# Fiber Bragg Grating-Based Optical Signal Processing: Review and Survey
## 2 Review of Design Methods for Fiber Bragg Gratings
### Synthesis of Fiber Bragg Gratings from a Targeted Reflection Specifications
#### 2.3.1 CMT-Based Synthesis Method  
The synthesis method based on CMT delivers the local coupling coefficient \(\kappa(z)\) from the specification of the target reflection spectral response \(H_{R}(f)\). As previously seen, the resulting \(\kappa(z)\) provides the needed information to write the grating structure on the fiber, i.e., the refractive index envelope \(\Delta n(z)\) and the period variation \(\Lambda(z)\) (see Equation (9)). This method has a straightforward description, since it makes use of the direct solution of exactly the same coupled-mode equations of the grating analysis [35; 36; 37].  
Once again, the propagation problem has to be discretized. Hence, the entire grating is virtually divided in a series of \(N\) sections of length \(\delta L\), which shall be modeled by uniform FBGs. \(N\) is selected so that each section contains several periods, \(\delta L\gg\Lambda_{0}\) (recall that the grating period for reference, \(\Lambda_{0}\), can be obtained from the central frequency of the target spectral response and the Bragg condition). The transfer matrix of a uniform Bragg grating  
Figure 3: Representation of the division of a non-uniform FBG into layers suitable for the analysis using ML-TMM. The dependence of waves with the optical frequency is omitted for simplicity.  
is \(M_{U,i}\), which is given by Equations (11)-(14). The fields before the first section \((z_{1}=0)\) are given by [36]:  
\[\left[\begin{array}{c}E^{+}(z=0,f)\\ E^{-}(z=0,f)\end{array}\right]=\left[\begin{array}{c}1\\ H_{\rm R}(f)\end{array}\right] \tag{21}\]  
From the matrix \(M_{U,i}\), it is possible to define a discrete, complex reflection coefficient associate to each segment, which is related to the coupling coefficient as [37]  
\[\rho=-\tanh(|\kappa|L)\frac{\kappa^{*}}{\kappa} \tag{22}\]  
From Fourier analysis, we know that the reflection coefficient of the first layer is  
\[\rho_{1}=h_{R}(t=0)=\frac{1}{N}\int_{0}^{\infty}H_{\rm R}(f)df \tag{23}\]  
The reason is that the impulse response for \(t=0\) coincides with the case where only the first reflector is present. The coupling coefficient of the first layer \(\kappa_{1}\) is obtained from Equation (23). At this point, we can generate the first matrix \(M_{U,1}\), which provides information of the grating structure from \(z=0\) to \(z=\delta L\). The spectral response yielded from the remaining grating is obtained by substituting Equation (21) into Equation (10), and calculating [36]  
\[H_{R}^{zL}(f)=\frac{E^{-}(z=\delta L,f)}{E^{+}(z=\delta L,f)}=\frac{-M_{U,1,2 1}+M_{U,1,11}\cdot H_{R}(f)}{M_{U,1,22}-M_{U,1,12}\cdot H_{R}(f)} \tag{24}\]  
Next, the reflection coefficient of the next layer is obtained following the same reasoning as in Equation (23), using the spectral response of the remaining grating structure \(H_{R}^{zL}(f)\),  
\[\rho_{2}=\frac{1}{N}\int_{0}^{\infty}H_{R}^{zL}(f)df \tag{25}\]  
These steps are employed successively until the entire grating structure is determined. This algorithm embodies a computationally efficient layer-peeling algorithm for the design of the broad majority of fiber gratings of practical interest, including long gratings (e.g., tens of cm).