## 2 Scanners with Pairs of Rotational Risley Prisms  
The principle scheme of a laser scanner with a pair of rotational Risley prisms is presented in Figure 1a. Figure 1 shows _the principle of the modelling process of the rotation of prisms_ performed using CATIA V5R20 and how this allows for the simulation of the scanning process, i.e., of the refraction of a laser beam (considered reduced to a single central ray) through the prism system. A simplifying hypothesis (and characteristic) of this entire study is that we perform it within the boundaries of geometrical optics. Thus, all the (Gaussian) laser beams are considered reduced to their central rays, and the dependence of the refractive indexes of the prisms materials with the laser wavelengths are not considered. As pointed out in the Introduction, there are studies focused on dispersion issues that occur especially when optical prisms are utilized, instead of wedges [15]. Figure 1: _Cont._  
Figure 1b,c show the results of the simulation, with two ray bundles, generated for the two prisms rotating in the same direction (Figure 1b) and in opposite directions (Figure 1c), respectively. The ray bundles are intersected with a plane (the screen in Figure 1a) situated at a certain distance from the scanner. The scan patterns thus obtained are shown in dotted lines at the end of the light bundles. This simple procedure is utilized further on in Section 4 to obtain different shapes of the scan patterns. The characteristic parameters of the system are (Figure 2): \(\theta_{1}\) and \(\theta_{2}\), the angles of the prisms; \(n_{1}\) and \(n_{2}\), the refractive indexes of the materials of Prism 1 and 2, respectively (indexes that are considered constant, as we do not take into account dispersion in the present work, as pointed out above); \(\omega_{1}\) and \(\omega_{2}\), the rotational speeds of the prisms; \(e\), the distance between the two prisms (measured between the diopters perpendicular on the symmetry/optical axis (O.A.) of the components of the system); and \(L\), the distance from the (final diopter perpendicular on the O.A. of the) scanner to the screen. The analysis of such scanners should also be based on [PERSON]'s synthetic and elegant parameters: \(k\), the ratio of the angles of the prisms (initially defined in [4] in a more particular way, as ratio of the deviation angles \(D=(n\!-\!1)\,\theta\) of each prism, obtained using the small angles approximation); and \(M\), the ratio of the rotational speeds [4]:  
\[k=\theta_{2}/\theta_{1}\text{ and }M=\omega_{2}/\omega_{1}. \tag{1}\]  
In Figure 1b,c, the two ray bundles were obtained for a certain modulus of \(M\), but (b) for \(M>0\) (when the two prisms rotate in the same direction), and (c) for \(M<0\) (when the two prisms rotate in opposite directions). As a color convention in all such figures in the paper, bundles and patterns obtained for \(M>0\) are drawn in blue, while those obtained for \(M<0\) are drawn in red. Four possible combinations of the relative positions of the two prisms can be obtained by considering all different possible orientations of the prism diopters [6]. By denoting with '\(a\)' a diopter perpendicular on the O.A. and with '\(b\)' a tilted one, these configurations are as follows (Figure 2): (a) _ab-ab_; (b) _ab-ba_; (c) _ba-ba_; and (d) _ba-ab_. Figure 1: (a) Pair of rotational Risley prisms, modelled with a commercially-available mechanical design program, CATIA V5R20. (b,c) Two light bundles generated with CATIA V5R20 in the simulation of scanning with a pair of rotational Risley prisms. As a color convention throughout the entire study, scan patterns obtained for prisms rotating in the same direction are presented in blue, while patterns of prisms rotating in opposite directions are presented in red. Details are presented in the text. Figure 2: The four possible configurations of a laser scanner with a pair of rotational Risley prisms: (**a**) _ab-ab_; (**b**) _ab-bu_; (**c**) _ba-bu_; (**d**) _ba-ab_.