LETTERS
PUBLISHED ONLINE: 28 JUNE 2009 | DOI: 10.1038/NMAT2489
An omnidirectional retroreflector based on the
transmutation of dielectric singularities
Yun Gui Ma1, C. K. Ong2, Tomáš Tyc3and Ulf Leonhardt4*
Transformation optics1–6 is a concept used in some
metamaterials7–11 to guide light on a predetermined path. In this
approach, the materials implement coordinate transformations
on electromagnetic waves to create the illusion that the waves
are propagating through a virtual space. Transforming space by
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 par-
ticular, 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
nreaches 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(r0) in spherical coordinates
{r0,ϑ 0,φ 0}. We use primes to distinguish virtual space from real
space. Then we represent the virtual coordinates in real space
{r,ϑ ,φ}by ϑ0=ϑ,φ0=φand the continuous function r0(r). We
require that r0(r) obeys
n(r0)dr0
dr=n0or, equivalently, r=r(r0)=1
n0Zn(r0) dr0(1)
where n0is a constant chosen such that beyond some radius a
the virtual and the real coordinates coincide, r0=rfor r≥a. The
radius adefines 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
εi
j=µi
j=diagn2r02
n0r2,n0,n0(2)
in spherical coordinates xi= {r,ϑ ,φ }. The tensors of the electric
permittivity20 εi
jand the magnetic permeability20 µi
jdescribe 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, CzechRepublic, 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(r0) does not diverge faster than r0−1. 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
a metallic rod in planar illumination. The Eaton lens is characterized
by the index profile18
n=r2a
r0−1 for r0<aand n=1 for r0≥a(3)
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
Ein the potential U, where U−E= −n2/2 (in dimensionless units).
The Eaton lens corresponds to the Kepler potential21 U= −a/r0for
r0<aand 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 coordi-
nate transformation
r=2a
n0"arcsinrr0
2a+sr0
2a1−r0
2a#,n0=1+π
2(4)
for r0<aand r=r0for r0>a, because this r(r0) solves the
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