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Recent advances in transformation optics

Yongmin Liuaand Xiang Zhang*ab

Received 9th May 2012, Accepted 27th June 2012

DOI: 10.1039/c2nr31140b

Within the past a few years, transformation optics has emerged as a new research area, since it provides

a general methodology and design tool for manipulating electromagnetic waves in a prescribed manner.

Using transformation optics, researchers have demonstrated a host of striking phenomena and devices;

many of which were only thought possible in science fiction. In this paper, we review the most recent

advances in transformation optics. We focus on the theory, design, fabrication and characterization of

transformation devices such as the carpet cloak, ‘‘Janus’’ lens and plasmonic cloak at optical

frequencies, which allow routing light at the nanoscale. We also provide an outlook of the challenges

and future directions in this fascinating area of transformation optics.

1. Introduction

Nobody would disagree that the better understanding, manipu-

lation and application of light, or electromagnetic waves in a

more general respect, play a crucial role in advancing science and

technology. The underlying driving force is the long-standing

interest and attention of human beings concerning novel elec-

tromagneticphenomenaand devices. Without persistent

pursuits, it is impossible to develop a more efficient and direc-

tional radar antenna, a brighter light source, or an instrument

with higher imaging resolution. One of the central aims of these

devices is to control and direct electromagnetic fields. For

instance, by optimizing the curvature of glass lenses in a micro-

scope, we intend to focus light to a geometrical point with less

aberration so that the imaging resolution could be improved.

Alternatively, the technique of gradient index (GRIN) optics has

been applied to design lenses by shaping the spatial distribution

of the refractive index of a material rather than the interface of

lenses. The resulting lenses can be flat and avoid the typical

aberrations of traditional lenses.

Yongmin Liu

Yongmin Liu received his Ph.D.

degree from the University of

California, Berkeley in 2009,

under the supervision of Prof.

Xiang Zhang. Currently he is a

postdoctoral researcher in the

same group. Dr. Liu will join the

faculty of Northeastern Univer-

sity in August 2012, with a joint

appointment in the departments

of Electrical & Computer Engi-

neering and

Industrial Engineering.

Liu’s research interests include

nanoscale materials and engi-

neering, nano photonics, nano

quantum optics

Mechanical&

Dr.

devices,

nanostructures.

andnonlinear andof metallic

Xiang Zhang

Xiang Zhang received his Ph.D.

degree from the University of

California, Berkeley in 1996. He

is Ernest S. Kuh Endowed Chair

Professor at UC Berkeley and

the Director of NSF Nano-scale

Science and Engineering Center.

He is also a Faculty Scientist at

Lawrence Berkeley

Laboratory. Prof. Zhang is an

elected member

Academy of

(NAE) and Fellow of four

scientific societies:

Association for the Advance-

ment of Science

National

of National

Engineering

American

(AAAS),

American Physical Society (APS), Optical Society of America

(OSA), and the International Society of Optical Engineering

(SPIE). His research interests are nano-scale science and tech-

nology, materials physics, photonics and bio-technologies.

aNSF Nanoscale Science and Engineering Center (NSEC), 3112

Etcheverry Hall, University of California, Berkeley, CA 94720, USA.

E-mail: xiang@berkeley.edu

bMaterials Science Division, Lawrence Berkeley National Laboratory, 1

Cyclotron Road, Berkeley, CA 94720, USA

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FEATURE ARTICLE

Page 2

In fact, it has been long known that a spatially changing

refractive index modifies light propagation characteristics. Based

on the early work of the ancient Greek mathematician Hero of

Alexandria (10–70 AD) and the Arab scientist Ibn al-Haytham

(965–1040 AD), Pierre de Fermat formulated the famous Fer-

mat’s principle to determine how light propagates in materials.1

This principle states that light follows the extremal optical paths

(shortest or longest, although mostly shortest), where the optical

path is measured in terms of the refractive index n integrated

along the light trajectory. If we replace one material with another

one with a different refractive index in the space where light

propagates, the light path will be bent or even curved instead of a

straight line. Many optical phenomena, such as refraction of a

straw at the interface of air and water, and the mirage effect in a

desert due to the air density (refractive index) variation, can be

explained by Fermat’s principle.

Fermat’s principle tells us how light propagates, if we know

the distribution of the refractive index in space. The emerging

field of transformation optics enables us to solve the inverse

problem, that is, how to realize a specific light path by designing

the variation of material properties.2–4Apparently, this is one

significant step moving forward. With transformation optics, we

have the most general and powerful method to realize almost all

kinds of novel optical effects and devices, some of which only

existed in science fiction and myths. Tremendous progress has

been achieved in the field of transformation optics during the

past a few years, thanks to the new electromagnetic theory and

modelling software, state-of-the-art fabrication tools as well as

greatly improved characterization and analysis techniques. In

this review article, we will first outline the general theory of

transformation optics and metamaterials that allow for the

realization of transformation optical designs. Then we will focus

on the most recent advances, both theory and experiment, in

transformation optics at optical frequencies and at the nano-

scale. Finally, the perspective of transformation optics will be

presented.

2. Theory of transformation optics

The fundamental of transformation optics arises from the fact

that Maxwell’s equations, the governing equations for all elec-

tromagnetic effects, are form invariant under coordinate trans-

formations. Assuming no free current densities, in a Cartesian

coordinate system Maxwell’s equations can be written as

?V ? E ¼ ?m$vH=vt

where E (H) is the electric (magnetic) field, and 3 (m) is the electric

permittivity (magnetic permeability) of a medium that can be a

tensor in general. It can be rigorously proved that after applying

a coordinate transformation x0¼ x0(x), Maxwell’s equations

maintain the same format in the transformed coordinate

system,2–4that is,

?V0? E0¼ ?m0$vH0=vt

In eqn (2), the new permittivity tensor 30and permeability tensor

m0in the transformed coordinate system are related to the orig-

inal 3 and m given by5,6

V ? H ¼ 3$vE=vt

(1)

V0? H0¼ 30$vE0=vt

(2)

8

>

>

>

>

>

>

:

<

30¼L3LT

detjLj

m0¼LmLT

detjLj

(3)

where L is the Jacobian matrix with components defined as Lij¼

vx

i/vxj. The Jacobian matrix characterizes the geometrical vari-

ation in the original space x and the transformed space x0. The

corresponding electromagnetic fields in the new coordinate are

given by

(

0

E0¼?LT??1E

H0¼?LT??1H

(4)

Eqn(1)–(4)formthebasisoftransformationoptics.Wecandesign

and manipulate the light trajectory by an arbitrary coordinate

transformation. Consequently, the material properties and field

components need to be rescaled according to the form invariance

ofMaxwell’sequations.Thisguaranteesthephysicalcharacteristic

oflightpropagationtobepreservedatdifferentscales.Infact,such

a correspondence between coordinate transformations and mate-

rials parameters has been noticed for a long time.

Probably the most remarkable transformation optical device is

the invisibility cloak, which can render an object unperceivable

although the object physically exists. One seminal design of such

a cloak was proposed by Sir John Pendry et al.2They considered

the hidden object to be a sphere of radius R1and the cloaking

region to be contained within the annulus R1# r # R2. By

applying a very simple coordinate transformation

8

:

<

r0¼ R1þ rðR2? R1Þ=R2

q0¼ q

f0¼ f

(5)

the initial uniform light rays in the central region (0 # r # R2) are

squeezed into a shell (R1#r0# R2), while the rest of the light rays

(in the region r > R2) are maintained. Waves cannot penetrate

into and hence interact with the core region (0 # r0# R1),

because it is not part of the transformed space. No matter what

object is placed inside the core, it appears to an observer that

nothing exists; that is, the object is concealed or cloaked. Based

on eqn (3), we can calculate the required material properties for

the cloaking device. In the region of r0# R1, 30and m0can

take any values and do not cause any scattering. In the region of

R1# r0# R2,

8

>

>

>

Finally, for r0$ R2the properties of materials are unchanged.

Under the short wavelength limit (R1, R2[ l), the ray tracing

results confirm the performance of the invisibility cloak as shown

in Fig. 1(a). The rays, which represent the Poynting vector or

energy flow, are numerically obtained by integration of a set of

Hamilton’s equations taking into account the anisotropic,

>

>

>

>

>

>

>

:

>

<

>

>

>

>

>

>

>

>

>

3

0

r0 ¼ m

0

r0 ¼

R2

R2? R1

?r0? R1

R2

R2? R1

R2

R2? R1

r0

?2

3

0

q0 ¼ m

0

q0 ¼

3

0

f0 ¼ m

0

f0 ¼

(6)

5278 | Nanoscale, 2012, 4, 5277–5292This journal is ª The Royal Society of Chemistry 2012

Page 3

inhomogeneous material properties in the compressed region

(R1# r0# R2).2,6Light is smoothly wrapped around the core,

and the propagation characteristic is preserved outside the cloak.

This implies that any object placed in the interior region appears

to be concealed, since there is no diffracted or scattered light in

the presence of the object. Full-wave simulations without

geometric optics approximation also verify the cloaking effect.7

It is worth mentioning that different approaches have been

proposed to realize invisibility cloaks. One example is the

conformal mapping technique to design the refractive index

profile that guides light around an object.8,9We can introduce a

new coordinate w described by an analytic function w(z) that

does not depend on z*, where a complex number z ¼ x + i ? y is

used to describe the spatial coordinate in a two-dimensional (2D)

plane and z* stands for the conjugate of z. Such a function

defines a conformal mapping that preserves the angles between

the coordinate lines. For a gradually varying refractive index

profile, both the electricand magnetic fields satisfy the Helmholtz

equation. In the new coordinate w, the Helmholtz equation has

the same format with a transformed refractive index profile that

is related to the original one as n0¼ n$

Helmholtz equation in the coordinates w as the Schr€ odinger

equation of a quantum particle in the Kepler potential, Leon-

hardt designs a dielectric invisibility cloak with refractive index

ranging from 0 to about 36 (Fig. 1(c)). Different from the

transformation optics approach, the conformal mapping tech-

nique is strictly two-dimensional. However, the conformal

mapping idea can be extended to non-Euclidean geometry to

realize three-dimensional (3D) cloaks, and eliminate the extreme

values of materials parameters that often appearin the method of

transformation optics.10The conformal mapping method has

been used for designing a variety of devices in addition to the

invisibility cloak.11–15Moreover, it has been shown that

conformal mapping performs well even in the regime beyond

geometrical optics.16Another type of approximate invisibility

cloaking is a core–shell structure. It has been shown that a

negative-permittivity shell can significantly reduce the scattering

cross-section of a small positive-permittivity core in the quasi-

static limit.17,18By exploiting the frequency dispersion of metals

and their inherent negative polarizability, it is shown that

????

dw

dz

????

?1

. By interpreting the

covering a dielectric or conducting object of a certain size with

multilayered metallic shells may reduce the ‘‘visibility’’ of the

object by several orders of magnitude simultaneously at multiple

frequencies.19

Meanwhile,researchers

exploring the interesting physics associated with invisibility

cloaks,20–24or trying to detect an invisibility cloak.25,26

The invisibility cloak has triggered widespread interest in

transformation optics. Many other novel effects and devices,

such as illusion optics,27–30optical black holes,31–34beam shifters

and rotators,35–37lossless waveguide bends38–41as well as various

lenses42–50have been proposed. In particular, combining the

concept of complementary medium51with transformation optics,

Yang et al. proposed a superscatter which can enhance the

electromagnetic wave scattering cross section, so that it appears

as a scatter with a larger dimension.52Subsequently, Chan’s

group theoretically conceived and numerically demonstrated a

general concept of illusion optics: making an arbitrary object

appear like another object with a completely different shape and

material constituent.27Cloaking can be considered as the crea-

tion of an illusion in free space. The principle behind illusion

optics is not light bending but rather the cancellation and

restoration of the optical path of light by using negative-index

materials. The key of an illusion device lies in two distinct pieces

of materials, that is, a complementary medium and a restoring

medium. The complementary medium annihilates the adjacent

space and cancels any light scattering from an object itself. Then

the restoring medium recovers the cancelled space with a new

illusion space that embraces another object chosen for the illu-

sion. Numerical simulations confirm the performance of the

illusion device, which transforms the field distribution scattered

from a dielectric spoon into the scattering pattern from a metallic

cup. More interestingly, the illusion device can work at a distance

from the object. It is shown that this ‘‘remote’’ feature enables the

opening of a virtual aperture in a wall so that one can peep

through the wall. Lai et al. also numerically demonstrate a

remote invisibility cloak that can cloak an object at a certain

distance outside the cloaking shell rather than encircled by the

cloaking shell.53

Unlikeprevious

devices,2–9the constitutive parameters of illusion devices do not

need a complex spatial distribution. However, materials with a

negative refractive index are required in the design, which are not

obtainable in nature.

havebeen actively

light-bendingcloaking

3.

optical designs

Metamaterials for realizing transformation

Although transformation optics provides the most general means

to design exotic optical effects and elements, the experimental

realization of them is far from trivial. As shown in eqn (3), both

electric permittivity and magnetic permeability need to be

spatially and independently tailored. Moreover, the resulting

material properties are anisotropic in general, and may require

unusual values (negative, zero or infinity). We are limited in

natural materials to fulfil such demands. For example, natural

materials only show magnetism (m/m0 s 1) up to terahertz

frequencies. Fortunately, the emerging field of metamaterials

offers an entirely new route to design material properties at will,

so that the transformation optical design could be experimentally

realized.54–61Different from natural materials, the physical

Fig. 1

from ref. 2 with permission. The rays, representing the Poynting vector,

divert within the annulus of the cloak region (R1 < r < R2), while

emerging on the far side without any scattering and distortion. (b) Ray

tracing results for an invisibility cloak in a three-dimensional view,

reprinted from ref. 2 with permission. (c) Ray propagation in the

dielectric invisibility device, reprinted from ref. 8 with permission. The

light rays (shown in yellow) smoothly flow around the interior cloak

region (shown in black). The brightness of the green background indi-

cates the refractive index profile taken from the Kepler profile.

(a) A schematic of a cloak in a two-dimensional view, reprinted

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Page 4

properties of metamaterials are not primarily dependent on the

chemical constituents, but rather on the internal, specific struc-

tures of the building blocks of metamaterials. These building

blocks function as artificial ‘‘atoms’’ and ‘‘molecules’’, in analogy

to those in natural materials. Through regulated interactions

with electromagnetic waves, they can produce extraordinary

properties that are difficult or impossible to find in naturally

occurring or chemically synthesized materials.

Metamaterials consist of periodically or randomly distributed

artificial structures, whose size and spacing are much smaller

than the wavelength of electromagnetic waves. As a result, the

microscopic detail of individual structures cannot be sensed by

electromagnetic waves. What matters is the average result of the

collective response of the whole assembly. In other words, we can

homogenize such a collection of inhomogeneous objects and

define effective material properties at the macroscopic level. This

is effective media approximation, which has been well known.62

The most attractive aspect of metamaterials, however, is that the

material properties can be controlled by properly engineering the

structures. For instance, metallic wire arrays63and metallic split-

ring structures64can produce effective 3 and m, respectively, with

tunable values ranging from positive to negative within a certain

wavelength range. By combining the two basic structures with

simultaneously negative 3 and m, we can even create materials

possessing a negative refractive index that enable negative

refraction65–69and perfect imaging.70–74Furthermore, meta-

materials allow us to achieve unusual anisotropy75–77and

chirality.78–80We refer readers to recent review papers and books

for more insights in the field of metamaterials.54–61

The complete control over electric permittivity and magnetic

permeability offered by metamaterials turns transformation

optical design into reality. In 2006, Smith’s group demonstrated

the first invisibility cloak in the microwave region.81To mitigate

the fabrication and measurement challenges, a 2D cylindrical

cloak instead of a 3D spherical one was implemented. Since the

electric field is polarized along the z axis of the cylindrical

coordinate, in the transformed 30and m0tensors only 3

are relevant. After a further renormalization, the reduced

material parameters are

?

0

z, m

0

rand m

0

q

3

0

z¼

R2

R2? R1

?2

;m

0

r¼

?r0? R2

r0

?2

;m

0

q¼ 1(7)

where the interior and exterior radium of the cloaking device is

R1and R2, respectively. The advantage of using reduced material

properties is that only one parameter (m

the other two are constant throughout the structure. This

parameter set is realized in a metamaterial structure consisting of

split-ringresonators withcarefully

(Fig. 2(a)). In the experiment, a field-sensing antenna is used to

record the field amplitude and phase inside the cloak and in the

surrounding free-space region. The experimental results show

that the cloak can significantly decrease scattering from the

hidden object and also reduce its shadow. From Fig. 2(b) and (c),

one can clearly see that electromagnetic waves smoothly flow

around the cloak, and propagate to the far side with only a

slightly perturbed phase front,which is mainly dueto the reduced

parameter implementation. In comparison, a bare Cu cylinder

without the cloak produces much stronger scattering in both the

forward and backward directions.

0

r) spatially varies while

designed geometries

4.

optical wavelengths

Broadband transformation optical design at

The pioneering work on transformation optics in 2006 (ref. 2, 3

and 6) stimulated the global attention of researchers in different

disciplines. Ever since then, tremendous effort has been devoted

to the field of transformation optics. Considering the great

application potential, one prime direction of transformation

optics is to implement designs working in the optical regime.82

However, most transformation optical devices rely on meta-

materials, in which the building blocks are normally much

smaller than the wavelength of interest. This indicates that the

feature size of the device should be precisely controlled at the

scale of a few hundred or even below one hundred nanometers.

More importantly, metamaterials are usually resonant structures

with narrow operation bandwidth and high loss. These two

factors impose severe challenges on the implementation of

transformation optical devices with broad bandwidth and low

loss at near-infrared and visible frequencies. New designs and

creative fabrication techniques are imperative to tackle the

challenges.

In the following, we will concentrate on the discussion of the

carpet cloak introduced by Jensen Li and John Pendry,83

although other designs, such as one-dimensional (1D) cloaks84,85

Fig. 2

the cloak. The split-ring resonator of layer 1 (inner) and layer 10 (outer) are shown in the transparent square insets. (b) Simulated and (c) experimentally

mapped field patterns of the cloak. Reprinted from ref. 81 with permission.

(a) An image of a 2D microwave cloak made of split-ring resonators. The background plots the values of the prescribed material properties for

5280 | Nanoscale, 2012, 4, 5277–5292 This journal is ª The Royal Society of Chemistry 2012

Page 5

and cloaks using non-Euclidean geometries,10can also operate

over a relatively wide range of wavelengths. Different from the

original complete cloak that essentially crushes the object to a

point and works for arbitrary incident angles, the carpet cloak

crushes the object to a sheet and the incident angle is limited

within the half space of a 2D plane. However, the carpet cloak

does not require extreme values for the transformed material

properties. Moreover, by applying the quasi-conformal mapping

technique, the anisotropy of the cloak can be significantly

minimized. Consequently, only isotropic dielectrics are needed to

construct the carpet cloak (Fig. 3(a)), implying the device could

be broadband and practically scalable to operate in the optical

regime. Full-wave simulations confirm that the carpet cloak

successfully imitates a flat reflecting surface. As shown in

Fig. 3(b), the light reflected from a curved reflecting surface, on

top of which is covered with the carpet cloak, well maintains the

flat wavefront without any distortion. It seems that the light is

reflected by a flat ground plane. Therefore, it renders an object

placed underneath the curved bump invisible. In contrast, if the

cloak is absent, the incident beam is deflected and split into two

different angles (Fig. 3(c)).

Soon after the demonstration of a microwave carpet cloak

based on non-resonant metallic metamaterials,86three groups

independently realized the carpet cloak in the near-infrared

region.87–89Interestingly, all of them utilized the same dielectric

platform (silicon-on-insulator (SOI) wafer) to achieve the broad-

band and low-loss carpet cloak, although the configurations are

different. In the design of Zhang’s group,87the carpet cloaking

device consists of two parts (Fig. 4(a) and (b)): a triangular

region with a uniform hole pattern that acts as a background

medium with a constant effective index (1.58), and a rectangular

region with varying hole densities to realize the spatial index

profile similar to Fig. 3(a). The holes with a constant diameter

(110 nm) were made through the Si layer by focused ion beam

(FIB) milling. Under the effective medium approximation, the

desired spatial index profile can be achieved by controlling the

density of holes through the relation 3eff¼ 3airrair+ 3SirSi, where

r is the volumetric fraction and 3 is the effective dielectric

constant of each medium. In addition, two gratings were fabri-

cated in order to couple light into and out of the Si slab wave-

guide. Finally, directional deposition of 100 nm gold was carried

out using electron beam evaporation to create the reflecting

surface. In the experiments, the authors characterize the reflected

beam profile of a Gaussian beam in three scenarios: (1) a flat

surface without a cloak, (2) a curved surface with a cloak and (3)

a curved surface without a cloak. It is observed that in both case

(1) and (2), the reflected beam preserves the Gaussian profile,

similar to the incident waves. In a sharp contrast, the light

Fig. 3

(refractive index n). The grey lines represent the transformed grid after

the quasi-conformal mapping. All of the square cells in the original

Cartesian coordinate are transformed to nearly squares of a constant

aspect ratio after the quasi-conformal mapping. Consequently, the

anisotropy of the material property is minimized to a negligible degree.

Comparing with the background (SiO2with n ¼ 1.45), the resulting

refractive index is higher in the region above the curved bump, while it is

lower at the two shoulders of the bump. (b) The electric field pattern for a

Gaussian beam launched at 45?towards the ground plane from the left,

with the spatial index distribution given in (a). (c) The electric field

pattern when only the curved bump is present without the cloak. The

wavelength is 500 nm for simulation in (b) and (c).

(a) The colour maps show the transformed material properties

Fig. 4

of a near-infrared carpet cloak, which is realized by milling holes with

different densities in a SOI wafer. Reprinted from ref. 87 with permission.

In the schematic figure (a), the rectangular cloak region marked as C1has

a varying index profile given by the transformation design, and the

triangular region marked as C2has a uniform hole pattern, serving as a

background medium with a constant effective index of 1.58. (c) and (d)

SEM images of another near-infrared cloaking device by etching silicon

posts in a SOI wafer. Light is coupled into the device via an input

waveguide and reflected by the Bragg mirror towards the x–z plane.

Reprinted from ref. 88 with permission. (e) Schematic and (f) cross-

sectional SEM image of a 3D carpet-cloak structure working in the near-

infrared region, reprinted from ref. 90 with permission. The 3D cone of

light in (e) corresponds to the NA ¼ 0.5 microscope lens.

(a) Schematic and (b) scanning electron microscope (SEM) image

This journal is ª The Royal Society of Chemistry 2012 Nanoscale, 2012, 4, 5277–5292 | 5281