# A Tutorial on Retroreflectors and Arrays Used in Satellite and Lunar Laser Ranging
## 2 SLR Link Equation  
During the first four decades of SLR (1964 to 2004), ground stations were characterized by a moderate-to-large (submeter-to-meter-class) receiver telescope, and the laser beam was transmitted via a parallel, small-diameter tube mounted on the side of the telescope. Since the laser beam divergence is proportional to the ratio of the laser wavelength to the beam radius (\(\lambda\)/\(\omega\)), this approach resulted in an unnecessarily high divergence beam that required much higher-energy lasers to obtain an adequate satellite return. These high-energy beams posed a potential eye safety hazard to personnel in low-flying aircraft and to ground personnel during ground calibrations. As a result, all early SLR stations were equipped with dedicated aircraft radars and safety personnel to man them. However, in 1994, the present author proposed a new system, SLR2000, to NASA management [6], which optimally sized and transmitted a low-divergence laser beam via a common transmit-receive telescope to achieve maximum illumination of the satellite array while simultaneously reducing the laser energy and the threat to eye safety. The optimum SLR2000 design drew heavily from earlier studies on optical antenna gain carried out in support of early NASA laser communications studies [7,8]. The NASA SLR2000 funding started slowly in 1998, but by 2003, the newly named Next-Generation Satellite Laser Ranging (NGSLR) system had tracked its first satellite and continued tracking until 2007, when it was sadly destroyed by a lightning bolt. NASA's replacement system, Space Geodesy Satellite Laser Ranging (SGSLR), adds additional automation and is currently being installed at three sites--(1) the NASA Goddard Space Flight Center in Greenbelt, MD, USA, (2) the McDonald Observatory at the University of Texas, USA, and (3) Ny Alesund, Norway [9]. The SLR link equation defines the number of photons detected by the ground station and is given by  
\[n_{s}=\frac{E}{h\tau}\eta_{t}\ \frac{2}{\pi(\theta_{d}R)^{2}}exp\left[-2\left( \frac{\Delta\theta_{p}}{\theta_{d}}\right)^{2}\right]\left[\frac{1}{1+\left( \frac{\Delta\theta_{j}}{\theta_{d}}\right)^{2}}\right]\left(\frac{\sigma A_{ r}}{4\pi R^{2}}\right)\eta_{r}\eta_{d}T_{a}^{2}T_{c}^{2} \tag{1}\]  
where  
\(n_{s}=\) the number of detected satellite photoelectrons per laser pulse  
\(E_{t}=\) the laser pulse energy  
\(h\
u=\) the laser photon energy = 3.73 \(\times\) 10\({}^{-19}\) J @ 532 nm (Doubled Nd:YAG)  
\(\eta_{t}=\) the transmitter optical throughput efficiency  
\(\theta_{d}=\) the Gaussian beam divergence half angle  
\(R=\) the slant range between the station and satellite  
\(\Delta\theta_{p}=\) the laser beam pointing error  
\(\Delta\theta_{j}=\) the RMS tracking mount jitter  
\(\sigma=\) the satellite optical cross-section  
\(A_{r}=\) the telescope receiving area. \(\eta_{r}\)= the receiver optical throughput efficiency  
\(\eta_{c}=\) the detector counting efficiency  
\(T_{a}=\) the one-way atmospheric transmission  
\(T_{c}=\) the one-way cirrus cloud transmission  
We note from Equation (1) that the received signal strength varies as \(1/R^{4}\) and that the satellite optical cross-section, \(\sigma\), is the sole contribution of the space segment to the link equation. Retroreflectors, also referred to as \"cube corners\", can be either hollow or solid, and the solid retros can be coated or uncoated. Table 1 provides a summary of the various retro types, their unique characteristics, and their historical usage. Note also that the satellite optical array cross-section, \(\sigma\), is the sole link contribution of the space segment. Retroreflectors (or \"cube corners\") are designed to reflect light back to the point of origin in a narrow beam. Increasing the size or number of retroreflectors increases the return signal strength. Figure 1 illustrates the two basic types of cube corner, i.e., solid or hollow.