I. INTRODUCTION
Gradient index lenses such as Luneburg and half Maxwell fisheye lenses have been widely used to focus the radiation of antennas. Their broadband behavior, ability to form multiple beams (especially for the Luneburg lens), and reasonable weight (as the frequency increases) make them suitable for a host of applications [Reference Mosallaei and Rahmat-Samii1–Reference Fuchs, Lafond, Rondineau and Himdi3]. This paper focuses on a gradient index lens that has, despite its many merits, not been exhaustively reported in the literature: the Mikaelian lens [Reference Mikaelian4, Reference Mikaelian5]. In 1951, Prof. Mikaelian proposed a self-focusing cylindrical dielectric waveguide represented in Fig. 1(a). In this medium, the refractive index decreases from the center to the outer of the cylinder. Inside this waveguide, all ray paths have the same length which in terms of wave theory means that all the waveguide modes have the same velocity of propagation. This device is now well known as an optical fiber. It is easy to see that a section of such self-focusing waveguide is a focusing lens as drawn in Fig. 1(b).
Fig. 1. Ray tracing inside (a) a self-focusing waveguide and (b) a section of this waveguide that is a focusing lens known as Mikaelian lens.
The main intrinsic differences of the Mikaelian lens with respect to the well-known Luneburg and half Maxwell fisheye lenses are the following:
• The shape in contact to the primary source, i.e. the base of the cylinder, is flat (as opposed to central) which allows an easy integration with a primary source.
• The refractive index distribution has an axial symmetry (as opposed to central) which may be simpler to manufacture.
The crucial issue with gradient index lenses is the manufacturing complexity. For that purpose, the innovative technique recently introduced and described in [Reference Merlet, Le Bars, Lafond and Himdi6–Reference Bor, Lafond, Merlet, Le Bars and Himdi8], is employed to manufacture a Mikaelian lens. The idea is to change the foam density in order to create the desired refractive index law.
To the best of our knowledge, Mikaelian lens antennas have not been previously investigated in the microwave frequency range. Therefore, the aim is here to assess its focusing properties both numerically and experimentally by comparisons with the well-known Luneburg lenses.
The paper is organized as follows. The theoretical properties of the Mikaelian lens are first presented. The different ways to manufacture it are briefly reviewed and a foam-based technique to create the gradient index is described. The focusing performances of the Mikaelian lens are investigated and compared to the Luneburg lens ones in Section III where experimental results of the Mikaelian lens antenna are also shown. Conclusions are drawn in Section IV.
II. THEORETICAL PROPERTIES AND MANUFACTURING TECHNOQUE OF THE MIKAELIAN LENS
The optical and electromagnetic characteristics of the Mikaelian lens are detailed, the ways to manufacture such gradient index lens are reviewed, and an innovative foam-based technique is described.
A) Theoretical properties of Mikaelian lenses
The Mikaelian lens is a cylindrical lens of thickness T and radius R (see Fig. 2(a)). Its refractive index varies with the radial distance r according to the following equation:
$$n\lpar r\rpar =\displaystyle{{n\lpar 0\rpar } \over {\cosh \left({\displaystyle{\pi \over {2T}}r} \right)}}\quad {\rm with}\quad 0 \leq r \leq R\comma \;$$where n(0) is the refractive index along the cylinder axis. This lens is also called “hyperbolic cosine lens” in [Reference Lo and Lee9] because of its refractive index distribution or “constant thickness Luneburg lens” in [Reference Jasik10], since the thickness of the lens (the height of the cylinder) is fixed contrary to the classical cylindrical Luneburg lens used for two-dimensional (2D) focusing.
Fig. 2. Geometrical optics ray tracing inside (a) a Mikaelian lens and (b) a generalized Mikaelian lens with the associated notations.
It is important to note that the ratio T/R has a direct impact on the refractive index dynamic. As plotted in Fig. 4, the thicker the lens with respect to its radius, the smaller the refractive index (and therefore permittivity) variation. Indeed, the ray path length differences are then less important. For a thin lens (T = R), the permittivity inside the dielectric varies from a factor 1 to more than 5, whereas for a thick lens (T = 5R), it goes only from 1 to 1.11.
Fig. 3. Relative permittivity distribution of a dielectric Mikaelian lens as a function of the normalized radius: influence of the ratio thickness over radius of the lens T = R.
Fig. 4. Near-electric field mappings computed by CST Microwave Studio of a plane wave impinging on (a) a Mikaelian lens and (b) a generalized Mikaelian lens. The white square shows the location of the lens.
Regarding the properties of the lens, geometrical optics predicts that the Mikaelian lens transforms a point source into a beam of parallel rays as illustrated in Fig. 1(a). The near-electric field mapping of Fig. 3(a) shows that a plane wave impinging on one side of the cylinder is focused on a point.
The behavior of the lens can be adjusted by modifying the refraction index distribution. To have a focal point at a given distance F from the lens, the law of the so-called generalized Mikaelian (given in [Reference Mikaelian5]) is:
$$n\lpar r\rpar =\displaystyle{{n\lpar 0\rpar T - F\left({\displaystyle{1 \over {\cos \gamma }} - 1} \right)} \over {T\cosh \left({\displaystyle{\pi \over {2T}}\lpar r - {r_1}\rpar } \right)}}\comma \;$$where
$$\displaystyle{\pi \over {2T}}\lpar r - {r_1}\rpar ={\tanh ^{ - 1}}\left({\displaystyle{{T\sin \gamma } \over {n\lpar 0\rpar \, T - F\left({\displaystyle{1 \over {\cos \gamma }} - 1} \right)}}} \right).$$The angle γ and the radial distance r 1 are defined in Fig. 2(a). The near-electric field mapping of a plane wave impinging on an ideal generalized Mikaelian lens with F = R = T/2 in Fig. 3(b) illustrates this focusing property. Note that the refraction index distribution (2) is very close to the one of the Kelleher lens [Reference Goatley and Parker11].
B) Gradient index manufacturing
Traditional gradient index lenses have been fabricated using homogeneous dielectric shells to approximate by steps the continuous refractive index distribution [Reference Peeler and Coleman12]. In [Reference Peeler and Archer13], the dielectric distribution of a planar Luneburg lens is achieved by varying the thickness of a dielectric between parallel plate waveguide working in the TE10 mode. The desired permittivity profile of Luneburg lenses can also be obtained by controlling the hole density in a dielectric as done with Teflon in [Reference Rondineau, Himdi and Sorieux14]. All these techniques are expensive in terms of fabrication time and money.
More recently, printed planar Luneburg lenses have been proposed. The refractive index variation is performed by etching holes on a printed circuit board in [Reference Xue and Fusco15]. In [Reference Pfeiffer and Grbic16], the refractive index of the Luneburg lens is controlled through a combination of meandering crossed microstrip lines and varying their widths.
Finally, a planar Luneburg lens has been realized using a variable printed metasurface in [Reference Bosiljevac, Casaletti, Caminita, Sipus and Maci17]. The desired refractive index is obtained by modulating the surface impedance inside a parallel-plate structure.
Recently, an ingenious technological process has been developed to control the permittivity of foam materials [Reference Merlet, Le Bars, Lafond and Himdi6, Reference Bor, Lafond, Merlet, Le Bars and Himdi7]. Foam is composed of a core material into which gas is injected the ratio between the core material and the gas, i.e. by pressing more or less the foam; it is possible to control up to a certain extent the permittivity of the foam. This permittivity can in principle range from the initial foam permittivity to the one of the core material composing the foam. More details on the pressing process and the relation between the foam density and its permittivity are given in [Reference Bor, Lafond, Merlet, Le Bars and Himdi7]. This process is a cheap way to easily manufacture very lightweight gradient index lenses. It has already been successfully implemented for Luneburg lenses [Reference Bor, Lafond, Merlet, Le Bars and Himdi8] and is here applied to Mikaelian lenses.
III. NUMERICAL PERFORMANCES AND EXPERIMENTAL VALIDATIONS IN MILLIMETER WAVES
The radiation performances of Mikaelian lens are investigated and compared numerically and experimentally to those of Luneburg lenses. The 3D full-wave commercial software CST Microwave Studio is used to compute the radiation characteristics of the lens antennas.
A) Focusing performances
The directivity and aperture efficiency of ideal Mikaelian and Luneburg lenses are compared for various lens diameters in Fig. 5. Let us recall that the aperture efficiency is the ratio between the directivity of the lens antenna and the one of a constant field circular aperture of the same diameter. Both lenses are composed of 20 homogeneous dielectric shells to approximate the ideal continuous gradient index. The Mikaelian lens has a central relative permittivity ε r(0) of 2 and a thickness T equal to 2R. The lenses are fed by a circular open-ended waveguide of diameter 2.85 mm whose directivity alone is 8.1 dB. These radiation characteristics show that, roughly speaking, both lenses behave similarly. It seems that the Luneburg lens antenna performs better than the Mikaelian lens for large diameters (above 20 λ), whereas the opposite happens for small diameter (below 15 λ). However, caution must be taken when interpreting these results since the directivity and consequently the aperture efficiency are relatively sensitive to the position of the antenna with respect to the lens.
Fig. 5. Directivity and aperture efficiency as a function of the lens diameter for the ideal Mikaelian (solid line) and Luneburg (dashed line) lens fed by a circular waveguide.
B) Experimental results
A foam-based planar Mikaelian lens of thickness T = 2R = 56 mm and height 3 mm has been manufactured. The lens diameter (or rather width for the planar version) is equal to 11.2 λ at 60 GHz. The lens is made from the foam material Airex PXc245 [18]. Without being pressed the properties of the foam are at 60 GHz: ε r = 1.31 and tan δ = 0.008. The permittivity law of the Mikaelian lens is approximated by 10 values given in Table 1 which correspond to the 10 heights before pressing shown in Fig. 6(a). The lens is fed by a WR15 open-ended waveguide operating in the V-band (see Fig. 6(b)).
Fig. 6. Pictures of the Mikaelian lens: (a) before being pressed and (b) pressed inside its mechanical support, including the feeding waveguide. (c) Full-lens antenna system.
Table 1. Characteristics of the manufactured Mikaelian lens
The measured and simulated radiation patterns of the Mikaelian lens antenna at 60 GHz are plotted in Figs 7(a) and (b). A fairly good agreement is obtained between simulation and measurements for both E- and H-planes. In the H-plane, the measured half-power beamwidth is equal to 4.4°, sidelobes are below −19 dB and the cross-polarization level is below – 25 dB. Note that the asymmetry in the E-plane is due to the non-symmetry of the mechanical support. The gain patterns in the H-plane are plotted as a function of the frequency in Fig. 7(c). They confirm the expected broadband behavior of the lens antenna and show that the Mikaelian permittivity law is properly respected. The measured gain and loss efficiency (gain over directivity) are plotted as a function of the frequency in Fig. 8. The gain increases from about 14 dB at 57 GHz to 15 dB at 66 GHz, whereas the loss efficiency is between 35 and 39%. These low values of the loss efficiency are due to the relatively high losses of the foam especially when pressed (see the loss tangent in Table 1).
Fig. 7. Radiation performances of the Mikaelian lens antenna. Normalized far-field pattern at 60 GHz in the (a) H- and (b) E-planes: measurements (solid line), simulation (dashed line). (c) Measured gain as a function of the frequency.
Fig. 8. Measured gain and loss efficiency of the Mikaelian lens antenna as a function of the frequency.
IV. CONCLUSION
An investigation on the Mikaelian lens antennas in the millimeter wave region has been presented. The design principles and theoretical properties of this cylindrical gradient index lens are recalled. A novel way to create a cheap and lightweight permittivity distribution from foam is applied to manufacture the lens. The focusing properties of Mikaelian lenses are similar to the Luneburg lens ones as shown numerically. The experimental results in the V-band validate the numerical simulations and the interest of the foam technique. Although the Mikaelian lens does not present the same ability as Luneburg lenses to form multiple beams, its cylindrical shape makes it very attractive for integrated designs.
ACKNOWLEDGEMENT
The authors thank Hervé Merlet and Dr. Philippe Le Bars for fruitful discussions.
Jonathan Bor was born May 10, 1987. He received his Electronics Engineering degree and French DEA (Masters) degree in Signal Processing and Telecommunications from the Ecole Supérieure d'Ingénieurs de Rennes (ESIR), Rennes, France, in 2010, and is currently working toward his last year of Ph.D. degree in Signal Processing and Telecommunications at the Institut d'Electronique et de Télécommunications de Rennes (IETR), University of Rennes 1, Rennes, France. He holds one patent and three journal papers. His research interests are millimeter-wave focusing and multibeam devices. His focus is on inhomogeneous lenses.
Benjamin Fuchs received his M.S. and Electrical Engineering degrees in 2004 from the National Institute of Applied Science of Rennes, France. He received his Ph.D. degree in 2007 from the University of Rennes 1, France. During his Ph.D., he was a visiting scholar at the University of Colorado at Boulder, CO, USA. In 2009, he joined the Institute of Electronics and Telecommunications of Rennes (IETR) as a researcher at the Centre National de la Recherche Scientifique (CNRS). He has spent 3 years (2008 as Postdoctoral Research Fellow and 2011–2012 on leave from CNRS) at the Swiss Federal Institute of Technology of Lausanne (EPFL) in Switzerland. His research interests include millimeter-wave antennas, focusing devices (lens antennas), and array synthesis methods.
Olivier Lafond received his M.S. degree in Radar and Telecommunications from the University of Rennes, Rennes, France, in 1996, and his Ph.D. degree in Signal Processing and Telecommunications from the University of Rennes 1, Rennes, France, in 2000. Since October 2002, he has been an Associate Professor with the Institute of Electronics and Telecommunications of Rennes (IETR), University of Rennes 1. He has authored or co-authored of 25 journal papers and 50 papers in conference proceedings. He has also authored/co-authored three book chapters. He holds six patents in the area of antennas. His research activities deal with passive and active millimeter-wave multilayer antennas and circuits, reconfigurable antennas, inhomogeneous lenses for shaping radiation patterns with active devices, and imaging antenna systems.
Mohamed Himdi received his Ph.D. degree in Signal Processing and Telecommunications from the University of Rennes 1, Rennes, France, in 1990. Since 2003, he has been a Professor with the University of Rennes 1, and is currently the Head of the High Frequency and Antenna Department, Institut d'Electronique et Télécommunications de Rennes (IETR), Unité Mixte de Recherche, Center National de la Recherche Scientifique. He has authored or co-authored 76 journal papers and over 180 papers in conference proceedings. He has also authored/co-authored two book chapters. He holds 24 patents in the area of antennas. His research activities concern passive and active millimeter-wave antennas. His research interests also include theoretical and applied computational electromagnetic, development of new architectures for printed antenna arrays, and new three-dimensional (3D) antenna technologies.
