## 2 Principles  
A corner cube retro-reflector (CCR) is based on groups of three perpendicular planes, as shown in Figure 2. Conventionally, the dihedral angle between any pair of reflecting faces is made to be 90\({}^{\circ}\)[1], so that the reflected beam is exactly contradictory directional to the incident beam [13]. If the angles differ from 90\({}^{\circ}\) by a specific amount, the reflected beam will be converged or diverged in multiple beams to achieve the required application [1]. We can move the three retro-reflected rays on the top into one direction and the other three into another direction by using the different angles between each pair of facets. In this way, the optical performance of a corner cube can be improved as the retro-reflected energy in a given direction is tripled; this is called superpin technology [24]. The superpin technology is characterized by the unique structure of corner cubes to concentrate the retroreflected light into two zones insteads of six zones, shown in Figure 2c. The SuperPin Plus structure was developed based \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{3}{*}{\
\begin{tabular}{c} **Regulation** \\ **Requirement** \\ \end{tabular} } & \multirow{3}{*}{
\begin{tabular}{c} **Observation** \\ **Angle (\({}^{\circ}\))** \\ \end{tabular} } & \multicolumn{4}{c}{**Minimum Coefficient of Luminous Intensity \(\mathbf{R_{I}}\)**} \\ \cline{3-6}  & & & \multicolumn{4}{c}{**(Med/Incident Lux)**} \\ \cline{3-6}  & & & \multicolumn{4}{c}{**Entrance Angle (\({}^{\circ}\))**} \\ \cline{3-6}  & & **0\({}^{\circ}\)** & **10\({}^{\circ}\) Up** & **10\({}^{\circ}\) Down** & **20\({}^{\circ}\) Left** & **20\({}^{\circ}\) Right** \\ \hline SAE & 0.2 & 1680 & 1120 & 1120 & 560 & 560 \\ ECE & 0.33 & 1680 & 1120 & 1120 & 560 & 560 \\ \hline \end{tabular}
\end{table} 
Table 1: Minimum mcd (millicandela)/incident lux for a white reflex reflector. Figure 1: The formula of retro-reflected light by (**a**) the EU Economic Commission for Europe (ECE) regular in ([16], Figure 1), (**b**) the ECE SuperPin in ([24], Figure 1), and (**c**) the ECE and US Society of Automotive Engineers (SAE) SuperPin Plus reflex reflector, an extension of SuperPin [24]. on the superpin technology with the same specification of the pins. The experiment beam pattern retro-reflected from the superpin plus corner is shown in the next section. Through an array of pins, corner cube retro-reflectors can be produced, as shown in Figure 2. The direction of the reflected rays after the reflections are given by [PERSON]'s formula:  
\[\overset{\rightarrow}{t}=\overset{\rightarrow}{q}+2\overset{\rightarrow}{a}( \alpha\overset{\rightarrow}{a}-\beta\overset{\rightarrow}{b}+\gamma\overset{ \rightarrow}{c}) \tag{1}\]  
where \(\overset{\rightarrow}{t}\) is the final direction; \(\overset{\rightarrow}{q}\) is the original direction; \(\alpha\), \(\beta\), and \(\gamma\) are the small angles, by which the angles between the three mirrors exceed right angles; and \(\overset{\rightarrow}{a}\), \(\overset{\rightarrow}{b}\), and \(\overset{\rightarrow}{c}\) are normal to the three mirrors taken in the order in a right-hand sense. The unit normals to the faces can be computed as follows, as shown in Figure 3. Let the normals to the faces without dihedral-angle offsets be the unit vectors \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\) along the three coordinates x, y, and z, respectively.