## 4 Results of Modeling and Simulations: Shapes of the Scan Patterns
(**1**) Left column, \(M>0\); (**2**) right column, \(M<0\). The values on the axes are in millimeters. Figure 6: Study of the scan patterns produced by the configuration _ba-ab_ of rotational Risley prisms (Figure 2d), for \(L=1\) m, \(e=25\) mm, and \(|M|=6\), with: (**a**) \(k=1/2\); (**b**) \(k=2\); (**c**) \(k=3\); (**d**) \(k=5\). (**1**) Left column, \(M>0\); (**2**) right column, \(M<0\). The values on the axes are in millimeters. Another relevant, simple equation can be extracted from this discussion regarding the number of loops of a scan pattern, obtained for the same \(\left\lvert M\right\rvert\) :  
\[\mu(M<0)=\mu(M>0)+2. \tag{38}\]  
One can also observe (easier for smaller \(\left\lvert M\right\rvert\)), as a specificity of these Risley prism curves, the way inward loops are generated for \(M>0\), while outward loops are generated for \(M<0\). In Figure 5, scan patterns obtained for \(\left\lvert M\right\rvert=6\) are shown, for increasing values of \(k\): from (a) \(k=2/4=1/2\) (obtained for \(D_{1}=4^{\circ}\) and \(D_{2}=2^{\circ}\)) to (b) \(k=4/2=2\) (for \(D_{1}=2^{\circ}\) and \(D_{2}=4^{\circ}\)), (c) \(k=6/2=3\) (for \(D_{1}=2^{\circ}\) and \(D_{2}=6^{\circ}\)), and (d) \(k=10/2=5\) (for \(D_{1}=2^{\circ}\) and \(D_{2}=10^{\circ}\)). The intermediate \(k=2/2=1\) case for this \(\left\lvert M\right\rvert\) was already shown in  
Figure 8: Comparison between the scan patterns of the four scanner configurations presented in Figure 2, for \(\left\lvert M\right\rvert=8\) and \(k=1\) (the latter obtained for \(D_{1}=D_{2}=10^{\circ}\)). (a) Left column, \(M>0\); (b) right column, \(M<0\). The values on the axes are in millimeters. Figure 7: Scan patterns produced by the configuration _ba-ab_ of Risley prisms (Figure 2d), for \(k=1\), \(M=6\), and (a) \(L=10\) m; (b) \(L=100\) m. The values on the axes are in millimeters. Figure 4c. As a remark, these specific values have been chosen because such Risley prisms are commercially available, for example from Thorlabs [63]. From Figure 5, by making a comparison along each of the two columns, one can see that as \(k\) increases, the loops intersect more and more. This is due to the variation of the minimum angular deviation that can be obtained as \(\theta_{2}\) increases--in the case of the scanner configuration in Figure 2d--from Equation (5). This discussion is made in detail in Section 6. Figure 6 shows the differences between the scan patterns, for four possible values of the distance \(e\) between the two identical Risley prisms, and for \(M=6\) (column 1) and \(M=-6\) (column 2). These values of \(e\) are different for each pair of prisms, considering their (individual) deviation angles: (a) \(D=2^{\circ}\), (b) \(D=4^{\circ}\), and (c) \(D=6^{\circ}\). The \(D=10^{\circ}\) case is not shown here anymore, as it is similar to the latter, but the obtained values are analyzed in Section 6 for such a case as well. Specifically, the two values of \(e\) are considered starting from a small one which (barely) avoids the situation when the prisms would touch, up to a maximum value \(e_{\text{max}}\) specific to each \(D\). For the latter the condition to still have the refractions inside the prisms for their \(2b\) diameter was considered for each pair of prisms. The ascertainment of these values \(e_{\text{max}}\) was done purely graphically, using the mechanical design program (as an example of the utility of this fast-to-operate method). Figure 7 presents a study of the scan patterns when varying significantly the values of the distance \(L\) from the scanner to the screen. Thus, a 10-times increase was considered for \(L\) from one pattern to another, i.e., from \(L=10\) to \(100\) m. The \(L=1\) m case was already considered in Figure 4c. These examples refer to scan patterns obtained for \(M=6\) and \(k=1\).