# 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
#### 2.3.2 ML-Based Synthesis Method  
The ML-TMM-based synthesis algorithm offers the advantages of generality and accuracy over the previously described CMT-based method. This method delivers the required refractive index profile, \(n_{FBG}(z)\), (instead of the coupling coefficient) from the specified response in reflection. Its high sampling rate makes this method a practical tool to synthesize gratings with discontinuities or local defects of the order of the local period, or even to detect defects in a fabricated device. In particular, the spatial resolution of the recovered local refractive index \(n_{FBG}(z)\) is below the local period (\(\delta l\ll\Lambda\)) [29; 30]. This minimum discretization step impedes working with an equivalent low-pass spectral response, forcing us to work with a spectral response \(H_{R}(f)\) defined from \(f=0\) to \(f=c/\left(4n_{eff}\delta l\right)\). The maximum value of the optical frequency is given by the sampling rate of the impulse response of the grating in reflection, \(\Delta t\), which is equal to half the round-trip delay in a layer of length \(\delta l\), \(\Delta t=n_{eff}\delta l/c\) (see Figure 4a). The sampled reflection impulse response, \(h_{RS}(p)\) (targeted reflection specifications) is obtained as:  
\[h_{R}(t)=\Im^{-1}\{H_{R}(f)\}=\sum_{p=0}^{\infty}h_{RS}(p)\cdot\delta(t-2p \Delta t) \tag{26}\]The steps to obtain the required grating's refractive index modulation are the following [35]. Let us assume \(n_{0}\) as the refractive index of the unperturbed waveguide. The value of \(n_{1}\) is calculated from the first sample of the impulse response \(h_{RS}(0)\):  
\[n_{1}=n_{0}\bigg{(}\frac{1-h_{RS}(0)}{1+h_{RS}(0)}\bigg{)} \tag{27}\]  
Observing Figure 4a, we have that  
\[h_{RS}(1)=t_{01}\cdot r_{12}\cdot t_{10} \tag{28}\]  
where \(t_{xy}\) and \(r_{xy}\) represent the field transmission and reflection coefficients (respectively) for the transition between the dielectric media x and y, assuming normal incidence. The coefficient \(r_{12}\) can be then calculated from the sampled impulse response in reflection as  
\[r_{12}=\frac{h_{RS}(1)}{t_{01}\cdot t_{10}}=\frac{(n_{0}+n_{1})^{2}h_{RS}(1)}{ 4n_{0}n_{1}} \tag{29}\]  
The refractive index in the second layer \(n_{2}\) is obtained as  
\[n_{2}=n_{1}\bigg{(}\frac{1-r_{12}}{1+r_{12}}\bigg{)} \tag{30}\]  
As represented in Figure 4b, it must be noted that, from the third coefficient of the sampled impulse response, \(h_{RS}(2)\), \(h_{RS}(p)\) can be split into a recursive contribution (left part) and a non-recursive one (right part) [35]. The non-recursive part can be written as  
\[h_{RS}^{nr}(2)=t_{01}t_{12}r_{23}t_{21}t_{10}=r_{23}\prod_{i=0}^{1}\left( \frac{4n_{i}n_{i+1}}{(n_{i}+n_{i+1})^{2}}\right) \tag{31}\]  
where the only unknown factor is \(r_{23}\) which is needed to obtain \(n_{3}\) since  
\[n_{3}=n_{2}\bigg{(}\frac{1-r_{23}}{1+r_{23}}\bigg{)} \tag{32}\]  
Considering that the grating structure finishes at the second layer, i.e., the second layer is infinitely long; the recursive part \(h_{RS}^{r}(2)\) can be readily calculated by applying the ML-TMM analysis technique. Then, \(r_{23}\) is obtained as  
\[r_{23}=\frac{h_{RS}^{nr}(2)}{\prod_{i=0}^{1}\left(\frac{4n_{i}n_{i+1}}{(n_{i} +n_{i+1})^{2}}\right)}=\frac{h_{RS}(2)-h_{RS}^{r}(2)}{\prod_{i=0}^{1}\left( \frac{4n_{i}n_{i+1}}{(n_{i}+n_{i+1})^{2}}\right)} \tag{33}\]  
Figure 4: (**a**) Scheme of the division into layers of the FBG and the space-time paths contributing to the total sampled impulse response; (**b**) Decomposition of the space-time paths into the recursive path and the non-recursive path. The value of the refractive index of each layer is calculated from the non-recursive component of the impulse response in that layer, where the total impulse response and the non-recursive component are known. From \(r_{23}\), the value of \(n_{3}\) is solved using Equation (32). This last step must be repeated until the last sample of \(h_{RS}(N)\).