# Optomechanical Analysis and Design of Polygon Mirror-Based Laser Scanners
## 3 Multi-Parameter Optical Analysis and Design of PM Scanning Heads
### Spot Variations Due to the Finite Diameter of the Laser Beam  
The variation of the dimensions of the laser spot on a lens or a scanned plane (as a consequence of the scanning process) is an issue that has to be approached for any type of scanner [1,2]. Therefore, the study in Figures 2 and 3b is extended in Figure 8 by considering the finite diameter (\(2\rho\)) of the laser beam incident on a PM facet. The discussion performed so far referred to a single ray (i.e., the center axis of the laser beam), represented with a solid line in Figure 8. For the left and right margins of the beam in the meridian plane, the scanning functions in Table 1 can be written by considering, in Equation (4), \(L-\rho\) and \(L+\rho\), respectively (instead of \(L\)), as well as \(e+\rho\) and \(e-\rho\)  
Figure 8: **(a)** Reflection of a laser beam (of finite diameter \(2\rho\)) on a PM facet for \(\theta=\theta_{1}\), \(\pi/4\), and \(\theta_{2}\)— as considered in Figure 2b for the axis of the beam; **(b)** reflection of the laser beam at the beginning and end of a PM facet. **(c)** The scanning function for the entire beam—generalized case of Figure 3b1, with **(d)** the variation \(\Delta y(\theta)\) of the spot on a direction perpendicular to the O.A. respectively (instead of \(e\)). With these considerations, for the left and right margins of the beam, one has  
\[h_{left,right}(\rho,\theta)=R\sqrt{2}-(e\mp\rho)-\frac{R}{\cos\theta}+(e\mp\rho) \cdot tan\theta-\frac{L\pm\rho}{\tan 2\theta} \tag{32}\]  
However, one must note that these two functions, \(h_{left}(\theta)\) and \(h_{right}(\theta)\), are measured from Figure 8 with regard to the positions \(h_{left}(\pi/4)\) and \(h_{right}(\pi/4)\) from the O.A., which are equal to \(-\rho\) and \(\rho\), respectively. Therefore, to return to the Oyz system of coordinates, the left and the right margins of the beam are characterized (with regard to the Oz axis) by  
\[ \begin{array}{l}h_{left}(\theta)-\rho=R\sqrt{2}-e-\frac{R}{\cos\theta}+(e- \rho)\cdot tan\theta-\frac{L+\rho}{\tan 2\theta}\\ h_{right}(\theta)+\rho=R\sqrt{2}-e-\frac{R}{\cos\theta}+(e+\rho)\cdot tan\theta -\frac{L-\rho}{\tan 2\theta}\end{array}  \tag{33}\]  
Therefore, the dimension of the laser spot on the \(y\)-axis is  
\[\Delta y(\theta)=\left(h_{right}(\theta)+\rho\right)-\left(h_{left}(\theta)- \rho\right)=2\rho/\sin 2\theta. \tag{34}\]  
One can remark that, from Equations (4) and (33) one has  
\[(h_{right}(\theta)+\rho)-h(\theta)=h(\theta)-(h_{left}(\theta)-\rho)=\rho(1/ \sin 2\theta-1), \tag{35}\]  
therefore the laser spot is symmetrical with regard to the axis of the beam, as it is also observed from Figure 8b. The graph of the \(\Delta y\) function is presented in Figure 8c. At the margins of the scanning domain, the variation of the spot can be evaluated as  
\[\Delta y(\theta_{min,max})=2\rho/\sin 2\theta_{min,max} \tag{36}\]  
However, the problem at the margins of the FOV is more complex, as it can be seen from Figure 8d: At the beginning and at the end of a PM facet, the beam is actually split in two, and one has a down-reflected part of the beam, corresponding to \(\theta=\theta_{\min}\), and an up-deflected part, corresponding to \(\theta=\theta_{\max}\). This particular consequence of the initial theory in [29; 32] recently led to a development of a PM-based surface cleaning system [35]. A numerical example related to the discussion on the spot variation is presented in Figure 8c for a PM with \(n=5\) facets, \(R=15\) mm, and \(e=13.58\) mm. For a beam with the radius \(\rho=0.15\) mm and for \(\theta_{1}=\pi/6\) and \(\theta_{2}=\pi/3\) (symmetrical with regard to \(\theta=\pi/4\)), one obtains \(\Delta y(\theta_{1})=\Delta y(\theta_{2})=0.346\) mm. From Equation (12), \(\theta_{\min}=0.104\) rad and \(\theta_{\max}=1.36\) rad, while the corresponding dimension of the spot on the \(y\)-axis would be \(\Delta y(\theta_{\min})=0.736\) mm and \(\Delta y(\theta_{\max})=1.446\) mm if the spot was entirely contained on the PM facet for the beginning and end of a facet scan.