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INVITED
PAPER
Leaky-Wave Theory,
Techniques, and Applications:
From Microwaves to
Visible Frequencies
The theory of electromagnetic leaky waves can explain a variety of electromagnetic
phenomena and scenarios, from microwave to visible frequencies.
By Francesco Monticone, Student Member IEEE, and Andrea Alu
`,Fellow IEEE
ABSTRACT |Leaky waves have been among the most active
areas of research in microwave engineering over the second
half of the 20th century. They have been shown to dominate
the near-field of several open wave-guiding structures, of great
interest to tailor their radiation, guidance and filtering prop-
erties. The elegant theoretical analyses and deep physical in-
sights in this area, developed in an era in which computational
resources were limited, represent a fundamental scientific le-
gacy that is still extremely relevant in today’s engineering so-
ciety and beyond. In this regard, the relevance of leaky-wave
concepts has been increasingly recognized in recent times over
a broader scientific community, including optics and physics
societies. In this paper, after revisiting the fundamental con-
cepts of leaky-wave theory, we discuss and connect different
relevant research activities in which leaky-wave concepts have
been applied, with the goal of facilitating multidisciplinary in-
teractions on these topics. In addition to the canonical micro-
wave applications of leaky waves, particular attention is
devoted to a few areas of interest in modern optics, such as
directive optical antennas, extraordinary optical transmission,
and embedded scattering eigenvalues, in which leaky waves
play a fundamental role.
KEYWORDS |Antennas; leaky waves; metamaterials; optics;
plasmonics
I. INTRODUCTION
In wave physics, the damping of harmonic oscillations in
time and/or space is intuitively associated with losses in a
dissipative system. However, it is well known that in open
systems the oscillation energy can also be gradually lost in
the form of radiation toward the remote boundaries of an
open region, hence reducing the amplitude of oscillations
even when the system is ideally nondissipative. Examples
of this phenomenon are ubiquitous in several areas of
physics and engineering, such as radioactive states in
quantum mechanics, damped resonances in open acoustic
or electromagnetic cavities, and leaky waves in open wave-
guiding structures [1]. Although the concepts of ‘‘radiation
loss’’ and ‘‘energy leakage’’ appear rather intuitive at first,
the analysis of wave localization and guiding in open sys-
tems has proved to be very interesting and at the same time
theoretically challenging, revealing surprising and coun-
terintuitive features, as we discuss in the following. In this
paper, we specifically focus on the fundamental and ap-
plied aspects of electromagnetic leaky waves, an exciting
research topic that traces back its origins to the early stages
of microwave engineering, and whose importance is now
being increasingly recognized in many scientific commu-
nities. In the last decades, leaky-wave concepts have been
successfully applied to design radiating systems (leaky-
wave antennas) and to explain and interpret several phe-
nomena throughout the electromagnetic spectrum, such as
Cherenkov radiation, Wood’s anomalies, and extraordinary
optical transmission. The goal of this paper is to review and
connect these areas of research, spanning several dis-
ciplines, and provide the reader with the fundamental
background information to understand and apply leaky-
wave concepts in different electromagnetic scenarios, from
microwave to visible frequencies. We hope that this effort
Manuscript received September 29, 2014; accepted December 23, 2014. Date of
publication May 11, 2015; date of current version May 22, 2015. This work
has been supported by the Air Force Office of Scientific Research with Grant
No. FA9550-13-1-0204, the Army Research Office with Grant No. W911NF-11-1-0447,
the Office of Naval Research with MURI Grant No. N00014-10-1-0942, and the
Welch foundation with Grant No. F-1802.
The authors are with the Department of Electrical and Computer Engineering, The
University of Texas at Austin, Austin, TX 78712 USA (e-mail: alu@mail.utexas.edu).
Digital Object Identifier: 10.1109/JPROC.2015.2399419
0018-9219 Ó2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
Vol. 103, No. 5, May 2015 | Proceedings of the IEEE 793
may be beneficial to the microwaves, optics, engineering,
and physics communities, so that the connections of leaky-
wave theory and applications to recent advances in physics
and optics may be appreciated in the context of the wide
background of discoveries carried out by the microwave
community in the past decades.
II. FUNDAMENTALS OF
LEAKY-WAVE PHYSICS
In the early days of microwave field theory and engineer-
ing, during the first half of the 20th century, most of the
attention was devoted to guided waves in closed systems,
such as metallic waveguides [2], with many of these early
research efforts brilliantly summarized in Marcuvitz’s
handbook [3]. In closed lossless systems, any field distribu-
tion satisfying the boundary conditions can be represented
as the superposition of source-free wave eigensolutions
(i.e., self-sustained oscillations, or eigenmodes), which
form an orthogonal and complete discrete spectrum of
modes. Each of these modal, or proper, solutions is charac-
terized by finite energy everywhere, i.e., it is absolute
square integrable. In a waveguideVtransversely closed in
one or two directionsVthese eigenmodes correspond to
pole singularities of an appropriate characteristic Green’s
function in the transverse-wavenumber complex plane [1].
From the engineering standpoint, this characteristic
Green’s function can be interpreted as the voltage (or
current) in a transmission line along one of the transverse
directions of the waveguide [4], [5]. Within this network
formalism, the pole singularities correspond to resonances
of the transverse network model, which can be conve-
niently calculated using analytical methods. For instance,
as shown in Fig. 1(a) and (b), for a parallel-plate waveguide
the transverse cross-section can be modeled as a transmis-
sion-line segment terminated by lumped impedances
ðZL1¼ZL2¼0 for a waveguide with perfectly conducting
walls). The transverse wavenumbers of the source-free
solutions are then found by solving the transverse reso-
nance equation
Z
þZ
!¼0(1)
where Z
!and Z
are the impedances looking at the two
sides of an arbitrary reference plane at x¼x0.Inclosed,
nondissipative wave-guiding structures, the eigenmodes
aretypicallydescribedbypurelyrealorpurelyimaginary
transverse wavenumbers, which represent, respectively,
propagating waves with constant amplitude, or evanescent
waves with constant phase along the waveguide axis [1].
When material losses are considered, these eigenmodes
become complex, taking into account absorption and the
corresponding modal decay. Conversely, open systems
can commonly support complex eigenmodes even in the
Fig. 1. Examples of closed and open wave-guiding structures. (a) A closed parallel-plate waveguide and (b) its transverse network model
for the application of the transverse resonance method. If the walls of the waveguide are perfectly conducting (PEC), the load impedances are
ZL1 =Z
L2 =0.Z
dindicates the characteristicimpedance of thetransmissionline, which dependson the materialproperties ofthe waveguide andthe
polarization underconsideration.The position of thereference planex = x0is arbitrary. (c) A grounded dielectric slab and(d) a grounded slabwith
a partially reflecting-surface (PRS) cover (e.g., a periodic arrangement of metal strips). The open structures in (c) and (d) can also be analyzed
using the transverse resonance method, by applying the network model in (b) with ZL2 =0andZ
L1 consisting of a reactive part, modeling the
discontinuity (assumed to be lossless), and a resistive part, representing the semi-infinite free space region where radiation may occur.
Monticone and Alu
`: Leaky-Wave Theory, Techniques, and Applications: From Microwaves to Visible Frequencies
794 Proceedings of the IEEE |Vol.103,No.5,May2015
nondissipative scenario, due to radiation losses, as dis-
cussed in the following.
1
Open waveguides as radiating systems were first inves-
tigated in the pioneering work of Hansen in the late 1930s
[6], who proposed an antenna structure realized by open-
ing a longitudinal slit in the side of a rectangular wave-
guide. The early development of these concepts, however,
were fundamentally hinderedbythelimitedunderstand-
ing of the underlying physical mechanism of leaky waves.
Soon, in fact, it was recognized that a leaking waveguide
mode is characterized by a complex longitudinal wave-
number, with attenuation constant due to radiation losses.
This fact is, in some instances, associated to seemingly
unphysical results: a longitudinal attenuation may in fact
correspond to a wave amplitude increase in the transverse
plane toward infinity. Consider, for example, the cases
depicted in Fig. 1(c), (d), and Fig. 2, with longitudinal
direction z(parallel to the waveguide axis) and transverse
direction x,inwhichthetopportionofthewave-guiding
structures is now partially or fully open, such that a portion
of the energy can leak out in the upper semi-infinite re-
gion. The leaking field on the aperture will have complex
longitudinal (horizontal) wavenumber kz¼j(as-
suming an ej!ttime-harmonic convention), where the real
quantities and are, respectively, the phase and atte-
nuation constant. The field above the waveguide aperture
is then characterized, using Helmholtz equation, by a
transverse (vertical) wavenumber
kx¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k2
0k2
z
q¼xjx(2)
where k0is the free-space wavenumber. For a mode carry-
ing power along the positive z,>0 due to passivity. A
leaky mode, for which the phase also flows in the same
positive zdirection, i.e., >0, necessarily requires the
branch cut choice xG0 in (2), implying that the eigen-
mode fields exponentially increase toward infinity in the
transverse xdirection [4], [7], [8], a result that puzzled the
early pioneers in the field of leaky-wave theory, since it
violates the conventional radiation condition for accept-
able solutions of Helmholtz equation. In contrast, for a
backward leaky mode, for which the phase flows backward
with respect to the power flow, i.e., G0, as in some
periodic leaky-wave structures, it follows that x>0and
the field properly decays towards infinity [8]. The
seemingly unphysical behavior of forward leaky modes
originally led to serious skepticism toward the very
existence of leaky waves, despite the fact that energy
leakage was experimentally measurable [9]. It was not
until the late 1950s that this issue was solved, thanks to
deeper theoretical understanding and physical insights
provided by the fundamental works of Marcuvitz [1] and
Oliner [4], [5].
While in closed systems the eigenmodal spectrum is
purely discrete, in open systems a continuous eigenmodal
spectrum arises, determined by the branch cuts of (2) that
emanate from the branch points kz¼k0.Forawave-
guiding structure open on one side, the sketch in Fig. 3
shows the complex plane of the longitudinal wavenumber
kz, with the typical choice of branch cuts. A proper spectral
solution of the source-excited electromagnetic problem
implies an integration over the top Riemann sheet associated
with waves decaying at infinity, namely, Im½kxG0, or
x>0, (radiation condition). However, as indicated in
Fig. 3, pole singularities may also be present in the bottom
Riemann sheet, and in this case they are denoted as nonmodal,
or improper source-free solutions of the field equation. As
shown by Marcuvitz [1], these complex poles can actually
correspond to leaky modes. Although they do not directly
contribute to the proper spectral solution, and can therefore
Fig. 2. Sketch of leaky-wave radiation from a waveguide partially
open (z > 0). The contour plot on the xz plane depicts a time-snapshot
of the field distribution of a guided fast wave, which becomes leaky
when the wave-guiding structure is partially opened. The radiation
is confined in the region x < z tan q, as discussed in Section 2. The
plots at different cross-sections show the field amplitude along the
transverse x direction (the case considered here is 2D; therefore, the
fields are constant along y). The red curves highlight the exponential
growth(a< 0) of the field amplitude alongthe transverse direction, and
the exponential attenuation (a> 0) along the longitudinal direction.
Roman numbers indicate the field amplitude on the
aperture at three different values of z.
1
For the sake of completeness, we should also mention that also in a few
closed geometries (e.g., dielectrically loaded or, more generally, inhomo-
geneous closed waveguides) complex modes can exist in the lossless limit.
These modes appear in pairs, having opposite phase constants, and they
contribute to the local reactive power storage [200]–[202]. In addition, as
discussed in details in [10], also complex modes in lossless open structures,
i.e., leaky modes, always come in pairs, but only one of the waves in a pair
contributes to the steepest-descent representation for the total field, leading
to real power transport in the backward or forward direction (according to
whether the leaky mode is proper or improper, respectively).
Monticone and Alu
`: Leaky-Wave Theory, Techniques, and Applications: From Microwaves to Visible Frequencies
Vol. 103, No. 5, May 2015 | Proceedings of the IEEE 795
be characterized by an unphysical growth toward infinity,
they can nevertheless accurately describe the radiation field
in limited spatial regions. In fact, leaky waves are defined
only in a wedge-like region of space dependent on the
location of the source, as shown in Fig. 2. The radiation angle
is generally given by ¼tan1ðRe½kx=Re½kzÞ, which is
usually approximated (for small values of and sufficiently
long radiating structures) as [8]
2
¼cos1
k0
:(3)
For a forward wave, the amplitude of the leaky mode in-
deed is expected to exponentially increase in the vertical
(transverse) direction x,asrequiredby(2),butthemodeis
defined only in the spatial region xGztan , for a source
placed at z¼0, implying that the leaky wave has always
finite amplitude and, under suitable conditions, it domi-
nates the near-field of the source and it becomes a valid
representation of the radiation field [4]. From the mathe-
matical standpoint, the direct relevance of these improper
poles becomes evident when applying the steepest-descent
method to solve for the integral contribution of the radia-
tion continuum to the overall radiation pattern [8], [10],
[11]. As the steepest-descent path passes close to the leaky
poles, they can significantly contribute, or even dominate
under certain conditions, the overall radiation pattern of
an open waveguide in the region xGztan .Froma
physical perspective, it is easy to picture why the field
distribution of the emerging leaky wave grows in the
transverse direction, and solve the apparent paradox of the
violation of the radiation condition, by inspecting the
fields on different transverse cross-sections, as shown in
Fig. 2. In the case of a forward leaky-wave, for any value of
z, larger values of x‘‘feel’’ the radiation originating from
points where the guided-wave amplitude is larger. In other
words, the exponential attenuation along the positive zaxis
directly implies an exponential growth of the field along
the xdirection. However, as seen in Fig. 2, the field
amplitude is always finite in the wedge-like region of space
over which the leaky wave exists, since no radiation can
reach the region x>ztan . Equation (3) also reveals that
leaky-wave radiation is only supported by fast waves along
the waveguide, namely, waves with phase velocity faster
than the speed of light in the background medium, since
(3) gives a real angle of radiation only if the phase constant
jjGk0, i.e., if the leaky mode can couple to radiation
modes in the background. In other words, due to momen-
tum conservation, or phase matching, fast waves nec-
essarily couple to the propagating plane waves in the
background medium for jjGk0, while no plane wave can
match the momentum of a slow mode with jj>k0.
Similar to proper eigenmodes, leaky poles can be found
by applying the transverse resonance method to open
structures, based on their network representation. This
approach is accurate in predicting the response of open
wave-guiding structures, and it has been successfully
applied to design several antenna structures of practical
interest [7], as further described in [4], [5], [10], [11]. We
should stress, however, that the physical significance of
specific leaky poles and their relevance to the continuous
spectrum of radiation modes from a given structure can be
rigorously assessed only through a careful analysis in the
complex plane [8], [10], [11]. Moreover, in various realistic
cases, the transverse network representation of the open
structure may be rather involved, and the radiation impe-
dance at the aperture may be difficult to determine, mak-
ing the transverse resonance method not necessarily
2
It should be noted that, while for leaky-wave radiation it is common
to define the radiation angle from the surface, as in Figs. 1 and 2, in
different situations, e.g., for some of the scattering problems considered in
the following sections, the angle is defined from the surface normal.
Fig. 3. Singularities on the two-sheeted complex plane of the
longitudinal wavenumber kz=b-jafor a wave-guiding structure
open on one side, as in Fig. 1(c), (d), and Fig. 2 (if the waveguide
were open on both sides, the complex plane would consist of
four Riemann sheets). Also shown are the branch cuts (green lines)
and the branch points kz=
+
-
koexpressedby (2). In particular, withthis
choice of branch cut, the lower half of the kx-plane (which complies
with the radiation condition, i.e.,
lm[
kx
]
< 0) maps on the top Riemann
sheet of the kz-plane; conversely, the upper half of the kx-plane
(
lm[
kx
]
> 0) maps on the bottom Riemann sheet of the kz-plane [10].
In the lossless case,a pole singularity[resonance ofthe network model
in Fig. 1(b)] lying on the real axis (blue stars), with
|Re[
kz
]| >
k0
(slow wave region), corresponds to a bound surface mode, whereas
a complex pole with
|Re[
kz
]| <
k0(fast wave region, indicated by the
dotted area) represents a leaky mode. Due to passivity, the upper half
of the kz-plane (light-red area) is forbidden for any mode carrying
power along the positive z axis. For forward phase propagation,
Re[
kz
]>
0, the leaky pole (red dot) lies on the bottom Riemann sheet
(
lm[
kx
]
> 0), and the leaky mode is said to be ‘‘improper’’ (it does
not respect the radiation condition), whereas for backward phase
propagation
Re[
kz
]<
0, the leaky pole (orange dot) is ‘‘proper,’’ as it
lies on the top Riemann sheet with
lm[
kx
]
< 0 (here, the ‘‘forward’’
and ‘‘backward’’ nature of a mode is defined according to whether
its energy and phase velocities are in the same, or in the opposite,
direction). In this plot,we show only one pole for each kind of complex
modes; in reality, however, complex modes (proper, or improper)
always come in pair (having opposite phase constant b), but only
<softreturnone contributes to real power transport [10].
Monticone and Alu
`: Leaky-Wave Theory, Techniques, and Applications: From Microwaves to Visible Frequencies
796 Proceedings of the IEEE | Vol. 103, No. 5, May 2015
convenient. In such situations (e.g., the relevant case of
open microstrip structures, or, more generally, strip-
loaded dielectric slabs), full-wave numerical methods can
be applied to calculate the complex propagation constants
of leaky modes, including the mode-matching technique
[12], the finite-difference time-domain (FDTD) method
[13], different variations of the method of moments
[14]–[16] often combined with the matrix-pencil method
[17]–[19], among many others.
In this section, we have discussed the fundamental
difference between closed and open electromagnetic sys-
tems, and the interpretation of leaky waves as source-free
modal solutions in open waveguides. It is interesting to
note that, while at microwave frequencies it is easy to
define closed systems using impenetrable metallic walls
(whose thickness can remain deeply subwavelength), in
the optical range, most micro- and nanostructures are
electromagnetically open, since materials change their
conduction properties at high frequencies and ideal con-
ductors are not available in optics, implying that field
screening is more challenging at these scales. Therefore,
leaky waves are expected to play a significant role at optical
frequencies and several concepts and designs developed at
microwaves may be translated to the infrared and visible
range to realize interesting coupling effects between guid-
ance and radiation. In the next section we overview the
general principles and recent developments in the field of
leaky-wave antennas at microwave frequencies, which will
allow us to explore the connections and relevance to optics
in the following sections.
III. LEAKY-WAVE ANTENNAS: GENERAL
PRINCIPLES AND RECENT TRENDS
Systematic research on leaky-wave antennas may be traced
back to the theoretical foundations of complex guided
modeslaiddowninthelate1950s,asbrieflyoutlinedin
Section II. The field has been in continuous development
since then, and leaky-wave antennas have become increas-
ingly popular in the microwave range thanks to their ap-
pealing features, especially the possibility of realizing
highly directive antennas without the need for complicated
feeding networks typical of phased arrays. Since the body
of literature on conventional leaky-wave antennas is re-
markably large, and this review paper does not want to be a
comprehensive survey of this classical field of antenna
technology, we refer the reader to [7], [8] for several ex-
amples of microwave leaky-wave antennas and their design
principles.
The advent of metamaterials [20], or artificial materials
with unusual electromagnetic properties, has further
boosted in the last fifteen years the interest in leaky-
wave antennas, as the new concepts unveiled by these
materials have offered novel inspiration for the design of
leaky-wave structures, overcoming some of the limitations
of conventional designs. In this section, we discuss recent
advances in leaky-wave antennas, particularly at micro-
wave frequencies, beneficial to introduce, in Section IV,
the application of these concepts to the optical range. In
order to fully appreciate the latest developments in this
research area, we start by briefly reviewing the funda-
mental properties of leaky-wave antennas, highlighting the
main challenges in their design and operation.
A. Basic Properties of Leaky-Wave Antennas
A leaky wave traveling along an open wave-guiding
structure realizes an effective antenna aperture that ra-
diates energy to the far field. As for any other antenna, the
far-field radiation pattern may be obtained by performing
the Fourier transform of the aperture field. The complex
wavenumber of the leaky wave on the aperture directly
determines the main features of its radiation pattern: di-
rection of the main beam, beamwidth, and sidelobe level.
We already discussed in the previous section how the angle
of leakage of the main beam is determined by the leaky-
wave phase constant , and how it can be approximately
calculated using (3) if the phase and attenuation constants
are invariant along the aperture. The attenuation constant
controls the angular width of the main beam in the
farfield, i.e., the antenna directivity. If the leaky wave-
guide is sufficiently long to avoid end reflections, the
leakage rate directly determines the size of the effective
antenna aperture: a large (small) implies a short (long)
effective aperture, which correspondsVafter a Fourier
transformationVtoawide(narrow)beaminthefarfield.
For one-dimensional (1D) uniform leaky-wave antennas,
the half-power beamwidth is linearly proportional to =k0
through the approximate formula [8]
BW ¼2cscðÞ
k0
:(4)
Typical 1D leaky-wave antennas with a small attenu-
ation constant produce narrow beamwidth in one plane
(the xz plane in Fig. 2), but a much wider beam in the
orthogonal plane (i.e., a fan beam). However, if is small,
the slow exponential decay of the aperture field implies
poor sidelobe features [21]. This problem is usually tackled
by tapering the antenna aperture, in such a way to main-
tain constant, while slowly varying in order to achieve
the desired performance.
Another fundamental property of leaky-wave antennas
is their inherent ability to frequency scan the main lobe
direction, due to the frequency dependence of the leaky-
wave phase constant . For many conventional leaky-wave
antennas, however, beam scanning close to the broadside
ð¼=2Þand endfire ð¼0;Þdirections has proven to
be particularly challenging. Therefore, the possibility of
achieving continuous frequency scanning from backward
endfire to forward endfire, with constant beamwidth, has
Monticone and Alu
`: Leaky-Wave Theory, Techniques, and Applications: From Microwaves to Visible Frequencies
Vol. 103, No. 5, May 2015 | Proceedings of the IEEE 797
been an important research direction in recent years, as
discussed in the following. Different categories of leaky-
wave antennas exhibit drastically different frequency-
scanning responses, related to their individual geometry
and operation principle. For many broadband antenna ap-
plications, frequency scanning of the main beam may ac-
tually be undesirable. In order to achieve constant beam
direction over a wide bandwidth, the antenna structure
should support a leaky mode with low dispersion of the
phase constant . For example, almost nondispersive leaky-
wave radiation can be supported by a long slot in a PEC
sheet between different semiinfinite dielectrics, a behavior
that can be exploited to design ultrawideband leaky-wave
antennas [22]–[24].
Given the wide range of different designs proposed in
the last sixty years, it is not an easy task to categorize leaky
wave antennas in different classes, and any possible classi-
fication will inevitably have elements of arbitrariness. In
several papers, reviews and books on this topic, leaky-wave
antennas are typically classified depending on whether
the geometry of the guiding structure is uniform, quasi-
uniform, or periodic. Another important distinction can be
made between antennas with 1D, or two-dimensional (2D)
guiding structures. In the present paper, we also adopt these
classifications, with the caveat that the boundaries between
different categories are not exactly defined, and there may
be exceptions that do not fall in one specific class.
One-dimensional uniform leaky-wave antennas have a
constant geometry along the length of the structure (al-
though in some cases the antenna opening may be grad-
ually tapered, as mentioned above). A typical example is
the case of a rectangular waveguide with an open slit in the
longitudinal direction. The fundamental mode of 1D uni-
form leaky-wave antennas is a fast wave (i.e., with phase
constant 0 GGk0), which can directly couple to a pro-
pagatingplanewaveiftheguidingstructureisopen.When
fed at one end, the main beam can be scanned in the
forward quadrant, from broadside to endfire by increasing
the frequency of operation (backward-radiating homoge-
nous leaky-wave antennas have also been proposed, based
on biased ferrite materials [25]). In most designs, the per-
formance typically deteriorates when reaching the ex-
tremes of this angular range. For example, for a leaky-wave
antenna consisting of a slotted waveguide, broadside ope-
ration is difficult because it implies working at the wave-
guide cutoff frequency, while radiation exactly at endfire is
forbidden due to the radiation null of an equivalent mag-
netic dipole at the waveguide slit, consistent with (4). A
few solutions to overcome these problems have been
discussed in the literature, e.g., [7] and [8].
In 1D periodic leaky-wave antennas, a periodic modula-
tion of the guiding structure is introduced along the longi-
tudinal direction of the antenna. In contrast to uniform
structures, the fundamental mode is a slow wave ð>k0Þ,
which therefore would not radiate even if the structure is
electromagnetically open. However, due to the periodic
corrugation, the guided wave consists of an infinite num-
ber of space harmonics (Floquet modes), with longitudinal
wavenumber
kz;n¼kz;0þ2n=p(5)
where kz;0is the wavenumber of the fundamental mode
(slightly different compared to the case without periodic
perturbation), nis the harmonic order, and pis the period
in the longitudinal direction. Although the fundamental
mode is a slow wave, it is possible to design the periodic
structure such that one of the space harmonics (typically
the n¼1) is fast and, thereby, it will radiate. In many
realistic situations (e.g., periodically patterned microstrip
lines [18], [19]), the periodic perturbation may also effi-
ciently excite higher-order modes, and some of their space
harmonics may significantly contribute to the overall
antenna radiation.
One of the main advantages of 1D periodic leaky-wave
antennas, compared to uniform structures, is the fact that
the phase constant of the leaky mode can assume either
positive or negative values at will, which allows scanning
the main beam from the backward to the forward quadrant
as the frequency is increased. However, the antenna per-
formance generally degrades when we approach broadside,
due to the presence of an open stopband of the periodic
structure. From a physical viewpoint, at the open stopband
frequency the traveling wave supported by the periodic
leaky-wave antenna becomes a standing wave, behaving
similar to an antenna array in which all the radiating ele-
ments are excited with same phase [8]. As a result, the
attenuation constant drops to zero and all the reflections at
the periodic discontinuities add in phase back to the input
port, determining a purely reactive input impedance and
large mismatch.
3
The issue of poor broadside radiation in
periodic leaky-wave antennas has been one of the main
motivations to pursue novel metamaterial-inspired de-
signs, as we discuss in the following.
Another important category of leaky-wave antennas is
represented by quasi-uniform structures, which are also
characterized by a periodic modulation of their geometry.
In this case, however, the fundamental mode is a fast wave,
as in uniform structures, and the period is chosen to be
small enough such that radiation comes only from the
3
The attenuation constant drops to zero at broadside only when the
broadside point corresponds to an open stopband, i.e., a stopband of an open
periodic structure where one of the space harmonics is radiating. This is
different compared to the behavior in closed stopbands (occurring when the
mode is bound), or below the guided-mode cutoff, in which cases there is no
leaky-wave radiation, but the attenuation constant may be nonzero, since it
corresponds to reactive evanescent decays of the fields. Consider, for ex-
ample, an air-filled rectangular waveguide with a longitudinal slit, operating
with the fundamental TE10 mode, which is leaky. As the frequency is
lowered, the main beam tends to broadside, as the phase constant goes to
zero at cutoff. At the same time, the attenuation constant grows, which
corresponds to an increase of reactive attenuation, not of leakage [7].
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fundamental mode, and is not coupled to other space
harmonics [7]. In general, quasi-uniform structures with
subwavelength period can be conveniently modeled with
effective homogenous material parameters or surface
impedance concepts. Although in quasi-uniform leaky-
wave antennas the periodicity does not play a direct role in
determining the radiation, since the fundamental mode is
already a fast wave, the periodic modulation can be used to
control the attenuation and phase constants of the leaky
mode. A basic example of quasi-homogenous leaky-wave
antenna is the so-called holey waveguide, first introduced in
[26], in which a series of closely spaced holes is realized on
the short side of a rectangular waveguide. The resulting
radiating structure is quasi-uniform, as the leakage comes
from the fundamental space harmonic of the periodic
structure. Compared to a uniform slitted waveguide, this
design has the advantage of providing smaller attenuation
constants (and thereby a narrower beam), since the
periodic holes do not completely disrupt the current lines
on the waveguide wall, as a long slit would do [8]. A recent
example of quasi-uniform radiating structures are the
leaky-wave antennas based on transmission-line metama-
terials, which allow controlling the complex propagation
constant of the leaky modes to a large extent [25], as we
discuss in the next subsection.
While historically 1D leaky-wave antennas have been
the most explored geometries at microwave frequencies,
2D geometries have been attractingincreasingattentionin
the past years, since several designs based on metamaterial
concepts belong to this category. 2D leaky-wave antennas
consist of a planar guiding structure, e.g., a parallel-plate
waveguide with a partially reflective wall, supporting a
cylindrical leaky wave radially propagating outward from
the source (which may be a short dipole embedded in the
open guiding structure). Notably, 2D leaky-wave antennas
with homogenous or quasi-homogenous geometry can
realize, at a given frequency, a directive pencil beam at
broadside, with maximum broadside radiation when the
phase and attenuation constants of the leaky mode are
nearly equal [7], [8]. At other frequencies, the radiation
will be in the form of a conical beam with axis parallel to
the surface normal. Typical 2D leaky-wave structures are
based on partially reflective metallic screens, or grounded
dielectric and metamaterial slabs [7], [8]. A few examples
involving metamaterials will be discussed in the following
subsection.
Recent trends in leaky-wave antenna research include
the planarization, miniaturization and tunability of the
antenna structure, as well as the possibility to achieve
continuous frequency scanning over the entire angular
range, including broadside, which may be facilitated by
exploiting metamaterial concepts, as discussed next.
Another important trend is the investigation of leaky-
wave antennas for frequencies above the millimeter-wave
region, in particular the optical frequency range, which
will be the main subject of Section IV.
B. Leaky-Wave Antennas Based on Artificial Surfaces
and Transmission-Line Metamaterials
Planar leaky-wave antennas often involve periodically-
modulated surfaces and artificial surfaces, e.g., patterned
metallic screens, which offer further degrees of freedom in
thedesignoftheirleakageproperties.Theinvestigationof
such artificial surfaces is an important research direction,
also in relation to the rising field of metasurfacesVthe
planarized version of metamaterialsVparticularly at opti-
cal frequencies [27]–[29].
Leaky waves on periodic surfaces were first investi-
gated by Oliner and Hessel in their studies of guided waves
on sinusoidally-modulated reactance surfaces [30]. Al-
though this work was mainly motivated by improving the
performance of endfire surface wave antennas, it has in-
spired many leaky-wave antenna designs (e.g., [31], [32]),
in which the sinusoidal modulation allows an indepen-
dent control of the phase and attenuation constants of the
leaky mode. Interestingly, these ideas have also been re-
cently applied to the THz frequency range, in the form of
sinusoidally-modulated graphene leaky-wave antennas
[33], [34]. In these designs, the complex conductivity of
the graphene sheet can be modulated by applying DC bias
voltages at different gating pads along the structure, as
showninFig.4,orlaunchingan acoustic wave traveling
along the surface, allowing a unique dynamic control of the
leaky-wave radiation from the surface.
Another important related concept is the one of
high-impedance surfaces,introducedbySievenpiper,
Yablonovitch, and co-workers [35]. While a perfect electric
conducting surface allows propagation of transverse mag-
netic (TM) surface waves, but forbids transverse electric
(TE) ones, a high impedance surface behaves as an artificial
magnetic conductor, which provides the dual operation,
forbidding TM surface waves, but supporting TE propaga-
tion in the form of leaky waves. High-impedance surfaces
owe their interesting properties to periodic structures with
a resonant unit cell, as shown in Fig. 5(a) and (b), which
corresponds to a lumped inductor-capacitor (LC) resona-
tor. The dispersion diagram for a typical artificial surface of
this kind is depicted in Fig. 5(c), within the first Brillouin
zone of the periodic structure. As seen from this diagram,
the high impedance surface can be employed as a leaky-
wave antenna by using the leaky portion of the TE mode
above the ‘‘light line,’’ namely, for kkG!=c,wherekkis the
parallel wavenumber of the surface wave, !is the angular
frequency and cthe speed of light in vacuum. As usual, the
main beam can be scanned with frequency, following the
dispersion of the leaky mode; alternatively, it is also pos-
sible to steer the beam at a fixed frequency by changing the
resonance frequency of the LC unit cell, which results in a
modification of the modal dispersion as shown in Fig. 5(c).
Based on these principles, several tunable and reconfigur-
able leaky-wave antennas have been proposed, which ex-
ploit a modification of the capacitance and/or inductance of
the unit cell obtained with different mechanisms, e.g.,
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mechanically [36], or electronically [37]. Notably, artificial
impedance surfaces have also been used to realize holo-
graphic surfaces [38]–[41], which have been interestingly
connected to leaky-wave antennas [42]. Moreover, artifi-
cial surfaces with subwavelength resonant unit cells in a
leaky-wave antenna configuration have been recently
Fig. 5. High impedance surfaces for tunable leaky-wave antennas. (a) Side and (b) top view of a high impedance surface. Panel (a) highlights
the unitcell of the periodicstructure, inwhich the gap betweenpatches and thevertical vias actas the capacitanceand inductance,respectively,in
a lumped LC resonator. (c) Dispersion diagram for surface waves propagating on the structure. The leaky portion of the first TE mode can
be exploited to realize leaky-wave antennas. The angle of radiation can be tuned, at a fixed frequency wA, by changing the resonance frequency
of the LC unit cell in (a), which determines a modification of the wavenumber k
||.
[Panels (a) and (b) are reproduced with permission
from [35], panel (c) from [36].]
Fig. 4. 1D leaky-waveantenna based on a sinusoidally modulated reactance surface,implemented with a biasedgraphene sheet, foroperation at
THz frequencies. (Reproduced with permission from [33].)
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800 Proceedings of the IEEE |Vol.103,No.5,May2015
exploited as ‘‘metamaterial apertures’’ for computational
imaging [43], [44].
Other important advances for leaky-wave antennas
have come from the field of transmission-line metamater-
ials, introduced independently by Caloz and Itoh [45],
Eleftheriades, and co-workers [46]–[48], and Oliner [49],
[50], in the early 2000s. The application of composite
right/left handed (CRLH) transmission-line metamaterials
to leaky-wave antennas has led to several important break-
throughs in this area of research and technology, the most
important being the possibility of continuously scanning
themainbeamthroughbroadside[51],[52].
As discussed above, conventional periodic leaky-wave
antennas suffer from the open stopband problem, which
leads to beam degradation when approaching the broadside
direction. From a transmission-line point of view, it was
realized that any periodic structure with only series or
shunt radiating elements would always exhibit an open
stopband at broadside [25]. A CRLH metamaterial is com-
posed of a transmission-line structure (e.g., a microstrip
line) altered by periodically loading it with so-called ‘‘left-
handed elements,’’ namely capacitances in series and in-
ductances in parallel, which are combined with the
elements of a conventional transmission line, i.e., per-
unit-length series inductances and shunt capacitances [25].
The unit cell of a CRLH metamaterial and an example of its
practical implementation are shown in Fig. 6(a) and (b).
When the unit cell periodicity is subwavelength, the
structure is quasi-uniform, and radiation occurs from the
fundamental n¼0mode,whichisafastwave.The
dispersion diagram in Fig. 6(c) (blue curve) shows that the
fundamental mode has indeed a branch with negative phase
velocity (backward radiation; antiparallel phase and group
velocity) at lower frequencies, and a branch with positive
phase velocity (forward radiation) at higher frequencies,
separated by a gap at ¼0, which corresponds to the open
stopband of the periodic structure. The edges of this
bandgap are determined by the resonance frequency of the
series and parallel branches of the unit cell, which are
generally different [25]. These considerations reveal that
the open stopband at the broadside point can be completely
closed (Fig. 6(c), red curve) by designing a ‘‘balanced’’
structure with identical series and shunt resonance
frequencies, corresponding to the following condition for
the inductances and capacitances of the unit cell [53], [54]
LRCL¼LLCR:(6)
If this condition is fulfilled, a 1D leaky-wave antenna based
on CRLH metamaterials can scan the main beam through
broadside without degradation, as the frequency is in-
creased [Fig. 6(d)]. Interestingly, it has been noted that the
series and shunt radiating elements must contribute
equally in order to obtain efficient broadside radiation, a
Fig. 6. CRLH leaky-wave antennas. (a) Equivalent circuit of the unit cell of a CRLH transmission-line metamaterial. (b) Practical implementation
of a 1D CRLH leaky-wave antenna at microwave frequencies. The antenna is fed at one end of the structure. (c) Typical dispersion diagram
of a CRLH structure, in the unbalanced (blue curves) and balanced (red) configurations. In the latter case, the open stopband at broadside b=0
is completely closed. (d) Radiation pattern at different frequencies (in GHz) for the leaky-wave antenna in panel (b), demonstrating continuous
scanning through broadside. [Panels (a), (b), and (d) are reproduced with permission from [25], panel (c) from [61].]
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situation achieved under the balanced condition (6); in-
stead, broadside radiation would be poor if radiation came
mostly from only series or shunt elements [55], [56].
Extensions of these concepts to 2D and 3D geometries
have been discussed in [52]–[54]. It is also worth noting
that the high-impedance surfaces discussed above (Fig. 5)
may be interpreted as a 2D precursor of CRLH transmis-
sion-line metamaterials [54], in which the vertical vias
provide the shunt inductances, while the gaps between
plates introduce series capacitances, as seen in the unit cell
in Fig. 5(a).
CRLH leaky-wave antennas have been an active area of
research over the last few years. Recent advances include
electronically tunable [57], active [25] and nonreciprocal
designs [58]. Moreover, the dispersion of leaky modes in
CRLH structures has also been interestingly exploited to
realize microwave analog real-time spectrum analyzers
[59]. Several other leaky-wave antenna designs based on
transmission-line metamaterials have been extensively
reviewed in [25], [60], [61].
C. Leaky-Wave Antennas Based on Plasma Layers and
Plasmonic Metamaterials
2D leaky-wave antenna geometries typically consist of a
partially reflective screen that covers a grounded dielectric
slab. An interesting alternative is based on grounded
plasma layers, which have been shown to support weakly
attenuated leaky waves sustaining directive radiation [8].
In particular, guided waves in plasma slabs were originally
investigated by Tamir and Oliner in the 1960s [62], [63],
when the topic of electromagnetic radiation in plasmas
was starting to be of strategic importance for military and
space applications. They demonstrated that a plasma layer
supports leaky modes only above its plasma frequency,
where the permittivity is small and positive. At these fre-
quencies, a 2D uniform leaky-wave antenna can therefore
be realized by grounding the plasma slab on one side and
embedding a source in it [62], [63]. Below the plasma
frequency, instead, the slab is opaque and it supports TM
surface waves at the plasma-air interface, corresponding to
surface plasmon modes [64].
Interestingly, these concepts may be readily extended
to optical frequencies by replacing the plasma layer with
natural plasmonic materials, in particular noble metals,
which have their plasma frequency in the visible/ultraviolet
range [64]. At microwaves, instead, natural plasmas exist
only in the form of ionized gases, which are not necessarily
practical for antenna applications; however, an artificial
plasma slab can be realized using engineered structures,
such as dense arrays of conducting wires, which may be
designed to exhibit an effective low permittivity at the
frequency of interest [65]. Leaky-wave antennas based on
grounded wire arrays have been first proposed in the
pioneering work of I. Bahl and K. Gupta [65], [66]. With
the advent of metamaterials, these ideas have been redis-
covered and extended to more practical geometries, stimu-
lating intensive research efforts on artificial plasmas by
several research groups worldwide [67]–[74]. An example
of low-permittivity leaky-wave antenna for microwave fre-
quencies, based on an effective wired medium, is shown in
Fig. 7(a).
Grounded metamaterial slabs with low positive permit-
tivity (or permeability) have been shown to be particularly
appealing to realize narrow beams at broadside, with
increasing directivity as the permittivity (or permeability)
is lowered, which can be intuitively interpreted as a
‘‘lensing effect’’ due to the low refractive index of the
structure [75]–[77], as further discussed in Section VI.
Interestingly, this phenomenon can also be explained as
the result of the excitation of a polaritonic resonance
4
in
the low-permittivity planar slab, which determines a
strong redirection of the power flow inside the structure,
such that the emerging wavefront is almost planar [77],
Fig. 7. Metamaterial-based leaky-wave antennas. (a) Directive
leaky-wave antenna consisting of a grounded metamaterial slab
(wire medium) realizing low positive permittivity at microwave
frequencies. The antenna is excited by a line source embedded in
the slab.(b, bottom) Grounded metamaterial bilayer(i.e., paired layers
with oppositely-signed constitutive parameters) for low-profile and
highly directive leaky-wave antennas. (b, top) Example of radiation
pattern, with directive beam at broadside. [Panel (a) is adapted with
permission from [67], panel (b) from [80].]
4
In the solid-state physics literature, polaritonic resonances, or material
polaritons, indicate scattering resonances due to the coupling of impinging
photons with collective excitations of the material, such as phonons,
plasmons, excitons, etc., resulting in field distributions mainly concentrated
in the material object, and strong redirection of the power flow.
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802 Proceedings of the IEEE |Vol.103,No.5,May2015
[78]. The same principle has been exploited to design
metamaterial slabs that efficiently tailor and reshape the
radiation pattern of arbitrary sources [78]. In this context,
it is important to note that material polaritons and leaky
modes supported by slabs having near-zero permittivity (or
permeability) are strongly related, representing different
descriptions of the same resonance phenomenon [77].
A quantitative comparison between the performance of
metamaterial-based and conventional designs for 2D pla-
nar leaky-wave antennas is presented in [75], [76], and we
refer the interested reader to these papers for more tech-
nical details on this topic. One of the main disadvantages
of low-permittivity metamaterial slabs used as directive
antennas is the fact that their optimal thickness for direc-
tive radiation is significantly larger than conventional de-
signs based on partially reflecting screens, as pointed out
in [8], [75], and [76], due to the long wavelength in near-
zero-index materials. In fact, consistent with the interpre-
tation of these antennas as Fabry-Perot resonators, the
optimum slab thickness for broadside radiation is given
by [8]
h¼
2ffiffiffiffi
"r
p(7)
which becomes large for small values of the relative per-
mittivity "rof the material (is the wavelength in free
space). An interesting solution to overcome this limitation
has been proposed in [79], [80], based on grounded meta-
material ‘‘bilayers’’ [Fig. 7(b)], composed of paired ‘‘com-
plementary’’ materials with oppositely-signed constitutive
parameters. These structures have been shown to support
subdiffractive polaritonic resonances at the interface be-
tween the two layers, associated with weakly attenuated
leaky modes. These features allow greatly reducing the
transverse size of 2D leaky-wave antennas, while main-
taining high directivity, as seen in Fig. 7(b). In particular,
the dispersion relations for TE and TM leaky modes
supportedbythegeometryinFig.7(b),assuming
subwavelength thicknesses, are given by [79]
TE :d1d22
k2
y21
TM :d1d2"2
k2
y1"1
(8)
where kyi ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
!2"iik2
x
pand kxis the longitudinal com-
plex wavenumber of the leaky mode. Equation (8) can be
used as design formulas for metamaterial leaky-wave
antennas, once the desired radiation angle is selected.
In general, the 2D leaky-wave antennas presented here
are able to produce directive radiation, even if fed by lo-
calized sources with low directivity, such as a small dipole
or a small aperture in a ground plane. Interestingly, this
may be directly related to the possibility of drastically en-
hancing the transmission through subwavelength holes in
metal screens, and the realization of coherent thermal
emitters, topics that have raised large interest at optical
frequencies, treated extensively in Sections VI and VII.
IV. LEAKY WAVES IN OPTICAL
ANTENNAS AND WAVEGUIDES
In the millimeter-wave region of the electromagnetic spec-
trum, above the GHz range, losses in metal structures be-
come significant due to skin effects. In fact, the skin depth
in conductors shrinks as the frequency increases, following
the relation ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2=!
p,whereis the magnetic per-
meability of the medium and is its conductivity (the latter
assumed to be constant and purely real at these frequen-
cies). As a result, the resistance of any real metal increases
at higher frequencies, implying that metallic waveguides
exhibit nonnegligible Ohmic losses at millimeter wave-
lengths. To limit this issue, mm-wave leaky-wave antennas
have been designed based on structures that are already
open, such as dielectric waveguides, groove guides, micro-
strip lines, etc., avoiding as much as possible the presence
of metal. These structures generally support a fundamental
mode that is purely bound (slow wave), which is coupled to
radiation modes by introducing proper perturbations in the
geometry, e.g., asymmetries or periodic corrugations. Sev-
eral designs have been proposed to realize mm-wave leaky-
wave antennas with minimized losses and we refer the
reader to [7], [81], and references therein, for an exhaustive
treatment of this topic.
At even higher frequencies, in the infrared and optical
ranges, the conduction properties of metals drastically
change, as the real part of the conductivity decreases, and
its imaginary part becomes dominant. This modification
results in much larger field penetration in the metal, which
becomes characterized by a dispersive permittivity with a
finite negative real part. Therefore, the conventional de-
sign principles of waveguides and leaky-wave antennas,
which exploit metals as impenetrable conductors to con-
fine and guide electromagnetic fields, can no longer be
directly applied above the millimeter-wave range.
Historically, the realization of guiding structures at
optical frequencies has been mainly based on dielectric
materials, while metals have been generally avoided due to
their inherent losses. Given the open nature of dielectric
structures, the concept of leaky waves plays an important
role in the analysis and design of their guidance and ra-
diation properties (see, e.g., [82]). Within a vast literature
on this topic, notable examples include the design of pla-
nar dielectric strip waveguides [83] and the rigorous anal-
ysis of multilayered and periodic dielectric structures, e.g.,
dielectric gratings, in terms of surface and leaky modes
[84], [85]. These research works have been mainly moti-
vated by the increasing interest in integrated optical sys-
tems, in which a clear understanding of leakage and
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radiation phenomena is important to design efficient
optical couplers, or conversely, to avoid unwanted cross-
talk among different parts of a photonic circuit. Leaky-
wave concepts have also been applied to the important
topic of optical fibers, particularly in the work of Snyder
and co-workers [86]–[88]. By studying the complex solu-
tions of the source-free field problem in dielectric cylin-
drical fibers, they demonstrated the existence of ‘‘leaky
rays’’ that geometrical optics would predict to be trapped
by total internal reflection. Families of leaky rays form
weakly leaky modes, which are important in the accurate
analysis of multimode optical fibers.
More recently, dielectric structures have also been
used to realize directive leaky-wave radiation at optical
frequencies, based, e.g., on photonic quasi-crystals [89], or
silicon perturbations in a dielectric waveguides [90], [91].
As an example, the optical leaky-wave antenna proposed in
[90] and shown in Fig. 8 is based on the excitation of the
fundamental mode of a silicon nitride waveguide, a slow
wave, coupled to radiation by periodic silicon perturba-
tions, following similar design principles as for periodic
leaky-wave antennas at microwave frequencies. Leaky-
wave radiation is then obtained by exciting the antenna at
one end of the dielectric waveguide, which produces a
directive beam thanks to the low attenuation constant of
the leaky mode in this structure.
A common problem of waveguides and leaky-wave an-
tennas based on dielectric materials is the fact that their
transverse dimension needs to be comparable to the
wavelength in order for the field to efficiently interact with
the periodic corrugations. More in general, this problem is
fundamentally associated with the diffraction limit in
optical structures, which implies that the electromagnetic
energy guided in any open structure cannot be easily
confined in a subwavelength volume, but it tends to spread
over a region with transverse dimensions comparable to
the wavelength [79]. At low frequencies, diffraction can be
beaten by exploiting the high conductivity of metals, which
can be used to shield and guide electromagnetic waves. As
an example, a coaxial cable can confine power (carried by
its TEM mode) in a region with cross-section significantly
smaller than the signal wavelength. Also at optical fre-
quencies the diffraction limit can be overcome using
metals, but based on different principles, since metallic
materials exhibit drastically different properties in the
optical range, as aforementioned.
At sufficiently high frequencies, typically in the near-
infrared range, the finite carrier density nein metals causes
the electrons to respond to the electromagnetic excitation
with increasing time delay, which can no longer be neg-
lected. Because of this noninstantaneous response, metals
become characterized by a frequency-dispersive permit-
tivity function, which is generally well approximated by a
classical Drude model
"¼"01!2
p
!ð!þj1Þ
!
(9)
with collision frequency 1and plasma frequency !p¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
nee2=m"0
p(for a cold plasma), where "0is the permit-
tivity of free space, eis the electron charge and mis the
effective mass of the electron (determined by the specific
band diagram of the material) [92]. For noble metals, like
gold and silver, the plasma frequency lies in the visible or
near-ultraviolet range, implying that their permittivity has
a small negative real part at infrared and visible frequen-
cies,upto!p. This property leads to the onset of plasmonic
effects in metallic structures, associated with the excita-
tion of collective oscillations of the electron gas, or
Fig. 8. Dielectric 1D leaky-wave antenna for operation at optical frequencies. The structure consists of a dielectric waveguide, with
periodic silicon perturbations, which produces directive leaky-wave radiation from the n = –1 space harmonics of the waveguide mode.
(Reproduced with permission from [90].)
Monticone and Alu
`: Leaky-Wave Theory, Techniques, and Applications: From Microwaves to Visible Frequencies
804 Proceedings of the IEEE |Vol.103,No.5,May2015
plasmons. When impinging photons couple to these
electron oscillations, the resulting plasmon polaritons
can sustain particularly strong light-matter interaction and
field enhancement [64], [93]. In particular, the interface
between a material with positive permittivity, e.g., air, and
a material with negative real permittivity, e.g., a noble
metal at optical frequencies, may support anomalous
resonances (surface plasmon resonances), which deter-
mine light localization in deeply subwavelength regions,
hence overcoming the diffraction limit. Such interface
resonances may therefore be exploited to largely reduce
the resonant dimensions of several electromagnetic
systems. Based on these principles, many plasmonic
optical nanoantennas have been recently proposed, which
allow converting free-space propagating optical radiation
into subwavelength localized, or guided, optical fields, and
vice versa [94], [95]. Although most plasmonic nanoan-
tenna geometries proposed to date belong to the category
of standing-wave resonant antennas, such as nanodipoles
or nanodimers, a few interesting traveling-wave and leaky-
wave designs have also been put forward.
A first notable example of plasmonic leaky-wave nano-
antenna is based on planar complementary bilayers, as
discussed in Section III-C [Fig. 7(b)], in which one layer is
chosen to be plasmonic, while the other is dielectric or
insulating. As aforementioned, leaky-wave antennas of this
kind can be very low-profile, while retaining high directi-
vity, thanks to the subdiffractive interface resonance be-
tween complementary ‘‘oppositely signed’’ materials,
which can readily be obtained at optical frequencies with
plasmonic media. Besides, the directivity can be further
enhanced exploiting the interesting properties of materials
with low permittivity [79]. An interesting extension of
these concepts to conformal structures has been proposed
in [96], in which a cylindrical plasmonic shell with sub-
wavelength cross section is shown to support a circularly
symmetric resonant leaky wave. The resulting radiation
pattern is omnidirectional in the azimuthal plane, but
highly directional in the elevation angle, and can be
scanned with frequency in the forward quadrant.
Another interesting category of subdiffractive plasmo-
nic leaky-wave antennas at optical frequencies is based on
linear arrays of plasmonic nanoparticles [Fig. 9(a)], which
have been extensively studied in recent years (see, e.g.,
[97]–[100] and references therein). In particular, it has
been shown that nanoparticle arrays with subwavelength
transverse cross-section may support both slow and fast
guided modes (eigenmodes with real and complex
Fig. 9. Optical leaky-wave antennas based on linear arrays of plasmonic nanoparticles. (a) Linear array of polarizable particles supporting
longitudinal (top) or transverse (bottom) eigenmodes. The dipole moment induced in the particle located at the origin is p0=aeeEloc,
where Eloc is the local field at the origin and aee the particle electric polarizability. (b) Three-dimensional radiation pattern of the linear chain
of nanoparticles (longitudinal mode), at visible frequencies (690 nm). The 1D array is fed at one end, with a small dipole source. (c) Yagi-Uda
nanoantenna consisting of a linear array of plasmonic nanodipoles with different length, on a glass substrate. The antenna is fed by a
localized source, e.g., a quantum dot (QD), indicated by the red square. (d) Radiation pattern of the nanoantenna in (c). The slow wave
supported by the particle array becomes fast at the interface with the substrate, hence producing leaky-wave radiation at an oblique angle.
[Panels (a) and (b) are adapted with permission from [98], panels (c) and (d) from [103].]
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`: Leaky-Wave Theory, Techniques, and Applications: From Microwaves to Visible Frequencies
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wavenumbers), which may be exploited, respectively, to
realize subdiffractive nanotransmission lines [97], or opti-
cal leaky-wave antennas [98]. Interestingly, under the di-
polar approximation for the nanoparticles forming the
array (namely, the particles can be modeled as polarizable
dipoles, neglecting higher-order multipolar contributions),
the guidance and leakage properties of a linear array are
fullydeterminedbythearrayperioddand the real part of
the particle electric polarizability Re½ee, which depends
on the geometrical and material parameters of the
particles, while the imaginary part Im½eeis related to
the background Green’s function and the local material
loss for power conservation considerations.
The complete eigenmode spectrum of such linear ar-
rays can be split into longitudinal and transverse polarized
modes, as shown in Fig. 9(a), which determine different
leaky-wave radiation properties, as exhaustively investi-
gated in [98]. The longitudinal polarization has been found
to be most suitable for leaky-wave operation, since it gua-
rantees directive conical radiation, as seen in Fig. 9(b), as
well as scanning capability in the forward direction. In-
stead, the transversal polarization supports backward-wave
radiation, but it is intrinsically less efficient than the lon-
gitudinal mode. In general, periodic arrays of nanoparti-
cles may offer more flexibility than thin plasmonic layers
in the design of efficient subdiffractive leaky-wave an-
tennas at optical frequencies.
A different example belonging to the broad category of
linear nanoparticle arrays aimed at tailoring directive ra-
diation is the optical Yagi-Uda antenna shown in Fig. 9(c)
[102], [103]. Drawing inspiration from its radio-frequency
counterpart, this optical antenna is composed of a finite
array of nanoelements, only one of which is driven by a
localized optical source, such as a quantum dot or a fluo-
rescent molecule, while the other ‘‘parasitic’’ nanoele-
ments direct and shape the radiated beam. Although the
Yagi-Uda array is usually considered a slow wave antenna,
supporting a surface wave that radiates towards endfire
[101], modifications of the geometry or environment may
transform it into a leaky-wave antenna. For example, in the
experimental demonstration of an optical Yagi-Uda nano-
antenna reported in [103], the presence of the glass sub-
strate causes the antenna to radiate not at endfire, but at an
oblique angle, as seen in Fig. 9(d), because the traveling
wave supported by the array continuously leaks into the
substrate.
Broadband optical leaky-wave antennas can also be
realized relying on nonresonant structures, such as plasmo-
nic stripes, slot, or groove waveguides [104], [105], often
inspired by wideband leaky-wave antennas at microwaves
[22]–[24]. For example, it has been shown in [104] that a
long and narrow slot in a plasmonic sheet deposited on a
silicon substrate supports a weakly dispersive leaky mode
that radiates a directive beam into the substrate, similar to
the behavior of its low-frequency counterpart [22]. Inter-
estingly, the beam remains directive over a large fractional
bandwidth (50% around the central wavelength of
1550 nm), and the nonresonant nature of this setup makes
it more robust to fabrication tolerances.
The field of optical nanoantennas has attracted increas-
ing attention from different scientific communities, as it
holds the potential for unprecedented subdiffractive light-
matter interactions at the nanoscale, efficient coupling
between far-field radiation and localized nanosources, as
well as the exciting possibility of realizing point-to-point
wireless links in optical nanocircuits [94], [95]. For many
of these applications, a directive beam is highly desirable,
which however cannot be realized with single resonant
optical antennas, such as nanodipoles. In this context, the
optical leaky-wave antennas discussed in this section offer
an ideal platform to achieve high directivity with a simple
and compact structure (simpler than, for example, optical
phased arrays [106]). For these reasons, we believe that the
topic of optical leaky-wave antennas will gain increasing
attention in the next years, as more ideas and techniques
developed at microwave frequencies are translated and
adapted to optical frequencies.
V. INTERPRETATION OF OTHER
ESTABLISHED PHENOMENA
IN TERMS OF LEAKY WAVES
Leaky waves represent a fundamental concept in wave phy-
sics and, although they are generally associated with antenna
technology, it is easy to realize that evidence of these waves is
ubiquitous in the physical word. In particular, it has been
shown that leaky waves play a key role in several diverse
phenomena. Some examples, such as Cherenkov radiation,
Smith-Purcell effect, Wood’s anomalies and Goos-Hanchen
effect, are briefly discussed in this section, while the more
recent areas of extraordinary optical transmission (EOT),
coherent thermal emission and embedded photonic eigen-
states will be the subject of the following sections.
A. Cherenkov Radiation
A charged particle, or a beam of charged particles (us-
ually electrons), traveling in, or near, a dielectric medium,
emits radiation if the velocity of the particles exceeds the
speed of light in the dielectric medium. This effect, known
as Cherenkov radiation, was first observed by Cherenkov
and Vavilov, and theoretically interpreted by Tamn and
Frank in the 1930s (see, e.g., [107]). Cherenkov radiation
occurs in the form of a radiation cone, around the particle
beam, at an angle ¼cos1ðc=nvÞ,wherecis the speed of
light in vacuum, nis the dielectric refractive index and vis
the particle velocity. Such radiation can be considered the
electromagnetic analogous of bow waves in acoustics and
fluid dynamics.
Interestingly, it has been shown by I. Palocz and
A. A. Oliner that Cherenkov radiation, at least in some
forms, can be conveniently interpreted in terms of leaky
waves [108]. In particular, they rigorously studied the
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problem of radiation from a realistic electron beam in the
vicinity of a dielectric medium. Instead of starting from
idealized current distributions for the moving electrons,
as was usually done, Palocz and Oliner considered the
more realistic scenario of a modulated electron beam of
finite thickness and investigated its guided space-charge
modes. Assuming that the electrons move along the lon-
gitudinal direction zand their velocity is weakly modu-
lated [108], it is possible to prove that the total current
density along zcan be described by an effective displace-
ment current density in a material with nonlocal relative
permittivity
"b¼12
P
ðekzÞ2(10)
where kzis the longitudinal wavenumber, e¼!=v0is the
electronic propagation wavenumber, v0is the average
electron velocity and Pis the plasma propagation wave-
number, as defined in [108]. Since there is no current in
the transverse xand ydirections, the electron beam region
can then be substituted by an anisotropic (and spatially
dispersive) material with tensor permittivity
"¼^x^xþ^y^yþ"b^z^z. Therefore, in order to study the
space-charge eigenmodes of the realistic electron beam,
it is possible to apply a transverse resonance procedure
based on this equivalent anisotropic slab. It follows that,
when the electron beam is surrounded by free-space, all
the space-charge waves are bound and decay exponentially
in the surrounding region, as depicted in Fig. 10(a) (left
panel). However, when a dielectric region is brought in
close proximity to the beam, under certain conditions, a
few space-charge modes may become leaky and radiate
energy into the dielectric (red arrows in Fig. 10(a), right
panel). As shown in [108], this energy leakage occurs,
approximately, when the average electron velocity v0is
larger than the phase velocity c=nin the dielectric
medium, which indeed corresponds to the condition for
Cherenkov radiation.
This elegant analysis based on leaky space-charge waves
represents a self-consistent solution of the Cherenkov ef-
fect in the considered geometry, which fully takes into
account the influence of radiation on the beam itself, as
well as the realistic properties of the beam, such as its
thickness. Furthermore, the leaky-wave solution provides
new information and physical insights: it shows that the
Cherenkov radiation is not determined by a single leaky
space-charge wave, ideally emerging at one angle; instead,
multiple leaky waves with similar, yet not identical,
wavenumbers contribute to the radiation, determining a
small angular spread (about 1 degree) [108]. The ability to
predict this fine structure of the radiation confirms the
power and elegance of the leaky-wave analysis applied to
this problem.
B. Smith–Purcell Effect
In 1953, Smith and Purcell predicted and experimen-
tally verified that a beam of electrons traveling close to a
metal diffraction grating radiates electromagnetic waves
(typically in the optical range) due to the periodic motion
of charges induced on the metallic surface [109]. A crude
model based on the Huygens principle predicts that the
relation between the radiation angle and the wavelength
is given by ¼dðc=vcos Þ(for the fundamental
space harmonic), where dis the grating period and vis the
electron velocity.
It is clear that the Smith–Purcell effect resembles the
phenomenon of Cherenkov radiation discussed above, and
indeed they both belong to the general category of
‘‘Cherenkovian effects’’ [110]. The fundamental connection
of the Smith–Purcell effect with leaky waves was again
elucidated by Palocz and Oliner [111]. Following a similar
approach as described above, they studied the guided
space-charge modes of a realistic electron beam in
different environments, based on the transverse resonance
method. In particular, when the electron beam passes be-
tween two parallel conducting plates [Fig. 10(b), left
panel], all the space-charge eigenmodes are bound. How-
ever, if periodic openings are defined in the metal plates
(Fig. 10(b), right panel), space harmonics arise, some of
which may be leaky, hence radiating energy at specific
directions and frequencies that depend on the grating.
As shown in [111], this leaky-wave interpretation of the
interaction of electron beams with periodic metallic
Fig. 10. Models for the interpretation of (a) Cherenkov radiation and
(b) Smith–Purcell effect in terms of leaky waves, as explained in the
text. In all the cases, the electron beam (blue), propagating along z,
is modeled as an anisotropic and spatially-dispersive slab. The guided
space-charge modes are then calculated by applying the transverse
resonance method to thedifferent configurations.Red arrows indicate
that leaky-wave radiation may be allowed. [Panel (a) is adapted with
permission from [108], panel (b) from [111].]
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gratings correctly predicts the main features of the Smith-
Purcell effect. Again, the application of leaky-wave theory
allows an accurate and self-consistent solution of the
problem.
C. Enhanced Goos–Ha
¨nchen Effect, Optical Beam
Couplers and Guided-Mode Resonances
When a light beam of finite size (e.g., a Gaussian beam)
is totally internally reflected by a dielectric interface, the
reflected beam may exhibit a lateral displacement, called
Goos–Hanchen shift, due to the finite penetration and la-
teral power flow associated to the evanescent waves ex-
cited at the interface [112]. This effect, first experimentally
observed by Goos and Hanchen in 1947 [113], is generally
very weak. In fact, at the interface between two conven-
tional dielectrics, the lateral displacement is typically a
small fraction of the incident beam width.
A way to largely enhance the Goos–Hanchen shift has
been proposed by Tamir and Bertoni [112], based on struc-
tures that support leaky waves, such as plane-stratified
media, or periodic structures. In particular, for specific
incident angles and frequencies that guarantee phase
matching between the incident wave and a leaky mode of
the structure, a large portion of the incident energy pene-
trates the structure and is guided laterally as a leaky wave.
As the wave travels along the structure, it radiates back
energy, which forms a reflected beam laterally shifted from
the position predicted by geometrical optics. Such an ef-
fect, due to the excitation of leaky modes, may be much
more pronounced than the conventional Goos–Hanchen
shift, leading to lateral displacements in the order of the
beam width.
The external excitation of guided leaky waves of the open
structure corresponds to complex poles of the reflection
coefficient on the transverse wavenumber plane. However,
any propagating plane wave possesses a real wavenumber, and
therefore it cannot directly excite the complex eigenmodes of
the structure (consistent with the fact that the plane-wave
reflection coefficient never goes to infinity, due to power
conservation). Nevertheless, whenever the real wavenumber
of the incident plane wave is close to the real part of the
complex wavenumber of a leaky pole (corresponding to a
phase matching condition), and its imaginary part is
sufficiently small, the leaky wave can be strongly excited (as
discussed in [114], this is essentially a forced resonance
phenomenon). Moreover, in the analysis of the enhanced
Goos–Hanchen effect, the width of the incident beam is
assumed to be large (at least tens of wavelengths), such that,
in a plane wave expansion (assuming an incidence angle
from the surface normal), the principal contribution comes
from the plane wave with transverse wavenumber kt¼
k0sin , which can strongly couple with the leaky pole.
Another interesting feature of the enhanced Goos–
Hanchen effect due to leaky waves is the fact that, while in
stratified structures the displacement is generally in the
forward direction, in periodic structures it may be either
forward or backward [negative Goos–Hanchen effect, as
depicted in Fig. 11(a)]. In the periodic case, if there are
several radiating space harmonics, the incident beam may
strongly couple to a leaky mode of the structure whenever
Fig. 11. Anomalous effects due to the external excitation of leaky modes. (a) Negative Goos-Hanchen shift at the interface with a structure
supporting backward-propagating leaky waves. The thick red arrow indicates the energy flow within the structure. The dashed lines denote the
positionof the reflected beampredicted by geometrical optics. (b) Guided-mode resonances in a periodicslab waveguide undernormal incidence.
As shown in the bottom panel, sharp reflectivity peaks are observed at frequencies close to the lower edge of the open stopbands at b=0.
[Panel (a) is adapted with permission from [112], panel (b) from [131].]
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it is phase matched to any of these space harmonics. Then,
as shown in [112], all of the diffracted beams scattered by
the periodic structure (including the specular reflected
beam) undergo a forward or backward displacement, ac-
cording to whether the incident beam is phase matched to
a space harmonic of forward or backward type. Interest-
ingly, similar considerations have also been applied to
explain negative displacement of reflected beams in
acoustics [115].
Besides periodic gratings, negative Goos–Hanchen shift
can also be observed in homogeneous plasma slabs with
negative permittivity in specific geometries, thanks to the
excitation of backward-propagating leaky waves [112]. This
effect has been predicted [116] and experimentally verified
[117] at optical frequencies in metallic films coupled to a
dielectric prism, which may support leaky surface-plasmon
waves in the so-called Otto configuration (i.e., prism-air-
metal) or Kretschmann configuration (i.e., prism-metal-air).
Furthermore, in recent years several engineered structures,
such as photonic crystals [118], negative-index metamaterials
[119] and stratified hyperbolic media [120], have also been
shown to support giant and negative Goos–Hanchen effect,
based on different mechanisms.
The same physical mechanism underlying the enhanced
Goos–Hanchen shift, namely, the coupling of an incident
beam to a leaky mode of an open structure, plays a key role in
the design of efficient beam-to-surface wave couplers (see,
e.g., [112], [121]–[123]). For example, it has been shown in
[112] that, in light of the large lateral displacement Dof an
incident beam due to leaky waves, it may be possible to
trap a large portion of the incident energy by avoiding the
leakage after a distance D=2. This can be accomplished by
modifying the structure such that the attenuation constant
is suppressed, while the wave impedance is almost unaf-
fected, hence allowing the energy to continue propagating
laterally in the form of a bound surface mode. These
considerations are particularly relevant today in the analysis
of the electromagnetic wave interaction with graded
metasurfaces, particularly in reflection mode (e.g., [124]–
[126]), which can often be interpreted in terms of well-
established leaky-wave theory.
Finally, the interaction of an incident plane wave with
the leaky modes of a structure has also been exploited to
realize total absorption in lossy layered media [127] and
anomalous filtering effects in periodic slab waveguides
[128]–[131]. Notably, this latter category is based on so-
called guided-mode resonances (or leaky-mode resonances),
which occur when the incident plane wave excites a leaky
waveguide mode due to phase matching [Fig. 11(b)], deter-
mining pronounced resonant peaks/dips in the reflection/
transmission spectra, especially when the incidence is very
close to the surface normal (or other special angles, such as
the Brewster angle [129]). These resonant effects have
been studied extensively in recent years to realize several
functionalities, such as broadband and narrowband filter-
ing and polarization control [130], [131].
D. Wood’s Anomalies and Fano Scattering Resonances
In 1902, Wood discovered anomalous sharp amplitude
variations (i.e., narrow bright and dark bands) in the
spectrum of an optical metallic reflection grating, under an
illumination with almost constant spectral intensity. Since
these features were not predicted by ordinary grating
theory, they started attracting large attention in the scien-
tific community. Lord Rayleigh found that the occurrence
of the anomalies corresponds to the emergence of a new
diffraction order at grazing angle, which determines a re-
arrangement in the amplitude of the other diffraction
orders. However, this explanation accounted only for a
specific class of anomalous spectral features (now known
as Rayleigh anomalies), while it did not explain many ex-
perimental observations (for further details, see, e.g., [114]
and [133]). The first theoretical breakthrough in the
modern understanding of Wood’s anomalies came from
the work of Fano in the 1930s [134], who recognized that
some of these features arise from forced resonances asso-
ciated with guided modes of the gratings. Interestingly, the
asymmetric lineshape of Wood’s anomalies was one of the
first observations of Fano resonances, later been shown to
be ubiquitous in quantum and classical systems [135],
[136], and now particularly popular in the optics literature
[137]–[144]. These resonances are essentially interference
phenomena occurring between a discrete oscillating state
and a continuum, which, in the case of a grating, corre-
spond, respectively, to a guided mode of the structure and
the continuum of radiation modes of free space.
The explanation of Wood’s anomalies in terms of
scattering resonances due to guided modes was made more
quantitative in the work of Hessel and Oliner [114], [145],
who elucidated the phenomenon in light of the modern
concept of leaky waves. In particular, drawing inspiration
from their previous work on guided waves on periodic
structures [30], Hessel and Oliner rigorously studied the
scattering from a periodic reactance surface representing
the grating. By representing the fields in a Floquet-type
expansion and solving the inhomogeneous system of equa-
tions associated with the boundary conditions, they cal-
culated the amplitude spectra of all diffraction orders
scattered by the periodic surface under a specific plane
wave incidence. Interestingly, these amplitude spectra ex-
hibit sharp asymmetric features, consistent with Wood’s
anomalies in realistic gratings. Most importantly, the ano-
malous features always occur near a frequency value that
guarantees phase matching between one of the diffraction
orders of the incident plane wave and a leaky mode of the
grating [114]. Therefore, in line with other phenomena
discussed in Section V-C, also Wood’s anomalies can be
explained as a form of forced scattering resonance arising
from the external excitation of leaky waves supported by a
periodic open structure.
The analysis of Wood’s anomalies by Hessel and Oliner
did not consider realistic gratings, but rather an ideal
periodic surface reactance; nevertheless, it correctly
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predicted and explained, for the first time, important pro-
perties observed in optical experiments, such as the possi-
bility of anomalies for both incident polarizations, and the
peculiar reluctance of different anomalies to merge. It was
later discovered that the leaky modes of metallic gratings
at optical frequencies are associated to surface plasmon
polaritons, which become leaky due to the periodicity of
the grating. Although more accurate analyses of Wood’s
anomalies have been recently developed, which take into
account the realistic geometry and material properties of
optical gratings [133], the elegant interpretation in terms
of scattering resonances and leaky waves explains the fun-
damental physical mechanism and captures the main fea-
tures of this phenomenon.
As an aside, we note that the recently studied anoma-
lous Fano scattering resonances in metallic nanoparticles at
optical frequencies [135]–[144] may be considered, in a
sense, the equivalent of these leaky-wave anomalies for
bounded 3D structures. These sharp scattering features, in
fact, are also a form of forced scattering resonances in-
volving a damped oscillatory state. As noted in [1], in fact,
both leaky waves in open waveguides and radiation-damped
oscillations in open cavities correspond to complex pole
solutions of a source-free boundary-value problem. Besides,
these Fano scattering resonances share some of the peculiar
features of Wood’s anomalies, such as the asymmetric and
narrow lineshape and the reluctance of different scattering
resonances to merge [143], [144], features that can be
exploited in several nano-optics scenarios.
VI. EXTRAORDINARY OPTICAL
TRANSMISSION
In the field of optics, one of the most popular break-
throughs of the last couple of decades has been the
discovery of extraordinary transmission of light through
subwavelength apertures in metallic films [146], [147],
which attracted large interest in the scientific and engi-
neering communities. To better appreciate these findings,
we should consider first the idealized case of a single hole in
an infinitely thin perfectly conducting metal screen. For
this configuration, Bethe showed long ago that the trans-
mission through a hole of radius ris proportional to ðr=Þ4,
where is the wavelength [148]; therefore, the transmis-
sion becomes rapidly very weak for subwavelength dimen-
sions. In addition, if the thickness of the screen is taken into
account, the transmission decreases even further, with
exponential dependence on the hole depth if rG=4(i.e.,
when no propagating modes are allowed in the hole).
Several years after Bethe’s seminal paper, Ebbesen and co-
workers showed experimentally that, when several sub-
wavelength holes are arranged in an array [146], or a single
hole is surrounded by a periodic texture [147], power
transmission can be dramatically enhanced, by several
orders of magnitude compared to Bethe’s limit. In addition,
it was observed that the transmitted energy can become
much larger than the energy impinging on the holes, im-
plying that also the light incident on the metal surface was
‘‘funneled’’ through the subwavelength apertures. Since
this effect cannot be predicted by conventional diffraction
theory and is somewhat counterintuitive, large attention
was devoted by scientists and engineers to investigate its
physical mechanism. It is interesting to note that the phe-
nomenon of resonant transmission in arrays of apertures in
metallic screens has been known for quite some time in the
microwave engineering community, particularly in the
context of frequency selective surfaces (FSSs) and filters
[149]–[151]. However, resonant transmission in conven-
tional FSSs typically involves aperture sizes comparable to
half wavelength, whereas EOT effects have been observed
for significantly smaller apertures [152]. It should be also
noted that the narrow EOT peak always appears at fre-
quencies close to the onset of the first grating lobe, whereas
the broader resonant transmission peak associated with
normal FSS operation is generally at lower frequency [153].
The narrowness of the EOT peak (especially for very thin
metal screens) and the fact that it occurs in a region of small
practical interest for FSS designers, due to the presence of
undesired grating lobes, partially explain why this kind of
transmission resonances was not observed (or noticed) in
the microwave community before the work of Ebbesen and
co-workers in 1998 [146], despite the fact that similar
geometries were commonly investigated by FSS designers,
as further discussed in [153].
Since the earliest attempts to interpret the EOT pheno-
mena observed in optical experiments, there has been a
general agreement among researchers that surface plas-
mons play a key role in the transmission enhancement
(see, e.g., [146], [147], [154]–[159]); however, the details
of the enhancement mechanismwereinitiallynotfully
understood. A very important step in the understanding of
the extraordinary optical transmission (EOT) effect was
the recognition of the fundamental role played by leaky
waves supported by the patterned metal screen. As dis-
cussed in Section IV, noble metals at optical frequencies
behave like plasmas, characterized by a dispersive permit-
tivity following a classical Drude model, as in (9). As
mentioned above, a smooth planar interface between a
metal with negative permittivity and air can support a TM
surface wave, known as surface plasmon, propagating
along the interface with longitudinal wavenumber [64]
kp¼k0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
"rð!Þ
1þ"rð!Þ
s(11)
where "rð!Þistherelativepermittivityofthemetalwith
Drude dispersion. It is clear from (11) that, for any value
"rð!ÞG1, a surface plasmon on a smooth interface is a
slow wave with kp>k0and, thereby, it does not radiate. In
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particular, for low frequencies ["rð!Þvery large and nega-
tive], the dispersion curve of the surface plasmon follows
the light line, i.e., kpk0, and the surface wave is only
weakly bound to the interface. Instead, at frequencies ap-
proaching the value !¼!p=ffiffiffi
2
p["rð!Þ¼1, according to
(11)], the plasmon wavenumber kpgoes to infinity, imply-
ing that the surface wave is tightly bound to the interface,
rapidly decaying on both sides, and corresponding to a so-
called surface-plasmon resonance.Asanaside,itisinterest-
ing to recognize the similarities and differences between
surface plasmon waves at optical frequencies and the so-
called Zenneck surface waves supported at the interface
between free space and a medium with finite conductivity
(for example, the poorly conducting surface of the Earth),
which have been extensively studied since the pioneering
days of wireless telegraphy. Indeed, like surface plasmons,
Zenneck waves are TM surface waves decaying away from
the conductor-vacuum interface, and their dispersion rela-
tion is consistent with (11) [160], with the relevant differ-
ence that, at the low frequencies usually considered in the
analysis of Zenneck waves, the complex permittivity of the
conductor can be assumed nondispersive. Another impor-
tant difference is that, in realistic situations, Zenneck
waves are significantly less localized on the interface than
surface plasmons. For further details on the relation be-
tween Zenneck waves and surface plasmons, and the lin-
gering controversy about the actual excitation of Zenneck
waves, we refer the interested reader to [160]–[162].
As discussed in Section III in the context of leaky-wave
antennas, a slow wave can be made to radiate by introduc-
ing a periodic modulation of the guiding structure (in this
case the metallic interface), which determines the ap-
pearance of infinite space harmonics of the fundamental
mode, with wavenumber given by (5). Therefore, in ana-
logy with periodic leaky-wave antennas, a periodic modu-
lation of the metallic surface can be designed such that a
selected space harmonic of the plasmon mode is a fast
wave and, as a result, the overall guided mode becomes a
leaky surface plasmon that can efficiently couple energy to
free-space radiation. D. R. Jackson, A. A. Oliner, and co-
workers applied these considerations based on leaky-wave
theory to provide the basis for a consistent explanation of
the EOT effect [76], [163]–[166]. Consider, for example,
the case of a single subwavelength aperture in a metallic
screen surrounded by periodic corrugations, which has
been shown to exhibit largely enhanced transmission com-
paredtoBethe’slimit[147].Fig.12(a)depictsa1Dversion
of this geometry, with periodic corrugations on the exit
Fig. 12. Interpretation of directive beaming and EOT in terms of leaky waves. (a) Sketch of a subwavelength aperture in a metallic film
surrounded by periodic corrugations on the exit face. Radiation is produced directly by the small aperture, in the form of a broad ‘‘space wave,’’
and by the leaky waves traveling on the corrugated surface. (b) Example of farfield radiation pattern, produced by a magnetic line source
representing the subwavelength aperture in (a), for an optimized corrugated surface (blue curve) and a smooth surface (red). (c) Magnetic
field intensity on the surface, as a function of the distance from the source, demonstrating the exponential decay expected for a leaky mode.
[Panel (a) is adapted with permission from [76], panels (b) and (c) from [166].]
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face. The subwavelength aperture essentially acts here as a
magnetic line source, which radiates part of the power
directly into free space [denoted as a space wave in
Fig. 12(a)]; however, the associated radiation pattern is
expected to be broad, due to the subwavelength size of the
aperture. The source can in addition excite surface plas-
mons on the metallic surface, which, in the absence of
corrugations, are bound and would radiate only at discon-
tinuities (e.g., at the end of the metal screen). Instead, as
discussed above, by applying a suitably tailored periodic
perturbation, directive leaky radiation can be induced
from one of the space harmonic of the fundamental plas-
mon mode, typically the n¼1, which becomes a leaky
wave with complex wavenumber kp;1¼1j(note
that the attenuation constant does not have an index, since
it is the same for all space harmonics). Therefore, the
surface plasmons launched by the subwavelength hole can
become leaky due to the periodic corrugations, and radiate
two beams at angles 1¼sin1ð1=k0Þ,asdepictedin
Fig. 12(a). As the frequency varies, the beam direction is
scanned, as in conventional leaky-wave antennas, until, at a
certain frequency value, the two beams merge into a single
beam around broadside. Although radiation at exactly
broadside is forbidden due to the open stopband of the
periodic structure, as discussed in Section III, it has been
shown that optimum broadside radiation can be obtained
for a phase constant 1slightly detuned from broadside,
corresponding to the ‘‘merged beam’’ being on the verge of
splitting [165]. Interestingly, this optimum point has been
shown to correspond to the condition [76], [167]
j1j¼; (12)
which guarantees maximum power radiated at broadside
for the geometry in Fig. 12(a). Similar considerations can
be extended to the 2D scenario [165].
Leaky surface plasmons can efficiently collect the
power emerging from the subwavelength aperture and ra-
diate it directively to free space, as they propagate laterally.
Theeffectiveaperturefromwhichthepowerisradiatedis
therefore much larger than the subwavelength hole, hence
allowing a large directivity. If the corrugations are properly
designed, as discussed above, at a certain frequency the
energy will be radiated very efficiently within a narrow
beam at broadside. As an example, Fig. 12(b) shows the
numerically computed farfield radiation pattern for a mag-
netic line source (which models the aperture) on a silver
film, with (blue line), or without (red line) optimized
corrugations (taking into account the losses of the metallic
material). The graph shows an impressive difference be-
tween the two cases, particularly at broadside, clearly
demonstrating the directive beaming effect. To further
confirm the leaky-wave explanation of this phenomenon,
numerical simulations [166] reported in Fig. 12(c) show
that, indeed, the field along the surface decays exponen-
tially from the aperture (even in the lossless case), as ex-
pected for a traveling leaky mode.
The enhancement of power radiated at broadside, as
described above, is exactly equivalent, by reciprocity, to the
enhancement of power transmitted through the hole, when
the periodic corrugations are placed on the entrance plane
[165]. Therefore, the leaky-wave explanation of directive
beaming also provides a consistent and accurate explana-
tion of the EOT effect through a subwavelength aperture.
Analogous considerations can also be applied to the case of
hole arrays, in which the periodic corrugations for an in-
dividual hole are provided by the rest of the array. Inter-
estingly, the fundamental role of leaky waves in the EOT
effect was also recognized independently by other authors,
although from different perspectives (see, e.g., [168]).
The leaky-wave theory of EOT provides practical design
guidelines to optimize the apertures and periodic corruga-
tions for enhanced transmission, based on well-established
leaky-wave antenna principles. Moreover, this viewpoint
reveals that the EOT phenomenon is actually just an ex-
ample of the directive beaming effect based on leaky waves
[76] and it is not restricted to optical frequencies or to plas-
monic materials. For example, enhanced transmission
effects have been demonstrated in subwavelength hole ar-
rays at microwave and mm-wave frequencies [173]–[175],
and beam collimation has been achieved in quantum cas-
cade lasers at THz frequencies by patterning periodic cor-
rugations on the laser facet [176]. Furthermore, given the
universality of leaky waves, similar concepts can be applied
to the acoustic and quantum realms. Notably, extraordi-
nary acoustic transmission (EAT) and sound collimation
have recently been demonstrated [169]–[172], which can
indeed be explained in terms of excitation of leaky acoustic
guided modes supported by the structure [172]. Moreover,
in the domain of quantum physics, leaky matter waves
have been shown to play a fundamental role in the pheno-
menon of enhanced transmission and directive beaming of
atoms through apertures much smaller than the atomic de
Broglie wavelength [177], [178], which may suggest intri-
guing applications of leaky-wave theory into the quantum
regime.
Since leaky waves can also be supported by homoge-
neous structures, the presence of a periodic modulation is
not a necessary requirement to achieve directive beaming
and enhanced transmission. For example, we discussed in
Section III-C that directive radiation at broadside can be
obtained with planar leaky-wave antennas based on
grounded homogeneous metamaterial slabs, where the
source is a small dipole embedded in the slab, or a slot in
the ground plane. Therefore, by reciprocity, these designs
can be used to enhance the transmission from subwave-
length apertures in a metallic screen. Notably, strong EOT
and directive beaming effects have been obtained when a
subwavelength hole in a perfectly conducting screen is
covered, on both sides, by homogenous slabs with low
Monticone and Alu
`: Leaky-Wave Theory, Techniques, and Applications: From Microwaves to Visible Frequencies
812 Proceedings of the IEEE | Vol. 103, No. 5, May 2015
positive permittivity (and/or permeability) [77]. The prin-
ciple of operation can be explained intuitively using ray-
optics,asshowninFig.13.Consider,forexample,theexit
face: since the wavenumber in the metamaterial slab is
much smaller than in free space, i.e., jkjk0, the phase
accumulated by the different rays will be only slightly
different on the exit plane and all the rays will refract in
free space at almost the same angle, producing a narrow
beam at broadside (in the ideal case of k!0, the radiation
pattern tends to a delta function). Interestingly, this
‘‘lensing’’ effect, explained with ray-optics, has been shown
to be fundamentally related to the weakly attenuated leaky
waves supported by the slab [67], [77]. The interpretations
in terms of ray-optics and leaky waves become equivalent
for sufficiently thick slabs [179].
For the setup with metamaterial slabs on both sides, as
in Fig. 13, reciprocity guarantees that the incident energy
will be collected, transmitted through the subwavelength
aperture, and radiated at broadside with very high effi-
ciency. Compared to the other approaches to EOT dis-
cussed above, in this case no plasmonic effects and periodic
corrugations are required, but the fundamental physical
mechanism is still based on the excitation of leaky modes in
an open guiding structure. Moreover, since no periodicity
is involved, this solution avoids the open stopband problem
discussed above. However, radiation at exactly broadside is
still problematic, because it would require the permittivity
(or permeability) to be identically zero, consistent with the
ray-optics interpretation in Fig. 13 [77]. A practical disad-
vantage of the low-permittivity metamaterial design may be
the large optimal thickness of the slabs [given by (7)],
which may be reduced using metamaterial bilayers, as
discussed in Section III-C.
VII. COHERENT THERMAL EMISSION
The ability of leaky-wave structures to realize highly direc-
tive radiation from low-directivity sources, which is at the
basis of the EOT effect discussed above, also explains
another intriguing optical effect, namely, the realization of
coherent thermal emission with microstructured surfaces.
Thermal sources, such as the filament of an incandescent
bulb, typically emit light over a broad angular range, as well
as a broad bandwidth determined by Planck’s law of ther-
mal radiation. These are incoherent light sources, in which
the emitted infrared light from different points of the
structure does not interfere, in contrast with, for instance,
the electromagnetic radiation from an antenna array.
Nevertheless, the technological need for low-cost light
sources in the mid-infrared range has recently stimulated
extensive research efforts to develop thermal sources that
are spatially and temporally coherent, i.e., directional and
frequency selective. In this context, it was shown that, by
simply introducing periodic corrugations on the surface of
a material, as shown in Fig. 14(a), the thermal emission
can be made highly directional and narrowband [180],
[181]. In particular, when such a grating is heated, the
resulting thermal emission pattern exhibits two directive
beams at opposite oblique directions, and strong frequency
dependence [Fig. 14(b)]. It is evident that the underlying
physical mechanism behind coherent thermal radiation is
indeed related to the excitation of leaky modes on the
corrugated structure supported within the Planck spec-
trum. To clarify this fact, consider the surface of a planar
thermal source without corrugations. When the structure
is heated, each volume element can be modelled as a ran-
dom point source, which may excite a guided surface wave,
such as a surface phonon, or a surface plasmon, in addition
to free-space radiation. Guided waves do not directly con-
tribute to thermal radiation, since they cannot couple to
propagating plane waves. Thermal radiation is therefore
determined by the low-directive radiation of random point
sources, which contribute incoherently (i.e., without in-
terference) to the overall emission. If the surface is pe-
riodically perturbed, as in Fig. 14(a), leaky-wave radiation
can be induced from one of the space harmonics of the
guided modes, consistent with (5). Such leaky waves, con-
tinuously radiating along the surface of the thermal source,
determineamuchwidereffectiveaperture(inother
words, a much longer ‘‘coherence length’’), which results
in directive radiation, as seen in Fig. 14(b). For a given
temperature, the effect is particularly strong if the pe-
riodicity of the corrugations is selected such that leaky
waves appear in the frequency range where Planck’s black
body radiation is maximum for the given temperature.
Interestingly, leaky-wave theory connects the seem-
ingly unrelated optical effects of extraordinary optical
Fig. 13. Ray-theory interpretation of enhanced transmission
and directive beaming based on metamaterial covers with low
wavenumber k. At the exit face, the rays emerging from the
metamaterial slab bend toward the surface normal, producing
directive radiation at broadside (represented by the red ellipse).
(Adapted with permission from [77].)
Monticone and Alu
`: Leaky-Wave Theory, Techniques, and Applications: From Microwaves to Visible Frequencies
Vol. 103, No. 5, May 2015 | Proceedings of the IEEE 813
transmission and coherent thermal emission, showing that
they have a common underlying mechanism. This simi-
larity is even clearer in a different implementation of a
coherent thermal source, based on the ‘‘bull’s eye’’
geometry shown in the inset of Fig. 14(c) [182], [183]. In
this case, concentric periodic corrugations on a metallic
sheet are suitably designed to support directive leaky-wave
radiation at broadside, very similar to the directive beaming
effectdescribedinSectionVI.AsseeninFig.14(c),atthe
design frequency the main beam can indeed be made very
narrow near broadside, thanks to the large effective
aperture of the thermal source. As the frequency is varied,
the beam is expected to shift and change significantly, due
to the frequency dispersion of the leaky plasmon modes. By
fixingtheobservationangleandvaryingthewavelength
[Fig. 14(d)], it is therefore possible to appreciate the
narrow bandwidth of the emissivity peak, resembling the
response of a coherent source, such as a laser, despite being
based on a thermal process. Moreover, owing to Kirchhoff’s
law of thermal radiation, which states that the emissivity of
a body is equal to its absorptivity, a structure designed to
work as a coherent thermal source can also be used as an
absorber characterized by high selectivity in both angle and
frequency.
As in the case of extraordinary optical transmission,
leaky-wave theory reveals that plasmonic effects and pe-
riodicity are not necessary to obtain coherent thermal
emission. In fact, any structure supporting leaky modes
in the spectrum where Planck’s radiation is near its
maximum can be exploited for coherent thermal emission
engineering, including photonic crystal slabs [184], meta-
material wire medium slabs [185], leaky-wave frequency-
selective surfaces [186], multilayered dielectric slabs on a
metallic substrate [187], among many other examples.
VIII. EMBEDDED PHOTONIC
EIGENVALUES
As we have discussed throughout this paper, a guided mode
with phase constant jjGk0, supported by an open wave-
guiding structure, is a leaky wave that radiates energy into
free space as it travels along the structure. In other words,
the leaky mode can couple to the radiation modes of
free space, namely, outgoing plane waves. Surprisingly,
Fig. 14. Coherent thermal emitters. (a) Periodic corrugations on the surface of a SiC substrate, supporting leaky-wave radiation in
the mid-infrared range when heated above 700 K. (b) Corresponding emissivity pattern at three wavelengths: blue, l=11.04mm;
red, l=11.36mm; green, l=11.86mm [180]. (c) Angular dependence of the emissivity at the design frequency, and (d) emissivity spectra
at different angles, for a tungsten bull’s eye structure shown in the inset of panel (c). [Panels (a) and (b) are adapted with permission
from [180], panels (c) and (d) from [182].]
Monticone and Alu
`: Leaky-Wave Theory, Techniques, and Applications: From Microwaves to Visible Frequencies
814 Proceedings of the IEEE |Vol.103,No.5,May2015
however, recent theoretical and experimental findings
have shown that, under specific conditions, a fast wave
traveling along an open structure can avoid radiation and,
instead, it can behave as a purely bound mode without
coupling to propagating waves in the background [188]–
[190]. This phenomenon has been experimentally ob-
served at optical frequencies in a photonic crystal slab
[189],showninFig.15(a),formedbyadielectricslab
waveguide with periodic corrugations (cylindrical holes).
The dispersion diagram for this 2D periodic structure
along the irreducible Brillouin zone is shown in Fig. 15(b).
As usual, the structure supports slow and fast waves, re-
spectively, below and above the light lines. In particular,
consider the first TM mode (indicated by the green line):
when it lies within the light cone (namely, the region
delimited by the light lines in different directions), the
mode is expected to be leaky, since it can couple with the
continuum of radiation modes of free space (in other
words, phase matching can be fulfilled). However, at a
specific point along the dispersion curve of the TM1mode
[red circle in Fig. 15(b)], it has been found that the leakage
rate actually vanishes, despite the presence of available
radiation modes to which the energy can couple. At this
specific condition, the guided mode becomes ideally bound
and confined in the slab, as seen in the numerically com-
puted field distribution in Fig. 15(c). In particular, the
mode does not exhibit longitudinal attenuation, since
there is no longer energy leakage (the attenuation constant
of the leaky mode is suppressed), which corresponds to an
oscillatory state with infinite quality factor (i.e., infinite
lifetime), as seen in Fig. 15(c). Peculiar bound states of this
kind, existing within the radiation continuum (the light
cone), fall into the category of embedded eigenvalues [189],
and can be considered the electromagnetic analogue of
anomalous localized electron states in quantum mechanics
[191], [192].
In leaky-wave theory, it is known that leaky modes may
have anomalous responses at certain particular points of
the dispersion diagram. For example, at the transition re-
gion in which a bound mode evolves into a leaky mode
(across the light line), a so-called ‘‘spectral gap’’ may be
present, in which the modal response may be quite com-
plicated, and even become nonphysical [193], [194]. More-
over, in the dispersion diagram of a periodic structure, an
open stopband may appear at the ¼0 point [denoted as
Gin Fig. 15(b)], as we frequently mentioned in this paper,
where leaky-wave radiation is forbidden, even though the
mode lies within the radiation cone. Interestingly, this
open stopband can be interpreted in terms of symmetry
incompatibilities, which prevent the guided wave to couple
Fig. 15. Embedded photonic eigenvalues. (a) Geometry of the photonic crystal slab (top) and layout of the fabricated structure (bottom).
(b) Dispersion diagram along two directions of the irreducible Brillouin zone (shown in the inset), showing the different TE and TM modes
of the 2D periodic structure. The yellow shaded region denotes the light cone. The position of the embedded eigenvalue of the TM1mode
is indicated by a red circle. (c) Quality factor of the first leaky mode (inversely proportional to its radiation losses), as a function of
the longitudinal wavenumber kx. Note that the qua lity factor goes to i nfinity a t the Gpoint (kx= 0; broadside radiation), realizing a
‘‘symmetry-protected’’ bound state, and at a seemingly unremarkable value of kx, corresponding to the embedded eigenvalue indicated
in panel (b). (d) The inset at the right shows the electric field distribution of the embedded eigenstate. (Adapted with permission from [189].)
Monticone and Alu
`: Leaky-Wave Theory, Techniques, and Applications: From Microwaves to Visible Frequencies
Vol. 103, No. 5, May 2015 | Proceedings of the IEEE 815
to radiation modes [195]. As a result, a ‘‘symmetry-pro-
tected bound state’’ appears at the Gpoint, characterized
by absence of radiation and an infinite quality factor, as
seen in Fig. 15(c).
Both the spectral gap and the open stopband, however,
occur at ‘‘exceptional’’ points of the dispersion diagram,
where an anomalous leaky-wave response may be expected.
Instead, as seen in Fig. 15(b), the embedded eigenvalues
described earlier appear at seemingly unremarkable wave-
numbers, not intuitively related to any specific physical
mechanism. Experimentally,ithasbeenshownthatthe
reflectivity spectra exhibit asymmetric Fano resonances
due to the external excitation of the leaky guided modes
of the slab [189], consistent with our discussion in
Section V-C and D. However, when the angle and fre-
quency of the incident wave are scanned closer to the
position of the embedded eigenvalue on the dispersion
diagram, the resonance gets sharper and sharper, as its
quality factor tends to infinity, and eventually vanishes,
since the leaky mode becomes a purely bound state, decou-
pled from free-space radiation. From a physical viewpoint,
the disappearance of leakage has been generally explained
as the result of destructive interference among different
‘‘radiation channels’’ [189], or the coupling between differ-
ent guided mode resonances, which can be studied with
coupled-wave theory [190]. From the perspective of leaky
waves, it appears that, as we approach the frequency of the
embedded eigenvalue, the complex leaky pole on the wave-
number plane moves closer and closer to the real axis, until
it becomes purely real in the ideal limit, which corresponds
to a bound surface mode with no attenuation, despite being
in the fast-wave region. The details of the physical mechanism
in terms of leaky-wave theory, however, are still not fully
unveiled, and represent an exciting open area of research.
Interestingly, analogous bound states in the radiation
continuum have also been observed in crystal acoustics
[196], [197]. In particular, surface acoustic modes that are
normally leaky have been shown to become purely bound
under specific circumstances. As in the electromagnetic
case, the disappearance of leakage in these anomalous
bound states, known as ‘‘secluded supersonic surface
waves,’’ is not determined by symmetry incompatibilities.
Given the analogy between leaky waves in open guiding
structures and radiation-damped oscillations in open cavi-
ties [1], recently the concept of embedded photonic eigen-
value has also been extended to three-dimensional open
structures [198], [199]. It has been shown that composite
plasmonic spheres of finite size, and possibly subwave-
length dimensions, can be designed to support oscillatory
states without radiation loss, even when surrounded by
available and compatible radiation modes (outgoing
spherical waves). Analogous to the leaky-wave scenario
describedabove,thecomplexpoleassociatedwith
radiation-damped oscillations moves closer and closer to
the real axis of the complex frequency plane (instead of the
wavenumber plane in the leaky-wave case). In the ideal
limit, this three-dimensional embedded eigenvalue corre-
sponds to an eigenmode (self-sustained oscillation) of the
open cavity ‘‘embedded’’ along the real frequency axis, re-
sulting in the disappearance of radiation loss and ideal
light confinement, as demonstrated in [199].
IX. CONCLUSION
In this paper, we have presented and discussed the general
theoretical principles and practical applications of electro-
magnetic leaky waves, with several relevant connections to
microwave and optical physics and engineering. After over
sixty years since its inception in the context of microwave
and antenna engineering, we have discussed how this area
of research is still very relevant today, and the importance
of leaky-wave concepts is becoming increasingly more re-
cognized in different scientific communities, beyond mi-
crowave engineering.
As we have discussed extensively along this paper, the
theory of leaky waves has proved to be of fundamental
significance in the understanding of different phenomena,
including several anomalous optical effects that cannot be
explained using conventional diffraction theory and geom-
etrical optics. Moreover, since most micro- and nanostruc-
tures are electromagnetically open at optical frequencies
(field screening is more challenging at these scales and
wavelengths), leaky-wave phenomena are ubiquitous in
nanooptics and play a key role in a variety of nano-optical
components and systems. Of particular importance today
is the application of leaky-wave concepts to design and
optimize optical radiation with high directivity, as well as
structures that exhibit enhanced optical transmission and
embedded eigenvalues, which may lead to unprecedented
manipulation of light at the subwavelength scale.
In conclusion, we believe that the alliance of well-es-
tablished leaky-wave concepts with the new areas of meta-
materials, plasmonics, and nanophotonics may open
exciting new research directions, with particular emphasis
to practical applications, extending the reach of leaky-wave
theory and bringing to new relevance the pioneering
studies of the scientists and engineers that have laid the
foundationsofthisexcitingresearcharea.h
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Monticone and Alu
`: Leaky-Wave Theory, Techniques, and Applications: From Microwaves to Visible Frequencies
820 Proceedings of the IEEE |Vol.103,No.5,May2015
ABOUT THE AUTHORS
Francesco Monticone (Student Member, IEEE)
received the Laurea degree (B.Sc.) and Laurea
Specialistica degree (M.Sc., summa cum laude)
from Politecnico di Torino, Italy, in 2009 and 2011,
respectively.
After developing part of his graduate re-
search work in Macquarie University, Sydney,
Australia, he joined Dr. Alu
`’s research group in
2011. He is a graduate research assistant in the
Metamaterials and Plasmonics Research Labora-
tory of Dr. Andrea Alu
`at The University of Texas at Austin, where he is
currently pursuing the Ph.D. degree in electrical and computer engi-
neering. He has authored and coauthored more than 50 scientific contri-
butions published or under review in peer-reviewed journal papers, book
chapters, and peer-reviewed conference proceedings. His current re-
search interests are in the areas of applied electromagnetics, metama-
terials, plasmonics, and nanophotonics, spanning a broad range of topics
including extreme scattering engineering, cloaking and invisibility,
nanoparticles, nanocircuits, nanoantennas, and advanced metasurfaces,
with particular emphasis on translating and exploiting well-established
methods and concepts from microwave/antenna engineering and circuit
theory to the realm of optics, photonics and nanotechnology.
Mr. Monticone’s first-author papers have appeared in several high-
impact journals, including Physical Review Letters (three times selected
as ‘‘Editor’s Suggestion’’), Nature Nanotechnology,andScience.Someof
his recent research work has been picked up by national and interna-
tional media outlets, such as the BBC, NBC News, and Time Magazine.He
has won several student awards, including the Best Paper Award (1st
prize) at the conference Metamaterials’2013, an IEEE Antennas and Pro-
pagation Society Doctoral Research Award, three ‘‘honorable mentions’’
at student paper competitions at the IEEE AP-S symposium, and multiple
travel grants. Recently, he has also been awarded the ‘‘H. L. Bruce Grad-
uate Fellowship’’ from the Graduate School of The University of Texas at
Austin. He is serving as a reviewer for several journals and international
conferences, and he is currently a member of the organizing committee
of Metamaterials’2015. He is a student member of the American Physical
Society (APS), the American Association for the Advancement of Science
(AAAS), and the Optical Society of America (OSA).
Andrea Alu
`(Fellow, IEEE) received the Laurea,
M.S., and Ph.D. degrees from the University of
Roma Tre, Rome, Italy, in 2001, 2003, and 2007,
respectively.
From 2002 to 2008, he was with the University
of Pennsylvania, Philadelphia, where he has also
developed significant parts of the Ph.D. degree
and postgraduate research. After spending one
year as a postdoctoral research fellow at the Univ-
ersity of Pennsylvania, he joined the faculty of the
UniversityofTexasatAustinin2009.HeisalsoamemberoftheApplied
Research Laboratories and of the Wireless Networking and Commu-
nicaions Group at The University of Texas at Austin (UT Austin). Currently,
he is an Associate Professor and David & Doris Lybarger Endowed Faculty
Fellow in Engineering at UT Austin. In 2015, he is also a KNAW Visiting
Professor at AMOLF, The Netherlands. He is the coauthor of an edited
book on optical antennas, more than 270 journal papers, and more than
20 book chapters. His current research interests span over a broad range
of areas, including metamaterials and plasmonics, electromangetics,
optics and photonics, scattering, cloaking and transparency, nanocircuits
and nanostructures modeling, miniaturized antennas and nanoantennas,
RF antennas and circuits, and acoustics.
Dr. Alu
`has organized and chaired various special sessions in inter-
national symposia and conferences. He has served as Technical Program
Committee Chair of Metamaterials’2013 and Metamaterials’2014. He is
currently on the Editorial Board of Physical Review B,Scientif ic Reports ,
and Advanced Optical Materials. He serves as Associate Editor of four
journals, including the IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS
and of Optics Express. He has been a Guest Editor for special issues of the
Journal of the Optical Society of America B, the IEEE JOURNAL OF SELECTED
TOPICS IN QUANTUM ELECTRONICS,Optics Communications,Metamaterials,
and Sensors on a variety of topics involving metamaterials, plasmonics,
optics, and electromagnetic theory. Over the last few years, he has
received several honors and awards for his research output, including the
NSF Waterman award (2015), the KNAW Visiting Professorship from the
Royal Netherlands Academy of Arts and Sciences (2015), the OSA Adolph
Lomb Medal (2013), the IUPAP Young Scientist Prize in Optics (2013), the
IEEE MTT Outstanding Young Engineer Award (2014), the Franco
Strazzabosco Award for Young Engineers (2013), the URSI Issac Koga
Gold Medal (2011), the SPIE Early Career Investigator Award (2012), the
NSF CAREER award (2010), the AFOSR, and the DTRA Young Investigator
Awards (2010, 2011), Young Scientist Awards from URSI General
Assembly (2005) and URSI Commission B (2010, 2007, and 2004). His
students have also received several awards, including two student paper
awards at IEEE Antennas and Propagation Symposia (in 2011 to Y. Zhao, in
2012 to J. Soric). He has been serving as OSA Traveling Lecturer since
2010, IEEE AP-S Distinguished Lecturer since 2014, and as the IEEE joint
AP-S and MTT-S chapter for Central Texas. He is a full member of URSI, a
fellow of OSA, a senior member of SPIE, and a member of APS and MRS.
Monticone and Alu
`: Leaky-Wave Theory, Techniques, and Applications: From Microwaves to Visible Frequencies
Vol. 103, No. 5, May 2015 | Proceedings of the IEEE 821
