3.3        Calculated FBG spectral and impulse responses

3.3.1        Homogeneous FBG examples

b)   Constant refractive index modulation amplitude

The spectral and impulse responses for a FBG at the constant refractive index modulation Dn_ac of 10^-4 have been calculated for FBG lengths of 0.1, 1, 10 and 100 mm. The impulse response amplitudes are presented in Fig. 3-12, where the OPLD is given in a logarithmic scale. The phase difference is not presented as only the p-shifts are observed. The amplitude responses for positions inside the gratings are perfectly superposed due to the fact that all gratings have the same Dn_ac. This means that the impulse response for a given OPLD is only influenced by the grating part between the grating entrance and a position inside the grating at a distance OPLD/(2n_eff) from this entrance. This is due to the causality principle that states that energy cannot be reflected at a grating position before the light has reached this region. It will be shown in the next section that this causality principle is the fundamental argument of the layer-peeling reconstruction method (§3.2). After the grating output position, the amplitude level increase with the grating length due to the fact that more photons are trapped inside the grating for several longer round-trips. For the smallest grating, the impulse response amplitude after the grating drops under -130 dB while for the 10 mm length grating, the output level is nearly as high as the grating entrance level.

Fig. 3-12 Impulse response amplitude of homogeneous FBGs with constant refractive index modulation amplitude and a length of 0.1 (solid line), 1 (dashed line), 10 (dashed-dotted line) and 100 mm (dotted line)

Fig. 3-13 Spectral response amplitude [dB] and time delay [ps] as a function of the wavelength [nm] for the four FBGs with constant Dn_ac = 10^-4 and length of 0.1, 1, 10 and 100 mm

The corresponding spectral responses are presented in Fig. 3-13. There is a wavelength range scaling factor that is inversely proportional to the grating length (100 nm at 0.1 mm, 10 at 1, 1 at 10 and 0.1 at 100). The reflection amplitude level increases with the grating length up to the saturation reflectivity of 0 dB. For the 100 mm grating, the saturation is observed even for some of the side-lobes, resulting in a much wider band-gap. The time delay behavior also shows a scaling factor, which in this case is proportional to the grating length.