Free space microwave focusing by a negative-index gradient lens

APPLIED PHYSICS LETTERS 88, 081101 共2006兲 Free-space microwave focusing by a negative-index gradient lens T. Driscolla兲 and D. N. Basov University of California San Diego, Physics Department, 9500 Gilman Drive, La Jolla, California 92093 A. F. Starr SensorMetrix, Incorporated, 5965 Pacific Center Boulevard, Suite 701, San Diego, California 92121 P. M. Rye and S. Nemat-Nasser University of California, San Diego, CEAM group, MAE Department, 9500 Gilman Drive, La Jolla, California 92093-0416 D. Schurig and D. R. Smith Duke University, ECE Department, Box 90291, Durham, North Carolina 27708 共Received 15 August 2005; accepted 23 December 2005; published online 21 February 2006兲 Metamaterial structures designed to have simultaneously negative permittivity and permeability are known as left-handed materials. Their complexity and our understanding of their properties have advanced rapidly to the point where direct applications are now viable. We present a radial gradient-index lens with an index of refraction ranging from −2.67 共edge兲 to −0.97 共center兲. Experimentally, we find that the lens can produce field intensities at the focus that are greater than that of the incident plane wave. These results are obtained at 10.3 GHz and in excellent agreement with full-wave simulations. We also demonstrate an advanced fabrication technique using conventional printed circuit board technology which offers significant design, mechanical, and cost advantages over other microwave lens constructions. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2174088兴 Since their first realization in 1999 by Smith et al.1—guided by the earlier theoretical work of Veselago2—understanding of negative index materials 共NIMs兲 has advanced rapidly. Metamaterial-style structures with negative index of refraction 共n = 冑␧冑␮兲 were initially constructed and tested at microwave frequencies.3,4 They have since advanced through infrared,5,6 and are now at a stage where direct applications are within reach. To date, these structures have largely been of the “wine-crate” construction,7 which is an array of split-ring resonators 共SRRs兲 and wires. This construction, while useful in demonstrating the basic phenomenon, lacks structural integrity and is time consuming to construct, and limits design complexity.8,9 Limitations in design tolerance and material in wine-crate metamaterials have also hindered the progress of experimental positive-gain lensing using NIMs. Given the recent evidence showing the superiority of NIM lenses in many situations,10 there is much interest in pushing NIM lenses toward applications. Here, we present a radial GRIN lens, built from elements similar to the SRRs and wires of previous experiments,7 but integrated into a fabrication technique using traditional printed circuit board 共PCB兲 technology.11 Our durable, lightweight, and modular lens operates as a positive gain spherical lens, focusing in two dimensions to achieve a focal spot amplitude +7 dB over incident. This represents a significant advance toward applications, as well as steps forward in design complexity and construction technique. Using ray-tracing software written in MATHMATICA, we designed a biplanar 共geometrically “flat”兲 lens with a radially varying gradient given in a兲 Electronic mail: [email protected] ␧共r, ␻兲 = ␮共r, ␻兲 = n共r, ␻兲 = − 0.97 − 7.30r2 + 0.18r4 . 共1兲 A 2 mm thick disk, 15 cm in radius, with this gradient is shown to behave as an f / 9 lens. In transferring this lens design to our metamaterial, the radial gradient is mapped onto a Cartesian array of unit cells with 50 steps over the radius. Each unit cell is a SRR and wire, shaped to have a specific magnetic and electric resonance. The result is a disk of ⬃8000 unit cells 关see Fig. 1共a兲兴, nearly one-quarter of which are unique. This large number of steps gives an excellent approximation of a continuous gradient.12 To meet the required optical parameters at each unit cell, many variations of the geometry shown in Fig. 1共c兲 were simulated for S parameters in the commercial software package HFSS by Ansoft. The geometry variations involved changing the wire thickness and location and the capacitorgap pad radius. A standard retrieval method11 was used to obtain real and imaginary electric permittivity ␧共␻兲 and magnetic permeability ␮共␻兲 from the simulation S parameters. Ten geometries were optimized to span the desired index range at an operating frequency of 10.1 GHz. The optimization goals, in addition to correct index, included matched impedance, low loss tangents, and insensitivity to small geometric changes. The effects of wire thickness, wire location, and capacitor area on the electromagnetic parameters are intertwined, making the design a process relying largely on experience and iterative refinement. We can, however, still identify some variables as heavily influential on particular electromagnetic properties. The parametric graph in Fig. 2 illustrates the dependence of the index on the wire spacing and capacitor pad radius. The circular data points are structures which were simulated as described above, and the square data points interpolated points. The impedance 共Z = 冑␧ / ␮兲 throughout is almost constant, at 1.06± 0.05. This is achieved by additionally varying the wire thickness, which 0003-6951/2006/88共8兲/081101/3/$23.00 88, 081101-1 © 2006 American Institute of Physics Downloaded 14 May 2007 to 132.239.174.219. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp 081101-2 Driscoll et al. Appl. Phys. Lett. 88, 081101 共2006兲 FIG. 2. Parametric plot of index of refraction as a function of wire spacing and capacitor pad radius. The wire thickness 共not plotted兲 is varied to keep the impedance constant 共z = 1.06± 0.05兲 throughout. Simulation points shown in outlined circles, squares are interpolated. will reduce diffractive effects by minimizing the difference in optical path length between two rays passing through and by the edge of the lens: FIG. 1. Three tier diagram showing 共a兲 actual picture of a lens disk, 共b兲 blow-up illustrating unit-cell array, 共c兲 further blown up single unit cell with SRR and wire elements. Magnetic field is applied in along Y direction, electric field is along X. predominantly affects the impedance. Each data point in Fig. 2 represents one possible unit-cell design, like the one shown in Fig. 1共c兲. These unit cells were then arranged in a twodimensional array in the X-Y plane, as shown in Fig. 1共b兲. This creates a PCB disk 2 mm thick and 30 cm in diameter 共pictured in Fig. 1共a兲兲, that responds to a single polarization. These disks can be stacked up to decrease the lens f/ number approximately linearly, resulting in a highly modular design. For testing, a microwave compact antenna test range was developed using an off-axis parabolic reflector coupled with a standard feed horn. The polarized plane wave coming off the dish shows moderately good amplitude uniformity—to within ⬃2 dB over the lens face. The lens is embedded in a styrofoam barrier with ␧ = 1.03, ␮ = 1, and is oriented normal to the plane wave in such a way that the incident electric and magnetic fields lie in the plane of the lens and are aligned to excite the wires and rings, respectively. Our plane wave is measured to have very low cross polarization 共⬍30 dB兲 which allows us to neglect effects of the structure anisotropy in this experiment. The image side of the lens is scanned in volume using a dipole detector, guided by a xyz translation stage as sketched in Fig. 3. The source and detector are connected to a Vector network analyzer, which performs a frequency sweep at each spatial point. Data are collected in a four-dimensional array: In XYZ volume over 24 cm by 30 cm by 54 cm, and between 8 GHz and 12 GHz every 0.04 GHz. These data are taken for various thicknesses of lens. In analyzing these data, we take into consideration that at ⬃10 GHz, our lens is only ⬃10␭ in diameter—and, the diffractive edge effects may play an important role in its behavior. We wish to consider those lens thicknesses which ⌬␾OPL = 2␲d 共n0 − n1兲 . ␭0 共2兲 Solving Eq. 共2兲 at the lens edge 共n1 = −2.67兲 the first two thicknesses which result in an integer 2␲ phase difference are d1 = 8.1 mm and d2 = 16.2 mm. Each lens disk is 2 mm thick, so a lens constructed of four or eight layers will minimize the influence of diffractive effects in the focal field. Here, we present these two thicknesses; the interesting behavior of highly diffractive small lenses will be considered in a future work. Shown below in Fig. 3 are data for an eight-layer lens 共in black兲 and a four-layer lens 共in gray兲. Dashed lines are experimental data for the root-mean-square 共rms兲 field amplitude along the optical axis 共x = 0, y = 0, translate z兲 at 10.3 GHz, normalized to incident field amplitude. Solid FIG. 3. rms electric-field amplitude in both simulation 共solid兲 and experiment 共dash兲 along the optical axis for GRIN lenses comprised of eight layers 共black兲 and four layers 共red兲. Downloaded 14 May 2007 to 132.239.174.219. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp 081101-3 Driscoll et al. curves are Microwave Studio simulation results for the same eight- layer and four-layer lenses, the details of which will be discussed presently. Returning to Fig. 3, we find good agreement between the experimental and simulation results for both the eight- and four-layer lenses. We also find good agreement with the predictions of geometric optics, as determined by our raytracing software. Our eight- layer and four-layer lenses have f-numbers of 1.13 and 2.25, respectively. The focal lengths predicted by these f-numbers are marked as vertical lines in Fig. 3. The eight-layer lens focuses almost exactly where predicted, and though the physical limitation of our scanning stage does not allow enough z-axis translation to reach the four-layer predicted focus, the line shape is still in good agreement with simulation. Microwave Studio simulations are done in the form of a two-dimensional “slice” of the lens along its diameter in order to reduce simulation size. Simulations use 50 unit cells across the lens radius, where each unit cell has been assigned a Lorentzian dispersion in both ␧共␻兲 and ␮共␻兲, designed such that the index and loss-tangent at 10.1 GHz are very close to the design values. We find only a 2% difference between the design frequency of 10.1 GHz and what we determine to be the best operation frequency at 10.3 GHz. This small shift is likely due to tolerances in both the PCB construction, and the dielectric. Our unit-cell designs call for a tolerance of 1 mil 共25 ␮m兲, which is at the current limit of technology offered by board fabrication vendors. Fabrication errors, such as registration misalignment, would cause a shift in the parameters of all unit cells. Similarly, an increase in the board dielectric of only 4% would cause a shift in operation frequency near that observed. The amplitudes in Fig. 3 are with respect to incident rms amplitude 1, with simulation and experiment normalized separately and not relative to each other. As well as showing strong support for the accuracy of our design and fabrication, this illustrates another important feature of this lens: It produces field intensities at the focus that are much greater than that of the incident plane wave. Many researchers have been concerned that losses in existing left-handed materials would preclude their use in real applications. Here, we have demonstrated otherwise. The intentional impedance matching of each unit cell to air, and low loss tangents, result in decreased back reflection and absorption respectively. The peak amplitude of the eight-layer lens focus is nearly 7 dB greater than the incident wave amplitude. A cross-sectional display of the field amplitude in the XY plane at the focal length 共Fig. 4 checkered surface兲 nicely illustrates this gain peak above the average incident amplitude 共translucent plane兲. The solid black line is the beam focus one-dimensional profile from the Microwave Studio simulations, showing fairly good agreement of the expected airy profile. In conclusion, we have constructed a biplanar gradient lens whose gradient ranges from n = −0.97 to n = −2.67, and shown the focusing characteristics to be in excellent agreement with simulation. As well as providing experimental confirmation of the behavior of negative-index gradient lenses, the adaptation of conventional PCB fabrication to the wires and SRR elements immensely simplifies the construction of metamaterial lenses. Our gradient lens with nearly Appl. Phys. Lett. 88, 081101 共2006兲 FIG. 4. Two-dimensional profile of the E-field at the eight-layer lens focal plane 共solid colored checker兲. Simulation one-dimensional focus profile overlayed 共black line兲. No-lens averaged background 共transparent plane兲 shown to emphasize magnification. 2000 unique unit cells represents the ultimate test of this recently pioneered construction technique.8 Experimental demonstration of positive-gain lensing using metamaterials represents an important step towards dealing with the losses often inherent in these materials, highly suitable for aerospace and radar application. This work was supported by DARPA through an ONR MURI, and also through ARO MURI No. DAAD19-00-10525 to UC San Diego. One of the authors 共D.S.兲 would like to acknowledge support by the IC Postdoctoral Research Fellowship Program. 1 D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, Phys. Rev. Lett. 84, 4184 共2000兲. 2 V. G. Veselago, Sov. Phys. Usp. 10, 509 共1968兲. 3 A. A. Houck, J. B. Brock, and I. L. Chuang, Phys. Rev. Lett. 90, 137401 共2003兲. 4 P. Vodo, P. V. Parimi, W. T. Lu, and S. 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