Scanning 60 degrees with a reflector antenna.
Many attempts have been made to develop a simple reflector antenna with a wide field of view. The most common scanning single reflectors, the torus and spherical cap, are reviewed, and a novel surface-shaping technique that provides high quality scanned beams with superior illumination efficiency is described. Both a symmetric and an offset configuration for this new scanning reflector are discussed. These designs are presented in a descriptive manner, with a mathematical discussion of the surface generation algorithm.[1-3] Both designs scan beams 30 wavelength in aperture diameter through [+ or -] 30 [degrees] with only 0.2 dB peak gain reduction from ideal and with highest sidelobe levels as low as - 11.5 to - 14 dB below peak. Also, a smaller field of view case ([+ or -] 5 [degrees]) 60 wavelength aperture with an 0.8 F/D ratio showing perfectly focused beams for all scan angles is presented.
An important requirement for many antennas is the ability to direct beams in different directions at the same time. An antenna with a wide field of view can communicate simultaneously with several targets without moving, and thus, provide a continuous link between the targets, for example, a geosynchronous satellite. A geosynchronous satellite beams to earth stations on different continents, as shown in Figure 1. Being able to form high quality beams in both the Eastern and Western hemispheres with a single antenna often is a very desirable capability.
Many types of antennas have wide angle scanning characteristics. Both phased arrays and microwave lenses have potentially large fields of view. However, the simplest, lightest, least expensive and most reliable microwave antenna is the reflector.
Reflector antennas can be thought of as mathematical surfaces that redirect microwave rays from a point source feed to an aperture and beyond, into the radiation farfield. In order for the surface to be useful as an antenna, it must have a special shape that reflects all rays so that they leave in a parallel or collimated beam and traverse the same total distance from the feed to a perpendicular planar wavefront. Only one surface accomplishes these two constraints simultaneously, the parabola of revolution.
When the feed source of a paraboloid is moved away from the focus on the axis of symmetry, the rays reflecting from the surface point in the direction of the scan angle, measured from the axis or boresight direction, are no longer exactly parallel. This leads to phase errors across the wavefront. For small source displacements, the deviations from parallel are small, and these nonplanar errors only slightly degrade the farfield beam. As the feed displacement and scan angle increase, the aberrations begin to lower the peak gain and raise the sidelobe levels of the antenna. The primary aberrations that degrade the beam are known as quadratic, coma, astigmatism and spherical aberration. Of these, only the quadratic aberration can be corrected with feed refocusing. Eventually, the other three aberrations, as well as high order errors, make the antenna's beam pattern so inferior that the reflector can no longer be used.
The factors that affect the acceptable field of view of a parabolic reflector are the ratio of the focal length to the reflector diameter (F/D) and the electrical size of the aperture. As the F/D ratio of a reflector increases, the surface curvature decreases, so the individual ray variations are not as great as for a deep, small F/D reflector. Thus, large F/D reflectors have large maximum scan angles. Also, as the electrical diameter D/[Lambda] of the antenna increases, the geometrical errors are electrically magnified, resulting in greater phase errors and a smaller field of view. The field of view often is measured by the total number of beamwidths scanned. While this measure is important for practical spot-beam applications, specifying the absolute angular scanning maximum is more restrictive. Designing a reflector with a field of view of 60 beamwidths is easier than one with 60 degrees of scanning.
Currently Used Scanning Reflectors
For large angle scanning, even paraboloidal reflectors with large F/D ratios fail to generate high quality beams. Other types of reflectors, specifically spherical cap and torus reflectors, generate much better scanned beams. The nonplanar aberration that is most damaging to scanned paraboloid beams is coma, the only odd-order primary aberration. Coma produces a large first sidelobe, the coma lobe, which quickly grows to within a few dB of the gain peak.
The two specialized reflector shapes, the shpere and the torus, have circular cross sections. Torus reflector surfaces are circular in the plane of scan, hereafter assumed to be the azimuth plane, and parabolic in the orthogonal or elevation plane. Since the radius of curvature of a parabola at its vertex is double its focal length, the torus azimuth circle radius is chosen nominally to be twice the elevation parabola focal length. For small aperture diameters, the shape of the torus surface is very close to that of a paraboloid, thus producing beams very similar to those of paraboloids. The same holds for spheres with focal points at half of the radius from the sphere surface. Because of circular symmetry, the surface of every illuminated aperture of the torus is the same in the plane of scan. Two possible scanned beams from a torus or sphere are shown in Figure 2. There is no coma aberration with either the torus or the spherical cap.
The important condition for these reflectors with circular profile sections is that each feed illuminates only the small portion of the reflector for which the circle does not deviate from a parabola too much. As the illuminated portion gets bigger, the astigmatism and spherical aberration grow. As with paraboloids, the geometrical difference between the actual circle and the desired parabola becomes increasingly significant with increased frequency.
The greatest problem with circular profile reflectors is that for large scan angles the illuminated aperture portion is only a small fraction of the entire reflector. As shown in Figure 2, if the frequency is high enough to restrict the useful aperture of each beam to just the highlighted regions, the two scanned regions do not overlap at all. It would be preferable to use two completely separate paraboloids instead of one extra wide torus. For any torus, the illumination efficiency is inversely related to its field of view.
Novel Scanning Reflector Design Method
A new type of reflector surface has been developed that makes use of some of the design philosophy of the torus reflector, yet has much higher illumination efficiency. The first step in the new algorithm is to examine a torus with elliptical rather than circular cross section. Then the profile is improved with a general polynomial function and the mathematical surface dependence in the elevation plane. Finally, the optimum source points and illuminated portions of the reflector are chosen for all scan angles throughout the field of view.
An ellipse reflects rays from one focus to the other. A circle is a degenerate ellipse, so a circular profile torus is a special case of the generalized elliptical profile torus. The advantage of using an ellipse instead of a circle is that the ellipse reflects rays from a source near one focus to the general direction of the other focus, while the circle will only reflect rays toward its center. Therefore, the elliptical torus is much more compact than its circular counterpart for scanned beams. However, finding the reflector source point is more complicated for the elliptical torus than for the circular one. The section of a parabola that most closely fits the ellipse must be found and then the focus of this parabola can be found.
There are some restrictions that guide the choice of the parabola. Its focal point should lie on the line between the semi-minor vertex (x,z) = (0, -b) of the ellipse and one of its foci (c,0). Also, for the reflector to work well for boresight beams in addition to scanned beams, it must be smooth at the vertex. That is, the first derivative of the best-fitting parabola at the vertex must be the same as that of the ellipse. With these constraints, the family of best-fitting parabolas is entirely parameterized by the focal point distance from the vertex t.
It is possible to find a cross-section profile that is even better than an ellipse. Design ideas for fitting parabolas to a profile also can be applied to a general polynomial. Figure 3 shows two parabolas tilted with respect to the boresight axis. Each focal point is a distance t along a line inclined at [+ or -] 30 [degrees] from the vertex to the imaginary ellipse focus on the x-axis. Both parabolas are perpendicular to the z-axis. Since Snell's law requires the ray from the focal point to the vertex to reflect at [+ or -] 30 [degrees], the axis of symmetry of each parabola also must be inclined at [+ or -] 30 [degrees]. Joining the two parabolas with a smooth fourth-order even polynomial z(x) =-b + [r.sub.1] [x.sup.2] + [r.sub.2] [x.sup.4] results in the desired profile. The coefficients b, [r.sub.0] and [r.sub.2] are found by using a least squares technique to minimize the difference between the polynomial profile and the two parabolas over a portion of the profile. The design parameters are the scan angle, the vertex distance and the matching region for the polynomial.
The parameter t corresponds roughly to the focal length of this profile. Increasing t allows for a more exact fit of the polynomial, but increases the F/D ratio. Keeping t as small as possible, good reflector profiles are possible with t = 0.95 times the vertex to imaginary ellipse focus distance a. It turns out that the best matching portion for this choice of t for the + 30 [degrees] reflector is -0.1a [is less than] x [is less than] 0.5a. Only the portion of the profile in this matching region will be illuminated by the scanned feed. The illuminated aperture centers for the scanned beams will be at x = [+ or -] 0.2a, respectively. The unscanned boresight beam will be centered at x = 0.
Next, this profile must be extended into a surface. This is accomplished by adding even order terms in x and y to the polynomial profile. Terms of the form P [y.sup.2] and R [y.sup.4] determine the surface shape along the elevation plane of symmetry (x = 0). The coefficients P and R are chosen by trial and error, balancing the nonplanar phase errors for the 30 [degrees] scanned beam with those of the unscanned beam. The errors for these two cases are in opposite directions from each other; deforming the surface to reduce one set of errors increases the other.
Further nonplanar phase errors can be reduced by adding terms of the form Q [x.sup.2] [y.sup.2] and S [x.sup.4] [y.sup.2]. The illuminated aperture centers must be different for scanned and unscanned beams since the shape along the elevation plane of symmetry is more prominent in the unscanned beam aperture than in the scanned aperture, while the Q and S coefficients have a greater effect on the scanned aperture.
Figures 4 and 5 show the nonplanar phase errors from rays reflected off the best scanning surface for 30 wavelength diameter illuminated apertures for the scanned and unscanned beams. The coma, quadratic and spherical aberrations for both error surfaces are absent. Even astigmatism, the saddle-shaped aberration, is relatively low for both plots. The total error also is very low at 0.18 [Lambda] and 0.28 [Lambda] (65 [degrees] and 100 [degrees]) across the illuminated aperture for scanned and unscanned beams, respectively. It should be noted that further surface deformation to decrease the error in one plot would increase it in the other.
An offset version of this scanning reflector also is designed using the same algorithm. Like an offset paraboloid, the reflector aperture is displaced above the plane of symmetry so that reflected rays miss the feed sources. As with the symmetric design, the scan profile in the azimuth plane is found by using least squares to best fit two tilted parabolas. However, this profile is chosen in a plane y = [y.sub.0] above the x-z plane, so that reflected rays from the sources lie in the plane above source points. Least squares also are used to find the surface derivatives that best fit those of the same tilted parabolas. The optimum unscanned source point now lies below the x-z plane, necessary to ensure the proper focal point location of the offset paraboloid that best fits the reflector surface near the offset vertex. The other polynominal terms that extend the profile into a surface are of the forms P [y.sup.2], Q [x.sup.2] [y.sup.2], R [y.sup.4], S [x.sup.4] [y.sup.2], NY, T [x.sup.2] y, U [x.sup.4] y, V [y.sup.3] and W [x.sup.2] [y.sup.3], with N, T and U determined by the surface derivatives. These other coefficients are found by trial and error to minimize the nonplanar phase error surface, as with the symmetry design.
Reflector Simulation and Analysis Procedure
Although precisely calculating the scattering from conducting objects is a difficult task, standard geometric optics approximations for computing the radiation pattern are fairly accurate near the beam peak and are easy to use. Rays are traced from the feed source to the reflector. Snell's law is used to find the reflected ray. The ray is traced to an aperture plane to calculate phase and amplitude of the reflected wave and then this phase and amplitude distribution at the aperture plane is integrated to find the farfield pattern. The phase at the aperture plane is simply the ray path length times the wavenumber (2 pi/[Lambda]). The integration step is simplified further by using the fast Fourier transform (FFT) algorithm.
Using the FFT speeds up the integration, but requires that the phase and amplitude distribution be specified on a regular rectangular grid. Since rays are first traced to rectangular grid points on the reflector, but then do not reflect exactly parallel to one another, they will not land on rectangular grid points on the aperture plane. An algorithm to adjust the rays to ensure that they land on the proper grid is shown in Figure 6.
The general idea is that the reflector point is moved a small amount in x and y until the traced and reflected ray hits the aperture at the desired location. The adjustments [Delta] x, [Delta] y are set to be the difference between the actual and desired values in the aperture plane. After less than 15 iterations, in most cases, the rays end up on a rectangular grid to within one part in [10.sup.-7]. Once the data are available on the grid, the FFt generates the farfield radiation pattern, which is then plotted in contour or cut plot form.
Picking Optimum Scanned Source Points
The source points for the [+ or -] 30 [degrees] scanned beams already are chosen. The unscanned source is derived by picking the point on the z-axis a distance 1/4 [r.sub.1] from the vertex, but choosing the source points for the other scanned angles between 0 [degrees] and 30 [degrees] needs additional work.
The optimization procedure for source point selection is similar to methods used for standard paraboloids. First, a ray from the aperture center is extended at the desired scan angle with respect to the z-axis. Next, points on this ray are tested successively as source points, traced to the aperture, and the nonplanar phase errors are determined. Once the source point with the lowest error is found, the aperture center point on the reflector is moved, and the procedure is repeated to see if a better source point is available. Although there does not seem to be an automatic way to find the best source point for each intermediate scan angle, the search is limited by the 0 [degrees] and 30 [degrees] apertures and source points.
Using the described algorithm, the farfield radiation patterns of some of the surfaces were derived. For a 60 [degrees] field of view reflector with a 30 [Lambda] diameter illuminated aperture, both a symmetric and an offset reflector are simulated. In addition, a 10 [degrees] field of view design with a 60 [Lambda] aperture and 76 percent illumination efficiency is simulated.
Figures 7 and 8 show the farfield power contour patterns for the symmetric 30 [Lambda] illuminated aperture diameter reflector at 30 [degrees] and 0 [degrees]. Figures 9 and 10 show the cut patterns through the beam peak in the elevation plane of the same scan cases as Figures 7 and 8. For comparison, the farfield power patterns for ideal uniform 30 [Lambda] apertures also are shown on these plots. The peak gain values are only 0.1 dB below the ideal peak circular aperture gain, and the highest sidelobe levels are 14 dB below the peak gain level. For intermediate scan angles the performance is no worse than at the edges and center of the field of view. The entire reflector is 54 [Lambda] wide X 30 [Lambda] high, so the illumination efficiency is slightly higher than 50 percent. The F/D ratio of the entire reflector surface is 0.95 for a comparably sized paraboloid, gain at 30 [degrees] falls by more than 2 dB, and the coma lobe is as high as 7 dB down from peak gain. While the paraboloid is better for on-axis beams, clearly it is unacceptable at 30 [degrees].
The offset version of this reflector produces very similar radiation patterns with sidelobe levels slightly higher, at 11.5 dB down for the 30 [degrees] scanned beam. None of the beams suffers from any feed blockage.
Figures 11, 12 and 13 show the nonplanar phase error plots for 5 [degrees] and 0 [degrees], and the farfield contour plot for the 5 [degrees] beam for the [+ or -] 5 [degrees] field of view reflector. Figure 14 shows the front view of this reflector with aperture centers indicated. This reflector can be formulated in a more compact, overall package than the previous [+ or -] 30 [degrees] cases, since the scan requirements are one-sixth as wide. The highest sidelobe levels are 17.6 dB down from peak. In effect, this reflector has no scan loss at all over the full 10 [degrees] field of view. A torus wide enough to scan [+ or -] 5 [degrees] would have to be almost 40 percent bigger than this reflector.
Scanning reflectors must sacrifice beam quality or illumination efficiency to maintain reasonable performance at the extreme edges of their fields of view. Similar to designing a flat amplifier frequency response, wide field of view reflectors are not the optimum design at any given scan value, but perform well across all scan angles. Circular cross-section surfaces yield excellent scanned beams and are simple to specify but have low illumination efficiency. For wide scan angles, using two reflectors is preferable.
The new design concept of fitting polynomials to tilted parabolas is a compromise between the overly wide circular surfaces and inefficiently scanning paraboloids. The reflector width is much smaller than a comparably performing torus, yet the phase errors and farfield beams are acceptable. The described algorithm can be used to generate a reflector for any limited scanning application. In the limit of 0 [degrees] field of view, the resulting degenerate surface is just a paraboloid. For any nonzero field of view, the resulting surface will always yield beams that are better than those of a paraboloid at the extreme scan angles.
The offset configuration of this new reflector performs almost as well as the symmetric form, but avoids all blockage by the source feeds. Offset torus designs have been built, but for spherical reflectors, every feed structure always blocks its own beam. The compact package and high quality scanned beams of this new reflector should make it a natural choice for many applications that require high performance antennas.
Much of the work reported in this paper was performed by William Craig in the course of pursuing his master's thesis. Many optimization results were computed by Jeffrey Mason. Ann Morgenthaler produced many of the figures. The author gratefully acknowledges these important contributions. [Figures 1 to 14 Omitted]
W.P. Craig, "A High Aperture Efficiency, Wide-Angle Offset Reflector Antenna," master's thesis, Northeastern University, June 1990. C. Rappaport and W. Craig, "High Aperture Efficiency, Symmetric Reflector Antennas with up to 60 Degrees Field of View," to appear in IEEE Transactions on Antennas and Propagation, March 1991. W. Craig and C. Rappaport, "A High Aperture Efficiency, Wide Angle Scanning Offset Reflector Antenna," in preparation for publication in IEEE Transactions on Antennas and Propagation.
Carey M. Rappaport received five degrees from the Massachusetts Institute of Technology, including his SB in mathematics, his SB, SM and EE in electrical engineering in June 1982 and his PhD in electrical engineering in June 1987. He has worked as a teaching and research assistant at MIT from 1981 to 1987, and during the summers at COMSAT Labs in Clarksburg, MD and The Aerospace Corp. in El Segundo, CA. He is assistant professor of electrical and computer engineering at Northeastern University. He has consulted with Bolt, Beranek and Newman Inc. and A.J. Devaney Associates, and has co-founded a biomedical device company, Berry and Rappaport Associates. His current research interests are specialized electromagnetic antenna design for biological and communication applications and EM computational methodology. He was awarded the IEEE Antenna and Propagation Society's H.A. Wheeler Award for best applications paper of 1985. Rappaport is a member of the Eta Kappa Nu and Sigma Xi.
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|Title Annotation:||microwave antennas|
|Author:||Rappaport, Carey M.|
|Date:||Feb 1, 1991|
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