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Telescope and Photodetector Array Field-Of-Views

Radobs 20

 

The following is a discussion on the form of focal plane silicon PIN
(positive-intrinsic-negative) photodetector array that will be used in the
optical (visible and near-infrared wavelength) heterodyne SETI receiver.
The unobstructed linear field-of-view (FOV) of a typical large telescope is
given by:
FOV = tan-1(1/f#)  degrees                                               (1)
    = 1/f#  radians
where f# = focal length/diameter (150/10).
For the new 10 meter diameter Great Keck telescope on Mauna Kea in Hawaii,
with an f/15 (f-number or focal ratio) secondary mirror effective focal
length of 150 meters:
                            FOV = 3.8 X 3.8 degrees
            *
               * Cone of light
                  *                                   .
                     *                                .
                        *                             *
                           *                       *  .
                              *                 *     .
  -------->                    / *           *        .
                              /     *     *            -  Photodetector
  -------->                THETAt      *       THETAa | |     Array
                              \     *     *            -   N X N Pixels
  -------->                    \ *           *        .
                              *                 *     .
                           *                       *  .
                        *                             *
                     *                                .
                  *                                   .
               *                                 Focal Plane
            *
                               THETAa << THETAt
Diagram (not to scale) illustrating how photodetector array only occupies a
small fraction of the telescope's field-of-view.
This is the theoretical FOV if there are no obstructions, limiting apertures
in the optical path, or considerations of peripheral field quality. 
However, because of limits on the Keck's field-of-view, the actual
(specified) FOV is 0.33 X 0.33 degrees.  The FOV of the array will be very
much less, and at any instant of time after being centered on a target star,
is likely to contain only a single star that is within 1,000 light years. 
We would probably want to use the rest of the telescope's field-of-view for
other purposes, such as laser guide-star sensors, course spectrographic
systems, cameras, and perhaps a two-dimensional (incoherent) photon-counting
array in addition.  The latter could be large 2048 X 2048 pixel devices. 
This is really no different to what is done in the Hubble Space Telescope
(HST), where the field-of-view is divided up into sectors for different
instrumentation packages.  There are some very good system reasons involving
telescope repositioning or slew time overheads, and which will be elaborated
upon later, for having a much larger telescope FOV that can be scanned with
a mirror across the array.
We now arrive at the basic design for the array through the following
analysis.  It is assume than an adaptive or space-based telescope will be
employed so that diffraction limited performance is achieved.  In this
situation, the photodetector element size, or pixel size, corresponds to the
minimum spot size, and may be set to one-half the diameter of the Airy disk.
This is a factor of 1.2 times the Full Width Half Maximum (FWHM) spot size,
and is half the size of the Airy disk which contains 84% of the energy (see
RADOBS.14).  If we set the pixel size to correspond to the diameter of the
Airy disk to include more signal energy, we would likely degrade the
sensitivity by having some of the local-oscillator power fall outside the
strong part of the spot, thus producing extra shot-noise with no
corresponding contribution to heterodyne signal gain.
This pixel size is given by the Rayleigh criterion:
pix = (1.22).Wl.(f#) microns                                             (2)
where Wl  = wavelength (656 nm).
                               pix = 12 microns
Now we need to determine the number of linear photodetector elements that we
need in order to efficiently observe the sky.  We note that the Rayleigh
criterion produces an angular resolution given by:
        (1.22).Wl
THETA = ---------  radians                                               (3)
            d
where  Wl = wavelength (656 nm),
       d  = diameter (10 m).
                           THETA = 0.0165 arcseconds
Thus, the FOV of each pixel, 12 microns in diameter, is 0.0165", and the
corresponding Image Scale is 1.38" per mm.  As previously indicated, for the
moment we shall assume that for the highest received signal-to-noise ratio
(SNR), the pixel size should be matched to the Rayleigh criterion size; this
being slightly larger than the FWHM (Full Width Half Maximum) diffraction
limited beamwidth (0.0138").  Again, see RADOBS.14 for a more detailed
discussion on beamwidths.  A rigorous analysis may show that for heterodyne
detection, a slightly different pixel size will maximize the SNR.  The
optimum pixel size will also depend on whether we illuminate the entire
array with a collimated local-oscillator (L.O.) laser, or scan a focused
(Gaussian beam waist) L.O., row by row or pixel by pixel.
For the nearest stars, say at range R = 10 light years, we would want the
entire stellar biosphere to fall within this field-of-view.  The beamwidth
at that distance based on the Rayleigh criterion, is given by:
          (1.22).Wl
D = R.tan[---------]                                                     (4)
              d
where the angle is in radians.
                               D =   0.0506 A.U.
Assuming that the maximum outer diameter of a stellar biosphere is 5 A.U.,
this means that the linear dimensions of the beam must illuminate at least
5 A.U.  The ratio of 5 A.U. to 0.0506 A.U. is 99.  Thus, a 128 X 128 pixel
array would allow us to simultaneously view the entire stellar biosphere
when the image of the star was centered on the array.  The angular FOV of
this array would be given by 128 X 0.0165", i.e., 2.1" X 2.1", which is
about 3.2 X 10^-6 of the area corresponding to the telescope's 0.33 X 0.33
degrees actual field-of-view.  The array FOV at 10 light years range would
have dimensions 6.5 X 6.5 A.U.  This gives us some margin for the relatively
few stars at ranges 5 to 10 light years.  We might possibly want to consider
a 256 X 256 pixel array with a 4.2 X 4.2 arcseconds field-of-view, but this
would probably be the largest we would want to use.  See RADOBS.22 for
concerns about array optical and electric power dissipation.
Note that to improve the efficiency of a targeted optical search, the
computer controlling the receiving system will be programmed to avoid
sampling pixels which are very unlikely to contain an ETI signal.  When we
target the more distant stars, the stars and their biospheres will be
contained within the array's central pixels.  Hence, there is no point in
spectrum analyzing the outputs of the peripheral pixels.  At a distance of
several hundred light years, Planckian starlight will arrive at the same
pixel as the signal from the ETI transmitter, and they will no longer be
separately resolved.  In this case, if the adaptive optics works very well,
only one pixel need be sampled.  See RADOBS.21 for a discussion about using
the array for space-diversity reception.
Let us assume that an imaging lens is used that magnifies (linear
dimensions) the focal plane image by a factor of 16.7.  If the linear
dimensions of the array are increased in the same ratio, the angular FOV
remains the same, the corresponding Image Scale drops to 0.083" per mm, and
the new spot size is given by:
                                 pix = 0.2 mm
This pixel diameter ought to provide enough active area to avoid photo-
detector current saturation effects and help dissipate heat, yet at the same
time present a very small capacitive load to minimize the RC time-constants.
Clearly, we can only optimize the physical pixel size for one wavelength. 
However, we could change the imaging lens and hence the magnification (and
array FOV) for different parts of the optical spectrum.
As far as present-day commercial photodetectors are concerned, Ortel
Corporation manufactures a state-of-the-art single photodetector device
which has an active region 25 microns in diameter, and a -3 dB bandwidth of
12 GHz with a 50-ohm load.  Optoelectronics Inc. also manufactures single
element 50-ohm Si photo-detectors with active areas equivalent to about
100 microns in diameter, and bandwidths of the order of 10 GHz.  Antel
Optoelectronics Inc. produces devices with bandwidths up to 20 GHz.  There
are no 10 GHz photodetector arrays commercially available today, though we
may expect that they will be available in a few years for fiber-optic
communication applications, and for internal high-speed communications
within large computers.
Since we propose to use a 128 X 128 element array, with a total of 16,384
pixels, the minimum size of the array is:
                               25.6 mm X 25.6 mm
                                       1"
                    -------------------------------------- 
                   |o o o o                               |
                   |o o    0.2 mm dia. pixels             |
                   |o                                     |
                   |o                                     |
                   |                                      |
                   |                                      |
                   |    Alien Transmitter (100 L.Y.)      |
                   |                  .                   |
                1" |                   *                  |
                   |                 Star                 |
                   |                                      |
                   |                                      |
                   |                                      |
                   |                                      |
                   |        Alien Transmitter (10 L.Y.)   |
                   |                      .               |
                   |                                      |
                    -------------------------------------- 
                                128 X 128 Array
                       Field-Of-View = 2 X 2 arcseconds
                   <----- 6.5 A.U. at 10 Light Years ----->
                          65 A.U. at 100 Light Years
Diagram showing the image of a nearby star system superimposed on the
outline of the heterodyning photodetector array.

This 1" square array is quite a reasonable size, though it has linear
dimensions an order of magnitude greater than typical CCDs with the same
number of pixels.  In this custom array, the pixels could be square or
round, with a small inactive barrier (mask) between adjacent pixels.  The
array size could be increased further if it is desired to increase the
masked area, provide more room for more complex interface circuitry, or just
to improve heat dissipation.
January 24, 1991
RADOBS.20
BBOARD No. 333
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
* Dr. Stuart A. Kingsley                       Copyright (c), 1991        *
* AMIEE, SMIEEE                                                           *
* Consultant                            "Where No Photon Has Gone Before" *
*                                                   __________            *
* FIBERDYNE OPTOELECTRONICS                        /          \           *
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* skingsle@magnus.ircc.ohio-state.edu         ..    ..    ..    ..    ..  *
* CompuServe: 72376,3545                                                  *
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