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Broadband 3D Luneburg Lenses Based onMetamaterials of Radially Diverging Dielectric Rods

Andrey Sayanskiy, Stanislav Glybovski, Valeri Akimov, Dmitry Filonov, Pavel Belov and Igor Meshkovskiy

Abstract—In this work we describe and study microwaveLuneburg lenses using a broadband metamaterial composedof radially diverging dielectric rods. The gradual Luneburg’spermittivity profile is achieved in this lens through arrangingidentical thin rods with a spatially variable cross-section invarious radial directions with a subwavelength separation. Theperformance of the considered metamaterial lenses was studiedby full-wave numerical simulation and measurements in ananechoic chamber. The results reveal that, although anisotropyof the structure causes aperture phase errors, the structurecomposed of diverging dielectric rods may still operate as acylindrical or spherical Luneburg lens. Advantageously, theconsidered design is attractive for mass production of large-scaledbeam-steering antennas because the lens is composed of identicalshaped dielectric parts.

Index Terms—Luneburg lens, Artificial dielectric, Metamate-rial, Radiation pattern.

I. INTRODUCTION

Luneburg lenses have the unique capability to focus aplane wave impinging from an arbitrary direction to a focalpoint on the opposite side of the lens. Antennas based onLuneburg lenses with rotational symmetry allow full-anglebeam steering without distortion of a radiation pattern. Theclassical Luneburg lens [1] having a focal point on its surfacecan be realized as a non-uniform dielectric sphere with agradual variation of permittivity in the radial direction. In thiscase the permittivity depending on the distance to the centerρ is given by the closed-form expression

εr = 2 − ρ2/R2, (1)

where R is the sphere radius. Also one can consider acylindrical lens, which concentrates the incident plane waveinto a focal line. In this two-dimensional problem the lens isan infinite cylinder of the radius R and the permittivity profileis given by the same expression as (1), but with ρ meaningthe distance to the cylinder’s axis. If the focal point is locatedoutside of the lens, the permittivity profile requires numericalintegration to be calculated [2].

Conventional practical realizations of Luneburg lenses usemultiple uniform shells made of different foams providinga stepwise approximation of the theoretical permittivity pro-file. Multi-layer lenses reach high aperture efficiencies up to60% [3], but their manufacturing is challenging due to therequirement of high shape accuracy and precision of the layer

A. Sayanskiy, S. Glybovski, D. Filonov, P. Belov, I. Meshkovskiy are withITMO University, St.Petersburg, Russia

V.P. Akimov is with the Peter the Great St. Petersburg Polytechnic Univer-sity, St. Petersburg, Russia

permittivity values. To solve this problem several techniqueswere suggested. Thus artificial composites were designed byusing ceramic powders to produce the required range ofpermittivity values for manufacturing of a 3D transformed X-band lens [4]. Also, non-uniformly perforated dielectric layers[5], metamaterial structures inside a parallel-plate waveguide[6], spatially-nonuniform PCB meshes [7] or metasurfacessupporting surface waves [8], [9] have been proposed asrelatively simple realizations of two dimensional Luneburglenses. However, the mentioned techniques are inconvenientfor designing spherical lenses. Therefore, there is an interestin spherical Luneburg lenses, which could be easily manu-factured using only one material. To construct an artificialspherical lens stacks of perforated dielectric slabs of the samepermittivity were employed [10], [11]. Such stacks form non-resonant metamaterial structures and are capable of producingtransformation-optics flattened lenses. More recently a three-dimensional X-band Luneburg lens with the efficiency of upto 55% was realized by controlling the filling ratio of apolymer/air-based unit cells. The 120-mm lens was fabricatedusing the polymer-jetting rapid prototyping technique [12]. Fi-nally a 3D-printed transformation-optics Luneburg lens havinga flattened and extended focal surface has been experimentallydemonstrated in Ku-band [13]. It is worth mentioning thathigh-gain microwave lens antennas may require much largeraperture dimensions. For the lens diameters larger than 300-500 mm 3D printing becomes cost inefficient. However, inmass production lenses should be preferably composed ofidentical smaller parts, all made of the same material.

In this work we study an alternative Luneburg lens based ona broadband all- dielectric metamaterial providing the requiredeffective permittivity profile. In this design (see Fig. 1) thespatially non-uniform permittivity prescribed by the theory [1],[2] is reached by engineering the local cross-section radius rof radially diverging dielectric rods. This local radius appearsto be a function of the distance ρ to the center of the lensproviding the required permittivity through engineering thevolume filling ratio. If such shaped and electrically thin rodsare arranged with subwavelength separation in various radialdirections, the structure behaves as a homogeneous lens withthe effective permittivity εeff(ρ) approximating the theoreticalLuneburg’s profile. This approach was previously proposed bythe authors [14], [15]. The proposed design is cheap and easyto manufacture in mass production, since all the multiple rodsmay be produced by casting under pressure. In the presentwork we describe the lens’ design technique in details andillustrate the performance of the lens with numerical andexperimental results.

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IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. XX, NO. XX, MMMM YYY 2

2r(ρ)

R

Rc

εr

εc

Rods

Core

Δφ

ρ

S(ρ)

Fig. 1. Cross-section view of the considered metamaterial Luneburg lens ofradially diverging rods. Inset shows microscopic structure of the medium ofdielectric cylinders for local description of the lens’ properties.

II. DESIGN AND NUMERICAL SIMULATIONS

In accordance with the theory [2], if the focal point isoutside of the lens (at the distance F ≥ R from the center),the permittivity reaches its maximum εmax ≤ 2 at the center(ρ = 0) being slightly higher than unity at the surface (ρ = R).To realize these values with the lens design depicted in Fig.1 the permittivity of dielectric rods εr must be larger or equalto 2. In this work we consider the rods made of polystyrenewith εr = 2.5. All the rods are mounted at the periphery ofan internal spherical core of the radius Rc. The core is usedfor mechanical support of the radially diverging rods and, ingeneral, may have another permittivity εc.

Since the local periodicity of rods is assumed to be elec-trically small throughout all the lens, the required rod radiusfunction r(ρ) can be estimated using the effective mediummodel. Also, since the lens is electrically large, the structureof rods can be locally considered as a medium of regularparallel rods. In other words, in a vicinity of each point theradially diverging structure can be described as a uniform andanisotropic medium of parallel infinite dielectric cylinders [16]depicted in inset of Fig. 1. If the cylinders are arranged in asquare lattice, this medium is uniaxial [16] and one shoulddistinguish two unequal permittivity tensor components: onedescribing polarization along the axis of rods (ε‖) and the otherdescribing the orthogonal polarization (ε⊥). These componentsare given by the formulas [16]

ε⊥ = 1 + 2 ·(

1

fV

εr + 1

εr − 1− 1

)−1; (2)

ε‖ = 1 +

(1

fV

1

εr − 1− k2 sin2 γ

β2p

)−1, (3)

where k is the free space wave number, γ is the local angle be-tween the wave vector and the rod’s axis and βp is the plasmawave number [16]. The volume filling fraction fV = πr2/Sis calculated as the ratio between the cross-section area ofrods and the area of the lattice unit-cell. In fact, the actualeffective permittivity in the proposed lens depends on boththe tensor components and, therefore, on the local polarization

and propagation angles with respect to the radial direction.In order to simplify the design procedure of the proposedlens here we neglected the anisotropy effects optimizing therod shapes only with respect to the transverse polarization. Inother words, we assumed effective permittivity at each pointto be equal to ε⊥. This is precisely valid if the electric vectoris perpendicular to the local radial direction of rods in eachpoint. In the spherical lens of diverging rods it takes place inonly one symmetry plane (H-plane of the feed horn). In therest of the lens the anisotropy may lead to phase errors. Dueto complexity of the spherical lens, its numerical simulationbecomes too complicated. Therefore we only studied thiscase experimentally. In numerical simulations it is ratherconvenient to consider a two-dimensional (cylindrical) lens.This lens is composed of similar unit slices with divergingrods depicted in Fig. 1, which are periodically repeated alongthe orthogonal to the figure direction (z-axis). In this case theassumption εeff = ε⊥ is correct at each point, but only forthe TMz-polarization. For the TEz-polarization phase errorsdue to anisotropy are expected. In the numerical model due toperiodicity of the slices, it was sufficient to consider only oneslice of diverging rods mounted on a cylindrical core betweentwo boundaries (PEC or PMC depending on polarization ofthe parallel-plate horn feed with respect to the axis z ofthe cylinder: TMz or TEz). For the simulations FrequencyDomain Solver of CST Microwave Studio 2015 was used.We note that this simulation is enough to show correctness ofour approach to determine the required rod shapes providingproper Luneburg lens operation for TMz-polarization. Due tothe requirement of sub-wavelength periodicity and sufficientthickness of rods for mechanical robustness, the divergenceangle and the periodicity of slices along z in the model werechosen ∆ϕ = 1.6◦ and b = 2.2 mm respectively. In order toreach high directivity in the X-band the following parametersof the lens were chosen: radius R = 230 mm, Rc = 60mm and the focal length F = 290 mm. Our cylindricallens can be approximated in each point as a regular latticeof rods with a rectangular unit cell instead of a square unitcell considered in [16] (the rods are periodic in z-directionwhile being diverging in the plane of a slice). However, theparticular permittivity tensor component (2) used for the lensoptimization depends only on the volume filling factor fV.Therefore, the same formula was applied for the cylindricallens, where fV = πr2/S with the local unit-cell area of thelattice of rods S = ρ∆ϕ · b. As shown in the following, sucha modification leads to correct results.

The calculated rod cross-section radius (dashed line) to-gether with the goal effective permittivity [2] (solid line) areplotted in Fig. 2 as functions of the normalized distance ρ/R tothe cylindrical axis. The core material in the simulation had thepermittivity of εc = 1.67, which could be practically realizede.g. by periodical perforating a solid polystyrene sphere [5].Fig. 2 shows that the calculated local radius of rods reachesits maximal value of rmax = 0.85 mm at ρ = 0.65R. At theperiphery the rod radius drops down to zero again. For theTMz polarization (E-field is perpendicular to the slice plane)the εeff = ε⊥ and a plane wave front at the lens apertureis created by the feed, which is illustrated by the simulated

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IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. XX, NO. XX, MMMM YYY 3

0.4 0.6 0.8 1

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

1.55

1.6

1.65

1.7

Normalized distance to center, ρ/R

0.10.20.30.40.50.60.70.80.911.11.21.31.41.51.6

εeff

(goal)

r, cylindrical lensr, spherical lens

Goa

l effe

ctiv

e pe

rmiti

vity

Loca

l rad

ius

of w

ires,

mm

1.051

Fig. 2. The goal effective permittivity εeff (solid curve) and the correspondinglocal radius r of rods for the cylindrical lens (dash curve) and for the sphericallens (dash-dot curve) vs. the radial distance ρ to the cylindrical axis.

−200 0 200

−200

−100

100

↑H

(a)

−1 −0.5 0 0.5 1

−400 −200 0 200

−200

−100

100

(c)

E

−30 −20 −10 0 10 20 30−40

−30

−20

−10

0

Angle, degrees

Normalized Rad. Pattern, TMZ, dB

8 GHz10 GHz12 GHz

(b)

−30 −20 −10 0 10 20 30−40

−30

−20

−10

0

Angle, degrees

8 GHz10 GHz12 GHz

(d)

Normalized Rad. Pattern, TEZ , dB

Fig. 3. Results of numerical simulation: distributions of the electric fieldmagnitude and normalized radiation patterns: (a,b) polarization parallel to thecylindrical axis (TMz); (c,d) polarization orthogonal to the cylindrical axis(TEz) with the increased distance to feed.

normalized E-field pattern at 10 GHz in Fig. 3(a) (unit levelcorresponds to 900 V/m for 0.5 W of accepted power). Thesimulated radiation patterns for the TMz polarization for 8,10 and 12 GHz are shown in Fig. 3(b). The horn aperturesize in this simulation was equal to 40 mm. For the TEz

polarization (E-field is oriented in the plane of the slice ofwires) the effective permittivity is not equal to ε⊥ and dependson the local angle between the electric field and the radialrod direction. As a result, phase errors in the aperture appear,so that the lens fails to produce a directive radiation pattern.However, the aperture phase deviation due to anisotropy inthis case was found to be mainly parabolic and could becompensated to some extent by increasing the distance to thefeed. Thus, with the focal distance increased by 63 mm asdepicted in Fig. 3(c) and the horn aperture increased to 70 mmfor optimal illumination, the cylindrical lens is again capableof beamforming even for the TEz polarization. However, some

Fig. 4. (a) Manufactured spherical lens and experimental setup for far-fieldmeasurements including the lens, X-band pyramidal horn and foam holders;(b) optimized shape of a single rod.

remaining phase errors can be seen in Fig. 3(c) as a curvedphase front. As a result, the radiation patterns shown in Fig.3(d) have high side-lobe levels and frequency dependence.Therefore, the considered cylindrical structure works perfectlyas a Luneburg lens only for the TMz polarization, and at leastwith pattern distortions for TEz polarization of the source.

III. MEASUREMENTS

In this section experimental results for a manufacturedspherical X-band lens of diverging rods are presented. Themeasured radiation patterns demonstrate effects of anisotropyin the spherical case that are hard to analyse numerically. Inthe spherical case the volume fraction in (2) can be foundas fV = πr2/(ρ2∆ϕ2) given that the angular period of rodsis the same for both the spherical angles. The calculatedcross-section radius for the spherical lens is shown in Fig.2 with the dash-dot line. In fact, since the polarizability ofdielectric rods is non-resonant and their period is deep sub-wavelength, the periodic order does not play an importantrole. For the random angular positions of rods an averagedvolume filling fraction should be put in the expression (2). Wehave manufactured the lens having the radius R = 230 mmand the focal length F = 290 mm containing approximately10000 identical polystyrene rods of the length 172 mm. All therods were produced on a casting machine under hot pressure.The rods were manually assembled by sticking to pits of thedepth 2 mm prepared by drilling the solid spherical core at itssurface. For simplicity, the core of the radius Rc = 60 mm wasmade of the same polystyrene as the rods. The lens togetherwith the used X-band pyramidal horn with the aperture 59×62mm and the length 70 mm are shown in Fig. 4(a). The pitsform a rhombic lattice on the core’s surface. Adjacent rods(see the inset in Fig 4(a)) have the averaged angular distanceof ∆ϕ = 2◦ (the averaged separation between the rod ends isR∆ϕ = 8.1 mm). In order to further reduce the periodicity ofrods at ρ = R, the shape or rods was modified (each rod splitsinto four identical branches at the distance 100 mm from thecore). The final rod shape is shown in Fig. 4(b) (the verticaland the horizontal scales are in the ratio 10:1).

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−30 −20 −10 0 10 20 30−40

−30

−20

−10

0

Angle, degrees

Nor

mal

ized

Rad

iatio

n P

atte

rn, d

B8 GHz10 GHz12 GHz

(a)

H-plane

−30 −20 −10 0 10 20 30−40

−30

−20

−10

0

Nor

mal

ized

Rad

iatio

n P

atte

rn, d

B

8 GHz10 GHz12 GHz

(b)

Angle, degrees

-planeE

Fig. 5. Measured normalized radiation patterns: (a) in H-plane; (b) in E-plane.

The radiation pattern was measured at the distance of 4 m inan anechoic chamber. The feed aperture plane was located 55mm away from the lens corresponding to maximal measureddirectivity. The normalized radiation patterns measured inthe H-plane and in the E-plane are presented in Fig. 5(a,b)correspondingly. It can be seen that the H-plane pattern has apronounced main lobe and the side lobe level of better than -11.4 dB in the range 8-12 GHz, which is in agreement with thesimulated behavior of the cylindrical lens for TMz polarization(see Fig. 3(b)). In contrast, the E-plane radiation pattern hasvery high side lobes (from -5 to -2 dB in the frequency range),similar to ones obtained in the simulation of the cylindricallens for the TEz polarization (see Fig. 3(d)). Similarly, thesedistortions of the measured E-plane radiation pattern canbe explained anisotropy effect causing aperture phase errors.Also we note that according to simulations the optimal feedpositions are different for the E- and H-planes. However forthe spherical lens we experimentally found a compromise feedposition maximizing the directivity. The estimated values ofdirectivity coming from the measured radiation patterns in E-and H-planes [17] are: 23.9 dB at 8 GHz; 23.0 dB at 10 GHzand 24.0 dB at 12 GHz.

IV. CONCLUSION

Having observed the above presented numerical and exper-imental results one can conclude that the proposed lens in thecylindrical case can be optimized and perfectly operate as aLuneburg lense for the TMz polarization. On the other handthe same cylindrical lens suffers from anisotropy of divergingrods in the case of the TEz polarization. As follows from themeasured radiation patterns, the manufactured spherical lenswith the radial arrangement of rods had a radiation patternwith a pronounced single lobe in the H-plane, while in the E-plane its pattern had high side lobe radiation due to the sameeffects of anisotropy. In fact, the effective permittivity seen

by a propagating wave inside the lens depends on the localpolarization angle with respect to a rod direction. However, inour simple design procedure the shape of rods was optimizedfor only one polarization state (orthogonal to the rods). An-other reason for the pattern distortion in the experiment wasthe permittivity 2.5 of the used solid polystyrene core. Thisvalue was slightly higher than the theoretical value of 1.67.

As an advantage, the manufactured spherical lens operatingin the wide band from 8 to 12 GHz consists of only twokinds of parts: a spherical core and a shaped rod. In the furtherwork, the radiation pattern distortion due to anisotropy will besuppressed by modification of rod shapes. The local anisotropyeffect could be achieved e.g. by skewering multiple electricallysmall dielectric spheres on each rod.

V. ACKNOWLEDGEMENTS

This work was supported by the Government of the RussianFederation (Grant 074-U01) and the Russian Foundation forBasic Research (Project No. 16-32-00796). The authors aregrateful to Ivan Deyneka for his help.

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[14] I. K. Meshkovsky and D. V. Shannikov, “Focusing device for tranceiveof radiowaves in samtimeter wave band,” 1999, patent RU, No. 2159487.

[15] ——, “Phase distortions in an anisotropic Luneberg lens,” TechnicalPhysics Letters, vol. 28, no. 11, pp. 925–926, 2002.

[16] M. G. Silveirinha, “Nonlocal homogenization model for a periodic arrayof ε-negative rods,” Phys. Rev. E, vol. 73, p. 046612, Apr 2006.

[17] C.-T. Tai and C. Pereira, “An approximate formula for calculating thedirectivity of an antenna,” IEEE Trans. Antennas Propag., vol. 24, no. 2,pp. 235–236, Mar 1976.