# Optomechanical Analysis and Design of Polygon Mirror-Based Laser Scanners
## 3 Multi-Parameter Optical Analysis and Design of PM Scanning Heads
### Scanning Function and Velocity  
The first study of the scanning function of a PM versus a GS is shown in Figure 3b--for a certain set of the parameters \(R\), \(x\), and \(L\). The asymmetry of the PM functions (in contrast to the symmetry of the dotted GS functions) can be observed, as well as the fact that the minimum scanning speed for GSs is reached for \(\theta=\pi/4\) (i.e., from Equation (3), with the acceleration \(a_{\text{GS}}(\theta=\pi/4)=0\)), while for PMs, the condition \(v=\min\). provides, by using Equation (5) in Table 1, the following equation:  
\[R\sin^{5}\theta-2e\sin^{4}\theta+R\sin^{3}\theta-2L\sin^{2}\theta+L=0. \tag{25}\]  
It can be demonstrated numerically that this equation has solutions for values higher than (but, in general, close to \(\theta=\pi/4\). From \(a(\pi/4)\), using Equation (6), the condition \(v(\pi/4)=\min\). is fulfilled, in a particular case, for  
\[x=3\sqrt{2}/4\cong 1.06. \tag{26}\]  
However, in general, for PMs  
\[v(\pi/4)=2\omega\big{(}L+e+R\sqrt{2}\big{)}\
eq v_{min}, \tag{27}\]  
with two more terms with regard to the GS case, for which \(v(\pi/4)=\min\). \(=2\omega L\). The main aspect of the scanning functions of both GSs and PMs is their non-linearity. Even for PMs, one may observe that the scanning velocity is variable, as \(a(\theta)\
eq 0\) regardless of the relationship between the \(e\), \(R\), and \(L\) parameters. Moreover, given the common values of this set of parameters, it can be easily observed that the condition \(v(\theta)>0\) is always fulfilled, therefore the scanning function is strictly increasing; there is no turning back of the scanning beam. A multi-parameter study of the PM scanning function and velocity in this respect, with regard to all these three characteristic parameters, is performed in Figure 7:  
1. The apothem \(R\) is considered from 15 to 40 mm, although smaller values are also utilized in practice (e.g., for square PMs in printers), while higher values of \(R\) are necessary for large PMs utilized in high-end applications for which high \(f_{\text{s}}\) values are acquired. To achieve the latter, the PMs must have a high \(n\), therefore \(R\) must be large to keep the facet aperture over a useful threshold (to accommodate a laser beam of a certain diameter and minimize the negative effect of vignetting). It can be observed that a larger \(R\) means a better linearity of the scanning function \(h(\theta)\) and an approximate constant scan velocity \(v(\theta)\) over a slightly larger domain of \(\theta\), therefore such an \(R\) is advantageous. However, it also implies a trade-off with structural aspects, as approached using FEA in Section 5. 2. The parameter \(x=e/R\) is considered with three values, corresponding to the three characteristic cases considered in Figure 6, as well for \(x<1\) or \(x>1\), and with their in-between limit case \(x=1\). As at the previous point, a slightly more linear \(h(\theta)\) and more constant \(v(\theta)\) over a larger domain of \(\theta\) can be obtained with a smaller \(x\), i.e., for the \(x<1\) case. This result is in good agreement with the discussion on the scan angles and the most advantageous case from that point of view--Figure 6(b1). 3. The distance \(L\) is considered from the small value of 20 mm, which allows one to capture a large part of the graph of the scanning function and to highlight its non-linearity--as observed in the experimental study in [32], as well. The largest considered dimension of \(L\) is 200 mm, which may require folding the beam, as addressed further in Section 4. As \(L=f\) (Figure 2), a more common value of \(L\) would be approximately 40 mm, as considered in Figure 3b, to have a reasonable dimension of the scanning head. However, for such a value, the non-linearity of \(h(\theta)\) is already manifested for common values of the scan angles (i.e., FOV), therefore the solution developed in Section 4 is useful, as well.