## Abstract

Luneburg lens is a marvellous optical lens but is extremely difficult to be applied in any practical antenna system due to its large spherical shape. In this paper, we propose a transformation that reduces the profile of the original Luneburg lens without affecting its unique properties. The new transformed slim lens is then discretized and simplified for a practical antenna application, where its properties were examined numerically. It is found that the transformed lens can be used to replace conventional antenna systems (i.e. Fabry-Perot resonant antennas) producing a high-directivity beam with low side-lobes. In addition, it provides excellent steering capabilities for wide angles, maintaining the directivity and side-lobes at high and low values respectively.

© 2011 OSA

## 1. Introduction

For the past few decades, various studies have been developed to increase significantly the directivity 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 compensate for 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 were reported in the literature, that manipulate an antenna beam. Some of them use a cylindrical superstrate for an omnidirectional antenna [12–14], shift the patch antenna from the centre of the cavities [15], and introduce an electrically phase-varying partially reflecting superstrate in front of the antenna [16, 17]. Despite the fact that all these significant advances in antenna performances, all the proposed antenna designs are rather complex or have drawbacks in either the 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 the past decade due to the development of metamaterials [18–20], and especially since the realization [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 the spherical wave from a point source to a plane wave. The Luneburg lens belongs to a family of 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 examples of lenses that belong to this family and are also free from spherical aberations. By performing a geometric transformation to the Luneburg lens, we manage to reshape it to a slimmer design, which is more applicable for antenna systems. Since all the properties of the traditional Luneburg 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 homogeneous dielectrics are used as the sub-units (i.e. non-dispersive structures) of the lens, then the lens functions very well within a very broadband range as well. Therefore, the lens proposed in this paper combines a number of significant and desirable antenna properties, without compromising or restricting the performance of the antenna system significantly, due to its simple design.

The paper is organised as: In section 2, our proposed transformation of the Luneburg lens is discussed both analytically and numerically using a Finite Difference Time Domain (FDTD) algorithm [25], explaining its performance, properties and limitations. In section 3, the continuously spatially variant transformation lens is discretized (in order to produce a more realistic design 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 steerability of 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 manages to focus a plane wave, or transform a circular wave from a point source to a plane wave, as shown in Figs. 1(a)–1(c). The spatially variant permittivity and refractive index of the lens, are functions of the lens’ radius *r* and given by:

*R*is the radius of the lens, and the lens is magnetically inactive (i.e.

*μ*= 1). It varies from

*ɛ*(

*r*= 0) = 2 in the centre of the lens, to

*ɛ*(

*r*=

*R*) = 1 at the circumference, such that there are no reflections at the interface of lens-vacuum. Using an FDTD algorithm [25], a 2D study of the lens is performed by placing a point source on the circumference of the Luneburg lens. The circular wave produced by the point source is transformed by the lens to a plane wave, as shown in Fig. 1(b). Also, note that for a source placed at any point on the circumference of the lens, the produced plane wave has the wavevector parallel to the line connecting the point source and the centre of the lens, as shown in Figs. 1(b)–1(c).

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

*δ*takes any real integer, which leads to:

*ɛ*′-map of Eq. (3) has

*ɛ*′

*< 1 component, which can be realised with metamaterials. However, ‘meta-atoms’ effective permittivities smaller than unity are resonant, which limits the frequency bandwidth of the lens and increases losses. Therefore, since the fields are focused in the high refractive index region of*

_{yy}*ɛ*′-map, and avoid the low index regions (except for the highest angles of incidence [26,27]), the approximation

*ɛ*′

*= 1 can be taken without severely affecting the performance of the lens. Also,*

_{yy}*μ*′

_{yy}*μ*′

*= 1, therefore*

_{zz}*μ*′ ∼ 1 is also a benign assumption. Hence, the lens is all-dielectric with

*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 *ɛ*′-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 be seen in Figs. 1(e)–1(f) (where *R* ∼ 13*λ*) that the wave emerging from the lens has plane-wave properties. Also, when the source is moved at a different point on the circumference of the new lens, the transformed plane wave has an angle with the z-axis, as expected. The plane wave is produced by delaying the phase of the circular wavefront inside the lens as it can be seen in Figs. 1(e) and 1(f). If we assume a source emitting a wave at *f* = 10*GHz* ⇒ *λ* = 30*mm*, then the lens in Figs. 1(d)–1(f) has Δ*y*′ ∼ 406*mm* ∼ 13.5*λ* (where Δ*y*′ is the height of the lens along the y-axis), which is extremely large and very difficult to be computationally modelled and also not desirable for most application purposes. Therefore, the lens needs to be scaled down by considering *R* ∼ *λ*.

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

There are several GRIN lenses discussed in the literature [8, 9, 24] and some of them have a thin width (such as Tarhanov’s and Ilinsky’s lenses [24]) that can also be used for antenna applications. However, the slim Luneburg lens proposed in this paper, has a significant advantage compared with the Tarhanov’s lens and Ilinsky’s lenses. The slim Luneburg lens has a smooth variation of the *ɛ*-map, with high values at the centre and low values at the edges of the lens (i.e. identical to vacuum at edges of the lens). Therefore, the wave enters the lens with negligibly small reflections and exits the lens in a similar fashion, avoiding (multiple) internal reflections. Hence, the power of the wave is not significantly compromised by passing through the slim Luneburg lens. However, Tarhanov’s and Ilinsky’s lenses have a high value of *ɛ* at one of their interface with vacuum, reflecting part of the wave, and limiting the performance of any antennas system they are implemented in. This significant advantage of the slim Luneburg lens is particularly important for all high-directive antennas, but more particular for high-power antenna 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 map from previous section (i.e. any cross-section of the 3D lens gives the permittivity map shown in Fig. 1(d)). In this paper, we consider as feeding source antenna a patch antenna emitting a wave at 10*GHz* (i.e. *x _{patch}* =

*y*= 6.95

_{patch}*mm*) and fed with a coaxial wire, such that the emitted wave has an E-field polarized along the x-axis (i.e.

*E*-polarization). Notice that, because of the revolution symmetry, the operating principle of the lens is independent of the actual polarization of the emitted wave propagating along z.

_{x}Furthermore, the transformed lens is discretized, to account for manufacturing issues. The discretized lens is shown in Fig. 3(b) and was based on the permittivity map produced analytically. The discretization was performed such that the width of each layer to be much smaller than the wavelength at 10*GHz*, ensuring minimum reflections from the layers. The effect of discretization on the performance of the lens was studied by illuminating the lens with a plane wave and observing its focusing properties. In Fig. 4, the |*E*|^{2} along the optical axis of the lens is plotted, where it can be seen that the discretization slightly increases the focal depth, but without significantly compromising the behaviour of the antenna. Also, Fig. 4 shows that the discretization level chosen (i.e. 6 layers) is the lowest without significantly limiting the performance of the lens, ensuring easier manufacturing process. The small increase on the focal depth for the 6-layer discretized lens (used for the rest of the paper) compared with the uniform lens, makes the antenna system of Fig. 3(a) less sensitive on the value of *b*, without affecting significantly the performance of the lens (which can be found useful for realistic applications of the lens).

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

The patch antenna emits a wave with frequency 10*GHz* and *E _{x}*-polarization. The wave reaches the discretized lens, and while it passes through, it experiences the appropriate phase delay, in order to be transformed to a plane wave. Figures 5(a)–5(b) show that the lens manage to produce a very conformal beam with the maximum main lobe directivity at 17.1

*dBi*and extremely low first side-lobe levels (FSLL) with values lower than ∼ −15

*dB*, where (FSLL is defined as the directivity (D) of the first side lobe with respect to the main lobe-i.e.

*FSLL*=

*D*–

_{firstsidelobe}*D*). These results were observed when the patch antenna was placed at

_{mainlobe}*b*= 31

*mm*∼

*λ*distance from the lens, where

*b*is the distance of the patch antenna from the lens (as defined in Fig. 3(a)).

The focal length of the lens was examined by moving the lens closer and/or further away from the patch antenna (i.e. along the z-axis). The directivity patterns calculated numerically are shown in Figs. 4 and 3(c). As it was expected, the directivity of the main lobe is low when the 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* = 31*mm* ∼ *λ*, as expected. The side-lobe levels are also reduced for larger values of *b*, and reach optimum values around *b* = 31*mm*. Note that despite several simplifications from the initial transformation of Eq. (5), the antenna system shows amazingly conformal directivities with high values for the main lobe, and extremely low values for FSLL. Also, the antenna system is not highly sensitive to *b*, unlike other high-directive antennas, such as Fabry-Perot, FSS or EBG antennas.

However, the most significant advantage of the slim Luneburg lens as discussed above is its steering properties. The steerable behaviour of the slim lens was examined by moving the patch antenna along the y-axis by a distance *c* (defined in Fig. 3(a)). It was found that the lens steered 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 system is steerable for ±20°, for directivity values higher than 15*dBi*. Also, the FSLL for *ϕ* = 0° and *ϕ* = 90° are plotted against *c*, and which are kept at very low values for ±20°. It is observed 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 *ɛ*′* _{yy}* = 1 was taken analytically in order to avoid using materials with

*ɛ*′

*< 1 and conclude to an easily-manufactured lens. However, the 40 ° steering angle range for main lobe directivities higher than 15*

_{yy}*dBi*is rather wide. Finally, the

*E*-fields for various values of

_{x}*c*are shown in Fig. 8, where the extreme steering of the beam is observed, along with the conversion of the spherical wavefront to planar. Note that there is no ground plane and that the reflections from the lens’ layers are insignificantly small as it can be seen in Figs. 8(a)–8(h).

Note that the steerability is observed only for the *ϕ* = 90° in Fig. 6, since the antenna emits a *E _{x}*-polarized wave and moves along the y-axis. If we move the patch antenna along the x-axis by

*c*, then the same behaviour will be observed only for the

*ϕ*= 0° directivity pattern, since the lens is symmetric in the yz-plane. If on the other hand, the main lobe needs to be steered at any other 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 low and to the best of the authors’ knowledge lower than any other antenna system reported in the literature and based on Fabry-Perot resonant cavities incorporating EBG, FSS and PRS structures. 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 low FSLL, 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 numerically using a FDTD algorithm. Several assumption were taken in order to simplify the material and design requirements for the lens and exclude dispersive media from the manufacturing process. Eventually, an all-dielectric, discretized lens was realised, with dielectric layers forming a ‘discus’-shaped design, which is relatively simple to be manufactured. The performance and properties of the lens were numerically examined, for a patch-antenna source. It has 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 similar lens and superstrate antennas. It is worth noting that the proposed slim Luneburg lens combines a number of significant and desirable antenna properties, without any major compromises or restrictions 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 International Cooperative Opportunities (NICOP) for the funding support with Grant No. N00014-09-1-1013.

## References and links

**1. **G. V. Trentini, “Partially reflecting sheet arrays,” IRE Trans. Antennas Propag. **4**, 666–671 (1956). [CrossRef]

**2. **D. R. Jackson and N. G. Alexopoulos, “Gain Enhancement methods for printed circuit antennas,” IEEE Trans. Antennas Propag. **33**, 976–987 (1985). [CrossRef]

**3. **D. Jackson and A. Oliner, “A leaky-wave analysis of the high-gain printed antenna configuration,” IEEE Trans. Antennas Propag. **36**, 905–910 (1988). [CrossRef]

**4. **A. Feresidis and J. Vardaxoglou, “High gain planar antenna using optimised partially reflective surfaces,” IEE Proc. Microwaves, Antennas Propag. **148**, 345–350 (2001). [CrossRef]

**5. **Y. Lee, J. Yeo, R. Mittra, and W. Park, “Design of a high-directivity electromagnetic band gap resonator antenna using a frequency-selective surface superstrate,” Microwave Opt. Technol. Lett. **43**, 462–467 (2004). [CrossRef]

**6. **M. Thevenot, C. Cheype, A. Reineix, and B. Jecko, “Directive photonic-bandgap antennas,” IEEE Trans. Antennas Propag. **47**, 2115–2122 (1999).

**7. **C. Cheype, C. Serier, M. Thevenot, T. Monediere, A. Reineix, and B. Jecko, “An electromagnetic bandgap resonator antenna,” IEEE Trans. Antennas Propag. **50**, 21285–21290 (2002). [CrossRef]

**8. **A. Goncharov, M. Owner-Petersen, and D. Puryayev, “Intrinsic apodization effect in a compact two-mirror system with a spherical primary mirror,” Opt. Eng. **41**, 3111–3115 (2002). [CrossRef]

**9. **O. Guyon, “Phase-induced amplitude apodization of telescope pupils for extrasolar terrestrial planet imaging,” Astron. Astrophys. **404**, 379–387 (2008). [CrossRef]

**10. **R. Gardelli, M. Albani, and F. Capolino, “Array thinning by using antennas in a Fabry-Perot cavity for gain enhancement,” IEEE Trans. Antennas Propag. **54**, 1979–1990 (2006). [CrossRef]

**11. **A. Weily, K. Esselle, T. Bird, and B. Sanders, “Dual resonator 1-D EBG antenna with slot array feed for improved radiation bandwidth,” IET Microwave Antennas Propag. **1**, 198–203 (2007). [CrossRef]

**12. **G. Palikaras, A. Feresidis, and J. Vardaxoglou, “Cylindrical electromagnetic bandgap structures for directive base station antennas,” IEEE Antenna Wirel. Propag. Lett. **3**, 87–89 (2004). [CrossRef]

**13. **H. Boutayeb, T. Denidni, K. Mahdjoubi, A. Tarot, A. Sebak, and L. Talbi, “Analysis and design of a cylindrical EBG based directive antenna,” IEEE Trans. Antennas Propag. **54**, 211–219 (2006). [CrossRef]

**14. **A. Feresidis, M. Maragou, G. Palikaras, and J. Vardaxoglou, “Cylindrical-conformal resonant cavity antennas using passive periodic surfaces,” in International Conference on Electromagnetics in Advanced Applications (2007), pp. 165–168.

**15. **Y. Hao, A. Alomainy, and C. Parini, “Antenna-beam shaping from offset defects in UC-EBG cavities,” Microwave Opt. Tech. Lett. **43**, 108–111 (2004). [CrossRef]

**16. **A. Ourir, S. Burokur, and A. de Lustrac, “Phase-varying metamaterial for compact steerable directive antennas,” Electron. Lett. **43**, 493–494 (2007). [CrossRef]

**17. **A. Ourir, S. Burokur, and A. de Lustrac, “Electronically reconfigurable metamaterial for compact directive cavity antennas,” Electron. Lett. **43**, 698–700 (2007). [CrossRef]

**18. **V. Veselago, “The electrodynamics of substances with simultaneously negative values of *ɛ* and *μ*,” Soviet Phys. Ups. **10**, 509–514 (1968). [CrossRef]

**19. **J. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. **85**, 3966–3969 (2000). [CrossRef] [PubMed]

**20. **R. Shelby, D. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science **292**, 77–79 (2001). [CrossRef] [PubMed]

**21. **J. Pendry, D. Schurig, and D. Smith, “Controlling electromagnetic fields,” Science **312**, 1780–1782 (2006). [CrossRef] [PubMed]

**22. **D. Schurig, J. Mock, B. Justice, S. Cummer, J. Pendry, A. Starr, and D. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science **314**, 977–980 (2006). [CrossRef] [PubMed]

**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 (2000). [CrossRef]

**25. **C. Argyropoulos, Y. Zhao, and Y. Hao, “A radially-dependent dispersive finite-difference time-domain method for the evaluation of electromagnetic cloaks,” IEEE Trans. Antennas Propag. **57**, 1432–1441 (2009). [CrossRef]

**26. **D. Schurig, “An aberration-free lens with zero f-number,” N. J. Phys. **10**, 115034 (2008). [CrossRef]

**27. **N. Kundtz and D. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. **9**, 129–132 (2010). [CrossRef]