# A Tutorial on Retroreflectors and Arrays Used in Satellite and Lunar Laser Ranging
## 8 Concluding Remarks
We can estimate the number of detected photons reflected by the lunar surface by applying the lidar equation (Equation (1) in [20]), which applies to a Lambertian-diffuse-reflecting surface that fills the receiver Field of View (FOV). Under the latter assumption, the number of laser photons received from the lunar surface is estimated at  
\[ \begin{array}{ll}n_{LS}&=nt(\eta_{t}\text{ }\eta_{r}\eta_{d})T_{ atm}^{2}\left(\frac{4\pi A_{p}}{\lambda^{2}}\right)gT\rho_{LS}\frac{cos \theta}{\pi R^{2}}A_{p}g_{R}\\ &=n_{t}(\eta_{t}\text{ }\eta_{r}\eta_{d})T_{ atm}^{2}gTgR\rho_{LS}cos\theta\left(\frac{\pi^{2}}{4}\right)\left(\frac{D_{p} ^{4}}{\lambda^{2}R^{2}}\right)\end{array}  \tag{18}\]  
where \(\rho_{LS}=0.12\) is the nominal reflectivity of the lunar surface at visible and near-infrared wavelengths and \(\theta\) is the nominal surface slope as viewed by the Earth station's line of sight. Dividing Equation (18) by Equation (16), we obtain an estimated ratio of lunar surface returns vs. retro returns, i.e.,  
\[\frac{n_{LS}}{n_{s}}=\frac{\rho_{LS}}{\rho}\ cos\theta\left(\frac{4}{\pi^{2}} \right)\frac{\lambda^{2}R^{2}}{D_{cc}^{4}} \tag{19}\]  
The latter equation indicates that the overall lunar surface return greatly exceeds that of the reflector since \((\lambda R)^{2}>>D_{cc}^{4}\) because, even when one uses the full aperture of the ground telescope to minimize laser beam divergence, the diameter of the illuminated surface typically spans hundreds of meters. However, the retro produces a \"hot spot\" in the return, which can be isolated spatially via a pinhole in the receiver FOV or, better yet, a multipixel SPAD (Single-Photon Avalanche Diode) array, which can detect the \"hot spot\" during target acquisition and drive it to the central pixel of the array in order to maximize the retro signal strength while simultaneously assigning the vast majority of the surface photons to other pixels in the array. Locating the Earth-facing retro near the perimeter of the lunar disk would drive the \(cos\theta\) term in Equation (19) close to zero. The Southern lunar pole region might be a good choice since it is believed to be a potential source of water for future manned missions. In order to further minimize the impact of lunar surface returns on range accuracy to the target retro, the pixel FOV should be chosen as small as is practically possible for rapid target acquisition. It may even be advantageous to elevate the retro a known distance above the lunar surface to more clearly separate the retro returns from local surface returns that fall within the common pixel FOV. Finally, a recent article by the LLR staff in Grasse, France, provides an interesting history of their evolving technology over decades of operations and their experiences with near-infrared wavelengths [21]. It should also be mentioned in closing that the LLR station in Grasse successfully tracked the Lunar Reconnaissance Orbiter (LRO) satellite using a 4 \(\times\) 3 retroreflector array mounted on the anti-nadir side of the spacecraft [22]. **Laser ranging throughout the solar system** using conventional single-ended ranging to passive reflectors is unrealistic due to the \(R^{-4}\) signal loss in the link equation. However, precise ranging (as well as time transfer and wideband communications) can be accomplished using two-way asynchronous laser transponders equipped with transmitters and receivers on both ends of the link [23; 24]. In this instance, the signal strength at either terminal varies only as \(1/R^{2}\) as opposed to \(1/R^{4}\). Furthermore, since most of the link burden (laser power or telescope aperture) can be carried by the Earth station, the space terminal can be relatively modest in size (submeter), weight, and power consumption [25]. The feasibility of the interplanetary transponder approach has already been demonstratedin experiments conducted at the 1.2 m telescope at NASA Goddard Space Flight Center with a satellite enroute to Mercury at a distance of 22 million km [24], which produced a range accuracy estimated at 20 cm [25].