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Optical SETI Observations during the Day -Theory

Radobs 05

See companion document for discussion about this theory.

The formula I used for the calculations is:

Pb = (PI.dWL.THETA^2.Ar/4).N(WL)

where Pb    = background power detected per pixel (W),
      dWL   = optical bandpass (nm),
      THETA = diffraction limited receiver beamwidth (radians),
      Ar    = receiving antenna aperture (m^2), 
      N(WL) = spectral radiance (W/m^2.nm/sr).

For visible radiation I have assumed a Spectral Radiance of:

0.01 W/cm^2.sr.micron., or
0.1 W/m^2.sr.nm.

For infrared radiation I have assumed a Spectral Radiance of:

0.0002 W/cm^2.sr.micron., or
0.002 W/m^2.sr.nm.

I was expecting 10,600 nm to be better than the visible because of the lower
Spectral Radiance, but the lower level of quantum noise (-12.1 dB with
respect to that at 656 nm) and the fact that while a bandwidth of
1 nm = 6.97 X 10^11 Hz at 656 nm, a bandwidth of 1 nm = 2.67 X 10^9 Hz at
10,600 nm.  The greater number of Hz in 1 nm at visible wavelengths helps
considerably in lowering the spectral density when quoted on a Hz basis.

As we reduce the mirror size, THETA^2 increases at the same rate as Ar
decreases.  Thus, while an aperture size reduction will reduce the signal
detected at a rate proportional to 1/d^2, it will not increase the skylight
detected by each photodetector, if each photodetector's diameter is matched
to the new diffraction limited pixel size.

Assume visible optical detection bandwidths less than 10 MHz.  For a 1 kW
10-meter 656nm symmetrical telescope system at 10 light years:

Received power per pixel Pr = 1.60 X 10-^15 W.
Quantum noise per pixel = 6.37 X 10^-19 W/Hz (post-detection electronic
Planck radiation per pixel = 1.86 X 10^-22 W/Hz (optical pre-detection
CNR = 34 dB re 1 Hz.

At 656 nm, a bandwidth of 1 nm = 6.97 X 10^11 Hz.  Thus, the Spectral
Radiance = 1.43 X 10^-13 W/m^2.sr.Hz.  I have a paper which gives the night
time background sky as = 10^-21 W/m^2.sr.Hz, i.e. a factor of about 82 dB
less.  For a 10 meter telescope with say a 0.1 degree field-of-view, the
total received night time background = 1.9 X 10^-25 W/Hz, while the
radiation per spatial mode or pixel could be many millions less.  This is
negligible compared to Planckian starlight radiation.

I have calculated that the received daylight background power per pixel or
spatial mode:

Pb = 2.9 X 10^-14 W/nm, or
Pb = 4.2 X 10^-26 W/Hz.

This you will notice is significantly below the Planckian starlight level,
i.e., at -36.5 dB, and is at -71.8 dB with respect to quantum shot noise. 
The -36.5 dB figure seems strange (counter-intuitive) since we can't see a
2nd magnitude star (as the Sun would appear at 10 light years) in daylight
with the naked eye.  Either my calculations are in error, or the effect of
the very small pixel field of view of a large telescope (compared to the
eye) is to substantially cut down the reception of scattered light in the
atmosphere and improve the contrast ratio between the Planckian starlight
and it?

The question is "Can we see (detect) a 2nd magnitude star with large
telescope focal plane arrays in daylight"?  If the answer is yes, then the
calculations would appear correct.  Of course, we would expect the daylight
background level to be further reduced for telescopes on high mountains
(deep blue sky).  With a narrow optical detection bandwidth, an ETI beacon
or signal would appear to stand out in broad daylight.  Note that if
1 million pixels (photodetectors) or spatial modes make up the focal plane
array, then the total daylight background received by the array
= 4.2 X 10^-20 W/Hz.

I think I have found the solution to the apparent paradox that my
calculations on Planckian starlight visibility in daylight, seemed to
indicate a much greater visibility or contrast than we know to be the case.
The problem is that present-day ground-based telescopes are theoretically
diffraction limited, but unless they have deformable mirrors for adaptive
seeing, they fall far short of diffraction limited performance.

The 71.8 dB quantum noise to background margin would disappear when the
optical bandwidth is increased to about 15 MHz.  Thus, very high-Q
Fabry-Perot spectrum analyzers could be used for tunable optical filtering
purposes in photon-counting systems without degradation in SNR.  If we
choose to use incoherent photon-counting front-ends, then I envisage using a
tandem system of filters, which can be mechanically switched into the
optical path.  A selection of narrow band interference pre-filters or
grating spectrometers would be used with one or more Fabry-Perot
Interferometers.  Burleigh sells electronically tunable Fabry-Perot
Interferometers covering the wavelength range 400 nm to 1,000 nm, with
spectral resolutions as high as 0.5 MHz.  If we want really narrow optical
filtering and be able to do the optical search in daylight, we cannot beat
optical heterodyne detection.

If the same calculations are done for the CO2 far-infrared wavelength, the
day and night temperatures of the sky are more or less equal (about 300 K)
and this determines the "sky noise".  Calculations indicate that it is only
about 29 dB below quantum noise.  So if the optical bandwidth is increased
to only 400 Hz, the sky noise will begin to degrade the SNR.

My comparative analysis has been done on a Continuous Wave (CW) basis.  It
was done this way not only to keep the analysis simpler, but also to be more
conservative with respect to some of the so-called Optical SETI problems. 
Clearly, if we assume pulsed laser systems, the very high peak power of the
pulses can overcome Planckian starlight problems and even daylight
background problems.  Under matched filter conditions
(bandwidth = 1/pulse period), the recovered SNR is given by the ratio of the
energy in the signal and the white-noise spectral density.  A pulse system
with a 10% duty cycle, and hence 10 dB increase in peak power (constant mean
power), would produce a 10 dB increase in SNR over quantum, Planckian or
background daylight noise power.  Perhaps 1 GW mean powers are a bit
excessive, but if the duty cycle was 1%, a peak power of 1 GW would be very
reasonable; implying a mean power of only 10 MW.  This is just another
reason why Optical SETI is not as bad as the microwave SETI people would
have us believe, and why perhaps we should be looking for signals with
greater bandwidths.

December 16, 1990
BBOARD No. 269

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
* Dr. Stuart A. Kingsley                       Copyright (c), 1990        *
* AMIEE, SMIEEE                                                           *
* Consultant                            "Where No Photon Has Gone Before" *
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