Detectability Of Pulsed Laser Beacons

9501-001

9501-001a

Peak Power

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

       Pav
Ppk = -----  W
      Tau.r

where:

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

P_pk = 10¹⁸ W.

9501-001b

Diffraction-Limited Telescope Gain

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

    (pi.D)2
G = -------
      Wl2

where:

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.

9501-001c

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:

EIRP_laser = 3.2 x 10³³ W.

For a star like the sun:

EIRP_star = 3.9 x 10²⁶ W.

Note that EIRP_laser is the peak EIRP of the laser, while EIRP_star is the mean EIRP of the star.

9501-001d

Received Intensity

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:

I_laser = 2.8 x 10^-2 W/m².

I_star = 3.5 x 10^-9 W/m².

9501-001e

Detected Power

The optical power appearing at the photodetector is given by:

                  (pi.d2)
S = Tatm.Aeff.Feff.-------.I  W
                     4

where:

For the ETI laser:

S_laser = 8.9 x 10^-5 W.

For a solar-type star:

S_star = 1.1 x 10^-11 W.

9501-001f

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)]

where:

Substituting the above values for a range of 10 light years, we find that:

m_laser = -15.

m_star ~ 2.

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

9501-001g

Photon Detection Rate

The photon detection rate is given by:

    eta.S
N = -----
      hf

where:

Substituting the values in parentheses for a center wavelength of 550 nm, we find that the "signal" photon detection rate N_laser:

Signal 44,000 counts per pulse.

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

Noise 6,000,000 counts per second.

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Conclusions

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.

Related materials:

9308-001

9308-002

9512-001