To: The Savvy Microwave Reflector Gang.
From: Dick, K2RIW. 10/10/00
Re: Dish Underillumination, F/D, G/T, Phase Center and Related Topics.

Warning: This is not a brief memo (I've got lots of strong feelings on this subject). However, after my errors are corrected, I hope my opinions on the operation of Parabolic Antennas will be helpful to those who care to read this. For those who's server will not allow a 5 page E-mail, I'll break it up into multiple E-mails.

Introduction -- I have had a rather close and hands-on relationship with Parabolic Dish antennas since 1968, when I scratch-built my first 12 footer -- "A 12 Foot Stressed Parabolic Dish," QST, August, 1972, page 16 and on the cover, by K2RIW.

On 10/06/00 at 8:35 am Zack, W1VT, asked a simple, but provocative question about under illuminating a dish as a method of beam broadening. I call this a "Zoom Control." I have used a Zoom Control for 4.5 years (with great pleasure) on my 432 MHz array of 16 yagis (19 elements each = 304 elements total) on a 100 foot tower. The use of electronically selectable beamwidth broadening during a contest operation can greatly increase the fun, and eliminate many of the azimuthal ambiguities.

Zack's fascinating question kicked off a series of 19 Reflector responses by K2TXB, G4BRK, K5TR, WA5VJB, W0EOM, KD7TS, KJ4SO, W6CWN, K0CQ, WA2SAY, W7CS, and AL7EB, in that order. Each of the responses contained vital "pieces of the puzzle," but few of the responses would give a newbie a "warm feeling" for what is happening within that modified Parabolic Antenna. Arguably, the most important and popular type of Microwave antenna is a Parabola.

We all want more, new and skilled Microwave operators -- so as to preserve our valuable spectrum, as well as other reasons. For this to happen we must remove some of the mystery about this valuable antenna type. Once learned, many new and crafty operators will "see" new antenna possibilities within every hardware store (such as Home Depot). This memo is my attempt to fill some of that void. Please read also, W1GHZ's On-Line Microwave Antenna Book.

This E-mail is broken up into seven sections that try to explain (I) The Focusing Action of a Parabola (and it's cousins, the Offset Parabola, Ellipse, and Circle), (II) The two meanings of The Parent Parabola, (III) The effects of Axial Horn Movement (Prime Fed Dish), (IV) Axial Horn Movement (Offset Dish), (V) Non-Axial Horn Motion (two types), (VI) Gain Maximization, and (VII) The W2IMU Horn Modification Problem.

PARABOLIC DISH ANTENNAS, Part 1.

By K2RIW, 10/10/00.

(I) Focusing RF Energy -- During reception, every antenna we use has a certain "capture area" wherein the antenna gathers and focuses the RF energy onto a "driven element," "feed horn," or similar structure that supplies the "gathered RF energy" to the transmission line. In the case of a Parabola, that "gathering," and "Focusing" (if done efficiently) involves at least TWO coherent functions -- (1) having the correct surface orientation (angle of incidence equals angle of reflection), and (2) obeying the correct phase length (path length). If used correctly, the so-called "conic sections" do this very well. I'll first give three conic section examples (ellipse, circle, and parabola) to illustrate this, as a kind of thought experiment:

Example (1), Ellipse -- Assume I have a (slightly isotropic) point source RF emitter that's located at the transmitter site, in my back yard, and I wish to capture all it's output at a receiving site, that's 10 feet away. There is one surface that does this perfectly, it's an ellipsoidal surface; it would look like a large, reflective, egg, that's placed around the two sites. That's a three dimensional (3D) ellipse, or an ellipse of revolution about the major axis. I'll boil this down to a 2D discussion. Assume I have placed a vertical sheet of metal through the two sites and I'll analyze the shape of the ellipsoid that touches the sheet (it's a 2D ellipse).

If the ellipse was placed at the coordinate center, the equation of the 2D elliptical reflector would be (X/a)^2 + (Y/b)^2 = 1, where a and b are constants that define the shape and size of the ellipse. The ellipse has two foci (focuses); each one is located at one of the sites (transmission and reception). The ellipse has a property called eccentricity, (e = 1 - (b/a)^2), which is less than 1 and somewhat proportional to the distance between the foci.

The magic of the elliptical surface is that it meets the TWO conditions -- (1) every square inch has the proper surface orientation to reflect the energy in the correct direction, AND (2) every possible path taken from the transmitter site to the receiver site (with one bounce) has exactly the same path length. That means that the ellipse (ellipsoid) will gather ALL the transmitted energy (from all of 3D space) and focus it (all in phase) at the receiver site. To me, that's kinda neat. By the way, ellipsoidal reflecting surfaces have often been used to focus almost all of the exciter energy of a laser pump onto a laser rod.

Example (2), Circle (Sphere) -- Now let me slowly move the transmitter closer to the receiver, while continuously changing the reflecting surface, so that I can keep gathering all the RF energy. In the limit, the transmitter and receiver will be collocated, and the reflecting surface will become a sphere (a circle on my sheet of metal), where a = b in the ellipse equation -- the eccentricity has gone to zero, and the two foci have moved together.

Example (3), Parabola -- Now let me move the transmitter site off to an infinite distance, while continuously changing the reflecting surface, so as to keep gathering all the RF energy. In the limit, that surface will be a Parabola, which is merely an ellipse where the (a) dimension has gone to infinity in the ellipse equation; the eccentricity is 1.0. With a little rearrangement, the equation takes on the familiar form, Y^2 = 4(f)X , where f equals the focal length and Y is the dish radial dimension.

Notice that a parabola (for an infinite distance) and an ellipse (for a close distance) are related, but slightly different curves. This may give you some feeling for why there is such a thing as a "Near Field Range" for a parabola (R = 2D^2/lambda), where D is the dish diameter and R is the range. If you attempt to make Parabolic Antenna measurements within that range (or closer), the surface shape is sufficiently far from the correct elliptical shape that the RF energy starts focusing slightly out of phase and you start getting a noticeable degradations in pattern and gain. The first sign of this is that the first null in the antenna pattern (between the main lobe and the first sidelobe) disappears.

Moving the feed focal length further away from the dish gives you a slight improvement in the Near-Field Focusing errors, but it does not accomplish a complete correction -- only reshaping the reflector into an ellipsoid would do a perfect job.

(II) The "Parent" Parabola -- This phrase has at least two meanings:

Meaning (1) -- Notice that the basic equation of a parabola always goes to infinity in the multiple directions (X and Y). When we decide to build a Parabolic Dish Antenna, we are deciding to build a reasonable portion of that complete Parabolic curve. By choosing the focal length (f) of our Parabola we are choosing whether the portion we build will be a deep dish or a shallow one -- each has it's advantages. Selecting the focal length (or the F/D) merely selects the radius of curvature (at the apex, for instance). If we cut away some of the reflector, or electrically do a similar function (by underilluminating), we are not changing the true focal length (radius of curvature) of the Parent Parabola.

In all cases, a well-constructed Parabolic Reflector has ONE FOCUS for all frequencies and dish diameters (when using the same Parent Equation [focal length] ). Many amateurs erroneously think that the focal length changes with the frequency or the portion of the surface that is constructed (or illuminated).

Meaning (2), Offset Reflectors -- When we construct a round shaped Parabolic Reflector with the Parabola Apex in the center, this too can be called the "Parent Parabola." We can then choose to "cut away" an off-centered portion of the reflector, leaving a round (or slightly oval) "Offspring" that includes the apex and one side of the perimeter. That new surface is an Offset Parabolic Reflector. Notice that the Offspring now has a non-symmetric surface that's more curved at the position of the apex of the Parent Parabola (this difference is rather subtle to an unaided eye). Also, there is only one position (spot in 3D space) where the reflector focus is located; you can not rotate the Offset Reflector and leave the feed horn in the same position -- the non-symmetrically curved surface will not focus properly.

One of the well kept secrets of almost all the manufactured Offset Fed Parabolic antenna systems, is that the Offspring Reflector includes the Parent Parabola's apex (center of the original Parent Parabola. The result of this is that you can now easily determine the elevation aiming point. Merely sight from the apex edge of the dish (the part closest to the feed) through the phase center of the horn; that's the antenna's boresight. If that turns out to not be true, than most likely you are using the wrong focal position. This became apparent when a number of 10 GHz operators started using the same 18" Offset Fed Dish on 24 GHz. The dish efficiency was quite low, until they determined the "true" focal position; the shorter wavelengths made this more critical. When they then went back to 10 GHz, they discovered a slight increase in efficiency with the "corrected" focal position.

An Offset Fed Parabolic antenna system has the feed horn phase center placed at the same focal point as the Parent Parabola (it didn't change because of the off-centered cut away). However, the Primary Feed Pattern would now be illuminating areas where the Parent Parabola, was cut away (that would be wasteful). Thus, the feed is re-aimed (but, not translated) at approximately the center of the remaining Offspring Reflector -- keep the phase center in the same place. This gives the best (and a higher) efficiency, because now the feed horn is not causing a blockage and the spill-over (and feed sidelobes) are usually illuminating cold space, and thus not contributing much to the system's noise temperature.

III Shifting The Feed, Forward and Aft (Prime Fed Dish) -- When we move the feed horn along the axis of transmission (in a prime-fed Parabola) we are shifting the focal distance, but we are creating a spherical aberration (a kind of defocusing). Moving the horn outward causes the emitted signal to have a concave wavefront (viewed from a position that's in front of the dish system), and it "Focuses" at a distance closer than infinity (it becomes near sighted). I use the word "Focus" in quotes because, as previously explained (in Section I), this is only a partial phase correction. It makes an improvement for a close emitter, but the technique can only be carried so far. You can't use this technique to get a good focus on a 10 GHz 3 foot dish at an emitter distance of 10 feet (mathematics to be supplied later).

Moving the horn inward of the calculated focal distance causes the emitted signal to have a convex wavefront and now the dish is "Focused" beyond infinity (it becomes far sighted).

The horn inward moving technique is a coarse method of "Beam Defocusing," or "Beam Broadening" that could be used as a Zoom Control. It causes a more spherical wavefront (lowers the gain), and simultaneously causes the desired underilluminating function required for a broader beamwidth. There will be some sidelobes developed, but they may be very tolerable; their magnitude is partially dependent on the system's F/D ratio, as well as the Feed Horn characteristics.

IV Shifting The Feed, Forward and Aft (Offset Fed Dish) -- However, the axial motion (along the boresight) of the feed horn will have a different effect on an Offset Fed Parabolic system. It will simultaneously cause Squint. That's a fancy way of saying that the boresight will shift. The direction of shift (squint) is fairly easily predicted (with slight inaccuracy) by knowing that (in general) the angle of incidence equals the angle of reflection.

V Shifting The Feed, Non-Axially (Offset & Non-Offset Fed Dish) -- But, I believe that there is a way of intentionally de-focusing an Offset Fed Parabolic system by moving the horn inward along the axis of the horn. This will slowly (because of the larger F/D ratio) cause a convex wavefront (front view) and, again, it will cause underillumination -- both are in the desired direction for beamwidth broadening. There probably are past references on this subject. But, even if there aren't any (for an Offset Fed system), this savvy group is becoming quite proficient with fancy modeling programs; this would be a great place to do some pattern/gain modeling. Or, simply make a "Leap of Faith" -- go do it (try it, you'll like it)!

For a Prime Focus Dish, there is a considerable amount of beam steering that can take place by only moving (translating) the horn in a transverse manner. In the Radiation Laboratory Series of Books ("Microwave Antenna Theory and Design," #11, McGraw-Hill Book Co., NY, 1949, by S. Silver) the author states that the boresight can be steered by six beamwidths, before the gain falls off by 1 dB, if a 0.6 F/D reflector is in use. The problem becomes worse for lower F/D ratios. This suggests that a cluster of almost 6 horns, side by side, is possible if a 0.6 or greater F/D is in use.

I believe that an Offset Fed Parabolic Antenna system will be even more forgiving, when transverse horn motion is used (in azimuth or elevation). This is because of the large F/D ratios that most of them posses. This suggests that a whole cluster of horns is possible, with a rather small gain sacrifice. The only disadvantage may be that a predictable amount of dish re-orientation will be required when a change in horns is initiated.

VI Gain Maximization -- Step One in the process of maximizing the gain of a parabolic Reflector antenna system is understanding the Geometry. This means:

(A) Is the reflector the correct shape and smooth enough for the wavelength in use? If not, can it be re-shaped; can the dents be hammered out, etc.? The Johnson, "Antenna Engineering Handbook", McGraw-Hill, 1992, has curves that predict the rate of gain fall off versus reflector errors (bumps).

(B) Is the reflector's mesh fine enough for the wavelength in use? Should it be covered over with a finer mesh? Johnson (ibid) Handbook has prediction curves.

(C) Has the proper feed horn focal position been found for the reflecting surface in use? By measuring the diameter (D) and the depth (d), and applying the formula, F = (D^2)/16d , this can be found for a Prime Focus Parabola. For an Offset Fed Parabola, the problem is a little more difficult. W1GHZ's web site will help. Even by pure experimentation, it can be found.

(D) Has the system's F/D been correctly determined (this is required to design/ build the proper feed horn)?

Step Two -- The next step is choosing the best Feed Horn to properly illuminate that Parabolic-shaped Reflecting surface -- this is a slight compromise. Do you want the maximum Gain, maximum Efficiency, or best Gain to Temperature (G/T) Ratio?. Most of us want the first two (they're very close). An EME'er will want the best G/T.

Well, where does the best Gain come from? If this was the best of all worlds, your Feed Horn design would apply an equal amount of RF energy (in phase) to every square inch (or square cm) of the reflecting surface (including the extra path loss to the dish perimeter). That Primary Feed Pattern would abruptly fall to zero at the edge of the dish -- there would be no spill-over energy. Such a Primary Feed Pattern would yield a reflecting surface with 100% aperture efficiency, and the G/T would be ideal.

Believe it or not, those feed horn characteristics are almost achievable. BUT, to have that high a Directivity and Pattern Control, such a horn (or a cluster of horns in a Phased Array) would be larger than most of the Parabolic Reflectors we have ever used. In a Prime Focus Parabolic System, that ideal horn would block out the whole reflector! Even in an Offset Fed Parabolic system, if your horn has more gain than the reflector, skip the reflector and simply aim the horn at the target!

So, for a bunch of reasons, we choose the best realistic horn we know of (Chaparral or Dual Mode [W2IMU] if your dish is near a 0.6 F/D) and adjust them for either a -10 dB, or -20 dB edge illumination taper. The two horn types (Chaparral and W2IMU) are really a multi-element type of feed that have a very desirable pattern, almost no edge currents outside the horn, almost no sidelobes, and they have a nearly constant point source phase center versus azimuth, elevation and diagonal observation angles.

Bear in mind that a subtle shift in the emitted phase from your horn, as a function of a change in observation angle, causes the same system degradation as if your reflector had a big dent in that area. It's hard to believe that some beautiful-looking Parabolic Dish Antenna systems can have a serious error that an untrained eye can not see.

A lot of experience has shown that the best gain occurs at about a -10 dB edge taper (nearly the best illumination taper), and the best G/T occurs at about a -20 dB edge taper (a kind of underillumination). The gain changes rather slowly (at first) as the illumination taper is changed. The most sensitive characteristic is the first side lobe levels -- they can be as low as -25 dB (or better) with a -20 dB edge taper.

VII W2IMU Horn Modification Problem -- I have read of enthusiastic builders who changed the length or diameter of the large-diameter section, or the diameter of the smaller section, of the W2IMU Dual Mode Horn, as a way of controlling the horn's beamwidth, so as to adapt it to a dish that has an F/D that's quite far from (the designed) 0.6. This can be a disappointing endeavor. Yes, the Dish is not going to crash and burn, but it may have a disappointing efficiency (or pattern) result.

What Dr. Dick Turrin, W2IMU, did in that horn design was to set up the two circular waveguide modes (TE11 and TM11) in just the right amplitude ratio and phase relationship that the resultant at the horn throat has virtually no edge currents to cause sidelobes and back lobes. This also causes the horn to have a constant point source phase center versus all observation angles.

If you were to change the 30 degree flare angle or the larger diameter, that would change the amplitude ratio of the higher mode generation (TM11). If you change the diameter or length of the larger section, that will change the phase relationship between the modes, because they have different cut-off wavelengths and thus different phase velocities in the larger diameter section. The length and diameter of this section is really a phase corrector between the two waveguide modes.

I may be overly-conservative, but here is my opinion. If you are very skilled at 3D modeling of higher-order waveguide mode generation techniques, and if you are skilled at calculating Bessel-Neumann and Hankel Functions, than have at it and please let me know about your results. For the rest of us mere mortals, I recommend that you follow one of W2IMU's two designs exactly, and only scale every dimension, proportional to your particular wavelength.

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I hope these thoughts are helpful to those who were brave enough to read all of this. Please feel free to correct the errors.

73 es Good UHF/SHF/EHF DX,
Dick, K2RIW.
World Grid: FN30HT84DC27.