Transformation Optics and the Geometry of Light

TLDR: In this article, the geometry of light and the concepts of transformation optics are described and connections between geometry and electromagnetism in media that is as consistent and elementary as possible.

Abstract: Publisher Summary This chapter describes the geometry of light and the concepts of transformation optics. Transformation optics is beginning to transform optics. The chapter introduces connections between geometry and electromagnetism in media that is as consistent and elementary as possible. This chapter analyzes four examples of transformation media—cloaking devices, perfect lenses, vortices, and horizons; these four cases illustrate characteristic non-trivial topologies, each one with different physics, and they have been experimentally verified. This chapter focuses on the main ideas and some connections between optics, in particular, transformation optics, and other areas of physics and mathematics. It also addresses the classical optics of transformation media.

Figures

Figure 3: Fermat’s principle. Light takes the shortest optical path from A to B (solid line) which is not a straight line (dotted line) in general. The optical path length is measured in terms of the refractive index n integrated along the trajectory. The greylevel of the background indicates the refractive index. The figure illustrates the ray trajectories involved in forming a mirage.

Figure 18: Aquatic analogue of the event horizon. (From Ref. [25]. Reprinted with permission of Peter Hoey.) Current can stop fish moving upstream and mimic an event horizon. A moving optical medium captures the physics at the event horizon, too. A pulse in an optical fibre creates such a moving medium [106].

Figure 11: Invisibility for waves. The device transforms waves emitted in physical space as if they propagate through empty space in transformed coordinates. (a) Transformed space. The figure shows the propagation of a spherical wave emitted at the point (−1.5, 0, 0). (b) Physical space. The invisibility device turns the wave of figure (a) into physical space by the coordinate transformation illustrated in Fig. 10. The cloaking layer with radii R1 = 0.5 and R2 = 1.0 deforms electromagnetic waves such that they propagate around the invisible region and leave without carrying any trace of the interior of the device.

Figure 14: Perfect lens. Negatively refracting perfect lenses employ transformation media. The top figure shows a suitable coordinate transformation from the physical x axis to the electromagnetic x′, the lower figure illustrates the corresponding device. The inverse transformation from x′ to x is either triple or single-valued. The triple-valued segment on the physical x axis corresponds to the focal region of the lens: any source point has two images, one inside the lens and one on the other side. Since the device facilitates an exact coordinate transformation, the images are perfect with a resolution below the normal diffraction limit [11]: the lens is perfect [101]. In the device, the transformation changes right-handed into left-handed coordinates. Consequently, the medium employed here is left-handed, with negative refraction [144].

Figure 8: Parallel transport around a closed loop on the sphere. A vector at the North Pole is parallel transported to the equator along the geodesic to which it is tangent. Since a geodesic parallel transports its tangent vector, the vector is still tangent to the geodesic at the equator. The vector is then parallel transported a quarter of the distance around the equator (also a geodesic). Parallel transport preserves the vector, as much as the curvature allows, so it stays at the same angle to the equator. Finally the vector is parallel transported back to the North Pole along a geodesic. On return to the North Pole the vector is rotated by 90 degrees with respect to the initial vector. This rotation is completely specified by the Riemann curvature tensor. Note that on a plane (zero Riemann tensor) this kind of operation does not rotate the vector.

Figure 4: Fermat’s principle: longest optical path. The figure shows a simple example where light takes the longest optical path: light beyond the focal point of a lens, traveling along a straight line from A to B. To see why the optical path from A to B is the longest, compare the solid line of the actual path with an example of a virtual path, with the dashed and dotted paths, and note that the optical path length between the focal points A and A′ of the lens is always the same, regardless whether light travels along a straight line or is refracted in the lens. Therefore the optical path length taken differs from the virtual path length by the difference between the two short sides and the long side of the triangle from L to A′ and B. The sum of the two short sides of a triangle is always longer than the long side: light has taken the longest optical path.

Full text

Transformation Optics and the Geometry of Light
Ulf Leonhardt
1
and Thomas G. Philbin
2
1
School of Physics and Astronomy, University of St Andrews,
North Haugh, St Andrews KY16 9SS, UK
2
Max Planck Research Group Optics, Information and Photonics,
G¨unther-Scharowsky Str. 1, 91058 Erlangen, Germany
June 7, 2008
To appear in Progress in Optics (edited by Emil Wolf).
1
arXiv:0805.4778v2 [physics.optics] 7 Jun 2008

1 Introduction
Metamaterials are beginning to transform optics and microwave technology thanks
to their versatile properties that, in many cases, can be tailored according to prac-
tical needs and desires [20, 31, 32, 48, 57, 80, 86, 120]. Although metamaterials
are surely not the answer to all engineering problems, they have inspired a series of
significant technological developments and also some imaginative research, because
they invite researchers and inventors to dream. Imagine there were no practical
limits on the electromagnetic properties of materials. What is possible? And what
is not? If there are no practical limits, what are the fundamental limits? Such
questions inspire taking a fresh look at the foundations of optics [11] and at con-
nections between optics and other areas of physics. In this article we discuss such
a connection, the relationship between optics and general relativity, or, expressed
more precisely, between geometrical ideas normally applied in general relativity and
the propagation of light, or electromagnetic waves in general, in materials [74].
Farfetched as it may appear, general relativity turns out [74] to have been put to
practical use in the first working prototype of an electromagnetic cloaking device
[122], it gives perhaps the most elegant approach to achieving invisibility [72, 104],
and [74], general relativity even works behind the scenes of perfect lenses [101, 143].
The practical use of general relativity in electrical and optical engineering may
seem surprisingly unorthodox: traditionally, relativity has been associated with the
physics of gravitation [97] and cosmology [99] or, in engineering [142], has been
considered a complication, not a simplification. For example, the Global Positioning
System would not be as accurate as it is without taking relativistic corrections into
account that are due to gravity and the motion of the navigation satellites. However,
here we are not concerned with the influence of the natural geometry of space and
time on optics, the space-time curvature due to gravity, but rather we show how
optical materials create artificial geometries for light and how such geometries can
be exploited in designing novel optical devices.
Connections between geometry and optics are nothing new; the ideas we ex-
plain here are rather, to borrow a phrase of Sir Michael Berry, “new things in old
things”. These ideas are based on Fermat’s principle [11] formulated in 1662 by
Pierre de Fermat, but anticipated nearly a millennium ago by the Arab scientist Ibn
al-Haytham and inspired by the Greek polymath Hero of Alexandria’s reflections on
light almost two millennia ago. According to Fermat’s principle, light rays follow
extremal optical paths in materials (shortest or longest, mostly shortest) where the
length measure is given by the refractive index. Media change the measure of length.
This means that any optical medium establishes a geometry [12, 119, 121]: the glass
in a lens, the water in a river or the air creating a mirage in the desert. General
relativity has cultivated the theoretical tools for fields in curved geometries [61, 97].
In this article we show how to use these tools for applications in electromagnetic or
optical metamaterials.
Metamaterials are materials with electromagnetic properties that originate from
man-made sub-wavelength structures [80, 86, 120]. Perhaps the best known metama-
terials are the materials used in the pioneering demonstrations of negative refraction
[127] or invisibility cloaking [122] of microwaves, see Fig. 1, or for negative refraction
2

Figure 1: Cloaking device. (From Ref. [122]. Reprinted with permission from AAAS.)
Two-dimensional microwave cloaking structure (background image) with a plot of the ma-
terial parameters that are implemented. The cloaking device is made of circuit-board with
structures that are about an order of magnitude smaller than the wavelength. The struc-
tures are split-ring resonators with tunable magnetic response. The split-ring resonators
of the inner and outer rings are shown in expanded schematic form (transparent square
insets).
of near-visible light [129]. These materials consist of metallic cells that are smaller
than the relevant electromagnetic wavelength. Each cell acts like an artificial atom
that can be tuned by changing the shape and the dimensions of the metallic struc-
ture. It is probably fair to regard microstructured or photonic-crystal fibres [118] as
metamaterials as well, see Fig. 2. Here sub-wavelength structures — airholes along
the fibre — significantly influence the optical properties of the fused silica the fibres
are made of. Metamaterials have a long history: the ancient Romans invented ruby
glass, which is a metamaterial, although the Romans presumably did not know this
concept. Ruby glass [148] contains nano-scale gold colloids that render the glass
neither golden nor transparent, but ruby, depending on the size and concentration
of the gold droplets. The color originates from a resonance of the surface plasmons
[7] on the metallic droplets. Metamaterials per se are nothing new: what is new is
the degree of control over the structures in the material that generate the desired
properties.
The specific starting point of our theory is not new either. In the early 1920’s
Gordon [40] noticed that moving isotropic media appear to electromagnetic fields as
certain effective space-time geometries. Bortolotti [12] and Rytov [119] pointed out
3

Figure 2: Photonic-crystal fibres. (From Ref. [118]. Reprinted with permission from
AAAS.) An assortment of optical (OM) and scanning electron (SEM) micrographs of
photonic-crystal fibre (PCF) structures. (A) SEM of an endlessly single-mode solid core
PCF. (B) Far-field optical pattern produced by (A) when excited by red and green laser
light. (C) SEM of a recent birefringent PCF. (D) SEM of a small (800 nm) core PCF
with ultrahigh nonlinearity and a zero chromatic dispersion at 560-nm wavelength. (E)
SEM of the first photonic band gap PCF, its core formed by an additional air hole in a
graphite lattice of air holes. (F) Near-field OM of the six-leaved blue mode that appears
when (E) is excited by white light. (G) SEM of a hollow-core photonic band gap fiber.
(H) Near-field OM of a red mode in hollow-core PCF (white light is launched into the
core). (I) OM of a hollow-core PCF with a Kagom´e cladding lattice, guiding white light.
that ordinary isotropic media establish non-Euclidean geometries for light. Tamm
[134, 135] generalized the geometric approach to anisotropic media and briefly ap-
plied this theory [135] to the propagation of light in curved geometries. In 1960
Plebanski [112] formulated the electromagnetic effect of curved space-time or curved
coordinates in concise constitutive equations. Electromagnetic fields perceive media
as geometries and geometries act as effective media. Furthermore, in 2000 it was un-
derstood [67] that media perceive electromagnetic fields as geometries as well. Light
acts on dielectric media via dipole forces (forces that have been applied in optical
trapping and tweezing [29, 94]). These forces turn out to appear like the inertial
forces in a specific space-time geometry. This geometric approach [71] was used to
shed light on the Abraham-Minkowski controversy about the electromagnetic mo-
mentum in media [2, 75, 90, 100]. Geometrical ideas have been applied to construct
conductivities that are undetectable by static electric fields [43, 44], which was the
precursor of invisibility devices [5, 38, 72, 73, 87, 104, 123] based on implementations
4

of coordinate transformations. From these recent developments grew the subject of
transformation optics. Here media, possibly made of metamaterials, are designed
such that they appear to perform a coordinate transformation from physical space
to some virtual electromagnetic space. As we describe in this article, the concept
of transformation optics embraces some of the spectacular recent applications of
metamaterials.
Transformation optics is beginning to transform optics. We would do injustice
to this emerging field if we attempted to record every recent result. By the time
this article goes to press, it would be outdated already. Instead we focus on the “old
things in new things”, because those are the ones that are guaranteed to last and to
remain inspiring for a long time to come. This article rather is a primer, not a typical
literature review. We try to give an introduction into connections between geometry
and electromagnetism in media that is as consistent and elementary as possible,
without assuming much prior knowledge. We begin in §2 with a brief section on
Fermat’s principle and the concept of transformation optics. In §3 we develop in
detail the mathematical machinery of geometry. Although this is textbook material,
many readers will appreciate a (hopefully) readable introduction. We do not assume
any prior knowledge of differential geometry; readers familiar with this subject may
skim through most of §3. After having honed the mathematical tools, we apply them
to Maxwell’s electromagnetism in §4 where we develop the concept of transformation
optics. In §5 we discuss some examples of transformation media: perfect invisibility
devices, perfect lenses, the Aharonov-Bohm effect in moving media and analogues
of the event horizon. Let’s begin at the beginning, Fermat’s principle.
2 Fermat’s principle
In a letter dated January 1st, 1662, Pierre de Fermat formulated a physical principle
that was destined to shape geometrical optics, to give rise to Lagrangian and Hamil-
tonian dynamics and to inspire Schr¨odinger’s quantum mechanics and Feynman’s
form of quantum field theory and statistical mechanics. Fermat’s principle is the
principle of the shortest optical path: light rays passing between two spatial points
A and B chose the optically shortest path, see Fig. 3. In some cases, however, light
takes the longest path; in any case, light rays follow extremal optical paths, see Fig.
4. The optical path length s is defined in terms of the refractive index n as
s =
Z
n dl =
Z
B
A
n
p
dx
2
+ dy
2
+ dz
2
(2.1)
in Cartesian coordinates. If the refractive index varies in space — for non-uniform
media — the shortest optical path is not a straight line, but is curved. This bending
of light is the cause of many optical illusions. For example, picture a mirage in the
desert [35]. The tremulous air above the hot sand conjures up images of water in
the distance, but it would be foolish to follow these deceptions; they are not water,
but images of the sky. The hot air above the sand bends light from the sky, because
hot air is thin with low refractive index and so light prefers to propagate there.
5

Frequently asked questions

Q1. What are the contributions in "Transformation optics and the geometry of light" ?

In this paper, the relationship between optics and general relativity is discussed, expressed more precisely between geometrical ideas normally applied in general relativity and the propagation of light, or electromagnetic waves in general, in materials. 

Q2. what is the phase modulation in the optical Aharonov-Bohm effect?

Assuming that in the optical Aharonov-Bohm effect the phase modulation is proportional to the integral of ∫ u ·dr the authors guess a space-time coordinate transformation that should describe the wave propagation. 

Q3. What is the reason why the hot air above the sand bends light from the sky?

The hot air above the sand bends light from the sky, because hot air is thin with low refractive index and so light prefers to propagate there. 

Q4. how many indices does a tensor have?

It transforms as a tensor with four indices (3.37), because the left-hand side of Eq. (3.102) is a tensor with three indices and the right-hand side contains a contraction over a vector. 

Q5. What is the main idea that inspired the surge of interest in metamaterials?

the idea that inspired the surge of interest in metamaterials in the first place, the perfect lens [101], turned out to represent an example of transformation optics as well [74]. 

Q6. What is the reason these texts use the more complicated non-coordinate bases?

The reason these texts use the more complicated non-coordinate bases is that to exploit the simplicity of coordinate bases requires a little knowledge of tensor analysis. 

Q7. What is the significance of the coordinate transformation in general relativity?

In this theory, the coordinate transformation is physically significant, it describes completely the macroscopic electromagnetic properties of the material, and differential geometry is just as useful for these purposes as it is in general relativity. 

Q8. How did the authors compensate for the shortcomings and omissions?

The authors hope to have compensated for these shortcomings and omissions by being clear and pedagogical in the main ideasand by focusing on the “new things in old things” and explaining “old things in new things”, by telling the aspects of the story that the authors believe are already guaranteed to last and to remain inspiring for a long time. 

Q9. What is the ratio of the tunneled to the total spectral intensity?

Therefore the ratio N of the tunneled to the total spectral intensity isN = exp( −2πωα ) 1− exp ( −2πωα) = [exp(2πω α ) − 1 ]−1 . (5.60)According to the quantum field theory at horizons [10, 16, 68, 107], positive- and negative-frequency photon pairs are spontaneously created from the quantum vacuum, because they do not cost any energy. 

Q10. How can the authors compute the dot products of a general tensor?

For the spherical polar basis (3.29) the authors can compute the dot products using the right-hand sides of Eqs. (3.29) or, much more simply, by using the scalar product (3.38) and the metric (3.23) in spherical polar coordinates.