# Enhancement of ECE SuperPin Curved Reflex Reflector by the Use of Double Pins with Corner Cubes
## 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),  
\[\widehat{n_{1}}=\hat{t}+\frac{\delta_{1}}{2}\ \hat{t};\ \widehat{n_{2}}=\hat{t}+ \frac{\delta_{1}}{2}\ \hat{t};\widehat{n_{3}}=\hat{k} \tag{2}\]  
\[\widehat{n_{1}}=\hat{t}+\frac{\delta_{2}}{2}\ \hat{t};\ \widehat{n_{2}}=\hat{t}+ \frac{\delta_{2}}{2}\ \hat{t};\widehat{n_{3}}=\hat{k} \tag{3}\]  
\[\widehat{n_{1}}=\hat{t}+\frac{\delta_{3}}{2}\ \hat{t};\ \widehat{n_{2}}=\hat{t}+ \frac{\delta_{3}}{2}\ \hat{t};\widehat{n_{3}}=\hat{k} \tag{4}\]  
For small angles \(\delta\), the above expressions are quite adequate. Offsets in the other two dihedral angles can be similarly represented. It is desirable to have the unit normals given in the coordinate system of the symmetry axis of the corner cube since the incidence angle of the laser beam is given with respect to this axis. The symmetry axis is in the direction of the vector x = y = z = 1, as shown in Figure 3c; we see that \(\cos\theta_{A}\)= \(\frac{1}{\sqrt{2}}\); \(\sin\theta_{A}=\frac{1}{\sqrt{2}}\); \(\cos\lambda_{A}=\frac{\sqrt{2}}{\sqrt{3}}\); \(\sin\lambda_{A}=\frac{1}{\sqrt{3}}\). The normals in the xyz coordinate system can be given in the coordinate system of the symmetry axis by rotating the original coordinate system about the z axis by  
Figure 2: (a) A flat corner-cube retro-reflector composed of groups of pins; and (b) a group of pins and its reflecting surfaces of the pin groups (red inner region). \(\theta_{\rm A}\) and about the y axis by \(-\lambda_{\rm A}\). This brings the x axis along the axis of the matrix form, and the total rotation is given by:  
\[\left( \begin{array}{c}x^{\prime}\\ y^{\prime}\\ z^{\prime}\end{array} \right)=\left( \begin{array}{ccc}\cos\lambda_{A}&0&\sin \lambda_{A}\\ 0&1&0\\ -\sin\lambda_{A}&0&\cos\lambda_{A}\\ \end{array} \right)\left( \begin{array}{ccc}\cos\theta_{A}&\sin\theta_{A}&0\\ -\sin\theta_{A}&\cos\theta_{A}&0\\ 0&0&1\end{array} \right)\left( \begin{array}{c}x\\ y\\ z\end{array} \right) \tag{5}\]  
Substituting the values of the sines and cosines and multiplying the matrices, we obtain \(x^{\prime}\)=\(\frac{1}{\sqrt{3}}\)\((x+y+z)\); \(y^{\prime}\)=\(\frac{1}{\sqrt{2}}\)\((y-x)\); \(z^{\prime}\)=\(\frac{1}{\sqrt{6}}\)\((2z-x-y)\). In Figure 4a, the unprimed axes represent the original coordinate system and the primed axes are the rotated coordinates. The incident beam after reflection at the front face is in a direction given by the angles \(\theta^{\prime}\) and \(\phi^{\prime}\) in the primed coordinate system, as shown in Figure 4b. A second rotation of the coordinate system must be performed to get the normals to the faces in the coordinate system of the laser beam. By rotating the coordinate system about the x\({}^{\prime}\) axis by \(\theta^{\prime}\) and then about the new z\({}^{\prime}\) axis by \(\phi^{\prime}\), we obtain:  
\[\left( \begin{array}{c}x^{\prime\prime}\\ y^{\prime\prime}\\ z^{\prime\prime}\end{array} \right)=\left( \begin{array}{ccc}\cos\phi^{\prime }&\sin\phi^{\prime}&0\\ -\sin\phi^{\prime}&\cos\phi^{\prime}&0\\ 0&0&1\end{array} \right)\left( \begin{array}{ccc}1&0&0\\ 0&\cos\theta^{\prime}&\sin\theta^{\prime}\\ 0&-\sin\theta^{\prime}&\cos\theta^{\prime}\\ \end{array} \right)\left( \begin{array}{c}x^{\prime}\\ y^{\prime}\\ z^{\prime}\end{array} \right) \tag{6}\]  
The relationship of the primed and double-primed coordinate axes is given in Figure 4c. The x\({}^{\prime}\) axis is the symmetry axis of the reflector, the y\({}^{\prime}\), z\({}^{\prime}\) plane is parallel to the front face, and the x\({}^{\prime\prime}\) axis is parallel to the beam after it enters the corner cube.