## 2 Materials and Methods
### The Estimation of Optical Radiation Losses with EIM and the Payne-Lacey Model  
As it shown in [29], the cross-sections of real (manufactured) integrated optical waveguides after the reactive ion etching stage differ from ideal ones (simulated). For the complex analysis of optical characteristics of real waveguides, the sidewall roughness was taken into account and optical radiation losses were evaluated. An RMS roughness and a correlation length \(L_{c}\) typically characterize a random waveguide sidewall perturbation and are used for estimation of the radiation losses along the waveguide [35]. The LER parameter is commonly used to characterize the technological processes of waveguide fabrication. RMS might be calculated as LER/3. Commonly, the sidewall roughness of the waveguide is inherited during reactive ion etching from the edge roughness of the mask [36]. For this reason, in the paper, the LER of the waveguide was determined as the LER of the mask. As shown in [29], the chromium hard mask was used for the fabrication of waveguides on an LNOI substrate. For LER analysis, a top view image of the chromium mask was obtained using a scanning electron microscope (SEM) Tescan Mira3 (Figure 3). In addition, the same analysis of the sidewall roughness of the fabricated waveguide on the LNOI substrate is not possible due to its charging during SEM investigation. SEM images were obtained at 100 kx magnification at three different points for one chromium strip. Then, pixel-by-pixel image processing was carried out with the OpenCV2-Python module. The chromium mask edges were defined as the intensity maxima in each pixel line of the grayscale image obtained from the secondary electron detector. Such image processing eliminates the human factor in determining the edge of the chromium mask. The illustrative example of the SEM image and the plot of mask edges of the chromium strip with high LER value is shown on Figure 4. \begin{table}
\begin{tabular}{c c c} \hline
**Etching Depth, \(\mu\)m** & **Waveguide Width, \(\mu\)m** & **Sidewall Angles, \({}^{\circ}\)** \\ \hline
0.1 & 2.1 & \\
0.2 & 1.5 & \\
0.3 & 1.1 & \\
0.4 & 0.9 & 60–90 \\
0.5 & 0.8 & \\
0.6 & 0.7 & \\ \hline \end{tabular}
\end{table} 
Table 1: The parameters of the cross-section of the studied waveguides based on X-cut LNOI. For the \(LER\) calculation, a middle line was built for the left and right edges of the chromium strip. The \(LER\) was calculated as [37]:  
\[LER_{k}=3\cdot\sqrt{\frac{\sum_{i=1}^{n}(x_{ik}-\overline{x_{k}})^{2}}{n}}, \tag{6}\]  
where \(i\)--point at which deviation is calculated, \(k\) = 1, 2 are the left and right edge lines of the chromium mask. It is assumed that the LER value is within the range of three standard deviations from the profile midline (calculated separately for the left and right edges). For the entire mask edges of the chromium strip, LER is calculated using Equation (7):  
\[LER=\sqrt{\frac{LER_{1}^{2}+LER_{2}^{2}}{2}} \tag{7}\]  
For conversion of pixels to nanometers, the final LER value was multiplied by a factor of 2.5, which corresponded to ratio of 2.5 nm/pixel for images at 100 kx magnification and 1024 \(\times\) 1024 resolution. The \(LER\) values of the hard chromium mask, formed with the process described in [29], ranged from 20 to 40 nm. Figure 4: The illustrative example of image processing for the LER analysis: (**a**) SEM image (Red line denotes the image scale); (**b**) plot of mask edges of the chromium strip. Figure 3: The top view of chromium strip used as the mask for reactive ion etching of LNOI for the optical waveguide fabrication (Red line denotes the image scale). The method of photolithography used determines the mask edge roughness. In [29] the contact lithography was applied for waveguide pattern transfer to the LNOI substrate. According to [38], the typical correlation length \(L_{c}\) for the contact lithography patterning is 160 nm.