## 5 Multi-Parameter Analysis of Angular and Linear Deviations
### Multi-Parameter Analysis of Linear Deviations  
Figure 11 shows a study of the minimum linear deviations/radiuses \(\Delta_{j}(\theta),\,j=a,\,b,\,c,\,d\) of the blind spot, i.e., of the central disk of the scan patterns. As discussed in Section 3, these radiuses refer to the case of two identical prisms (with \(n_{1}=n_{2}=n\,\) and \(\theta_{1}=\theta_{2}=\theta\), therefore \(k\)=1), when emergent beams are in this case parallel to the O.A., as marked with dotted lines in Figure 2(a2,b2,c2,d2). For the four scanner configurations (Figure 2), these radiuses are given by: (a) Equation (15), (b) Equation (21), (c) Equation (27), and (d) Equation (35). The characteristic parameters of this study are the prism angles \(\theta\), the distance \(e\) between them, their refractive index \(n\), and the radius \(b\) of each circular prism. In each of the four steps of the following analysis, three of these parameters were kept constant, while the other ones were considered with two or three values to determine the variance of the \(\Delta_{j}(\theta),\,j=a,\,b,\,c,\,d\) functions. A maximum limit of \(45^{\circ}\) for the prism angle \(\theta\) was considered, although considering a value of around \(18^{\circ}9^{\prime}\approx 0.314\) rad for the optical wedge angle (to produce an individual prism deviation \(D\) equal to \(10^{\circ}\)) [60] is also useful when referring to the most utilized devices. Considering Figure 11, on its columns are the following:  
(1) Exact radiuses \(\Delta_{j}(\theta),\,j=a,\,b,\,c,\,d\) and approximate/linearized ones \(\Delta_{j}^{lin}(\theta),\,j=a,\,b,\,c,\,d\), the latter given by (a) Equation (16), (b) Equation (22), (c) Equation (28), and (d) Equation (36) were determined for the most common parameters \(n=1.517,\,e=25\) mm, and \(b=12.7\) mm (the latter corresponding to commercially-available prisms of \(1^{\prime\prime}\) in diameter [60]). The difference \((\Delta_{j}-\Delta_{j}^{lin})(\theta)\) between these exact and approximate radiuses for each scanner was also determined. One can see the differences between the four configurations, with configuration (b) providing the smallest values of \(\Delta\), and configuration (d) the largest ones. From the point of view of the linearity, the approximation \(\Delta_{j}^{lin}\) works up to \(D=10^{\circ}\), and even up to \(\theta=\pi/8\). The limit angles calculated in Equations (40)-(47) apply here as well. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Scanner** & \multicolumn{2}{c}{**Refractive Index**} \\ \cline{2-3}
**Configurations** & \(\mathbf{n=1.517}\) & \(\mathbf{n^{\prime}=1.7}\) \\ Figure 2a & \(\theta_{a}^{lin}(n)=0.523=\pi/6\) & \(\theta_{a}^{lim}(n^{\prime})\cong 0.43\) \\ Figure 2b & \(\theta_{b}^{lim}(n)=0.72\) & \(\theta_{b}^{lim}(n^{\prime})\cong 0.6\) \\ Figure 2c & \(\theta_{c}^{lim}(n)=0.96\) & \(\theta_{c}^{lim}(n^{\prime})\cong 0.785=\pi/4\) \\ Figure 2d & \(\theta_{d}^{lim}(n)=0.53\) & \(\theta_{d}^{lim}(n^{\prime})\cong 0.45\cong\pi/7\) \\ \hline \hline \end{tabular}  
* Remark: One must highlight that these (exact) limit angles must be used in general, while the approximate values obtained in Equation (42), (44), and (47) are restricted to the small angle domain (i.e., for \(\theta<10^{\circ}\)).  
\end{table} 
Table 2: Limit prism angle \(\theta_{j}^{lin}(rad),\,j=a,\,b,\,c,\,d\) with regard to refractive index (the identical prisms case, i.e., \(k=1\)). (2) Exact radiuses \(\Delta_{j}(\theta)\), \(j=a,\ b,\ c,\ d\) were studied for three values of the distance between the prisms \(e_{1}=25\), \(e_{2}=50\), and \(e_{3}=75\) mm (with \(n=1.517\) and \(b=12.7\) mm), with the remark that much larger values of \(e\) can be considered, as studied in Figure 6. The same interval (of up to 20 mm) was set here for \(\Delta\), and the same conclusion can be reached for the smallest value of \(e\). For the two larger \(e\), \(\Delta\) gets closer for the four configurations, (a) to (d); therefore, if one aims to minimize \(\Delta\), going for the smallest possible \(e\) is the solution.