# A Tutorial on Retroreflectors and Arrays Used in Satellite and Lunar Laser Ranging
## 8 Concluding Remarks
A one-meter telescope, on the other hand, would require a minimum of 195 \(\upmu\)J to realize a single photon return. It should be noted that the use of a 100 mm-diameter cube, as proposed by several international LLR investigators, increases the signal from a single 38 mm Apollo cube by a factor of (100 mm/38 mm)\({}^{4}\) = 48, or roughly a factor of 6 lower than the 300-retro Apollo 15 array response. It would take a single 158 mm cube to roughly match the Apollo 15 array cross-section. This value is in excellent agreement with Table 1 in [15]. Based on their simulations, the latter authors also concluded that only the y-axis of the hollow retro's three retro axes would have to be corrected by between 0.65 and 0.8 arcsec offset for velocity aberration. Chinese researchers, on the other hand, describe the design, manufacture, and testing of a larger 170 mm hollow retroreflector [17] with all three axes deviating from 90\({}^{\circ}\). **Millimeter-Accuracy LEO-to-MEO satellites** use large-radius spherical satellites to better match the incoming plane wave, minimize pulse spreading, and allow for more reflectors within the active area to increase the cross-section. One can minimize satellite pulse broadening by reducing the range of accepted incidence angles through the use of hollow cubes or by recessing the cubes (hollow or solid) in tubes drilled into the satellite. It should also be noted that incidence angles < 17\({}^{\circ}\) do not leak light in solid TIR reflectors. Finally, one must select cube diameters and clocking angles that best match the \"\(\alpha\) annulus\"while favoring the response at high zenith (low elevation) angles, which is key to efficient array design. **GNSS and GEO satellites** typically have a nadir face pointed near the Earth's center due to their other operational functions, e.g., Earth observation, communications, navigation, etc. Therefore, flat panels perform reasonably well but still exhibit reduced cross-sections and several hundred picoseconds of temporal spread at lower satellite elevation angles. Flat circular (rather than square or rectangular) arrays would largely eliminate azimuthal range biases, but replacing current flat panels by a sphere segment of comparable cross-section would further improve range accuracy at all observation angles. **Lunar Laser Ranging (LLR)** The single large cube concept put forward by various international teams of investigators clearly has the beneficial effects of (1) better defining the point of reference on the lunar surface and (2) reducing the target-induced, angular-dependent, temporal spread of the return pulse. However, achieving the same (or larger) target cross-sections than are currently available with the Apollo 15 array would require single-retro diameters in excess of 160 mm. Furthermore, as the size of the single retroreflector grows, the divergence of the reflected beam decreases, and careful attention must be paid to velocity aberration effects due to lunar orbital motion and Earth rotation. Furthermore, Earth's rotation effects decrease with the increasing latitude of the ground station and peak at the point of closest approach to the lunar target. Finally, unlike Earth-orbiting satellites, where the target of interest is surrounded by a vacuum, lunar arrays are surrounded by a lunar surface, which can potentially introduce additional unwanted laser or solar photons to the overall return signal and thereby potentially degrade the range measurement accuracy. Even if one uses the entire 3.5 m aperture of the largest Apollo Observatorytelescope to achieve the least divergent laser beam, the central lobe of the transmitted beam will span a diameter of roughly \(2\lambda R/D_{p}=2\left(532\text{ nm}\right)\left(384.4\times 10^{6}\text{ m}\right)/3.5\text{ m}=117\text{ m}\), corresponding to an angular FOV of 0.3 \(\upmu\)rad.