Multifocal Bootlace Lens Design Concepts: a Review Carey M. Rappaport* Northeastern University, Boston, MA 02115,
[email protected] Amir I. Zaghloul Virginia Polytechnic Institute and State University,
[email protected]
Reflectors and lenses provide two optical mechanisms to magnify small apertures. They also provide limited scanning of the narrow beams that are produced by the magnified aperture. In both cases, scanning is achieved by switching the feed element location. However, there are a number of basic differences between the two structures. While the reflector system is less complex, it may occupy more volume for the same features and suffers from blockage losses. The reflector has a single perfect focal point where secondary radiation has a planar phase front. Other elements in the feed array produce non-planar wave fronts or planar fronts with significant phase variations. The bootlace or Rotman lenses, on the other hand, are more complex, have three multiple-element radiating structures, and offer more degrees of freedom that help in realizing multiple focal points. The three radiating structures are the feed array, the receiving array on one side of the lens, and the radiating array on the other side of the lens. The traditional bootlace lens is two dimensional, with feed points, receiving, and transmitting arrays all lying on a plane. While this provides for easy strip-line or printed circuit fabrications, there is no physical restriction from extending the formulations to three-dimensional structures. The degrees of freedom include the feed element locations and the transmission line lengths between the receiving and radiating sides of the lens. These allow for choosing a simple planar array surface on the radiating side and for optimizing the parameters to produce multiple perfect focal points with little phase errors for points that fall in between the perfect foci. The optimization process, aiming at increasing the number of focal points and minimizing the phase errors at other points, has produced lens designs with two to five focal points. Some of these are: the 2-dimensional Rotman Lens [1], with a straight radiating aperture and three perfect focal source points; the Planar McGrath Lens [2], with straight inner surface as well and two focal points; the Rao Bifocal Lens [3] with two focal points, the Quadrufocal Lens [4], with four perfect focal points, and the three-dimensional Quintafocal Lens which can scan beams in two dimensions [5]. Bootlace Lens Formulation and Analysis In the general configuration of the microwave bootlace lens, shown in Figure 1, the transmission line lengths between the receiving and the radiating sides of the lens are key parameters in optimizing the lens performance. They offer a number of control parameters that are not available in a solid dielectric lens. This translates into a larger number of degrees of freedom that can be used to reduce the phase errors on the wave front as the direction of radiation changes. For predefined surfaces for the two sides of the lens, which may include a planar radiating side, the transmission line length and the feed locations constitute the parameters for the lens performance optimization. The general bootlace lens design equations are derived by solving for the path lengths from each feed point to it respective aperture. Rays leaving the lens 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, then adding the distance to an inclined wavefront in the predetermined beam direction, must give the same constant value for any inner surface point.
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Figure 1: Cross sectional geometry of a general bootlace lens, showing scanning focal point and axial (Rotman Lens) focal point.
Referring to Figure 1, which shows the central plane (Y = V = 0) of a lens, the inner lens surface denoted by Z(X,Y), outer lens surface by W(U,V), transmission line lengths connecting the inner element at (X,Y) to the outer element at (U,V) by L(U,V), and with a focal point at (x,y,z)=(Fsinα 0, -Fcosα), transmitting rays which leave the lens at an angle α with respect to the w-axis, establish the geometry of the lens. Note that 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. Equating the general ray with the vertex ray gives the fundamental bootlace lens equation:
X − F sin 2 α + Y 2 + ( Z + F cos 2 α ) + L + U sin α − W cos α = F
There will be 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. For planar lenses, the V and Y dependencies are omitted from the lens equations; while for threedimensional lenses, the mapping of the y-coordinate from inner to outer lens surface V is trivial: V(X, Y) = Y. There exists a formulation of three dimensional bootlace lenses with five degrees of freedom, and five perfect focal points, which incorporates a specific mapping from (X,Y) to [U(X,Y), V(X, Y)] [5]. The equations for the four primary bootlace lens types are summarized in the following table, which gives, in terms of the outer lens surface transverse positions U and V, the functional dependences of the inner surface transverse and longitudinal positions X and Z, the outer surface longitudinal position W, and the transmission line lengths L. The Bifocal and Planar lenses each have a pair of perfect focal points, located a distance F from the inner lens vertex at angles ±α (but with no focus at distance G). The Rotman Lens has a third focus at distance G, while the Quadrufocal Lens has third and fourth focal points each at a distance G, but at angles ±β . Figure 2 gives the cross-sectional shapes and focal arcs of the four lenses. Only the Quadrufocal lens has a non-planar outer surface (top curve). The Planar Lens has no curvature for either lens surface. The cases shown in Figure 2 are for F = 1 and G = 1.137 (optimal for the Rotman lens [1]), and G = 1.0683 (for the Quadrufocal, making its greatest extent the same as for the Rotman lens), and α = 40° and β = 27°. For the particular choice of focal lengths G = F, the Quadrufocal lens simplifies to a variant with a circular focal arc, a planar output surface and a parabolic cylinder inner surface (similar to the threedimensional Rotman lens). Alternatively, setting G = F cosβ /sinα generates a focal arc that lies on the circle passing through the origin with diameter d = F /sinα, a circular inner lens profile that also lies on
Bootlace Lens Type Bifocal (Rao) Planar (MaGrath)
Rotman Trifocal D0= F-G D1= F-Gcosα D2= G-Fcosα
Quadrufocal (Rappaport) D0= F-G D1= Fcosβ - Gcosα D2= Fcosα - Gcosβ
{(cos 2
X
Z
W
L
F 2 − U 2 sin 2 α + V 2 F2 −U 2
D 1W D0
− U 4 sin 4 α / 4 D 22
c = U 2 D1 cos α / D 2 − V 2
U 2 ( 2 / F + D 0 sin 2 α / D 22 )
b = 2 FG (cos α − 1) / D 2 +
b + b 2 − 4ac − , 2a a = 1 − D 02 / D 22 − U 2 / F 2
F−F
0
D0 ( X / U − 1) cosα − cos β
0
0
D U 2 sin 2 α +L 0 2D2 D2
− F cos α + ( F 2 − U 2 ) cos 2 α − V 2
−
]}
U
{
1 D2U 2 + F 2 − G2 D1 GF − (U 2 − G 2 )(U 2 − F 2 ) + V 2 (G 2 − F 2 )2 / D12
−
[
0
F 2 − G2
0
/
L U 1 − F
L U 1 − F
1/ 2
}
α − cos 2 β )U 4 +
( 2 ZD 2 + F 2 − G 2 )U 2
Figure 2: Outer lens profile, inner lens profile, and optimized focal arc for the four types of bootlace lenses.
that circle: X 2 + (Z + d / 2)2 = d 2 / 4 ; and a circular outer lens profile with double the radius:
U 2 + (W + d ) 2 = d 2 . The Quadrufocal thus simplifies to an R-2R lens, with zero phase error between the perfect focal points. Careful specification of the focal arc connecting the perfect focal points minimizes phase errors for source points in between them. Using perturbation analysis to minimize even-order phase aberration, the optimal focal arc for a two-dimensional lens is given by the function: f (θ ) =
FG cos 2 θ (cos α − cos β ) F cos 2 β (cos α − cosθ ) − G cos 2 α (cos β − cosθ )
Figure 3 shows the relative scan performance of the four lenses depicted in Figure 2. Plotted is the normalized maximum deviation of path length across the one-dimensional aperture line for each scan angle feed position. Because of the significantly higher errors for the Planar and Bifocal lens cases, their perfect design scan angles were chosen to be α = 10°. Even so, the error at boresight is greater than the worst error for the Rotman lens across the entire ±42° field of view. The Quadrufocal lens which fits within the same area has about 8 times less error than the Rotman lens.
Figure 3: Maximum path length error, normalized to focal length F, as a function of scan angle for ), Planar ( ), Rotman (****), and Quadrufocal ( ). the four lenses: Bifocal (
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References
1. Rotman, W. and Turner, R.F., “Wide Angle Microwave Lens for Line Source Applications," IEEE T. Ant. Prop, Vol. Ap-11, Nov. 1963, pp. 623-632. 2. McGrath, D., “Planar 3 Dimensional Constrained Lenses,” IEEE T. Ant. Prop., Jan. 1986, pp. 46-50. 3. Rao, J.B.L., “Multifocal Three-Dimensional Bootlace Lenses," IEEE T. Ant. Prop, Vol. AP-30, Nov. 1982, pp. 1050-1056. 4. Rappaport, C. and Zaghloul, A., “Optimized Three Dimensional Lenses for Wide-Angle TwoDimensional Scanning," IEEE T. Ant. Prop., Nov. 1985, pp. 1227-1236. 5. Rappaport, C. and Mason, J.,”A Five Focal Point Three-Dimensional Bootlace Lens with Scanning in Two Planes,” IEEE Ant. Prop./URSI Sym. Digest, July 1992, pp. 1340-1343.
