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# Specimen OSETI Calculations

```                            DETECTABILITY OF PULSED LASER BEACONS

Readers are encouraged to check the relationships given below, and the specimen calculations.

Peak Power

The relationship peak power Ppk and average power for a pulsed laser is given by:

Pav
Ppk = -----  W                                                (1)
tau.r

where  Pav = average power (1 GW),
tau = pulse width (1 ns),
r   = pulse repetition rate (1 Hz).

Substituting the values in parentheses for a pulsed ETI laser beacon system with a 1 Hz repetition rate, we find that:

Ppk = 10^18  W.

Diffraction-Limited Telescope Gain

The gain of a diffraction-limited dish or telescope is given by:

pi.D
G = [----]^2                                                  (2)
Wl

where  D  = diameter of transmitter aperture (10 m),
Wl = wavelength (550 nm).

Substituting the values in parentheses for an ETI uplink transmitting at the center of the human photopic response (and taking 10.log of the resulting gain), we find that:

G = 155.1  dB.

The Effective Isotropic Radiated Power is the power that the transmitter appears to have if it radiated isotropically (uniformly) in all directions.  It is given by:

EIRP = P.G  W                                                 (3)

Adding the appropriate subscripts and substituting the values for P and G given above, we find that:

EIRPlaser = 3.2 X 10^33  W.

For a star like the sun:

EIRPstar = 3.9 X 10^26  W.

Note that EIRPlaser is the peak EIRP of the laser, while EIRPstar is the mean EIRP of the star.

The intensity of the received signal and stellar background noise is given by the general expression:

EIRP
I = --------  W/m^2                                           (4)
4.pi.R^2

where R = range (9.461 X 10^16 m).

Adding the appropriate subscripts, and substituting the values in parenthesis for a range of 10 light years, we find that just outside the atmosphere:

Ilaser = 2.8 X 10^-2  W/m^2.

Istar = 3.5 X 10^-9  W/m^2.

Detected Signal Power

The signal power appearing at the photodetector through a V-type optical filter is given by:

pi.d^2
S = Tatm.Aeff.Feff.[------].I  W                              (5)
4

where  Tatm = atmospheric transmission (0.25),
Aeff = telescope aperture efficiency (0.5),
Feff = optical filter efficiency (0.5),
d    = diameter of receiver aperture (0.254 m).

Adding the appropriate subscripts and substituting the values in parentheses, we find that:

Slaser = 8.9 X 10^-5  W.

For a solar-type star:

Sstar = 1.1 X 10^-11  W.

Magnitude

For a solar-type star and a laser centered on the human visual response, the apparent magnitude may be expressed in terms of its intensity I:

m = -[19+2.5log(I)]                                           (6)

where  Ilaser = 2.8 X 10^-2  W/m^2,
Istar  = 3.5 X 10^-9  W/m^2.

Adding the appropriate subscripts and substituting the above values for a range of 10 light years, we find that:

mlaser = -15.

mstar = 2.

During each brief pulse, the laser is brighter than the ETIs' star by a factor of nearly 10 million!

Photon Detection Rate

The signal photon detection rate is given by:

eta.S
N = -----                                                     (7)
h.f

where  eta = photodetector quantum efficiency (0.17),
h   = Planck's constant (6.63 X 10^-34 J.s),
f   = optical frequency (5.45 X 10^14 Hz).

Again, adding the appropriate subscripts and substituting the values in
parentheses for a center wavelength of 550 nm, we find that the "signal"
photon detection rate:

Nlaser (Signal) = 44,000 counts per pulse.

For a solar-type star, we find that the stellar background "noise" photon
detection rate:

Nstar (Noise) = 6,000,000 counts per second.

The "signal" is buried in the noise and the ratio between the "signal" and
"noise" photons is approximately -20 dB.  However, during each one
nanosecond laser pulse, the SNR is positive and nearly 70 dB!  This is
the very important benefit of searching for very short pulses in adjacent
time slots corresponding to the expected pulse duration, even if the
"signal" consists of only one or two detected photons per pulse.  Another
important benefit is that knowledge of the "magic frequency" is not
required.

Note that the "National Ignition Facility" upgrade to the NOVA laser at
the Lawrence Livermore National Laboratories will increase the peak
power output from 10^14 W to 10^15 W, albeit at only one pulse per
day.  By the year 2002, we humans, over a period of 40 years, will have
increased peak laser output powers on this planet from 3 kW to
10^15 W.  How long will it take to increase the peak output power from
10^15 W at one pulse per day to 10^18 W at one pulse per second?
The answer, of course, is no time at all on the cosmic time scale.
```

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