
EJASA  Part 12Page 80 Field Of View (FOV): The relationship between the solid angle occupied by each star and the area of the celestial sphere "occupied" by a typical star is: A OMEGAs =  sr (17) R^2 where A = area of the celestial sphere, i.e., 4.PI.R^2/N; N being the number of stars being considered (10^6). 4.PI OMEGAs =  sr (18) N Let us assume that sky survey is done out to a distance of 1,000 light years. This means that we are searching the entire celestial sphere around the Sun with a radius of 1,000 light years. This sphere of 4.PI steradians (sr), contains about 10 million stars of which approximately 1 million are solartype. Assuming that for a sphere of this size, these 1 million stars are distributed fairly uniformly: OMEGAs = 1.26 X 10^5 steradian For small angles, the solid angle FOV OMEGAs and the linear angle FOV THETAs, are related by: PI.THETAs^2 OMEGAs =  sr (19) 4 THETAs = 0.23 degrees Array Field Of View: Figure 10 shows the typical fieldofview (FOV) for a 10 meter diameter telescope. It has a usable Telescope FieldOfView of about 0.33 X 0.33 degrees. At 656 nm, the diffraction limited FOV for each pixel, and based on the Rayleigh criterion (1.22)Wl/d radians, is 8 X 10^8 radians (0.0165"). For a 128 X 128 diffraction limited twodimensional array, the array has a linear fieldofview = 1.02 X 10^5 radians (2.1"). The corresponding array FOV is: FOV = 2.1" X 2.1" Thus, at any instant of time, the average number of stars in the 2.1" X 2.1" array fieldofview is approximately: 6.4 X 10^6 Page 81         *       2.1 arc seconds   >o<   Array FOV           * 0.23 degrees *   <>   Telescope FOV = 0.33 degrees <> Figure 10  Typical FOVs for a large optical telescope. The diagram (not to scale) illustrates the fact that the optical telescope's array fieldofview generally observes empty space; the array itself occupying just a small fraction of the telescope's usable (focal plane) fieldofview. Number Of Received Beams: The number of directions resolved by a telescope (with a maximum off axis loss of 1 dB) is stated in the Cyclops report [5] as being given approximately by: Nd = 4.G (20) where G = gain. For a 10 meter diameter telescope at 656 nm, G = 2.3 X 10^15. Thus: Nd = 9.2 X 10^15 beams An alternative expression has been given [8] where Nd = G. In this paper, for the purposes of roughly estimating the search time for an All Sky Survey, Equ. 20 has been used. Nd has been taken as being 10^16 beams or directions. Page 82 The Search Time For the Targeted Search, the time to scan a single star with the heterodyning array, is given by: Inor.Npix.(fufl).Td Ts =  s (21) Imin.Nmsca.Bmsca.Bbin where Inor = normalized flux (8.12 X 10^16 W/m^2), Imin = minimum detectable flux (8.12 X 10^16 W/m^2), Npix = number of pixels (16,384 photodetectors), Nmsca = number of parallel multichannel spectrum analyzers (MCSAs), {<= Npix} (1), Bmsca = total bandwidth of MCSA (10 GHz), Bbin = minimum MCSA bin bandwidth (100 kHz), fu = upper optical frequency (8.57 X 10^14 Hz), fl = lower optical frequency (4.29 X 10^14 Hz), Td = dead time overhead factor per array scan (1.0). The dead time overhead factor is >= 1, and for this estimate, has been taken to be unity, i.e., implying zero overhead. The normalized flux is defined as that flux level that causes the normalized CNR (SNR) (dB re 1 Hz) to fall to 0 dB. Note that if the pilottone maximal ratio predetection combining system described later is employed, the number of pixels (Npix) is effectively reduced to unity. Also, the number of receiver beams Nd is assumed relatively constant over the band fufl. If we substitute the values given in parentheses into Equ. (21), for the visible optical bandwidth between 350 nm and 700 nm, and a minimum detectable flux level of about 150 dBW/m^2, we find that: Ts = 2 hours The time to do an All Sky Survey of this type is increased by a factor (10^16/16,384), so that Ts = 136 million years! If we wanted to store all the data collected, the number of bits would be, to say the least, astronomical. Clearly, we would need to be very selective in the wave lengths scanned. i.e., fufl would have to be very small, so that a guess of the magic optical frequencies would be mandatory. This rough optimistic search time estimate, shows that it would be ridiculous to consider a Visible SETI All Sky Survey modelled on the one being employed for the Microwave Observing Project (MOP). [4045] Optical Heterodyne Detection: In an optical heterodyne receiver (Figure 2, Page 15), the signal current I is proportional to the product of the signal electric field and the localoscillator electric field, and a difference or Inter mediate Frequency (I.F.) is produced because the photodetector is a squarelaw device. [7178,8182] Let us see how this heterodyne beat Page 83 signal is created. Consider two optical beams mixing on a photodiode (squarelaw detector). Let the beams be given by: Received signal beam electricfield component = Er.cos(wrt+phi), Localoscillator beam electricfield component = Eo.coswot. The photodetector current is given by: I = k(Er+Eo)^2 (22) where k = a constant of proportionality relating the current respon sivity of the photodetector (Ri) to the electricfield. I = k[Er.cos(wrt+phi)+Eo.coswot]^2 I = kEr^2.cos2(wrt+phi)+2kEr.Eo.cos(wrt+phi).coswot+kEo^2.cos2wot I = 0.5kEr^2[1+cos2(wrt+phi)] + kEr.Eo[cos{(wrwo)t+phi}]+kEr.Eo[cos{(wr+wo)t+phi)}] + 0.5kEo^2[1+cos2wot] Rejecting all but the difference frequency term, I = kEr.Eo[cos{(wrwo)t+phi}] (23) where (wrwo)/(2.PI) = frfo = Bif, is the difference, beat or intermediate frequency. Thus, the signal detected is proportional to the product of the received signal and localoscillator electricfields. In an optical homodyne receiver, wo = wr, and the intermediate frequency is zero. The optical mixing efficiency factor H, which is not indicated here (Equ. 32 & 33) and accounts for wavefront distortion and beam misalignment, is typically somewhat less than 50%. PilotTone Maximal Ratio Predetection Combining: The pilottone technique has been previously applied to radio frequency diversity receivers to overcome deep fades. [84] It has also been employed by the author on multimode fiber homodyne and heterodyne systems with a 4quadrant photodetector acting as an optical space diversity receiver. [81,82] The spatial incoherence of the radiation pattern from a multimode optical fiber is very similar to that of a freespace optical beam received by a large telescope within an atmosphere. The theory behind the terrene pilottone method is as follows, and makes no specific assumption about modulation techniques employed by ETIs, i.e., whether intensity, polarization, frequency or phase modulation, analog or digital. With reference to Figure 1 (Page 10): Page 84 Let the pilottone carrier at fp be given by: Ep(t).sin[wpt+dphi] (24) and the modulated information signal at fs be given by: Es(t).sin[wst+phi(t)+dphi] (25) where dphi = phase disturbance caused by the transmitter laser (jitter) or Earth's atmosphere, phi(t) = represents possible phase or frequency modulation. The phase disturbances dphi, are essentially common to both the signal and the pilottone, as they are almost identical optical frequencies and travel the same optical path. However, dphi generally differs at each photodetector.  sin[(wswo)t+phi(t)+dphi]   cos[(wswp)t+phi(t)] > Mixer > LPF > 1st I.F (1.1 GHz)    2nd I.F (100 MHz)  ^ To Summer >  sin[(wpwo)t+dphi]   2nd L.O. (1 GHz) Figure 11  Maximal Ratio Precombining. The bandpassfiltered signal from each photodetector provides two separatelyfiltered 1st I.F and 2nd L.O. signals to an electronic mixer. The 2nd I.F. produced after the low pass filter (LPF), has all the laser localoscillator and atmospheric induced phase noise dphi eliminated. The frequencies given in brackets in Figure 11 are arbitrary, and used to help clarify the technique. Each pixel of the 128 X 128 array has one of these circuits, whose inphase outputs are simply added (in a summer) and taken to a single MCSA. If we heterodyne a localoscillator laser operating at frequency wo with both these signals, we obtain the difference frequency signals or 1st I.F. from the photodetector proportional to: Ep(t).Eo.sin[(wpwo)t+dphi] (26) Es(t).Eo.sin[(wswo)t+phi(t)+dphi] (27) where dphi now also includes the effects of localoscillator jitter. Page 85 The pilottone signal as stated by Equ. (26), may be passed through a narrowband filter and amplifier, to produce what is effectively a strong electrical second local oscillator (2nd L.O.) signal for an electrical mixer. It may also be used to lock a narrowband Phase Locked Loop (PLL) whose Voltage Controlled Oscillator (VCO) is used as the strong, amplitudestable and clean 2nd local oscillator. The information signal as stated by Equ. (27), may be passed through a wideband filter and applied to the other port of this electrical mixer. The 2nd I.F. output of the electrical mixer is proportional to: Ep(t).Es(t).Eo(t)^2.cos[(wswp)t+phi(t)] (28) The phase disturbances dphi introduced by the atmospheric turbulence and laser jitters have been eliminated by the process of electrical mixing. Thus, if the image of the transmitter is instantaneously or sequentially smeared out over many pixels, all the second I.F. contri butions are in phase, and may be simply summed to provide predetection diversity combining and a substantial reduction in amplitude instability (scintillation). It also provides the best type of predetection summation in the form of MaximalRatio Combining. Although the system appears to implement EqualGain Combining, the effect of the electronic mixer is to cause the weakest signals to be automatically weighted downwards, and hence cause Maximal Ratio Combining of the photodetector signals. Those pixels producing the weakest signal also produce the lowest quantum, Planckian or background noise contributions to the input of the electrical mixer, so that the summed electrical signal power is not degraded by noise from pixels with little or no optical signal. This occurs because when no optical signal is present, the noise output of each electronic mixer is essentially that due to a noise^2 term, and hence is very small. Only a single MCSA would be required, which would be effectively continuously "looking" at the combined outputs of all 16,384 pixels. We would have only one MCSA, but 16,384 electronic frontend systems for predetection combining of the photodetector outputs, based on the mixing technique illustrated in Figure 11. A predetection combining system with a single MCSA would not detect directly any Planckian starlight noise from a star in the array field ofview alone, only that which overlapped and mixed (downconverted) with an ETI signal on one or more pixels. However, for nearby stars where the transmitter and star are separately resolved, we would lose any Planckian suppression effect of a (single pixel) diffraction limited telescope. Also, if there are significant interstellar or atmospheric groupdelay dispersion effects between the signal and pilottone, the technique would not work. This consideration may affect the choice for the value of (fsfp) and may itself limit modulation bandwidth to be less than a few GHz, notwithstanding SNR considerations. Of course, to use this technique will require the cooperation of the ETI. Would they be so obliging? It would be difficult to justify building such a receiving signal processing system without foreknowledge that Page 86 ETIs employ this technique  this could be said to be putting the cart before the horse! Anyway, before implementing such a system, assuming ETIs would use such a modulation format, we would have had to previously detect this modulation format to know what electrical filters to use! Radio Frequency SignalToNoise Ratio: The CarrierToNoise Ratio (CNR) in the Microwave Heterodyne [5,8,85] 100meter diameter, 1 kW dish system operating at 1.5 GHz over a range of 10 light years: Pr CNR =  (29) kTBe where Pr = received power (1.72 X 10^22 W), T = effective system temperature (10 K), Be = electrical intermediate frequency bandwidth (1 Hz). CNR = 1 dB A symmetrical Cyclops array system [5] with 900 such dishes at both the transmitter and receiver would have a CNR = 60 dB. Optical SignalToNoise Ratio: The dimensions of all signal and noise components the following optical expressions are in units of amperes^2, and by multiplying by the photodetector load impedance, may be turned into units of power. The numerators are representative of the electrical signal power in the photodetector load, while the denominators represents the electrical noise power in the photodetector load. [7178] For coherent receivers, dualbalanced photodetection is assumed so that all the received signal power is utilized, and the noise floor is not raised by excess intensity noise on the localoscillator laser. It is further assumed that the linewidths of the received signal and local oscillator laser are sufficiently small compared to the modulation bandwidths, as to not raise the noise floor. The effective system noise temperature of an optical receiver may be expressed in the form: h.f Teff =  K (30) eta.k where h = Planck's constant (6.63 X 10^34 J.s), f = frequency (4.57 X 10^14 Hz). Teff = 43,900 K Page 87 Incoherent SignalToNoise Ratio: Direct Detection and PhotonCounting Pr^2(MRi)^2 SNR =  (31) [2e{Ri(Pr+NbBo)+Ib}M^(2+x)+2eIs+2Nb{Pr+NbBo}(MRi)^2+4kTF/RL]Be where Pr = received optical power (W), Po = local oscillator power (W), M = avalanche gain, eta = photodetector quantum efficiency (0.5), Ri = unity gain responsivity (W/A), e = electronic charge (1.6 X 10^19 C), Nb = background radiation spectral density (W/Hz), Ib = bulk dark current at unity gain (A), Is = surface dark current (A), x = excess noise factor, k = Boltzmann's constant (1.38 X 10^23 J/K), T = frontend amplifier temperature (K), F = frontend amplifier noise figure, RL = frontend load (Ohms), Bo = optical predetection bandwidth (Hz), Be = noise equivalent electrical bandwidth, which for a single pole filter = PI/2 x maximum modulation frequency (Hz). The electrical signal power is proportional to Pr^2, and the noise components proportional: 1. To the quantum noise produced by the signal photons. 2. To the fluctuation noise produced by the background radiation Pb (NbBo). Notice that this noise is proportional to the optical bandwidth, and the ratio of this noise to the quantum noise component is inversely proportional to the received optical power. 3. To the shot noise produced by the bulk dark current in the photo detector. 4. To the shot noise produced by the surface leakage dark current. 5. To the background radiation beating with the signal, which is independent of optical bandwidth. The noise spectral density is the important factor here. 6. To the noise beating with noise, which is proportional to both the noise spectral density squared and the optical bandwidth. The latter two noise components are insignificant and may be safely omitted for this application where the background is very small. 7. To the thermal kT noise in the photodetector load and frontend amplifier, and may be neglected for shot noise limited direct detection receivers, and ideal photoncounting receivers.
