## 5 Interplanetary Laser Communications
For example, with 14 photons received per bit, only 1 in a million (MHz) “ones” would be missed, while with 21 photons received per bit, only 1 in a billion (GHz) “ones” would be missed. To account for internal instrument or background light noise, one can set a photon threshold below which a “zero” bit is highly likely. Table 1 provides a list of planetary characteristics relevant to interplanetary communications from which one can estimate the approximate range of link distances between the Earth and the planetary probe as well as the relative velocities of the two terminals and the angular uncertainties during initial acquisition and subsequent tracking of the opposite terminal. Choosing telescope (optical antenna) designs that minimize the transmitter beam divergence and ensure that the received light is fully captured by the detector array is crucial and has been previously studied in detail [17] and will be used in the upcoming link analysis. The number of laser photons received at the Earth station is given by the equation  
\[n_{rE}=n_{tP}\eta_{tP}G_{tP}G_{rE}\eta_{rE}\eta_{dE}R_{PE}^{-2} \tag{1}\]  
where  
\(n_{tP}=\) the number of photons per pulse/bit emitted by the planetary laser  
\(\hat{\eta}_{tP}=\) the optical efficiency of the planetary transmit optics  
\(G_{tP}=\) the transmitter gain of the planetary telescope (optical antenna)  
\(G_{rE}=\) the receiver gain of the Earth telescope (optical antenna)  
\(\eta_{rE}=\) the optical efficiency of the Earth receive optics  
\(\eta_{dE}=\) the optical efficiency of the Earth detector  
\(R_{PE}=\) range between the Earth and planetary terminals  
The corresponding equation for the Earth to planet optical uplink is obtained by interchanging the _Es_ and _Ps_ in Equation (1) to yield  
\[\eta_{rP}=n_{tE}\eta_{tE}G_{tE}G_{rP}\eta_{rP}\eta_{dP}R_{PE}^{-2} \tag{2}\]  
According to Figure 4, aluminum or silver coatings applied to the primary and secondary mirrors of both terminals would result in a total transmission of about 72% at three candidate wavelengths of 1550, 1064, and 532 nm, but the shortest wavelength laser at 532 nm (frequency-doubled Nd:YAG) would produce the highest gain. Additional antireflection or high reflectivity coatings on other optical elements in the path would add only about 0.25% loss per surface. For our sample link calculations, we will assume three additional transmitting or dielectrically-coated reflective elements in each of the two telescopes of an additional total loss of 3%. The Earth receiver also includes a narrowband filter whose transmission can be as high as 95% assuming no direct illumination of the detector by the Sun, which would require a much narrower bandwidth with a lower transmission. Thus, we will assume an estimated total coating loss of \(\eta_{tot}=\) 72 (0.97) (0.95) = 0.66. The gain of a transmitting optical antenna is given by [17]  
\[G_{T}=\frac{4\pi A_{T}}{\lambda^{2}}g_{T}(\alpha_{T},\beta_{T},\gamma_{T},X_{T}) \tag{3}\]  
where \(A_{T}\) is the area of the telescope primary mirror, \(\lambda\) is the laser wavelength, \(\alpha_{T}=a/\omega\) where \(a\) is the physical radius of the transmitting primary mirror and \(\omega\) is the Gaussian beam radius at the telescope exit aperture, \(\gamma_{T}=b/a\) where \(b\) is the physical radius of the secondary mirror, \(X_{T}=2\pi a_{T}\sin\varphi/\lambda=\pi D_{T}\sin\varphi/\lambda\) where \(\varphi\) is the pointing error, \(D_{T}\) is the diameter of the transmitting telescope, and \(\beta_{T}=0\) unless the outgoing beam is purposely defocused to create a wider beam angle as might occur during initial acquisition of the opposite terminal. Thus, once the two terminals are locked onto each other, we can set \(X_{T}=\beta_{T}=0\) and the gain equation simplifies to  
\[