## 2 Materials and Methods  
### Calculations of Optical Waveguides with EIM  
Optical waveguides based on the X-cut of LNOI are anisotropic; consequently it is necessary to solve the vector wave equation for analysis of radiation propagation along these waveguides. Additional difficulties appear during the modes simulation if the waveguide axis does not coincide with one of the crystallographic axes of LN, since the permittivity tensor included in the wave equation will not be diagonal [12; 13; 14; 15; 16; 17; 18; 19]. The problem of solving the vector wave equation for anisotropic waveguides might be simplified by considering the optical waveguide directed along one of the crystallographic axes, when the permittivity tensor becomes diagonal. In this case, if we assume that the optical modes in the waveguide are polarized strictly along the transverse axes, then the electric displacement vector will be parallel to the electric field vector. Then the vector wave equation should be [30]:  
\[\
abla^{2}E+k^{2}n^{2}\cdot E=\
abla(\
abla\cdot E) \tag{1}\]  
where \(E\) is the electric field vector, \(k\) is the wave vector in vacuum and \(n\) is the diagonal refractive index tensor. Equation (1) might be represented as a simplified vector wave equation:  
\[\Big{(}\
abla_{t}^{2}+k^{2}n_{t}^{2}-\beta^{2}\Big{)}e_{t}=-\
abla_{t}\Big{(} e_{t}\cdot\
abla_{t}\ln n_{t}^{2}\Big{)} \tag{2}\]  
where \(\
abla_{t}^{2}\) is the transverse nabla operator, \(n_{t}=n_{x}\) is \(xx\) component of the refractive index tensor, \(e_{t}=e_{x}\) is \(x\) component of the electric field for modes polarized along the \(x\) axis, \(n_{t}=n_{y}\) is \(yy\) component of the refractive index tensor, \(e_{t}=e_{y}\) is \(y\) component of the electric field for modes polarized along the \(y\) axis and \(\beta\) is the propagation constant. Equation (2) describes the propagation of quasi-TE and quasi-TM modes and could be represented as two scalar wave equations with polarization corrections characterizing the structural anisotropy of the waveguide (Equation (3)):  
\[ \begin{array}{l}\frac{\partial^{2}e_{x}}{\partial x^{2}}+\frac{\partial^{2} e_{y}}{\partial y^{2}}+\big{(}k^{2}n_{x}^{2}-\beta^{2}\big{)}e_{x}=-\frac{ \partial}{\partial x}\Big{(}\frac{1}{n_{x}^{2}}\frac{\partial n_{x}^{2}}{ \partial x^{2}}e_{x}\Big{)},\\ \frac{\partial^{2}e_{y}}{\partial x^{2}}+\frac{\partial^{2}e_{y}}{\partial y^ {2}}+\Big{(}k^{2}n_{y}^{2}-\beta^{2}\Big{)}e_{y}=-\frac{\partial}{\partial y }\Big{(}\frac{1}{n_{y}^{2}}\frac{\partial n_{y}^{2}}{\partial y}e_{y}\Big{)}.\end{array}  \tag{3}\]  
In addition, in Equation (2), the term responsible for the interaction between the transverse components of the electric field is omitted, which follows from the assumption of a strong polarization of the TE and TM modes along the transverse axes [30]. Since it is assumed that the electric fields of the TE and TM modes are strongly polarized along the \(x\) and \(y\) axes, the solution of Equation (3) can be obtained using methods of separation of variables. According to [31], the wave Equation (3), which takes into account polarization effects, could be solved using EIM. EIM is a well-developed semi-analytical method for solving wave equations based on separation of variables. Widespread application of EIM is explained by its simplicity even among semi-analytical methods. The idea of EIM is that the cross-section of a real 3D waveguide is sequentially substituted by equivalent slab 2D waveguides. It is known that EIM works well when the propagating fundamental TE- and TM-modes are far from the cutoff condition [31; 32; 33]. To apply EIM, the waveguide must be slab-like. A waveguide is slab-like when it is strongly structurally anisotropic, for example, when the waveguide width is greater than its height. In addition, if a shallow-etched waveguide is considered, then a smaller etching depth will correspond to greater slab-likeness, since the refractive index gradient along the horizontal axis of the waveguide cross-section will be less expressed.