# Comparative Analysis of the Methods for Fiber Bragg Structures Spectrum Modeling
## 3 Results and Discussion
### Results of a \(\pi\)-Phase-Shifted FBG Modeling  
As it was mentioned before, the fiber Bragg structures comprising a phase shift (Figure 1e) are of particular interest for sensing applications since their spectral response includes an ultra  
Figure 5: Deviations of the homogeneous FBG spectra from the reference (OptiGrating) spectrum (left axis): deviation of the Layer Sweep method (red line), deviation of the Transfer Matrix method (green line), reference OptiGrating spectrum (black line, right axis). Figure 6: Values of the root-mean-square deviation (RMSD) of the homogeneous FBG spectrum modeled using the LS method (blue dots) depending on the partition interval \(\Delta\)z, and their approximation (blue solid line). narrow transparency window, the wavelength position of which can be measured with higher accuracy than the shift of the entire FBG spectrum. Such a fiber Bragg structure can be represented as two identical FBGs located sequentially with the distance between them equal to the phase \(\pi\)-shift of their periodic refractive index variation. The modeling parameters of the sub-gratings are listed in Table 2. The simulated spectra of the \(\pi\)-phase-shifted FBG are shown in Figure 7, while the wavelength dependences of the spectra deviations from the OptiGrating data are presented in Figure 8. The partition interval was equal to \(\Delta z=0.04\times\Lambda\). In this case, \(\text{RMSD}_{\text{LS}}=3.89\times 10^{-3}\) and \(\text{RMSD}_{\text{TM}}=3.68\times 10^{-3}\) for the LS and TM methods, respectively. As can be seen from the Figures 7 and 8, the results of both methods are also very close to the spectrum obtained using the specialized software. The RMSD values were obtained for various partition intervals as well, which is demonstrated in Figure 9.  
Unlike the previously considered example, the dependence of the RMSD\({}_{\mathrm{LS}}\) values on the partition interval \(\Delta z\) in the case of \(\pi\)-phase-shifted FBG modeling is not monotonical and cannot be approximated with an acceptable precision. There are certain values of the partition interval, at which the RMSD\({}_{\mathrm{LS}}\) significantly increases. In order to establish the cause of such phenomenon, the modeled FBG refractive index profiles near the phase shift with different partition intervals \(\Delta z\) and the resulting spectra were compared to each other, as it is illustrated in Figure 10, where the intervals \(\Delta z\) = 0.13 \(\times\)\(\Lambda\) and \(\Delta z\) = 0.12 \(\times\)\(\Lambda\) are considered as an example. As it can be seen from Figure 10, the spectrum obtained using the LS method with the partition interval \(\Delta z\) = 0.12 \(\times\)\(\Lambda\) (Figure 10a, red solid line) has an asymmetric shape, which results in the increased RMSD in comparison with the spectrum acquired with the interval \(\Delta z\) = 0.13 \(\times\)\(\Lambda\) despite the lower value of the partition interval. In this case, the displacement of transparency window of  
Figure 10: Comparison of modeling results of the \(\pi\)-phase-shifted FBG using the Layer Sweep method at different partition intervals: (**a**) resulting spectra: at \(\Delta z\) = 0.12 \(\times\)\(\Lambda\) (red solid line), at \(\Delta z\) = 0.13 \(\times\)\(\Lambda\) (green solid line), OptiGrating spectrum (blue dotted line); (**b**) refractive index profile of the grating at \(\Delta z\) = 0.12 \(\times\)\(\Lambda\); and (**c**) refractive index profile of the grating at \(\Delta z\) = 0.13 \(\times\)\(\Lambda\). Figure 9: Values of the root-mean-square deviation (RMSD) of the \(\pi\)-phase-shifted FBG spectrum modeled using the LS method (blue dots) depending on the partition interval \(\Delta z\). the phase-shifted FBG is comparable to its FWHM, which is not acceptable, since such structures are often used for high-resolution sensing [6].