## 2 Scanners with Pairs of Rotational Risley Prisms
Notations: _‘a’_ denotes a diopter perpendicular to the optical axis (O.A.) and _‘b’_ a tilted one. Each configuration is shown in two extreme characteristic positions of the device, namely with a relative rotation angle between the two prisms \(\varphi=0\) (column **1**) and \(\varphi=\pi\) (rad) (column **2**). The extreme positions of the two rotational prisms are considered in Figure 2, namely for a relative rotational angle \(\varphi\) between the two prisms equal to 0 (on column 1) and equal to \(\pi\) (on column 2), respectively. The trajectories of the laser beam (considered reduced to its center ray) are pointed out in Figure 2 for each scanner configuration. From [PERSON]'s refraction law, as well as from simple geometric considerations, the prism equations for each of the four scanner configurations (considering the two relative positions of the prisms in Figure 2) are provided in Table 1. They are necessary to proceed to simulations, but also to deduce and analyze angular and linear deviations of the beam through the system. As a necessary remark, these equations can be utilized in simulations to trace the rays through the prism system--by 'rotating' each prism using the mechanical design program into different positions. However, they do not provide analytically the directions of the emerging rays to generate the scan patterns. For the latter, the (rather) complex mathematical approaches pointed out in the Introduction, for example [5; 6; 7], must be considered. _The minimum and maximum deviation angles \(D_{j}^{max,min}\), \(j=a\), \(b\), \(c\), \(d\)_ can be easily deduced by combining (on each of the four columns in Table 1) the equations of each scanner configuration. From Figure 2a, for scanner (a) _ab-ab_  
\[D_{a}^{max,min}=arcsin\bigg{\{}n_{2}sin\bigg{[}\pm\theta_{2}+arcsin\frac{sin [arcsin(n_{1}sin\theta_{1})-\theta_{1}]}{n_{2}}\bigg{]}\bigg{\}}\mp\theta_{2}. \tag{2}\]  
From Figure 2b, for scanner (b) _ab-ba_  
\[D_{b}^{max,min}=arcsin\bigg{\{}n_{2}sin\bigg{[}\pm\theta_{2}+arcsin\frac{sin[ arcsin(n_{1}sin\theta_{1})-(\theta_{1}\pm\theta_{2})]}{n_{2}}\bigg{]}\bigg{\}}. \tag{3}\]  
From Figure 2c, for scanner (c) _ba-ba_  
\[D_{c}^{max,min}=arcsin\left\{n_{2}sin\bigg{[}\pm\theta_{2}+arcsin\frac{sin \Big{\{}arcsin\Big{[}n_{1}sin\Big{(}\theta_{1}-arcsin\frac{sin\theta_{1}}{n_{ 1}}\Big{)}\Big{]}\mp\theta_{2}\Big{\}}}{n_{2}}\bigg{]}\bigg{\}}. \tag{4}\]  
From Figure 2d, for scanner (d) _ba-ab_  
\[D_{d}^{max,min}=arcsin\bigg{\{}n_{2}sin\bigg{[}\pm\theta_{2}+arcsin\bigg{(} \frac{n_{1}}{n_{2}}sin\bigg{(}\theta_{1}-arcsin\frac{sin\theta_{1}}{n_{1}} \bigg{)}\bigg{)}\bigg{]}\bigg{\}}\mp\theta_{2}. \tag{5}\]  
\begin{table}
\begin{tabular}{c c c} \hline
**Configuration (Figure 2)** & **(a) Scanner _ab-ab_ & **(b) Scanner _ab-ba_** & **(c) Scanner _ba-ba_** \\ \hline _Incidence angle_ & \(\varepsilon_{1}=0\) & \(\varepsilon_{1}=-\theta_{1}\) \\ Point I\({}_{1}\) & \(\varepsilon_{1}^{\prime}=\varepsilon_{1}\) & \(\text{ }\text{  
By considering the first order/Gaussian approximation, from Equations (2)-(5) _the approximate (linearized) deviation angles_ have, for all four configurations, the following general (and well-known) equation:  
\[D_{lin}^{max,\ min}=(n_{1}-1)\theta_{1}\pm(n_{2}-1)\theta_{2}. \tag{6}\]  
Therefore, for the particular case of identical wedges, when \(n_{1}=n_{2}=n\) and \(\theta_{1}=\theta_{2}=\theta\),  
\[k=\theta_{2}/\theta_{1}\text{ and }M=\omega_{2}/\omega_{1}. \tag{1}\]0