Multifocal microwave bootlace lenses.
Throughout history man has sought to magnify images of distant objects. Telescopes with reflectors and lenses allow an observer to see much farther than with the unaided eye, but only in one direction at a time. There has always been a trade-off between magnification and field of view. A telescope or antenna that strongly magnifies and also scans across the sky without physically moving is extremely useful for communications.
The bootlace lens is a passive antenna that magnifies microwave signals using transmission rather than reflected signals. Unlike a dielectric lens, it consists of an input array connected to an output array by transmission lines. The input field couples to the elements of the inner surface array. Each element sends its signal through its transmission line to its corresponding element of the outer lens surface, then the output array couples to the output field.
The goal of wide field-of-view scanning antennas is to collimate source rays from a variety of incident angles within a large angular sector of space. These antennas should produce minimally distorted beams in all directions within this sector, which is possible only if the phase errors across the antenna aperture are less than a particular maximum value, which depends on the application.
With multifocal antennas, there are several scan angles with no phase errors and ideal undistorted beams. In between these perfect beam angles, the errors are nonzero, but can be kept acceptably small with proper design. Bootlace lenses can be developed with up to four perfect focal points, giving a much greater useful field of view than reflector antenna systems, which are limited to approximately 16 [degrees] of scan.[1,2] Yet they are less complex, less expensive, far more reliable than phased-array antennas and require no processing of the individual element signals.
These lenses can provide a means of accurately focusing plane waves arriving from sources separated by as much as 90 [degrees]. They generally have longer focal lengths than reflectors but suffer no blockage. Each source in the focal region of the lens produces a well-formed beam, so combinations of sources give shaped power contour patterns that can be designed to suit particular illumination requirements. Furthermore, by distributing small amplifiers, one per transmission line between the inner and outer lens surfaces, the lens becomes a high power, high efficiency, high reliability antenna.
Bootlace lenses can be either three-dimensional, with a radiating surface aperture, or two-dimensional planar types with a linear aperture. The two-dimensional planar type has been more popular because it is easy to fabricate in microstrip.
There are several adjustable constraints on the bootlace lenses, including the shape of the inner surface, the shape of the outer surface, the electrical length of each transmission line connecting the elements of the two, and the mapping of the position of each element on the inner surface to the position of the corresponding element on the outer surface. These constraints can be manipulated to produce a variety of lenses, including the two-dimensional Rotman lens with a straight radiating aperture and three perfect focal source points; the McGrath lens with a straight inner surface and two focal points; the Rao bifocal lens with two focal points; and the Rappaport quadrufocal lens[6,7] with four perfect focal points.
General Bootlace Lens Equations
All bootlace lenses are formulated by solving the path length equations for rays traveling from the design source points to the intended scanned transmitted phase front. If the electrical path lengths are the same for all rays from a given source, the wave represented by the rays will leave the lens with no phase errors, and hence, will be perfectly focused. Since there are several degrees of freedom in the shapes and transmission line lengths of a bootlace lens, there can be multiple perfect focal points. If the foci are not too far apart, it is possible to have a wide scanned field of view with no beams having large phase errors.
Upon leaving the lens, the rays from any given focus must be parallel and must have equal phase along any wavefront perpendicular to the outgoing beam direction. That is, adding the length between a given focus and the inner lens curve to the transmission line length at that contact point, and then adding the distance to an inclined wavefront in the predetermined beam direction, must give the same constant value for any inner surface point.
As shown in Figure 1, the inner lens surface, denoted by Z(X,Y), outer lens surface, denoted by W(U,V), transmission line lengths connecting the inner element at (X,Y) to the outer element at (U,V), denoted by L(U,V), and a given focal point at (x,y,z) = (F sin [Alpha],0, - F cos [Alpha]), transmitting rays that leave the lens at an angle [Alpha] with respect to the w-axis establish the geometry of the lens. The coordinate system for the inner surface is parallel to and has the same origin as the outer surface coordinate system in the figure. In practice, however, the inner and outer surfaces can be separated and tilted from one another, as long as the lengths of the cables, fibers or printed circuit lines connecting them are all within the same constant electrical length from the design specifications. For planar lenses, there are no Y and V coordinates and the values of Y and V in the subsequent equations are set to zero.
The distance on the input side, from the focal point to a point on the inner surface, is [square root of [(X - F sin [Alpha]).sup.2] + [Y.sup.2] + [(Z + F cos [Alpha ]).sup.2]] The distance on the output side, from the outer surface along a scanned ray to the inclined phase front, which passes through the origin (u,v,w) = (0,0,0), can be shown to be U sin [Alpha] - W cos [Alpha]. Adding these distances to the relative line length L gives the total path length for any ray leaving the given focal point. In particular, for the ray that traces to the inner surface origin and vertex (X,Y) = (0,0) and Z = 0, the line length is L = 0, assumed to be zero at the origin, and the ray leaves at the outer surface point W(0,0) = 0 to give a total path length of F. Equating the general ray with the vertex ray gives the fundamental bootlace lens equation (1) [Mathematical Expression Omitted] There is one of these equations for each focal point. Combining the set of equations with further geometric constraints gives the wide variety of lens types currently used.
The Rotman Lens
The three characteristic specifications for the flat Rotman lens are the inner lens array, the mapping function from inner to outer array elements and the lengths of the transmission lines connecting these elements. The outer array is a straight line, defined by (2) W = 0 Three focal points are used with the Rotman lens, two symmetrical ones about the negative z-axis and one on the negative z-axis at z = -G. The three design parameters for this type of bootlace lens are the focal lengths F and G, and the design scan angles [+ or -] [Alpha] (with a fixed boresight beam). Solving for the three fundamental equations in the form of Equation 1 give the algebraic equation for the line lengths (3) L = -b - [square root of [b.sup.2] - 4ac]/2a where a = 1 - [(G - F).sup.2]/[(G - F cos [Alpha]).sup.2] - [U.sup.2]/[F.sup.2] b = [Mathematical Expression Omitted] c = [Mathematical Expression Omitted] The mapping of the outer vertical coordinate U to inner vertical coordinate X is (4) X = U (1 - L/F) and for the inner curve the formula is (5) Z = - 1/2 [U.sup.2] [sin.sup.2] [Alpha]/G - F cos [Alpha] - L G - F/G - F cos [Alpha] Given W(U), L(U), X(U) and Z(U), the Rotman lens geometry is specified in terms of the outer vertical coordinate.
Although the original reference to the Rotman lens only specifies a planar version, a three-dimensional formulation also exists. The only change in the design equations is to add the term -[V.sup.2] to the c coefficient of Equation 3, and add one further equation (6) V = Y Now, all design parameters are functions of (U,V) instead of just U.
The proper choice of design parameters depends on the antenna application. The useful field of view is limited by the design scan angle, with phase errors becoming significant for scan angles greater than 5 percent or 10 percent beyond [Alpha]. The phase errors within the field of view at the nonperfect source points are usually greatest at the lens ends (or edge, for a three-dimensional lens). As with a reflector, the errors are directly dependent on the focal length to diameter ratio F/D. A long focal length makes the errors smaller, but also makes the antenna larger. Once the F/D is selected, which is based on packaging requirements, the ratio of the focal lengths G/F can be optimized to so that the worst phase errors across the aperture have minimum magnitude for the worst scanned beam. This is readily accomplished by comparing the error curves for various choices of parameters.
The McGrath Planar Lens
The McGrath lens starts with the assumption that the inner as well as the outer lens surfaces are planar, defined by (7) W = 0 and (8) Z = 0 While the design constraint greatly simplifies lens fabrication, it gives up a degree of flexibility, and therefore, does not have as large a field of view as the Rotman lens. There are only two possible path length equations in the form of Equation 1. Solving these equations gives (9) L = F - F [square root of [F.sup.2] - [U.sup.2] [sin.sup.2] [Alpha]/[F.sup.2] - [U.sup.2]] (10) X = U (1 - L/F) For the three-dimensional version of the McGrath lens, Equation 9 is replaced by (11) [Mathematical Expression Omitted] with the additional mapping equation (12) V = Y The equations for the mapping from outer to inner element points, Equations 4 and 6, are the same as Equations 10 and 12.
The Rao Bifocal Lens
Like the McGrath lens, the Rao lens has two perfect foci and a planar outer surface. The distinctive feature of the Rao lens is that the transmission line lengths are constant. It also turns out that the mapping function is particularly simple. The inner surface transverse shape of this lens is circular, generated by rotating the two-dimensional curve about the line connecting the foci. The equations for this lens are (13) W = 0 (14) L = 0 (15) X = U (16) Z = -F cos [Alpha] + [square root of ([F.sup.2] - [U.sup.2]) [cos.sup.2] [Alpha] - [V.sup.2]] and, as before (17) Y = V Equations 15 and 17 show all the transmission lines to be parallel and Equation 14 shows that the transmission lines have the same length as the vertex line. Thus, the Rao lens is perhaps the simplest three-dimensional lens to build because only the shape of the inner surface needs be specially constructed, and it is a surface of revolution.
The Rappaport Quadrufocal Lens
There are two forms of the Rappaport quadrufocal lens.[6,7] The first is the most general, while the second has a flat outer surface as with the previous bootlace lenses.
If all of the geometric constrains are removed, allowing for both a curved inner and outer surface, and adjustable line lengths and element mapping, then four focal points and the four corresponding equations can be specified. The resulting quadrufocal lens has the widest field of view, but is the most difficult to fabricate.
It is assumed that the focal points are in two symmetric pairs with focal lengths [F.sub.1] and [F.sub.2] at angles [+ or -] [Alpha] and [+ or -] [Beta], respectively. Solving the four equations in the form of Equation 1 in terms of these four lens parameters results in (18) [Mathematical Expression Omitted] where [D.sub.1] = [F.sub.1] cos [Beta] - [F.sub.2] cos [Alpha] [D.sub.2] = [F.sub.1] cos [Alpha] - [F.sub.2] cos [Beta] (19) [Mathematical Expression Omitted] (20) W = [F.sub.1] - [F.sub.2]/cos [Alpha] - cos [Beta] (X/U -1) and (21) L = [D.sub.1]/[F.sub.1] - [F.sub.2] W again, as before, (22) Y = V The second version of the quadrufocal lens assumes that the four focal points lie on a circle, centered at the lens' inner vertex of [F.sub.1] = [F.sub.2] = F. Expanding Equation 18 in terms of powers of [F.sub.2] - [F.sub.1], and then setting the difference to zero yields a new equation for the inner surface (23) [Mathematical Expression Omitted] This equation is independent of V, unlike the other three-dimensional lens formulations. This inner surface is a parabolic cylinder.
To find the formula for the mapping function, the four fundamental lens equations of Equation 1 must be used in conjunction with Equation 23. The mapping equation does depend on V as well as U, so it is the most complicated aspect of this lens (24) [Mathematical Expression Omitted] (25) W = 0 (26) L = F (1 - X/U) and (27) Y = V.
The two-dimensional version of this four focal point lens, like all the other lenses, results from setting Y = V = 0.
Equation 25 indicates that the outer surface is flat, which would seem to lead to a contradiction, since there are four focal points but only three varying lens constraints, which are X(U,V), Z(U,V) and L(U,V). However, since there is only one focal length, there are only three lens parameters, F, [Alpha] and [Beta]. In effect, the Rotman lens on-axis focal point is split into a symmetric pair. The Rotman lens with G = F is a special case of the Rappaport quadrufocal with [Beta] = 0, as can be seen by reducing Equations 3 and 5 into Equations 24 and 23, respectively.
The addition of the fourth focal point with the quadrufocal lens further reduces the aperture phase errors of the in-between beams in the field of view. Thus, the maximum scan angle can be chosen to be greater than for the Rotman lens with the same quality beams throughout the field of view.
Lens Shape and Performance
The inner surface profiles in the y = 0 plane for the discussed bootlace lens and the nominal focal arcs for the Rotman lens and the Rappaport planar quadrufocal lens are shown in Figure 2. For all cases, the design scan angle is [Alpha] = 40 [degrees], so each lens has roughly the same maximum scan angle. The on-axis focal length for the Rotman lens is G = 1, while the scanned focal length is F = 0.92. For the Rappaport quadrufocal, the second pair of foci are at [Beta] = 28 [degrees], which is the optimum angle to minimize the phase errors across the entire field of view from -42 [degrees] to +42 [degrees].
The inner surfaces are plotted for lenses with an aperture length (or diameter for three-dimensional versions) of 1.6 units. The F/D ratio is 0.63. These nominal focal arcs are portions of circles. It is occasionally possible to reduce the errors of the in-between beams by refocusing the corresponding focal points and reshaping the focal arc.
Refocusing is a well-known method of improving distorted reflector antenna beams. The source is moved in or out along its central ray pointing to the inner surface vertex. This adjustment does not alter the beam direction and has very little effect on the other odd-order phase aberrations, such as coma. The errors for the Rotman lens tend to be dominated by odd-order functions, so refocusing gives only limited improvement for this bootlace lens.
For the two-dimensional quadrufocal lens, the on-axis beam has the greatest errors in the field of view, but all these errors are even-order aberrations. Refocusing to minimize the error for this beam gives the focal arc as a function of scan angle [Theta] (28) [Mathematical Expression Omitted] producing the best field of view performance of any two-dimensional lens.
After refocusing, the path length errors across the aperture at the worst scan angles in the field of view for each of the described lenses are shown in Figure 3. These errors represent nonplanar phase variation of phase along the scanned wavefront. The effect of these errors on the radiated beam depend on the electrical size of the antenna. For example, a 32 wavelength wide lens aperture would have a nominal focal length of 20 [Lambda], so a normalized path length error of 0.005 would correspond to 0.1 [Lambda], or 36 [degrees] of phase error. This level of maximum phase error causes distortion of about 0.1 dB of peak gain reduction and slight increases of the second and subsequent sidelobe levels.
The asymmetry in the Rotman lens error curve indicates that the dominant aberration is clearly odd-order. The two bifocal designs have error values at the edges of the aperture that are so large they exceed the limits of the graph. For the McGrath lens, the amount of refocusing needed to minimize the error near the axis is quite large. The best on-axis source point, for the given set of parameters is at f = -2.25. This is an excessively long focal length, corresponding to a focal-length-to-diameter ratio F/D = 1.4. For the Rao bifocal lens, the best refocusing occurs for f = -0.67.
Figure 4 shows the worst error, which occurs at U = 0.8 on the aperture, for the two-dimensional Rotman and Rappaport lenses. In both cases, the error is zero at [Theta] = 40 [degrees], and also at [Theta] = 0 [degrees] for the Rotman lens and at [Theta] = 28 [degrees] for the Rappaport quadrufocal lens. The quadrufocal lens has significantly lower maximum phase error for the entire field of view.
Reciprocal Quadrufocal Lens
A new formulation of the bootlace lens concept takes advantage of the four available degrees of freedom to specify a reciprocal bifocal, with the same shape outer and inner surfaces and parallel transmission lines. With these constraints, the resulting lens can have feeds on both sides and transmit on both sides. Further, by translating and rotating the outer coordinate system relative to the inner coordinate system, it is possible to have the transmitted beams from one surface be adjacent to transmitted beams from the other, giving a continuous scanned field of view twice as big as a one-sided lens.
Figure 5 shows the geometry of the inner and outer surfaces with a pair of focal points symmetrically placed about the axis of each surface. Since the transmission lines are merely cables or microstrip lines, they easily can be bent to fit this geometry.
Two equations in similar form to Equation 1 are used to determine the reciprocal quadrufocal, one for + [Alpha], and one for - [Alpha]. In addition, since the outer surface must have the same shape as the inner surface and the lines are parallel, (29) X = U (30) Y = V (31) W(U,V) = -Z(X,Y) so the two equations become (32) [Mathematical Expression Omitted] Equation 32 is rearranged, squared, and the second equation for the - [Alpha] focal point is subtracted from the first equation for + [Alpha] resulting in (33) L = -Z cos [Alpha] Equation 33 is now substituted back into Equation 33 to give a formula for the inner surface (34) Z = -F cos [Alpha] + [square root of ([F.sup.2] - [X.sup.2]) [cos.sup.2] [Alpha] - [Y.sup.2] which is the same as the Rao inner surface, given in Equation 16.
After refocusing, the best focal arc is found to be (35) f ([Theta]) = F [cos.sup.2] [Theta]/cos [Alpha] (2 cos [Theta] - cos [Alpha]) which is the optimum focal arc of the quadrufocal lens, given in Equation 28, with [Beta] = [Alpha]. Results of this design are not as good as for the Rotman lens or Rappaport quadrufocal lens, but the maximum field of view can be much larger, even up to 180 [degrees]. Figure 6 shows the phase error across the profile curve of the two-dimensional version (with Y = V = 0) of a reciprocal quadrufocal with design scan angle [Alpha] = 33 [degrees] (relative to the inner and outer surface axis of symmetry) at 0 [degrees] and 45 [degrees], the worst scan angles in the field of view. The 33 [degrees] design angle was chosen so that the worst errors of the most distorted beams have the same magnitude. The path length error magnitude of 0.025 F is the worst error on both ends of the aperture for both beams.
Tilting the inner and outer surfaces, as shown in Figure 5, causes the rays leaving the sources on the left to be collimated and transmitted up and to the left, and rays leaving from the right to transmit up and to the right. Choosing the up (broadside) direction to be 0 [degrees], the field of view is now -90 [degrees] to +90 [degrees].
Figure 7 is a graph of the worst path length error for all beams within this 180 [degrees] field of view, which is the widest of any reported bootlace lens. The error is zero at [+ or -] (45 [degrees] [+ or -] 33 [degrees]), or at -78 [degrees], -12 [degrees], 12 [degrees] and 78 [degrees] and reaches its worst values at 0 [degrees], [+ or -] 45 [degrees] and [+ or -] 90 [degrees]. The best Rotman lens for this field of view has design scan angle [Alpha] = 77 [degrees], yielding maximum error magnitude more than 10 times the worst error of this lens at [+ or -] 66 [degrees] and [+ or -] 90 [degrees]. An important feature of this reciprocal lens is that since there are two separate inclined radiating apertures, the projection of the scanned aperture is never < cos 45 [degrees] times the full aperture size. For planar phased arrays, a beam can in principal be scanned [+ or -] 90 [degrees], but the projected aperture shrinks to zero at these extremes.
One difficulty of this two-dimensional design is that the feeds on one side block the radiated rays from feeds on the other side. Although blockage is a minor problem for the three-dimensional version, it can be significant if the two-dimensional application requires many adjacent scanned beams. Special microwave transmission line design, such as employing polarization rotators or directional couplers to separate incoming and outgoing signals at the lens surfaces, might help alleviate the blockage problem.
This survey of bootlace lens antennas has shown the design constraints, optimization considerations and performance comparisons of two, three and four focal point systems. Performance comparisons between these lenses have been made holding the lens size and F/D constant. In general, as the number of focal points increases, the performance improves but the design and fabrication becomes more complex. A new bootlace lens design, the reciprocal quadrufocal, has been described, which has the same shape inner and outer surfaces, has parallel transmission lines so that it can transmit equivalently from either surface, and achieves a 180 [degrees] field of view.
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Antennas and Propagation, April 1991, pp. 64-72. 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 Eta Kappu Nu and Sigma Xi.
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|Author:||Rappaport, Carey M.|
|Date:||Feb 1, 1992|
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