## 8 Conclusions  
A multi-parameter study of the scan patterns of a pair of rotational Risley prisms was performed. Theoretical aspects of the (minimum and maximum, exact and approximate) angular and linear deviations were approached. Simulations of the scan patterns were performed with the easy-to-use, exact graphical method that, to our knowledge, we have introduced [60] and developed [49]. This multi-parameter analysis of both deviations and scan patterns was carried out considering all the constructive parameters of the scanner, as well as [PERSON]'s parameters [4]. A correlation between analytical aspects and simulations was made, as pointed out with examples. Thus, it was demonstrated that theory and simulations validate each other. An experimental validation of the performed simulation was also made, as shown in this study, but also in (other) different examples in [49; 61; 65]. The multi-parameter analysis of the shapes and of the (outer and inner) dimensions of the (ring-shaped) scan patterns allowed for extracting rules-of-thumb to support the optimal design of optomechanical scanners with Risley prisms for the requirements of different applications. Future work in this direction of research comprises applying the newly-developed graphical method for other types of scanners with Risley prisms, with tilting, rotational and tilting, as well as with three prisms or doublets. In addition, the Gaussian profile of laser beams must be considered to extend the present study beyond the simplifying hypothesis of beams reduced to their central rays. The dependence of the refractive indexes of each prism material with the laser wavelength, to account for dispersion issues, is another topic of interest that can take advantage of the developed graphical method. While to our knowledge no other approach has been made so far to develop a graphical method to study Risley prisms scanning, future work in our group includes graphical approaches with dedicated, common ray-tracing programs such as Zemax to assess, for example, ease-of-operation and analysis capabilities compared to mechanical design programs such as the one utilized in the present study. Applications of Risley prisms are also considered in our group, especially for biomedical imaging using confocal microscopy and OCT. However, the range of applications is much larger, and the designing rules deduced here may serve for laser manufacturing, industrial metrology, Remote Sensing, as well as Security and Defense scanners. Conceptualization, methodology, and theory, V.-F.D. ; simulations, A.-L.D. ; modelling and data analysis, V.-F.D. ; experiments, V.-F.D. and A.-L.D. ; writing--original draft, V.-F.D. and A.-L.D. ; writing--review and editing, V.-F.D. ; funding, supervision, and project administration, V.-F.D. All authors have read and agreed to the published version of the manuscript. This research was supported by the Romanian Ministry of Research, Innovation and Digitization, CNCS/CCCDI-UEFISCDI, project PN-III-P4-ID-PCE-2020-2600, within PNCDI III ([http://3 om-group-optomechatronics.ro/](http://3 om-group-optomechatronics.ro/), accessed on 1 August 2021). Data Availability Statement:Data supporting reported results can be obtained from the corresponding author. \begin{table}
\begin{tabular}{|c|c|c|c|c|} \hline
**Ratio of the Rotational Speeds, Equation (1)** & \(M=4\) & \(M=-4\) & \(M=8\) & \(M=-8\) \\ \hline Mean value of the FOV radius: & & & & \\ \(\overline{y_{max}^{\prime}}=\frac{1}{\mu}\sum\limits_{j=1}^{k}y_{max}^{\prime}\) & & & & \\ \hline Standard deviation of the FOV radius: & & & & \\ \(\sigma=\sqrt{\frac{1}{\mu}\sum\limits_{j=1}^{k}\left(\overline{y_{max}}-y_{max}^ {\prime}\right)^{2}}\) & 0.591 & 0.645 & 1.336 & 1.708 \\ \hline Relative error: \(\varepsilon^{\infty}=\left(\overline{y_{max}}-y_{max}\right)\cdot 100/y_{max}\) & 2.027 & 2.192 & 2.207 & 3.548 \\ \hline \end{tabular}
\end{table}