# Optomechanical Analysis and Design of Polygon Mirror-Based Laser Scanners
## 3 Multi-Parameter Optical Analysis and Design of PM Scanning Heads  
### Scan Angles and FOV  
From Equation (12), Table 1, the two extreme scan angles \(\theta_{min}\) and \(\theta_{max}\) in Figure 5a are functions of two constructive parameters: The number \(n\) of PM facets (i.e., of the angle \(\alpha=\pi/n\)) and the unitless parameter \(x=e/R\). The latter allows for a comparison of the scan angles regardless of the PM dimension (given by the apthom \(R\)). The former influences both the scan angles and the angular FOV (equal to \(4\alpha\)). From the study of the scan angles with regard to these two parameters in Figure 6, several conclusions can be drawn:  
* From the study with regard to the parameter \(x\), Figure 6a:
* Three values of \(n\) are considered, from \(n=5\) (i.e., for a pentagonal PM utilized, for example, in commercial or industrial applications such as printers or handheld probes) to \(n=10\), and finally, to \(n=100\), with the latter being specific, for example, to high-end applications such as PM-based SSs [19; 20; 21; 22; 23; 24; 25; 26]. One can observe the way the \(2\alpha={\rm FOV}/2\) difference between the two scan angles decreases with \(n\). Thus, the FOV = \(4\alpha\) is \(144^{\circ}\) for \(n=5\), \(72^{\circ}\) for \(n=10\), but decreases to \(7^{\circ}12^{\prime}\) for \(n=100\). This is a major issue in designing PM scanning heads: While for small \(n\) commercial PMs there is a large FOV available to place the lens, for high-end applications (such as OCT), which require fast PMs with a high \(n\) (to maximize \(f_{\rm s}\)), the FOV is extremely small, therefore it might be difficult to place the objective lens within its limits. Furthermore, in the latter case, the lens must have a small diameter \(D\), therefore the linear FOV is small, as well. The positive aspect is that in applications such as OCT, this level of FOV is common, as the scan amplitude is small to avoid distortions but also to maximize the resolution in the time-limited imaging procedure specific to in vivo investigations, for example. Figure 6: Study of the extreme scan angles of the PM, \(\theta_{\rm min}\) and \(\theta_{\rm max}\) (for which the ray reaches the margins of a PM facet) with regard to (**a1**–**a3**) the angle \(\alpha=\pi/n\)) and (**b1**–**b3**) the unitless parameter \(x=e/R\) for different intervals of each of the two parameters. Regardless of the particular values of \(n\), the possible domain of the parameter \(x\) has a lower limit equal to \(\tan\alpha\) (for which \(\theta_{min}=0\) and \(\theta_{max}=2\alpha\)) and an upper limit equal to \(\sec\alpha\) (from which both \(\theta_{min}\) and \(\theta_{max}\) are not defined anymore). At the limit, \(\theta_{min,max}(x=\sec\alpha)=\pi/2\mp\alpha\). If \(x<\tan\alpha\), one would have \(\theta_{min}<0\), and this corresponds to the situation when the down-reflected ray in Figure 5a would be directed opposite to the (positive sense of the) Oz axis. One might exploit values of \(x<\tan\alpha\), depending on the required FOV, as for \(x>-\tan\alpha\), a small FOV can be still obtained. However, in this situation, the O.A. of the lens cannot be perpendicular to the incident beam, as the \(\pi/4\) value of \(\theta\) is no longer contained in the \([\theta_{min},\theta_{max}]\) interval. Actually, to contain the \(\theta=\pi/4\) value in the \([\theta_{min},\theta_{max}]\) interval, one must have \(\theta_{max}<\pi/4\), which gives, using Equation (12),  
\[x<(1-\tan\alpha)/\sqrt{2} \tag{20}\]  
(b) From the study with regard to the angle \(\alpha\), Figure 6b:  
* Two specific intervals of the parameter \(x\), smaller and larger than 1, are considered, as well as the particular case \(x=1\). Although \(\theta_{min}\) and \(\theta_{max}\) are considered as functions of the angle \(\alpha=\pi/n\), the rows of values of the two angles are actually relevant, as only integer values of the number \(n\) of facets correspond to reality.