## 2 Principles
If the angle between the zx plane, and the zy plane is \(\alpha=(\pi/2)+\delta_{1}\), the xz plane and the xy plane is \(\beta=(\pi/2)+\delta_{2}\), the yz plane and the yx plane is \(\gamma=(\pi/2)+\delta_{3}\), shown in Figure 3b; this can be expressed by Equations (2)-(4),  
\[\hat{r_{1}}=\hat{i}+\frac{\delta_{1}}{2}\hat{j};\;\;\hat{r_{2}}=\hat{j}+\frac{ \delta_{1}}{2}\hat{i};\;\;\hat{r_{3}}=\hat{k} \tag{2}\]  
\[\hat{r_{1}}=\hat{i}+\frac{\delta_{2}}{2}\hat{j};\;\;\hat{r_{2}}=\hat{j}+\frac{ \delta_{2}}{2}\hat{i};\;\;\hat{r_{3}}=\hat{k} \tag{3}\]  
\[\hat{r_{1}}=\hat{i}+\frac{\delta_{3}}{2}\hat{j};\;\;\hat{r_{2}}=\hat{j}+\frac{ \delta_{3}}{2}\hat{i};\;\;\hat{r_{3}}=\hat{k} \tag{4}\]  
Figure 3: (**a**) The dihedral-angle offsets; (**b**) the angle between each plane; and (**c**) the symmetry axis direction. Figure 2: (**a**) A 3D design of flat corner cube SuperPin Plus retro-reflector; (**b**) a group of pins and its reflecting surfaces (blue line inner region); and (**c**) SuperPin retroreflected light pattern [24]. The retro-reflected beam spread angle at normal beam incidence when all dihedral angles are offset by the same amount is given by the formulas (5)-(7),  
Figure 1:0 Figure 1:1 (6) Figure 1:2  
where \(\delta\) is the dihedral angles of corner cubes and \(\varepsilon\) is the angle between the incident ray and the reflected rays, thus there will be several retro-reflected rays generated, once one ray is incident on CCR. The direction of the incident beam is determined by \(\theta^{\prime}\) and \(\phi^{\prime}\) in the primed coordinate system. By adjusting the angles between each pair of facets, we can set the six retro-reflected rays with equal angle intervals or let them propagate in almost contradictory directions, such as the simulation results performed by regular, SuperPin, and SuperPin Plus retro-reflectors, as shown in Figure 4a-c.  
After analyzing the values of the angles \(\alpha\), \(\beta\), and \(\gamma\), we concluded the influence of incident angle on the value and position of the reflected beams. In case 1: \(\alpha=\beta=\gamma\), the reflected beam is divided into six small beams that are symmetrical with each other in a circle, known as regular structure, shown in Figure 4a. In case 2: \(\alpha=\beta<\gamma\), the reflected beams converged into two symmetric groups on the top and bottom (three beams on the top and three beams on the bottom), known as the SuperPin structure, shown in Figure 4b. In case 3: \(\alpha=\beta>\gamma\), the reflected beams converged to four beams at the top and bottom positions (two beams on the top and two beams on the bottom) and two beams at the left and right locations (one beam on the left and one beam on the right), known as the SuperPin Plus structure, shown in Figure 4c. The SuperPin Plus structure, which is accomplished by our proposal, will reduce manufacturing costs as well as be more advantageous as it can be used in both the European and American markets. When the dihedral angles \(\delta_{1}=0.12^{\circ}\); (\(\alpha=90.12^{\circ}\)); \(\delta_{2}=0.12^{\circ}\); (\(\beta=90.12^{\circ}\)); and \(\delta_{3}=0.01^{\circ}\) (\(\gamma=90.01^{\circ}\)), the CCR is called the SuperPin Plus type, where the reflected beam angle can be calculated as \(\varepsilon_{1}=\varepsilon_{2}=0.33^{\circ}\), \(\varepsilon_{3}=0.2^{\circ}\), and the luminous intensity \(\mathbf{R_{1}}\) meets EU ECE and US SAE regulation requirements. In Figure 5, the simulation results through the TracePro software (Lambda Research Corporation of Littleton, Massachusetts, USA) can verify the above \(\varepsilon\) and \(\delta\) relation formulas. In Figure 5b,d, the retro-reflected light can reach up to \(0.2^{\circ}\), and up to \(0.33^{\circ}\) through the CCR shown in Figure 5a and the \(45^{\circ}\) rotated CCR, shown in Figure 5c respectively. Therefore, the SuperPin Plus design  
Figure 1:3 Relationship between dihedral angles and the reflected beam direction. (**a**) The reflected intensity distribution by a regular retro-reflector.