## 5 Conclusions  
The simulation results obtained by FDFD and EIM showed that EIM is applicable to the analysis of anisotropic waveguides on X-cut of LNOI directed along the Y crystallographic axis when the considered modes are strongly polarized along the transverse axes. The relative error is less than 1% for the set of cross sections with sidewall slope angles from 60\({}^{\circ}\) to 90\({}^{\circ}\) and etching depths from 0.1 to 0.3 \(\upmu\)m for 1.55 \(\upmu\)m wavelength. Typically, the waveguide is designed for the specific central wavelength. It can be assumed that if the mode is far from cutoff condition at the certain wavelength, then the EIM will provide accurate effective refractive index values over the wide wavelength range, as can be seen in Figure 5. However, if the waveguides are considered with sidewall angles less than 70\({}^{\circ}\) and etching depths of more than 0.3 \(\upmu\)m, the relative error increases. The increase of the relative error is due to the fact that in the work, the sidewalls slope angle was not taken into account in the EIM simulations. However, the general EIM approach demonstrated in [29] could provide better accuracy when the sidewalls slope angle is approximated by the set of planar waveguides of different heights. In addition, the EIM can be applied for simulations of higher order modes. In these cases, the calculating accuracy of the effective indexes will also be determined by the proximity of the mode to the cutoff condition, the sidewall slope angle, and the etching depth as well as for the fundamental modes' simulations. Our research group is developing the fabrication process of ridge waveguides based on LNOI and our preliminary results show that it is possible to obtain waveguides with a sidewall slope angle of about 80\({}^{\circ}\) (we intend to publish these results in the near future). Thus, we intend to apply the EIM for the fast estimation of mode characteristics in fabricated waveguides.  
Moreover, EIM is a method based on the assumption that x and y electromagnetic field components are separable. It means that a three-dimensional problem of field propagation in a photonic integrated element can be reduced to a two-dimensional problem with the real refractive indexes replaced by the effective refractive indexes of the media, if the considered photonic element is parallel to one of the crystallographic axes. In addition, as shown in [28], EIM could be integrated with an S-matrix simulator. Thus, it is possible to reduce calculation time for fast circuit simulation.  
**Author Contributions:** D.M.: methodology, numerical simulation, investigation, analysis and writing--original draft; A.K.: methodology, investigation, SEM analysis and review; U.S.: conceptualization, review and editing; V.K.: supervision and review; A.V.: supervision and review. All authors have read and agreed to the published version of the manuscript.  
**Funding:** The research was funded by the Ministry of Science and Higher Education of the Russian Federation (Project No. FSNM-2023-0005)[PERSON]  
**Institutional Review Board Statement:** Not applicable[PERSON]  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The data reported in this manuscript are available on request from the corresponding author.  
**Acknowledgments:** This work was performed, in part, using the equipment of the Physics department of the Perm State National Research University. The authors would like to thank [PERSON] for the results discussion.  
**Conflicts of Interest:** The authors declare no conflict of interest.