# Designing a Novel High-Performance FBG-OADM Based on Finite Element and Eigenmode Expansion Methods
## 5 Result and Analysis of the FBG-OADM  
FBG-OADM simulations, designs, and research were made using the combined FEM-EEM approach. The parameters in Figure 3 are explained as follows: \(a_{1}\) is the core radius; \(a_{2}\) is the cladding layer radius; \(n_{1}\) is the RI of the core layer; \(n_{2}\) is the RI of the cladding layer; \(n_{3}\) is the RI of the surround layer; \(\delta n\) is the peak induced-index change by the UV light; \(d_{pml}\) is the thickness of the PML; \(L_{1}\) and \(L_{3}\) are the noncoupling areas of the FBG-OADM; \(L_{2}\) is the mode-power coupling area; \(d_{\text{WPC}}\) and \(d_{\text{HPC}}\) are the width and height between the cladding layer and the PML, respectively; W and H are the width and height of the overall structure, respectively. The parameters were set as follows: \(a_{1}=2.25\)\(\upmu\)m, \(a_{2}=12.25\)\(\upmu\)m, \(a_{1}=1.454\), \(n_{2}=1.43\), \(n_{3}=1.415\), \(\delta n=1.454\times 10^{-3}\), \(d_{pml}=1\)\(\upmu\)m, \(L_{1}=1\)\(\upmu\)m, \(L_{3}=1\)\(\upmu\)m, \(L_{2}=\Lambda_{FBC}\times N_{p}\), \(d_{WPC}=13.7\)\(\upmu\)m, \(d_{HPC}=11.25\)\(\upmu\)m, \(W=78.4\)\(\upmu\)m, and \(H=49\)\(\upmu\)m, where \(N_{p}\) is the number of period in the FBG-OADM. Moreover, the distance between the two optical fibers was set to zero and the operating wavelength \(\lambda=1550\) nm. Figure 4 shows the side view (i.e., \(X\)-\(Z\) plane) of the FBG-OADM during the numerical simulations. Figure 3: Geometric structure of the FBG-OADM: (**a**) sectional view from the \(X\)-\(Y\) plane; and (**b**) side view from the \(Y\)-\(Z\) plane. According to optical coupling theory, the only guided modes that are subjected to the influence of the 2 \(\times\) 2 optical fiber coupler and could subsequently transmit and couple power from Fiber 1 to Fiber 2 are the cladding modes. In addition, because of the perturbation effect of the FBG, core modes couple only to cladding modes with a powered core. In other words, of the four guided modes shown in Figures 5-8, only the one in Figure 5 (\(\
u\) = 9) achieved acceptable FBG-OADM drop and add functionality. Mathematically, all guided modes must display pairwise orthogonality. However, in numerical simulations, because of memory and computation time constraints, it is impossible to achieve pairwise orthogonality between modes. Therefore, in this study, the OMM and BMM were combined to determine the appropriate segmentation size and formulate a review standard for the orthogonal value. Figure 9 shows the orthogonal values between the 80 modes, which confirms that the results satisfied the review standard. The FBG-OADM was designed after determining the optimal mesh resolution and number of modes. The ratio of the drop power to input power for the FBG-OADM must equal 1. In other words, its drop power should be 0 dB. Thus, \(\Lambda_{FBG}\) and \(N_{P}\) values that satisfied this condition had to be identified to design an FBG-OADM that achieved zero insertion loss and minimal heterodyne and homodyne crosstalk. Figure 10 shows the scanning diagram of the optimal \(\Lambda_{FBG}\). According to the power transmission curve in Figure 10, the power transmission approached 0 dB when \(\Lambda_{FBG}\) = 0.5396 \(\upmu\)m. Figure 11 illustrates the scanning diagram of the optimal FBG-OADM length, showing that the power transmission approached 0 dB when \(L_{2}\) = 0.9908 cm. Next, from Equation (3), \(N_{P}\) (i.e., the number of period in the FBG-OADM) was calculated and found to be 18,361. \[L_{2}=\Lambda_{FBG}\times N_{P} \tag{3}\]  
Figure 8: 2D power distribution plot for guided-mode \(\
u\) = 79 (\(n_{neff}^{\
u=79}\) = 1.420274). Figure 9: Relationships between the orthogonal values of 80 guided modes for the proposed FBG-OADM. According to the EEM descriptions in Section 4, a small amount of power loss can be observed in all modes during transmissions; to address this, the power loss review standard was established.