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Diffraction Limited Beams and Gaussian Optics

Radobs 14

 
This document clarifies and expands on information given on expected beam
diffraction angles and beam sizes at targeted star systems with simple
single aperture telescope systems.  When dealing with diffraction-limited
beams, there are two different (extreme) assumptions that can be made:
1.   The aperture of the transmitting telescope is filled with collimated
     laser radiation so that the beam intensity (W/cm^2) is essentially
     constant across the entire aperture.  This beam has very definite
     finite boundaries.  For the Optical SETI comparative analysis, the
     simplifying assumption has been that the aperture illumination is as
     shown below.  Generally, microwave dishes are illuminated by a feed
     horn that produces a cosine or cosine^2 amplitude taper.
                                      |
                             *********|*********
                             *        |        *
                             *        |        *
                             *        |        *
                             *        |        *
                             *        |        *
                             *        |        *
                      ---------------------------------
                             <-----Aperture---->
             Beam intensity (density) constant across aperture.
2.   The aperture of the transmitting telescope is filled with collimated
     laser radiation in such a way that the beam density across the aperture
     closely matches the Gaussian TEMoo single transverse mode profile of a
     laser beam.  A pure Gaussian TEMoo beam does not have finite
     boundaries.  In practice, a laser beam has boundaries set by the output
     aperture of the laser.  However, these boundaries are usually at
     positions where the beam intensity is negligible.  For the following
     discussions it will be assumed that the output aperture of the
     telescope is the limiting aperture.
                                      |
                                      *
                                    * | *
                                   *  |  *
                                 *    |    *
                               *      |      *
                            *.        |        .*
                         *   .        |        .   *
                      *      .        |        .      *
                  -----------------------------------------
                             <-----Aperture---->
            Beam with Gaussian intensity profile across aperture.

These two extremes of aperture illumination produce slightly different
answers for beam divergence, beam profile and half-power beamwidth, but make
a big difference to the level of the sidelobes.  In practice, the excitation
of a transmitting telescope is likely to lie between the two extremes of
uniform illumination and a pure Gaussian TEMoo illumination.  The former
makes better use of the entire telescope aperture, but throws some energy
into the sidelobes.  The latter may increase the beamwidth and degrade the
antenna gain, but generates very little sidelobe energy.  A reasonable
compromise for a Gaussian beam, as will be illustrated later, is to match
the 1/e^2 collimated beam diameter to the telescope's output aperture. 
Firstly, let us now look in more detail at the relationships for diffraction
at a circular aperture, assuming each of the above conditions.
Uniformly Illuminated Apertures:
Generally, when discussing beam widths and beam shapes of microwave dishes
and optical telescopes, it is usual to assume the simple case of uniform
illumination, i.e., constant beam intensity across the aperture, with a
sudden cut-off of the beam at the extremes of the aperture.  The directivity
or polar response pattern of such a transmitting telescope is identical to
that of the same size of receiving telescope (without an aperture taper). 
In the Optical SETI comparative analysis, uniform illumination has been
assumed, hence the antenna aperture efficiency has been taken as unity, with
the corresponding antenna gains and Effective Isotropic Radiated Powers
(EIRPs) being at a maximum.
                                                                *
                                                           *    .
      beam diameter = telescope aperture = d          *         .
                                                 *              .
      **********>**************||***********                    .
                               ||                               .
                             d ||                   far-field   . D
                               ||                               .
      **********>**************||***********                    .
      collimated beam             near-field      *             .
                                      Rr              *         .
                                                           *    .
                                                                *
  Beam divergence for a fully illuminated aperture, constant density beam.
The above case amounts to diffraction at a circular aperture, and theory
states that the Full Width Half Maximum (FWHM) diffraction angle (where the
intensity is reduced by 3 dB) is given by:
         (1.02).Wl
THETA3 = ---------  radians                                              (1)
             d
         (58.5).Wl
       =  ---------  degrees
             d
where  Wl = wavelength (656 nm),
       d  = diameter of aperture and beam (10 m).
For a fully-illuminated 10 meter diameter telescope system operating at a
wavelength of 656 nm, the FWHM diffraction angle is:
                       THETA3 = 0.0138 seconds of arc
The polar response (PR) or intensity diffraction pattern is given by:
     [2.J1{(PI.d/Wl).sin(THETA/2)}]^2
PR = --------------------------------                                    (2)
        [(PI.d/Wl).sin(THETA/2)]^2
where J1 is the Bessel Function of the first kind.
                                      |
                                0 dB  *
                                    * | *
                               --->*  |  *<-- -3 dB FWHM
                                  *   |   *
                                 *    |    *
                                *     |     *
                -17.6 dB  * *  *      |      *  * *
                        *    **       |       **    * First Sidelobe
                       *     **       |       **     *
                      *      **       |       **      *
                     *       **       |       **       *
                    *        **       |       **        *
                   ---------------------------------------
                               <--Airy Disk-->
   Polar response for a fully illuminated aperture, constant density beam.
The angular distance between the first zeroes of this polar response or
diffraction pattern, corresponds to the width of the central bright zone. 
This zone is known as the Airy disk, and the radius of this zone corresponds
to the situation where {(PI.d/Wl).sin(THETA/2)} = (1.220).PI = 3.833.  This
occurs when:
         (2.44).Wl
THETAa = ---------  radians                                              (3)
             d
         (139.8).Wl
       = ----------  degrees
             d
For the 656 nm, 10 meter telescope, the Airy angle is:
                       THETAa = 0.0330 seconds of arc
This is slightly wider than twice the FWHM beamwidth previously calculated. 
This zone within the first dark ring, contains 83.8% of the energy in the
beam.  THETAa/2 also corresponds to the classic Rayleigh criterion for the
resolving power of a telescope, i.e., the Rayleigh criterion for the angular
resolution of a telescope is half the above value of THETAa, which amounts
to 0.0165 seconds of arc.  This criterion is based on the ability to resolve
two equally intense objects.  Thus, to an approximation (within a factor
of 1.2), the Rayleigh resolution corresponds to the FWHM beamwidth.
In the far-field, the FWHM diffraction limited beam diameter is given by:
D = 2.R.tan(THETA3/2)                                                    (4)
where R = range (10 L.Y.).
For a fully-illuminated 10 meter diameter telescope system, operating at a
wavelength of 656 nm at a range of 10 light years (632,420 A.U.), for which
THETA3 = 0.0138":
                               D = 0.0381 A.U.
The beam diameter at 10 L.Y. defined by the Rayleigh criterion for which
THETAa/2 = 0.0165":
                               D = 0.0506 A.U.
At 100 L.Y.
                               D = 0.506 A.U.
At 1,000 L.Y.
                                D = 5.06 A.U.
At 10,000 L.Y.
                                D= 50.6 A.U.
The latter diameter is about as large as our planetary system.  Whether you
believe that an Advanced Technical Civilization (ATC) could hit planet Earth
with such beams is another matter.  Out to about 100 L.Y., it should be
possible for a space-based 10 meter diameter telescope to separately resolve
the Earth and the Sun, as long as there are efficient means to block the
intensity of the Sun (7th magnitude at 100 L.Y.) which would be over one
billion times brighter (22.5 magnitude factor), and/or measure differential
Doppler shifts between the direct Planckian starlight and reflected
planetary starlight!  Of course, if we assume the use of much larger optical
arrays, as could well be available to a technical civilization just a
hundred years more advanced than us, then the problem of resolving the Earth
and learning all about its orbital motion is very much eased.
Gaussian Illuminated Apertures:
The Gaussian TEMoo single transverse mode laser beam has a normalized
intensity profile transverse to the direction of propagation given by:
I = exp(-2x^2/wo^2)                                                      (5)
where  x  = is the displacement from the center of the beam,
       wo = is the Gaussian radius of the beam defined at its 1/e^2 points
            (0.135).  For a focused or collimated Gaussian beam, this is
            also known as the beam waist.  It is a property of a Gaussian
            beam that the far-field radiation pattern is the same as the
            near-field.
The Rayleigh range is the near-field range over which the beam shows little
divergence.  It is approximately the range over which the beam diameter
doubles.  It is given approximately by:
      PI.wo^2
Rr  = -------                                                            (6)
        Wl
For wo = 5 m, and Wl = 656 nm:
                       Rr = 1.2 X 10^8 m (0.0008 A.U.)
This is a good reason for steering well clear of the beam when in the
vicinity of the 10 meter diameter transmitter, i.e., within about
200,000 km, as the power densities will be extremely high!  For the Rayleigh
range to be greater than 10 light years, the aperture diameter (2wo) must be
greater than 281 km!.  To be greater than 5,000 light years, the aperture
must be greater than 6,283 km - definitely a planet-sized array, and the
product of a very advanced technical civilization.  Note that this
expression may also be used for uniformly illuminated apertures.
      beam diameter 2wo < telescope aperture d                  *
                                                           *    .
                               ||                     *         .
                               ||                *              .
      *********>***************||***********                    .
                               ||                               .
   TEMoo Beam              2wo ||                   far-field   . D
                               ||                               .
      *********>***************||***********                    .
      collimated beam          || near-field     *              .
                               ||     Rr              *         .
                                                           *    .
                                                                *
         Beam divergence for a beam with a Gaussian density profile.
As I was putting the finishing touches to this document, I received an
advanced copy of a paper on interstellar laser communications by Dr. John
Rather of NASA HQ.  A description of this paper will be found in the
document RADOBS.15 to follow.  One of the main features of Dr. Rather's
communication system, is to use large planetary-sized phased arrays which
allow the formation of beams that have a very long Rayleigh (near-field)
range.  In this way, a beam could be formed that is essentially the same
diameter at 5,000 light years as it is at 100 light years.  The implication
of this is that the beamwidth would be optimized to enclose typical stellar
biospheres, i.e., several astronomical units in diameter, and would have
fixed dimensions for any range of target.  This option is available to
technically advanced civilizations.
I have previously suggested that for a puny single aperture (element)
10 meter diameter telescope, the beamwidth might be "modulated" to defocus
the beam at nearby stars to ease the targeting problem.  Very slightly
adjusting (modulating) the optical path length between optical elements in
the telescope to change the collimation, would provide this facility.  While
the beam would be a far-field beam, it could be dynamically-modulated to
keep the beamwidth the same at both near and far stars.  In both cases, over
the range where the beam has a fairly constant diameter and hence intensity,
the Signal-To-Planck Ratio (SPR) would, as Dr. John Rather points out,
actually increase with range because the inverse square law would not apply
to the signal.  It should be understood here that for both techniques, it is
implied that we do not make the beam as intense as we could at short ranges.
Whether ETIs or we would wish to do this is another matter, but then with
very large arrays and an abundance of optical power, there is a lot of
latitude, even with signal bandwidths as high as 10 GHz!
                                      |
                                 0 dB *
                                    * | *
                               --->*  |  *<-- -3 dB FWHM
                                  *   |   *
                                 *    |    *
                                *     |     *
                               *      |      *
                              *       |       *
                            *         |         *
                      * * *           |           *  * * Weak Sidelobe
                    *    **           |            **    *
                  *      **           |            **      *
                 --------------------------------------------
                              <------2wo------>
         Polar response for a beam with a Gaussian density profile.
At the 1/e^2 points, the intensity I is 13.5%.  This diameter contains 86.5%
of the power in the beam, not so very different to the 83.8% in the Airy
disk stated earlier for uniform illumination over the entire aperture.  The
FWHM points occur where the intensity I = 0.5, i.e., where x/wo = 0.59.  The
circle formed by this radius "x" also contains half the power in the beam.
In recent years, a term called M^2 has come into use for describing the
quality of a focused or collimated Gaussian beam.  It is a measure of how
far a beam departs from an ideal Gaussian profile, and is equal to unity for
a perfect single-mode Gaussian beam.  It is defined as:
w(z) = wo.[1 + {(Wl.M^2)/(PI.wo^2)}^2.{z - zo}^2]^0.5                    (7)
where  wo  = waist radius of the beam,
       zo  = waist location along the propagating (Z) axis,
       M^2 = beam quality factor.
For the situation where the beam intensity profile across the aperture of
the transmitting telescope follows a Gaussian TEMoo profile, and the
aperture is greater than 4wo, theory gives the far-field 1/e^2 beam
diffraction angle as:
         2.Wl
THETAg = -----  radians                                                  (8)
         PI.wo
         (115).Wl
       = --------  degrees
          PI.wo
At x = 2wo, or twice the Gaussian radius, the intensity is only 0.03% of its
value on the axis.  So, if the aperture diameter is at least 3wo to 4wo,
then very little diffraction will occur at the telescope output aperture and
there will be hardly any sidelobe energy or modification to the central
(main) lobe.  Let us compromise a bit and assume that the 10 meter diameter
telescope is illuminated such that the aperture diameter d = 2wo, then using
Equ. (8) to an approximation, we can show that:
                       THETAg =  0.0172 seconds of arc
This means that the corresponding FWHM beamwidth would be 0.59 times this
value, i.e., THETA3 = 0.0102 seconds of arc.  This is not that different
from the 0.0138 seconds of arc previously calculated for the FWHM beamwidth
of a uniformly illuminated 10 meter diameter aperture, though it should be
noted that in this situation with d = 2wo, the FWHM Gaussian beamwidth is
actually smaller than the case for uniform illumination.  However, if the
beam was even less truncated by the telescope output aperture, so that
d > 3wo, the beamwidth would be larger than for the case of uniform
illumination.  Note that I have not done a rigorous analysis for the 2wo
aperture illumination, so I am not quite sure how valid the 0.0102" FWHM
beamwidth is for this amount of Gaussian beam truncation.
From Equ. (4), and THETA3 = 0.0102", the corresponding FWHM diameter of the
Gaussian beam at 10 L.Y. is:
                              D =  0.0313 A.U.
The corresponding (1/e^2) Gaussian beam diameter for THETAg = 0.0172" is:
                              Dg =  0.0527 A.U.
This is very similar to the Rayleigh spatial resolution of 0.0506 A.U.
calculated earlier.
                                Summary Table
 -------------------------------------------------------------------------- 
| Beam specifications for 10 meter diameter diffraction limited telescope  |
| operating at a wavelength of 656 nm over a range of 10 light years.      |
|--------------------------------------------------------------------------|
| Illumination:                               Uniform       Gaussian (2wo) |
| FWHM Beamwidth                              0.014"        0.010"         |
| Rayleigh Resolution/Gaussian Beamwidth      0.017"        0.017"         |
| FWHM Beam Diameter                          0.038 A.U.    0.031 A.U.     |
| Rayleigh Resolution/Gaussian Beam Diameter  0.051 A.U.    0.053 A.U.     |
 -------------------------------------------------------------------------- 
Conclusions:
Thus, we see that the beamwidth and beam size are dependent on what we
assume is the aperture taper in the transmitting telescope.  In the Optical
SETI analysis done to date, I have been perhaps a little sloppy and have
tended to mix and match definitions, although it doesn't make much
difference to a comparative performance analysis, as long as one is
consistent.  The FWHM beamwidths for both types of illumination are very
similar, and the antenna gains and Effective Radiated Isotropic Power
(EIRPs) are also expected to be similar, as long as the telescope aperture
(d) for the Gaussian beam is no larger than about 2wo.  However, the first
sidelobes for Gaussian illumination are much weaker, i.e., typically more
that 30 dB below the main beam.  It appears not unreasonable to define the
Gaussian (1/e^2) beamwidth as being equivalent to the Rayleigh resolution.
Perhaps the greatest impact this analysis has, is on our appreciation of the
beamwidth, and the problems posed to an advanced technical civilization
(ATC) in obtaining detailed knowledge about its target, and then targeting
such a beam.
January 8, 1991
RADOBS.14
BBOARD No. 313
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
* Dr. Stuart A. Kingsley                       Copyright (c), 1991        *
* AMIEE, SMIEEE                                                           *
* Consultant                            "Where No Photon Has Gone Before" *
*                                                   __________            *
* FIBERDYNE OPTOELECTRONICS                        /          \           *
* 545 Northview Drive                          ---   hf >> kT   ---       *
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* Tel. (614) 258-7402                     .  .  .  .  .  .  .  .  .  .  . *
* skingsle@magnus.ircc.ohio-state.edu         ..    ..    ..    ..    ..  *
* CompuServe: 72376,3545                                                  *
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