## 5 Interplanetary Laser Communications
\begin{array}{l}G_{T}=\frac{4\pi A_{T}}{\lambda^{2}}g_{T}(\alpha_{T},0, \gamma_{T},0)\\ =\frac{4\pi A_{T}}{\lambda^{2}}\frac{2}{\alpha_{T}^{2}}\left\{\exp\!\left(-2 \alpha_{T}^{2}\right)+\exp\!\left(-2\alpha_{T}^{2}\gamma_{T}^{2}\right)^{-}-2 \exp\!\left[-\alpha_{T}^{2}\!\left(1+\gamma_{T}^{2}\right)\right]\right\}\\ \end{array}  \tag{4}\]For a given obscuration ratio, the value of \(\alpha_{T}\) which maximizes the transmitter antenna gain, is given by  
\[\alpha_{T}^{opt}=1.12-1.30\gamma_{T}^{2}+2.12\gamma_{T}^{4} \tag{5}\]  
Similarly, the gain of the receiving antenna is given by [18]  
\[G_{R}=A_{R}\Big{(}1-\gamma_{R}^{2}\Big{)}\eta_{RD} \tag{6}\]  
where \(A_{R}\) is the area of the receive telescope primary mirror, \(\gamma_{R}\) is the receiver obstruction ratio of the secondary mirror, and \(\eta_{RD}\) has a maximum value of 0.95 if \(\pi R_{D}/\lambda F_{E}>0.7\) where \(R_{D}\) is the detector radius and \(F_{E}\) is the Earth telescope F-number, i.e., the ratio of the receive telescope focal length over the primary diameter [18]. If we multiply both sides of Equation (1) by the photon energy \(h\
u\) and the desired bit frequency \(f_{b}\), we generate the equation for the transmitted laser power from the planetary platform needed to generate the desired number of received photons per bit at the Earth station, i.e.,  
\[P_{rE}=n_{rE}(f_{b})h\
u f_{b}=P_{tp}\eta_{tot}\eta_{dE}\frac{\pi^{3}}{4}g_{tp }\Big{(}1-\gamma_{rE}^{2}\Big{)}\frac{(D_{tp}D_{rE})^{2}}{(\lambda R_{PR})^{2}} \tag{7}\]  
From Equation (7), we obtain  
\[P_{tp}(D_{tp}D_{rE})^{2}=P_{rE}(\lambda R_{PE})^{2}\frac{4}{\pi^{3}\eta_{tot} \eta_{dE}g_{tp}(1-\gamma_{rE}^{2})} \tag{8}\]  
In Equation (8), \(P_{rE}\) is the desired received power at the Earth station. For example, if we want to limit the Bit Error Rate (BER) to one per thousand in a KHz system, one per million in a MHz system, or one per billion in a GHz system, then, based on Figure 3, the desired laser power received at the Earth terminal, \(P_{rE}\) is equal to the photon energy \(h\
u\) multiplied by the desired received photons per bit \(n_{rE}\) and the bit frequency \(f_{b}\), e.g., 7 thousand (KHz), 14 million (MHz), or 21 billion (GHz). We will now insert our previously determined values of \(\eta_{tot}=0.66\) and \(\eta_{dE}=0.7\) into Equation (8) and further assume that a secondary mirror is not required for the smaller planetary telescope. The latter assumption implies \(\gamma_{IP}=0\) and \(g_{tp}=0.81\) from Figure 3. If  
Figure 4: Maximizing the far field gain of the transmitting antenna. The left hand graph gives the ratio of the primary mirror radius to the laser Gaussian beam radius, i.e., \(\alpha_{T}=\alpha_{T}/\omega\), which yields maximum on-axis far field gain as a function of the secondary mirror obscuration ratio \(\gamma_{T}=b_{T}/\alpha_{T}\). The second graph shows the resulting reduction in the peak far field transmitter gain relative to \(4\pi A_{T}/\lambda^{2}\) where \(A_{T}\) is the area of the primary mirror. The relatively minor effects of secondary mirror support struts on the far field pattern can be minimized by mounting the secondary mirror on an AR-coated window the same diameter as the primary or alternatively using thin supporting blades that are attached to and run parallel to the telescope tube. we further assume an orbiting Earth telescope, like NASA's 2.4 m diameter Hubble Space Telescope, the value of \(\gamma_{rE}\) = 0.3 m/2.4 m = 0.125 or (1 \(-\)\(\gamma_{rE}\)\({}^{2}\)) = 0.984. In Figure 5, the product of the transmitted laser power emitted by the planetary probe and the two telescope diameters squared (\(y\)-axis) is plotted as a function of the planetary distance (\(x\)-axis) for four candidate bit rates: 1 MHz (black), 10 MHz (orange), 100 MHz (brown), and 1 GHz (pink) for a laser wavelength of \(\lambda\) = 532 nm.