3.2        FBG synthesis and reconstruction

3.2.2       Discrete Layer-peeling

The layer-peeling method has an origin in the geologic field. It has been developed to retrieve the ground properties from the seismic impulse response measured when a explosive charge is activated. The method is based on the propagation description of the fields through a discrete structure with simultaneous retrieval of the material impedance based only on causal arguments. The rigorous mathematical description of the method has been achieved for one-dimensional systems [3-17, 3-18] and its application in various other fields has been found. Application for the synthesis of FBG has been proposed by Faced et al. [3-19] and a simpler formulation has been proposed by Skaar et al. [3-5], where a continuous version of the layer-peeling is also presented but with no advantage over the discrete method. The method is briefly explained hereafter. It is based on the formulation of Skaar et al. and a modified version is also proposed for FBGs where some losses are observed during the propagation (for example tilted gratings).

The layer-peeling method is a backscattering method based on the complex impulse response of an unknown structure (Fig. 3-4).

Fig. 3-4 Backscattering problem

The structure is divided in N layers of a physical thickness D. For a given impulse time t, only the part of the grating that has been illuminated during the time interval of t/2 can contribute to the impulse response (causality principle). This is represented by the white layers in Fig. 3-5.

Fig. 3-5 Layers and causality principle

The causality principle imposes that all reflections in a layer occur at a single point (Fig. 3-6).

Fig. 3-6 Single point reflection approximation

It was assumed in the causal T-matrix that for a small enough layer, the FBG can be represented by a single, localized and complex reflector, as defined in equation (3-10).

The complex reflection amplitude of the grating r₁(n) is given by the Fourier transform of the impulse response h₁(t) (and vice-versa). For t = 0, only the first layer contribute to the impulse response and the complex reflector r₁ is described by the impulse response for t = 0, h₁(0), as seen in Fig. 3-7.

Fig. 3-7 First layer case

The calculation of r₁ from the discrete form of the spectral response is given by the discrete Fourier transform of r₁(d_m) for t = 0 for which the exponential factor is canceled and then r₁ is given by

where the number of spectral points M must be greater than the number of layers N and where the detuning range |d| is determined from the layer thickness :

From the complex reflectivity r₁ and the complex reflector r₁, it is possible to use the equation (3-13) to calculate the reflectivity r₂ = v₂/u₂ for the FBG without the first layer (peeled-off), which is represented in Fig. 3-8 by the gray layers (u and v are the forward and backward propagating modes).

Fig. 3-8 Field propagation through the first layer

Only considering the remaining grating constituted of the layers 2 to N, we observe that its spectral response is known, namely r₂. Then the same calculation process can be performed to retrieve the complex reflector value r₂ of the second layer and the reflection response r₃ of the grating constituted of layers 3 to N (Fig. 3-9). The whole grating complex reflector r_j are thus recursively reconstructed.

Fig. 3-9 Second layer reconstruction

The layer-peeling reconstruction algorithm is the counterpart of the causal T-matrix method (§3.1.5).

In summary, from the starting reflection amplitude r₁(d)=r(d), the grating is reconstructed in an iterative way. At each step, r_j is calculated for the first layer of the remaining structure at the step j and a new reflection amplitude r_j+1(d) is calculated for the structure without the layer j (peeled off) :

where r_j(m) is the discrete form of r_j(d) for , and is the wavenumber detuning, the light wavenumber and the Bragg design wavenumber.