Slim Luneburg lens for antenna applications

Slim Luneburg lens for antennaapplications

Angela Demetriadou1,2,∗ and Yang Hao2,3

1Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2AZ, UK2Electronic Engineering, Queen Mary, University of London, E1 4NS, London, UK

[email protected]∗[email protected]

Abstract: Luneburg lens is a marvellous optical lens but is extremely dif-ficult to be applied in any practical antenna system due to its large sphericalshape. In this paper, we propose a transformation that reduces the profileof the original Luneburg lens without affecting its unique properties. Thenew transformed slim lens is then discretized and simplified for a practicalantenna application, where its properties were examined numerically. It isfound that the transformed lens can be used to replace conventional antennasystems (i.e. Fabry-Perot resonant antennas) producing a high-directivitybeam with low side-lobes. In addition, it provides excellent steeringcapabilities for wide angles, maintaining the directivity and side-lobes athigh and low values respectively.

© 2011 Optical Society of America

OCIS codes: (080.2740) Geometric optical design; (080.3620) Lens system design; (080.3630)Lenses; (160.3918) Metamaterials; (220.3620) Lens system design; (220.3630) Lenses;(260.2110) Electromagnetic optics.

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#150086 - $15.00 USD Received 29 Jun 2011; revised 1 Sep 2011; accepted 1 Sep 2011; published 27 Sep 2011(C) 2011 OSA 10 October 2011 / Vol. 19, No. 21 / OPTICS EXPRESS 19925

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cloak at microwave frequencies,” Science 314, 977–980 (2006).23. R. Luneburg, Mathematical Theory of Optics (Brown University, 1944).24. R. Ilinsky, “Gradient-index meniscus lens free of spherical aberration,” J. Opt. A: Pure Appl. Opt. 2, 449–451

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1. Introduction

For the past few decades, various studies have been developed to increase significantly thedirectivity and lower the side-lobe levels of an antenna systems. In previous designs of highly-directive antennas [1–4], several schemes including the use of frequency selective surfaces(FSS) [5], electromagnetic band-gap (EBG)superstrates [5–7], partially reflecting superstrates(PRS) and phase-induced apodization methods [8,9] were demonstrated in order to compensatefor the phase difference caused by different path lengths of the rays. For broadband operations,array feeds were suggested instead of single feeds [10, 11]. Also, several important ways werereported in the literature, that manipulate an antenna beam. Some of them use a cylindricalsuperstrate for an omnidirectional antenna [12–14], shift the patch antenna from the centre ofthe cavities [15], and introduce an electrically phase-varying partially reflecting superstrate infront of the antenna [16, 17]. Despite the fact that all these significant advances in antennaperformances, all the proposed antenna designs are rather complex or have drawbacks in eitherthe frequency bandwidth, the side-lobe levels or their steering capabilities.

The method discussed in this paper is a new and simpler approach to create a broadband,high-directivity, low side-lobe and steerable antenna, by using a dielectric lens as a super-strate. The idea is to use transformation optics to produce a lens that ‘directs’ and ‘transforms’the beam from a feeding antenna as we wish. Transformation optics flourished rapidly in thepast decade due to the development of metamaterials [18–20], and especially since the realiza-tion [21] and experimental verification [22] of a cloaking device.

We are inspired by the Luneburg lens [23], which can focus a plane wave or convert thespherical wave from a point source to a plane wave. The Luneburg lens belongs to a familyof GRIN (GRadient INdex) lenses that feature spherical iso-surfaces for their refractive index.Mikaelian lens, Tarhanov plane convex lens and Ilinsky’s meniscus lens [24] are other examplesof lenses that belong to this family and are also free from spherical aberations. By performinga geometric transformation to the Luneburg lens, we manage to reshape it to a slimmer de-sign, which is more applicable for antenna systems. Since all the properties of the traditionalLuneburg lens are preserved, then the new lens can create a high-directivity beam with very low


levels of side-lobes and steering capabilities for relatively wide angles. Furthermore, if homo-geneous dielectrics are used as the sub-units (i.e. non-dispersive structures) of the lens, then thelens functions very well within a very broadband range as well. Therefore, the lens proposedin this paper combines a number of significant and desirable antenna properties, without com-promising or restricting the performance of the antenna system significantly, due to its simpledesign.

The paper is organised as: In section 2, our proposed transformation of the Luneburg lens isdiscussed both analytically and numerically using a Finite Difference Time Domain (FDTD)algorithm [25], explaining its performance, properties and limitations. In section 3, the contin-uously spatially variant transformation lens is discretized (in order to produce a more realisticdesign for future manufacturing purposes) and Finite Integration simulations are performed.The lens is fed by a patch antenna and the directivity, first side-lobe levels and the steerabilityof the overall system are computed, where the excellent performance of the lens is evident.

2. Transformed Slim Luneburg

The Luneburg lens is a spatially variable refractive-index spherical structure, that managesto focus a plane wave, or transform a circular wave from a point source to a plane wave, asshown in Figs. 1(a)–1(c). The spatially variant permittivity and refractive index of the lens, arefunctions of the lens’ radius r and given by:

ε(r) = 2− r2

R2 = n2(r) (1)

where r =√

y2 + z2 at a given point (assuming a 2D case in a yz-plane) and R is the radius ofthe lens, and the lens is magnetically inactive (i.e. μ = 1). It varies from ε(r = 0) = 2 in thecentre of the lens, to ε(r = R) = 1 at the circumference, such that there are no reflections at theinterface of lens-vacuum. Using an FDTD algorithm [25], a 2D study of the lens is performed byplacing a point source on the circumference of the Luneburg lens. The circular wave producedby the point source is transformed by the lens to a plane wave, as shown in Fig. 1(b). Also, notethat for a source placed at any point on the circumference of the lens, the produced plane wavehas the wavevector parallel to the line connecting the point source and the centre of the lens, asshown in Figs. 1(b)–1(c).

However, due to the large spherical (or cylindrical in 2D) shape of the lens, it is very un-suitable for antenna applications. Also, it is difficult to spatially control an antenna, since it hasto be moved on the circumference of such a large lens. Therefore, we propose a transforma-tion of the Luneburg lens, which creates a slimmer lens, while all the properties of the originalLuneburg lens are maintained. The transformation equations create a ‘discus’-shaped lens, by‘squeezing’ or ‘slimming’ the spherical lens and are given by:

y′ = y (2)

z′ = z/δ

where δ takes any real integer, which leads to:

ε ′ = ε(

1/δ 00 δ

)=

(2− y′2 +(δ z′)2

R2

)(1/δ 0

0 δ

)(3)

and

μ ′ =(

1/δ 00 δ

)(4)


Fig. 1. For R ∼ 13λ , (a) ε(r) for the conventional Luneburg lens. (b) Field configurationfor a conventional Luneburg lens with a point source at (−R,0) and (c) with a point sourceat (−Rcos(π/4),−Rcos(π/4)). (d)ε ′ for the transformed slim Luneburg lens for δ = 6.(e) field configuration for the slim Luneburg lens for a point source excited at (R/6,0) and(f) at (R/6,−0.5R).

The ε ′-map of Eq. (3) has ε ′yy < 1 component, which can be realised with metamaterials. How-ever, ‘meta-atoms’ effective permittivities smaller than unity are resonant, which limits the fre-quency bandwidth of the lens and increases losses. Therefore, since the fields are focused in thehigh refractive index region of ε ′-map, and avoid the low index regions (except for the highestangles of incidence [26,27]), the approximation ε ′yy = 1 can be taken without severely affectingthe performance of the lens. Also, μ ′

yyμ ′zz = 1, therefore μ ′ ∼ 1 is also a benign assumption.

Hence, the lens is all-dielectric with

ε ′ = ε ′zz = δ(

2− y′2 +(δ z′)2

R2

)(5)

which is plotted in Fig. 1(d) for δ = 6. These assumptions simplify the manufacturing process,open a broad range of suitable materials and ensure that the lens can be non-dispersive (i.e.broadband).

Using a FDTD code, the performance of the new lens is initially examined in 2D for δ =6 and Eq. (5) describing the ε ′-map (for the rest of the paper, consider δ = 6 and Eq. (5)).A point source is placed on two different point on the circumference of the lens. It can beseen in Figs. 1(e)–1(f) (where R ∼ 13λ ) that the wave emerging from the lens has plane-waveproperties. Also, when the source is moved at a different point on the circumference of the newlens, the transformed plane wave has an angle with the z-axis, as expected. The plane wave isproduced by delaying the phase of the circular wavefront inside the lens as it can be seen inFigs. 1(e) and 1(f). If we assume a source emitting a wave at f = 10GHz ⇒ λ = 30mm, thenthe lens in Figs. 1(d)–1(f) has Δy′ ∼ 406mm ∼ 13.5λ (where Δy′ is the height of the lens alongthe y-axis), which is extremely large and very difficult to be computationally modelled and alsonot desirable for most application purposes. Therefore, the lens needs to be scaled down byconsidering R ∼ λ .


Fig. 2. For R ∼ λ , field configurations for the slim Luneburg lens (a) for a point sourceexcited at (R/6,0), (b) (R/6,−R/2) (c) (R/2,0) and (d)(R/2,−R/2).

However, the proposed transformation equations hold for all frequencies where λ is abovethe diffraction limit of the lens, as happens with all conventional lenses. Therefore, we wouldexpect that for Δz′ > (λ/2) the lens to behave as in Figs. 1(d)–1(f), where Δz′ is the width ofthe lens. For Δz′ < (λ/2), the lens becomes too small with respect to λ and stops operatingas it was designed, but as shown in Figs. 2(a) and 2(b). It is clear that the width of the lensis not large enough to cause the required phase delay of the wave and convert the wavefrontfrom circular to plane. Therefore, the wavefront emerging from the lens still has significantcurvature. We can address this problem by simply placing the point source at a distance fromthe lens. Then, the wavefront reaching the lens has smaller curvature and the lens is now able todelay the wave phase enough in order to have a flat wavefront emerging from the other side ofthe lens, as it can be seen in Figs. 2(c) and 2(d). Therefore, the focal length of the lens becomesproportional to λ of the wave.

There are several GRIN lenses discussed in the literature [8, 9, 24] and some of them havea thin width (such as Tarhanov’s and Ilinsky’s lenses [24]) that can also be used for antennaapplications. However, the slim Luneburg lens proposed in this paper, has a significant advan-tage compared with the Tarhanov’s lens and Ilinsky’s lenses. The slim Luneburg lens has asmooth variation of the ε-map, with high values at the centre and low values at the edges ofthe lens (i.e. identical to vacuum at edges of the lens). Therefore, the wave enters the lens withnegligibly small reflections and exits the lens in a similar fashion, avoiding (multiple) internalreflections. Hence, the power of the wave is not significantly compromised by passing throughthe slim Luneburg lens. However, Tarhanov’s and Ilinsky’s lenses have a high value of ε atone of their interface with vacuum, reflecting part of the wave, and limiting the performance ofany antennas system they are implemented in. This significant advantage of the slim Luneburglens is particularly important for all high-directive antennas, but more particular for high-powerantenna applications.

3. Antenna application for the slim Luneburg lens

A 3D lens of ’discus’-shape is generated by rotating around the z-axis the 2D all-dielectric mapfrom previous section (i.e. any cross-section of the 3D lens gives the permittivity map shown inFig. 1(d)). In this paper, we consider as feeding source antenna a patch antenna emitting a waveat 10GHz (i.e. xpatch = ypatch = 6.95mm) and fed with a coaxial wire, such that the emittedwave has an E-field polarized along the x-axis (i.e. Ex-polarization). Notice that, because of therevolution symmetry, the operating principle of the lens is independent of the actual polarizationof the emitted wave propagating along z.

Furthermore, the transformed lens is discretized, to account for manufacturing issues. Thediscretized lens is shown in Fig. 3(b) and was based on the permittivity map produced analyt-ically. The discretization was performed such that the width of each layer to be much smaller


Fig. 3. (a)The cross-section of the antenna system set-up. (b) The yz-cross-section of thediscretized lens showing the dielectric layers and dimensions. (c)The maximum directivity(red square dots), the FSLL (first side lobe level) for φ = 0◦ (green x-points) and for φ =90◦ (blue star-point) plotted against the distance b from the patch antenna, where FSLL =Dfirstsidelobe −Dmainlobe.

than the wavelength at 10GHz, ensuring minimum reflections from the layers. The effect ofdiscretization on the performance of the lens was studied by illuminating the lens with a planewave and observing its focusing properties. In Fig. 4, the |E|2 along the optical axis of the lensis plotted, where it can be seen that the discretization slightly increases the focal depth, butwithout significantly compromising the behaviour of the antenna. Also, Fig. 4 shows that thediscretization level chosen (i.e. 6 layers) is the lowest without significantly limiting the per-formance of the lens, ensuring easier manufacturing process. The small increase on the focaldepth for the 6-layer discretized lens (used for the rest of the paper) compared with the uniformlens, makes the antenna system of Fig. 3(a) less sensitive on the value of b, without affectingsignificantly the performance of the lens (which can be found useful for realistic applicationsof the lens).

Table 1. The dimensions of the discretized lens, as they were used for the CST MicrowaveStudio simulations, where ε is the dielectric permittivity of each layer, Ry and Rz are definedin Fig. 3(b).

Structure ε Ry (mm) Rz (mm)1 12 15.7 1.662 10 26.73 3.313 8 34.9 4.44 6 38.43 5.55 4 44.63 6.26 2 47.93 6.9

Note that the value of δ dictates the geometry and ε ′-map of the lens and can be varied tosuit various manufacturing techniques and material property restrictions. The value δ = 6 cho-sen for this work along with the discretization level, allow for reasonably easy manufacturingtechniques (since all layer widths are larger than 1mm). Also, for all ε values needed, naturalmaterial can be found with extremely low loss tangent. The set-up of the antenna system isshown in Fig. 3(a) and the dimensions of the discretized lens used from now on in this paperare shown in Fig. 3(b) and Table 1 obtained after optimization of the y-axis dimensions of thelens to allow for the non-point-like patch antenna wave. The following results were obtainedusing CST Microwave Studio (CST GmbH, Darmstadt, Germany).

The patch antenna emits a wave with frequency 10GHz and Ex-polarization. The wave


0

0.2

0.4

0.6

0.8

1

-1.4 -1.2 -1 -0.8 -0.6

norm

aliz

ed |E

|2 (a.

u.)

z/R

uniform lens8 layers6 layers4 layers

Fig. 4. |E|2 along the optical axes of the lens for different levels of discretization (i.e.number of layers), when the lens is illuminated with a plane wave. The set-up of the lensis identical to Fig. 2, where the lens’s centre is at the origin point and the plane wave isincident from the right towards the left.

reaches the discretized lens, and while it passes through, it experiences the appropriate phasedelay, in order to be transformed to a plane wave. Figures 5(a)–5(b) show that the lens man-age to produce a very conformal beam with the maximum main lobe directivity at 17.1dBiand extremely low first side-lobe levels (FSLL) with values lower than ∼ −15dB, where(FSLL is defined as the directivity (D) of the first side lobe with respect to the main lobe-i.e. FSLL = Dfirstsidelobe −Dmainlobe). These results were observed when the patch antenna wasplaced at b = 31mm ∼ λ distance from the lens, where b is the distance of the patch antennafrom the lens (as defined in Fig. 3(a)).

Fig. 5. The directivity pattern for (a) φ = 0◦ and (b)φ = 90◦, plotted against θ for variousvalues of b.

The focal length of the lens was examined by moving the lens closer and/or further awayfrom the patch antenna (i.e. along the z-axis). The directivity patterns calculated numericallyare shown in Figs. 4 and 3(c). As it was expected, the directivity of the main lobe is low whenthe lens is too close to the patch antenna. However, as the lens moves further away from the


patch antenna, the maximum directivity increases gradually and saturates at b = 31mm ∼ λ , asexpected. The side-lobe levels are also reduced for larger values of b, and reach optimum valuesaround b = 31mm. Note that despite several simplifications from the initial transformation ofEq. (5), the antenna system shows amazingly conformal directivities with high values for themain lobe, and extremely low values for FSLL. Also, the antenna system is not highly sensitiveto b, unlike other high-directive antennas, such as Fabry-Perot, FSS or EBG antennas.

Fig. 6. The directivity pattern for (a) φ = 0◦ and (b)φ = 90◦, plotted against θ for variousvalues of c.

However, the most significant advantage of the slim Luneburg lens as discussed above isits steering properties. The steerable behaviour of the slim lens was examined by moving thepatch antenna along the y-axis by a distance c (defined in Fig. 3(a)). It was found that the lenssteered the main lobe, while it maintained high values for the directivity and low side-lobes.The directivity plots for various c values are shown in Fig. 6 for φ = 0◦ and φ = 90◦. In Fig. 7,the directivity of the main lobe is plotted against c, where it can be seen that the antenna systemis steerable for ±20◦, for directivity values higher than 15dBi. Also, the FSLL for φ = 0◦ andφ = 90◦ are plotted against c, and which are kept at very low values for ±20◦. It is observed

Fig. 7. (a)The directivity and steered angle of the main lobe plotted against c for b= 31mm.(b) The FSLL for φ = 0◦ and φ = 90◦, and steered angle plotted against c for b = 31mm.

in Fig. 7 that the directivity of the main lobe is reduced and FSLL increases for higher steering


angles. This is expected, since the assumption that ε ′yy = 1 was taken analytically in order toavoid using materials with ε ′yy < 1 and conclude to an easily-manufactured lens. However, the40◦ steering angle range for main lobe directivities higher than 15dBi is rather wide. Finally,the Ex-fields for various values of c are shown in Fig. 8, where the extreme steering of the beamis observed, along with the conversion of the spherical wavefront to planar. Note that there isno ground plane and that the reflections from the lens’ layers are insignificantly small as it canbe seen in Figs. 8(a)–8(h).

Fig. 8. The Ex-field for b = 31mm and (a)c= 0mm, (b)c= 2mm, (c) c = 4mm, (d) c = 6mm,(e) c = 10mm, (f) c = 16mm, (g) c = 20mm and (h) c = 24mm.

Note that the steerability is observed only for the φ = 90◦ in Fig. 6, since the antenna emits aEx-polarized wave and moves along the y-axis. If we move the patch antenna along the x-axisby c, then the same behaviour will be observed only for the φ = 0◦ directivity pattern, since thelens is symmetric in the yz-plane. If on the other hand, the main lobe needs to be steered at anyother direction, then the patch antenna can be moved accordingly on the xy-plane.

Observing the results discussed in this paper, one may notice that FSLL is extremely lowand to the best of the authors’ knowledge lower than any other antenna system reported inthe literature and based on Fabry-Perot resonant cavities incorporating EBG, FSS and PRSstructures. Additionally, it maintains high directivity values for wide steering angles. Therefore,a very simple design for a superstrate manages to create a broadband, high-directivity, and lowFSLL, steerable antenna.

4. Conclusions

A transformed Luneburg lens is proposed that can be applied in practical antenna systems.Initially, the transformation equations are introduced and discussed both analytically and nu-merically using a FDTD algorithm. Several assumption were taken in order to simplify thematerial and design requirements for the lens and exclude dispersive media from the manufac-turing process. Eventually, an all-dielectric, discretized lens was realised, with dielectric layersforming a ‘discus’-shaped design, which is relatively simple to be manufactured. The perfor-mance and properties of the lens were numerically examined, for a patch-antenna source. Ithas been shown that the antenna system, composed of the feed and the slim Luneburg lens,has high directivity, low FSLL and steering capabilities for wide angles, outperforming similarlens and superstrate antennas. It is worth noting that the proposed slim Luneburg lens combinesa number of significant and desirable antenna properties, without any major compromises orrestrictions to the performance of the antenna system, due to its simple design.


Acknowledgments

The authors are grateful to the Office of Naval Research Global (ONRG) under Naval Interna-tional Cooperative Opportunities (NICOP) for the funding support with Grant No. N00014-09-1-1013.


Slim Luneburg lens for antenna applications

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