# Optomechanical Analysis and Design of Polygon Mirror-Based Laser Scanners
## 6 Optomechanical Design Scheme  
The calculus of some of the main constructive parameters of PM scanning heads is rather straightforward, as described in [1]. The synthesis of such an algorithm is shown in Figure 16. Essentially, the imposed scan frequency \(f_{\mathrm{s}}\) provides the number of PM facets  
\[n=2\pi\cdot f_{\mathrm{s}}/\omega, \tag{48}\]  
By considering technological, but also cost, limitations that provide the maximum possible rotational velocity \(\omega\). On the other hand, the calculus starts from the necessary  
Figure 16: Optical scheme for dimensioning a PM—from the brief classical calculus in [1]—with optical and mechanical issues left to be solved by designers and manufacturers. Figure 15: FEA of a hexagonal PM for \(\omega=120\) krpm, \(R=25\) mm, \(r=5\) mm, and \(b=10\) mm. Variation with regard to the radius of the circle on which the centers of the supplemental holes are placed, \(R_{\mathrm{h}}\), from (**a**) 13 mm to (**b**) 14 mm; with regard to the diameter \(\delta\) of these holes from (**a**) \(\delta=2\) mm to (**c**) \(\delta=3\) mm; with regard to the situation when these holes are placed in the direction of the PM apex (**a**) versus in the direction of the middle of PM facets (**d**). laser beam diameter \(2\rho\) that must be accommodated on the PM facet with a maximum admissible level of vignetting (e.g., with a common value of 50% [2,26]). This beam diameter and vignetting level give the necessary width \(a\) of a PM facet--Figure 2a. Therefore, using Equation (48) and \(\alpha=\pi/n\), the PM apothem is  
\[R=0.5\cdot a/\tan\alpha. \tag{49}\]  
The issue is that all other optical and mechanical aspects are left to be solved by the designer and manufacturer, respectively. Simplified, approximate (i.e., linearized) equations have been utilized for aspects such as scanning function and velocity, or the PM position, as well as (mostly) experimental approaches in obtaining functional setups [18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. To fulfil the scope of this study to offer the community a tool to approach the optical aspects (pointed out in the left-bottom block in Figure 16) in a rigorous, but as simple as possible way, they are detailed in the left-side track in Figure 17, based on the equations deduced and discussed in this work. The most important aspect is the off-axis position \(e\) of the PM and the impact it has on the deduced characteristic functions and parameters, as discussed in Sections 2 and 3. Figure 17: Optomechanical designing scheme of PM scanning heads connecting optical and mechanical aspects. On the other hand, a more overlooked part of a PM design refers to mechanical aspects, as pointed out in the left-bottom block in Figure 16. These are mostly left to manufacturers, and they make progress pushed by ever-increasing requirements of high-end applications such as OCT, evolving in a trade-off between performances and costs. Based on the present work, in the more detailed optomechanical scheme proposed in Figure 17, these aspects are listed in its right-side track and linked to the optical track in its left-side track. Such links in this concluding scheme include the following:  
* The scan frequency \(f_{\rm s}\), in connection to the maximum chosen level of the rotational speed \(\omega\), imposes, through the FEA, the type of material for the PM. * From this step, the manufacturing method is imposed, in relation to the type of scanner that results, from a range of optical parameters (for a PM, with normal or inverted, prismatic and pyramidal, simple or double polygons [1; 2]). * From the type of scanner, one can chose the scanner architecture: Pre- or post-objective, with an on- or off-axis PM, telescope- or telescope-less, with different types of lenses (or prisms), with single- or double-pass, etc. * The previous point, together with the optical parameters calculus, leads to the optical design and then to the geometric parameters of PM, including eccentricity e and distance L--with the functions developed in the present study.