A Printed, Broadband Luneburg Lens Antenna

Transcription

1 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 9, SEPTEMBER A Printed, Broadband Luneburg Lens Antenna Carl Pfeiffer and Anthony Grbic Abstract The design of a D broadband, Luneburg lens antenna imlemented using rinted circuit board techniques is detailed. The refractive index of the lens is controlled through a combination of meandering crossed microstri lines and varying their widths. The 1 4 diameter lens is designed to oerate in the transverse electromagnetic (TEM) mode at 13 GHz. The lens antenna was designed, fabricated, and measured. The measured half ower beamwidth of the exerimental antenna is Index Terms Broadband antennas, Luneburg lens, multibeam antennas, rinted circuit fabrication. Fig. 1. To view of a network of crossed transmission lines above a ground lane. The network has a roagation constant times larger than an individual line. I. INTRODUCTION Luneburg lenses have the ability to transform a oint source excitation on the edge of the lens into a lane wave on the oosite side. This rovides Luneburg lens antennas with high gain and radial symmetry, which makes them attractive for use in wide angle scanning alications. Luneburg lenses have refractive index (n) rofiles given by the relation [1], n(r) = 0 r (1) where r is the normalized radius. Traditionally, these lenses have been fabricated using discrete layers of materials with different ermittivities that aroximate the refractive index variation given by (1) [], [3]. In [3], it was shown that using 10 shells was sufficient to emulate the continuous refractive index variation needed to design a 45.7 cm diameter lens at X band. The refractive index rofile of a Luneburg lens has also been achieved by changing the thickness of a arallel late waveguide oerated in the TE 01 mode [4]. In [5], [6], the dielectric within a arallel-late waveguide was loaded with small holes in order to control the index of refraction. Adding small holes lowered the effective ermittivity of the substrate by decreasing the amount of dielectric er unit volume. All these methods of fabrication are labor intensive and exensive, since they involve binding different dielectrics, contouring a dielectric, or drilling numerous holes. More recently, a Luneburg lens was made by etching small holes into one side of a rinted circuit board (PCB) to vary the index of refraction [7]. This lens oerated in the TE 01 mode, and the etched holes varied the inductance of the waveguide, or equivalently the index of refraction. However, the rectangular sacing of the holes resulted in anisotroy. Also, the TE 01 mode is inherently bandwidth-limited. In this communication we detail the design, fabrication and measurement of a D Luneburg lens antenna that was first reorted by the authors in [8]. The antenna oerates in a quasi-tem mode around 13 GHz. The desired refractive index rofile of the lens is achieved through Manuscrit received February 09, 010; revised March 09, 010; acceted March 17, 010. Date of ublication June 14, 010; date of current version Setember 03, 010. This work was suorted by Presidential Early Career Award for Scientists and Engineers (FA ) and an NSF Faculty Early Career Develoment Award (EECS ). The authors are with the Radiation Laboratory at the Deartment of Electrical Engineering and Comuter Science, University of Michigan, Ann Arbor, MI, USA ( carlfei@umich.edu; agrbic@umich.edu). Color versions of one or more of the figures in this communication are available online at htt://ieeexlore.ieee.org. Digital Object Identifier /TAP a network of crossed microstri lines with varying degrees of meander and line width. This allows the structure to be constructed using standard PCB techniques. Since the lens oerates in the TEM mode, it maintains a wide bandwidth of oeration. The lens design is shown to be isotroic and broadband. The lens antenna is simly fed using taered microstri transmission lines. II. CONCEPT In order to realize the refractive index rofile given by (1), a attern was etched onto a coer-clad substrate with a relative ermittivity r 1. At the center of the lens, a refractive index of was achieved by rinting a network of crossed transmission lines onto the substrate. Let s consider roagation along one of the rincile axes, as shown in Fig. 1. By symmetry, magnetic walls can be laced on either side of the line. The line then looks like a transmission line loaded with oen-circuited, transverse stubs. Assuming the sacing between lines is much less than the guided wavelength, the roagation constant of the loaded line is times larger than that of an unloaded line, given the added caacitance of the oen-circuited stubs [9] [11]. Since the transmission lines used in this lens are not ideal, some degree of meandering was needed to increase the index of refraction to. A refractive index n 1 at the edge of the lens was achieved with simly a arallel-late waveguide. Gradual transitioning between the meandered, crossed transmission lines and arallel-late waveguide was used to vary the index from n to n 1. The transition between the center and the edge consisted of reducing the meander and increasing the width of the lines. Wire grid Luneburg lens antennas such as in [9] oerate using similar rinciles. However, these antennas accomlish a transition from n to n 1 by increasing the distance between the uer grid conductor and a ground lane, instead of simly rinting microstri lines on a substrate with a uniform thickness as is done here. III. LENS DESIGN The diameter of the Luneburg lens antenna was chosen to be 8.6 cm, which corresonds to 1:4 ( is the free sace wavelength) to achieve a half ower beamwidth of 5 at the design frequency of 13 GHz. Since the Luneburg lens design is based on geometrical otics, the diameter of the lens needed to be many wavelengths in diameter to limit diffraction effects. A 13 GHz frequency of oeration was chosen since this was the lowest frequency (largest hysical lens size) our fabrication rocess would allow. The lens was discretized into square unit cells 1.6 mm (/10.) in dimension, where = =. With unit cells <=10 in size, the lens can be described as a medium with a definable effective index of refraction. As a result, effective medium theory can be used to design the lens. The substrate used for the lens was 1 mm X/$ IEEE

2 3056 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 9, SEPTEMBER 010 Fig.. Unit cells comrising the rinted Luneburg lens. (a) Unit cell near the center of the lens. (b) Unit cell near the edge of the lens. thick Rohacell 31HF, which has a ermittivity of and loss tangent tan() =0:00. To ease the fabrication tolerances and minimize conductor loss, the minimum width of the microstri lines was chosen to be at least m (4 mils). The commercial finite element solver Ansoft HFSS was used to otimize the unit cell designs. Since the unit cell dimension and the substrate height are comarable, the microstri lines are not ideal transmission lines. Consequently, the effective index of a unit cell could not be increased to n = by merely crossing the microstri lines. Therefore, the lines at the center of the lens were meandered in a sine wave attern (see Fig. (a)) to increase the hase delay across the unit cell, or equivalently increase the index of refraction. The amlitude of the sine wave was reduced as the distance from the center of the lens increased. This allowed for a gradual decrease in refractive index. Once the amlitude of the sinusoidal meander was reduced to zero, the width of the crossed microstri lines was increased in order to further decrease the refractive index and achieve n 1 at the edge of the lens (see Fig. (b)). To determine how the amlitude of the sine wave affects the refractive index of a unit cell, a arametric swee was erformed, as shown in Fig. 3(a). To see how the line widths of the crossed microstri lines (without a meander) affect the refractive index, another arametric swee was needed (see Fig. 3(b)). In order to model the refractive index versus the swet variable, a olynomial fit was alied to the data. Although an oerating frequency of around 13 GHz is used in this communication, this design can be scaled to higher frequencies by decreasing the line width and substrate thickness. Standard PCB lithograhy techniques allow the line width to decrease by a factor of 10. The thickness of the substrate can also be further decreased. However, the minimum thickness of coer-clad Rohacell substrate that can be achieved still needs to be investigated. To rovide insight into the maximum ossible index of refraction that can be achieved by this meander line technique, simulation of a cell with larger sine wave amlitude and narrower line width was erformed. It was shown that it is ossible to achieve an index of refraction of n =1:7 if the line width is decreased to 34 m and sine wave amlitude is increased to 700 m. If the substrate height is additionally decreased to 50 m, an index of refraction of n =:34 can be achieved. A 1 mm thick substrate was chosen in this communication because it is the minimum substrate thickness that is commercially available. IV. DISPERSION ANALYSIS Once the unit cells comrising the Luneburg lens were designed, their satial and frequency disersion was analyzed and modelled. In order for the Luneburg lens to oerate roerly, its cells must be satially nondisersive and isotroic. To ensure that the cells are isotroic and exhibit minimal satial disersion, the cell in the very center of the lens was simulated. This cell was chosen since it exhibits the largest hase delay, and therefore naturally the greatest satial disersion of all Fig. 3. Results of arametric swees showing how the geometry of a unit cell affects its index of refraction at 13 GHz. (a) Amlitude of the sine wave shown in Fig. (a) vs. index of refraction. (b) Width of the cross shown in Fig. (b) vs. index of refraction. the cells. In addition, it is the least symmetric cell in the lens. An eigenmode analysis of the cell was erformed using the finite element electromagnetic solver Ansoft HFSS. For a secified wave vector, the solution frequency for the quasi-tem mode suorted by the unit cell was found. The resulting equifrequency contours are shown in Fig. 4(a). For frequencies below aroximately 0 GHz, the equifrequency contours are circular; the frequency is constant for a fixed magnitude wave vector irresective of its roagation direction. Above this frequency, the contours are no longer circular and the cell exhibits satial disersion. To gain a simlified understanding of the unit cell, a transmission-line model was develoed. The unit cell was modelled as a crossed network of orthogonal transmission lines [10] [1], which has the following disersion equation, sin kxd + sin kyd = sin d where k x and k y are the wavenumbers along the x and y directions, is the roagation constant along each transmission line section, and d is the unit cell dimension. The roagation constant was found through a scattering arameter simulation at a frequency of 13 GHz. The knowledge of at a single frequency was then used to find the hase velocity v =!=. Given the hase velocity, () was used to model the satial and frequency disersion of the unit cell. In the limit that d aroaches 0, the disersion equation reduces to () k x + k y = : (3) This equation shows that in the limit of d! 0, the unit cell is comletely isotroic and that the magnitude of the wavevector is as exected. As can be seen in Fig. 4(a), the crossed transmission-line circuit of the unit cell models the simulated data very well at low frequencies,

3 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 9, SEPTEMBER Fig. 4. Analytical and full wave simulation equal frequency contours of unit cells. (a) Equal frequency contours of the innermost cell. (b) Equal frequency contours of a cell near the edge. The cell simulated has no meander in the lines and the line widths are 50 m. Fig. 5. Measured return loss and ort-to-ort isolation. (a) Measured return loss is around 10 db or greater over the oerating frequency. (b) Measured ort-toort isolation between any two orts is around 0 db or more over the oerating frequency. desite the fact that the substrate height is comarable to the unit cell size, and that the lines are meandered. At higher frequencies the shaes of the analytical and simulated equifrequency contours match well, but the frequencies are slightly off. This difference between the analytical and simulated results only starts to occur for frequencies greater than 0 GHz, which is well beyond the oerating bandwidth of the lens. For comarison uroses, a unit cell without meandering and microstri line width equal to 50 m was also simulated in the same manner. The simulated equifrequency contours for the unit cell without meandering are shown in Fig. 4(b). They remain circular for all frequencies. This is because the effective index of refraction for this cell is very close to that of the substrate (n 1). In other words, the cell is closely modelled by a arallel-late waveguide. The variation of the index of refraction of the two cells with frequency was also found. The index of refraction varies by % between 10.3 GHz and GHz for the cell corresonding to Fig. 4(a). For the cell corresonding to Fig. 4(b), the maximum variation of the index of refraction between.5 GHz and 7.5 GHz is 1.6%. This variation of the index of refraction results in added frequency disersion which is not modelled above, and is resonsible for the difference between the simulated and analytical models at higher frequencies. V. FABRICATION AND MEASUREMENTS The Luneburg lens was fabricated using standard PCB techniques. The coer cladding was bonded to the Rohacell substrate by the Rogers Cororation using the adhesive R/Flex Jade [13]. A hydraulic ress alied ressure to adhere coer to the Rohacell with the adhesive. To revent the hydraulic ress from deforming the substrate, a sturdy frame was laced around the outside of the Rohacell when bonding. The R/Flex Jade also revented the rigid foam substrate from absorbing the chemicals used in the etching rocess. To interface the Luneburg lens to free sace, a 10 cm flare was added to the lens by soldering thin coer-clad laminates to its edge (see Fig. 6(a)). The flare was designed to rovide a return loss greater than 1 db from GHz. To feed the lens, exonentially taered microstri feed lines were added to the lens. The taered lines were designed to rovide a return loss greater than 16 db over the same frequency range. The measured return loss and ort-to-ort isolation are shown in Fig. 5. Both the flare and and the feed lines are shown in Fig. 6(a). Seven feeds were laced along the lens erihery from 045 to 45 at 15 increments to rovide beam switching caabilities. It should also be mentioned that the hase center of the feed was not at the edge of the lens. Therefore, a defocused Luneburg lens was designed by a modification of (1), according to [14]. The hase at the feed to arallel-late waveguide interface was numerically comuted and comared to that of a cylindrical wave. It was found that the hase at Fig. 6. Entire Luneburg lens antenna (a), and a closer look at the center (b), and the edge (c) of the lens. (a) Fabricated Luneburg lens antenna. (b) Meandered microstri lines at the center of the lens. (c) Transition from meandered microstri lines to arallel-late waveguide. the interface was most closely aroximated by a oint source located 1 mm from the lens erihery. To verify that the fabricated Luneburg lens was working as exected, the vertical electric field directly above the lens was robed in a similar manner to [15]. Plots of the electric field are shown in Fig. 7. The robe was made by exosing mm of the inner conductor of a. mm diameter semi-rigid coaxial cable. This robe was connected to one ort of an Agilent vector network analyzer, and the feed line of the Luneburg lens was connected to the other ort. The measured transmission coefficient, which is roortional to the vertical electric field directly above the lens, was measured. Fig. 7 shows time snashots (comuted from hase and amlitude measurements of the transmission coefficients) of the steady-state electric field above the lens. The oint source radiation from the feed is transformed into a lane wave on the oosite side of the lens. Fig. 7(a) (c) are fed at 0, 045, and 45 resectively. The far field attern at 1 GHz is shown in Fig. 8. The radiation from the back of the antenna is attributed to radiation from the edge mount connectors. Measurements yielded an H-lane half ower beamwidth of 5.97 and 4.34 at 9 GHz and 1 GHz, resectively. The measured gain was 16 db and 16.7 db at 9 GHz and 1 GHz, resectively. Between 9 and 1 GHz, cross olarization levels remained below 00 db, similar to other Luneburg lens antennas [16], [17]. The rinted Luneburg lens is 1.4 wavelengths in diameter and each unit cell is =10: in size. As a result, the lens consisted of over 5,000 unit cells and could not be accurately simulated with HFSS given the comutational resources available to the authors. Therefore the antenna was simulated by aroximating the structure using a

4 3058 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 9, SEPTEMBER 010 Fig. 8. Measured and simulated H-lane radiation atterns at 1 GHz. Fig. 7. Time snashots of the measured vertical electric field directly above the lens at 1 GHz. (a) Measured vertical electric field directly above the lens when fed at 0. (b) Measured vertical electric field directly above the lens when fed at 045. (c) Measured vertical electric field directly above the lens when fed at 45. Fig. 9. Measured radiation atterns when fed between 045 and 45 at 1 GHz. Luneburg lens discretized into 18 concentric rings. The index of refraction of each ring was set to that of an ideal Luneburg lens. The dielectric loss tangent of each ring was determined from the simulated comlex roagation constant of the unit cells. Simulations showed a similar half ower beamwidth of 5.60 and 4.5 at 9 GHz and 1 GHz resectively, and a gain of 15.9 db and 18.3 db at 9 GHz and 1 GHz resectively. The simulated directivity was 18.3 db at 9 GHz and 1. db at 1 GHz. The discreancy between the simulated and measured gain is attributed to reflections from the lens-flare interface. This is suggested by the slight standing wave within the lens which can be observed in Fig. 7(a) (c). This standing wave is not resent in simulations. The standing wave reduces the gain since some ower is absorbed and radiated from the microstri lines during multile reflections within the lens, instead of being radiated by the flare after the first ass. Since the Luneburg lens is radially symmetric and oerates in the TEM mode, it rovides beam switching caabilities and a large bandwidth. In Fig. 9, the radiation atterns of the antenna can be seen when fed between 045 and 45. Fig. 10 demonstrates that the antenna rovides nearly identical radiation atterns in the broadside direction between 9 GHz and 13 GHz, which suggests at least a 35% bandwidth. Fig. 11 shows the measured and simulated E-lane radiation atterns. A sidelobe is observed in the measured attern at 90, which is directly above the antenna. This is attributed to radiation from the microstri transmission lines. Since the rinted Luneburg lens has a refractive index which varies between n 1 and n =, the waves guided by the lens are inherently slow wave and should not radiate. However, a standing wave within the lens can cause them to radiate. Fig. 10. Fig. 11. Measured radiation atterns at 9 GHz, 1 GHz, and 13 GHz. Measured and simulated E-lane radiation atterns at 1 GHz. VI. COMPLETE ANTENNA COVERAGE The feeds used in the fabricated lens were large and saced far aart to maximize the return loss and ort-to-ort isolation. However, to enhance the antenna s coverage, more feeds need to be inserted. If the feeds are inserted such that adjacent beams have crossover levels of 03 db, the antenna could rovide full coverage from 045 to 45. Adding more feeds requires using smaller feed lines, which degrades

5 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 9, SEPTEMBER ower beamwidth of 4.34 and can be switched between 045 to 45. This highly directive antenna is simle to fabricate and is attractive for switched-beam alications. ACKNOWLEDGMENT The authors would like to thank S. M. Rudolh for his technical assistance, and M. Kuszaj at Rogers Cororation for metallizing the Rohacell substrate. Fig. 1. Simulated Luneburg lens antenna with 19 feeds. (a) Return loss and worst case ort-to-ort isolation of the center ort. (b) Simulated radiation attern when individually fed at the different orts. the return loss of the antenna. Placing feeds closer together also degrades the ort-to-ort isolation. For these reasons, we further investigated feed designs to verify that the lens antenna could rovide comlete coverage from 045 to 45. The newly designed feeds were still chosen to be exonentially taered in order to rovide a large bandwidth. The chosen feed design has an exonential taer from 4.5 mm to 7.86 mm across an 8 mm line. Since the 3 db beamwidth of the antenna is 5, the feeds were laced 5 from each other to rovide beam crossover levels of 03 db. It should be noted that changing the shae of the feeds also changes the location of the hase center from the feeds. The new feeds have a hase center 10 mm from the lens erihery. Therefore, (1) again needed to be modified according to [14] in order to rovide the necessary focusing. Fig. 1(a) shows the simulated Luneburg lens, worst case ort-to-ort isolation, and return loss. Having a simulated return loss and ort-to-ort isolation of greater than 10 db over the oerating frequency demonstrates the caability of this Luneburg lens antenna to rovide comlete coverage from 045 to 45. The directivity, gain, and half ower beamwidth of the simulated lens with 19 feeds at 1 GHz are 1.1 db, 18.4 db and 4.37, which differs very little from the simulation used to model the fabricated lens antenna. The various beams are lotted in Fig. 1(b). REFERENCES [1] R. K. Luneburg, Mathematical Theory of Otics. Providence, RI: Brown Univ. Press., [] L. C. Gunderson and G. T. Holmes, Microwave Luneburg lens, Alied Otics, vol. 7, no. 5, , [3] G. D. M. Peeler and H. P. Coleman, Microwave steed-index Luneburg lenses, IRE Trans. Antennas Proag., vol. 6, no.,. 0 07, Ar [4] G. Peeler and D. Archer, A two-dimensional microwave Luneberg lens, IRE Trans. Antennas Proag., vol. 1, no. 1,. 1 3, Jul [5] L. Xue and V. F. Fusco, 4 GHz automotive radar lanar Luneburg lens, IRE Trans. Antennas Proag., vol. 1, no. 3, , Jun [6] K. Sato and H. Ujiie, A late Luneberg lens with ermittivity distribution controlled by hole density, Electro. Commun. Jn., vol. 85, no. 9, t. 1,. 1 1, Ar. 00. [7] L. Xue and V. F. Fusco, Printed holey late Luneburg lens, Microw. Ot. Technol. Lett., vol. 50, no., , Dec [8] C. Pfeiffer and A. Grbic, A D broadband, rinted Luneburg lens antenna, in Proc. Antennas Proag. Society Int. Sym., Jun. 009, [9] R. L. Tanner and M. G. Andreasen, A wire-grid lens antenna of wide alication art I: The wire-grid lens-concet and exerimental confirmation, IRE Trans. Antennas Proag., vol. 10, no. 4, , Jul [10] P. B. Johns and R. L. Buerle, Numerical solution of -dimensional scattering roblem using a transmission-line matrix, Proc. IEE, vol. 118, no. 9, , Se [11] C. R. Brewitt-Taylor and P. B. Johns, On the construction and numerical solution for transmission-line and lumed network models of Maxwell s equations, Int. J. Numer. Methods Eng., vol. 15, no. 1, , Oct [1] A. Grbic, Suer-resolving negative-refractive-index transmission-line lenses, Ph.D. dissertation, University of Toronto, ON, Canada, 006. [13] R/Flex JADE Halogen-Free Adhesive System Rogers Cororation. [14] D. Cheng, Modified Luneberg lens for defocused source, IRE Trans. Antennas Proag., vol. 8, no. 1, , Jan [15] A. Grbic and G. V. Eleftheriades, Overcoming the diffraction limit with a lanar left-handed transmission-line lens, Phys. Rev. Lett., vol. 9, no. 11, , 4, Mar [16] L. Xue and V. Fusco, Patch-fed lanar dielectric slab waveguide Luneburg lens, Microw. Antennas Proag., vol., no., , Mar [17] X. Wu and J. J. Laurin, Fan-beam millimeter-wave antenna design based on cylindrical Luneburg lens, IEEE Trans. Antennas Proag., vol. 55, no. 8, , Aug VII. CONCLUSION In this communication, a D Luneburg lens antenna fabricated using standard PCB techniques is resented. The lens oerates in the TEM mode to rovide a broadband erformance. The design methodology outlined in Section III makes it ossible to accurately vary the effective index of refraction from n =1to n >. Use of a thinner Rohacell substrate can be used to decrease losses and increase the frequency oeration. The exerimental lens antenna exhibits a half