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An omnidirectional retroreflector based on the transmutation of dielectric singularities

Yun Gui Ma1, C. K. Ong2, Tomáš Tyc3 and Ulf Leonhardt4*

Transformation optics1–6 is a concept used in some metamaterials7–1 toguidelightonapredeterminedpath.Inthis approach,thematerialsimplementcoordinatetransformations on electromagnetic waves to create the illusion that the waves arepropagatingthroughavirtualspace.Transformingspaceby appropriately designed materials makes devices possible that have been deemed impossible. In particular, transformation optics has led to the demonstration of invisibility cloaking for microwaves12,13, surface plasmons14 and infrared light15,16. Here, on the basis of transformation optics, we implement a microwave device that would normally require a dielectric singularity, an infinity in the refractive index. To fabricate such a device, we transmute17 a dielectric singularity in virtual space into a mere topological defect in a real metamaterial. In particular, we demonstrate an omnidirectional retroreflector18,19, a device for faithfully reflecting images and for creating high visibility from all directions. Our method is robust, potentially broadband and could also be applied to visible light using similar techniques.

Dielectric singularities are points where the refractive index n reaches infinity or zero, where electromagnetic waves travel infinitely slow or infinitely fast. Such singularities cannot be made in practice for a broad spectral range, but one can transmute them into topological defects of anisotropic materials17. Here is a brief summary of the underlying theoretical results17: imagine an isotropic index profile with a singularity in virtual space. Suppose that around the singularity the index profile is spherically symmetric, described by the function n(r′) in spherical coordinates {r′,ϑ′,φ′}. We use primes to distinguish virtual space from real space. Then we represent the virtual coordinates in real space {r,ϑ,φ} by ϑ′ = ϑ, φ′ = φ and the continuous function r′(r). We require that r′(r) obeys

where n0 is a constant chosen such that beyond some radius a the virtual and the real coordinates coincide, r′ = r for r ≥ a. The radius a defines the boundary of the device. Theory4,17 shows that both the coordinate transformation and the virtual index profile are implemented by a material with the dielectric tensors

in spherical coordinates xi = {r,ϑ,φ}. The tensors of the electric permittivity20 εij and the magnetic permeability20 µij describe an

1Temasek Laboratories, National University of Singapore, Singapore 119260, Singapore, 2Centre for Superconducting and Magnetic Materials, Department of Physics, National University of Singapore, Singapore 117542, Singapore, 3Institute of Theoretical Physics and Astrophysics, Masaryk University, Kotlarska 2, 61137 Brno, Czech Republic, 4School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews, KY16 9SS, UK. *e-mail:ulf@st-andrews.ac.uk.

impedance-matched20, anisotropic dielectric20 with varying radial component. Such media are said to be anisotropic, because they respond to different electromagnetic-field components differently, as described by the tensors (2), although the device they constitute is spherically symmetric. As real space and virtual space coincide for r ≥ a, the device with the properties (2) has the same physical effect as the index profile n, but the dielectric tensors remain finite, as long as n(r′) does not diverge faster than r′−1. The direction of dielectric anisotropy, however, is not defined at r = 0, the position of the virtual singularity: the singularity has been transformed into a topological defect.

Consider, for example, the Eaton lens18,19 illustrated in Fig. 1.

The Eaton lens would reflect light back to where it came from, while faithfully preserving any image the light carries (apart from inverting the image). The lens is spherically symmetric, and so it would retroreflect light regardless of direction: it would make a perfect omnidirectional retroreflector. In contrast, conventional optical retroreflectors—‘cat’s eyes’—are made of mirrors and have a finite acceptance angle. A metallic sphere is an omnidirectional retroreflector as well, but only for rays that directly hit its centre. The sphere scatters off-centre light and distorts images, and so does ametallicrodinplanarillumination.TheEatonlensischaracterized by the index profile18

To understand why the profile (3) acts as a perfect retroreflector, one can use the following analogy21: a light ray corresponds to the trajectory of a fictitious Newtonian particle that moves with energy E inthepotentialU,whereU −E =−n2/2(indimensionlessunits). The Eaton lens corresponds to the Kepler potential21 U =−a/r′ for r′ <a and U =−1 outside the device, with total energy E =−1/2. The fictitious particle draws a half Kepler ellipse around the centre of attraction. So it leaves in precisely the opposite direction it came from19: light is retroreflected.

One sees that the profile (3) diverges with the power −1/2 and therefore an Eaton lens has never been made. However, we can transmute the singular isotropic profile into the regular but anisotropic material with the properties (2) by the coordinate transformation

LETTERS NATUREMATERIALSDOI:10.1038/NMAT2489 a b

Figure 1 | Eaton lenses. a, Spherical lens. b, Cylindrical lens. An artist’s impression of the retroreflection of light that carries an image, the letter ‘E’ for ‘Eaton’. In the outgoing light, the image is inverted, but preserved (in a: flipped and upside down, in b: flipped). The implementation of an Eaton lens would require a singularity in the refractive index profile where the index tends to infinity, unless the singularity is transmuted into a harmless topological defect, as we demonstrate in this letter for the cylindrical lens with metamaterials for microwaves.

b c

Original Transformed medium

Rescaled medium

Figure 2 | Simulation of Eaton lenses. a–c, The left pictures show the distribution of the electric field at the original (a) and transformed Eaton lenses (b,c), with dielectric functions (5) and (6), respectively; the right pictures show the corresponding biscattering diagrams. The dotted circles in the left pictures mark the boundary of the device at radius a (40mm); the dotted lines refer to an absorbing sheet that separates the incident and reflected electromagnetic waves, with their direction indicated by the arrows. The wavelength is a/4 and the scale is in millimetres for both axes. The biscattering diagrams refer to the electric field infinitely far away from the device. They show the ratio of the magnitude of the electric field as a function of angle, normalized by the largest value.

NATUREMATERIALSDOI:10.1038/NMAT2489 LETTERS

Figure 3 | The device. The split-ring resonators constitute a metamaterial for microwave radiation with the designed radial magnetic permeability (6). The rings were then filled with a white dielectric powder that generates the

required electric permittivity of n20 (not shown here).

integral (1) and approaches r′ at r′ =a. The singularity of the Eaton lens has been transformed away.

We found that this transmutation of singularities is remarkably robust: Fig. 2a,b compares numerical simulations of electromagnetic wave propagation in an ideal and a transmuted cylindrical Eaton lens. The cylindrical lens of Fig. 1b is characterized by its diagonal dielectric tensors in cylindrical coordinates xi = {r,ϕ,z}, but we use the same functions (2)–(4) for the diagonal components εi and µi as in the case of the spherical lens,

We study the propagation of polarized electromagnetic waves in the {r,ϕ} plane with the electric field pointing in the z direction. Such waves are not influenced by the other non-zero tensor components εr, εϕ and µz. As Fig. 2a,b shows, the wave propagations in the two devices are nearly identical, although the theory (2)–(4) presumes a spherical transformation, not a cylindrical one. Moreover, the corresponding biscattering diagrams indicate that the transformed medium seems to be a better retroreflector than the numerically simulated Eaton lens, because of numerical problems associated with the singularity in the Eaton profile (3). Furthermore, Fig. 2c shows the effect of another modification of the ideal case (2)–(4) that is useful in practice: in this simulation, we rescale the dielectric functions in the device, but not in the air around it. Maxwell’s equations are not changed if one rescales the dielectric functions (5) by the constant n0 as

Consequently, the wave propagation in such media is the same as in materials with the dielectric functions (5). However, in reality the rescaling can be done only in the device, but not in air where ε=µ=1. So the impedance20 is mismatched at the interface, which causes reflections20—scattering—but, as Fig. 2c shows, the phase profile of the outgoing wave is preserved, although the intensity is reduced. The associated biscattering diagram clearly indicates the loss in intensity, but not in directionality. As any incident wave can be represented as a superposition of plane waves, images will be faithfully reflected. The rescaled medium maintains the principal functionality of the Eaton lens.

Figure 3 shows our device. We built the transmuted Eaton lens from 10 concentric rings of low-loss printed circuit board (Rodgers RT6006 with 17.5µmcopper coated on one side of 0.127mm dielectric substrate with ε = 6.2−i0.01). The rings have a height of 10mm and are spaced with 4mm distance until the radius of the outer ring reaches 40mm. The rings carry three rows of split-ring resonators etched out from the copper of the circuit μμϕ r μ r

Figure 4 | Effective electromagnetic properties. The blue curve shows the index profile (3) of the original Eaton lens as a function of the radius r; the other curves show the rescaled dielectric functions (6) of the transmuted

Eaton lens. The inset shows µr with magnified scale. The symbols are the dielectric functions calculated for the corresponding layers of the metamaterial.

board (lengths 3.14mm and heights 3.3m). The split-ring resonators create the required profile (6) of the radial magnetic permeability µr for microwave radiation at 8.9GHz frequency (34mm wavelength). They constitute a metamaterial7–1, a material with dielectric properties created by subwavelength structures—the split-ring resonators in our case. The circuit-board rings were glued on a 0.1-m-thick aluminium sheet that was prepatterned with concentric circular prints exactly fitting the rings. Then the rings were filled with dielectric powder (Emerson & Cuming Microwave

Products)togeneratetherequiredelectricpermittivity(6)ofn20. Figure 4 shows the curves of the dielectric functions. We designed the split-ring resonators by reverse-engineering where we calculate their dielectric response and modify their parameters until the calculated µr matches the required profile (6). However, each split-ring resonator interacts with its neighbours and cannot be considered on its own. For each individual resonator, we calculated the effective dielectric properties using Ansoft HFSS, assuming the resonator to constitute the elementary cell of an infinite lattice22 (lattice constants 3.14, 4.0 and 3.3m in accordance with the dimensions of the resonators and the spacing of the circuit-board rings). In this way, we embed, in the modelling, the split-ring resonator into a lattice of its own kind. This procedure presumes that the dielectric properties do not vary much and it also ignores the curvature of the circuit-board rings, even at the inner ring of the device where the profile (6) contains a topological defect. Nevertheless, our experimental data prove that such a crudely transmutedEatonlensworksremarkablywellinpractice.

Figure 5 shows the results of our measurements. Our microwave source is a vector network analyser (HP8722D, Agilent Technologies) that we also use for detection. It has two ports connected by two coaxial cables (Megaphase 1GVT4), the feeding and the detecting port. To confine and scan the microwave radiation, we apply the parallel-plate electromagnetic field mapping system described elsewhere23. The microwave radiation is confined in the horizontal direction by two large aluminium boards at 1m spacing. The field is scanned by a sub-wavelength monopole probe antenna, a thin coaxial cable, inserted through a hole at the centre of the top plate. The bottom plate, carrying the device, is mounted on a computer-controlled stage that can move in two dimensions. The total scanning area covers 120×240mm2 with a step resolution of 1m. On the bottom plate, pieces of 10-m-thick microwave-absorbing foam were placed to define the incidence channel and suppress scattering at the boundaries. An absorbing

LETTERS NATUREMATERIALSDOI:10.1038/NMAT2489 a

Measurement Simulation

Figure 5 | Results. a,b, Measurement results (a) compared with simulation (b). Description as for the left pictures of Fig. 2, except that here the wavelength is 34mm (0.85a).

sheet segregates the incidence channel from the reflected radiation. The sheet was made of a 0.1-m-thick copper tape sandwiched between two 1.5-m-thick layers of magnetically loaded silicone absorber (ECCOSORB).

As Fig. 5a shows, the measured field inside the device is perturbed by the graininess of the metamaterial and its ring structure, but the outgoing wave is in very good agreement with the numerical simulation for a smooth dielectric profile shown in Fig. 5b. In simulations, we even reduced the number of rings to 5, and still observed a good performance of the transmuted Eaton lens. There we approximated the device by five equidistant uniform layers with constant dielectric properties (6). Our crudely approximated prototype preserves the functionality of the ideal Eaton lens, which indicates that the transmutation of singularities17 can be remarkably robust and reliable in practice. As the required dielectric properties (2) lie within a finite range, such devices can, in principle, work over a broad range of the spectrum and similar devices15,16,24–28 could even operate in the visible.

Received 17 March 2009; accepted 27 May 2009; published online 28 June 2009

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Acknowledgements

Y.G.M. and C.K.O. are supported by the Defense Science and Technology Agency under the Defense Innovative Research Program, Singapore (DSTA-NUS-DIRP/2004/02), T.T. acknowledges the grants MSM0021622409 and MSM0021622419 and U.L. is supported by a Royal Society Wolfson Research Merit Award.

Author contributions

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