# A Tutorial on Retroreflectors and Arrays Used in Satellite and Lunar Laser Ranging
## 5 Science Applications of SLR Impact the Satellite Array Design  
The current SLR constellation spans a wide range of altitudes (500 km to 36,000 km) and inclinations depending on mission requirements, and each retroreflector array has to be designed accordingly based on orbital altitude, mission goals, desired signal strength, etc. The artificial satellites typically fall into four altitude realms with very different scientific goals, e.g.,  
1. **Low-Earth Orbiting (LEO: h < 1500 km):** Earth gravity field studies and/or orbital support for spaceborne radars or lidars observing terrain heights, sea/ice levels, tree mass, etc. 2. **Medium Earth Orbiting (MEO: h\(\sim\)6000 km):** Low-drag MEO satellites such as LAGEOS 1 and 2 are ideal for accurately determining relative station positions, tectonic plate motion, regional crustal deformation, etc. 3. **Global Navigation System Satellites (GNSS: h-20,000 km):** SLR provides precise orbital support to International Navigation Constellations, such as GLONASS (Rus  
Figure 4: The normalized cross-section as a function of beam incidence angle for hollow and solid fused silica retroreflectors. The 50% and 0% efficiency points for fused silica (\(n\) = 1.455) are 13\({}^{\circ}\) and 45\({}^{\circ}\), respectively, whereas the 50% and 0% efficiency points for a hollow cube (\(n\) = 1) are 9\({}^{\circ}\) and 31\({}^{\circ}\), respectively, i.e., hollow cubes have a narrower angular response range than solid cubes. sia), GALILEO (EU), and COMPASS/BeiDou (China). All of the satellites in the aforementioned constellations are, or will be, equipped with laser retroreflectors, thereby enhancing the accuracy of GNSS orbits and their associated ground networks and transportation guidance. Only two of the US GPS satellites carry retroreflectors. 4. **Geosynchronous satellites (GEO: h-36,000 km):** in equatorial orbits provide continuous monitoring of roughly a third of the Earth's surface. Besides obvious defense and military applications, civilian applications include global communications, weather forecasting, television broadcasting, as well as communications between ground stations and/or relaying information between other satellites or spacecraft. For maximum accuracy in orbit determination, the distance of the effective light reflection point from the satellite center of mass is ideally independent of the viewing angle. Thus, passive LEO and GEO satellites are typically spheres embedded with retros. Furthermore, since the signal strength decreases with satellite range as \(1/R^{4}\), the sphere diameter typically increases with altitude to accommodate more retros in order to meet cross-section (signal strength) requirements. To improve detection probability and range precision, modern SLR stations typically employ fast, single-photon-sensitive, ultralow-timewalk detectors such as Micro Channel Plate Photo Multiplier Tubes (MCP/PMTs) or Single Photon Avalanche Diode (SPAD) arrays. When operated at or near single-photon-return levels, these detectors have allowed satellite detection at few kHz rates with low laser pulse energies as opposed to the historical few pulses per second with high pulse energies [6]. In the kHz low-signal scenario, each retroreflector in the array has a diminishing probability of providing the detected photon in a given measurement as it is further displaced from the laser line-of-sight (retro #7) and the laser beam incidence angle increases (Figure 5). In Figure 6, \(\tau\) is a time normalized to the time it takes a light pulse to travel the diameter of the satellite, i.e., \(2R_{s}/c\), where \(R_{s}\) is the radius of the sphere. Increasing the radius of the satellite will increase the cross-section by allowing the impinging laser beam to illuminate more cubes at low incidence angles. At the same time, however, the larger satellite will also broaden the impulse response, thereby degrading range precision.