# Optomechanical Analysis and Design of Polygon Mirror-Based Laser Scanners
## Appendix A  
The Myz system of coordinates is considered in Figure 2b. Its origin is the incidence point M = P(\(\theta=\pi/4\)), which characterizes the horizontal position of the reflected ray (on the O.A. of the lens). P(\(y,z\)) is the current point of incidence of the laser ray on the PM facet. To obtain the coordinate \(y\) of the point P, we write the equation of the PM facet on which the reflection is produced in the fixed system of coordinates O\(\mu\zeta\) (with the origin in the PM pivot O):  
\[\left\{ \begin{aligned} \zeta&=\zeta_{B}+(\mu-\mu_{B})tan \theta\\ \mu_{B}&=Rsin\theta;\zeta_{B}=-Rcos\theta\end{aligned} \right. \tag{1}\]  
where point B is the base of the perpendicular from O to the facet. As M is the incidence point for \(\theta=\pi/4\), from Equation (1)  
\[\left\{ \begin{aligned} \zeta_{P}&=-Rcos\theta+(e-Rsin \theta)tan\theta\\ &\zeta_{M}=e-R\sqrt{2}\end{aligned} \right. \tag{2}\]  
Therefore, as \(y=\zeta_{P}-\zeta_{M}\), using the above equations, the expression of the scanning function that can be deduced from Figure 2b  
\[h(\theta)=y(\theta)-L/tan2\theta \tag{3}\]  
gives the final Equation (4) in Table 1.  
* _[PERSON] et al. (2005)_ [PERSON]; [PERSON]; [PERSON]; [PERSON] 115 kHz tuning repetition rate ultrahigh-speed wavelength-swept semiconductor laser. _Opt. Lett._**2005**, _30_, 3159-3161. [CrossRef]
* [PERSON] et al. (2009) [PERSON]; [PERSON]; [PERSON]; [PERSON] High performance wavelength-swept laser with mode-locking technique for optical coherence tomography. _Opt. Commun._**2009**, _282_, 88-92. [CrossRef]
* [PERSON] et al. (2015) [PERSON]; [PERSON] [PERSON]; [PERSON]; [PERSON] Simple and versatile long range swept source for optical coherence tomography applications. _J. Opt._**2015**, _17_, 125301. [CrossRef]
* [PERSON]. High-speed polygon-scanner-based wavelength-swept laser source in the telescope-less configurations with application in optical coherence tomography. _Opt. Lett._**2008**, _33_, 1741-1743. [CrossRef]
* [PERSON] et al. (2009) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]. High-power wavelength-swept laser in Littman telescope-less polygon filter and dual-amplifier configuration for multichannel optical coherence tomography. _Opt. Lett._**2009**, _34_, 2814-2816. [CrossRef]
* [PERSON] et al. (2008) [PERSON]; [PERSON]; [PERSON] [PERSON]; [PERSON] Spectral narrowing effect by quasi-phase continuous tuning in high-speed wavelength-swept light source. _Opt. Express_**2008**, _16_, 21105-21118. [CrossRef] [PubMed]
* [PERSON] et al. (2017) [PERSON]; [PERSON]; [PERSON] Geometric & radiometric vignetting associated with a 72-facet, off-axis, polygon mirror for swept source optical coherence tomography (SS-OCT). _AIP Conf. Proc._**2017**, _1796_, 040004. [CrossRef]
* [PERSON] et al. (2007) [PERSON]; [PERSON]; [PERSON]. Double-reflection polygon mirror for high-speed optical coherence microscopy. _Opt. Lett._**2007**, _32_, 3528-3530. [CrossRef] [PubMed]
* [PERSON] (2005) [PERSON] On-line measurements with optical scanners: Metrological aspects. _Proc. SPIE_**2005**, _5856_, 606-618. [CrossRef]
* [PERSON] (2017) [PERSON] Analysis of polygonal mirror scanning heads: From industrial to high-end applications in swept sources for OCT. _Proc. SPIE_**2017**, _100560_, 121-131. [CrossRef]
* [PERSON] and [PERSON] (2010) [PERSON]; [PERSON] Mechanical Constraints and Design Considerations for Polygon Scanners. _Mech. Mach. Sci._**2010**, \(5\), 475-483. [CrossRef]
* [PERSON] and [PERSON] (2013) [PERSON]; [PERSON] Polygon mirror scanners in biomedical imaging: A review. _Proc. SPIE_**2013**, _8621_, 175-183. [CrossRef]
* [PERSON] (2017) [PERSON] Polygonal mirror laser scanning heads: Characteristic functions. _Proc. Rom. Acad. Ser. A_**2017**, _18_, 25-33. * [PERSON] et al. (2019) [PERSON]; [PERSON]; [PERSON] Minimization of mixed-color image noise caused by shape error of polygonal mirror in color-laser printing system. _Micusyst. Technol._**2019**, _26_, 25-32. [CrossRef]
* [PERSON] et al.