# A Tutorial on Retroreflectors and Arrays Used in Satellite and Lunar Laser Ranging
## 6 Velocity Aberration and \"Spoiled\" Retroreflectors  
If there is no relative velocity between the station and satellite, the beam reflected by the retroreflector will fall directly back onto the station. However, as illustrated in Figure 7, a relative velocity, \(v\), between the satellite and station causes the reflected beam to be angularly deflected from the station in the forward direction of the satellite motion by an angle \(\alpha\) = 2\(v/c\). This phenomenon is referred to as \"velocity aberration\". Since small-diameter cubes have small optical cross-sections but large-angle far-field diffraction patterns (FFDPs), the signal at the station is not significantly reduced by velocity aberration. On the other hand, large-diameter cubes with high cross-sections have small-angle FFDPs, and the signal at the station is therefore substantially reduced by velocity aberration. Figure 6: Impulse response in the large spherical satellite limit (\(R_{s}>>nL\)) for both hollow and solid quartz cube corners. The difference is due to the smaller acceptance angle in hollow cubes resulting in fewer cubes contributing to the return. tion. In general, the signal is reduced by half or more if the cube diameter, \(D_{cc}\), satisfies the inequality  
\[D_{cc}>D_{1/2}\ =\ \frac{1.6\ \lambda}{\pi\alpha}=\frac{0.8\ \lambda c}{\pi v} \tag{10}\]  
If there is a relative velocity between the satellite and the station, the coordinates of the FFDP are translated in the direction of the velocity vector. The magnitude of the angular displacement in the FFDP is given by  
\[\alpha(h_{s},\theta_{zen},\ \omega)=\alpha_{max}(h_{s})\sqrt{cos^{2}\omega+ \Gamma^{2}(h_{s},\ \theta_{zen})sin^{2}\omega} \tag{11}\]  
where the maximum and minimum angular displacement values are given by  
\[\alpha_{max}(h_{s})=\alpha(h_{s},0,0)=\frac{2v_{s}}{c}=\frac{2}{c}\sqrt{\frac{ gR_{E}^{2}}{R_{E}+h_{s}}}\]  
\[\alpha_{min}(h_{s})=\alpha(h_{s},\ 70^{\circ},90^{\circ})=\alpha_{max}(h_{s})\ \Gamma(h_{s},\ 70^{ \circ})\]  
\[\Gamma(h_{s}\theta_{zen})=\sqrt{1-\left(\frac{R_{E}\ sin\theta_{zen}}{R_{E}+h_ {s}}\right)^{2}} \tag{12}\]  
\[\omega=cos^{-1}\left[\left( \begin{array}{c}\cdot\\ r\ x\ p\end{array} \right) \begin{array}{c}\cdot\\ \ast v\end{array} \right]\]  
where \(v\) = the satellite velocity at an altitude \(h_{s}\) above sea level, \(R_{E}\) = the Earth's radius = 6378 km, \(g\) = the surface gravity acceleration = 9.8 m/s\({}^{2}\), \(c\) = the velocity of light = 3 \(\times\) 10\({}^{8}\) m/s, \(\theta_{zen}\) = the largest satellite zenith angle for tracking = 70\({}^{\circ}\), \(\hat{r}\) = the unit vector to the satellite from the geocenter, \(\hat{p}\) = the unit vector from the station to the satellite, and \(\hat{v}\) = unit vector in the direction of the satellite velocity. The maximum and minimum values of the parameter \(\alpha\) are plotted in Figure 8 as a function of the satellite altitude above sea level, \(h_{s}\). Figure 7: Velocity aberration causes the reflected beam to be angularly displaced from the ground station in the direction of the satellite velocity. \"Spooling\" of the retroreflector is used to compensate for velocity aberration and improve the signal return from the satellite over the full range of observation angles. If we offset one or more (\(N=1\) to 3) of the cube dihedral angles from \(90^{\circ}\) by an amount \(\delta\), the central lobe of the FFDP splits into \(2N\) spots, as in Figure 9. If \(n\) is the cube index of refraction, the mean angular distance of the lobe from the center of the original Airy pattern increases linearly with the dihedral angle offset, \(\delta\), according to  
\[\gamma=\frac{4}{3}\sqrt{6}n\delta=3.27\ n\ \delta \tag{13}\]  
As before, the angular size of any given lobe decreases as the cube diameter gets larger, and the FFDP of each lobe is the 2D Fourier transform of an individual \(60^{\circ}\) sector. The energy distribution is complex but has a hexagonal symmetry if all three \(\delta\)s are equal.