4.2        OLCR measurement of the complex impulse response

4.2.1        Overview

This section presents the main aspects of an OLCR and its application to the characterization of the FBG impulse response h(t). The OLCR technique is based on a scanning Michelson or Mach-Zender interferometer coupled with a broadband light source. Fig. 4-2a presents a simplified all-fiber Michelson type OLCR set-up for FBG characterization. The 3 dB-coupler (X) splits the low coherent source light (L) in two components that propagate in the so-called reference and test arms. Partial or total reflections occur on the moveable mirror (M) and inside the FBG (FBG). Half the reflected light from the reference and test arms is sent back to the detector (D) through the coupler. The position P_t in the test arm is located at the FBG entrance. The optical path length in the reference arm between the coupler and the position P_r corresponds to the optical path length in the test arm between the coupler and the position P_t. The physical distances from coupler to P_r and P_t are different due to the free-space part of the reference arm (i.e. a distance d in vacuum corresponds to a distance d/n_g in the fiber where n_g is the group index of the fiber). The optical path length matching implies that the input electrical fields E₀ at P_r and P_t have the same phase. As a consequence, the reflected fields at the detector are a delayed version of E_r and E_t at P_r and P_t, respectively, with the same delay time and with half the intensity due to the coupler ( in amplitude).

Fig. 4-2 Basic OLCR set-up for FBG characterization (a) and main interfering region in the FBG for a mirror position z

The measured optical intensity by the detector, I(z), can then be expressed as

where z is the mirror position (z = 0 coincides with P_r). The first term corresponds to the sum of the intensities of each signal and the second one is the interfering contribution. For a mirror position z, the interference signal can be seen as the superposition of the mirror reflected light and the reflection of the small FBG part located at z/n_g of width L_c (Fig. 4-2b) where L_c is the light coherence length (a few micrometers). This intuitive description does not take into account the input light attenuation along the grating and multiple reflections. We will show in the following sections that the measured OLCR signal corresponds to the interference intensity

where I_s is the light source intensity, g(t) the complex degree of coherence of the light source, h(t) the impulse response of the FBG, c₀ the light speed in vacuum and t = 2z/c₀. The impulse response h(t), in reflection, is defined at the FBG entrance, P_t, and corresponds to the Fourier transform of the complex reflection amplitude r(n). We will show that g(t) is the normalized Fourier transform of the light source spectral power density (equation (4-9)).

In the following subsections, the formal frame to describe the temporal coherence effects is introduced. The complete description of the OLCR signals of FBGs will be derived in the case of fibers with or without dispersion.