# 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  
From the point of view of the movement of the reflected/emergent laser beam (and of the corresponding spot on a scanned plane or objective lens), both types of scanners are characterized by several functions, as pointed out in Figures 1b and 2b:  
(i) _The scanning function_\(h(\theta)\) is defined by the (current) position of the laser spot on the scanned plane/lens or of the laser beam emerging the lens with regard to its optical axis (O.A.). To allow for a comparison of GS and PM scanning heads, the O.A. of both scanners is considered perpendicular on the fixed incident laser beam, in a common setup utilized in numerous applications [1]. Furthermore, to allow for a comparison of the functions of the two systems, the \(\theta=0\) position is considered for a horizontal position of the GS mirror, as well as for a horizontal PM facet (i.e., for the PM apothem on the vertical axis O\(Q\))--Figure 2b. (ii) _The scanning velocity_\(v(t)=\mathrm{d}h/\mathrm{d}t\) represents the sweeping velocity of the laser beam on the scanned plane or in the image space of the lens. (iii) _The scanning acceleration_\(a(t)=\mathrm{d}v/\mathrm{d}t\) represents the acceleration of the laser beam in the scan direction--perpendicular on the O.A. For the GS, the condition of having the reflected beam permanently parallel to the O.A. (i.e., with both \(v(t)\) and \(a(t)\) perpendicular to the beam emerging the lens) is fulfilled by placing the pivot O in the object focal point of the lens. For the PM, this condition can be fulfilled only approximately, as discussed in Section 2.3, but it also represents the desired aspect. These three functions are obtained for a GS from Figure 1b and for a PM scanning head from Figure 2b:  
* For a GS, the start position of the GS mirror is at \(\pi/4\) [rad] with regard to the horizontal axis Oz. As the mirror rotates with \(\theta\), the reflected beam rotates with \(2\theta\), therefore the scanning function would be \(h(\theta)=L\cdot\tan 2\theta\). While this simple equation is usually utilized for GSs [7,9,10], in the present study, to allow for a comparison to PMs, the \(\theta\) = 0 position is considered for a horizontal position of the GS mirror (when the reflected beam would be superposed to the incident one), as previously pointed out. In this case, the scanning function results in being \(h(\theta)=-L/\tan 2\theta\), therefore the scanning velocity and acceleration have Equations (2) and (3), respectively (Table 1). The graphs of these three functions are presented in Figure 3a. For a PM scanning head, a facet is represented in three positions in Figure 2b, considering \(\theta\) = 0 for the facet in a horizontal position. Similar to GSs, the rotational angle \(\theta\) = \(\pi/4\) defines the facet position in which the (fixed vertical) incident beam is reflected on the (horizontal) Oz axis, which defines the O.A. of the lens. For \(\theta_{1}\) and \(\theta_{2}\) (with \(\theta_{1}<\pi/4<\theta_{2}\)), the reflected beam reaches the inferior and superior margins of the lens in the meridian plan Oyz, therefore \(h(\theta_{12})=\mp H\) (where \(H<D/2\)), Figure 2. Points P1,