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EJASA - Part 12

                                                                     Page 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:

    OMEGAs = ---  sr                                                    (17)

    where A = area of the celestial sphere, i.e., 4.PI.R^2/N; N being the
              number of stars being considered (10^6).

    OMEGAs = ----  sr                                                   (18)

    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 solar-type.  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:

    OMEGAs = -----------  sr                                            (19)

                             THETAs = 0.23 degrees

     Array Field Of View:

     Figure 10 shows the typical field-of-view (FOV) for a 10 meter
     diameter telescope.  It has a usable Telescope Field-Of-View 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
     two-dimensional array, the array has a linear field-of-view =
     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 field-of-view 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 field-of-view
    generally observes empty space; the array itself occupying just a small
    fraction of the telescope's usable (focal plane) field-of-view.

    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:

    Ts = ----------------------  s                                      (21)

    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 multi-channel 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 pilot-tone 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 fu-fl.  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

                                 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., fu-fl 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). [40-45]

    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 local-oscillator electric field, and a difference or Inter-
    mediate Frequency (I.F.) is produced because the photodetector is a
    square-law device. [71-78,81-82]  Let us see how this heterodyne beat

                                                                     Page 83

    signal is created.  Consider two optical beams mixing on a photodiode
    (square-law detector).  Let the beams be given by:

    Received signal beam electric-field component = Er.cos(wrt+phi),
    Local-oscillator beam electric-field 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 electric-field.

    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{(wr-wo)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{(wr-wo)t+phi}]                                       (23)

    where (wr-wo)/(2.PI) = fr-fo = Bif, is the difference, beat or
    intermediate frequency.

    Thus, the signal detected is proportional to the product of the
    received signal and local-oscillator electric-fields.  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%.

    Pilot-Tone Maximal Ratio Predetection Combining:

    The pilot-tone 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 4-quadrant 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
    free-space optical beam received by a large telescope within an

    The theory behind the terrene pilot-tone 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 pilot-tone 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 pilot-tone, as they are almost identical optical frequencies
    and travel the same optical path.  However, dphi generally differs at
    each photodetector.

    sin[(ws-wo)t+phi(t)+dphi]|       |    -----    cos[(ws-wp)t+phi(t)]
    ------------------------>| Mixer |-->| LPF |-------------------------->
        1st I.F (1.1 GHz)    |       |    -----      2nd I.F (100 MHz)
                                 ^                        To Summer ------>
        sin[(wp-wo)t+dphi]       |
         2nd L.O. (1 GHz)

    Figure 11 -

    Maximal Ratio Precombining.  The bandpass-filtered signal from each
    photodetector provides two separately-filtered 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 local-oscillator 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 in-phase outputs are simply added (in a
    summer) and taken to a single MCSA.

    If we heterodyne a local-oscillator 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[(wp-wo)t+dphi]                                         (26)

    Es(t).Eo.sin[(ws-wo)t+phi(t)+dphi]                                  (27)

    where dphi now also includes the effects of local-oscillator jitter.

                                                                     Page 85

    The pilot-tone signal as stated by Equ. (26), may be passed through a
    narrow-band 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 narrow-band Phase
    Locked Loop (PLL) whose Voltage Controlled Oscillator (VCO) is used as
    the strong, amplitude-stable 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[(ws-wp)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
    Maximal-Ratio Combining.  Although the system appears to implement
    Equal-Gain 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
    front-end 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-
    of-view 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 group-delay dispersion effects between the signal and
    pilot-tone, the technique would not work.  This consideration may
    affect the choice for the value of (fs-fp) 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 Signal-To-Noise Ratio:

    The Carrier-To-Noise Ratio (CNR) in the Microwave Heterodyne [5,8,85]
    100-meter diameter, 1 kW dish system operating at 1.5 GHz over a range
    of 10 light years:

    CNR = ----                                                          (29)

    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 Signal-To-Noise 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. [71-78]

    For coherent receivers, dual-balanced 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 local-oscillator 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:

    Teff = -----  K                                                     (30)

    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 Signal-To-Noise Ratio:

    Direct Detection and Photon-Counting

  SNR = --------------------------------------------------------------  (31)

    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   = front-end amplifier temperature (K),
           F   = front-end amplifier noise figure,
           RL  = front-end load (Ohms),
           Bo  = optical pre-detection 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-

    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 front-end
        amplifier, and may be neglected for shot noise limited direct
        detection receivers, and ideal photon-counting receivers.


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