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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

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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.

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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)

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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

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LASER & PHOTONICS

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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

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Laser Photonics Rev. 6, No. 2 (2012)

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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.

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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

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Laser Photonics Rev. 6, No. 2 (2012)

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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-

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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-

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Laser Photonics Rev. 6, No. 2 (2012)

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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].

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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

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Laser Photonics Rev. 6, No. 2 (2012)

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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

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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].

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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

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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

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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-

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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,

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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-

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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

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Laser Photonics Rev. 6, No. 2 (2012)

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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-

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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

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Laser Photonics Rev. 6, No. 2 (2012)

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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-

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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-

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Laser Photonics Rev. 6, No. 2 (2012)

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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

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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

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Laser Photonics Rev. 6, No. 2 (2012)

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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]).

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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)

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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.

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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)

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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].

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LASER & PHOTONICS

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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)

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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.

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LASER & PHOTONICS

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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

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Laser Photonics Rev. 6, No. 2 (2012)

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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

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LASER & PHOTONICS

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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

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Laser Photonics Rev. 6, No. 2 (2012)

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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.

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214

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REVIEWS

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).

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