# Fiber Bragg Grating-Based Optical Signal Processing: Review and Survey
## 2 Review of Design Methods for Fiber Bragg Gratings
### Analysis of Fiber Bragg Gratings
#### 2.2.1 Coupled-Mode Theory (CMT)-Based Analysis Method  
The Coupled-mode theory is a widespread technique that relates counter-propagating electromagnetic waves within the grating structure using coupled differential equations [25; 27; 28]. CMT directly provides an analytic solution for the propagation of the electromagnetic fields through a uniform FBG, which is a grating with constant \(\Delta n(z)\) and \(\Lambda(z)\) along its length. In particular,  
\[n_{\text{FBG}}(z)=n_{eff}+\Delta n_{\text{max}}\cdot\cos\left\{\frac{2\pi}{ \Lambda}\right\} \tag{6}\]  
where \(n_{eff}\), \(\Delta n_{\text{max}}\) and \(\Lambda\) are all constant values. The reflective and transmissive coefficients of the uniform FBG are given by [6; 27]:  
\[H_{R}(f) =\frac{-j\kappa^{*}\text{sinh}(\kappa L)}{\gamma\cosh(\kappa L)+j( \Delta\beta/2)\text{sinh}(\kappa L)} \tag{7}\] \[H_{T}(f) =\frac{\gamma\cdot\exp(-j\beta_{0}L)}{\gamma\cosh(\kappa L)+j( \Delta\beta/2)\text{sinh}(\kappa L)} \tag{8}\]  
where \(*\) stands for complex conjugation, the optical waveforms \(E^{+}(z,f)\) and \(E^{-}(z,f)\) involved in these expressions are defined in Figure 1, the propagation constant is \(\beta_{0}=\pi/\Lambda\), and \(\gamma\) is defined as \(\gamma=|\kappa|^{2}-(\Delta\beta/2)^{2}\). The parameter \(\Delta\beta=\beta-\beta_{0}\). The coupling coefficient \(\kappa\) is defined as \(\kappa=-j\pi f\Delta n_{\text{max}}/c\). Now, let us assume the analysis of a non-uniform FBG. The induced refractive index can be expressed following Equation (1). In this equation, we have written the phase of the refractive index modulation as a function of the grating period variation (\(\Lambda(z)\)). In this case, the coupling coefficient varies along the grating's length and is given by:  
\[\kappa(z)=-j\frac{\pi f\Delta n(z)}{c}\exp\left\{-j2\pi\int_{0}^{z}\left(\frac {1}{\Lambda(z^{\prime})}-\frac{1}{\Lambda_{0}}\right)dz^{\prime}\right\} \tag{9}\]  
where \(\Lambda_{0}\) is a specific grating period for reference. The CMT-based algorithm leverages the transfer-matrix method (TMM) to implement a discretization process of a non-uniform grating [31; 33], as shown in Figure 2. Thus, the entire grating is split into \(N\) layers of length \(\delta L_{i}\), with \(i\in[1,N]\), each of them containing few periods. In each layer, we assign constant values to the parameters \(\gamma\), \(\kappa\) and \(\Delta\beta\), e.g., by selecting their values at the center of the section. Hence, the different grating layers are approximated as uniform structures [2; 27], which can be readily described by the transfer matrix of a uniform FBG [6; 27] of length \(\delta L_{i}\), i.e.,  
\[\left[ \begin{array}{c}E^{+}(z_{i},f)\\ E^{-}(z_{i},f)\end{array} \right]=\left[ \begin{array}{cc}M_{i,11}(f)&M_{i,12 }(f)\\ M_{i,21}(f)&M_{i,22}(f)\end{array} \right]\left[ \begin{array}{c}E^{+}(z_{i}+ \delta L_{i},f)\\ E^{-}(z_{i}+\delta L_{i},f)\end{array} \right]=[M_{\text{U},i}]\left[ \begin{array} []{c}E^{+}(z_{i}+\delta L_{i},f)\\ E^{-}(z_{i}+\delta L_{i},f)\end{array} \right] \tag{10}\]  
Figure 2: Representation of the division of a non-uniform FBG into layers that are approximated by uniform FBG. The resulting structure can be readily analyzed via CMT and TMM. The dependence of propagating waves with the optical frequency is omitted for the sake of simplicity.