impulse responses is very important. Two parameters can be adjusted for a given grating period : the grating length L and the refractive index modulation Dn_ac. The cases of constant L, constant Dn_ac and constant L×Dn_ac are presented hereafter.

a)   Constant length

The grating length is set to 10 mm and the refractive index modulation amplitude to the following values : 10^-6, 10^-5, 5×10^-5 and 2×10^-4. The spectral response is presented in Fig. 3-10. The reflectivity amplitude shows saturation in the stop-band for refractive index modulation above 5×10^-5 and for 2×10^-4, even the side-lobes positions are slightly moved. We observe that the reflection amplitude slopes are very close in the ripples regions for refractive index modulation under 10^-4.

Fig. 3-10 Spectral reflectivity amplitude response in dB scale (top) and time delay (bottom) for homogeneous gratings of 10 mm length and refractive index modulation amplitude of 10^-6 (dotted lines), 10^-5 (dashed-dotted lines), 5×10^-5 (dashed lines) and 2×10^-4 (solid lines)

Fig. 3-11 Impulse response amplitude (top) and phase difference with the Bragg wavelength propagation phase (bottom) for homogeneous gratings of 10 mm length and refractive index modulation amplitude of 10^-6 (solid lines), 10^-5 (dashed lines), 5×10^-5 (dashed-dotted lines) and 2×10^-4 (dotted lines)

The time delay t is defined as the derivative of the reflective amplitude phase with respect to the angular frequency w

The reflection phase (of the amplitude signal) exhibits p shifts that induce discontinuities in the delay time simulation. For this reason, the time delay is calculated from the reflection phase of the intensity signal, which is twice the time delay obtained from the amplitude signal. The p shifts in the amplitude response become 2p shifts in the intensity and disappear in the unwrapping process (and then also the discontinuities). We observe that, away from the Bragg wavelength, the time delay asymptotically tends to the same value of 48.36 ps and independently from the grating strength Dn_ac. (Fig. 3-10 bottom), corresponding to the time needed to travel back and forth in the grating.

The corresponding impulse responses are presented in Fig. 3-11. Two regions are identified. The first one is the grating zone, with the OPLD inside the grating, that is OPLD < 2n_gL, where n_g is the group refractive index. The second one is related to the region after the grating output. The impulse response in the grating region is dominated by the reflections occurring at the corresponding position in the grating. In the region after the grating output, the impulse response is given by light that has been reflected several times in the structure, as for a Fabry-Perot resonator.

At the grating entrance, all the light energy is available and the amplitude of the reflected signal is proportional to the refractive index modulation amplitude. While propagating in the grating, a part of the energy is gradually reflected for selected wavelengths and the amount of energy decreases.

For small Dn_ac, the pulse attenuation is also small and nearly constant impulse amplitude is observed (Dn_ac < 10^-5). The impulse amplitude after the grating is very small indicating negligible multiple reflections. In this case, the complex coupling coefficient is directly the complex impulse response within the grating. This approximation is known as the Fourier approximation where only the first reflection is considered [3-3].

When the refractive index modulation increases, this approximation breaks and the impulse amplitude shows a more or less important decrease in the grating region and even total amplitude annihilation for a more important Dn_ac (two times for Dn_ac = 2×10^-4). In the region after the grating, the signal amplitude is important and multiple reflections are observed.

The phase difference between the impulse phase and the phase for propagation at the Bragg wavelength is constant except at amplitude poles and at the grating output where p-shifts are observed. This can be seen at the bottom of Fig. 3-11. This effect is similar to the phase shift observed for a reflection at a mirror interface.