## 3 Factors Affecting Accuracy and Accuracy-Improvement Methods
### Effect of the Laser Spot Characteristic of the LD  
Usually, the laser spot that projects onto a QD is approximated as a circular spot with a Gaussian intensity distribution. The intensity distribution of the laser beam can be described by:  
\[I_{c}(x,\ y)=I_{0}\cdot\exp[-\frac{2(x^{2}+y^{2})}{r^{2}}], \tag{2}\]  
where \(I_{0}\) represents the light intensity of the centroid and \(r\) is the Gauss radius of the spot-intensity distribution. Most of the LDs, however, have output elliptical laser spots with different beam radii \(r_{x}\) and \(r_{y}\) in the \(X\) and \(Y\) directions and an inclined angle \(\varphi\) with respect to the \(X\) axis. The ratio between \(r_{x}\) and \(r_{y}\) is defined as ellipticity \(e\). The intensity distribution of the LD can be expressed by:  
\[I_{e}(x,\ y)=I_{0}\cdot\exp[-(\frac{2(x\cos\varphi+y\sin\varphi)^{2}}{r_{x}^{2 }}+\frac{2(-x\sin\varphi+y\cos\varphi)^{2}}{r_{x}^{2}}\cdot e^{2})], \tag{3}\]  
Using the normalization and differential algorithm of the photodetector to calculate \(\delta x\) and \(\theta x\), the measurement errors (\(\Delta\delta x\) and \(\Delta\theta x\)) induced by the elliptical laser spot can be expressed by:  
\[\left\{ \begin{array}{l}\Delta\delta x=\Delta x_{ir}-\Delta x_{ell}=K_{\delta x }\frac{(I_{\delta x+1}+I_{0-1})-(I_{\delta x+1}+C_{-1})}{I_{\delta x+1}+I_{ \delta x-1}+C_{-1}+I_{0-1}}-K_{\delta x}\frac{(I_{\delta x+1}+I_{0-1})-(I_{ \delta x-1}+C_{-1})}{I_{\delta x-1}+I_{0-1}+C_{-1}+I_{0-1}}\\ \Delta\theta x=\frac{\Delta x_{ir}}{J}-\frac{\Delta x_{ir}}{J}=\frac{1}{J} \left(K_{\delta x}\frac{(I_{\delta x+1}+I_{0-2})-(I_{\delta x+1}-2)}{(I_{ \delta x+1}+I_{0-2}+C_{-2})}-K_{\delta x}\frac{(I_{\delta x+1}+I_{0-2})-(I_{ \delta x-2}+I_{0-2})}{(I_{\delta x-2}+I_{0-2}+C_{-2}+I_{0-2})}\right)\end{array} \right. \tag{4}\]  
where \(K_{\delta x}\), \(K_{\delta x}\) and \(I_{me\text{-}i}\) (\(m=A\), \(B\), \(C\) and \(D\); \(i=1\), 2) represent the \(X\)-directional sensitivities and the output currents of QD1 and QD2 when an elliptical laser spot projects onto QD1 and QD2. A group of simulations were first carried out to investigate the effect of the elliptical laser spot on the accuracy of the LD-based system. Figure 2 shows the effect of \(r_{x}\) on \(\Delta\delta x\) and \(\Delta\theta x\). \(e\) and \(\varphi\) were set to be 0.9 and 15\({}^{\circ}\), respectively. The radii of the laser spots for measuring \(\delta x\) and \(\theta x\) were 2.5 mm and 5 um, respectively, in the designed LD-based system. As can be seen from Figure 2a, \(\Delta\delta x\) was increased with the increase of \(r_{x}\). When \(r_{x}\) and the command distance were set to be 2.5 mm and 200 um, the maximum \(\Delta\delta x\) was evaluated to be 0.0038 um, which was small enough and could be ignored. However, \(\Delta\theta x\) was increased with the decrease of the spot diameter, as shown in Figure 2b. As can be seen, the maximum \(\Delta\theta x\) was evaluated to be 11.25 arcsec when \(r_{x}\) and the command angle were set to be 5 um and 200 arcsec. Therefore, the effect of the elliptical laser spot on the measurement accuracy of the angular errors cannot be ignored. It was verified that the elliptical laser spot of the LD will affect the measurement accuracy of the angular errors but not affect that of the straightness errors. Figure 3 shows the effects of \(e\) and \(\varphi\) on the angular error measurement accuracy. As can be seen from Figure 3, in which \(r_{x}\) was set to be 5 um, \(\Delta\theta x\) was increased as \(e\) and \(\varphi\) increased. \(\varphi\) and \(e\) were set to be 15\({}^{\circ}\)and 0.95, respectively, in Figure 3a,b. Therefore, in order to improve the angular error measurement accuracy of the LD-based measurement system, it is essential to shape the laser spot of the LD.