## 3 Results  
The FDFD simulation showed that fundamental TE- and TM-modes in the waveguide depicted on Figure 2 are strongly polarized along x and y axes, which is expressed in \(\gamma_{TE}\) values exceeding 98% (\(\gamma_{TE}\) is around 0 for TM-mode). Thus, the assumption that an anisotropic optical waveguide could be approximated by an isotropic waveguide with the refractive index corresponding to a certain polarization is correct for the considered cross-section (shown on Figure 2) with a known accuracy. Furthermore, for the specified cross section on Figure 2, simulations were carried out using the EIM and FDFD methods in the wavelength range from 1.5 to 1.6 \(\upmu\)m.  
The obtained results of the EIM simulations are in good agreement with the results obtained with the accurate FDFD method for the considered waveguide cross-section, and the relative error of the effective refractive index calculation does not exceed 1% for both polarizations (Figure 5). In simulations of single mode waveguides, the dependencies of the effective refractive index and polarization fraction along the \(x\) and \(y\) axes from the etching depth were studied at the wavelength 1.55 \(\upmu\)m (see the waveguides cross-section parameters in Table 1). The simulation showed that with increase of etching depth, the effective refractive indexes of the TE- and TM-modes decrease to values close to the substrate refractive index (SiO\({}_{2}\)), and the polarization fraction of the TE-mode along the transverse axis decreases while the value for the TM-mode increases. However, the smaller the slope angle of sidewalls, the faster the effective refractive index and the fraction of polarization of the TE-mode decrease. Figure 5: FDFD and EIM simulation results: (**a**) effective refractive indexes for the fundamental TE- and TM-modes versus wavelengths; (**b**) the relative calculation error of effective refractive indexes by FDFD and EIM versus wavelengths. In the case of the TM-mode, the slope angle of sidewalls affects the polarization fraction, limiting its maximum value (Figure 6). Moreover, at an etching depth of about 0.4 \(\upmu\)m, the values of the effective refractive indexes of the TE and TM modes become almost the same, which means that inter-mode hybridization occurs. According to results obtained with FDFD method, when the etching depth increases further, the effective refractive index of the TM-mode becomes larger than the effective refractive index of the TE-mode (Figure 7a). The same is observed in the results obtained by EIM, however, in case of EIM, the inter-mode hybridization region is shifted to an etching depth of 0.5 \(\upmu\)m (Figure 7b). Figure 6: The simulation results of set of single-mode waveguides with different sidewall slope angles: (**a**) effective refractive indexes of the TE-mode versus the etching depth; (**b**) the TE-mode polarization fraction versus the etching depth; (**c**) effective refractive indexes of the TM-mode versus the etching depth; (**d**) the TM-mode polarization fraction versus the etching depth. To evaluate the accuracy of EIM, the relative error of calculating the effective refractive indexes of the TE- and TM-modes by EIM and FDFD method was analyzed for the waveguides described in Table 1. As can be seen from Figure 8, the relative calculation error increases with an increase of the waveguide etching depth and with decrease of the sidewall slope angle. The plot shows that the calculation error of the effective refractive index by the EIM does not exceed 1% at the etching depths less than 0.3 \(\upmu\)m and at the sidewall slope angle at least 70\({}^{\circ}\). After the EIM accuracy estimation, optical radiation loss could be calculated with the Payne-Lacey method.