## 2 Materials and Methods  
### Numerical Simulations for Developing the Topology of Optical Waveguides on an LNOI Substrate  
The numerical simulation of an optical waveguide needs to be done to determine the typical sizes of basic and some test microstructures, etching depth, aspect ratio of structures, admissible slope angle, and other specifications of the technological process and the final result (the cross-section of the fabricated waveguide). Furthermore, the designs of an optical waveguide topology and all basic building blocks are based on the results of numerical simulation. In this study, the developed design (topology of optical waveguides) was used to fabricate a photolithographic mask.  
To design the topology, numerical simulation with the finite-difference frequency domain (FDFD) method was carried out. The main goal of the simulation process was an optimal waveguide parameters evaluation. The waveguide cross-section geometry for single-mode light propagation was optimized, and the critical bending radii were calculated.  
The simulation was carried out by taking into account the configuration of the LNOI wafers produced by NanoLN (China): the thickness of the SiO\({}_{2}\) layer was 4.7 \(\upmu\)m and the thickness of the LiNbO\({}_{3}\) layer was 0.6 \(\upmu\)m.  
Two types of integrated optical waveguides can be implemented on LNOI: deep-etched and shallow-etched waveguides.  
On the one hand, according to published articles, shallow-etched optical waveguides are characterized by smaller scattering losses induced by sidewall roughness in comparison with deep-etched waveguides [24; 25; 26; 27; 28].  
On the other hand, deep-etched optical waveguides are characterized by smaller critical bending radii in contrast to shallow-etched ones [24; 25; 27]. This opportunity is crucial in cases with a high element integration density.  
In order to reveal the single-mode condition of light propagation in waveguides, the etch depth and waveguide width were varied. In subsequent calculations, we used the parameters of the waveguide cross-section that are close to high-order mode excitation. Normally, waveguides with a large cross-section provide higher optical mode confinement and, therefore, lower scattering losses induced by sidewall roughness.  
The following parameters of the calculation window were used: calculation window width--9 \(\upmu\)m, calculation window height--5 \(\upmu\)m, discretization density--251 points per axis and a perfectly matched layer was used as a boundary condition.  
The dispersion in the medium was determined with the Sellmeier equations:  
\[n_{SiO_{2}}(\lambda)=\sqrt{1+\frac{0.6961663\lambda^{2}}{\lambda^{2}-0.068404 3^{2}}+\frac{0.4079426\lambda^{2}}{\lambda^{2}-0.1162414^{2}}+\frac{0.8974794 \lambda^{2}}{\lambda^{2}-9.896161^{2}}}, \tag{1}\]  
\[n_{LIN\text{O}_{3}}^{o}(\lambda)=\sqrt{1+\frac{2.6734\lambda^{2}}{\lambda^{2} -0.01764^{2}}+\frac{1.2290\lambda^{2}}{\lambda^{2}-0.05914^{2}}+\frac{12.614 \lambda^{2}}{\lambda^{2}-474.60^{2}}}, \tag{2}\]  
where \(\lambda\) is the wavelength of light (typically, 1550 nm) [29; 30].  
As a result, the optimal cross-section parameters were defined: waveguide width--1.4 \(\upmu\)m and etching depth--0.3 \(\upmu\)m (in the case when the cladding refractive index was equal to 1). This configuration of the waveguide supports the single-mode condition of light propagation with wavelengths in the range of 1.5 \(\upmu\)m to 1.6 \(\upmu\)m. In addition, the calculated critical bent radius was 50 \(\upmu\)m with bend optical losses less than 0.001 dB/90\({}^{\circ}\). Figure 1 shows the results of the numerical simulation: the distribution of the E-field in the cross-sections of both the straight and bent waveguides is presented.  
Based on the results of the numerical simulation, the photolithographic mask containing the design of the optical waveguides topology was fabricated.