The optical properties of artificial media structured at a ...

The optical properties of artificial media structured at
a subwavelength scale
Philippe Lalanne 1 , Mike Hutley 2
Abstract ⎯ In most optical materials the atomic or molecular
structure is so fine that the propagation of light within them may
be characterized by their refractive indices. When an object has
structure which is larger than the wavelength of light, its
influence on the propagation of light may be described by the
laws of diffraction, refraction and reflection. Between these two
extremes is a region in which there is structure that is too fine to
give rise to diffraction but is too coarse for the medium to be
considered as homogenous. For this, a full description can only
be achieved through a rigorous solution of Maxwell's
electromagnetic equations and resonance phenomena are often
observed. Recent developments in micro-lithography have
extended the possibility of generating sub-wavelength structures
and it is now possible to produce materials with remarkable new
optical properties.
Index Terms ⎯ Artificial dielectrics, subwavelength structures,
photonic crystals, effective medium theory.
I. INTRODUCTION
The basic principles of optical design and
the physics of reflection, refraction and diffraction
on which it is based, have been well understood for
a very long time. This knowledge has enabled the
successful development of optical science and
technology over the last couple of centuries and in
recent years has been developed to new levels of
sophistication with the availability of computer
programs for the optimization of all types of system.
However, until relatively recently the whole of
optical technology has been limited by the very
reasonable constraint that optical systems can only
be designed to be made from materials that are
actually available! Consider the very simple task of
designing an antireflection coating to work at one
wavelength at normal incidence. The theory tells us
that a single layer will generate two reflected waves,
one from the air/layer interface and one from the
layer/substrate interface. If the optical thickness of
the layer is such that the two are exactly out of phase
and that the refractive index of the layer is equal to
the square root of that of the substrate, the two
reflections will have the same amplitude and will
cancel exactly. So a single layer will behave as a
perfect antireflection coating. Unfortunately most
common optical glasses have a refractive index in
the region of 1.55, so the layer needs to have a
refractive index of 1.245. Sadly such a material
does not, as far as we know, exist so more complex
solutions have to be found.
Consider, however, what happens if we
introduce into a standard material a very fine
structure, such as for example a series of holes. If
the scale of the structure is substantially smaller
than the wavelength of light, it will not be resolved
by the light and the light "sees" a composite
material of which the optical properties are between
those of air and those of the base material. By
varying the fraction of material that is removed, it is
possible to control the effective refractive index and
add to the range of materials that are available to the
optical designer. This principle can be expanded to
include composites consisting of several
components. Strictly speaking all structures are
three dimensional, but in practice many examples
consist of relatively shallow modulations of an
optical surface. It is therefore customary to refer to
one, two and three dimensional structures when
describing for example a simple diffraction grating,
a crossed diffraction grating and a "photonic
crystal" respectively.
1 Laboratoire Charles Fabry de l’Institut d’Optique, BP 147, 91403 Orsay Cedex. Can be joined at
philippe.lalanne@iota.u-psud.fr
2 Floating Images Ltd., Hampton, TW12 3JU, United Kingdom
1

In the limit where the wavelength of light is
very much greater than the dimensions of the
structure it is possible to regard the material as
being homogeneous and possessing an appropriate
effective value of refractive index. When the
dimensions of the structure are close to or larger
than the wavelength of light the optical properties
are dominated by the effects of diffraction. However,
there is a region between these two extremes where
the dimensions are sufficiently small that no
diffracted orders propagate, but where it is not
possible to apply the simple approximations of an
homogeneous medium. This is often referred to as
the "subwavelength domain" for which
homogeneization techniques do not strictly apply
but give however a good physical understanding of
the medium properties. Developments in
microlithography and associated technologies now
make it possible to put these principles into practice
and in particular to produce "artificial media" which
operate in the resonance domain. As a result, the
subject of subwavelength structures now attracts a
great deal of research interest with a view to
extending the possibilities of waveguides, optical
fibres and electro-optical materials.
In the present article we shall review recent
developments in the understanding and technology
of sub-wavelength structures and artificial optical
media. We shall start by considering the underlying
physics and then describe a selection of real
examples by way of illustration.
II. LONG-WAVELENGTH LIMIT OF ARTIFICIAL MEDIA
The propagation of electromagnetic waves
in composite media with subwavelength
inhomogeneities is an old but still very active
subject. In principle, the inhomogeneity could be
randomly or regularly distributed. However, if it is
random, there will in fact be a full spectrum of
spatial frequencies present. Furthermore, if the
distribution of spatial frequencies is such that there
is a significant proportion at wavelengths which are
similar to that of the light, then the medium will
scatter. It is therefore preferable to produce
structures which are regular and periodic. In this
way it is possible to control the spatial frequencies
that are present and avoid random scattering.
Moreover the theory of composite materials is made
easier for periodic structures because a
reciprocal-lattice analogous to that found for Bloch
electron wave in crystals can be introduced to
drastically simplify the analysis.
Initially, various effective medium
approaches like the Maxwell-Garnett or
Clausius-Mossotti approximations 1 were used to
determine the dielectric constant of periodic
composite materials, like those of Fig 1. It was later
realized that those approaches which rely on a
spatial average but which ignore the fine geometry
of the inhomogeneity were inadequate even in the
long-wavelength limit, i.e. when the period is
infinitely smaller than the wavelength.Initially,
various effective medium approaches like the
Maxwell-Garnett or Clausius-Mosotti
approximations were used to determine the
dielectric constant of subwavelength composite
materials. It was later realized that these approaches
that rely on a spatial average but that ignore the fine
geometry of the heterogeneity were inadequate even
in the long-wavelength limit, i.e. when the period is
infinitely smaller than the wavelength.
z
y
x
1D 2D 3D
Fig. 1. Examples of 1D, 2D and 3D periodic structures. Black and white
regions might correspond to high and low refractive-index materials for
instance. For periods sufficiently small compared to the wavelength of the
illumination beam, these structured periodic structures may behave as
artificial homogeneous materials whose optical properties (refractive index,
birefringence, dispersion) are related to the fine geometry of the periodic
arrangement. We may talk of refractive index engineering by structuring
materials at a subwavelength scale.
The long-wavelength limit is a very
important case for which some analytical general
results are available from theories known as
"Effective Medium theory", "homogenization" or
"Mean-Field theory" in the literature. The first
important result is the genuine equivalence between
periodic artificial media and homogeneous material.
For instance, 1D periodic structures are equivalent
to uniaxial crystals with form birefringence 2 and 2D
or 3D periodic structures are in general equivalent
to biaxial crystals. This equivalence is far from
being trivial from a mathematical point of view, see
2

[3] for instance. For 1D periodic structures, the
ordinary no and extraordinary ne indices of
refraction take the simple forms :
no = 1/2 and ne = -1/2 , (1)
where ε denotes the relative permittivity of the
periodic structure and the brackets refer to spatial
averaging. For the lamellar two-medium structure
of Fig. 1, Eq. 1 becomes
no = [fεH+(1-f)εL] 1/2 and ne = [f/εH+(1-f)/εL] -1/2 . (2)
In Eq. (2), the fill factor f represents the
fraction of high-index material with relative
permittivity εH imbedded in the low-index material
with relative permittivity εL. Equation 2 can be
obtained by elementary considerations 2 . With the
exception of y-polarized waves propagating in the
xz-plane of 2D periodic structures for which the
effective index is equal to 1/2 , no closed-form
expressions for the effective indices are available
for 2D or 3D periodic structures. Note that, for 2D
periodic structures, simple expressions 4 are known
for the upper and lower bounds of the two principal
effective indices experienced by waves propagating
in the y-direction. When the index contrast is small,
these bounds are generally quite narrow and their
average is a good approximation for the effective
index.
III. REAL ARTIFICIAL MEDIA
The long-wavelength limit is an academic
case. Even with the more advanced nanofabrication
facilities, it is not possible to manufacture optical
periodic structures that operate in the
long-wavelength limit. With the present
state-of-the-art in nanofabrication, it is only
possible to manufacture structures with periods
slightly smaller than optical wavelengths. For these
real artificial media, one does not have in hand a
theorem of equivalence between periodic structures
and homogeneous media. On the contrary, the
physical properties of real composite materials may
sometimes strongly differ from those of
homogeneous media.
Two different approaches may be
distinguished. In the first approach, one looks for
closed-form expressions for the effective index by
expanding the effective index neff in a power series
of the period-to-wavelength ratio Λ/λ
n eff = n (0) + n (2) (Λ/λ) 2 + n (4) (Λ/λ) 4 + ... (3)
where n (0) represents the effective index in the
long-wavelength limit and n (2) and n (4) are
dimensionless coefficients depending on the
microgeometry. Since the pioneer work by Rytov 5 ,
closed-form expressions up to the order four are
available for 1D periodic two-layered media like
that of Fig. 1. The homogenization of
two-component layered media is simplified by the
fact that the permittivity is piecewise-constant and
thus that the modes supported by the structure are
analytically known. Closed-form expressions for
the effective index up to the second orders (and to
the fourth order for some specific directions and
polarizations) are now available 6,7 for arbitrary 1D
periodic structures. From these expressions, it is
concluded that the normal surface of ordinary
waves is still a sphere up to the second order
(deviation being observed in the fourth-order term
only) and that the normal surface of extraordinary
waves is no longer an ellipsoid of revolution up to
the second order, see [7] for the general case and
[5,8] for the two-component layered media of Fig. 1.
For 2D periodic structures, almost no results are
available: closed-form expressions (requiring
however a matrix inversion) up to the second order
were derived in [ 9 ] for the effective indices
experienced by waves propagating in the
y-direction.
In the second approach, one resorts to
computation. Because Maxwell's equations for
linear dielectric materials are exact, computation
plays a crucial role in the analysis and design of
periodic artificial media. In principle, any classical
numerical method in electromagnetism may be used
to compute the effective index; for instance,
boundary-matching methods 10 appear quite
attractive for special shapes like spheres which do
not overlap. In recent years, specific attention has
been devoted 11,12 to Fourier expansion techniques
which can be used to study any periodic
microgeometry and are therefore much wider in
scope than the previous approach. Fourier
expansion techniques are not exempt from
3

numerical difficulties 13,14 due to the problems of
accurately representing permittivity discontinuities
with Fourier series. The Fourier plane-wave method
of Ref. [ 15 ] that incorporates recent Fourier
Factorization theorems 16 on the product of
discontinuous functions appears particularly
efficient.
IV. SUBSTITUTION OF A SUB-WAVELENGTH GRATING
BY AN ARTIFICIAL MEDIUM
Let us consider the diffraction problem
shown in the left side of Fig. 2, where a
subwavelength grating of depth h is illuminated at
oblique incidence by a linearly polarized plane
wave with a free-space wavelength λ.
h
h
Λ
λ
λ
θ
θ
y
! ?
y
n 2
Fig. 2. Replacement of a grating whose permittivity is independent of the
y-direction by a homogeneous artificial thin film with an effective index n eff. Is
it legitimate?
The refractive index of the incident medium
is n1 and that of the substrate is n2. For the sake of
simplicity, we restrict the following discussion to
1D gratings illuminated under TE polarization (the
electric-field vector of the incident plane wave is
perpendicular to the plane of incidence), but the
general conclusions we will derive basically holds
for any structure including volume or crossed
gratings, for arbitrary polarization and for arbitrary
incidence (which may be out of the plane of
dispersion) as long as the grating permittivity is
independent of y. In this Section, we seek to answer
n 1
x
n L n H a)
n 2
n 1
x
n eff
b)
the following question: "under what conditions may
the complex diffraction problem of Fig. 2a be
approximated to by a simple refraction-reflection
problem on a homogeneous thin film with an
effective refractive index neff ?" Clearly, the
expected value for neff is that seen by a wave
propagating in the xy-plane of the 1D periodic
structure of Fig. 1 with a polarization along the
z-direction and with an oblique angle θ’ with
respect to the y-direction verifying the Snell’s law
n1 sin(θ) = neff sin(θ’). (4)
Note that for the general case, since the
homogeneous thin film is possibly anisotropic, two
effective indices experienced by the extraordinary
and ordinary waves may be defined for the
equivalent thin film. It is important to bear in mind
that the replacement of Fig. 2 can only ever be an
approximation because in no real situations are the
two problems of Fig. 2 strictly equivalent. See
[17-18] where the mathematical equivalence is
demonstrated in the long-wavelength limit
(Λ/λ→ 0).
The answer to the question is far from being
trivial, and many authors have contributed to
partially answer the question. For 1D lamellar
gratings with high modulation contrasts see
[ 19 ,9, 20 ], for slanted volume gratings with a
sinusoidal modulation, a situation for which the
grating permittivity depends on y, see [21,7], for 1D
gratings under conical mounts see [ 22], for 2D
gratings, see [9,23,24]. In general, three conditions
are necessary so that the two problems of Fig. 2 are
approximately equivalent.
First condition. The first condition is that only the
zeroth orders propagate in the substrate and in the
incident medium, all the other orders have to be
evanescent. Whether a diffraction order propagates
or not is given by the grating equation. If only the
zeroth transmitted and reflected orders are to
propagate, it is immediately deduced from the
grating equation that the period-to-wavelength ratio
must verify
Λ
1
<
λ max(n1,
n 2 ) + n1
sin( θ)
, (5)
where max holds for the maximum of the arguments.
4

The condition of Eq. 5 that provides a cutoff value
solely dependent on n 1 sin(θ) and on the refractive
indices of the incident medium and of the substrate
does not depend on the grating geometry. It is
necessary but not sufficient.
Second condition. The second condition is more
subtle and is related to the number of
non-evanescent modes that are able to propagate in
the xy-plane of the 1D periodic structure of Fig. 1
with a polarization along the z-direction and with an
oblique angle θ’. When only one mode propagates
in the periodic structure with a speed c/neff (all the
other mode are evanescent), this mode travels
backward and forward between the two grating
boundaries in the same way as multiple beam
interference occurs in a thin film with a refractive
index neff. Consequently, the zeroth-order reflected
and transmitted amplitudes are approximately those
of the thin film. If more than one mode propagates
in the grating, this picture is no longer valid, see the
discussion in [ 25 ]. The fact that for a given
frequency only one mode propagates in the grating
depends mainly on the grating geometry and weakly
on n1sin(θ), a quantity that determines the direction
of propagation in the grating (see Eq. 4). Figure 3
shows the domain for which only one mode
propagates into the lamellar grating of Fig. 2a for
normal incidence and for nL = 1 and nH = 2.3. Λs
represents the largest grating period below which
one can achieve a full range of effective indices
between nL and nH simply by varying the fill factor
while preserving the analogy with an artificial thin
film. It is called the structural cutoff in [25-26] to
emphasize that it only depends on the grating
structure and not on n1 and n2. This is an important
factor in practice because the larger it is, the more
easily can the grating be manufactured.
Third condition. The last condition is related to the
grating depth. As pointed out in [20, 27 ], the
one-mode picture above mentioned is no longer
valid for small grating depths. If the grating depth is
small enough, the evanescent modes that are created
at the top and bottom grating interfaces may tunnel
through the grating region and participate with the
fundamental to the multiple beam interference. For
dielectric gratings, the impact of evanescent modes
on the grating effective properties is significant for
grating depths smaller than a quarter wave 27 .
The set of the three above conditions is
rather complex since basically all the
diffraction-problem parameters are involved in the
validity of the replacement. The refractive indices
of the substrate and of the incident medium from
Eq. 3, the grating geometry from the second
condition, and the grating depth from third
condition. However, in general, if the three above
conditions are fulfilled in practice, the grating
diffraction problem of Fig. 2a can be seen, with a
good approximation, as a simple
refraction-reflection problem on a homogeneous
thin film. This drastic reduction in terms of
complexity has suggested in the past and recently
interesting applications 28 for artificial media
synthesized by etching surfaces at a subwavelength
scale. We note for example, applications exploiting
the form birefringence of binary gratings for
fabricating wave plates 29,30 or wire-grid polarizers,
and for characterizing subwavelength lamellar
gratings 31 , applications relying on continuous
grating profiles which mimic gradient-index
perpendicular to the substrate mainly for broadband
antireflection coating, and applications relying on
binary profiles which mimic gradient-index parallel
to the substrate mainly for fabricating blazed
diffractive elements. We shall now describe some of
these in a little more detail and the interested reader
will find a collection of articles on these subjects in
Ref. [32].
1/n L
Λ s /λ
Λ/λ
0 0.2 0.4 0.6 0.8 1
Fill factor
Fig. 3. Domain of grating parameters (fill factor and period) for which only
one mode propagates in the lamellar grating of Fig. 2a. The computation is
performed for TE polarization, for normal incidence and for n L = 1 and
n H = 2.3. The fill factor is defined as the fraction of high refractive index
material. The horizontal dashed line at abscissa Λs represents the largest
grating period below which one can achieve a full range of effective indices
from n L to n H simply by varying the fill factor from 0 to 1, while preserving a
good analogy with artificial thin films.
5

V. THE WIRE-GRID POLARIZER
The wire-grid polarizer was probably the
earliest device to exploit the form birefringence of
subwavelength metallic gratings as it was used by
Hertz to test the properties of the newly discovered
radio wave in the late 19 th century. It consists of a
fine grid of parallel metal wires with a spacing
which is less than that of the wavelength of light.
The oscillating electric field of the incident light
tends to induce electric dipoles at the surface and
the response of the material to this field determines
its optical properties. In a metal, there are electrons
which are free to vibrate under the influence of the
field, electric dipoles are induced which then
re-radiate in such a way that the light is reflected. In
a dielectric there are no free electrons and the light
is transmitted. In a wire-grid polarizer the electrons
are free to oscillate along the wires but not
perpendicular to them because the wire width is
smaller than the wavelength. Therefore for light
polarized parallel to the wires (TE polarization
case), it behaves as a conductor and reflects,
whereas for light of the orthogonal polarization it
behaves as a dielectric and transmits. This result
may simply be derived by inserting into equation 2 a
large negative value for the relative permittivity εH
of the metal while assuming f ≈ 0.5 and εH = 1. For
wires of high conductivity the transmission is
significantly greater than the proportion of the area
occupied by the gaps between the wires.
There are various ways of producing
wire-grid polarizers. For the far infrared it is
possible to wind a wire round a suitable former and
produce a free-standing grid. For shorter
wavelengths in the near infrared, it is necessary to
form them on a transparent substrate such as zinc
selenide or KRS5. In this case it is possible either to
rule a diffraction grating directly into the surface of
the substrate or to record in photoresist a fine,
straight line interference pattern. Either way this
provides a corrugated surface which may be
metallized by vacuum deposition at an oblique
angle in such a way that only the tips of the
corrugations are coated and the net result is a series
of very fine parallel wires 33 . It is also possible to
apply the techniques of photolithography combined
with laser or electron beam writing to generate a
pattern of metallic strips. In general, the greater the
wavelength-to-period ratio, the better the wire-grid
polarizer performs. For most applications a ratio of
ten to one would be desirable. As the resolution of
lithographic processes improves, the smaller it is
possible to make the period of the grid and the
shorter the wavelength at which it will function as a
polarizer. They are now operating in the
visible 34,35,36 and even in the ultraviolet 37 . However,
the performance is degraded by the low
conductivity of metals in the visible and the
ultraviolet regions of the spectrum.
VI. ANTI-REFLECTION COATING
The feature of an optical surface which gives
rise to unwanted Fresnel reflections is the sudden
transition from one optical medium to another. If the
transition can be made more gradual and extended
over at least a significant fraction of a wavelength,
the reflection can be significantly reduced. This has
been achieved in glass by treating the surface with
acid. Under the appropriate circumstances, material
may be leached out of the glass to leave a more open
structure in the region of the surface and a gradually
more dense structure as one penetrates into the glass.
As this occurs at a molecular level it is on a scale
much finer than the wavelength of light and is
equivalent to a gradual change of refractive index
and the reflection of the surface is reduced to a level
as low as that achieved with very complex
multilayer antireflection coatings.
An analogue of the leached glass
antireflection surface is to be found on the eye of the
night-flying moth. The cornea is covered with a fine
regular hexagonal array of protuberances which
have a period of about 200 nm and a similar depth
and a cross section that is approximately sinusoidal.
This was discovered by Bernhard in 1967 who
postulated 38 that this was had the effect of a gradual
change of refractive index which reduced the
reflection over a wide spectral and angular
bandwidth and significantly improved the moth's
camouflage. The effect was first copied by Hutley
and Clapham 39 by recording finely spaced optical
interference fringes in two orthogonal directions in
photoresist. This generated an "egg box" periodic
modulation in the surface. The antireflection
properties of such a surface match those of
6

multilayer interference coatings over a spectral
region which covers the visible and extends much
further into the infrared.
Although photoresist is a convenient
material in which to make the motheye structure, it
is very soft and highly absorbant in the blue so it is
unsuitable for most practical applications. However,
the structure may be copied by electroforming in
nickel. Such a copy appears very dark blue, almost
black, because almost all of the light is absorbed.
From the nickel copy it is possible to copy the form
of the surface into very highly polymerised plastic
through the process of UV embossing and into
slightly less hard plastics by hot pressing 40 .
Recently, Gombert 41 has developed the technology
to the stage where large areas (a substantial fraction
of one square-meter) are available and his team is
finding commercial applications such as on the
reverse side of Fresnel lenses in overhead
projectors.
The problem of surface reflection is
particularly acute for semiconductors which have
very high values of refractive index in the visible
and near infrared spectral regions. The reflection
losses at normal incidence are ≈40%; to reduce
them would be of interest for solar-cell applications
and other optoelectronic devices. Binary pillar
structures (with antireflection performance
comparable to those achieved by a single-layer
coating) have been investigated 42 for the 10-25 µm
region of the spectrum by conventional lithography
techniques and reflectivities as low as 0.5% have
been experimentally demonstrated 28 for the
10.6-µm line of the CO2 laser. For visible
antireflection coating, motheye-type structures with
a 260-nm period have been fabricated 43 on silicon
substrates with overall performance comparable to
two-layer coatings. More recently, using
electron-beam lithography, high aspect-ratio
pyramid profiles similar to optimal ones predicted
by electromagnetic analysis 44 have also been
produced 45 , see Fig. 4.
Quite apart from their optical performance,
structures of this type offer the additional
advantages that there are no problems with adhesion
or with diffusion of one material into another. Since
they are monolithic and introduce no foreign
material, they also tend to be more stable and
durable than multilayer dielectric materials
particularly when used with high-powered lasers 46 .
On the other hand they are generally more sensitive
to mechanical damage.
Fig. 4. Scanning-electron micrograph of an antireflection surface generated
with an electron-beam writer. The surface is composed of a square grid of
350-nm-high pyramids etched into silicon. The grating period is 150 nm.
Under illumination at normal incidence, the reflectivity of the antireflection
surface does not exceed 4% over a broad spectral domain from 200 nm to 1.1
µm. For comparison, the reflectivity of bare silicon substrate is above 40% in
this spectral domain. More details can be found in [45]. (Courtesy Yoshiaki
Kanamori, University of Tohoku)
VII. BLAZED GRATINGS
In order to maximize the efficiency with
which a grating or other diffracting component
directs light into a chosen order of diffraction, it is
necessary that it be "blazed". That is to say that the
form of the groove profile is that of a prism which
refracts light into the same direction as that of the
chosen order of diffraction. This may be explained
by considering a grating working in the first
diffracted order so there is an optical path difference
of one wavelength for light passing through
adjacent grooves. As we consider different parts of
the wavefront, the geometry of the diffraction
process introduces a variation of phase of 2π from
one side of a groove to the other. In order to achieve
maximum amplitude in the diffracted wave, it is
necessary to keep the phase constant, so that all
components interfere constructively, and then
introduce a sudden change of 2π at the boundaries
of the grooves. To do so, one has to introduce
across the groove a means of varying the phase to
compensate for the change of path length. A
triangular prism produces such a variation of
optical path across the grating period by varying the
relative proportion of the path in air and in the
7

material of the prism. Gratings with a triangular
profile are often referred to as échellettes and have
been the standard way of maximizing the efficiency
of spectroscopic gratings for the last 90 years.
Unfortunately, for gratings of high
dispersion (and for diffractive optical components
of high numerical aperture) the angles of diffraction
and hence the prism angles become quite high. It
then happens that each groove casts a shadow on its
neighbor, the groove is vignetted and energy is lost.
However, the same prism effect can be
achieved with a material of constant thickness but in
which there is a progressive variation of refractive
index. If such a structure were fabricated, vignetting
would still occur (the optical paths within the
grooves would be curved, as in a mirage). Such a
variation of effective index can also be achieved
with an artificial optical medium composed of
binary structures consisting of a series of
microscopic pillars of varying dimension, provided
that the inter-pillar distances are small enough
compared to the wavelength of light. For these
structures, if the inter-pillar distances are not to
small (let us say about the structural cutoff for the
considered wavelength) practically no vignetting is
observed 26 .
In fact, each pillar acts as a waveguide 47
which conducts the light from one side of the
grating to the other without shadowing, but
introduces a phase lag which depends upon the
pillar dimensions. At the exit surface of the grating
there is effectively an array of coherent phased
emitters. Binary blazed gratings of this form have
been made in small areas using electron beam
writing and have demonstrated efficiencies that are
superior to those of the equivalent conventional
blazed gratings. Gratings have been made with
periods as small as 0.99 µm blazed in transmission
for red light, which in a conventional transmission
grating would require facet angles in the region of
40 o . Not only have efficiencies in excess of 80%
been measured for both polarizations, but this has
been maintained over a much wider range of angles
of incidence than would have been possible with a
conventional grating 48 . Figure 5 shows an electron
micrograph of a typical structure.
Fig. 5. Scanning-electron micrograph located not far from a corner of a 20°
off-axis diffractive lens designed for operation at 860 nm. The lens has a
square aperture of 200 µm x 200 µm and a focal of 400 µm. The Fresnel-zone
widths are in between 1.7 µm and 9 µm. The pillars etched into a TiO 2 thin
film deposited on a glass substrate are 990-nm high and are located on a
square grid with a 405-nm period. The corresponding aspect ratio for the
thinner pillars is approximately 10. Tested with a
vertical-cavity-surface-emitting-laser emitting a nearly-gaussian beam, a 80%
first-order efficiency was measured. More details can be found in [26].
VIII. PHOTONIC CRYSTALS
In the practical examples considered so far,
waves were propagating in 1D or 2D structures
parallel or nearly parallel to one of the
translation-invariant directions of the artificial
media. Very interesting optical properties are also
obtained for waves propagating perpendicularly to
that direction. Consider for example propagation
along the x-direction in the 1D structure of Fig. 1.
The light sees a series of layers with alternating
high- and low-refractive indices. Multiple
reflections and refractions occur at the interfaces
and along with interference, light may be reflected
back for wavelengths approximately equal to twice
the period. The width of the reflectance band is
defined by the wavelengths between which the
reflectance increases as layers are added. In general,
low absorption and high reflectivity are obtained.
An other way to understand the reflectance band is
to consider that the medium, supposed infinite in the
x-direction, prevents the propagation of light in a
certain range of wavelengths. For these wavelengths,
the effective index computed with the above
mentioned methods is complex. Its imaginary part
does not imply any heat dissipation because the
alternate layers are made of transparent materials; it
just signifies that waves cannot propagate, and its
value is simply inversely proportional to the
penetration depth (see [49] for 1D structures and [15]
8

for 2D and 3D structures). The range of
wavelengths for which propagation is forbidden is
called the band gap 49 by analogy with the energy
gap experienced by electrons submitted to a
periodic potential in semiconductors. As the
propagation direction deviates progressively from
the normal to the layers, the gap shift in frequency
and becomes more and more narrow. By increasing
the degree of symmetry, a 2D bandgap can open for
all propagation directions in the xz-plane for the 2D
structure of Fig. 1. Ultimately, structures with 3D
periodicity may offer a full omnidirectional bangap:
in a given frequency range, no wave can propagate
in all directions for any polarization 50 . The potential
of 3D photonic crystal to fully control the
spontaneous emission of atoms and to influence
optoelectronic emitting devices, was first realized
by Yablonovitch 51 in 1987. It took some years for
the scientific community to react, but the topic is
now the subject of intense research activity.
Photonic crystal with 3D periodicity are
difficult to manufacture especially in the visible or
near infrared. Fortunately 2D periodic structures
which are easier to manufacture already exhibit
some useful properties. For instance, if a
single-point defect in the periodicity is created by
removing one cylinder in the 2D structure of Fig. 1,
a wave confined in the air gap by the
multidirectional Bragg reflector constituted by the
surrounding pillars, is able to propagate along the
y-direction without attenuation. In this case, the
light is actually confined in the low-refractive-index
region. This is the contrary of the standard
waveguide for which guidance is achieved by
total-internal reflection in the high-index material.
New generation of fibers working on that principle
have been actually fabricated by almost standard
preform and stretching techniques 52 , see Fig. 6.
Since the light sees essentially no material
(vacuum), transmissions of waves at power levels
not possible in conventional waveguides appear
feasible without damage, leading to greatly
increased threshold powers for nonlinear optics.
Such fibres are also able to conduct light round
relatively sharp bends without significant radiation
losses. This is due to the fact that the photonic
crystal structure surrounding the air core supports
only evanescent waves and therefore prevents the
loss of energy in radiative modes. This property can
also be exploited for compact optoelectronic
integrated circuits based on photonic crystal
waveguides supporting sharp turns without loss 50,53 .
Fig. 6. Scanning-electron micrograph of an air-core photonic-crystal fibre.
More details can be found in [52]. (Courtesy of P S-J Russell, University of
Bath)
IX. CONCLUSION
We have seen that in many cases it is
possible by controlling the structure of a medium on
a scale smaller than that of the wavelength of light,
to control its optical properties. By producing such
an artificial medium it is possible to extend the
range of possibilities that are available to the optical
designer. In some cases it is possible to consider a
material with subwavelength structure as
homogeneous and to characterize its performance
by an effective value of refractive index. In other
cases the situation is more complex and can only
satisfactorily be described by a rigorous solution of
Maxwell's equations.
We have seen that the principle of artificial
media is well established and have considered a
selection of practical examples. However, the
subject is currently receiving a considerable amount
of research and development interest because the
technology of microfabrication is advancing very
rapidly. It is becoming increasingly practical to
generate in a complete range of materials, arrays of
pillars, ridges and holes that are deep, narrow and
smooth. It therefore seems realistic to expect that in
the near future we shall see a range of high
performance optical and electro-optical devices
based on optical materials that have been tailored
specifically for their purpose.
9

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11