# A Tutorial on Retroreflectors and Arrays Used in Satellite and Lunar Laser Ranging
## 8 Concluding Remarks  
**A common transmit-receive telescope** maximizes the observed signal strength, allowing for the use of lower-power lasers and significantly higher measurement rates. Early SLR systems typically transmitted the quasi-Gaussian laser beam through a separate narrow parallel tube mounted on the receiving telescope. This resulted in a highly divergent transmitter beam with a full angular width equal to \(4\lambda/D_{b}\) in our link equation, Equation (1), where \(D_{b}\) is the exiting laser beam diameter. Using the full aperture of the ground telescope to transmit the beam increased the received signal strength by minimizing the beam divergence of the outgoing beam, thereby concentrating a much greater fraction of the laser pulse energy on the satellite. Adopting this approach in NASA's photon-counting SLR2000 system [6] increased measurement rates from under 10 Hz to 2 kHz and, due to significantly lower transmitted pulse energies and large transmitted beamwidths, greatly reduced eye safety hazards to aircraft. In this instance, using Equation (2), we can rewrite the original link equation, Equation (1), as [7; 8]  
\[n_{s} =n_{t}(\eta_{t}\;\eta_{t}\eta_{d})T_{atm}^{2}\left(\frac{4\pi A_{t }}{\lambda^{2}}\right)g_{T}\;\frac{\sigma A_{p}}{4\pi R^{2}}g_{R} \tag{16}\] \[=n_{t}(\eta_{t}\;\eta_{t}\eta_{d})T_{atm}^{2}\left(\frac{\pi D_{p} ^{2}}{\lambda^{2}}\right)g_{T}\;\frac{\pi^{3}\;\rho\;D_{c}\;4}{4\lambda^{2}} \frac{D_{p}^{2}}{4R^{2}}g_{R}\] \[=n_{t}(\eta_{t}\;\eta_{t}\eta_{d})T_{atm}^{2}\left(\frac{\pi}{2} \right)^{4}\left(\frac{D_{c}\;D_{p}}{\lambda R}\right)^{4}\rho g_{T}g_{R} \leq 0.034n_{t}T_{atm}^{2}\left(\frac{D_{c}\;D_{p}}{\lambda R}\right)^{4}\]  
where \(\rho\) is the reflectivity of the retro surfaces discussed in Section 2, \(n_{t}\) is the number of transmitted photons per laser pulse, \(\eta_{t}\cong\eta_{r}\cong 0.66\) are the nominal optical throughput efficiencies of a telescope with aluminum- or silver-coated primary and secondary mirrors (plus a few additional low-loss optical elements) in the transmit and receive paths [20], \(\eta_{d}\) (<0.70) is the detector's counting efficiency, \(T_{atm}\) is the one-way transmission of the atmosphere, \(D_{cc}\) is the retro diameter, \(D_{p}\) is the diameter of the telescope's primary mirror, \(\lambda\) is the laser wavelength, \(R\) is the target range, and \(g_{T}\) (\(\leq\)0.81) and \(g_{R}\) (\(\leq\)1) are the reductions in the optimized transmitter and receiver gains due to the truncation of the outgoing quasi Gaussian beam by the primary mirror perimeter and/or the presence of a secondary mirror that blocks the central portion of the outgoing and incoming radiation [7; 8]. Solving for \(n_{t}\) in Equation (16) and multiplying by the photon energy, \(hc/\lambda\), the laser pulse energy required to achieve the desired number of signal photons, \(n_{s}\), is  
\[E_{t}=n_{t}\frac{hc}{\lambda}\geq\frac{30}{T_{attn}^{2}}n_{s}\frac{hc}{\lambda} \bigg{(}\frac{\lambda R}{D_{cc}D_{p}}\bigg{)}^{4} \tag{17}\]  
From Equation (17), the advantages of utilizing shorter laser wavelengths (\(\lambda\)) and the full primary diameter (\(D_{p}\)) of the ground-based telescope to transmit the laser beam is immediately apparent [6]. This is especially true for LLR, where the product \(\lambda R=(532\times 10^{-9}\) m)(3.844 \(\times\) 10\({}^{8}\) m) =204.5 m\({}^{2}\) at the widely used 532 nm green wavelength of the frequency-doubled Nd:YAG laser. If we assume a lunar retro diameter \(D_{cc}\) = 100 mm and the Apollo telescope primary mirror diameter of \(D_{p}\) = 3.5 m used in both transmit and receive modes, then \(D_{cc}D_{p}\) = 0.35m\({}^{2}\). Ignoring atmospheric transmission losses, the minimum number of transmitted photons required to achieve a single photon return is 30 (\(\lambda R/D_{cc}D_{p}\))\({}^{4}\) = 30(204.5/0.35)\({}^{4}\) =3.5 \(\times\) 10\({}^{12}\) photons, corresponding to a minimum laser pulse energy of 1.3 \(\upmu\)J at 532 nm.