# Optomechanical Analysis and Design of Polygon Mirror-Based Laser Scanners
## 2 Characteristic Functions and Parameters of PMs versus GSs
### Characteristic Functions of GSs and PM Scanning Heads
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Characteristic Function/Parameter** & \multicolumn{2}{c}{**Galvanometer Scanner**} & \multicolumn{2}{c}{**Polygon Mirror (PM) Scanning Head**} \\ \hline Scanning function & \(h(\theta)=\frac{-L}{\tan 2\theta}\) & (1) & \(h(\theta)=y(\theta)-\frac{L}{\tan 2\theta}=\) \\  & \(h(\theta)\) & \(h(\theta)=\frac{-L}{\tan 2\theta}\) & (1) & \(h(\theta)=y(\theta)-\frac{L}{\cos\theta}=e\cdot\tan\theta-\frac{L}{\tan 2\theta}\) \\ \hline Scanning velocity & \(v(t)=\frac{dt}{d\theta}\cdot\frac{d\theta}{dt}\) & \(v(t)=\frac{2\omega L}{\sin^{2}2\theta}\) & (2) & \(v(t)=\frac{\omega}{\cos^{2}\theta}\left[e-R\cdot\sin\theta+\frac{L}{2\sin^{2} \theta}\right]\) & (5) \\ \hline Scanning acceleration & \(a(t)=\frac{dt}{d\theta}\cdot\frac{d\theta}{dt}\) & \(a(t)=\frac{-8\omega^{2}L\cdot\cos 2\theta}{\sin^{3}2\theta}\) & (3) & \(\frac{a(t)}{a(t)}=\) \\  & & & \(\frac{\omega^{2}}{\cos^{3}\theta}\left[2e\cdot\sin\theta-R\left(1+\sin^{2} \theta\right)-\frac{L\cdot\cos 2\theta}{\sin^{3}\theta}\right]\) & (6) \\ \hline \multicolumn{2}{c}{Transversal} & \multicolumn{2}{c}{Transversal} \\ Migration functions & - & \(y(\theta)=R\sqrt{2}-e-\frac{R}{\cos\theta}+e\cdot\tan\theta\) & (7) \\  & & & Longitudinal/axial & \\  & & \(z(\theta)=y(\theta)\cdot\tan 2\theta\) & (8) \\ \hline \multicolumn{2}{c}{\(\pi/2-\theta_{m}\) and \(\theta_{m}\),} & \multicolumn{2}{c}{where} \\ Scan angles, & \multirow{2}{*}{Extreme} & \(2\theta_{m}-\pi/2=\) & (9) & \(\theta_{min,max}=\arcsin\frac{e\cos\theta}{R}\mp\alpha\) \\ Figures 1a, 2a & & \(\mathrm{atan}(L/H)\), with \(H\leq\) & & \\ and 3 & & \(D/2\) & & \\  & & \(\pi/2-\theta_{a}\) and \(\theta_{a}\), & & \\  & & & where & \\  & & \(2\theta_{a}-\pi/2=\) & (10) & \\  & & \(\mathrm{atan}(L/x_{a})\), with \(x_{a}\leq\) & (13) & \\  & & \(H\) & & \\ \hline Field of View & Angular & \(Dom\theta=2\theta_{m}-\pi/2\) & (11) & \(Dom\theta=2(\theta_{max}-\theta_{min})=4\alpha\) \\ Field of View & \multirow{2}{*}{Linear/ Scanning domain} & \(Dom\ h=2h(\theta_{m})=\) & (15) & \(Dom\ h=h(\theta_{max})-h(\theta_{min})=\) \\  & & \(\frac{-2L}{tan2\theta_{m}}\) & (15) & \(\frac{\sin 2\alpha}{(1-x^{2})\cos^{2}\alpha}\left[2e+L\frac{\cos 2\alpha}{(2x^{2} \cos^{2}\alpha-sin^{2}\alpha)}\right]\), & (16) \\  & & & where \(x=e/R\) & \\ \hline Duty cycle, & Theoretical (of the input) & \(\eta_{t}=\frac{t_{s}}{t}=1-\frac{\Delta t}{t}\) & (17) & \(\eta=\frac{\theta_{2}-\theta_{1}}{\theta_{max}-\theta_{min}}=n\frac{\theta_{1} -\theta_{1}}{2\pi}\) & (19) \\  & & & & \\ and 2a & & output) & \(\eta=\frac{t^{\prime}_{s}}{t}\) & (18) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Overview of characteristic functions and parameters of GSs versus PM scanning heads. M, and P\({}_{2}\) are the intersections of the incident ray with the PM facet for these three rotation angles considered (i.e., \(\theta_{1}\), \(\pi/4\), and \(\theta_{2}\), respectively). For a current rotational angle \(\theta\), from Appendix A, the scanning function is deduced, with Equation (4), therefore the scanning velocity and acceleration are obtained with Equations (5) and (6), respectively (Table 1). An experimental validation of this PM scanning function was completed in [32]. * The comparison of each corresponding pair of functions (i.e., for GSs and PMs) is performed in Figure 3b, on a numerical example with \(L\) = 40 mm, \(R\) = 15 mm, \(e\) = 6 mm, and \(\omega\) = 10 krpm (the latter corresponding to an idle rotation of the PM). The asymmetry of the PM functions graphs can be remarked, in comparison to the symmetry of (dotted graphs of) GS functions--to be discussed in Section 3. Figure 3: Scanning function, velocity, and acceleration of (**a1–a3**) GS; (**b1–b3**) PM scanning head versus GS (for the latter the graphs are marked with dotted lines).