Detectability Of Pulsed Laser Beacons
The relationship between peak power Ppk and average power for a pulsed laser is given by:
Pav Ppk = ----- W Tau.r
Substituting the values in parentheses for a pulsed ETI laser beacon system with a 1 Hz repetition rate, we find that:
Ppk = 1018 W.
Diffraction-Limited Telescope Gain
The gain of a diffraction-limited dish or telescope is given by:
(pi.D)2 G = ------- Wl2
Substituting the values in parentheses for an ETI uplink transmitting at the center of the human photopic response, we find that:
G = 155.1 dB.
Effective Isotropic Radiated Power
The Effective Isotropic Radiated Power is the power that the transmitter appears to have if it radiated isotropically. It is given by:
EIRP = P.G W
Substituting the values for P and G given above, we find that:
EIRPlaser = 3.2 x 1033 W.
For a star like the sun:
EIRPstar = 3.9 x 1026 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:
EIRP I = ------- W/m2 4.pi.R2
where R = range (9.461 x 1016 m).
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/m2.
Istar = 3.5 x 10-9 W/m2.
The optical power appearing at the photodetector is given by:
(pi.d2) S = Tatm.Aeff.Feff.-------.I W 4
For the ETI laser:
Slaser = 8.9 x 10-5 W.
For a solar-type star:
Sstar = 1.1 x 10-11 W.
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)]
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 photon detection rate is given by:
eta.S N = ----- hf
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 Laboratories will increase the peak power output from 1014 W to 1015 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 1015 W. How long will it take to increase the peak output power from 1015 W at one pulse per day to 1018 W at one pulse per second? The answer, of course, is no time at all on the cosmic time scale.