## 4 Discussion  
Using the EIM and FDFD methods, the effective refractive indexes of the fundamental TE- and TM-modes in the waveguide with a cross section shown in Figure 2 were calculated for the wavelengths 1.5-1.6 \(\upmu\)m. The results demonstrate that as the wavelength increases, the mode localization in the waveguide core decreases. It is known that in this case the mode field is more sensitive to the waveguide-environment interface, as a result the hybridity of the mode becomes more expressed (the amplitude of longitudinal field components increases). Wave equations (Equation (3)) take into account the refractive index only along the axis parallel to direction of field polarization and do not consider the hybrid mode nature, which is expressed in the possible interaction of transverse electric field components. This leads to an increase in the calculating error in the effective refractive indexes with increase of wavelength (Figure 5b) by EIM. Nevertheless, the value of the relative error of calculations of the effective refractive index with EIM does not exceed 1.0% in wavelength range 1.5-1.6 \(\upmu\)m for the considered waveguide cross section. The executed calculations in this paper allow the estimation of the applicability limits of EIM for a simulation of anisotropic waveguides based on LNOI (Figure 8). We assume that if the calculating error of the effective refractive index does not exceed 1%, the EIM could be used for the rapid estimation of the parameters of anisotropic waveguides based on LNOI. It is known that the accuracy of EIM depends on the proximity to the cutoff condition and slab-likeness of the waveguide for the propagating mode. Slab-likeness could be defined as the polarization fraction of radiation along one of the transverse axes of the waveguide, as well as the absence of a sharp refractive index gradient. The higher the polarization fraction of radiation along the one of the axes, the smaller the real part of the amplitudes of electromagnetic field \(E_{z}\) and \(H_{z}\) components, and the closer the propagating modes to TEM-modes. This paper shows an evaluation of the accuracy of EIM applied to waveguides based on LNOI through an estimation of the calculating accuracy of the effective refractive indexes. That is why for the simulations with variation of the etching depth, the waveguide width was chosen to be slightly less than the width corresponding to the occurrence of the first-order propagating modes, i.e., studied modes were far from the cutoff condition. If the waveguide width is reduced, then the calculation accuracy of the effective refractive indexes will decrease as the propagating mode approaches the cutoff condition. Figure 9: Evaluation of optical radiation losses of TE- and TM-modes in waveguide showed in Figure 2 with different LER parameter and \(L_{c}\) = 160 nm. The results obtained with the FDFD method show that for the TE-mode the etching depth of 0.1 \(\upmu\)m corresponds to the maximum polarization fraction along the horizontal axis (\(\gamma_{TE}\) = 99.8%), while the TM-mode is highly hybridized (60% \(\leq\gamma_{TM}\leq\) 69%). Nevertheless, the calculating accuracy of the effective refractive index of the TM-mode with EIM remains comparable to the accuracy of the FDFD method for all studied sidewall slope angles of the waveguide. As the waveguide etching depth increases, the TE-mode becomes more hybrid, since the refractive index gradient becomes sharper along the horizontal axis. However, since in each calculation iteration the mode remains far from the cutoff condition, the polarization fraction drops only to values of about 96%. In turn, the polarization fraction of the TM-mode increases with increasing the waveguide etching depth to values close to 98%. In addition, an increase of the etching depth leads to a decrease in the effective refractive indexes of both modes, and in the range of etching depths from 0.4 \(\upmu\)m to 0.5 \(\upmu\)m, a hybridization region of the TE- and TM-modes appears when their effective refractive indexes become equal.