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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:
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 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:
PI.THETAs^2
OMEGAs = ----------- sr (19)
4
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
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------------------------------
| |
| |
| |
| * |
| |
| |
| 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:
Inor.Npix.(fu-fl).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 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
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., 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
atmosphere.
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:
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 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:
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 Signal-To-Noise Ratio:
Direct Detection and Photon-Counting
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 = 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-
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 front-end
amplifier, and may be neglected for shot noise limited direct
detection receivers, and ideal photon-counting receivers.
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