Deterministic aperiodic nanostructures for photonics and plasmonics applications

Abstract

This review focuses on the optical properties and device applications of deterministic aperiodic media generated by mathematical rules with spectral features that interpolate in a tunable fashion between periodic crystals and disordered random media. These structures are called Deterministic Aperiodic Nano Structures (DANS) and can be implemented in different materials (linear and nonlinear) and physical systems as diverse as dielectric multilayers, optical gratings, photonic waveguides and nanoparticle arrays. Among their distinctive optical properties are the formation of multi-fractal bandgaps and characteristic optical resonances, called critical modes, with unusual localization, scaling and transport properties. The goal of the paper is to provide a detailed review of the conceptual foundation and the physical mechanisms governing the complex optical response of DANS in relation to the engineering of novel devices and functionalities. The discussion will mostly focus on passive and active planar structures with enhanced light-matter coupling for photonics and plasmonics technologies.

Full text

LASER & PHOTONICS
REVIEWS Laser Photonics Rev. 6, No. 2, 178–218 (2012) / DOI 10.1002/lpor.201000046
Abstract
This review focuses on the optical properties and de-
vice applications of deterministic aperiodic media generated
by mathematical rules with spectral features that interpolate
in a tunable fashion between periodic crystals and disordered
random media. These structures are called Deterministic Ape-
riodic Nano Structures (DANS) and can be implemented in
different materials (linear and nonlinear) and physical systems
as diverse as dielectric multilayers, optical gratings, photonic
waveguides and nanoparticle arrays. Among their distinctive
optical properties are the formation of multi-fractal bandgaps
and characteristic optical resonances, called critical modes, with
unusual localization, scaling and transport properties. The goal
of the paper is to provide a detailed review of the conceptual
foundation and the physical mechanisms governing the complex
optical response of DANS in relation to the engineering of novel
devices and functionalities. The discussion will mostly focus on
passive and active planar structures with enhanced light-matter
coupling for photonics and plasmonics technologies.
Deterministic aperiodic nanostructures for photonics and
plasmonics applications
Luca Dal Negro*and Svetlana V. Boriskina
Introduction
Understanding optical interactions in aperiodic determinis-
tic arrays of resonant nanostructures offers an almost unex-
plored potential for the manipulation of localized electro-
magnetic fields and light scattering phenomena on planar
optical chips.
Periodic scattering media support extended Bloch eigen-
modes and feature continuous energy spectra corresponding
to allowed transmission bands. On the other hand, in the
absence of inelastic interactions, random media sustain ex-
ponentially localized eigenmodes with discrete (i. e., pure-
point) energy spectra characterized by isolated
δ
-peaks. A
substantial amount of work has been devoted in the past few
years to understand transport, localization and wave scatter-
ing phenomena in disordered random media
[1–5]
. These
activities have unveiled fascinating analogies between the
behavior of electronic and optical waves, such as disorder-
induced Anderson light localization [5,6], the photonic Hall
effect [7], optical magnetoresistance [8], universal conduc-
tance fluctuations of light waves [9], and optical negative
temperature coefficient resistance [10]. However, the tech-
nological and engineering appeal of multiple light scatter-
ing and disorder-induced phenomena in random systems,
such as the celebrated Anderson-light localization, are still
very limited. Random structures, while in fact providing
a convenient path to field localization, are irreproducible
and lack predictive models and specific engineering de-
sign rules for deterministic optimization. These difficul-
ties have strongly limited our ability to conceive, explore,
and manipulate optical resonances and photon transport in
systems devoid of spatial periodicity. On the other hand,
aperiodic optical media generated by deterministic mathe-
matical rules have recently attracted significant attention in
the optics and electronics communities due to their simplic-
ity of design, fabrication, and full compatibility with cur-
rent materials deposition and device technologies
[11–14]
.
Initial work, mostly confined to theoretical investigations
of one-dimensional (1D) aperiodic systems
[15–22]
,have
succeeded in stimulating broader experimental/theoretical
studies on photonic-plasmonic structures that fully leverage
on deterministic aperiodicity as a novel strategy to engi-
neer optical modes, devices, and functionalities. Beyond the
emerging technological implications, the study of aperiodic
structures in nanophotonics is a highly interdisciplinary and
fascinating research field whose conceptual underpinning
is deeply rooted in three highly interconnected research ar-
eas: mathematical crystallography
[23–25]
, dynamical sys-
Department of Electrical and Computing Engineering & Photonics Center, Boston University, 8 Saint Mary’s street, Room 825, Boston, MA 02215-2421,
USA
*e-mail: dalnegro@bu.edu
© 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim

Laser Photonics Rev. 6, No. 2 (2012)
REVIEW
ARTICLE
179
tems (i. e., specifically, symbolic dynamics and automatic
sequences) [26–28], and number theory [29–31].
The scope of this review is to discuss the foundation
along with the optical properties and the current device
applications of Deterministic Aperiodic Nano Structures
(DANS). DANS are optical structures in which the refrac-
tive index varies over length scales comparable or smaller
than the wavelength of light. They include dielectric and
metallic structures, metallo-dielectric nanostructures and
metamaterials. In all cases, DANS are designed by math-
ematical algorithms that interpolate in a tunable fashion
between periodicity and randomness, thus providing novel
opportunities to explore and manipulate light-matter interac-
tions at the nanoscale. DANS can be conveniently fabricated
using conventional nanolithographic techniques while dis-
playing unique transport and localization properties akin
to random systems. Differently from well-investigated frac-
tal structures, DANS may not possess self-similarity in di-
rect space although they exhibit a far richer complexity in
Fourier space resulting in distinctive diffraction patterns.
Most importantly, the Fourier space of DANS can be sim-
ply designed to range from a pure-point discrete spectrum,
such as for periodic and quasiperiodic crystals, to a dif-
fused spectrum with short-range correlations, as for disor-
dered amorphous systems. In addition, DANS diffraction
patterns can display non-crystallographic point symmetries
of arbitrary order as well as more abstract mathematical
symmetries [12].
The structural complexity of DANS profoundly influ-
ences the character of photon transport in the multiple scat-
tering regime, and results in a high density of discrete reso-
nances, known as critical modes, with multi-fractal spatial
patterns and large fluctuations of their photonic mode den-
sity (LDOS). As we will discuss, these are key attributes
to boost the frequency bandwidth and the strength of light-
matter coupling in photonic-plasmonic structures, offering
yet unexplored avenues to advance fundamental optical sci-
ences and device technology.
This review is organized as follows: Sect. 1 covers the
conceptual foundation and the different notions of aperi-
odic order, generation techniques and classification schemes
based on Fourier spectral properties. In Sect. 2 we review the
main achievements in the engineering of dielectric DANS
in one spatial dimension, such as photonic multilayers, and
their device applications. Section 3 focuses on the optical
properties of two-dimensional (2D) DANS in the linear and
nonlinear optical regime and surveys their primary device
applications. In Sect. 4 we introduce the emerging field of
Complex Aperiodic Nanoplasmonics (CAN) and its engi-
neering applications. Theoretical and computational meth-
ods for photonic and plasmonic DANS are briefly reviewed
in Sect. 5. The summary in Sect. 6 offers our thoughts on
the outlook for this technological platform based on the en-
gineering of aperiodic order. Finally, a word has to be added
about our list of cited papers. Given the enormous amount
of references in this diversified and fast-growing field, we
have chosen to cite only a few representative articles rather
than to attempt to be exhaustive. We apologize in advance
for those cases where our selection was defective.
1. Deterministic aperiodic order
1.1. Order with periodicity
Structural order is often exclusively associated with the
concept of spatial periodicity. Humans can easily recognize
periodic patterns even in the presence of substantial noise
or perturbations, and many of the beautiful regularity of
natural phenomena are manifestations of periodic systems.
In his book “An introduction to mathematics”, published in
1911, the English mathematician Alfred North Whitehead re-
marked that [32]: “The whole life of nature is dominated by
the existence of periodic events. . . The presupposition of pe-
riodicity is fundamental to our very conception of life”. The
rotation of the Earth, the yearly recurrence of seasons, the
phases of the moon, the cycles of our bodily life, are all fa-
miliar examples of periodic events. Periodic patterns repeat
a basic motif or building block in three-dimensional space.
In general, any vector function of position vector
r
satisfying
the condition
Φr R0Φr
describes spatially periodic
patterns because it is invariant under the set of translations
generated by the vector
R0
. As a result, we say that peri-
odic structures display a specific kind of long-range order
characterized by translational invariance symmetry along
certain spatial directions. The beautiful regularity of inor-
ganic crystals best exemplifies periodically ordered patterns
where a certain atomic configuration, known as the base,
repeats in space according to an underlying periodic lattice,
thus defining a crystal structure. In three-dimensional (3D)
space, crystal structures are mathematically described by
their 32 point-group symmetries
1
, which are combinations
of pure rotation, mirror, and roto-inversion operations fully
compatible with the translational symmetry of the 14 Bra-
vais lattices. The addition of translation operations defines
the crystallographic space-groups, which have been com-
pletely enumerated in 230 different types by Fedorov and
Schoenflies in 1890 [23–25]. One of the most fundamental
results of classical crystallography states that the combina-
tion of translation and rotation operations restricts the total
number of rotational symmetries to only the ones compati-
ble with the periodicity of the lattice. This important result
is known as the crystallographic restriction. We say that a
structure possesses an n-fold rotational symmetry if it is left
unchanged when rotated by an angle
2πn
, and the integer
n
is called the order of the rotational symmetry (or the order
of the symmetry axis). It can be shown that only rotational
symmetries of order
n234
and 6 can match the transla-
tional symmetry of 2D and 3D periodic lattices in Euclidean
space
[23–25]
. As a result, the pentagonal symmetry, which
is often encountered in the world of living structures as the
1
A point group is a group of geometric symmetries (i. e., isome-
tries) leaving a point fixed. Point groups can exist in a Euclidean
space of any dimension. The discrete point groups in two dimen-
sions, also called rosette groups, are used to describe the sym-
metries of an ornament. There are infinitely many discrete point
groups in each number of dimensions. However, the crystallo-
graphic restriction theorem demonstrates that only a finite number
are compatible with translational symmetry.
www.lpr-journal.org © 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim

180
LASER & PHOTONICS
REVIEWS
L. Dal Negro and S. V. Boriskina: Deterministic aperiodic nanostructures for photonics and plasmonics applications
Figure 1
(a) Periodic square lattice. (b) Reciprocal space of the
square lattice calculated via its Fourier transform amplitude. (c) Pe-
riodic hexagonal lattice. (d) Reciprocal space of the hexagonal
lattice calculated via its Fourier transform amplitude.
pentamerism of viruses, micro-organisms such as radiolar-
ians, plants, and marine animals (i. e., sea stars, urchins,
crinoids, etc.) was traditionally excluded from the mineral
kingdom until the revolutionary discovery of quasiperiodic
order.
A fundamental feature of the diffraction patterns of all
types of periodic lattices is the presence of well defined and
sharp (i. e.,
δ
-like) peaks corresponding to the presence of
periodic long-range order. As a result, the reciprocal Fourier
space of periodic and multi-periodic lattices is discrete (i. e.,
pure-point), with Bragg peaks positioned at rational mul-
tiples of primitive reciprocal vectors. As an example, we
show in Fig. 1 the square and hexagonal lattices (Fig. 1a,c)
with their corresponding diffraction spectra (Fig. 1b,d), or
reciprocal spaces, obtained by calculating the amplitudes of
the lattice Fourier transforms (i. e., the Fraunhofer regime).
Bright diffraction spots arranged in patterns with square
and hexagonal symmetries are clearly visible in Figs. 1(b,d).
Their intensities progressively decrease away from the cen-
ters of the diffraction diagrams due to the contributions of
the circular shape of finite-size particles, which filter the
diffraction pattern according to the envelope of a Bessel
function, and the size of the entire arrays, which determines
the shape/size of individual diffraction spots.
This picture best exemplifies the notion of periodic ar-
rangement of atoms which is at the origin of the traditional
classification scheme of materials into the two broad cate-
gories of crystalline and amorphous structures. According
to this simple classification scheme, long-range structural
order and periodicity are considered identical, leading to
the widespread conception of crystalline materials as the
paradigm of order in solid state electronics and optics (e. g.,
photonic crystals). We will see in the next section that this
simple picture proved to be inadequate after the discov-
ery that certain metallic structures exhibit long-range ori-
entational order without translational symmetry, forcing a
complete redefinition of our notion of crystalline materials.
1.2. Order without periodicity: quasicrystals
One of the great intellectual triumphs of the twentieth cen-
tury is the discovery of aperiodic order in the mathematical
and physical sciences. In a series of lectures on the applica-
bility of physical methods in biology, the physicist Erwin
Schr
¨
odinger envisioned an aperiodic crystal storing genetic
information in the geometric configuration of its covalent
bonds. Schr
¨
odinger noticed that information storage could
be more efficiently achieved [33] “. . . without the dull de-
vice of repetition. That is the case of the more and more
complicated organic molecule in which every atom, and
every groups of atoms, plays an individual role, not entirely
equivalent to that of every others (as it is the case in a
periodic structure). We might quite properly call that an
aperiodic crystal or solid and express our hypothesis by say-
ing: we believe a gene, of perhaps the whole chromosome
fiber, to be an aperiodic solid”.
In mathematics, quasiperiodic order was originally in-
vestigated by Harald Bohr who developed in 1933 a general
theory of almost periodic functions, including quasiperiodic
functions strictly as a subset [12, 23]. However, it was the
mathematical study of symmetry, planar tilings, and discrete
point sets (i. e., Delone sets) that paved the way to the dis-
covery of aperiodic order in geometry, leading to the first
application of quasiperiodic functions.
The study of tilings and the associated point sets, which
is among the oldest branches of geometry, has only recently
been formalized using the advanced group-theoretic meth-
ods of modern mathematical crystallography and provides
the most general framework to understand quasiperiodic and
aperiodic structures [23,24, 34]. Tilings or tessellations are a
collection of plane figures (i. e., tiles) that fill the plane with-
out leaving any empty space. Early attempts to tile planar
regions of finite size using a combinations of pentagonal and
decagonal tiles were already explored by Johannes Kepler,
arguably the founder of the mathematical theory of tilings,
in his book Harmonices Mundi published in 1619 [35].
However, it was not until 1974 when the mathemati-
cian Roger Penrose discovered the existence of two simple
polygonal shapes capable of tiling the infinite Euclidean
plane without spatial periodicity [36]. Three dimensional
generalizations of Penrose tilings were demonstrated in
1976 by Robert Ammann, who produced a pair of stretched
and squashed building blocks (i. e., Ammann rhombohedra)
filling the 3D space aperiodically [37].
It was realized in the early 1980s that the diffraction
patterns of aperiodic point sets consist of sharp diffraction
peaks with icosahedral point-group symmetry, which in-
cludes the “forbidden” pentagonal symmetry. In Fig.2a we
show a particle array displaying ten-fold rotational symme-
try in the arrangement of its interior Bragg peaks. This array
© 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim www.lpr-journal.org

Laser Photonics Rev. 6, No. 2 (2012)
REVIEW
ARTICLE
181
Figure 2
(a) Penrose particle array,
L11 9 μ
m, generation 12;
(b) Reciprocal space sampled within a pseudo-Brillouin zone.
Δ
is
the minimum center to center particle distance (
Δ400
nm); The
tenfold rotational symmetry of the diffraction spots is clearly ob-
servable.
is simply obtained by positioning particles at the vertices
of a Penrose tiling. In general, the symmetry of particle
arrays is best displayed by the corresponding diffraction
diagrams, obtained by calculating the amplitudes of their
Fourier transforms. However, while for periodic arrays all
the information contained in their reciprocal space can be
compressed into periodic Brillouin zones, aperiodic arrays
have non-periodic diffraction diagrams and therefore Bril-
louin zones cannot uniquely be defined. As a result, when
comparing diffraction patterns of different types of aperiodic
arrays, it is important to adopt an approach that guarantees a
homogeneous sampling of their aperiodic spectral features.
This can be done by restricting their reciprocal spaces to
the so-called pseudo-Brillouin zones, which contain spa-
tial frequencies in the compact interval
1Δ
, where
Δ
is
the minimum or the average inter-particle separation for
the specific type of aperiodic array. By using this conven-
tion, in Fig. 2b we display the pseudo-Brillouin zone of the
Penrose particle array shown in Fig. 2a. Throughout this
review, we will always compare pseudo-Brillouin zones of
aperiodic particle arrays of comparable interparticle separa-
tions. One of the main features of aperiodic arrays is their
ability to encode rotational symmetries in either discrete or
continuous Fourier spectra. It was recently discovered that
aperiodic tilings can be constructed with an arbitrary degree
of rotational symmetry using an algebraic approach [38]. In
addition, deterministic tilings with full rotational symmetry
of infinite order (i. e., circular symmetry) have also been
demonstrated [39] by a simple procedure that iteratively
decomposes a triangle into five congruent copies. The re-
sulting tiling, called Pinwheel tiling, has triangular elements
(i. e., tiles) which appear in infinitely many orientations,
and in the infinite-size limit, its diffraction pattern displays
continuous (“infinity-fold”) rotational symmetry. Radin has
shown that there is no discrete component in the Pinwheel
diffraction spectrum, but it is still unknown if the spectrum
is continuous [23]. In Figs. 3 we compare a particle array
with seven-fold symmetry (i. e., Danzer arrays [40]) and
the Pinwheel array along with their corresponding diffrac-
tion spectra (i.e., pseudo-Brillouin zones). We will see in
Sect. 3.2 that finite-size particle arrays with full circular
Figure 3
(a) Danzer array,
L26 6 μ
m, generation 4; (b) Pin-
wheel array,
L16 1 μ
m, generation 5; (c) Danzer reciprocal
space where
Δ
is the minimum center to center particle distance
(
Δ400
nm); (d) Pinwheel reciprocal space where
Δ
is the mini-
mum center to center particle distance (Δ400 nm);
symmetry in Fourier space can also be obtained by engi-
neering aperiodic spiral order in the form of Vogel’s spirals.
Despite no rigorous results exist on the spectral nature of
aperiodic spirals, they appear to best exemplify the concept
of a “turbulent crystals” discussed by Ruelle [41].
It is clear from our discussion that aperiodic structures
possess a rich and novel type of long-range order, described
by more abstract symmetries than simple translational in-
variance. A simple example of abstract symmetry in qua-
sicrystals was given by John Horton Conway for the case of
Penrose tilings. Conways’ theorem states that, despite the
global aperiodicity of Penrose tilings, if we select a local
pattern of any given size, an identical pattern can be found
within a distance of twice that size [42].
In 1984, Dan Shechtman et al [43] were the first to exper-
imentally demonstrate the existence of physical structures
with non-crystallographic rotational symmetries. When
studying the electron diffraction spectra from certain metal-
lic alloys (Al
6
Mn), they discovered sharp diffraction peaks
arranged according to the icosahedral point group symmetry
(i. e., consisting of 2-, 3-, 5-, and 10-fold rotation axes for a
total of 120 symmetry elements). The sharpness of the mea-
sured diffraction peaks, which is related to the coherence of
the spatial interference, turned out to be comparable with the
one of ordinary periodic crystals. Stimulated by these find-
ings, Dov Levine and Paul Steinhardt promptly formulated
the notion of quasicrystals in a seminal paper titled [44]:
“Quasicrystals: a new class of ordered structures”. It was
www.lpr-journal.org © 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim

182
LASER & PHOTONICS
REVIEWS
L. Dal Negro and S. V. Boriskina: Deterministic aperiodic nanostructures for photonics and plasmonics applications
subsequently discovered that three-dimensional icosahedral
structures can be obtained by projecting periodic crystals
from an abstract six-dimensional superspace, starting the
fascinating field of quasi-crystallography [23, 24]. In re-
sponse to these breakthrough discoveries, the International
Union of Crystallography (IUCr) established in 1991 a spe-
cial commission with the goal of redefining the concept of
crystal structures According to the IUCr report, the term
“crystal” should be associated to “any solid having an es-
sentially discrete diffraction diagram” [45], irrespective of
spatial periodicity. The essential attribute of crystalline order
(both periodic and quasiperiodic) is to display an essentially
discrete spectrum of Bragg peaks. As observed in [12], this
definition has shifted the main attribute of crystalline struc-
tures from real space to reciprocal Fourier space. This shift
directly affects the engineering of aperiodic systems, which
is best achieved in reciprocal Fourier space for structures
of low to intermediate refractive index contrast, as we will
later detail.
In the next two sections, we will significantly broaden
the notion of aperiodic order by discussing more general
approaches and methods for the generation of determinis-
tic systems with non-periodic spatio-temporal complexity
beyond quasicrystals.
1.3. Aperiodic order beyond quasicrystals
Deterministic aperiodic order plays today an important role
in several fields of mathematics, biology, chemistry, physics,
economics, finance, and engineering. Nonlinear dynamical
systems, continuous or discrete, often manifest an extreme
sensitivity to their initial conditions, rendering long-term
prediction impossible in general [45, 46]. This feature was
fully recognized by the mathematician Poincar
´
e in the con-
text of celestial mechanics in 1890 [47, 48]. In his research
on the three-body problem, Poincar
´
e discovered an Hamilto-
nian dynamical system with sensitive dependence on initial
conditions. This phenomenon was brought to worldwide
attention by the meteorologist Edward Lorenz who showed
in 1963 that a simple system of three coupled differential
equations, used to model atmospheric convection, can give
rise to chaotic behavior with extreme dependence on initial
conditions (i. e., the butterfly effect), laying the foundations
of the theory of chaotic dynamical systems [49]. Spatio-
temporal chaotic behavior has since then been discovered
in a large number of deterministic physical systems, includ-
ing coupled nonlinear oscillators, hydrodynamic turbulence,
chemical reactions, nonlinear optical devices (i. e., lasers)
and even low-dimensional conservative systems such as ge-
ometric billiards [45, 46, 50, 51]
2
. The long-term behavior
of deterministic chaotic systems is unpredictable, though
2
Chaotic dynamics is also displayed by elastic collision prob-
lems involving only few particles, such as the scattering of a small
mass by three disks [45]. Additional examples of deterministic
chaotic behavior can be found in the iterations of simple nonlinear
maps, such as one-dimensional quadratic functions (i. e., the logis-
tic map) and their generalizations in the complex domain [45, 46].
not random, due to the sensitive dependence on initial con-
ditions, which cannot be specified with enough precision.
The theory of chaotic dynamical systems has recently
found direct application to problems of condensed matter
physics, including the study of the excitation spectra of 1D
and 2D aperiodic optical systems
[52–54]
such as the ones
discussed in Sects. 2–4.
Numeric sequences and geometric patterns with deter-
ministic, though unpredictable behavior, are deeply rooted
in discrete mathematics and number theory. Number theory
is primarily concerned with the properties of integer num-
bers [30, 31], and provides algorithms for the generation of
various types of pseudo-random point sets and aperiodic
tilings. The connection between deterministic aperiodicity
and unpredictability
3
[55] is central to number theory and it
has motivated many engineering applications in fields such
as cryptography and coding theory
[56–58]
. The origin of
this type of unpredictability can be traced back to the dif-
ficulty (i. e., algorithmic complexity) of certain arithmetic
problems, such as factoring, the invertibility of one-way
functions (e. g., the discrete logarithm problem) [29, 56],
or to open number-theoretic questions related to the dis-
tribution of prime numbers and primitive roots. Number-
theoretic methods are ideally suited to generate aperiodic
point sets with different degrees of structural correlations
and geometrical complexity [29]. Moreover, these methods
provide simple algorithms, defined on finite number fields,
to construct aperiodic binary sequences with well-defined
Fourier spectral properties [29,31]. These properties have
been utilized to engineer pseudo-random number generators,
mostly based on the behavior of polynomial congruencies
or more advanced methods that have even been shown to
be cryptographically secure (i. e., Blum-Blum-Shub algo-
rithms) [56–58].
Deterministic pseudo-random generators (DPRG) pro-
duce numerical sequences and spatial patterns displaying
statistical randomness (i. e., no recognizable patterns or
regularities) and Fourier spectra approaching uncorrelated
white noise. Deterministic structures generated by number-
theoretic numerical sequences with flat-Fourier spectra have
recently found technological applications in different re-
search areas ranging from the engineering of acoustic dif-
fusers to radar abatement (stealth surfaces), spread spectrum
communication (jamming countermeasures, secure channel
sharing), and the design of minimum redundancy antenna
arrays (surveillance) in the RF regime [29]. However, these
techniques are still largely unexplored in the domain of
optical technologies.
In Fig. 4 we show two interesting examples of aperiodic
particle arrays constructed on Gaussian primes (Fig. 4a) and
finite Galois fields
4
(Fig. 4b). The corresponding pseudo-
3
The role of randomness in number theory and its relation with
computability,non-decidability and algorithmic complexity lies at
the core of the fascinating and highly interdisciplinary research
field, pioneered by G. Chaitin, R. Solomonoff, and A. N. Kol-
mogorov, known as Algorithmic Information Theory (AIT) [55].
4
Galois fields are finite number fields of prime order p, de-
noted as GF(p). They consist, for example, of the elements
© 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim www.lpr-journal.org

Laser Photonics Rev. 6, No. 2 (2012)
REVIEW
ARTICLE
183
Figure 4
(a) Gaussian Prime Array,
L19 4 μ
m, generation 6;
(b) Galois Array,
L12 2 μ
m, generation 5; (c) Gaussian Prime
reciprocal space where
Δ
is the minimum center to center particle
distance (
Δ400
nm); (d) Galois reciprocal space where
Δ
is the
minimum center to center particle distance (Δ400 nm)
Brillouin zones are shown in Figs. 4 (c-d). Gaussian primes
are integers that are prime in the complex field, and are
defined by
n im
, where
n
and
m
are integers and
i
is the
imaginary unit [29]. We notice that primes of the form
4k1
in the field of integers are still primes in the complex field,
but 2 and primes of the form
4k1
can be factored in the
complex field [29]. By plotting the Gaussian primes with
(
n m
integer coordinates in the plane we obtain the highly
symmetric pattern shown in Fig. 4a. Patterns with different
degree of rotational symmetries can be obtained by con-
sidering primes defined by
nαm
, where
α
is a complex
algebraic root of unity. When considering the complex cube
root of unity, which is the solution of the algebraic equation
1α α20
, we obtain the two-dimensional pattern of
Eisenstein primes, which displays hexagonal symmetry [29].
Finite Galois fields can also be utilized to generate binary-
valued periodic sequences with pseudo-random properties,
and particle arrays in two spatial dimensions [29], as shown
in Fig. 4b. Sequences with elements from
GF p
and with
coefficients determined by primitive polynomials in
GF pm
have unique correlation and flat Fourier transform proper-
ties which found important applications in error-correcting
012 p1
, for which addition, subtraction, multiplication and
division (except by 0) are defined, and obey the usual commutative,
distributive and associative laws. For every power
pn
of a prime,
there is exactly one finite field with
pn
elements, and these are
the only finite fields. The field with
pn
elements is called
GF pn
where GF stands for Galois field.
codes, speech recognition, and cryptography
[29, 56–58]
.
Moreover, in contrast to other number-theoretic sequences
(e. g., Legendre sequences) with flat Fourier spectra, Ga-
lois sequences can be generated by simple linear recur-
sions
[29, 56–58]
. Two-dimensional Galois arrays display
diffraction spectra with a high density of spatial frequen-
cies, theoretically a flat measure for infinite-size arrays. This
property has been used to improve the image resolution of
X-ray sources in astronomy [29]. Only recently, aperiodic
arrays of metal nanoparticles generated according to Gaus-
sian primes, and other number-theoretic functions have been
explored in the context of plasmon scattering and optical
sensing
[59–61]
. These structures are: (i) coprime arrays,
which are 2D distributions of particles with coprime coordi-
nates
5
, (ii) the prime number arrays, which are 2D arrays of
particles representing prime numbers in reading order; and
(iii) Ulam spirals, which consist of prime numbers arranged
on a square spiral. The direct and reciprocal Fourier spaces
of the prime number arrays are shown in Fig. 5.
1.4. Aperiodic substitutions
In optics and electronics, an alternative approach to gen-
erate deterministic aperiodic structures with controlled
Fourier spectral properties relies on symbolic substitu-
tions [12,26,27,62]. Not surprisingly, profound connections
exist between the theory of substitutional sequences, dynam-
ical systems, and number theory [26, 27]. Substitutions are
an essential component of every recursive symbolic dynam-
ical system formally defined on a finite symbolic alphabet
GA, B, C, . . .
. In physical realizations, each letter cor-
responds to a different type of building block (e. g., nanopar-
ticle, dielectric layer, etc). A specific substitution rule
ω
replaces each letter in the alphabet by a finite word, starting
from a given letter called an axiom or initiator. An aperiodic
(deterministic) sequence is then obtained by iterating the
substitution rule
ω
, resulting in a symbolic string of arbitrary
length. For instance, the Fibonacci sequence is simply ob-
tained by the iteration of the rule
ωF
:
A AB
,
B A
with
axiom A, as exemplified in the process:
A AB ABA
ABAAB ABAABABA ABAABABAABAAB
.
Symbolic dynamical systems are examples of L-systems or
Lindenmayer systems [63]. L-systems were introduced and
developed in 1968 by the Hungarian theoretical biologist
and botanist Aristid Lindenmayer (1925–1989). L-systems
are used to model the growth processes of plant develop-
ment, but also the morphology of a variety of organisms [63].
In addition, L-systems can be used to generate self-similar
fractals and certain classes of aperiodic tilings such as the
Penrose lattice.
A large variety of substitutions have been explored in the
study of deterministic aperiodic optical multi-layered sys-
tems
[12–14]
. It is possible to associate to each substitution
rule a substitution matrix
Sij Niωj
whose elements
5
Two numbers
n
and
m
that have no common factors are called
relatively prime, mutually prime or coprime. Their greatest com-
mon divisor (gcd) is equal to 1.
www.lpr-journal.org © 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim

184
LASER & PHOTONICS
REVIEWS
L. Dal Negro and S. V. Boriskina: Deterministic aperiodic nanostructures for photonics and plasmonics applications
Figure 5
(a) Prime Array,
L
12 6 μ
m, generation 5; (b) Co-
Prime Array,
L11 4 μ
m, gener-
ation 5; (c) Ulam Spiral Array,
L
19 2 μ
m, generation 50; (d) Prime
reciprocal space where
Δ
is the
minimum center to center parti-
cle distance (
Δ400
nm); (e) Co-
Prime reciprocal space where
Δ
is the minimum center to center
particle distance (
Δ400
nm);
(f) Ulam Spiral reciprocal space
where
Δ
is the minimum center
to center particle distance (
Δ
400 nm).
indicate the number of times a given letter
i A B
appears
in the substitution rule
ωj
, irrespective of the order in
which it occurs. The dimension of the matrix
Sij
is deter-
mined by the size of the letter alphabet
G
. For instance,
the substitution matrix of the Fibonacci sequence is:
11
10
.
The advantage of this substitutional approach over purely
geometrical or number-theoretic methods is that relevant
information on the characteristics of the diffraction spectra
can be directly obtained from the substitution matrix. As
discovered by Bombieri and Taylor, there is a fundamental
connection between the arithmetical nature of substitutions
and the presence/absence of Bragg peaks in the correspond-
ing Fourier transforms
[64–68]
. According to the Bombieri-
Taylor theorem [69, 70], if the spectrum of the substitution
matrix
S
contains a so-called Pisot-Vijayvaraghavan (PV)
number as an eigenvalue, then the sequence is quasiperi-
odic and its spectrum can be expressed as a finite sum of
weighted Dirac
δ
-functions, corresponding to Bragg peaks
that are indexed by integer numbers, otherwise it is not.
6
A
very relevant result, know as the gap-labeling theorem, re-
lates the positions of the diffraction Bragg peaks of substitu-
tional sequences with the locations of the gaps in the energy
spectra of elementary excitations supported by the structures
(e. g., optical modes, electronic states, etc) [64,65,71]. More
specifically, a perturbative analysis of the integrated den-
sity of states of aperiodic structures generated by Pisot-type
substitutions shows that both the positions and the widths
of the gaps in their energy or transmission spectra can be
“labeled” by the singularities of the Fourier transform as-
6
A PV number is a positive algebraic number larger than one
and such that all of its conjugate elements (i. e., the other solutions
of its defining algebraic equation) have absolute value less than one.
For instance, the golden mean, satisfying the algebraic equation
x2x1 0is a PV number.
sociated to the sequence of scattering potentials (optical
or electronic) [64, 71]. This approach, first introduced for
the 1D Schr
¨
odinger equation [54], is valid beyond perturba-
tion theory for quasiperiodic and almost-periodic structures,
and it has been recently extended in two dimensions for
sequences with more complex Fourier spectra [64,71], such
as the Thue-Morse and Rudin-Shapiro sequences, which we
will discuss in the next section.
1.5. Classification of aperiodic structures
Until recently, patterns were simply classified as either peri-
odic or non-periodic, without the need of further distinctions.
However, it should now be sufficiently clear that the word
“non-periodic” encompasses a very broad range of different
structures with varying degrees of order and spatial corre-
lations, ranging from quasiperiodic crystals to more disor-
dered “amorphous” materials with diffuse diffraction spec-
tra. Moreover, mixed spectra with both discrete peaks and
diffuse backgrounds can also frequently occur, as demon-
strated in Figs. 3–7. One of the most fascinating questions
in quasicrystals theory is whether there exist tilings with ab-
solutely continuous diffraction spectra, as opposed to mixed
ones [23]. The pinwheel tiling is a plausible candidate, and
we will now introduce others that can be obtained by 2D
generalizations of 1D substitution rules. Finally, we will
review important ideas and results that motivate a general
classification of deterministic structures, significantly ex-
panding our definition of aperiodic order.
Aperiodic structures can be rigorously distinguished
according to the nature of their diffraction patterns and
energy spectra, which correspond to mathematical spectral
measures [12, 27]. In optics and electronics, these spectral
© 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim www.lpr-journal.org

Laser Photonics Rev. 6, No. 2 (2012)
REVIEW
ARTICLE
185
measures are often identified with the characteristics of the
transmission/energy spectra of the structures. A spectral
measure is also associated with the Fourier transforms of
the structures, or lattice spectrum, providing information on
the nature of the corresponding diffraction patterns.
According to the Lebesgue’s decomposition theorem,
any measure can be uniquely decomposed in terms of three
kinds of primitive spectral components (or mixtures of
them), namely: pure-point (
μP
), singular continuous (
μSC
),
and absolutely continuous spectra (
μAC
), such that:
μ
μPμSC μAC
. For example, the diffraction spectrum of a
Fibonacci lattice is pure-point, featuring a countable set of
δ
-like Bragg peaks at incommensurate intervals. More com-
plex structures display singular continuous spectra, meaning
that the support of their Fourier transforms can be covered
by an ensemble of open intervals with arbitrarily small total
length [12]. For these structures, and in the limit of systems
with infinite size, individual Bragg peaks are not separated
by well-defined intervals, but tend to cluster forming “broad
bands” in the reciprocal space. As a result, the correspond-
ing eigenmodes (e. g., optical modes, electronic wavefunc-
tions, acoustic modes, etc) are generally more localized in
space compared to structures with pure-point spectra.
The chief example of a deterministic sequence with
singular continuous diffraction and energy spectra is the
Thue-Morse sequence [27, 72]. The Thue-Morse sequence
is generated by the substitution
ωTM
:
A AB
,
BBA
.
This binary sequence was first studied by Eug
`
ene Prouhet
in 1851, who applied it to number theory [73]. Axel Thue in
1906 used it to found the study of combinatorics on words.
The sequence was successively brought to worldwide atten-
tion by the differential topology work of Marston Morse in
1921, who proved that the complex trajectories of dynamical
systems whose phase spaces have negative curvature can be
mapped into a discrete binary sequence, the Thue-Morse
sequence
7
[74]. More recently, in the context of the algorith-
mic theory of finite automata, a number of results have been
demonstrated that connect the existence of palindromes of
arbitrary length in the Thue-Morse and similar binary se-
quences with singular continuous Fourier spectra [26].
Aperiodic substitutions can also give rise to the third
primitive type of spectral measure, the absolutely continu-
ous Fourier spectrum, akin to random processes. The chief
example of a deterministic structure with absolutely continu-
ous Fourier spectrum is the Rudin-Shapiro sequence [27,75,
76]. In a two-letter alphabet, the RS sequence can simply be
obtained by the substitution:
AA AAAB
,
AB AABA
,
BA BBAB
,
BB BBBA
[77]. The Fourier spectra of
the three main examples of 1D aperiodic sequences gen-
erated by binary substitutions are displayed in Fig. 6. The
Fourier spectrum of the Rudin-Shapiro sequence becomes,
in the limit of a system with infinite size, a continuous
function (i. e., flat spectrum), generating delta-correlated
pseudo-random noise (see Figs. 6,7). Rudin-Shapiro struc-
tures are expected to share most of their physical properties
with disordered random systems, including the presence of
7
Here is another instance of the deep connection between dy-
namical systems, number theory, and substitutions.
Figure 6
(a) Absolute value of the Fourier coefficients of a
quasiperiodic (Fibonacci) structure, (b) of an aperiodic (TM)
structure with singular continuous spectrum, (c) of an aperiodic
structure with absolutely continuous spectrum (RS structure).
From [62].
localized optical states (i. e., Anderson-like states). How-
ever, the abundance of short-range correlations, whose main
effect is to reduce the degree of disorder and localization,
favors the existence of resonant extended states in their en-
ergy spectra, and significantly complicates the theoretical
analysis of Rudin-Shapiro and other deterministic structures
with absolutely continuous Fourier spectra [77, 78]. Opti-
cal states in 2D photonic membranes with Rudin-Shapiro
morphology have been recently investigated experimentally,
and lasing from localized Rudin-Shapiro states has been
demonstrated for the first time [79]. These results will be
reviewed in Sect. 3.4.
The localization nature of the electronic and optical
states of a 1D Rudin-Shapiro system has been theoretically
investigated [71]. It was shown that the wavefunctions dis-
play a wide range of features ranging from weak to expo-
nential localization. Nevertheless, depending on the values
of scattering potential, extended (i. e., delocalized) states
have also been discovered in the spectrum of Rudin-Shapiro
structures [77, 78]. Another advantage of the substitutional
method is that it can be easily generalized to higher di-
mensions. A quasiperiodic Fibonacci 2D lattice has been
recently introduced by applying two complementary Fi-
bonacci substitution rules along the horizontal and the ver-
tical directions, alternatively [62]. This way, a square 2D
Fibonacci matrix was obtained. Following this approach,
2D generalizations of both Thue-Morse and Rudin-Shapiro
sequences have been recently implemented to design metal-
lic nanoparticle arrays of interest for nanoplasmonics device
technology [80]. Figure 7 shows the direct and recipro-
cal Fourier spaces of Fibonacci, Thue-Morse and Rudin-
Shapiro arrays of particles obtained by the 2D substitu-
www.lpr-journal.org © 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim

186
LASER & PHOTONICS
REVIEWS
L. Dal Negro and S. V. Boriskina: Deterministic aperiodic nanostructures for photonics and plasmonics applications
Figure 7
(a) Fibonacci Ar-
ray,
L13 4 μ
m, generation 7;
(b) Thue-Morse,
L12 6 μ
m, gen-
eration 5; (c) Rudin Shapiro,
L
12 6 μ
m, generation 5; (d) Fi-
bonacci reciprocal space where
Δ
is the minimum center to cen-
ter particle distance (
Δ400
nm);
(e) Thue-Morse reciprocal space
where
Δ
is the minimum center
to center particle distance (
Δ
400
nm); (f) Rudin Shapiro recipro-
cal space where
Δ
is the minimum
center to center particle distance
(Δ400 nm).
tion method. We notice that, despite more sophisticated
generation methods have been independently developed to
construct 2D aperiodic arrays, the character of the Fourier
spectra of the resulting arrays does not depend on the spe-
cific generation method
[81–83]
. E. Maci
´
a [12] has recently
proposed a classification scheme of aperiodic systems based
on the nature of their diffraction (in abscissas) and energy
spectra (in ordinates). According to this classification, the
rigid dichotomy between periodic and amorphous structures
is surpassed by a matrix with nine different entries, corre-
sponding to all the combinations of the possible types of
spectral measures, as shown in Fig. 8.
We are now ready to turn the focus of our review towards
the implementation of the powerful concept of aperiodic
order in photonics and plasmonics.
2. One-dimensional aperiodic structures
in photonics
The research on 1D quasiperiodic photonic structures started
in 1987 with the study of dielectric multilayers arranged in
a Fibonacci sequence [16]. Such structures possess a very
rich transmission spectrum with a multifractal organization.
It has been realized only after the breakthrough discovery
of quasicrystals and the fabrication of Fibonacci [19, 20]
and Thue-Morse semiconductor heterostructures [84, 85]
that physical systems can give rise to a category of energy
spectra not previously encountered in natural structures.
These energy spectra, named singular-continuous, exhibit a
fractal pattern similar to the one of self-similar Cantor sets.
In particular, they feature an infinite hierarchy of narrow
transmission pseudo-gaps with vanishingly small widths
(in the limit of infinite-size systems). Seminal work on the
Figure 8
Classification of aperiodic systems according to the
spectral measures of their Fourier transform and their Hamiltonian
energy spectrum. From [11].
nature of Fibonacci spectra and the corresponding eigen-
modes was performed by Kohmoto, who established an
exact isomorphism between the 1D Schr
¨
odinger equation
with arbitrary multiple scattering potentials and the opti-
© 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim www.lpr-journal.org

Laser Photonics Rev. 6, No. 2 (2012)
REVIEW
ARTICLE
187
Figure 9
Transmission spectrum versus the optical phase length
δ
of a layer for a Fibonacci multilayer with 55 layers. The indices
of refraction are chosen as nA2and nB3. From [16].
cal wave equation [15,54]. A transfer matrix method was
introduced, enabling the analytical treatment of 1D scatter-
ing problems for optical and electronic excitations on the
same footing.
A powerful approach was subsequently developed by
Kohmoto, Kadanoff and Tang
[15, 52–54, 86, 87]
that made
use of recursion relations connecting the Fibonacci trans-
fer matrices (i. e., their traces and anti-traces) of dielec-
tric layers in order to define a nonlinear dynamical system
(i. e., dynamical map) that governs wave propagation in
1D quasiperiodic structures. From the knowledge of the
phase-space trajectories of this dynamical system, complete
information on the energy spectra and eigenmodes of opti-
cal and electronic quasiperiodic structures can be obtained.
This method, known as the trace map approach, has been
subsequently generalized to more complex 1D aperiodic sys-
tems (Thue-Morse, Rudin-Shapiro, arbitrary substitutional
sequences) by Kolar and Nori [66], enabling the application
of the powerful methods of dynamical system theory to the
solution of optical and electronic scattering problems. Fol-
lowing this approach, Kohmoto and Oono [53] discovered
the Smale horseshoe mechanism in the dynamical system
associated to Fibonacci multilayer stacks, leading to the
original prediction of their Cantor-set energy spectrum of
nested pseudo-gaps, which was demonstrated experimen-
tally by Gellermann et al [19]. The distinctive scaling of
the transmission spectra of optical Fibonacci multilayers
is clearly visible in Figs. 9,10. The many pseudo-gaps of
Fibonacci optical multilayers are separated by strongly fluc-
tuating wavefunctions with power-law localization scaling,
called critical modes. The notion of critical wavefunctions
is still not rigorously defined in today’s literature, leading to
somewhat confusing situations. However, general character-
istics of critical wavefunctions are their complex oscillatory
behavior, which results in self-similar spatial fluctuations
best described by multi-fractal scaling and wavelet analy-
sis
[12, 88–90]
. The spatial oscillations of critical modes
originate from the self-similarity of the structures, through a
series of tunneling events involving the overlap of different
sub-systems, repeating at different length scales. Despite the
notion of an envelope function is clearly ill-defined for such
Figure 10
Optical transmis-
sion spectra for Fibonacci di-
electric multilayers of increas-
ing layer numbers from 5 to
55. Curves (a): experiments.
(b): theory. From [19].
www.lpr-journal.org © 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim

188
LASER & PHOTONICS
REVIEWS
L. Dal Negro and S. V. Boriskina: Deterministic aperiodic nanostructures for photonics and plasmonics applications
strongly fluctuating critical states it is possible, using simple
scaling arguments and the Conway’s theorem, to show that
critical modes decay in space with characteristic power law
localization [12, 91]. The rich physical behavior of critical
states, including the presence of extended fractal wavefunc-
tions at the band-edge energy of the Fibonacci spectrum,
and their relation to optical transport are analytically studied
by Kohmoto and Maci
´
a using rigorous discrete tight bind-
ing method and a transfer matrix renormalization technique,
respectively [87, 92]. Critical modes in quasiperiodic Fi-
bonacci systems were observed experimentally by Desideri
and co-workers [93] in the propagation of Rayleigh sur-
face acoustic waves on a quasi-periodically corrugated solid.
Characteristic spatial patterns with remarkable scaling fea-
tures were obtained from an optical diffraction experiment.
However, to the best of our knowledge, a direct experimen-
tal observation of multi-fractal critical modes in the optical
regime is still missing. The photonic dispersion of photons
transmitted through a 1D Fibonacci multilayer structure was
investigated experimentally by Hattori and co-workers [94].
In their paper, they deposited 55 layers of SiO
2
/TiO
2
to form
an optical multilayer stack on a glass substrate, and mea-
sured the spectrum of the amplitude and phase of transmitted
light using a phase-sensitive Michelson-type interferometer.
They experimentally obtained phase and amplitude spectra
clearly demonstrating the self-similar fractal nature of the
Fibonacci spectra.
The interplay between quasiperiodic order and mir-
ror symmetry has been investigated by Huang et al [95].
They discovered that the addition of internal symmetry
greatly enhances the transmission intensity of the many
Fibonacci peaks. In particular, the spectral positions and
the self-similar scaling of symmetry-induced resonances
with perfect unit transmission were discussed for a class
of generalized Fibonacci multilayers within the analyti-
cal trace map approach. These interesting features were
experimentally demonstrated in TiO
2
/SiO
2
Fibonacci op-
tical multilayers with internal mirror symmetry by Peng
and collaborators [96]. The authors envisioned the use of
symmetry-induced perfect transmission states in Fibonacci
multilayers for the engineering of multi-wavelength narrow-
band optical filters, wavelength division multiplexing sys-
tems, and photonic integrated circuits, where a high-density
of narrow transmission peaks is particularly desired. The
light transport properties of Fibonacci band-edge states in
porous silicon multilayers have been first investigated by
Dal Negro and collaborators by means of ultrashort pulse
interferometry [97]. A strongly suppressed group velocity
has been observed at the Fibonacci band-edge states with
fractal behavior, which resulted in a dramatic stretching
in the optical pulses (see Fig. 11). It was also found that
the thickness drift naturally occurring during the growth
of porous silicon Fibonacci multilayers induces the local-
ization of band-edge modes, without compromising their
characteristic self-similar patterns [98].
Non-linear optical phenomena have also been exten-
sively investigated in Fibonacci and other quasiperiodic mul-
tilayered systems demonstrating superior performances over
periodic structures in terms of optical multi-stability [99]
Figure 11
(online color at: www.lpr-journal.org) (top): Measured
transmission spectrum of a 12th order Fibonacci quasicrystal. The
inset shows three examples of the power spectrum of the incom-
ing laser pulses in the time-resolved experiment reported at the
bottom. (bottom): Experimental data and calculation of the trans-
mission through Fibonacci samples at four different frequencies.
Also the undisturbed pulse, which has passed through only the
substrate and not the Fibonacci sample, is plotted for comparison.
When the laser pulse is resonant with one band edge state the
transmitted intensity is strongly delayed and stretched. When two
band edge states are excited, mode beating is observed. Adapted
from [97].
and second/third harmonic generation due to a far richer
Fourier spectrum [100, 101]. In particular, the aperiodic na-
ture of quasiperiodic multilayers provides more reciprocal
vectors for the simultaneous quasi-phase-matching of differ-
ent nonlinear processes. Zhu and collaborators [100, 101]
showed that this approach leads to a direct third-harmonic
generation with high efficiency through a coupled paramet-
ric process (Fig. 12). Their results demonstrate that high-
order harmonics may be generated in a quadratic nonlinear
medium by a number of quasi–phase-matching processes,
and therefore, can result in important device applications of
quasiperiodic structures in the field of nonlinear optics. Us-
ing multiple quasi-phase-matching in aperiodic structures,
a 1D photonic quasicrystal which acts as a simultaneous
frequency doubler for three independent optical beams has
© 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim www.lpr-journal.org

Laser Photonics Rev. 6, No. 2 (2012)
REVIEW
ARTICLE
189
Figure 12
(a) Schematic dia-
gram shows a Quasi Periodic
Optical Superlattice (QPOS)
composed of two blocks,
A
and
B
, arranged in Fibonacci se-
quence and the polarization
orientation of electric fields in
the THG process with respect
to the superlattice. Below is
shown a schematic diagram
of the process of THG in a
QPOS material. (b) The SHG
and THG tuning curves for the
QPOS sample, (top) calculated
and (bottom) measured by us-
ing an ns-optical parametric os-
cillator. (c) The average powers
of second- and third-harmonic
fields versus the average power
of the fundamental field for the
QPOS sample. The light source
is an ns-optical parametric os-
cillator with a repetition of 10 Hz.
Adapted from [101].
been demonstrated experimentally using electric-field poled
LiTaO
3
crystals [101]. These devices have been shown ex-
perimentally to perform almost a factor of ten better than
comparable ones based on quasi-phase-matching in periodi-
cally poled structures. Generalized quasiperiodic structures
(GQPS) for the simultaneous phase-matching of any two
arbitrarily chosen nonlinear interactions have also been re-
cently demonstrated [102]. These quasiperiodic structures,
first fabricated in a KTP nonlinear crystal, can efficiently
phase-match multiple nonlinear interactions with arbitrary
ratio between their wave vector differences.
Recently, the optical properties of multilayered struc-
tures based on the Thue-Morse sequence, characterized
by a singular continuous Fourier spectrum, have also at-
tracted considerable attention [72, 103]. The first optical
Thue-Morse structure was fabricated by Dal Negro et al. us-
ing magnetron sputtering of Si/SiO
2
multilayer stacks [84].
In the same paper, optical gaps corresponding to the sin-
gularities of the Fourier spectrum have been discovered
and explained by the presence of short-range correlations
among basic periodic building blocks distributed across the
structure [84]. Similar results were reported shortly after
by Qiu and collaborators [104]. The presence of a self-
similar hierarchy of optical gaps scaling according to a
characteristic triplication pattern of symmetry-induced per-
fect transmission states was also established by Liu and
collaborators in Thue-Morse multilayers (Fig. 13) [103].
More recently, Jiang and collaborators demonstrated, both
theoretically and experimentally, that the photonic band
gaps in Thue–Morse aperiodic systems can be separated
into two types, the fractal gaps (FGs) and the traditional
gaps (TGs), distinguished by the presence or absence of a
fractal structure, respectively [105]. The origin of the two
kinds of gaps was explained by the different types of inter-
face correlations and this explanation was confirmed by the
gap width behaviors. They also found that the eigenstates
near the FGs have a cluster-periodic form with large magni-
tude fluctuations (i. e. a critical mode), while those near the
TGs resemble Bloch waves. Spectrally enhanced light emis-
sion from SiN
x
/SiO
2
Thue-Morse multilayer structures was
demonstrated by Dal Negro et al. (Fig.14) [106]. Significant
light-emission enhancement effects at multiple wavelengths
corresponding to critically localized states were experimen-
tally observed in a 64 layer thick Thue-Morse structure,
yielding a total emission enhancement of almost a factor
of 6 in comparison to homogeneous light-emitting SiN
x
samples (Fig. 14d) [106]. The self-similar fractal nature of
Thue-Morse modes was subsequently investigated by the
use of the multi-scale wavelet analysis [107].
The possibility to engineer an all-optical diode using
nonlinear Thue-Morse multilayers was recently investigated
by Biancalana [108]. In this paper, it was demonstrated
that the strong asymmetry of odd-order Thue-Morse lattices,
combined with a Kerr nonlinearity, gives rise to a highly non-
reciprocal transmission spectrum which is the major feature
of an all-optical diode. The proposed Thue-Morse design
allows for an unprecedented reduction in device size at rela-
tively low operational optical intensities, a consequence of
the intrinsic anti-symmetry of the considered structure, and
the localized nature of its transmission states.
Dielectric multilayers consisting of pairs of dielectric
materials with different refractive indices arranged accord-
ing to a fractal structure in the direct physical space have
also been investigated, and are referred to as fractal filters
(Fig. 15). These structures were introduced in optics by Jag-
gard and Sun in 1990 [109,110], and their distinctive optical
properties have been discussed in detail in
[111–114]
. Frac-
tal filters generated by Cantor set algorithms were found
www.lpr-journal.org © 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim

190
LASER & PHOTONICS
REVIEWS
L. Dal Negro and S. V. Boriskina: Deterministic aperiodic nanostructures for photonics and plasmonics applications
Figure 13
(a) The lattice like electric
field distributions in the Thue-Morse multi-
layers for a completely transparent fre-
quency for structures with 32 (
n5
),
64 (
n6
), 128 (
n7
) layers, respec-
tively. (b) The reduced frequency
Ω
giv-
ing rise to identical matrix vs generation
n
for
a b 0 25
(quarter-wavelength).
Around
Ω1
is a quasi-continuous band
I formed. (c) Localization index
γ
vs
the number of layers N for the lattice
like electric field distributions at a com-
pletely transparent frequency
Ω0 758
.
From [103].
Figure 14
SEM image of light-emitting Thue-Morse (TM) structure with SiN
x
and SiO
2
layers. (a) Experimental transmission spectrum
for the 64 layer T-M structure. (b) Room temperature PL intensity of TM64 sample (solid line), TM32 sample (dashed line), homogenous
SiN
x
reference sample (dash-dotdot line). The excitation wavelength was 488 nm and the pump power was 10 mW. (c) Comparison of
the TM64 transmission spectrum and the homogeneous reference sample emission rescaled according to the SiN
x
thickness ratio.
(d) Experimentally derived wavelength spectrum of the optical emission enhancement in the TM64 sample. Adapted from [106].
to exhibit a distinctive spectral scaling of their transmis-
sion spectra resulting from their geometrical self-similarity.
Moreover, it was recently demonstrated analytically by
Zhukovsky and collaborators that universal recurrence rela-
tions exist, valid for every self-similar multilayer structure,
between the intensity reflection and transmission coeffi-
cients of generations Nand N1 [112, 114].
According to this picture, the transmission/reflection
spectrum of every type of optical fractal structure of genera-
tion
N
contains embedded transmission/reflection spectra of
all the preceding generations squeezed along the frequency
axis by a characteristic scaling factor. Finally, the splitting
of the transmission bands of symmetrical fractal filters has
been discovered
[111–115]
, with a multiplicity that grows
with the generation number (see Fig. 15). In the case of frac-
tal filters made of layers with refractive indices
nA
and
nB
,
thicknesses
dA
and
dB
, and satisfying the Bragg condition
nAdAnBdBλ04
at a wavelength
λ0
, every period in
the transmission spectrum (of amplitude
2ω0
,
ω02πcλ0
,
c
the speed of light) will contain a number of peaks equal to
the number of layers in the fractal structure. For example, if
N
is the generation order of a triadic and pentadic Cantor fil-
ter, it can be shown that
3N
and
5N
peaks can be encoded in
the periods of the corresponding transmission spectra [112].
© 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim www.lpr-journal.org

Laser Photonics Rev. 6, No. 2 (2012)
REVIEW
ARTICLE
191
Figure 15
Scalability of optical spectra for Cantor multilayers of
order
G5
: the full period of a (3,4) spectrum (a) and the central
part of it magnified in the frequency scale by 5 (b) versus the
full period of a (3,3) spectrum (c). Comparing (b) and (c), one
can see that the spectra are scalable with the factor equal to
G
.
From [112].
Finally, the propagation of plasmon-polaritons in 1D
metal-dielectric aperiodic superlattices generated by vari-
ous substitutional sequences was thoroughly investigated by
Albuquerque and collaborators [116,117]. Based on an ana-
lytical transfer matrix approach, they investigated polariton
spectra and discovered self-similar characteristic patterns
with multifractal distributions of polariton bandwidths.
The unique structural and optical properties of deter-
ministic aperiodic nanostructures in one spatial dimension
provide unprecedented opportunities for the engineering of
novel optical devices that fully leverage on the control of
aperiodic order. Specific device applications of 1D aperiodic
structures will be discussed in the next section.
2.1. Optical devices based on one-dimensional
aperiodic structures
The application of aperiodic order to photonic devices has
resulted in the engineering of a number of novel components
and functionalities enabled by the unique structural and op-
tical characteristics of aperiodic dielectric multilayers. In
what follows, we will discuss the signatures of aperiodic or-
der that are most relevant to device engineering, and review
the main achievements.
a)
Critical modes: these are unique spatial wavefunc-
tions (i. e., optical fields) with strongly fluctuating envelopes
and power-law localization properties ideally suited to en-
hance light-matter interactions (linear and nonlinear) over
large areas in defect-free photonic structures. In Fig. 16 we
show a calculated band-edge critical mode of a Thue-Morse
structure, from [105]. The spatial localization, frequency
bandwidth, and intensity enhancement of critical modes can
be utilized to engineer multi-frequency light sources and
Figure 16
The electric field magnitude vs position
x
. (a) The gap-
edge state near the BFG in an
S10
TM lattice. The inset shows
the state at the same frequency in an
S12
TM lattice. (b) The
gap-edge state near the BTG in an S10 TM lattice. From [105].
optical sensors. Following this approach, light emission en-
hancement from Thue-Morse active multilayers [84,106]
and chemical sensing with Thue-Morse porous silicon lay-
ers have been recently demonstrated [118]. Moretti et al.
have provided a detailed comparison of the sensitivities of
resonant optical biochemical sensors based on both periodic
and aperiodic structures [118]. The shifts of the reflectivity
spectra of these devices upon exposure to several chemical
compounds have been measured and Thue-Morse aperiodic
multilayers were found more sensitive than periodic ones
due to the lower number of interfaces and enhanced mode
localization.
b)
Dense reciprocal space: this feature enables the en-
gineering of multiple light scattering phenomena in the
absence of disorder and, for nonlinear materials, it offers
the unique possibility of achieving quasi-phase-matching
of multiple nonlinear processes simultaneously. Multicolor
harmonic generation based on quasiperiodic and aperiodic
multilayers has been demonstrated as well as the genera-
tion of several harmonics with high efficiency
[100–102]
.
These effects are relevant to the engineering of a novel class
of tunable multiwavelength optical parametric oscillators
and nonlinear optical devices including multi-frequency
bistable elements and adiabatic shapers of quadratic soli-
tons. The high density of reciprocal wave vectors available
in quasiperiodic structures has recently been utilized for the
demonstration of the first quasi-crystal distributed feedback
(DFB) laser by Mahler and co-workers [119]. In particular,
they showed that, by engineering a quasi-crystalline struc-
ture in an electrically pumped device, several advantages
of random lasers, such as the very rich emission spectra,
can be combined with the predictability of traditional dis-
tributed feedback resonators. Using a Fibonacci sequence,
they fabricated a terahertz quantum cascade laser in the
one-dimensional grating geometry of a conventional dis-
turbed feedback laser (Fig. 17a,b). This device makes use of
www.lpr-journal.org © 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim

192
LASER & PHOTONICS
REVIEWS
L. Dal Negro and S. V. Boriskina: Deterministic aperiodic nanostructures for photonics and plasmonics applications
Figure 17
(online color at: www.lpr-journal.org) (a) Top view of the fabricated Fibonacci laser device. (b) Fourier transform of a
quasi-periodic grating of the Fibonacci type. The inset shows the corresponding refractive index profile. The Fibonacci sequence
features many additional resonances, which can be used to create distributed feedback. The work in [119] considers a DFB with
first-order diffraction on the Bragg peak indicated by the arrow. (c–d) Spatial intensity distribution of the modes on the high-energy and
low-energy band-edges of the photonic gap that emerges from the Bragg peak highlighted in panel (b). The quasi-crystalline structure
(blue) leads to a distribution very different from the periodic case (light grey). The x-axis gives the layer number. (e) Far-field pattern of
a device operating on the lower band-edge. The red and blue lines show the computed and measured far-fields, respectively. The inset
shows the interaction of the optical mode with the grating, both represented by their Fourier transform in black and cyan respectively.
Adapted from [119].
a quantum-cascade active region in the terahertz frequency
range, is electrically pumped, and therefore of great interest
for realistic applications. Single-mode emission at a specific
angle from the device surface was obtained along with dual-
wavelength operation. These results demonstrate that the
engineering of self-similar spectra of quasiperiodic gratings
naturally allows optical functionalities that are hardly possi-
ble with traditional periodic resonators, such as directional
output independent of the emission frequency and multi-
color operation (Fig. 17e). In particular, the dense fractal
spectra of quasi-crystals can be easily engineered to con-
trol independently the energy spacing and positions of the
modes, providing the opportunity of using more than one
Bragg resonance for feedback, which leads to a multi-color
laser operation at arbitrarily chosen frequencies within the
gain bandwidth. Finally, we note that in the weak scattering
regime, each Fibonacci Bragg peak leads to a bandgap in
the optical transmission spectrum, and therefore the emis-
sion properties of quasiperiodic DFB lasers can be entirely
designed a priori based on the structure of their Fourier
space, significantly broadening the engineering possibilities
compared to periodic laser devices.
c)
Fractal transmission spectra: the distinctive fractal
scaling and peak splitting behavior of the transmission spec-
tra of quasiperiodic and aperiodic multilayers have profound
significance for the engineering of novel devices and the
management of group velocity, light dispersion, and energy
conversion effects. In relation to dispersion engineering,
Gerken et al [120, 121] demonstrated that using a single
66-layer non-periodic thin-film stack enables the separa-
tion of four wavelength channels by spatial beam shifting
due to strong group velocity dispersion effects, similarly to
the superprism effect observed in photonic crystals. How-
ever, the use of aperiodic multilayers guarantees larger and
more controlled shifts including constant dispersion allow-
ing for equidistant channel spacing. Using this approach, a
nearly linear 100
μ
m shift over a 13 nm wavelength range
was achieved, paving the way to the fabrication of thin-film
filters that can be utilized to obtain compact, cost-effective
wavelength multiplexing and demultiplexing devices.
Using the complex transmission spectra of a fractal multi-
layer structures, Gaponenko et al [113] have recently sug-
gested the possibility to encode and identify numeric infor-
mation. This procedure could potentially be used in optical
data recording and read-out by coding Fibonacci numbers
in the transmission peaks of dielectric multilayers. This
approach can open unprecedented scenarios for perform-
ing arithmetic operations by the physical process of light
© 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim www.lpr-journal.org

Laser Photonics Rev. 6, No. 2 (2012)
REVIEW
ARTICLE
193
propagation through aperiodic multilayers. The systematic
analysis of the splitting behavior of certain transmission
peaks in Cantor dielectric multilayers has even revealed
fascinating correlations between the numerical values asso-
ciated to their transmission spectra and the corresponding
prime factors [112–115].
Another key application domain where the fragmentation of
aperiodic multilayers spectra could play a crucial role is in
the field of thermo-photovoltaics. Aperiodic and quasiperi-
odic dielectric and metal-dielectric multilayers have in fact
been utilized to selectively enhance thermal radiation over
broad frequency bands [122, 123]. Moreover, it has been
shown by Maksimovi and Albuquerque that the spiky ther-
mal emission profiles of the fractal multilayers can be sub-
stantially smoothed by incorporating metamaterials in the
dielectric layers, thus providing broad spectral ranges of
enhanced thermal emission [122, 123]. The highly frag-
mented nature of the energy spectra of quasiperiodic and
aperiodic low-dimensional electronic materials could also
play a crucial role in enhancing the generation efficiency
of photovoltaic cells by providing an intermediate struc-
ture of electronic multi-bands. As discussed by Peng and
collaborators in their original design study [124], multiple
electronic mini-bands can be engineered below the barri-
ers of semiconductor superlattices thus providing additional
channels for efficient photo-induced absorption over broader
frequency ranges. Their results are summarized in Fig. 18.
By designing the optimal energy band structure through the
control of aperiodic order, it appears possible to approach
the 93% absolute thermodynamic limit of work production
in light converters [125].
d)
Broader design space: as discussed in Sect. 2, the
Fourier space of deterministic aperiodic structures can be
flexibly engineered to span across all the possible spectral
measures. In addition, the design parameters can further
be expanded when following a novel approach pioneered
by Maci
´
a [21, 22]. This approach considers the blending
of periodically and aperiodically arranged multilayers, thus
defining a hybrid-order made of different kinds of subunits,
with different types of topological order present at different
length scales. The introduction of structural subunits with
different periodic/aperiodic order mimics the mechanisms
of structural colors and iridescence phenomena observed
in multi-scale natural systems such as biological photonic
nanostructures (e. g., butterfly’s wings) [126]), and complex
biological macromolecules (e. g., DNA). Optical multilayer
systems with hybrid-order can be designed to exhibit com-
plementary optical responses (i. e., high transmission-high
reflection), depending on the choice of the incident angle
(see Fig. 19) [21, 22]. These hybrid structures additionally
present fundamental questions in the theory of complex op-
tics. In fact, despite it has been shown that hybrid spectra
share characteristic features with both their periodic and
aperiodic subunits, the mathematical nature of their spectra
is still a fascinating open problem [12] that deserves intense
work in the near future. Finally, we note that additional
parameters can be introduced in the design space of aperi-
odic structures by: (i) arbitrary structural distortions (e. g.,
chirped quasiperiodic structures); (ii) suitably combining
Figure 18
Electronic miniband structures for the several periodic
and aperiodic In
0 49
Ga
0 51
P/GaAs superlattices below the barrier.
The origin of the energies is set at the center of the gap of well
material GaAs. The index equal to 1–6 stands for the following
structure: 1-periodic SL with
N21
,
a b 2 5
nm; 2-periodic
SL with
N21
,
a b 3 5
nm; 3-SL with two parts: the first part
with
N10
,
ab2 5
nm, and the second part with
N11
,
a b 3 5
nm; 4-Fibonacci SL:
N21
,
a2 5
nm,
b3 5
nm;
5-Thue–Morse SL:
N16
,
a2 5
nm,
b3 5
nm; 6-one kind of
random SL with
N21
,
a2 5
nm,
b3 5
nm, where
N
is the
total number of layers,
a
and
b
are two thicknesses of the wells (
a
for block
A
and
b
for block
B
, and the thickness of each barrier
of all SLs is the same as
d2 0
nm. The inset is a schematic
of the band-edge diagram of the In
0 49
Ga
0 51
P/GaAs interface.
From [124].
different types of aperiodic order within the same structure
(aperiodic heterostructures); (iii) abstract symmetry opera-
tions such as symbolic conjugation (e. g., by interchanging
the symbols A and B in binary sequences); mirror symmetry,
resulting in positional correlations and resonant transmis-
sion peaks [12].
3. Two-dimensional aperiodic structures
in photonics
In periodic photonic crystals (PhCs), optical bandgaps are
formed when coherent multiple scattering of photons in-
duced by periodic variations of the refractive index acts to
prevent propagation of electromagnetic waves along certain
directions and within certain frequency ranges [127, 128].
On the other hand, random multiple scattering of photons in
disordered structures can give rise, for large enough values
of refractive index contrast, to a purely interference effect
called Anderson localization of light [129]. In turn, multiple
light scattering in photonic structures with various degrees
of aperiodic order results in a much richer physical picture
and leads to the formation of frequency regions of forbid-
den propagation, known as “pseudo-gaps”, and distinctive
resonant states with various degrees of spatial confinement,
known as critical modes [11]. Although this physical pic-
www.lpr-journal.org © 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim

194
LASER & PHOTONICS
REVIEWS
L. Dal Negro and S. V. Boriskina: Deterministic aperiodic nanostructures for photonics and plasmonics applications
Figure 19
(a) Dependence
of the transmission coefficient
with the refractive index contrast
in hybrid-order dielectric mul-
tilayer of different thicknesses.
(b) Sketch or hybrid-order struc-
tures. (c) Dependence of the
transmission coefficient on the
angle of incidence for a periodic
multilayer and a Fibonacci di-
electric multilayer, demonstrat-
ing complementary optical re-
sponses. (d) Dependence of the
transmission coefficient with the
incidence angle for the conju-
gated order device sketched (b),
Adapted from [21, 22].
ture is perfectly general, deterministic aperiodic structures
in two spatial dimensions are unique at combining long-
range structural order with higher order rotational axes (i. e.,
forbidden symmetries) and more abstract symmetries (e. g.,
fractal inflations, spiral symmetry, etc.) leading to more
isotropic band gaps, omnidirectional reflection spectra, and
unusual localization of optical modes.
3.1. Origin of 2D photonic bandgaps and
eigenmodes
The frequently-invoked analogy between photonic bandgap
materials and condensed matter physics, where electronic
bandgaps are present even in the absence of the long-range
structural order (e. g. amorphous semiconductors), has in-
spired the recent study of long-range and short-range spatial
correlations in complex photonic structures. The formation
of photonic bandgaps in 2D structures with quasiperiodic
order has first been demonstrated for octagonal tilings by
Chan et al in 1998 [130]. A comparative analysis of the pho-
tonic bandgaps evolution in periodically-arranged square
supercells of progressively larger sizes led the authors to the
conclusion that the gaps formation in quasiperiodic struc-
tures is governed exclusively by short-range correlations.
The evidence of the major role played by short-range corre-
lations and local symmetry groups has later been observed
for photonic quasicrystals with up to 12-fold rotational sym-
metry
[131–134]
. However, all the above studies focused on
aperiodic structures consisting of high-refractive-index di-
electric rods embedded in a low-index host medium (cermet-
type structures), which favor formation of bandgaps for
transverse-magnetic (TM)-polarized light waves [128]. In
this case, usually referred to as the “strongly localized pho-
ton picture”, short-range coupling of sharp Mie resonances
supported by the individual rods is the underlying mecha-
nism that drives the formation of optical bandgaps. Accord-
ingly, cermet-type photonic structures (both periodic and
aperiodic) can be described as tight-binding systems of indi-
vidual Mie resonators with identical resonance frequencies
ωr
, analogous to the tight-binding description of electronic
gaps by the coupling of atomic orbitals in semiconductors.
Interactions of these resonators with each other and with
the continuum band of the surrounding medium results in
the formation of bandgaps at frequencies distributed around
ωr
[13]. Therefore, the spectral positions of bandgaps in
cermet-type aperiodic structures with different array mor-
phologies approximately coincide with each other, and also
with the positions of bandgaps in periodic crystals with
similar geometrical and material parameters [135, 136]. Fur-
thermore, these types of photonic bandgaps are quite robust
to positional disorder [137]. In addition to Mie resonance
coupling, there is another bandgap formation mechanism,
non-resonant Bragg scattering, which becomes dominant in
the case of low refractive index contrast, sparse or network-
like photonic structures [13, 137]. Bragg scattering is a re-
sult of the destructive/constructive interference of incident
field and the field scattered by the refractive index varia-
tions such as dielectric or air rods, lattice planes, etc. In the
first-order single-scattering approximation, analogously to
nearly-free-electron systems, photonic bandgaps are formed
in a one-to-one correspondence with the spectral positions
of Bragg peaks in reciprocal space of the structure, and the
gap widths scale with the intensities of the corresponding
Bragg peaks [138]. Accordingly, the bandgap positions can
be adjusted by tuning the nearest-neighbor separation in pho-
tonic structures and by designing the Fourier space of the
structures. As a result, the Bragg scattering regime is partic-
ularly relevant to the engineering of aperiodic systems since
simple design rules can be formulated directly by analyzing
the Fourier spectral properties of the structures. However,
© 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim www.lpr-journal.org

Laser Photonics Rev. 6, No. 2 (2012)
REVIEW
ARTICLE
195
Figure 20
(online color at: www.lpr-journal.org) Bandgaps scal-
ing with the size of the 2D Penrose-type quasicrystal. (a) LDOS
spectra of the periodic, and Penrose structures made of 576 and
530 cylinders, respectfully. (b) Minimum LDOS value in the three
bandgap scaling with the structure size (adapted with permission
from [140].
photonic structures in the nearly-free-electron regime are
more sensitive to structural disorder, which destroys long-
range phase coherence of the scattered waves [137]. On the
other hand, aperiodic structures in the low index regime have
been successfully engineered for optical sensing applica-
tions [60], and will be discussed in Sect. 3.4. Finally, Bragg
interference effects beyond first-order scattering
[139–141]
may also play a role in the gap formation, especially in
photonic structures with moderate or high index contrast,
but are less explored at present. The Mie-resonance and
the Bragg scattering regimes usually co-exist in any given
aperiodic photonic structure, as illustrated in Fig. 20, which
shows the structure of photonic bandgaps of a finite-size
Penrose array of high-index dielectric cylinders and their
evolution with the increase of the structure size [140]. The
bandgap formation is revealed via calculation of the local
density of states (LDOS) in the center of the quasicrystal.
The exponential decay of the LDOS in the central bandgap
of the Penrose structure is almost identical to that observed
in periodic photonic crystals (see Fig. 20b), which is an
indication that it originates from relatively short-range cor-
relations. However, the scaling behavior of the two lateral
bandgaps is quite different and shows the existence of a
minimum structure size necessary to establish exponential
decay, which demonstrates the role of long-range correla-
Figure 21
(online color at: www.lpr-journal.org) (a) The radiated
power spectra for the 12-fold photonic quasicrystal (shown in
the inset) of the lattice radial size
R2a
(black),
4a
(red),
6a
(green),
8a
(blue), and
10a
(purple);
a
is the distance between two
nearest cylinders. (b) The photonic bandedge evolution in large
quasicrystal lattices,
R24a
(black),
36a
(red),
48a
(green), and
60a
(blue). (c–f) The intensity maps at the resonances marked as
“A,” “H,” “B,” and “C” in (b). After [143].
tions in the formation of these gaps. A similar picture of
the bandgaps scaling with the structure size has also been
observed in 1D aperiodic multilayer stacks [84, 105, 142].
The complex interplay between short-range and long-
range interactions in aperiodic photonic structures is also
manifested by the scaling of the band-edge states with the
increase of the structural size. As shown in Fig. 21 new
states appear at the edges of the bandgaps whose forma-
tion is driven by the short-range coupling between Mie
resonances of individual scatterers. These new states re-
sult from the coupling between resonances of small clus-
ters, which repeat throughout the photonic lattice and act
as resonant scatterers [143, 144]. These effects are more
pronounced in low-index structures, where coupling be-
tween local and global symmetries becomes more dominant,
leading to larger deviations of their Mie-scattering-driven
bandgap positions/widths from those of periodic photonic
crystals [131]. Furthermore, the increase of the structure size
of aperiodic structures results in both the appearance of new
modes inside the bandgaps and the formation of smaller
gaps inside the transmission bands [105, 143, 145, 146],
as shown in Fig. 21. Such evolution of the transmission
spectra of aperiodic structures originates from the fact that,
with the increase of the system’s size, larger resonant clus-
ters absent at smaller scales are formed, which can sup-
port localized resonances inside the bandgaps. A further
increase of the lattice size results in coupling of such clus-
ter resonances and in the formation of new bandgaps. Ac-
www.lpr-journal.org © 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim

196
LASER & PHOTONICS
REVIEWS
L. Dal Negro and S. V. Boriskina: Deterministic aperiodic nanostructures for photonics and plasmonics applications
Figure 22
(online color at: www.lpr-journal.org) Evolution of the
radiated power frequency spectra (a,e) and the field intensity pat-
terns of selected modes (c–d, g–h) in the Thue-Morse (a–d) and
Rudin-Shapiro (e–h) aperiodic structures composed of high-index
(ε10) dielectric rods (adapted with permission from [146]).
cordingly, resonant modes with extended and self-similar
states have been observed in defect-free aperiodic struc-
tures
[143, 145–150]
. Analogously to 1D structures, the
modes of 2D aperiodic systems are strongly fluctuating criti-
cal modes with a self-similar structure and power-law local-
ization scaling [105, 145, 146, 151]. Localization properties
of critical modes in the structures with pure-point (quasiperi-
odic) or singular continuous Fourier spectra are usually ex-
plained by the interplay between the lack of periodicity,
which drives for localization, and global scale-invariance
(i. e., self-similarity), which drives for the coupling be-
tween localized cluster states and thus tends to establish
extended wave functions [130]. However, localized modes
can form in the structures with sparsely distributed high-
symmetry clusters [147, 152, 153] or in aperiodic structures
with high degree of disorder (characterized by flat Fourier
spectra), which can be viewed as composites of different
low-symmetry local clusters with varying resonant charac-
teristics [146, 154, 155]. The effect of the spectral Fourier
properties of aperiodic structures on their transmission spec-
tra and modal properties is illustrated in Fig. 22, which com-
pares the Thue-Morse (singular-continuous Fourier spec-
trum) and Rudin-Shapiro (absolutely-continuous spectrum)
arrays of dielectric cylinders. Clearly, coupling of wave
functions localized at the clusters that repetitively appear in
Thue-Morse arrays of larger size results in the formation of
mini-bands inside the bandgap (Fig. 22a) and is reflected in
the self-similar scaling of the modes intensity distribution
in progressively larger lattices ( Fig.22b–d). To the contrary,
increasing the size of the pseudo-random Rudin-Shapiro ar-
rays causes the appearance of new localized states (Fig. 22e),
which do not form bands and remain isolated within local
clusters of scatterers (Fig. 22f–h).
3.2. Bandgap engineering with
aperiodic structures
Traditional 2D photonic crystals have spatial arrangements
that correspond to one of the five Bravais lattices. Among
these lattices, only the triangular and the honeycomb possess
the highest order of rotational symmetry (
n6
), and their
Brillouin zones are the closest to a circle. As a result, these
structures exhibit the widest 2D complete bandgaps (gaps
for all polarizations and directions), but their band diagrams
are strongly dependent on the light propagation direction.
The study of light propagation in photonic structures
with quasiperiodic order
[134, 135, 140, 143, 153, 156–165]
was initially motivated by the expectation that the higher ro-
tational symmetries of certain classes of quasicrystals would
result in more angularly isotropic distributions of scattering
peaks in Fourier space, favoring the formation of highly
isotropic bandgaps. As discussed in Sect. 1, and in contrast
to periodic photonic structures, there is no upper limit to the
degree of global rotational symmetry in 2D quasiperiodic
and aperiodic crystals. Photonic quasicrystals with 8-, 10-,
and 12-fold rotational symmetries were demonstrated, and
their optical properties were indeed found to be much less
dependent on the propagation direction.
It is worth mentioning that an alternative approach
to isotropic photonic dispersion has been recently devel-
oped based on the engineering of more complex periodic
structures with multi-atom bases, known as Archimedean
tilings [25, 166]. These periodic structures can approxi-
mate “crystallographically forbidden” rotational symme-
tries (
n5
,
n6
) over a finite number of diffraction
peaks, and when the number of atoms per cell is increased,
their diffraction spots tend to distribute over a circle. A
detailed computational analysis of the bandgap and dis-
persion characteristics of 8-fold and 12-fold quasicrystals
in comparison to periodic Archimedean tilings has been
carried out in [166] using a standard plane-wave expan-
sion method. Archimedean tilings, in the case of strong
refractive index contrast, were found to exhibit the same
gap widths and degree of isotropy as 8-fold and 12-fold
quasicrystals, reflecting the fact that the light dispersion
properties are mostly determined by short-range interac-
tions. The opposite situation was found for structures with
moderate or weak refractive index modulation (e.g., sili-
con nitride in air), where the gap properties of quasicrystals
significantly deviate from Archimedean tiling structures
due to the presence of very long-range interactions [166].
In this situation, it was concluded that quasiperiodic struc-
tures present a decisive advantage for the engineering of
isotropic bandgaps compared to their periodic approximants
and the Archimedean tilings. It is important to mention
© 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim www.lpr-journal.org

Laser Photonics Rev. 6, No. 2 (2012)
REVIEW
ARTICLE
197
Figure 23
SEM micrographs of
(a) Golden spiral array; (b) Vogel’s
spiral obtained with divergence an-
gle
θ137 6
; (c) Vogel’s spi-
ral obtained with divergence angle
θ137 3
; the arrays consist of
Au particles with 200nm diameters.
(d–f) Calculated diffraction pat-
terns (pseudo-Brillouin zone rep-
resentation) of the spirals shown
in panels (a–c).
Δ
represents the
average center to center particle
separation. After [172].
that GaN ad GaAs-based light emitting devices incorpo-
rating omnidirectional photonic crystals and optimized de-
signs based on the engineering of Archimedean tilings have
been successfully demonstrated experimentally
[167–170]
.
Additionally, isotropic photonic structures have been ob-
tained by designing reflection Bragg vectors equispaced
in angle [171]. However, it is important to note that this
design approach does not allow the investigation of more
complex isotropic structures characterized by either contin-
uous (i. e., no well-defined Bragg peaks) or mixed Fourier
spectra. More recently, light scattering phenomena in pho-
tonic and plasmonic structures with diffuse, rotationally
symmetric Fourier spectra were discussed [162, 172]. As
mentioned in Sect. 1, the Pinwheel array exhibits a circu-
larly symmetric Fourier space with infinity-fold rotational
symmetry, but this feature only develops in the limit of
infinite-size arrays. On the other hand, Fourier spaces with
circular symmetry are beautifully displayed by finite-size
aperiodic Vogel’s spirals (Fig. 23), which are fascinating
structures where both translational and orientational sym-
metries are missing [172]. These structures have been in-
vestigated by mathematicians, botanists, and theoretical
biologists [63] in relation to the outstanding geometrical
problems posed by phyllotaxis
[173–175]
, which is con-
cerned with understanding the spatial arrangement of leaves,
bracts and florets on plant stems, most notably as in the
seeds of a sunflower. Vogel’s spiral arrays are obtained by
generating spiral curves and subsequently restricting the
radial (r) and angular variables (
θ
) according to the quan-
tization condition [12, 176, 177]:
r a n
,
θnα
, where
a
is a constant scaling factor,
n0 1 2
,
α137 508
is an irrational number known as the “golden angle” that
can be expressed as
α360 ϕ2
,
ϕ1 5 2 1 618
is the golden number, which can be approximated by the
ratio of consecutive Fibonacci numbers. Rational approxi-
mations to the golden angle can be obtained by the formula
α360 1 p q 1
where
p
and
q p
are consecutive
Fibonacci numbers. The angle
α
gives the constant aperture
between adjacent position vectors
r n
and
r n 1
of parti-
cles in the “sunflower spiral”, also called the “golden spiral”
(Fig. 23a). Additionally, since the golden angle is an irra-
tional number, the golden spiral lacks both translational and
rotational symmetry, as evidenced by its Fourier spectrum
(Fig. 23d). Interestingly, Vogel’s spirals with remarkably dif-
ferent structures can be obtained by choosing only slightly
different values for the aperture angle
α
, thus providing
the opportunity to control and explore distinctively differ-
ent degrees of aperiodic structural complexity. The struc-
tures and the Fourier spectra of the three most investigated
types of aperiodic Vogel’s spirals [172] (divergence angles
α137 508
,
α1137 3
, and
α2137 6
) are shown in
Fig. 23. Only diffuse circular rings are evident in the Fourier
spectra of Vogel’s spirals (Fig. 23d–f), potentially leading
to fascinating new lasing and photon trapping phenomena.
To the best of our knowledge, the energy spectrum of ele-
mentary excitations propagating through these spiral lattices
is still unknown. However, as illustrated in Fig. 24, it was
found that the angular dependence of bandgaps (of Bragg
scattering origin) in aperiodic structures critically depends
on the degree of rotational symmetry of the pseudo-Brillouin
zones or their Fourier transforms [162]. Accordingly, struc-
tures with higher degree of rotational symmetry, such as the
Vogel’s spirals, indeed provide wider and more isotropic
photonic bandgaps.
Aperiodic photonic structures are also promising can-
didates for providing not just isotropic but complete pho-
tonic bandgaps controlled by a wider range of parameters
than periodic PhCs. Complete and isotropic bandgaps in
2D and 3D photonic periodic structures cannot be easily
achieved [128,178, 179]. In many cases they occur for such
values of structural and material parameters that impose
stringent fabrication requirements
[171, 180–182]
. For ex-
www.lpr-journal.org © 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim

198
LASER & PHOTONICS
REVIEWS
L. Dal Negro and S. V. Boriskina: Deterministic aperiodic nanostructures for photonics and plasmonics applications
Figure 24
Transmitted electric field intensity (logscale) as a func-
tion of the in-plane angle and normalized frequency for (a) hexag-
onal, (b) dodecagonal, (c) 12-fold Stampfli-inflated, and (d) sun-
flower lattices of low-index (
ε2
) dielectric rods. Darker color
corresponds to lower transmission (bandgap). (e–h) The corre-
sponding diffraction patterns showing only the strongest Bragg
peaks (reprinted with permission from [162]).
ample, the 3D face-centered cubic (fcc) lattice does not
posses a complete bandgap [178], and complete bandgaps
that open up in the spectra of 2D hexagonal lattices require
high air filling fractions and a high refractive index con-
trast, which may be challenging from the fabrication point
of view [180,181]. This limitation of periodic PhCs stems
from the degeneracy of the PhC bands at the points of high
crystal symmetry, which prevents opening of the bandgaps.
It has been shown that by reducing the lattice symmetry
(e. g. by adding/removing some of the “building blocks”
that form the photonic crystal or by periodically modulating
their sizes) it is possible to increase the widths of exist-
ing bandgaps, to ease fabrication tolerances, and even to
open up new bandgaps where none existed in structures of
high-symmetry [178, 179, 183, 184].
The most celebrated examples of reduced-symmetry pe-
riodic structures that feature complete bandgaps are the
3D diamond lattice [178] and the 2D honeycomb (also
called graphite) lattice [128, 184, 185]. Another example
of a 2D periodic lattice with reduced-symmetry is a di-
atomic square lattice, which consists of two elements (e. g.,
dielectric rods or air-holes) of different sizes [179, 183].
Complete bandgaps in such a structure can form because
the degeneracy of the TE bands at the
M
point of the Bril-
louin zone is lifted by the introduction of smaller-radii rods
at the center of each square unit cell. The ratio of the two
radii can be varied to maximize the bandgap width; and the
honeycomb lattice is clearly a limit case of the diatomic
lattice with r2r10.
Although some controversy exists as to the lowest value
of the refractive index contrast necessary for the formation
of complete bandgaps in aperiodic lattices [131, 156, 157],
there are indications that complete Bragg-scattering induced
bandgaps can be realized in large-area aperiodic photonic
structures of low refractive index. For example, it has been
shown that by increasing the length of aperiodic multilay-
ered stacks, photonic bandgaps can be opened at an arbitrary
long wavelength, in the regime where periodic structures
behave as homogeneous effective media [186]. Analogously,
particle clusters in 2D and 3D aperiodic structures can act
as resonant scatterers at lower frequencies than individual
scatterers, which drive the formation of low-frequency pho-
tonic bandgaps [140, 145]. However, since such effects be-
come dominant only in large-size structures [140,145, 162],
they may not be revealed by performing numerical sim-
ulations on finite-size aperiodic systems using periodic-
supercell structures, which only retain smaller-size local
clusters [130,131,157, 187] (see Sect.5). It is also important
to note that, unlike the extended Bloch modes of periodic
PhCs, critical band-edge modes of different bandgaps in
the same aperiodic photonic structure may feature drasti-
cally different spatial electric field distributions [158,188].
This property of aperiodic lattices opens a way to separately
adjust the widths of individual bandgaps (without affect-
ing other gaps) by adding additional structural elements at
certain positions within the aperiodic lattice. For example,
it was shown that the width of the third bandgap of the
2D 12-fold quasicrystals lattice of dielectric rods can be
selectively manipulated by adding metal or dielectric rods
at pre-defined locations [158].
An alternative approach to engineer complete photonic
bandgaps in aperiodic structures was recently developed by
Florescu and collaborators [189]. Building on prior work
on the generation of “stealth” and equi-luminous materi-
als [190] with respectively zero and constant scattering in-
tensity over a range of wavelengths, they have designed the
first known example of amorphous optical structures of ar-
bitrary size supporting complete photonic bandgaps. In par-
ticular, they presented a simple constructive algorithm with
only two free parameters for the design of two-dimensional,
isotropic, disordered, photonic materials displaying com-
plete photonic band gaps blocking all directions and polar-
izations (Fig. 25).
The largest photonic band gaps were obtained in the
large refractive index contrast, using silicon and air, within
an optimization method that starts from a hyperuniform
disordered point pattern. The authors observed that there
is a strong correlation between the degree of hyperunifor-
mity for a variety of 2D crystal structures and the resulting
band gaps. Hyperuniform structures are distinguished by
their suppressed density fluctuations on long length scales,
and they consist in arrays of points whose number variance
© 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim www.lpr-journal.org

Laser Photonics Rev. 6, No. 2 (2012)
REVIEW
ARTICLE
199
Figure 25
(online color at: www.lpr-journal.org) (a) Protocol for
mapping point patterns into tessellations for photonic structure
design: a chosen point pattern (open circles) is partitioned by
using a Delaunay triangulation (thin lines). Next, the centroids
of the neighboring triangles (solid circles) of a given point are
connected, generating cells (thick lines) around each point, as
shown for the five (green) Delaunay triangles in the upper left
corner of the figure. (b) Realization of a stealthy hyperuniform
pattern. (c) Structure factor S(k) corresponding to the pattern
shown in (b). This structure exhibit a complete photonic bandgap.
(d) A plot showing how the PBG increases with the degree of
hyperuniformity and short-range geometric order. TM (red circles),
TE (orange squares), and complete (green diamonds) photonic
band gaps versus order the parameter for disordered, stealthy
hyperuniform arrays of Si rods in air. Adapted from [189].
within a spherical sampling window grows more slowly than
the volume. In this work, the authors demonstrate that hy-
peruniformity, combined with uniform local topology and
short-range geometric order, result in complete photonic
band gaps without long-range translational order, opening
novel pathways for the control and manipulation of elec-
tronic and photonic band gaps in amorphous materials.
3.3. Structural defects and perturbations of
aperiodic structures
It is well-known that localized states can be formed in the
bandgaps of periodic PhCs by introducing structural de-
fects [191]. These localized states are classified as either
donor or acceptor modes. Donor modes are pulled from the
higher-frequency air (conduction) band by introducing extra
dielectric material at the defect site. Acceptor modes are
pushed into the optical gap from the lower-frequency dielec-
tric (valence) band when dielectric material is removed from
one or several unit cells [191,192]. In photonic quasicrystals,
each cylinder is located in a different environment, so that
removing one cylinder from a different location can produce
defect states with different frequencies and mode patterns,
thus offering higher degree of flexibility and tunability for
defect mode properties [163, 193]. Some studies indicate
that for small-size structures, aperiodic geometries exhibit
superior defect-mode confinement properties with respect
to their periodic counterparts [194]. Waveguides can also
be created by introducing channel-type defects in aperiodic
lattices that support photonic bandgaps [153, 163, 193, 195],
which demonstrate more structured transmission spectra
than defect waveguides formed in periodic photonic crystals.
Furthermore, finite-size aperiodic structures may support
another type of defect modes (surface defect modes), whose
frequencies and mode profiles depend on the specific shape
of the truncated portion of the photonic lattice [150].
3.4. Device applications
Many of the proposed device applications of 2D aperiodic
photonic structures hinge on the characteristic icosahedral
group symmetry of quasicrystals, which results in more
isotropic photonic gaps. However, multiple light scattering
in 2D structures of controlled aperiodic order additionally
offers the opportunity to generate unique optical modes with
a broad spectrum of localization properties. In this section,
we will review photonic devices that rely on the unique
localization and spectral properties of critical and localized
mode patterns in 2D aperiodic structures, specifically focus-
ing on the engineering of novel light sources, colorimetric
biosensors and nonlinear elements for multi-wavelength
generation.
A photonic quasicrystal lasers has been recently demon-
strated by Notomi and co-workers [196]. In their work, they
fabricated Penrose lattices of 150 nm-deep holes in a 1
μ
m-
thick SiO
2
layer on a Si substrate by electron-beam lithogra-
phy and reactive ion etching. Subsequently, they evaporated
a 300 nm-thick active material (DCM-doped Alq
3
layer
on the patterned SiO
2
to form a quasiperiodic laser cavity
(see Fig. 26). When the samples were optically pumped by
a pulsed nitrogen laser at 337 nm pump wavelength, they
have observed coherent lasing action above a characteristic
pumping threshold (100 nJ/mm
2
. This laser action resulted
from the optical feedback induced by the quasiperiodicity
of the structures, exhibiting a variety of 10-fold-symmetric
lasing patterns associated to the extended critical modes
of the structure. The properties of these lasing modes, in-
cluding their reciprocal lattice representations and their de-
pendence on the geometrical characteristics of the Penrose
lattice, were all explained by diffraction effects induced by
the quasiperiodicity. The results of this study show that las-
ing action due to standing waves coherently extended over
the surface of bulk quasicrystals is possible, in contrast to
the lasing behavior of traditional photonic crystals lasers,
driven by defect-localized states. These results open the way
to the engineering of various lasing states and conditions,
considering that the wide variety of the reciprocal lattices
of quasicrystals can encode an arbitrary order of rotational
symmetry and density of spatial frequencies.
Very recently, Yang and co-workers demonstrated laser
action from multiple localized modes in deterministic ape-
www.lpr-journal.org © 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim

200
LASER & PHOTONICS
REVIEWS
L. Dal Negro and S. V. Boriskina: Deterministic aperiodic nanostructures for photonics and plasmonics applications
Figure 26
(online color at: www.lpr-journal.org) (a) Schematic
view of photonic quasicrystal laser; (b) scanning electron
micrographs; (c–f) emission spectra of samples with
a
540 560 580 600 nm. From [196].
riodic structures with Rudin-Shapiro (RS) pseudo-random
morphologies [79] (Fig. 27). As discussed in Sect. 2, RS
structures, unlike quasi-periodic systems with discrete
Bragg peaks, feature a large density of spatial frequency
components which form nearly-continuous bands. As the
system size increases, the spectrum approaches the contin-
uous Fourier spectrum of white-noise random processes.
These pseudo-random systems are therefore ideally suited
to solve the major limitation to device applications of tra-
ditional random lasers, namely the lack of control and re-
producibility of their lasing modes. Yang and collaborators
proposed to solve this problem by engineering lasing modes
in deterministic structures with pseudo-random aperiodic or-
der. They fabricated a free-standing GaAs active membrane
with an array of air holes arranged in a two-dimensional
RS sequence and found that pseudo-random RS arrays of
air holes can support spatially localized optical resonances
at well-reproducible frequency locations that exhibit clear
lasing behavior in the presence of gain.
The air holes were fabricated with a square shape with
the side length
d330
nm and an edge-to-edge separation
between adjacent holes of 50 nm. The total size of pattern
was 25 μm × 25 μm, containing 2048 air holes.
The pump spot, about 2
μ
m wide, was moved across
the sample to excite localized modes at different positions.
A numerical study (based on 3D-FDTD calculations) of
the resonances performed on passive systems and the di-
rect optical imaging of lasing modes in the active struc-
tures enabled the authors to interpret the observed lasing
behavior in terms of distinctive localized resonances in the
Figure 27
(online color at: www.lpr-journal.org) First demonstra-
tion of laser action from pseudo-random DANS in GaAs multi
quantum wells. (a) SEM picture of the nanofabricated aperiodic
membrane; (b) corresponding Fourier and (c–d) experimentally
measured spatial profiles of localized lasing modes and (e–f) las-
ing characteristics. After [79].
membrane-type RS structures. The reproducibility of these
lasing modes, and their robustness against fabrication im-
perfections, were proved by fabricating and testing three
identical RS patterns on the same wafer. The nanofabricated
pseudo-random lasers introduced by Yang et al. provide a
novel approach, alternative to traditional random media and
photonic crystals, for the engineering of multi-frequency
coherent light sources and complex cavities amenable to
predictive theories and device integration.
The light scattering and localization properties of aperi-
odic photonic structures may also provide new exciting
opportunities for the design of functional elements for
bio-chemical sensing applications [60, 61, 154]. In current
biosensing technology, 2D periodic lattices, (i.e. 2D opti-
cal gratings) provide a well-established approach for bio-
chemical colorimetric detection, which can yield label-free
sensing of various molecular analytes and protein dynam-
ics. Standard periodic grating biosensors provide a distinct
change either in the intensity of diffracted light or in the
frequency of optical resonances in response to changes in
the refractive index of the surrounding environment.
The physical mechanism at the base of these optical sig-
natures is the well-known phenomenon of Bragg scattering.
While this process provides frequency selective responses
that are useful for colorimetric detection, the ability of light
waves to interact with adsorbed or chemically bound ana-
lytes present on the surface of these sensors is intrinsically
limited. In fact, in the small perturbation theory of light scat-
tering from rough surfaces [197], Bragg scattering already
appears as a first-order contribution to the complete solu-
© 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim www.lpr-journal.org

Laser Photonics Rev. 6, No. 2 (2012)
REVIEW
ARTICLE
201
Figure 28
(online color at: www.lpr-journal.org) Calculated sensi-
tivities of the critical modes of the Rudin-Shapiro structure (blue
circles) as well as the air band-edge mode and a point-defect
monopole mode (red circles) of the periodic PhC as a function
of the filling fraction of the mode field energy in the host medium.
(b–e) Typical intensity distributions of the high-Q quasi-localized
critical modes. Adapted from [146, 154].
tion. Scattered photons easily escape from a periodic surface
within well-defined spectral bands and without prolonged
interaction with the sensing layer. Development of optical
bio(chemical) sensing platforms calls for the design of scat-
tering elements that simultaneously provide high sensitivity
to the environmental changes and high spectral resolution,
as both factors contribute to the improvement on the sensor
detection limit [198].
Detector sensitivity is conventionally defined as the mag-
nitude of the wavelength shift induced by the change of
the ambient refractive index
SΔλΔnh
(measured in
nm/RIU), and can be improved by enhancing the light-
matter interaction. In turn, the resolution in measuring wave-
length shifts inversely depends on the linewidth of the res-
onant mode supported by the structure. It has been shown
that quasi-localized critical modes of aperiodic photonic
structures can simultaneously feature high quality factors
and high field intensity distributions over large sensing ar-
eas. A combination of these factors results in the improved
sensitivity of aperiodic-order-based sensors over their peri-
odic PhC counterparts based either on localized point-defect
or extended band-edge modes [154]. This is illustrated in
Fig. 28a, where the sensitivity values of quasi-localized crit-
ical modes of the Rudin-Shapiro structure are compared to
those of point-defect and band-edge modes of the periodic
square-lattice PhC. The sensitivity is plotted as a function
of the analyte filling fraction, i. e., the fraction of the optical
mode energy that overlaps with the analyte [154,198]:
fa
analyte
εaE r 2dV
εr E r 2dV 0fa1 (1)
and the increased overlap of the high-intensity portion of
the modal field with the analyte is clearly shown to improve
the sensitivity of the device. Typical near-field intensity
portraits of four of the high-Q critical modes supported
by the Rudin-Shapiro structure are plotted in Fig. 28b–e
and feature characteristic quasi-localized intensity fluctua-
tions [146, 154, 155].
A novel approach to label-free optical biosensing has
recently been developed by Lee and collaborators based on
micro-spectroscopy and spatial correlation imaging of struc-
tural color patterns obtained by white light illumination of
nanoscale deterministic aperiodic surfaces [61]. In contrast
to traditional photonic gratings or photonic crystal sensors
(which efficiently trap light in small-volume defect states),
aperiodic scattering sensors sustain distinctive resonances
localized over larger surface areas. In particular, nanoscale
aperiodic structures possess a dense spectrum of critical
modes, which result in efficient photon trapping and surface
interactions through higher-order multiple scattering pro-
cesses thereby enhancing the sensitivity to refractive index
changes. The complex spatial patterns of critical modes in
these structures offer the potential to engineer structural
color sensing with spatially localized patterns at multiple
wavelengths, which have been called colorimetric finger-
prints (shown in Fig. 29).
The proposed approach is intrinsically more sensitive to
local refractive index modifications compared to traditional
ones due to the enhancement of small phase variations,
which is typical of the multiple light scattering regime.
Multiple light scattering from nano-patterned determin-
istic aperiodic surfaces, which occurs over a broad spectral-
angular range, leads to the formation of colorimetric fin-
gerprints [60], in their near and far-field zones, which can
be captured with conventional dark-field microscopy [60].
These colorimetric fingerprints have been used as transduc-
tion signals in a novel type of highly sensitive label-free
multiplexed sensors [60, 61]. In particular, both the peak
wavelength shifts of the scattered radiation as well as the
environment-dependent spatial structure of the colorimetric
fingerprints of aperiodic surfaces have already been uti-
lized to detect the presence of nanoscale protein layers [61]
(Fig. 30). Lee and collaborators recently proposed to quan-
tify the spatial modifications of the structural color finger-
prints induced by small refractive index variations using
image autocorrelation analysis performed on scattered radi-
ation (Fig. 30b,c). By engineering the intensity of backscat-
tered radiation from aperiodic surfaces, the refractive index
changes induced by the analytes can be detected by shifts
in the scattering intensity spectra [60, 61]. Combining Elec-
tron Beam Lithography (EBL), dark-field scattering micro-
spectroscopy, autocorrelation analysis and rigorous multiple
scattering calculations based on the Generalized Mie The-
ory (GMT) [199], S. Lee and collaborators have engineered
aperiodic arrays of Chromium (Cr) nano-particles on quartz
substrates, and showed that the information encoded in both
spectral and spatial distributions of structural colors can
be simultaneously utilized. The potential of the proposed
approach for rapid, label-free detection of biomolecular
analytes in the visible spectral range was experimentally
demonstrated by showing a distinct variation in the spectral
and spatial colorimetric fingerprints in response to mono-
layer increments of protein layers sequentially deposited on
the surface of aperiodic arrays of nanoparticles [61].
The unique properties of Fourier spectra of aperiodic
structures, which can be designed with any combination
of Bragg peaks (wave vectors), have also been successfully
www.lpr-journal.org © 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim

202
LASER & PHOTONICS
REVIEWS
L. Dal Negro and S. V. Boriskina: Deterministic aperiodic nanostructures for photonics and plasmonics applications
Figure 29
(online color at: www.lpr-journal.org) (A–D) Scanning electron
microscopy (SEM) images of 2D periodic and aperiodic arrays of 100-radius
and 40 nm-high cylindrical Cr nano-particles on a quar tz substrate and the
associated dark-field images were illuminated at a grazing incidence with white
light. The structural color patterns of the images vary by the numerical aperture
(N. A.) of the imaging objective, in which different diffractive order is included
into the collection cone. The periodic arrays in (A) were observed under 10
×
objective with an 1 mm iris of N. A. reduced to 0.1 and the aperiodic arrays
in (B) Thue-Morse lattice (nearest center-to-center separation
Λ
400 nm);
(C) Rudin-Shapiro lattice (
Λ
400 nm); (D) Gaussian prime lattice (
Λ
300 nm),
were observed under 50
×
objective with N. A. 0.5. The structure color patterns
also vary by increasing the grating period with a progressive red-shift of the
scattered wavelengths in (A) (clockwise from top-left) (E) A schematic of the
dark field scattering setup used in the measurements. From [61].
Figure 30
(online color at: www.lpr-
journal.org) SEM image of 2D Gaus-
sian Prime aperiodic array of 100 nm-
radius and 40 nm-high cylindrical Cr nano-
particles on a quartz substrate (a) and the
array dark-field images with (c) and with-
out (b) a thin layer of silk polymer on the
array surface. The sensitivity to different
thicknesses of silk monolayers is quanti-
fied by the spectral shift of the scattered
radiation peaks (d) and by monitoring the
spatial changes of patterns quantified by
the variances of their spectral correlation
functions (ACFs) (e). From [61].
exploited in the engineering of nonlinear photonic structures
for multiple-wavelength or broadband optical frequency con-
version. Such structures make use of the three-wave mixing
process when two incoming waves
ω1k1
and
ω2k2
can
interact via the medium second-order nonlinear susceptibil-
ity tensor
χ2
to produce an outgoing wave
ω3k3
with
the frequency
ω3ω1ω2
and the wave vector mismatch
Δk k1k2k3
[200]. The efficiency of the frequency con-
version process is proportional to the value of the Fourier
transform of the relevant component of
χ2
at
Δk
, and thus
is maximized if the reciprocal lattice of
χ2
features a Bragg
peak corresponding to the mismatch vector
Δk
. This ap-
proach, known as the “quasi-phase matching” [200] can be
implemented in the medium with periodic modulation of
© 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim www.lpr-journal.org

Laser Photonics Rev. 6, No. 2 (2012)
REVIEW
ARTICLE
203
the sign of the nonlinear susceptibility with the period cor-
responding to the phase mismatch. However, periodically-
modulated nonlinear structures are mostly limited to the
phase-matching of a single optical process, while simultane-
ous phase-matching of multiple processes is important for
a variety of applications including generation of multicolor
optical solitons and creation of multi-frequency optical and
THz sources. Multiple-wavelength frequency conversion in
nonlinear periodic lattices can only be achieved by using
reciprocal wave vectors that are integral multiples of the
primary one, thus severely limiting the number of differ-
ent wavelengths for which the quasi-phase matching can
be realized.
To the contrary, nonlinear aperiodic structures provide
a large number of reciprocal vectors, which make possible
multiple-wavelength frequency conversion [201]. Multiple-
wavelength second and third harmonic generation has been
successfully demonstrated, both theoretically and experi-
mentally, in a variety of 1D
[202–206]
and 2D
[207–212]
aperiodic lattices. An example of the effective generation
of red, green and blue light by frequency doubling at three
wavelengths in 2D decagonal LiNbO
3
nonlinear photonic
structure is demonstrated in Fig. 31a–c [207]. The reciprocal
space of the decagonal structure shown in Fig. 31a features
tenfold rotational symmetry and four concentric sets of most
strongly pronounced Bragg peaks. The three reciprocal vec-
tors (labeled in the symbolic five-dimensional vector nota-
tion) that provide phase matching of the frequency doubling
process at three different wavelengths are shown in Fig. 31b,
and the conversion efficiency for the generation of the red,
green and blue coherent radiation is plotted in Fig. 31c as a
function of the input power of the fundamental beam.
Another example of an aperiodic structure, which is
engineered to provide a broadband second harmonic gen-
eration, is shown in Fig. 31d. This structure is obtained by
arranging randomly-oriented identical unit cells (squares
or other polygons) in a 2D periodic lattice. The recip-
rocal space of this structure features both sharp Bragg
peaks reflecting the square-lattice arrangement of unit cells
and broad concentric rings related to their random rota-
tions (Fig. 31e). Continuously-distributed reciprocal vectors
within the rings provide the phase-matching conditions for
the broadband second harmonic generation with the spec-
trum of the generated light almost covering the whole visible
range (Fig. 31f). In general, aperiodic nonlinear structures
can be designed to feature any set of wavevectors required
Figure 31
(online color at: www.lpr-journal.org)
Multiple-wavelength second harmonic genera-
tion in nonlinear aperiodic structures. Decago-
nal quasiperiodic lattice (a) features a number of
sharp Bragg peaks in its reciprocal spectrum (b),
and provides phase-matching conditions for sev-
eral wave-mixing processes (c) (adapted with
permission from [207]). Short-range-ordered
structure (d) features broad continuous rings
in its Fourier space (e), making possible broad-
band second harmonic generation with nearly
constant conversion efficiency (f) (Adapted with
permission from [208]).
www.lpr-journal.org © 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim

204
LASER & PHOTONICS
REVIEWS
L. Dal Negro and S. V. Boriskina: Deterministic aperiodic nanostructures for photonics and plasmonics applications
to provide the condition for quasi-phase matching of several
wave mixing processes [206, 210, 211]. However, this is
only possible at the costs of lower conversion efficiency and
larger structure sizes.
4. Complex Aperiodic
Nanoplasmonics (CAN)
Nanoplasmonics is the science of collective oscillations of
metal conduction electrons, which occur at metal-dielectric
interfaces, metal nanoparticles, or nanoparticles aggre-
gates [213].
These oscillations, called surface plasmons, are mostly
electronic oscillations and therefore can be localized on the
nanoscale. The nanoscale localization of plasmonic reso-
nances creates high intensity electromagnetic fields called
“hot spots”, or “giant fields”, which are the basis of numerous
effects and applications to nanoplasmonics and nanopho-
tonics. Several approaches have been proposed to enhance
the field localization and intensity on the nanoscale, includ-
ing bow-tie nano-antennas, periodic arrays of nanoparti-
cles or nano-holes, and photonic-plasmonic band-gap sys-
tems
[214–219]
. However, today the best approaches to
generate strongly enhanced electromagnetic fields rely on
“roughening” of metal surfaces by etching or by colloidal
synthesis of nanoparticles [220]. This often results in ran-
dom aggregates of metal nanoparticles or surface corruga-
tions statistically described by fractal morphologies that
can lead to a dramatic “structural enhancement” of the
local electromagnetic fields sufficient for observing sin-
gle molecules by Surface Enhanced Raman Spectroscopy
(SERS) [221, 222]. Differently from the familiar shapes of
Euclidean geometry, such as squares, circles, etc, fractal
objects are characterized by a non-integer dimensionality
(i. e., Hausdorff dimension), which is always smaller than
the dimensionality of the space in which fractals are embed-
ded. The fractal dimensionality describes their distinctive
scale-invariant symmetry, which is also referred to as self-
similarity, meaning that the spatial structures observed on
one length scale appears identical when observed at succes-
sively smaller scales [223]. The physical principles, com-
putational methods, as well as the engineering aspects of
fractal electrodynamics for the design and implementation
of multiband antenna elements and arrays are beautifully
reviewed in [224].
4.1. Nanoplasmonics of fractal structures
The optical excitations of small-particle statistical fractal ag-
gregates have been abundantly investigated by Shalaev in re-
lation to surface-enhanced optical nonlinearities [225, 226].
Specific scaling laws and close-form analytical results for
enhanced Raman and Rayleigh scattering, four-wave mix-
ing, and Kerr nonlinearities along with important figures of
merits are obtained within the quasi-static dipole approxi-
mation and beautifully discussed in [225, 226].
Plasmonic nanostructures arranged according to deter-
ministic fractals (i. e., Sierpinski carpet) have also been
recently studied in a computational work that demonstrates
their sub-diffraction focusing properties [227]. A recent de-
sign paper has additionally investigated the potential of Ag
nanocylinders arranged in a Pascal triangle for the genera-
tion of controllable local field enhancement [228].
Stockman [229] developed a comprehensive theory of
the statistical and localization properties of dipole eigen-
modes (plasmons) of fractal and random non-fractal clus-
ters. Because of scale-invariance symmetry, the eigenmodes
of fractals cannot be extended running waves as for trans-
lational invariant (i.e., periodic) structures. On the oppo-
site, fractal clusters of small metallic particles support a
variety of dipolar eigenmodes distributed over wide spec-
tral ranges. The vibration eigenmodes of fractals, generally
known as fractons [91], tend to be spatially localized and
are characterized by strong fluctuations of their intensity
profiles (see Fig. 32). Differently from the case of random
systems, fractons have very inhomogeneous localization
patterns and very different coherence length can coexist
at the same frequency. As shown by Stockman [229], the
plasmon eigenmodes of metal-dielectric fractal structures
can even result in a distinctive chaotic behavior in the vicin-
ity of the plasmon resonance of individual particles. This
chaotic behavior consists of rapid changes of the phase of
the amplitude correlation in spatial and frequency domains,
and cannot be observed in random clusters with non-fractal
geometry [229].
The large fluctuations of the local fields characteris-
tic of self-similar (fractal) structures leads to an efficient
transfer of excitations towards progressively smaller length
scales of the aggregates where the electromagnetic enhance-
ment reaches the
1012
level needed to observe single mole-
cule SERS [222, 230, 231]. Statistical fractal aggregates and
rough metal surfaces led to successful applications in single
molecule spectroscopy [230], but they lack reproducibil-
ity, the hot-spots locations and frequency spectra cannot be
known a priori, and they cannot be reliably fabricated using
nanolithography approaches. It is in fact important to realize
that any physical realization of a self-similar fractal process
is necessarily limited by a cut-off length scale associated to
the specific fabrication technology, usually electron beam
nanolithography (EBL) for plasmonic nanostructures. There-
fore, the fascinating physical properties originating from the
distinctive scale-invariance symmetry of mathematical frac-
tal objects cannot be entirely displayed by experimental frac-
tal structures, or pre-fractals. Moreover, fractal objects obey
a power-law scaling of their mass density-density correla-
tion function
ρrρr R ∝RD d
, where
D
is the fractal
dimension and
d
the Euclidean dimension of the embedding
space (i. e., D d). This defining property constraints the
mass density
ρr
of any fractal object to quickly “rarefy”
when increasing its size
R
, therefore decreasing the density
of localized fracton modes [232]. For this reason, fractals
cannot display the high density of spatial frequencies associ-
ated to the continuous Fourier transforms of pseudo-random
structures such as the Rudin-Shapiro sequence. A direct
consequence of the power-law density scaling of fractal
© 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim www.lpr-journal.org

Laser Photonics Rev. 6, No. 2 (2012)
REVIEW
ARTICLE
205
Figure 32
Spatial dis-
tribution of the local-field
intensities for external
excitation of an individ-
ual N51500 CCA clus-
ter for the values of the
frequency parameter
X
and polarizations of the
exciting radiation shown.
From [229].
objects is that only a small fraction of the total area of
fractal aggregates of metal nanoparticles is covered by local-
ized electromagnetic hot spots. This limits the technological
potential of plasmonic fractals for the engineering of pla-
nar optical devices such as light-emitters, photo-detectors,
optical biosensors that require strong enhancements of elec-
tromagnetic fields over large chip areas. On the contrary,
engineering aperiodic resonant structures with plasmonic
nanoparticles arranged in deterministic patterns with a large
density of spatial frequencies could overcome the limitations
of both fractals and random media. By generalizing aperi-
odic substitutions in two spatial dimensions, deterministic
aperiodic arrays of metallic nanoparticles with pure-point,
continuous and singular continuous diffraction spectra have
been recently demonstrated by Dal Negro and collabora-
tors [62, 80] in the context of nanoplasmonics scattering
and field localization (Fig. 33). These structures are cre-
ated by mathematical rules amenable to predictive theories,
and provide a novel engineering approach for the control
of hot-spot positions, radiation patterns and localized field
states in photonic-plasmonic nanoparticle systems between
quasiperiodicity and pseudo-randomness.
Figure 33
Schematics of the nanofabrication
(EBL) process flow developed to fabricate vari-
ous DANS using metallic nanoparticles and cor-
responding SEM pictures of arrays of Au nano-
disks (200 nm diameter, 20 nm separation, 30 nm
thickness) with different deterministic aperiodic ge-
ometries: (a) square lattice; (b) Fibonacci array;
(c) Thue-Morse; (d) Rudin-Shapiro; (e) Gaussian
prime; (f) Penrose lattice; (g) co-prime array.
www.lpr-journal.org © 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim

206
LASER & PHOTONICS
REVIEWS
L. Dal Negro and S. V. Boriskina: Deterministic aperiodic nanostructures for photonics and plasmonics applications
4.2. Aperiodic nanostructures beyond fractals
New scenarios can emerge by combining deterministic ape-
riodic geometries with resonant metallic nanostructures sup-
porting surface oscillations of conduction electrons local-
ized on the nanoscale, known as Localized Surface Plas-
mons (LSPs).
Analogously to the coupling of atomic and molecular
orbitals in solid state and quantum chemistry, individual
LSPs can strongly couple by near-field quasi-static interac-
tions and by far-field multipolar radiative effects (known
in this context as diffractive coupling), giving rise to lo-
calized photonic modes in artificial nanoparticle arrays de-
signed on templates with deterministic aperiodic order. The
interplay between these two coupling regimes offers a tun-
able approach to engineer photonic-plasmonic resonances
in complex aperiodic media with deterministic order.
The optical properties of surface plasmon-polaritons in
localized quasi-crystal arrays of sub-wavelength nanoholes
fabricated in metallic thin films have been the subject of
intense research efforts in the last few years, leading to
the demonstration of novel phenomena such as resonantly
enhanced optical transmission, sub-wavelength imaging and
super focusing effects [233–236].
Recently, DalNegro and collaborators explored 1D and
2D deterministic aperiodic arrays of metal nanoparticle ar-
rays as a novel approach to design broadband electromag-
netic coupling and sub-wavelength plasmonic field enhance-
ment for on-chip applications. In particular, they initially fo-
cused on the spectral, far-field and near-field optical proper-
ties of nanoparticle arrays generated according to symbolic
substitutions such as Fibonacci, Thue-Morse, and Rudin-
Shapiro structures characterized by multifractal and diffuse
Fourier spectra [62, 80, 237–240].
This approach offers additional flexibility in the design
of the Fourier space of plasmonic devices beyond the lim-
itations of periodic and fractal systems. In particular, this
design flexibility enables to better engineer the interplay
between short-range quasi-static coupling (i. e., plasmon
field localization at the nanoscale) and long-range radiative
coupling (e. g., multiple scattering) over broad angular and
frequency spectra.
Dal Negro and collaborators showed [62, 237] that the
aperiodic sub-wavelength modulation of particle positions
in metallic chains and arrays results in a hierarchy of gaps
in their energy spectra, and in the formation of localized
modes. The full dispersion diagrams of plasmon excitations
in quasiperiodic and aperiodic metal nanoparticle arrays are
calculated in [237, 240]. In addition, a characteristic power-
law scaling in the localization degree of the eigenstates, mea-
sured by their participation ratio, was discovered [62], result-
ing in larger intensity enhancement effects with respect to
the case of periodic plasmon arrays. Using accurate multiple
scattering calculations (Generalized Mie Theory, T-matrix
null-field method approach) the scattering and extinction
efficiencies of periodic and deterministic aperiodic arrays of
metal nanoparticles were compared [239] for different ge-
ometries and lattice parameters, establishing the importance
of radiative coupling effects in the plasmonic response of
deterministic aperiodic structures. A rigorous analysis was
performed by Forestiere and co-workers [240] who devel-
oped a theory that enables the quantitative and predictive
understanding of the plasmon gap positions, field enhanced
states, scattering peaks of metallic quasi-periodic arrays of
resonant nanoparticles in terms of the discontinuities of their
Fourier spectra. This work extends the reach of the so-called
gap-labeling theorem [64] to aperiodic nanoplasmonics.
The role of nanoparticle shape/size and the uniqueness
of deterministic aperiodic arrays for the engineering of the
spatial localization of plasmonic modes are discussed fur-
ther in [59, 239, 240]. These works highlight the unique
advantages offered by the controllable density of spatial
frequencies in aperiodic Fourier space, and show that elec-
tromagnetic hot-spots with larger field enhancement values
cover a larger surface areas of aperiodic arrays with respect
to periodic structures.
Gopinath and collaborators have fabricated using Elec-
tron Beam Lithography (EBL) 2D arrays of Au nano-disks
in various deterministic aperiodic geometries [80], and they
demonstrated broad plasmonic resonances spanning across
the entire visible spectrum (Fig. 34). It was also discovered
in [80] that far-field radiative coupling in deterministic aperi-
odic structures leads to the formation of distinctive photonic
resonances with spatially inhomogeneous profiles, similarly
to the case of colorimetric fingerprints of aperiodic surfaces
discussed in Sect. 3.4. In addition, the interplay between
quasi-static plasmonic localization and photonic localiza-
tion of morphology-dependent optical modes associated to
radiative long-range coupling in aperiodic arrays has been re-
Figure 34
(online color at: www.lpr-journal.org) Measured extinc-
tion as a function of the wavelength for 2-D arrays of Au nanopar-
ticles with radii of 100 nm: (a) periodic, (b) Fibonacci, (c) Thue-
Morse, and (d) Rudin-Shapiro, and varied minimum separation
distances: 50 nm (black), 100 nm (navy), 150 nm (blue), 200 nm
(dark cyan), 300 nm (green), 400 nm (magenta) and 500nm (red).
The varying intensity plateaus at around 400 nm arise from differ-
ent filling fractions of the arrays. Taking into account the collection
angle of the objective (40.5 ) and the cone of total internal reflec-
tion (39.0 ), we estimate that we collect 17% of the scattered light.
From [80].
© 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim www.lpr-journal.org

Laser Photonics Rev. 6, No. 2 (2012)
REVIEW
ARTICLE
207
cently exploited by the Dal Negro group for the engineering
of substrates with large values (
107
–
108
of reproducible
SERS enhancement [241,242]. Engineered SERS substrates
with cylindrical and triangular Au nanoparticles of different
diameters, separations and aperiodic morphologies were fab-
ricated by EBL and experimentally characterized by molec-
ular SERS spectroscopy. Large, morphology-dependent val-
ues of average SERS enhancement (i. e., averaged over the
laser excitation area) in DANS arrays with 25nm minimum
separations [242] were reported (Fig. 35). The fundamen-
tal role of long-range radiative coupling in the formation
of local hot-spots was discussed [242], along with engi-
neering scaling rules for DANS with different degrees of
spectral complexity. Larger values of SERS enhancement
were recently obtained by Gopinath and collaborators us-
ing a combination of EBL and in-situ chemical reduction
giving rise to multi-scale aperiodic structures referred to as
“plasmonic nano-galaxies” [241]. Previous studies on the far-
field and near-field optical behavior of 2D Fibonacci lattices
fabricated by EBL demonstrated the presence of strongly
localized plasmon modes whose exact location can be accu-
rately predicted from purely structural considerations [238].
In particular, by performing near-field optical measurements
in collection mode and 3D FDTD simulations, Dallapiccola
at al [238] showed that plasmonic coupling in a Fibonacci
lattice results in deterministic quasi-periodic sub-lattices
of localized plasmon modes which follow a Fibonacci se-
quence. In addition, stronger field enhancement values were
experimentally observed in Fibonacci compared to peri-
odic nanoparticle arrays [238], unveiling the potential of
quasiperiodic gold nanoparticle arrays for the engineering
of novel nanoplasmonic devices. More recently, by engi-
neering the scattering properties of quasiperiodic Fibonacci
Au nanoparticle arrays, Gopinath and collaborators [243]
fabricated the first plasmonic-coupled quasiperiodic light
emitting device using Erbium doping of silicon nitride. In
this work [243], by engineering quasi-periodic structures
with near-infrared spectral resonances, they demonstrated a
3.6 times enhancement of the photoluminescence intensity
of Erbium atoms. In addition, due to the modification of
the local density of optical states (LDOS) at the 1.54
μ
m
emission wavelength, a substantial enhancement of the Er
emission rate was also observed [243] (see Fig. 36).
In the context of nanoplasmonics, aperiodic arrays of
Au nanoparticles with diffuse, circularly symmetric Fourier
space were recently investigated by Trevino et al [172]. By
studying light scattering from the three main types of Vo-
gel’s spirals fabricated by electron-beam lithography on
quartz substrates (Figs. 23 and 37 a,c), Trevino et al. showed
that plasmonic spirals support distinctive structural reso-
nances with circular symmetry carrying orbital angular mo-
mentum (Fig. 37b,d). Moreover, due to the distinctive circu-
lar symmetry of the Fourier space, polarization-insensitive
planar light diffraction was demonstrated in aperiodic spi-
rals across a broad spectral range, providing a novel strategy
for the engineering of diffractive elements that can enhance
light-matter coupling on planar surfaces over a broad range
of frequencies [172].
Figure 35
(online color at: www.lpr-
journal.org) SERS platforms based on Fi-
bonacci plasmonic nanoparticle arrays. Exper-
imental SERS spectra of pMA on lithographi-
cally defined arrays of (a) nanodisks, (b) nano-
triangles, and (c) nanodisks decorated by small
Au spheres (termed plasmonic nanogalaxy). Au
particles are 200 nm in size and are separated
by minimum interparticle gaps of 25 nm. The
insets show an in-plane electric field pattern
in the nanodisk array at the pump wavelength
(a), an SEM image of a nanotriangles array
(b) and an SEM image of a nanogalaxy array
together with a field distribution around one of
the electromagnetic hot spots in the structure
(c). (d) The scaling behavior of Raman enhance-
ment factor calculated from experimental data
in periodic (dash blue) and Fibonacci (solid red)
nanoparticle arrays. Circles, triangles and stars
correspond to the nanodisk, nanotriangles and
nanogalaxy arrays, respectively. (e) Experimen-
tal stokes SERS spectrum of E-coli bacteria on
the Fibonacci Au nanogalaxy array with 25nm
min interparticle gaps. The inset shows the SEM
image of bacteria on the SERS chip. Adapted
from [241, 242].
www.lpr-journal.org © 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim

208
LASER & PHOTONICS
REVIEWS
L. Dal Negro and S. V. Boriskina: Deterministic aperiodic nanostructures for photonics and plasmonics applications
Figure 36
(online color at: www.lpr-journal.org) Demonstration of
light emission enhancement from Erbium atoms coupled to quasi-
periodic plasmonic arrays of Au nanoparticles (200 nm diameters).
SEMs pictures of periodic (a) and quasiperiodic Fibonacci (b) Au
nanoparticle arrays fabricated on light emitting Er:SiNx substrates
of 80 nm thickness. (c) PL spectra excited at 488 nm through peri-
odic and aperiodic nanoparticle arrays with 50 nm min interparticle
separations; (d) PL time decay of Er atoms through unpatterned
substrate (black) and Fibonacci arrays with varying interparticle
separations indicated in the legend. From [243].
Figure 37
(online color at: www.lpr-journal.org) SEM micrographs
of a Au nanoparticle aperiodic
α2
-spiral (a) and golden angle
(c) spiral arrays. The spirals contain approximately 1,000 parti-
cles with a diameter of 200 nm. Dark-field microscopy images of
plasmonic golden angle (b) and
α2
-spiral (d) spirals on quartz
substrates. Adapted from [172].
The importance of non-crystallographic rotational sym-
metries in quasiperiodic hole arrays for enhancing the ab-
sorption of organic solar cells was recently demonstrated ex-
perimentally by Ostfeld et al [244]. In this recent work, spec-
trally broad, polarization-insensitive absorption enhance-
ment of a 24 nm-thick organic layer spin-cast on quasi-
periodic hole arrays (fabricated on silver films by Focused
Ion Beam) was measured (with 600% peak enhancement
at 700 nm) over that of a reference layer deposited on a
flat film. Moreover, in correspondence of the absorption en-
hancement, a significant fluorescence intensity enhancement
(up to a factor of 2) was observed as a result of the increased
excitation rate in the thin absorbing film [244]. These recent
results unveil the potential of engineered nanoplasmonic
structures with circularly symmetric Fourier space to en-
hance the efficiency of thin-film photovoltaic cells. In this
section, we have discussed how DANS technology could
provide an alternative route for the engineering of novel
nanoplasmonic devices with distinctive optical resonances
and field localization on the nanoscale. Furthermore, a par-
ticularly important advantage of this technological platform
is the possibility to enhance simultaneously optical cross
sections and nanoscale field intensities across broad fre-
quency spectra and over large device areas. A discussion of
these specific aspects will be presented in the next section.
4.3. Broadband enhancement of optical
cross sections
An important requirement for device applications of nanoplas-
monics, such as solar cells, optical biosensors, nonlinear
elements and broadband light sources is the ability to en-
gineer strong enhancement values of optical cross sections
and electromagnetic fields over a broad frequency range.
Resonant enhancement of nanoscale plasmon fields in pe-
riodic arrays of metal nanoparticles can be achieved at
specific wavelengths when the evanescent diffraction orders
spectrally overlap the broad LSP resonances, resulting in
strong Fano-type coupling and enhancement of optical cross
sections over a relatively narrow frequency range. Under
this condition, known as Rayleigh cut off condition for peri-
odic gratings, an incoming plane wave at normal incidence
is diffracted in the plane of the grating (i. e., at 90 degree
angle) and efficiently couples to near-field plasmonic reso-
nances enhancing the strength of local plasmonic fields. For
a plane wave incident at an oblique angle
θ0
(in a medium
of index
n0
and diffracted by a 1D periodic grating with
period
a
, the Rayleigh cut-off condition yielding strongest
plasmonic enhancement is given by:
λa
mn n0sinθ0(2)
By adjusting the angle of incidence, one can exactly con-
trol the wavelength of the strongest coupling to the LSP
resonance. This effect has previously been shown to pro-
duce extremely narrow LSP resonances, and significant en-
hancements of the LSP near-fields by Fano-type coupling ef-
fects [245].
© 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim www.lpr-journal.org

Laser Photonics Rev. 6, No. 2 (2012)
REVIEW
ARTICLE
209
Figure 38
(online color at: www.lpr-
journal.org) (a) Calculated spectral
dependence of the plasmonic near-
field enhancement in different DANS
structures of Ag nanospheres with
100 nm diameter and 25 nm sepa-
ration. (b) Maximum values of plas-
monic field enhancement in arrays
with different morphologies ranging
from periodic to quasi-periodic and
pseudo-random. Calculated Cumu-
lative Distribution of Field Enhance-
ment (CDFE) spectra for periodic
(c) and RS arrays (d). The color-
coded CDFE function measures the
geometrical fraction of the arrays
that is covered by plasmon fields
larger than a fixed value indicated
for each frequency by the left verti-
cal axis. Adapted from [59].
The two main ingredients that determine the bandwidth
and the strength (i. e., near-field enhancement) of the optical
response of plasmonic structures are: a) the linewidth of the
LSP near-field resonance, which is mostly broadened by
the metallic character (i. e., losses) of nanoparticles coupled
in the quasi-static regime; b) the availability of spatial fre-
quencies matching the Rayleigh cut-off condition over the
entire bandwidth of the LSP quasi-static response, which is
normally very broad (i. e., 50–100 nm).
Deterministic aperiodic arrays of metal nanoparticles
offer by construction a high density of spatial frequencies
and are ideally suited to feed into multiple LSP resonances
distributed across engineerable frequency bandwidths [80].
Moreover, the broadband plasmonic response of DANS can
be obtained using arrays of identical particles, differently
from random systems, where disorder in particles shapes
and sizes is often present. We recall here that the geometry
of DANS arrays is described by the spectral Fourier prop-
erties of their reciprocal space. This can be engineered to
encode large fluctuations in the spatial arrangement of differ-
ent clusters of identical particles (e. g., dimers, triplets and
other local particle configurations, Fig. 39a) which strongly
interact in the quasi-static sub-wavelength regime, broad-
ening the overall plasmonic response of the system. The
key aspect of aperiodic plasmon arrays is their ability to
further enhance the intensity of these plasmonic near-fields
by diffractive effects at multiple wavelengths, resulting in
“multiple Fano-type coupling” for structures with progres-
sively denser Fourier spectra. Moreover, the enhanced den-
sity of photonic states available in aperiodic systems re-
sults in stronger photonic-plasmonic Fano-type coupling ef-
fects compared to traditional periodic gratings (see Fig. 37).
These concepts have been recently addressed quantitatively
by Forestiere et al. who studied the plasmonic near-field
localization and the far-field scattering properties of non-
periodic arrays of Ag nanoparticles generated by prime
number sequences in two spatial dimensions [59]. In this
study, it was demonstrated that the engineering of dense ar-
rays characterized by large values of spectral flatness in the
Fourier space is necessary to achieve a high density of elec-
tromagnetic hot-spots distributed across broader frequency
ranges and larger surface areas with respect to both periodic
and quasi-periodic structures [59]. The varying degree of
structural complexity of the different arrays was quantified
by a parameter, called the spectral flatness (SF), associated
to their Fourier spectra (see Fig. 38). The SF is a digital
signal processing measure of how spectrally diffuse a signal
is. In our case, the different arrays are considered as dig-
itized spatial signals and the SF is calculated by dividing
the geometric mean and the arithmetic mean of their power
spectra, according to the definition [59]:
SF
NN1
∏
n0
DFT s n
N1
∑
n0
DFT s n
N
(3)
where
s n
is the value of the spatial signal (array) in bin
n
,
N
is the total number of bins in the array,
DFT
is the
Discrete Fourier Transform, and is the magnitude. For a
signal with a completely flat power spectrum, the geometric
mean will equal the arithmetic mean causing the SF to be
equal to one. This indicates that there is equal power in
every frequency band. If there are frequencies with zero
power, the geometric mean will be zero so SF will also be
zero indicating a band limited signal.
www.lpr-journal.org © 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim

210
LASER & PHOTONICS
REVIEWS
L. Dal Negro and S. V. Boriskina: Deterministic aperiodic nanostructures for photonics and plasmonics applications
Figure 39
(online color at: www.lpr-
journal.org) Geometry of the optimized sil-
ver nanoparticle array (a) and its Fourier
Transform (b) (log scale). The resulting
lattice is made up by 1506 nanoparticle,
featuring a particle density of 34
μ
m
2
.In
Fig. 1b
a125
nm is the minimum cen-
ter to center distance. The central peak
and the cross in the middle of the Fourier
space result from the square symmetry
of finite size array. (c) Maximum field en-
hancement
E
spectra of the optimized
arrays (triangles) of silver nanoparticles,
illuminated, at normal incidence, by a cir-
cularly polarized plane wave of unitary
intensity, compared to the performances
of periodic array (circles), in which all al-
lowed position are filled by a particle, and
with the single particle (squares). The
color-map (d) shows the cumulative distri-
bution function of the field enhancement
(CDFE) (logarithmic scale) versus wave-
length (
x
-axis) and field-enhancement (y-
axis). Adapted from [246].
On a subsequent computational study (Fig. 39) [246],
Forestiere et al. demonstrated the role and the importance
of aperiodic particle geometries for broadband plasmonic
near-field enhancement using an evolutionary computational
technique known as “particle swarm” optimization algo-
rithm. In their study, they aimed to find array geometries
suitable to achieve high field enhancement values spanning
across the 400 nm–900 nm spectral window when the struc-
ture was illuminated by a plane wave at normal incidence.
Interestingly, the structures “selected” by the optimization
algorithm turned out to be aperiodic arrays with almost ideal
spectral flatness resulting in many closely packed particle
clusters, similarly to the typical geometries of engineered
DANS with continuous Fourier spectra [246]. These re-
sults demonstrate that significant field-enhancement effects
in nanoplasmonics can be obtained within a specified fre-
quency bandwidth by engineering deterministic aperiodic
order with a large number of spatial frequencies (spectral
flatness), enabling the simultaneous coupling of critically
localized photonic modes and sub-wavelength plasmonic
resonances at multiple frequencies. However, it is clear that
aperiodic designs come at the additional cost of a larger
system’s size compared to narrow-band periodic structures,
requiring application-driven engineering trade-offs between
intensity enhancement, frequency spectra, and device di-
mensions in real space.
5. Electromagnetic design of
aperiodic systems
The lack of translational invariance in aperiodic photonic
structures makes impossible direct application of the ana-
lytical tools based on the concepts of the Brillouin zone
and the Floquet-Bloch theorem, which are well-established
in the design of conventional 2D and 3D periodic Bra-
vais lattices [128, 178, 180, 247]. However, a number of
semi-analytical and numerical techniques have been de-
veloped to calculate the dispersion diagrams, density of
optical states and light transmission characteristics of aperi-
odic photonic structures. One approach is based on study-
ing finite-size portions of infinite arrays, e. g., obtained
by performing only a few iterations of the inflation rule
used to define the aperiodic structures [80, 105, 111,142,
146, 151, 154, 155, 239, 242] or by truncating the size of
a quasiperiodic tiling obtained by the cut-and-projection
method [135,144,150, 158,165,248]. Scaling analysis of the
bandgap formation and modes localization properties can
then be performed by comparing structures of progressively
increasing size
[105, 140, 142, 143, 145–147]
. Alternatively,
infinite structures can be constructed by arranging finite-size
clusters (supercells) in a periodic arrays, which can then
be numerically simulated by imposing periodic boundary
conditions at the supercell edges [130–132, 187].
The simplest approximation that can be used to model
and design both periodic and aperiodic photonic structures
is the first-order Born approximation (also known as the
Rayleigh-Gans approximation), which only takes into ac-
count single scattering events. In the framework of the Born
approximation, the phase shift of a wave propagating inside
the particle is considered to be small. In essence, it is equiv-
alent to replacing the total field with the incident field in the
calculations of the constructive interference condition and
ignoring multiple-scattering effects. Although the validity
of the Born approximation is limited to sparse structures
and/or structures with low index contrast (
k2
0ε1V1
,
where
V
is the volume of the scatterer,
ε
is a relative per-
mittivity and
k0ωc
is the wavenumber in vacuum), it
© 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim www.lpr-journal.org

Laser Photonics Rev. 6, No. 2 (2012)
REVIEW
ARTICLE
211
Figure 40
(online color at: www.lpr-
journal.org) Fourier-spectrum analysis and
design of aperiodic structures. (a) Fourier
spectrum of the Penrose quasicrystal with
the solid circles corresponding to the Bragg
condition for the bandgaps shown in Fig. S1
(reproduced with permission from [140]).
Reciprocal lattice representation of lasing
conditions (b) and the out-of-plane emis-
sion patterns (c) in the photonic structure
with Penrose lattice (reproduced with per-
mission from [196]). (d) Fourier transform of
an aperiodic structure (shown in the inset)
designed to provide strong Bragg peaks re-
quired for the phase-matching of multiple
nonlinear frequency-conversion processes
(adapted with permission from [210]).
provides a useful tool for initial structure design. Indeed,
if the influence of the internal resonances of the scattering
object is minimized, the correlation between the Fourier
spectrum and the optical spectrum of the structure can be
very strong [144, 162], as rigorously proved by the gap-
labeling theorem in the case of 1D structures [64, 65, 71].
The spectral positions of the low-frequency bandgaps in
the optical spectra can be inferred from the locations of the
singularities in the lattice Fourier spectrum by using the
well-known Bragg law (constructive interference condition):
G2k0(4)
where
G
is a 2D reciprocal space vector of the lattice per-
mittivity profile. This is illustrated in Fig. 40a, where the
positions of the lower-frequency bandgaps in the Penrose
lattice spectrum are approximately explained by using the
Bragg condition (4) [140].
It has also been shown that the lasing modes in low-
index aperiodic structures can be analyzed and visualized
by using the reciprocal lattice representation as illustrated
in Fig. 40b,c [196]. The lasing condition (equivalent to the
standing-wave condition for a wave with a wavevector
k
in the simplest two-wave coupling case is
k k G 0
,
which is satisfied when the circle of radius
2k
intersects a
reciprocal lattice point
G
(see Fig. 40b). In sharp contrast to
periodic photonic crystals, for which the standing wave con-
dition can only be satisfied at symmetry points of the first
Brillouin zone, a large number of pronounced reciprocal
peaks in the Fourier transforms of aperiodic structures trans-
lates into a large number of supported lasing modes [196].
Furthermore, the properties of reciprocal lattices of aperi-
odic structures can be used to visualize the emission patterns
of lasing modes as shown in Fig. 40c. The phase-matching
condition between the in-plane lasing mode associated with
the reciprocal lattice point
GL
and the out-of-plane radiation
modes is satisfied if there are other major reciprocal points
GC
such that
GCGL
. Out-of-plane emission patterns
of the lasing modes can be reconstructed taking into account
that the lasing mode energy is emitted in the directions de-
fined by the
GLGC
vectors projections on the dispersion
surface of air (see Fig. 40c).
Finally, Fourier spectrum design can be used to engineer
aperiodic lattices for a specific application. One example
of the engineered nonlinear aperiodic structure is shown in
Fig. 40d, which features strong Bragg peaks at pre-defined
positions and has been used for simultaneous phase match-
ing of several optical frequency-conversion processes. In
particular, when illuminated by a single-frequency optical
wave this nonlinear structure generates a color fan – the
light output that consists of the second, third and fourth
harmonics each emitted in a different direction [210].
Photonic bandstructure of both periodic and aperiodic
lattices is investigated by solving for the eigenvalues of the
wave equation and plotting the resulting dispersion rela-
tion
ωk
[128]. The plane-wave expansion method (PWM)
has long been the main workhorse in the simulations of
the bandstructures of periodic photonic crystals owing to
its simplicity and flexibility [128, 178, 249, 250]. PWM is
a spectral method based on the expansion of both the di-
electric permittivity and the field amplitudes into Fourier
series (plane waves) on the reciprocal lattice and the use of
pseudo-periodic conditions to obtain Bloch waves that can
propagate in a given direction in an infinite structure. All
www.lpr-journal.org © 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim

212
LASER & PHOTONICS
REVIEWS
L. Dal Negro and S. V. Boriskina: Deterministic aperiodic nanostructures for photonics and plasmonics applications
possible direction of the wavevector
k
can be considered us-
ing crystallographic points. However, the spacing between
the basis vectors in the Fourier transforms of quasiperiodic
structures is described by irrational numbers, making direct
application of the PWM impossible. Yet, it has been shown
that a periodic (or rational) approximant to a quasiperiodic
structure can be constructed by replacing the basis vectors
with approximate vectors whose ratios can be expressed
by rational numbers [138, 139, 148, 149], making possible
application of PWM to the analysis of quasiperiodic lat-
tices. A complementary approach, which is an analog of
the density wave approximation in condensed matter the-
ory [251], relies on identifying a dominant set of Bragg
peaks in the reciprocal space of an aperiodic structure and
approximating the spatial distribution of the dielectric con-
stant by a Fourier-like series that involve only the reciprocal
vectors belonging to the dominant set [142,160, 252, 253].
Finally, the bandstructure of the photonic quasicrystals that
are defined by the cut-and-project construction method can
be calculated by solving Maxwell’s equations in periodic
higher dimensional crystals, to which a generalization of
Bloch’s theorem applies [254].
Other spectral methods that rely on the expansion of un-
known fields into a series of functions that form a complete
basis, e. g. “the multiple-scattering technique (MST)” [133,
135, 140, 146, 150, 155, 163, 249, 264] (also known as Gener-
alized multi-particle Mie Theory (GMT) [255]), the Transi-
tion Matrix (T-matrix) method [239, 256], and other spec-
tral methods based on surface or volume integral equa-
tions 257
–
261 can be used to study the scattering spec-
tra of aperiodic photonic structures or to probe the local
density of states. Multiple-scattering algorithms require in-
version of block-form or dense matrices and thus are best
suited for simulating finite-size aperiodic structures. For ef-
ficient simulations of large-size aperiodic lattices, multiple-
scattering formulation can be combined with the sparse-
matrix canonical-grid (CMCG) method, which makes possi-
ble calculating the interactions between far-away scatterers
via a canonical grid by using the Fast Fourier transform
algorithms [143, 145]. Finally, the Maxwell equations in
both periodic and aperiodic media with the specified bound-
ary conditions can be solved in the frame of the FDTD
method [163]. FDTD algorithms can be used to study finite-
size aperiodic structures [151, 156], and, by defining a su-
percell and imposing the periodic boundary conditions at
the supercell edges, infinite block-type lattices [130, 163]
that combine periodic and aperiodic structural properties.
6. Conclusions and future perspectives
While light transport and localization in periodic and ran-
dom structures have been investigated for decades, the study
of light scattering phenomena in deterministic aperiodic
systems is still in its infancy. In this paper, we have re-
viewed the conceptual foundation, the optical properties
and the major device applications of 1D and 2D photonics-
plasmonics optical systems with aperiodic index fluctua-
tions generated by algorithmic rules, referred to as Deter-
ministic Aperiodic Nano-Structures (DANS). The study
of DANS represents a novel, fascinating, and highly in-
terdisciplinary research field with profound ramifications
within different areas of mathematics and physical sciences,
such as crystallography and computational geometry, dy-
namical systems, and number theory. Due to the unprece-
dented complexity of their Fourier space, which can be
designed to span across all possible spectral singularity
measures, DANS provide unprecedented opportunities to
manipulate light states, diffraction diagrams, and optical
cross sections for nanophotonics and nanoplasmonics de-
vice technologies. In this paper, we specifically emphasized
structural-property relations leading to the formation of pho-
tonic pseudo-bandgaps, critically localized optical modes,
and multifractal energy spectra in aperiodic structures. The
fascinating new regime of isotropic multiple light scattering,
or “circular light scattering”, and its relevance for the for-
mation of large bandgaps, planar diffraction effects, and om-
nidirectional gaps in quasi-crystals and amorphous optical
structures was also discussed. The main device applications
of 1D and 2D photonic DANS in the linear and nonlin-
ear optical regimes, uniquely enabled by their distinctive
point-group symmetries, where also reviewed.
In the context of optical biosensing, we have discussed
nanoscale aperiodic surfaces and showed that they support
a dense spectrum of highly complex structural resonances,
i. e., colorimetric fingerprints, giving rise to efficient photon
trapping through higher-order multiple scattering processes
beyond traditional periodic Bragg scattering. These complex
colorimetric structures can be designed by simple Fourier
analysis and quantitatively modeled using existing analytical
multiple scattering theories, such as Generalized Mie The-
ory or the T-matrix null-field method theory. In the emerging
context of Complex Aperiodic Nanoplasmonics, we have
discussed the engineering of structural complexity in ape-
riodic metal-dielectric nanoparticle arrays and its potential
to boost the intensity of nanoscale localized optical fields
over large frequency spectra. Specifically, we discussed the
phenomenon of broadband photonic-plasmonic coupling,
and commented on recent device applications for the en-
hancement of linear and nonlinear optical processes on chip-
scale device structures. In the context of label-free optical
biosensing, we showed that the photonic-plasmonic cou-
pled modes of DANS have unique scaling and localization
properties that are ideally suited to enhance the sensitivity
and reproducibility of SERS substrates. Moreover, current
work on the DANS engineering of multi-frequency light
sources and lasers with tailored radiation diagrams and an-
gular spectra was also presented. The main numerical and
analytical approaches utilized to model aperiodic systems
have been briefly reviewed in this paper, with particular
attention to simple Fourier space design approaches. In this
review, we aimed at significantly broaden the engineering
perspective on Fourier space by considering various types of
aperiodic order and engineering design rules of aperiodic
structures beyond what already present in the mathematical
literature (e. g., Thue-Morse, Rudin-Shapiro, etc sequences,
etc.) or displayed by natural structures (i. e., quasicrystals).
The availability of nanoscale fabrication techniques and of
© 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim www.lpr-journal.org

Laser Photonics Rev. 6, No. 2 (2012)
REVIEW
ARTICLE
213
multi-scale computational methods with increasing predic-
tive power makes possible to take full advantage of aperiodic
Fourier spaces by directly engineering optical functionali-
ties, without reference to regularity and order in direct space.
The paradigm of DANS engineering therefore shifts the tra-
ditional engineering perspective from direct to reciprocal
space, enabling to transition flexibly between periodic and
aperiodic order uniquely driven by the need of optimizing
optical functions and performances in a much richer pa-
rameter space. Interestingly, we note that this engineering
approach, which explores the boundary between symmetry
and complexity, is often adapted by Nature in its fascinating
evolutionary strategies. In fact, more often than it is realized,
complex and multi-scale biological systems self-organize
according to information-rich, aperiodic patterns that beau-
tifully optimize specific functions (i. e., cell networks, tis-
sues and webs geometries, leaves arrangements, etc) in the
absence of directly recognizable symmetry or periodicity.
As amply discussed in this review, the absence of simple
symmetries (e. g., translational and rotational symmetry) in
complex optical media does not imply recourse to random-
ness or the associated stochastic optimization methods. On
the contrary, given their deterministic character, engineer-
ing design rules can still be established for DANS, within
clearly defined validity domains. However, the development
of a general theory, capable to expand our knowledge of
aperiodic deterministic systems beyond what presented in
this review paper, is still missing. It is our opinion that this
ambitious goal could be achieved by continuing to address
the fundamental themes in the optical physics of long-range
correlated (deterministic) aperiodic nanostructures, namely:
1)
The relation between topological order and photonic-
plasmonic modes: this effort will provide a better under-
standing of the relation between the geometrical structure
of DANS, determined by spectral measures, and their op-
tical spectra and critical resonances. Only a few classes
of aperiodic structures have been considered in the opti-
cal literature so far, limiting our ability to conceive novel
properties and functions. For instance, DANS structures
generated by number-theoretic methods, or possessing
combined rotational-translational symmetries (e. g., ran-
domized dot patterns, hyperuniform lattices, aperiodic
spirals, etc) exhibit fascinating Fourier properties, often
described by an elegant analytical approach, that are yet
almost completely unexplored in current device technol-
ogy. Moreover, the general connection between structural
complexity and anomalous optical transport still needs to
be properly formalized, despite its potential impact in the
engineering of aperiodic inhomogeneous environments
for slow-light and solar device applications.
2)
The role of structural perturbations and defects engineer-
ing: very little is currently known on the optical proper-
ties of defect-localized modes in DANS environments.
This topic is naturally connected to the understanding
of hybrid periodic-aperiodic and multi-scale order in
nanophotonics, and to the engineering of hierarchical op-
tical structures. For examples, a quantitative phase-space
redistribution model for the design of enhanced photonic
and plasmonic fields across broad frequency spectra is
still missing. Moreover, the fundamental physical and
engineering tradeoffs between field concentration, spec-
tral bandwidths, mode localization, and local symmetries
in photonic-plasmonic DANS of arbitrary geometries
still remain to be adequately addressed beyond the tradi-
tional toolsets of Fourier space analysis. More powerful
approaches better suited to understand the role of local
structural perturbations in aperiodic geometries, possi-
bly requiring the development of local-spectral analysis
and time-frequency decomposition tools (e. g., Wavelets,
Wigner transforms, phase space optics) still need to find
adequate applications in DANS engineering. We expect
that further studies will be addressing all these important
issues in the near future, leading to a more comprehen-
sive understanding of the complex optical physics of
photonic-plasmonic aperiodic nanostructures.
3)
The development of rigorous and predictive multi-scale
modeling tools: while several electromagnetic tech-
niques, both numerical and pseudo-analytical, are cur-
rently available to design specific aperiodic structures,
a general method capable of dealing with the intrinsic
multi-scale character and the large size of DANS is still
missing. Moreover, little is known on the solution of
multiple light scattering problems in nonlinear and op-
tically active (i. e., lasing, light emitting) DANS, espe-
cially in relation to nonlinear-enhanced wave localiza-
tion phenomena (e. g., aperiodic discrete breathers). The
fundamental interplay between aperiodic order and opti-
cal nonlinearity still needs to be addressed theoretically
and experimentally in photonic-plasmonic DANS, poten-
tially leading to the discovery of novel physical effects.
Moreover, novel methods for theoretical and computa-
tional research are needed for the efficient solution of
inverse scattering problems in aperiodic environments
with arbitrary Fourier spectral components. Advances in
computational methods capable of dealing with multiple
length scales in large aperiodic systems are essential in
order to leverage the unique design opportunities enabled
by the aperiodic Fourier space.
Finally, we believe that, despite the many challenges
still ahead, the engineering of DANS can provide signif-
icant advances in both fundamental optical sciences and
technological applications, potentially influencing diverse
fields such as solid-state lighting, solar cells and photon
detection, optical biosensing, and nonlinear nanophotonic
devices (i. e., modulators, switchers). Our expectation is
that optical DANS could become the platform of choice to
elaborate the architecture of the next generation of nanopho-
tonic devices capable to operate over significantly broader
frequency and angular spectra by tailoring enhanced light-
matter coupling over planar optical chips with controllable
degree of structural complexity.
Acknowledgements.
This paper is based upon the support of
the US Air Force program “Deterministic Aperiodic Structures for
on-chip nanophotonic and nanoplasmonic device applications”
under the Award FA9550-10-1-0019, the SMART Scholarship
Program, and the NSF Career Award No. ECCS-0846651.
www.lpr-journal.org © 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim

214
LASER & PHOTONICS
REVIEWS
L. Dal Negro and S. V. Boriskina: Deterministic aperiodic nanostructures for photonics and plasmonics applications
Received: 29 December 2010, Revised: 16 May 2011,
Accepted: 17 May 2011
Published online: 6 July 2011
Key words:
Photonics, plasmonics, aperiodic structures, light
localization.
Luca Dal Negro
received both the Lau-
rea in physics, summa cum laude, in 1999
and the Ph. D. degree in semiconductor
physics from the University of Trento, Italy,
in 2003. After his Ph. D. in 2003 he joined
MIT as a post-doctoral research asso-
ciate. Since January 2006 he has been
a faculty member in the Department of
Electrical and Computer Engineering and
in the Material Science Division at Boston University (BU). He
is currently an Associate Professor and a member of the Pho-
tonics Center at BU. Prof.Dal Negro manages and conducts
research projects on light scattering from aperiodic media,
nano-optics and nanoplasmonics, silicon-based nanophoton-
ics, and computational electromagnetics of complex structures.
Svetlana V. Boriskina
obtained M. Sc.
and Ph. D. degrees from Kharkov National
University (Ukraine). She is currently a
Research Associate at Boston University,
with interests in nanophotonics, plasmon-
ics, optoelectronics, metamaterials and
biosensing. Dr.Boriskina is a holder of
the 2007 Joint Award of the International
Commission for Optics and the Abdus
Salam International Centre for Theoret-
ical Physics, a senior member of the Institute of Electrical and
Electronics Engineers (IEEE), and a member of the Optical
Society of America (OSA).
References
[1] P.-E. Wolf and G. Maret, Phys. Rev. Lett. 55, 2696 (1985).
[2]
P. Sheng, Introduction to wave scattering, localization and
mesoscopic phenomena (Springer Verlag, Berlin, 2006).
[3]
M. P. V. Albada and A. Lagendijk, Phys. Rev. Lett.
55
, 2692
(1985).
[4]
D. S. Wiersma, M. P. van Albada, and A. Lagendijk, Phys.
Rev. Lett. 75, 1739 (1995).
[5]
A. Lagendijk, B. van Tiggelen, and D. S. Wiersma, Phys.
Today 62, 24–29 (2009).
[6] P. W. Anderson, Phil. Mag. B 52, 505–509 (1985).
[7] B. A. van Tiggelen, Phys. Rev. Lett. 75, 422 (1995).
[8]
A. Sparenberg, G. L. J. A. Rikken, and B. A. van Tiggelen,
Phys. Rev. Lett. 79, 757 (1997).
[9]
F. Scheffold and G. Maret, Phys. Rev. Lett.
81
, 5800 (1998).
[10]
D. S. Wiersma, M. Colocci, R. Righini, and F. Aliev, Phys.
Rev. B 64, 144208 (2001).
[11] E. Maci ´
a, Rep. Prog. Phys. 69, 397–441 (2006).
[12]
E. Maci
´
a, Aperiodic structures in condensed matter: Fun-
damentals and applications (CRC Press Taylor & Francis,
Boca Raton, 2009).
[13]
W. Steurer and D. Sutter-Widmer, J. Phys. D, Appl. Phys.
40 (2007).
[14]
A. N. Poddubny and E.L. Ivchenko, Phys. E
42
, 1871–1895
(2010).
[15]
M. Kohmoto, L. P. Kadanoff, and C. Tang, Phys. Rev. Lett.
50, 1870 (1983).
[16]
M. Kohmoto, B. Sutherland, and K. Iguchi, Phys. Rev. Lett.
58, 2436 (1987).
[17]
M. S. Vasconcelos and E.L. Albuquerque, Phys. Rev. B
59
,
11128 (1999).
[18]
M. Dulea, M. Severin, and R. Riklund, Phys. Rev. B
42
,
3680 (1990).
[19]
W. Gellermann, M. Kohmoto, B. Sutherland, and P. C. Tay-
lor, Phys. Rev. Lett. 72, 633 (1994).
[20]
R. Merlin, K. Bajema, R. Clarke, F. Y. Juang, and P. K. Bhat-
tacharya, Phys. Rev. Lett. 55, 1768 (1985).
[21] E. Maci ´
a, Appl. Phys. Lett. 73, 3330–3332 (1998).
[22] E. Maci ´
a, Phys. Rev. B 63, 205421 (2001).
[23]
M. Senechal, Quasicrystals and geometry (Cambridge Univ.
Press, Cambridge, 1996).
[24]
T. Janssen, G. Chapuis, and M. DeBoissieu, Aperiodic crys-
tals: from modulated phases to quasicrystals (Oxford Uni-
versity Press, Oxford, 2007).
[25]
M. De Graef, M. E. McHenry, and V. Keppens, J. Acoustical,
Soc. Am. 124, 1385–1386 (2008).
[26]
J. P. Allouche and J. O. Shallit, Automatic Sequences: The-
ory, Applications, Generalizations (Cambridge University
Press, New York, 2003).
[27]
M. Queff
´
elec, Substitution dynamical systems-spectral anal-
ysis (Springer Verlag, Berlin, 2010).
[28]
D. A. Lind and B. Marcus, An introduction to symbolic
dynamics and coding (Cambridge University Press, Cam-
bridge, 1995).
[29]
M. Schroeder, Number theory in science and communica-
tion: with applications in cryptography, physics, digital in-
formation, computing, and self-similarity (Springer Verlag,
Berlin, 2010).
[30]
G. H. Hardy, E. M. Wright, D. R. Heath-Brown, and J. H. Sil-
verman, An introduction to the theory of numbers (Claren-
don Press, Oxford, 1979).
[31]
S. J. Miller and R. Takloo-Bighash, An invitation to modern
number theory (Princeton University Press, Princeton, NJ,
2006).
[32]
A. N. Whitehead, An Introduction to Mathematics (H. Holt
and Company, New York, T. Butterworth, Ltd., London,
1939).
[33]
E. Schr
¨
odinger and R. Penrose, What is Life? (Cambridge
Univ. Press, Cambridge, 1992).
[34]
B. Gr
¨
unbaum and G. C. Shephard, Tilings and patterns (WH
Freeman & Co., New York, NY, USA, 1986).
[35]
J. Kepler, Harmonices mundi libri V (J. Plank, Linz.
Reprinted, Forni, Bologna, 1969).
[36] R. Penrose, Inst. Math. Appl. 7, 10 (1974).
[37]
R. Ammann, B. Gr
¨
unbaum, and G. Shephard, Discrete Com-
put. Geom. 8, 1–25 (1992).
[38]
N. G. DeBruijn, Proc. K. Ned. Akad. Wet. A
43
, 84 (1981).
[39] C. Radin, Ann. Math. 139, 661–702 (1994).
[40] L. Danzer, Discrete Math. 76, 1–7 (1989).
[41] D. Ruelle, Phys. A 113, 619–623 (1982).
[42] M. Gardner, Sci. Am. 236, 110–121 (1977).
[43]
D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, Phys.
Rev. Lett. 53, 1951 (1984).
© 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim www.lpr-journal.org

Laser Photonics Rev. 6, No. 2 (2012)
REVIEW
ARTICLE
215
[44]
D. Levine and P. J. Steinhardt, Phys. Rev. Lett.
53
, 2477
(1984).
[45]
H. J. Korsch, H. J. Jodl, and T. Hartmann, Chaos: a program
collection for the PC (Springer Verlag, Berlin, 2008).
[46]
R. C. Hilborn, N. Tufillaro, and T. American Association of
Physics, Chaos and nonlinear dynamics (Oxford University
Press, Oxford, 2000).
[47] H. Poincar ´
e, Acta Math. 13, A3–A270 (1890).
[48]
F. Diacu and P. Holmes, Celestial encounters: the origins of
chaos and stability (Princeton University Press, Princeton,
NJ, 1999).
[49] N. Lorenz Edward, J. Atmos. Sci. 20, 130–141 (1963).
[50]
S. Lynch, Dynamical systems with applications using
MAPLE (Birkhauser, Boston, 2010).
[51]
L. Lam, Introduction to nonlinear physics (Springer Verlag,
New York, 2003).
[52]
M. Kohmoto, in: Chaos and Statistical Methods, edited by
Y. Kuramoto (Springer, Berlin, 1984).
[53]
M. Kohmoto and Y. Oono, Phys. Lett. A
102
, 145–148
(1984).
[54] M. Kohmoto, Phys. Rev. B 34, 5043 (1986).
[55]
C. S. Calude, Randomness, and Complexity: from Leibniz
to Chaitin (World Scientific, Singapore, 2007).
[56]
J. Hoffstein, J. C. Pipher, and J. H. Silverman, An introduc-
tion to mathematical cryptography (Springer Verlag, New
York, 2008).
[57] C¸
.K.Ko
c¸
, Cryptographic Engineering (Springer US, New
York, 2009).
[58]
W. Trappe, L. Washington, M. Anshel, and K. D. Boklan,
Math. Intell. 29, 66–69 (2007).
[59]
C. Forestiere, G. F. Walsh, G. Miano, and L. DalNegro, Opt.
Express 17, 24288–24303 (2009).
[60]
S. V. Boriskina, S.Y. K. Lee, J.J. Amsden, F. G. Omenetto,
and L. Dal Negro, Opt. Express 18, 14568–14576 (2010).
[61]
S. Y. Lee, J.J. S. Amsden, S. V. Boriskina, A. Gopinath,
A. Mitropolous, D. L. Kaplan, F. G. Omenetto, and
L. Dal Negro, Proc. Natl. Acad. Sci. USA
107
, 12086–
12090 (2010).
[62]
L. Dal Negro, N. N. Feng, and A. Gopinath, J. Opt. A, Pure
Appl. Opt. 10, 064013 (2008).
[63]
P. Prusinkiewicz and A. Lindenmayer, The Algorithmic
Beauty of Plants (Springer-Verlag, New York, 1990).
[64] J. M. Luck, Phys. Rev. B 39, 5834 (1989).
[65] C. Godreche and J. M. Luck, Phys. Rev. B 45, 176 (1992).
[66] M. Kolar and F. Nori, Phys. Rev. B 42, 1062 (1990).
[67] G. Gumbs and M. K. Ali, Phys. Rev. Lett. 60, 1081 (1988).
[68]
X. Wang, U. Grimm, and M. Schreiber, Phys. Rev. B
62
,
14020 (2000).
[69]
E. Bombieri and J. E. Taylor, Contemp. Math
64
, 241–264
(1987).
[70]
E. Bombieri and J. E. Taylor, J. Phys. Colloq.
47
, C3-19–
C13-28 (1986).
[71]
M. Dulea, M. Johansson, and R. Riklund, Phys. Rev. B
45
,
105 (1992).
[72]
M. Kolar, M.K. Ali, and F. Nori, Phys. Rev. B
43
, 1034
(1991).
[73] M. E. Prouhet, CR Acad. Sci. Paris 33, 225 (1851).
[74] H. M. Morse, J. Math. 42 (1920).
[75]
H. S. Shapiro, Extremal problems for polynomials and
power series, Sc. M. thesis, Massachusetts Institute of Tech-
nology, Boston, MA (1951).
[76]
F. Axel and J. Allouche, J. Phys., Cond. Mat.
4
, 8713 (1992).
[77]
L. Kroon, E. Lennholm, and R. Riklund, Phys. Rev. B
66
,
094204 (2002).
[78] L. Kroon and R. Riklund, Phys. Rev. B 69, 094204 (2004).
[79]
J.-K. Yang, S. V. Boriskina, H. Noh, M.J. Rooks,
G. S. Solomon, L.D. Negro, and H. Cao, Appl. Phys. Lett.
97, 223101–223103 (2010).
[80]
A. Gopinath, S. V. Boriskina, N. N. Feng, B. M. Reinhard,
and L. D. Negro, Nano Lett. 8, 2423–2431 (2008).
[81]
A. Barb
´
e and F. Haeseler, J. Phys. A, Math. Gen.
37
, 10879
(2004).
[82]
A. Barb
´
e and F. Haeseler, J. Phys. A, Math. Gen.
38
, 2599
(2005).
[83]
A. Barb
´
e and F. Von Haeseler, Int. J. Bifurcation Chaos
17
,
1265–1303 (2007).
[84]
L. Dal Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan,
L. C. Kimerling, J. LeBlanc, and J. Haavisto, Appl. Phys.
Lett. 84, 5186 (2004).
[85]
Z. Cheng, R. Savit, and R. Merlin, Phys. Rev. B
37
, 4375
(1988).
[86] C. Tang and M. Kohmoto, Phys. Rev. B 34, 2041 (1986).
[87]
M. Kohmoto, B. Sutherland, and C. Tang, Phys. Rev. B
35
,
1020 (1987).
[88]
T. Fujiwara, M. Kohmoto, and T. Tokihiro, Phys. Rev. B
40
,
7413 (1989).
[89]
C. S. Ryu, G.Y. Oh, and M. H. Lee, Phys. Rev. B
46
, 5162
(1992).
[90]
J. M. L. C. Godreche, J. Phys. A, Math. Gen.
23
, 3769
(1990).
[91]
C. Janot, Quasicrystals: A Primer (Oxford University Press,
Oxford, 1997).
[92]
E. Maci
´
a, and F. Dom
´
ınguez-Adame, Phys. Rev. Lett.
76
,
2957 (1996).
[93]
J.-P. Desideri, L. Macon, and D. Sornette, Phys. Rev. Lett.
63, 390 (1989).
[94]
T. Hattori, N. Tsurumachi, S. Kawato, and H. Nakatsuka,
Phys. Rev. B 50, 4220 (1994).
[95]
X. Q. Huang, S. S. Jiang, R. W. Peng, and A. Hu, Phys. Rev.
B63, 245104 (2001).
[96]
R. W. Peng, X. Q. Huang, F. Qiu, M. Wang, A. Hu,
S. S. Jiang, and M. Mazzer, Appl. Phys. Lett.
80
, 3063–
3065 (2002).
[97]
L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson,
A. Lagendijk, R. Righini, M. Colocci, and D. S. Wiersma,
Phys. Rev. Lett. 90, 055501 (2003).
[98]
M. Ghulinyan, C. J. Oton, L. Dal Negro, L. Pavesi,
R. Sapienza, M. Colocci, and D. S. Wiersma, Phys. Rev.
B71, 094204 (2005).
[99] S. D. Gupta and D. S. Ray, Phys. Rev. B 38, 3628 (1988).
[100]
S.-n. Zhu, Y.-Y. Zhu, Y.-Q. Qin, H.-F. Wang, C.-Z. Ge, and
N.-b. Ming, Phys. Rev. Lett. 78, 2752 (1997).
[101]
S.-n. Zhu, Y.-Y. Zhu, and N.-B. Ming, Science
278
, 843–846
(1997).
[102]
K. Fradkin-Kashi, A. Arie, P. Urenski, and G. Rosenman,
Phys. Rev. Lett. 88, 023903 (2001).
[103] N.-h. Liu, Phys. Rev. B 55, 3543 (1997).
[104]
F. Qiu, R.W. Peng, X. Q. Huang, X. F. Hu, M. Wang, A. Hu,
S. S. Jiang, and D. Feng, Europhys. Lett. 68, 658 (2004).
[105]
X. Jiang, Y. Zhang, S. Feng, K. C. Huang, Y. Yi, and
J. D. Joannopoulos, Appl. Phys. Lett. 86, 201110 (2005).
[106]
L. Dal Negro, J. H. Yi, V. Nguyen, Y. Yi, J. Michel, and
L. C. Kimerling, Appl. Phys. Lett. 86, 261905 (2005).
www.lpr-journal.org © 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim

216
LASER & PHOTONICS
REVIEWS
L. Dal Negro and S. V. Boriskina: Deterministic aperiodic nanostructures for photonics and plasmonics applications
[107]
L. Dal Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan,
L. C. Kimerling, J. LeBlanc, and J. Haavisto, New Mater.
Microphotonics 871, 75–81 (2004).
[108] F. Biancalana, J. Appl. Phys. 104, 093113 (2009).
[109]
D. L. Jaggard and X. Sun, Opt. Lett.
15
, 1428–1430 (1990).
[110]
X. Sun and D. L. Jaggard, J. Appl. Phys.
70
, 2500–2507
(1991).
[111]
C. Sibilia, I. S. Nefedov, M. Scalora, and M. Bertolotti, J.
Opt. Soc. Am. B 15, 1947–1952 (1998).
[112]
A. V. Lavrinenko, S. V. Zhukovsky, K. S. Sandomirski, and
S. V. Gaponenko, Phys. Rev. E 65, 036621 (2002).
[113]
S. V. Gaponenko, S. V. Zhukovsky, A.V. Lavrinenko, and
K. S. Sandomirskii, Opt. Commun. 205, 49–57 (2002).
[114]
S. V. Zhukovsky, A.V. Lavrinenko, and S.V. Gaponenko,
Europhys. Lett. 66, 455 (2004).
[115]
S. V. Zhukovsky, S.V. Gaponenko, and A.V. Lavrinenko,
Nonlinear Phenom. Complex Syst. 4, 383–389 (2001).
[116]
E. L. Albuquerque and M. G. Cottam, Phys. Rep.
376
, 225–
337 (2003).
[117]
M. S. Vasconcelos and E.L. Albuquerque, Phys. Rev. B
57
,
2826 (1998).
[118]
L. Moretti, I. Rea, L. Rotiroti, I. Rendina, G. Abbate,
A. Marino, and L. De Stefano, Opt. Express
14
, 6264–6272
(2006).
[119]
L. Mahler, A. Tredicucci, F. Beltram, C. Walther, J. Faist,
H. E. Beere, D. A. Ritchie, and D. S. Wiersma, Nat. Photon.
4, 165–169 (2010).
[120]
M. Gerken and D. A. B. Miller, IEEE Photon. Technol. Lett.
15, 1097–1099 (2003).
[121]
M. Gerken and D. A. B. Miller, Appl. Opt.
42
, 1330–1345
(2003).
[122]
F.F. Medeiros, E. L. Albuquerque, M. S. Vasconcelos, and
P. W. Mauriz, J. Phys., Cond. Mat. 19, 496212 (2007).
[123]
M. Maksimovi and Z. Jak
ˇ
s,i, J. Opt. A, Pure Appl. Opt.
8
,
355 (2006).
[124]
R. W. Peng, M. Mazzer, and K.W. J. Barnham, Appl. Phys.
Lett. 83, 770–772 (2003).
[125] A. Luque and A. Marti, Phys. Rev. B 55, 6994 (1997).
[126]
S. Y. S. Kinoshita, J. Miyazaki, Rep. Prog. Phys.
71
, 1–30
(2008).
[127] E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[128]
J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and
R. D. Meade, Photonic crystals: molding the flow of light
(Princeton University Press, Princeton, NJ, 2008).
[129] S. John, Phys. Rev. Lett. 58, 2486 (1987).
[130]
Y. S. Chan, C. T. Chan, and Z.Y. Liu, Phys. Rev. Lett.
80
,
956 (1998).
[131]
S.-C. Cheng, X. Zhu, and S. Yang, Opt. Express
17
, 16710–
16715 (2009).
[132]
Y. Yang and G. P. Wang, Opt. Express
15
, 5991–5996
(2007).
[133]
E. Di Gennaro, C. Miletto, S. Savo, A. Andreone,
D. Morello, V. Galdi, G. Castaldi, and V. Pierro, Phys. Rev.
B77, 193104 (2008).
[134]
Y. Wang, Y. Wang, S. Feng, and Z.-Y. Li, Europhys. Lett.
74, 49–54 (2006).
[135]
C. Rockstuhl, U. Peschel, and F. Lederer, Opt. Lett.
31
,
1741–1743 (2006).
[136]
D. Sutter-Widmer and W. Steurer, Phys. Rev. B
75
, 134303
(2007).
[137]
E. Lidorikis, M. M. Sigalas, E. N. Economou, and
C. M. Soukoulis, Phys. Rev. B 61, 13458 (2000).
[138]
K. Wang, S. David, A. Chelnokov, and J.M. Lourtioz, J.
Mod. Opt. 50, 2095–2105 (2003).
[139]
A. Ledermann, D. S. Wiersma, M. Wegener, and
G. von Freymann, Opt. Express 17, 1844–1853 (2009).
[140]
A. Della Villa, S. Enoch, G. Tayeb, V. Pierro, V. Galdi, and
F. Capolino, Phys. Rev. Lett. 94, 183903 (2005).
[141]
A. Ledermann, L. Cademartiri, M. Hermatschweiler,
C. Toninelli, G.A. Ozin, D. S. Wiersma, M. Wegener, and
G. von Freymann, Nat. Mater. 5, 942–945 (2006).
[142]
M. Kaliteevski, V. Nikolaev, R. Abram, and S. Brand, Opt.
Spectrosc. 91, 109–118 (2001).
[143]
Y. Lai, Z.-Q. Zhang, C.-H. Chan, and L. Tsang, Phys. Rev.
B76, 165132 (2007).
[144]
D. Sutter-Widmer, S. Deloudi, and W. Steurer, Phys. Rev. B
75, 094304 (2007).
[145]
Y. Lai, Z. Q. Zhang, C. H. Chan, and L. Tsang, Phys. Rev.
B74 54305 (2006).
[146]
S. V. Boriskina, A. Gopinath, and L. DalNegro, Opt. Ex-
press 16, 18813–18826 (2008).
[147]
A. Della Villa, S. Enoch, G. Tayeb, F. Capolino, V. Pierro,
and V. Galdi, Opt. Express 14, 10021–10027 (2006).
[148] K. Wang, Phys. Rev. B 76, 085107 (2007).
[149] K. Wang, Phys. Rev. B 73, 235122 (2006).
[150]
C. Rockstuhl and F. Lederer, J. Lightwave Technol.
25
,
2299–2305 (2007).
[151]
L. Moretti and V. Mocella, Opt. Express
15
, 15314–15323
(2007).
[152]
Y. Wang, X. Hu, X. Xu, B. Cheng, and D. Zhang, Phys. Rev.
B68, 165106 (2003).
[153]
K. Mnaymneh and R. C. Gauthier, Opt. Express
15
, 5089–
5099 (2007).
[154]
S. V. Boriskina and L. DalNegro, Opt. Express
16
, 12511–
12522 (2008).
[155]
S. V. Boriskina, A. Gopinath, and L. D. Negro, Phys. E
41
,
1102–1106 (2009).
[156]
M. E. Zoorob, M. D.B. Charlton, G. J. Parker, J. J. Baum-
berg, and M. C. Netti, Nature 404, 740–743 (2000).
[157]
X. Zhang, Z.-Q. Zhang, and C. T. Chan, Phys. Rev. B
63
,
081105 (2001).
[158]
Y. Wang, S. Jian, S. Han, S. Feng, Z. Feng, B. Cheng, and
D. Zhang, J. Appl. Phys. 97, 106112–106113 (2005).
[159]
M. A. Kaliteevski, S. Brand, R. A. Abram, T. F. Krauss,
R. D.L. Rue, and P. Millar, Nanotech. 11, 274–280 (2000).
[160]
M. A. Kaliteevski, S. Brand, R. A. Abram, T. F. Krauss,
P. Millar, and R. M. De La Rue, J. Phys., Cond. Mat.
13
,
10459–10470 (2001).
[161]
R. Gauthier and K. Mnaymneh, Opt. Express
13
, 1985–1998
(2005).
[162]
M. E. Pollard and G. J. Parker, Opt. Lett.
34
, 2805–2807
(2009).
[163]
S. S. M. Cheng, L.-M. Li, C. T. Chan, and Z. Q. Zhang, Phys.
Rev. B 59, 4091 (1999).
[164] M. C. Rechtsman, H.-C. Jeong, P. M. Chaikin, S. Torquato,
and P. J. Steinhardt, Phys. Rev. Lett.
101
, 073902–073904
(2008).
[165]
J. Yin, X. Huang, S. Liu, and S. Hu, Opt. Commun.
269
,
385–388 (2007).
[166]
S. David, A. Chelnokov, and J. M. Lourtioz, IEEE J. Quan-
tum Electron. 37, 1427–1434 (2001).
[167]
A. David, T. Fujii, E. Matioli, R. Sharma, S. Nakamura,
S. P. DenBaars, C. Weisbuch, and H. Benisty, Appl. Phys.
Lett. 88, 073510–073513 (2006).
© 2012 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim www.lpr-journal.org

Laser Photonics Rev. 6, No. 2 (2012)
REVIEW
ARTICLE
217
[168]
H. B. M. Rattier, E. Schwoob, C. Weisbuch, T.F. Krauss,
J. M. Smith, R. Houdre, U. Oesterle, Appl. Phys. Lett.
83
,
1283 (2003).
[169]
J. D. H. Benisty, A. Telneau, S. Enoch, J. M. Pottage,
A. David, IEEE J. Quantum Electron. 44, 777 (2008).
[170]
K. B.C. Weismann, N. Linder, U. T. Schwartz, Laser Pho-
tonics Rev. 3, 262 (2009).
[171] D. R. D. P. L. Hagelstein, Opt. Letters 24, 708 (1999).
[172]
H. C. J. Trevino and L. Dal Negro, Nano Lett., DOI:
10.1021/nl2003736 (2011).
[173]
P. Ball, Shapes (Oxford University Press, New York, 2009).
[174]
R. V. Jean, Phyllotaxis (Cambridge University Press, New
York, 1995).
[175]
D. A. W. Thompson, On Growth, and Form (Dover, New
York, 1992).
[176] M. Naylor, Math. Mag. 75, 163–172 (2002).
[177] G. J. Mitchison, Science 196, 270–275 (1977).
[178]
K. M. Ho, C. T. Chan, and C. M. Soukoulis, Phys. Rev. Lett.
65, 3152 (1990).
[179]
C. M. Anderson and K. P. Giapis, Phys. Rev. Lett.
77
, 2949
(1996).
[180]
M. Plihal and A. A. Maradudin, Phys. Rev. B
44
, 8565
(1991).
[181]
A. Barra, D. Cassagne, and C. Jouanin, Appl. Phys. Lett.
72, 627–629 (1998).
[182]
H. Benisty and C. Weisbuch, Photonic crystals, in: Progress
in Optics, Vol. 49, edited by E. Wolf (Elsevier, Amsterdam,
2006), pp. 177–313.
[183]
C. M. Anderson and K. P. Giapis, Phys. Rev. B
56
, 7313
(1997).
[184]
D. Cassagne, C. Jouanin, and D. Bertho, Phys. Rev. B
52
,
R2217 (1995).
[185]
D. Cassagne, C. Jouanin, and D. Bertho, Phys. Rev. B
53
,
7134 (1996).
[186]
F. Zolla, D. Felbacq, and B. Guizal, Opt. Commun.
148
,
6–10 (1998).
[187]
H. Zhao, R. P. Zaccaria, J.-F. Song, S. Kawata, and H.-
B. Sun, Phys. Rev. B 79, 115118 (2009).
[188]
M. Florescu, S. Torquato, and P. J. Steinhardt, Phys. Rev. B
80, 155112–155117 (2009).
[189]
M. Florescu, S. Torquato, and P. J. Steinhardt, Proc. Natl.
Acad. Sci. USA 106, 20658 (2009).
[190]
R. D. Batten, F. H. Stillinger, and S. Torquato, J. Appl. Phys.
104, 033504–033512 (2008).
[191]
R. D. Meade, K. D. Brommer, A.M. Rappe, and
J. D. Joannopoulos, Phys. Rev. B 44, 13772 (1991).
[192]
E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe,
K. D. Brommer, and J. D. Joannopoulos, Phys. Rev. Lett.
67, 3380 (1991).
[193]
M. Bayindir, E. Cubukcu, I. Bulu, and E. Ozbay, Phys. Rev.
B63, 161104 (2001).
[194]
E. Di Gennaro, S. Savo, A. Andreone, V. Galdi, G. Castaldi,
V. Pierro, and M. R. Masullo, Appl. Phys. Lett.
93
, 164102–
164103 (2008).
[195]
J. Romero-Vivas, D. Chigrin, A. Lavrinenko, and C. So-
tomayor Torres, Opt. Express 13, 826–835 (2005).
[196]
M. Notomi, H. Suzuki, T. Tamamura, and K. Edagawa, Phys.
Rev. Lett. 92, 123906 (2004).
[197]
J. A. K. L. Tsang, K. Ding, Scattering of electromagnetic
waves: Theories, and Applications (John Wiley & Sons,
Inc., New York, 2000).
[198]
I. M. White and X. Fan, Opt. Express
16
, 1020–1028 (2008).
[199] Y. Xu, Appl. Opt. 34, 4573–4588 (1995).
[200]
J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Per-
shan, Phys. Rev. 127, 1918 (1962).
[201]
R. Lifshitz, Nanotechnology and quasicrystals: from self-
assembly to photonic applications, in: Silicon Versus Car-
bon (Springer Verlag, Dordrecht, 2009), pp. 119–136.
[202]
B.-Y. Gu, Y. Zhang, and B.-Z. Dong, J. Appl. Phys.
87
,
7629–7637 (2000).
[203]
Y. Qin, C. Zhang, D. Zhu, Y. Zhu, H. Guo, G. You, and
S. Tang, Opt. Express 17, 11558–11564 (2009).
[204]
Y. W. Lee, F. C. Fan, Y. C. Huang, B. Y. Gu, B. Z. Dong, and
M. H. Chou, Opt. Lett. 27, 2191–2193 (2002).
[205]
H. Liu, S. N. Zhu, Y.Y. Zhu, N. B. Ming, X. C. Lin,
W. J. Ling, A. Y. Yao, and Z. Y. Xu, Appl. Phys. Lett.
81
,
3326–3328 (2002).
[206]
A. Bahabad, N. Voloch, A. Arie, and R. Lifshitz, J. Opt. Soc.
Am. B 24, 1916–1921 (2007).
[207]
Y. Sheng, J. Dou, B. Cheng, and D. Zhang, Appl. Phys. B,
Lasers Opt. 87, 603–606 (2007).
[208]
Y. Sheng, J. Dou, B. Ma, B. Cheng, and D. Zhang, Appl.
Phys. Lett. 91, 011101–011103 (2007).
[209]
Y. Sheng, S. M. Saltiel, and K. Koynov, Opt. Lett.
34
, 656–
658 (2009).
[210]
R. Lifshitz, A. Arie, and A. Bahabad, Phys. Rev. Lett.
95
,
133901 (2005).
[211]
A. Bahabad, A. Ganany-Padowicz, and A. Arie, Opt. Lett.
33, 1386–1388 (2008).
[212] V. Berger, Phys. Rev. Lett. 81, 4136 (1998).
[213]
S. A. Maier, Plasmonics: fundamentals and applications
(Springer Verlag, New York, 2007).
[214]
J. Dai, F. Ccaronajko, I. Tsukerman, and M. I. Stockman,
Phys. Rev. B 77, 115419 (2008).
[215]
S. Enoch, R. Quidant, and G. Badenes, Opt. Express
12
,
3422–3427 (2004).
[216]
N. Felidj, J. Aubard, G. Levi, J. R. Krenn, M. Salerno,
G. Schider, B. Lamprecht, A. Leitner, and F. R. Aussenegg,
Phys. Rev. B 65, 075419 (2002).
[217]
D. A. Genov, A. K. Sarychev, V. M. Shalaev, and A. Wei,
Nano Lett. 4, 153–158 (2003).
[218]
B. Khlebtsov, A. Melnikov, V. Zharov, and N. Khlebtsov,
Nanotechnology 17, 1437 (2006).
[219]
K. Li, M. I. Stockman, and D. J. Bergman, Phys. Rev. Lett.
91, 227402 (2003).
[220]
K. Kneipp, M. Moskovits, and H. Kneipp, Surface-enhanced