# Improved Autocollimator for Roll Angle Measurement with an Enlarged Measuring Range
## 2 Methods
### Measurement of Roll Angle  
The classic autocollimator does not work for measuring the roll angle, since the light spot on the photodetector does not move as the PM rolls. To overcome this difficulty, the WP is inserted in front of the PM and rolls with a target to be measured, but the PM remains stationary. Figure 2a illustrates the roll angle measuring principle. A light beam can be represented by a directing vector following its propagation direction and with a length equal to the refractive index of the medium in which this beam propagates, whereas a reflecting or refracting surface can be represented by its normal vector pointing from the incident medium to the interface, with a length of 1. In this way, the vector form of the reflection and refraction laws can be applied to calculate the directions of reflected and refracted beams. Before the WP rolls, as the collimated beam passes through the WP's front and back, the PM, and the WP's back and front in sequence, the normal vectors of each refracting and reflecting surface can be expressed as  
\[\left\{ \begin{array}{l}N_{1}=\left[0\;0\;-1\right]^{\mathrm{T}},\\ N_{2}=\left[-\theta\;0\;-1\right]^{\mathrm{T}},\\ N_{3}=\left[0\;0\;-1\right]^{\mathrm{T}},\\ N_{4}=\left[\theta\;0\;1\right]^{\mathrm{T}},\\ N_{5}=\left[0\;0\;1\right]^{\mathrm{T}},\end{array} \right. \tag{1}\]  
where \(\theta\) signifies the wedge angle of the WP and takes a small value. The collimated beam with directing vector \(A_{0}= \begin{bmatrix}0&0&-1\end{bmatrix} ^{\mathrm{T}}\) propagates against the \(z\)-axis to the WP and is split into two beams by its front. According to the reflection law [23], the directing vector of the reflected beam is calculated as  
\[A_{1}=A_{0}-2(A_{0}\cdot N_{1})N_{1}=\left[0\;0\;1\right]^{\mathrm{T}}. \tag{2}\]  
Then, this beam converges on the origin of the CMOS and forms light spot 1, as marked with a red ring in Figure 2b. For the transmitted beam, according to the refraction law [23], its directing vector is calculated as  
\[A_{2}=A_{0}+\left(\sqrt{n^{2}-1+\left(A_{0}\cdot N_{1}\right)^{2}}-A_{0}\cdot N _{1}\right)N_{1}=\left[0\;0\;-n\right]^{\mathrm{T}}, \tag{3}\]  
Figure 1: Schematic of improved autocollimator for roll angle measurements. where \(n\) signifies the refractive index of the WP. Subsequently, this beam is refracted by the WP's back, reflected by the PM, and refracted again by the WP's back and front. The directing vectors of these refracted and reflected beams involved in the above process are calculated as  
\[\left\{ \begin{array}{l}A_{3}=A_{2}+\left(\sqrt{1-n^{2}+\left(A_{2}\cdot N_{2} \right)^{2}}-A_{2}\cdot N_{2}\right)N_{2}=[\theta(n-1)\ 0\ -1]^{\mathrm{T}},\\ A_{4}=A_{3}-2(A_{3}\cdot N_{3})N_{3}=[\theta(n-1)\ 0\ 1]^{\mathrm{T}},\\ A_{5}=A_{4}+\left(\sqrt{n^{2}-1+\left(A_{4}\cdot N_{4}\right)^{2}}-A_{4}\cdot N _{4}\right)N_{4}=[2\theta(n-1)\ 0\ n]^{\mathrm{T}},\\ A_{6}=A_{5}+\left(\sqrt{1-n^{2}+\left(A_{5}\cdot N_{5}\right)^{2}}-A_{5}\cdot N _{5}\right)N_{5}=[2\theta(n-1)\ 0\ 1]^{\mathrm{T}}.\end{array} \right. \tag{4}\]  
Ultimately, the beam refracted by the WP's front also focuses on the CMOS and forms light spot 2, as marked with the green ring in Figure 2b. When the WP rolls by an angle \(\gamma\), the rotation matrix [24] is written as  
\[R= \begin{bmatrix}\cos\gamma&-\sin\gamma&0\\ \sin\gamma&\cos\gamma&0\\ 0&0&1\end{bmatrix} . \tag{5}\]  
Correspondingly, the normal vectors of each refracting and reflecting surface become  
\[\left\{ \begin{array}{l}{N_{1}}^{\prime}=RN_{1}=[0\ 0\ -1]^{\mathrm{T}},\\ {N_{2}}^{\prime}=RN_{2}=[-\theta\cos\gamma\ -\theta\sin\gamma\ -1]^{ \mathrm{T}},\\ {N_{3}}^{\prime}=N_{3}=[0\ 0\ -1]^{\mathrm{T}},\\ {N_{4}}^{\prime}=RN_{4}=[\theta\cos\gamma\ \theta\sin\gamma\ 1]^{ \mathrm{T}},\\ {N_{5}}^{\prime}=RN_{5}=[0\ 0\ 1]^{\mathrm{T}}.\end{array} \right.