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Diffraction Limited Beams and Gaussian OpticsRadobs 14This 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.
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<-----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.
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*. | .*
* . | . *
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<-----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 * .
* .
**********>**************||*********** .
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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 *
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--->* | *<-- -3 dB FWHM
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-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 *
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*********>***************||*********** .
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TEMoo Beam 2wo || far-field . D
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*********>***************||*********** .
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! |
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--->* | *<-- -3 dB FWHM
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* * * | * * * Weak Sidelobe
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<------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 --- * * Columbus, Ohio 43209 \__________/ * * United States .. .. .. .. .. * * Tel. (614) 258-7402 . . . . . . . . . . . * * skingsle@magnus.ircc.ohio-state.edu .. .. .. .. .. * * CompuServe: 72376,3545 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
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