4.1         Methods for measuring the complex impulse response of a grating

It has been shown in chapter 3 that the complex coupling coefficient q(z) of a grating can be reconstructed from the grating complex impulse response in reflection h(t) (z is the position along the grating and t the time). The representation of a FBG in the time domain is not as usual as in the frequency domain. For such reason, we present here in Fig. 4-1 the spectral and the impulse responses of a homogeneous grating. The grating is 5 mm long with a refractive index modulation amplitude of 2×10^-4, an effective refractive index of 1.45 and a period corresponding to a Bragg wavelength of 1.3 mm (Fig. 4-1a). The reflection intensity |r|², where r is the reflection amplitude, and the time delay of the grating are shown in Fig. 4-1b. The impulse response amplitude |h(t)| (Fig. 4-1c) shows that the interesting time range is about 200 ps. Fig. 4-1d reports the real part Re(h(t)) of the FBG impulse respones. The period of Re(h) is 4.34 fs, which corresponds to a phase change of 2p for h(t). The impulse response can also be viewed in a distance scale x, corresponding to the travel distance in free-space for the given impulse time (x = c₀t where c₀ is the vacuum light speed). The distance for 200 ps is 6 cm and the phase period is the 1.3 mm of the Bragg wavelength.

Fig. 4-1 Parameters of a homogeneous FBG (a); Spectral reflection intensity and time delay (b); Impulse response amplitude (c) and real part (d)

There are three main ways to experimentally obtain h(t) :

-         Direct method : launch a light pulse at the entrance of the grating and collect the time response h(t) of the reflected light

-         OLCR (optical low coherence reflectometry) method : measure the interference signal between a low coherence light reflected by the FBG with a delayed part of the same source light

-         Spectral method : measure the complex reflection spectral response and then perform a Fourier transform

The direct method is very difficult to realize due to the very small time scales (200 ps duration with periodic modulation of 4 fs for the grating presented in Fig. 4-1). The light pulse, ideally a Dirac function of time, would need to be as short as a few femtoseconds. The detection system also would need to be extremely fast to measure the electric field variations. These constraining technical requirements explain why such FBG characterization method has not been used yet.

The OLCR method uses the property inherent to a broadband light source to interfere only with a very small delayed version of itself, corresponding to a travel distance of a few micrometers, which is defined as the coherence length [4-4]. The resulting interference signal corresponds to the impulse response h(t) smoothed over a few micrometers due to a convolution with the source coherence function. A detailed description of the OLCR method is presented in the following section 4.2.

The spectral method, consisting in the measurement of the FBG reflection amplitude and phase, has been published recently, but several drawbacks can be identified. Actually, the precise measurement of the spectral response phase is difficult and slow [4-2]. The measurements errors have important effects on the calculated Fourier transform that limits the h(t) dynamic range and introduce a large amount of noise in the reconstructed coupling coefficient (§3.4.5).