Content uploaded by Chia Wei Hsu
Author content
All content in this area was uploaded by Chia Wei Hsu on Apr 06, 2018
Content may be subject to copyright.
The partial or complete confinement of waves is ubiqui-
tous in nature and in wave-based technology. Examples
include electrons bound to atoms and molecules, light
confined in optical fibres and the partial confinement of
sound in musical instruments. The allowed frequencies
of oscillation are known as the wave spectrum.
To determine whether a wave can be perfectly con-
fined or not (that is, if a ‘bound state’ exists or not) in
an open system, a simple criterion is to look at its fre-
quency. If the frequency is outside the continuous spec-
tral range spanned by the propagating waves, it can exist
as a bound state because there is no pathway for it to
radiate away. Conversely, a wave state with the frequency
inside the continuous spectrum can only be a ‘resonance’
that leaks and radiates out to infinity. This is the conven-
tional wisdom described in many books. A bound state
in the continuum (BIC) is an exception to this conven-
tional wisdom: it lies inside the continuum and coexists
with extended waves, but it remains perfectly confined
without any radiation. BICs are found in a wide range of
material systems through confinement mechanisms that
are fundamentally different from those of conventional
boundstates.
The general picture is clear from the spectrum and
the spatial profile of the modes (FIG.1). More specifi-
cally, consider waves that oscillate in a sinusoidal way
as e−iωt in time t and at angular frequency ω. Extended
states (blue; FIG.1) exist across a continuous range of
frequencies. Outside this continuum lie discrete levels
of conventional bound states (green; FIG.1) that have
no access to radiation channels; this is the case for
the bound electrons of an atom (at negative energies),
the guided modes of an optical fibre (below the light
line) and the defect modes in a bandgap. Inside the
continuum, resonances (orange; FIG.1) may be found
that locally resemble a bound state but in fact couple
to the extended waves and leak out; they can be associ-
ated with a complex frequency, ω = ω0 – iγ, in which the
real part ω0 is the resonance frequency and the imagi-
nary part γ represents the leakage rate. This complex
frequency is defined rigorously as the eigenvalue of the
wave equation with outgoing boundary conditions1,2. In
addition to these familiar wave states, there is the less
known possibility of BICs (red; FIG.1) that reside inside
the continuum but remain perfectly localized with no
leakage, namely γ = 0. In a scattering experiment, waves
coming in from infinity can excite the resonances,
causing a rapid variation in the phase and amplitude
of the scattered waves within a spectral linewidth of 2γ.
However, such waves cannot excite BICs, because BICs
are completely decoupled from the radiating waves and
are invisible in this sense. Therefore, a BIC can be con-
sidered as a resonance with zero leakage and zero line-
width (γ = 0; or infinite quality factor Q = ω0/2γ). BICs
are sometimes referred to as embedded eigenvalues or
embedded trappedmodes.
In 1929, BICs were proposed by von Neumann and
Wigner 3. As an example, von Neumann and Wigner
mathematically constructed a 3D potential extending
to infinity and oscillating in a way that was tailored to
support an electronic BIC. This type of BIC-supporting
system is rather artificial and has never been realized.
However, since this initial proposal, other mechanisms
leading to BICs have been identified in different material
1Department of Applied
Physics, Yale University, New
Haven, Connecticut 06520,
USA.
2Research Laboratory of
Electronics, Massachusetts
Institute of Technology,
Cambridge, Massachusetts
02139, USA.
3Physics Department and
Solid State Institute, Technion,
Haifa 32000, Israel.
*These authors contributed
equally to this work.
Correspondence to
C.W.H. and B.Z.
chiawei.hsu@yale.edu;
bozhen@mit.edu
Article number: 16048
doi:10.1038/natrevmats.2016.48
Published online 19 Jul 2016
Bound states in the continuum
Chia Wei Hsu1*, Bo Zhen2,3*, A.Douglas Stone1, John D.Joannopoulos2
and Marin Soljacˇic´2
Abstract | Bound states in the continuum (BICs) are waves that remain localized even though
they coexist with a continuous spectrum of radiating waves that can carry energy away.
Their very existence defies conventional wisdom. Although BICs were first proposed in
quantum mechanics, they are a general wave phenomenon and have since been identified
in electromagnetic waves, acoustic waves in air, water waves and elastic waves in solids.
These states have been studied in a wide range of material systems, such as piezoelectric
materials, dielectric photonic crystals, optical waveguides and fibres, quantum dots,
graphene and topological insulators. In this Review, we describe recent developments in this
field with an emphasis on the physical mechanisms that lead to BICs across seemingly very
different materials and types of waves. We also discuss experimental realizations, existing
applications and directions for future work.
NATURE REVIEWS
|
MATERIALS ADVANCE ONLINE PUBLICATION
|
1
REVIEWS
Nature Reviews | Materials
Spectrum Mode profile
Space
BIC
Extended state
(continuum)
Regular
bound state
Resonance
(leaky mode)
Confinement
Frequency, ω
systems, many of which have been observed in experi-
ments in electromagnetic, acoustic and water waves. In
recent years, photonic structures have emerged as a par-
ticularly attractive platform owing to the ability to tailor
the material and structure, which is often impossible in
quantum systems. The unique properties of BICs have
led to numerous applications, including lasers, sensors,
filters and low-loss fibres, with many more possible uses
proposed and yet to be implemented.
Most theoretically proposed and all experimen-
tally observed BICs are realized in extended structures
because, in most wave systems, BICs are forbidden in
compact structures (BOX1). Among the extended struc-
tures that support BICs, many are uniform or periodic
in one or more directions (for example, x and y), and the
BIC is localized only in the other directions (for example,
z). In such systems, the concept of BICs requires careful
definition. More specifically, because translational sym-
metry conserves the wave vector, k// = (kx, ky), a state is
considered a BIC when it exists inside the continuous
spectrum of modes at the same k// but remains localized
and does not radiate in the z direction. These BICs are
typically found at isolated wave vectors.
In this Review, we present the concepts and physical
mechanisms that unify BICs across various material sys-
tems and in different types of waves, focusing on exper-
imental studies and applications. First, we describe BICs
protected by symmetry and separability; second, we dis-
cuss BICs achieved through parameter tuning (with cou-
pled resonances or with a single resonance); and third, we
describe BICs built with inverse construction (for example,
potential, hopping rate or shape engineering). We con-
clude with the existing and emerging applications ofBICs.
Bound states due to symmetry or separability
The simplest places to find BICs are in systems in which
the coupling of certain resonances to the radiation
modes are forbidden by symmetry or separability.
Symmetry-protected BICs. When a system exhibits a
reflection or rotational symmetry, modes of different
symmetry classes completely decouple. It is common
to find a bound state of one symmetry class embedded
in the continuous spectrum of another symmetry class,
and their coupling is forbidden as long as the symmetry
is preserved.
The simplest example concerns sound waves in air,
with a plate placed along the centreline of an acoustic
waveguide (FIG.2a). The fluctuation of air pressure, p,
follows the scalar Helmholtz equation with Neumann
boundary condition ∂p/∂n = 0 on the surfaces of the walls
and of the plate, where n is the direction normal to the
surface. The waveguide supports a continuum of waves
propagating in the x direction that are either even or odd
under mirror reflection in the y direction; the odd modes
(red; FIG. 2a, middle panel) require at least one oscillation
in the y direction and only exist above a cut-off frequency,
πcs/h, where cs is the speed of sound and h is the width of
the waveguide. The plate respects the mirror symmetry
and, as a result, modes localized near it are also even or
odd in the y direction, and an odd mode below the cut off
is guaranteed to be a bound state despite being embedded
in the continuum of even extended modes (FIG.2a). Parker
first measured4 and analysed5 such modes using a cascade
of parallel plates in a wind tunnel. These modes can be
excited from the near field and are audible with a steth-
oscope placed near the plates. This plate-in-waveguide
system has been studied by others6–8, and obstacles with
arbitrary symmetric shapes have also been considered9.
It should be noted that obstacles that are infinitesimally
thin and parallel to the waveguide are decoupled from the
fundamental waveguide mode even without mirror-in-y
symmetry10–13.
Similar symmetry-protected bound states exist in
canals as surface water waves14–20, in quantum wires21–23, or
for electrons in potential surfaces with antisymmetric cou-
plings24. A common setup is a 1D waveguide or lattice array
that supports a continuum of even-in-y extended states,
with two defects attached symmetrically above and below
this array to create an odd-in-y defect bound state. This
configuration has been explored with the defects com-
prising single-mode optical waveguides25–28, mechanically
coupled beads29,30, quantum dots31–41, graphene flakes42,43,
ring structures37,44,45 or impurity atoms46,47. Experimental
realizations are demonstrated in two of these studies27,28,
both using coupled optical waveguides. When the mirror
symmetry is broken, the bound state turns into a leaky
resonance. In one example, the mirror symmetry is broken
by bending the defect waveguides, which allows coupling
light into and out of the would-be BIC27. In another case,
a temperature gradient changes the refractive index of the
material and breaks the mirror symmetry, which induces
radiation in a controllable manner28 (FIG.2b).
Symmetry-protected BICs also exist in periodic
structures: for example, a photonic crystal (PhC) slab48
comprising a square array of cylindrical holes etched
into a dielectric material (FIG.2c). Because of the perio-
dicity in the x and y directions, the photonic modes can
be labelled by k// = (kx, ky). When the 180° rotational sym-
metry around the z axis (C2) is preserved (for example,
Figure 1 | Illustration of a BIC. In an open system, the frequency spectrum consists of
a continuum or several continua of spatially extended states (blue) and discrete levels
of bound states (green) that carry no outgoing flux. The spatial localization of the bound
states is a consequence of a confining structure or potential (black dashed line). States
inside the continuous spectrum typically couple to the extended waves and radiate,
becoming leaky resonances (orange). Bound states in the continuum (BICs; red) are
special states that lie inside the continuum but remain localized with no radiation.
REVIEWS
2
|
ADVANCE ONLINE PUBLICATION www.nature.com/natrevmats
at k// = (0, 0), commonly known as the Γ point), even
and odd modes with respect to C2 are decoupled. At
frequencies below the diffraction limit of ωc = 2πc/na
(where a is the periodicity, n is the refractive index of
the surrounding medium and c is the vacuum speed of
light), the only radiating states are plane waves in the
normal direction (z) with the electric and magnetic field
vectors being odd under C2; therefore, any even mode at
the Γ point is a BIC. Away from the Γ point, these states
start to couple to radiation, because they are no longer
protected by C2. This disappearance of radiation has
been observed in early experiments on periodic metal-
lic grids49, documented in theoretical studies on PhC
slabs50–56 and measured quantitatively from the Q of reso-
nances in large-area PhC slabs57 (FIG.2c). The suppressed
radiation has also been characterized in the lasing pat-
tern of 1D periodic gratings58,59. Such photonic BICs are
commonly realized in silicon photonics and with III–V
semi conductors, and have found applications in lasers,
sensors and filters (see the last section).
In crystal acoustics, symmetry-protected BICs exist
as the surface acoustic wave (SAW) in anisotropic solids,
such as piezoelectric materials. This phenomenon can be
used to enhance performance beyond the typical limit of
bulk materials. For example, the phase velocity, V = ω/|k//|,
of a SAW is limited to the speed of the slowest bulk wave,
otherwise it becomes a leaky resonance. However, along
high-symmetry directions, symmetry may decouple the
SAW from the bulk waves, turning the resonance into
a supersonic but perfectly confined SAW60–64, allow-
ing higher phase velocity than the bulk limit. A related
example exists in optics in uniform slabs with anisotropic
permittivity and permeability tensors65.
Separable BICs. Separability can also be exploited to
construct BICs. For example, consider a 2D system with
a Hamiltonian of the form
H = Hx(x) + Hy(y) (1)
where Hx acts only on the x variable and Hy acts only
on the y variable. It is possible to separately solve
the 1D eigen-problems HxΨx
(n)(x) = Ex
(n)Ψx
(n)(x) and
HyΨy
(m)(y) = Ey
(m)Ψy
(m)(y). If Ψx
(n)(x) and Ψy
(m)(y) are
bound states of the 1D problems, their product wave-
function, Ψx
(n)(x)Ψy
(m)(y), is bound in both dimen-
sions and will remain localized even if its eigenvalue,
Ex
(n) + Ey
(m), lies within the continuous spectrum of the
extended states for the 2D Hamiltonian; coupling to
the extended states is forbidden by separability. This
type of BIC was first proposed by Robnik66, and sub-
sequently studied in other quantum systems67–69 and
in Maxwell’s equation in 2D70–73. So far, separable BICs
have not been observed experimentally, but there are
promising examples in several material systems, includ-
ing photo refractive medium, optical traps for cold atoms
and certain lattices described by tight-binding models74.
Bound states through parameter tuning
When the number of radiation channels is small, tuning
the parameters of the system may be enough to com-
pletely suppress radiation into all channels. Generally,
if radiation is characterized by N degrees of freedom, at
least N para meters need to be tuned to achieve a BIC.
In many cases, this suppression can be interpreted as an
interference effect in which two or more radiating com-
ponents cancel each other. We describe three different
scenarios in the following subsections.
Fabry–Pérot BICs (coupled resonances). A resonant
structure coupled to a single radiation channel is known
to cause unity reflection near the resonance frequency,
ω0, when there are no other losses. This is because the
direct transmission and the resonant radiation interfere
and completely cancel each other75. Two such resonant
structures can act as a pair of perfect mirrors that trap
waves in between them. BICs are formed when the
resonance frequency or the spacing, d, between the
two structures is tuned to make the round-trip phase
shifts add up to an integer multiple of 2π (FIG.3a). This
structure is equivalent to a Fabry–Pérot cavity formed
between two resonant reflectors.
Temporal coupled-mode theory76 provides a simple
tool to model such BICs. In the absence of external driv-
ing sources, the two resonance amplitudes A = (A1, A2)T
evolve in time as i∂A/∂t = HA with Hamiltonian77–79
H = – iγ
κ
κω
0
ω0e
iψ
e
iψ
11
(2)
where κ is the near-field coupling between the two reso-
nators, γ is the radiation rate of the individual resonances
and ψ = kd is the propagation phase shift between the
two resonators, where k is the transverse wavenumber
(FIG.3a). The two eigenvalues of H are
ω± = ω0 ± κ – iγ(1 ± eiψ) (3)
Box 1 | Non-existence of single-particle BICs in compact structures
Most structures supporting bound states in the continuum (BICs) extend to infinity
in at least one direction. This is because BICs are generally forbidden in compact
structures for single-particle-like systems.
Consider a 3D compact optical structure in air, characterized by its permittivity ε(r)
and permeability μ(r), with R as the radius of a sphere that encloses the structure.
Outside the bounding sphere ε(r) = μ(r) = 1; therefore, the electric (E) and magnetic (H)
fields follow the Helmholtz equation and can be expanded in spherical harmonics and
spherical Hankel functions with wavenumber k = ω/c. A bound state must have no
radiating far field, but every term in the expansion carries an outgoing Poynting flux,
and hence, all terms must be zero, meaning that E and H must both vanish for |r| > R. If
ε(r) and μ(r) are neither infinite nor zero anywhere, continuity of the fields requires E
and H to be zero everywhere in space, and as a result, such a bound state cannot
exist254. The same argument applies to a 1D or 2D system.
This non-existence theorem does not exclude compact BICs when the material has
ε = ±∞, μ = ±∞, ε = 0 or μ = 0, which can act as hard walls that spatially separate the
bound state from the extended ones. Examples with ε = 0 have been proposed251,254,255
but are difficult to realize because typically the loss Im(ε) is significant at the plasma
frequency of a metal, where Re(ε) = 0.
The same argument applies to the single-particle Schrödinger equation. For an
electron with a non-vanishing effective mass (m; the m = 0 case is studied in REF.256)
in a compact and finite potential (V(r) = 0 for |r| > R, and V(r) ≠ ±∞ everywhere), a
bound state with positive energy E > 0 cannot exist. Similarly, this non-existence
theorem can be applied to acoustic waves in air and to linearized water waves in
constant-depth (z-independent) structures, because both systems are described by
the Helmholtz equation. However, this theorem does not apply to water waves in
structures with z dependence80, which follow the Laplace equation instead.
REVIEWS
NATURE REVIEWS
|
MATERIALS ADVANCE ONLINE PUBLICATION
|
3
a Acoustic waveguide
h
Extended c PhC slabb Coupled-waveguide array
0
Odd
Even
Angle (degrees)
2,000
4,000
6,000
8,000
10,000
2 μm
Even
Odd
a
Extended
Bound
Bound
0
Even
Odd
100
–100
0
Intensity
Radiation (a.u.)
115
125
120
130
Bound
Bound
0
0.2
0.4
0.6
0.8
1.0
Temperature difference (K)
Bound
y
x
Sound pressure
level (dB)
86 88 90 92 94 96
Frequency, ω (Hz)
x (μm)
y (μm)
–200 200
0
πcs/h
HzHzEzEz
TE1
TE1
TM1
TM1
TM4
TE2
β
z
β
z
0 20 40 60 80 100
Ex,y
Bound
2 πc/na
Qtot
0.0 0.3 0.6 0.9
Nature Reviews | Materials
… …
ω
ω
ω
ω
When ψ is an integer multiple of π (namely, when the
round trip phase shift is an integer multiple of 2π), one of
the two eigenmodes becomes more lossy with twice the
original decay rate, and the other eigenmode becomes a
BIC with a purely real eigenfrequency.
Fabry–Pérot BICs are commonly found in systems
with two identical resonances coupled to a single radi-
ation channel. They exist in water waves between two
obstacles80–85, as first proposed by McIver80 — these are
sometimes called sloshing trapped modes86. In quantum
mechanics, they are found in impurity pairs in a wave-
guide20,87, in time-dependent double-barrier structures88,
in quantum dot pairs connected to a wire89–93, in double
metal chains on a metal substrate94 or in double wave-
guide bends95. In photonics, Fabry–Pérot BICs exist in
structures ranging from stacked PhC slabs96–98 and dou-
ble gratings99,100, to off-channel resonant defects con-
nected to a waveguide or waveguide array25,27,45,77,101,102.
Such BICs have also been studied in acoustic cavities103.
A unique property of Fabry–Pérot BICs is that the two
resonators interact strongly through radiation even
when they are far apart. These long-range interactions
have been studied in cavities or qubits coupled through
a waveguide104–106 and for two leaky solitons coupled
through free-space radiation107.
The same principle applies when a single resonant
structure is next to a perfectly reflecting boundary, such
as a hard wall, lattice termination or a PhC with a band-
gap. For example, Fabry–Pérot BICs exist on the surface
of a photonic crystal108 and in a semi-infinite 1D lattice
with a side-coupled defect, which has been predicted109
and then experimentally realized110 using coupled optical
waveguides (FIG.3b). This principle can be extended to
polar or spherical coordinates111,112.
Friedrich–Wintgen BICs (coupled resonances). The
intuitive unity-reflection explanation of Fabry–Pérot
BICs applies when the two resonators are far apart.
However, equation 3 shows that a BIC can arise even
with no separation (d = 0). In other words, two reso-
nances at the same location can lead to a BIC through
interference of radiation — unity reflection is not a
requirement.
In temporal coupled-mode theory, when two reso-
nances reside in the same cavity and are coupled to
the same radiation channel, the resonance amplitudes
evolve with the Hamiltonian113,114
H
= – i
κ
κω1
ω2
γ1
γ2
√γ1γ2
√γ1γ2
(4)
Figure 2 | Symmetry-protected bound states. a | An acoustic waveguide with an obstructing plate (black) placed at the
centre. An odd bound state exists at the same frequency as an even extended state but cannot couple to it. Measuring the
sound pressure near the plate reveals the bound state (bottom panel)6. b | A coupled-waveguide system with two defects
placed symmetrically parallel to a linear array, which supports a similar odd bound state. The propagation constant, βz, has
the role of frequency. A temperature gradient can break the mirror symmetry by the thermo-optic effect and turn the
bound state into a leaky resonance (bottom panel)28. c | A photonic crystal (PhC) slab with a 180° rotational symmetry
around the z axis (C2). At the Γ point, modes that are even under C2 cannot radiate because plane waves in the normal
direction are odd under C2. Away from the normal direction, the bound states become leaky with finite quality factors (Qs),
as confirmed by reflectivity measurements (bottom panel)57. πcs/h, cut-off frequency (where cs is the sound speed and h is
the width of the waveguide); 2πc/na, diffraction limit (where c is the vacuum speed of light, n is the refractive index of the
surrounding medium and a is periodicity); a.u., arbitrary units; Ex,y, z, the x, y and z components of the electric field; h,
height; Hz, the z component of the magnetic field; Qtot, total quality factor; TE1,2, first and second transverse-electric-like
modes; TM1,4, first and fourth transverse-magnetic-like modes; ω, angular frequency.
REVIEWS
4
|
ADVANCE ONLINE PUBLICATION www.nature.com/natrevmats
Nature Reviews | Materials
1
0
Standing wave
a Mechanism b Realization
F
Theory
–
+
End
Experiment
Frequency
Reflectivity
Resonance 2Resonance 1
5 4 3 2 1
…
γ γ γ γ
ω0ω0
ψ
= kd
…
Intensity F
5 4 3 2 1
…
ω0
Here, we consider the scenario in which the two reso-
nances can have different resonance frequencies, ω1,2, and
different radiation rates, γ1,2. The two resonances radiate
into the same channel, and hence, interference of radia-
tion gives rise to the via-the-continuum coupling term
√γ1γ2
. As a result, when
κ (γ1 – γ2) = √γ1γ2 (ω1 – ω2) (5)
one eigenvalue becomes purely real and turns into a BIC
and the other eigenvalue becomes more lossy. This type of
BIC is named after those who first derived equation 5 —
Friedrich and Wintgen115. Note that when κ = 0 or when
γ1 = γ2, the BIC is obtained at ω1 = ω2; therefore, when κ ≈ 0
or γ1 ≈ γ2, Friedrich–Wintgen BICs occur near the fre-
quency crossings of the uncoupled resonances. Generally,
these BICs are possible when the number of resonances
exceeds the number of radiation channels116,117, but the
required number of tuning parameters also grows with
the number of radiation channels.
The first examples of Friedrich–Wintgen BICs were
proposed in atoms and molecules118,119, and their effects
have been observed experimentally as a suppressed auto-
ionization in certain doubly excited Rydberg states of
barium120. More recently, these BICs have been studied
in continuum shell models121, cold-atom collisions122, 2D
topological insulators with a defect123, and for quantum
graphs124, quantum billiards125 or impurity atoms126,127
attached to waveguides. In acoustics, they have been
studied in multiresonant cavities103,128. In optics, they have
been studied in multiresonant dielectric objects in micro-
wave waveguides129,130 and as ‘dark state lasers’ (REF.1 31).
Single-resonance parametric BICs. The preceding exam-
ples relate to two (or more) coupled resonances whose
radiations cancel to produce BICs. Meanwhile, a sin-
gle resonance can also evolve into a BIC when enough
parameters are tuned. The physical picture is similar to
the preceding examples; here, the single resonance itself
can be thought of as arising from two (or more) sets of
waves, and the radiation of the constituting waves can be
tuned to cancel eachother.
BICs tuned from a single resonance have been pre-
dicted and realized in a PhC slab132, as shown in FIG.4a. At
wave vectors away from k// = (0, 0), modes above the light
line (ω > |k//|c/n) radiate and form leaky resonances54.
When the PhC slab has C2 symmetry, up–down mirror
symmetry and time-reversal symmetry, the number of
radiation channels is reduced132. At a generic k point along
the Γ-to-X direction, the resonance turns into a bound
state, as shown by the diverging radiative quality factor,
Qr (FIG.4a). Qr can be determined through the reflectiv-
ity spectrum132, or through the photocurrent spectrum
by embedding a detector in the slab133. Such BICs also
exist in a linear periodic array of rectangles134,135, cylin-
ders136 or spheres137, and related BICs have been found
in time-periodic systems138. It is possible to analyse them
through spatial coupled-wave theory139. With the mode
expansion method, one can solve for the BICs efficiently
and also reveal which sets of waves interfere to cancel the
radiation134,135,140. Although these BICs are not guaranteed
to exist by symmetry, when they do exist they are robust
to small changes in the system parameters, and their gen-
eration, evolution and annihilation follow strict rules that
can be understood through the concept of topological
charges141, which also governs other types of BICs (BOX2).
These BICs can be described as ‘topologically protected’
and are known to exist generically if the system param-
eters (for example, the lattice spacing and thickness of
the PhC) can be varied over a sufficient range. The topo-
logical protection of BICs in a periodic structure has been
studied in quantum Hall insulators142 (BOX2).
Single-resonance parametric BICs can also exist in
non-periodic structures, as shown theoretically in acous-
tic and water waveguides with an obstacle143–148, in quan-
tum waveguides with impurities149–151 or bends95,152,153,
for mechanically coupled beads29,30 and mechanical res-
onators154, and in optics for a low-index waveguide on a
high-index membrane155.
These types of BICs also manifest themselves through
other types of SAWs in anisotropic solids. For example,
it was predicted156,157 that on the (001) plane of GaAs, the
leaky SAWs become true surface waves (that is, no leakage
into the bulk) at a propagation direction of ϕ ≈ 33° (where
ϕ is the angle from the [100] direction), in addition to
the more well-known symmetry-protected SAW at the
[110] direction of ϕ = 45° (FIG.4b). The reduced attenu-
ation near ϕ ≈ 33° was observed experimentally158,159.
Such SAWs exist in other solids160–168 and are sometimes
called secluded supersonic SAWs161. With a periodic
mass loading on the surface, secluded supersonic SAWs
may also be found in isotropic solids169–171. This type of
acoustic BIC was first reported in a piezoelectric material,
LiNbO3 (REF.172), and has been used in supersonic SAW
devices173–176 (see the last section).
Bound states from inverse construction
Instead of looking for the presence of BICs in a given
system, the problem can be turned around; if starting
with a desired BIC, it is possible to design a system that
Figure 3 | Fabry–Pérot BICs. a|A schematic illustration of the Fabry–Pérot bound state
in the continuum (BIC). Two identical resonances radiate into the same radiation channel,
and each resonance acts as a perfect reflector at the resonance frequency, ω0; therefore,
waves can be trapped in between the two resonances when the round-trip phase shift is
an integer multiple of 2π. b|Realization of a Fabry–Pérot BIC in a semi-infinite coupled
waveguide array, in which the defect waveguide (F) and its mirror image with respect to
the end are the two resonances. γ, radiation rate; ψ = kd, propagation phase shift between
two resonators (where k is the transverse wavenumber and d is distance). Part b is
adapted with permission from REF.110, American Physical Society.
REVIEWS
NATURE REVIEWS
|
MATERIALS ADVANCE ONLINE PUBLICATION
|
5
0
0.2
0.4
0.6
a
2.4
2.6
2.8
3.0
3.2
3.4
Linewidth (a.u.)
BICs BICs
Air
b
(001)
[100]
Continuum
Continuum
Nature Reviews | Materials
Liquid
Si3N4
SiO2
Si
θ
a
GaAs
crystal
k//
V// (km s–1)
Im(V//) (km s–1)
Angle ϕ
(degrees) Angle ϕ (degrees)
Angle
θ
(degrees)
0 10 20 30 40 20
00.0 0.5 20 40 60
25 30 35 40 45
Supersonic SAW (leaky)
Subsonic SAW
10–1
10–3
103
104
105
106
10–5
10–7
10–9
Experiment
Theory
Experiment
Theory
Qr
BICs
BICs
Resonance
Regular bound state
Frequency
ω
a/2
π
c
Wavevector kxa/2
π
PhC slab
SAW
ϕ
can support this bound state and the continuous spec-
trum containing it. This inverse construction is achieved
by engineering the potential, the hopping rate or the
boundary shape of the structure.
Potential engineering. The first proposal of BICs by von
Neumann and Wigner was based on potential engineer-
ing3. For a desired BIC with wave function Ψ and energy
E > 0, the corresponding potential V can be determined
by rewriting the Schrödinger equation (in reduced units):
–
∇2Ψ + VΨ = EΨ → V = E + ∇2Ψ
2Ψ
2
1 (6)
Ψ and E must be chosen appropriately so that the result-
ing V vanishes at infinity (to support the continuum)
and is well defined everywhere. There are many possi-
ble solutions. The example given by von Neumann and
Wigner is Ψ(r) = f(r)sin(kr)/kr with f(r) = [A2 + (2kr – sin
(2kr))2]−1, which has an energy E = k2/2 embedded in the
continuum E ≥ 0. This bound wave function and the cor-
responding potential V(r) from equation 6 is shown in
FIG.5a for A = 25,
k
= √8 and E = 4 (note that REF.3con-
tainsanalgebraicmistake177,178). More examples can be
found in REF.178, and this procedure has been general-
ized to non-local potentials179 and lattice systems180,181.
From a mathematical point of view, this inverse con-
struction is closely related to the inverse spectral theory
of the Schrödinger operator182 and the Gel’fand–Levitan
formalism of the inverse scattering problem, which
can also be used to construct potentials supporting a
finite183–186 or even infinite number of BICs182,187.
A related approach uses the Darboux transforma-
tion188 that is commonly used in supersymmetric (SUSY)
quantum mechanics to generate a family of potentials
that share the same spectrum. This transformation
can be applied to a free-particle extended state to yield
a different potential in which the corresponding state
keeps its positive energy (remaining in the continuum)
but becomes spatially localized189–191. In some cases,
this SUSY method is equivalent to the von Neumann–
Wigner approach and the Gel’fand–Levitan approach192.
The SUSY method has been applied to generate BICs in
point interaction systems193, periodic Lamé potentials194
and photonic crystals195. The SUSY method has also
been extended to non-Hermitian systems with material
gain and loss, in which BICs are found below, above and
at an exceptional point196–204.
Potential engineering allows for analytic solutions of
BICs. However, the resulting potentials tend to be un -
realistic — indeed, none have been realized experimen-
tally so far. In addition, perturbations generally reduce
such BICs into ordinary resonances205,206.
Hopping rate engineering. A more experimentally rele-
vant construction is to engineer the hopping rate between
nearest neighbours in a tight-binding lattice model.
Such construction can be carried out through the SUSY
transformation199,207 and has been demonstrated in a
coupled optical waveguide array, in which the hopping
rate is tuned by the distance between neighbouring wave-
guides208. Intuitively, this method can be understood as
‘kinetic energy engineering’.
The array of coupled optical waveguides208 comprises
a semi-infinite 1D lattice in which the on-site energy is
constant and the hopping rate, κn, between sites n and
n + 1 follows (FIG.5b):
n ≠ lN,
n = lN, (l = 1,2,3,...)
κ
n =
κ
lκ
β
l + 1 (7)
where N > 1 is an integer, β > 1/2 is an arbitrary real number
and κ is the reference hopping rate. This system supports
N – 1 BICs localized near the surface (n = 1). The particu-
lar case of N = 2 and β = 1 was experimentally realized
with 40 evanescently coupled optical waveguides208. The
Figure 4 | Single-resonance parametric BICs. a|Bound state in the continuum (BIC)
from a single resonance in a photonic crystal (PhC) slab. The left panel is a schematic
illustration of the system. The middle panel shows the photonic band structure. The leaky
resonance turns into two BICs at wavevectors kx = 0 (due to symmetry) and kxa/2π ≈ 0.27
(through tuning) as marked by red circles. The radiative quality factor, Qr, diverges to
infinity at the two BICs, as shown by the experimental data (red crosses) and theory (blue
line) in the right panel. b|BIC from the leaky surface acoustic wave (SAW) of GaAs. The
left panel is a schematic illustration of the (001) surface of GaAs. The middle panel
depicts the acoustic band structure. Radiation of the leaky SAW (orange line) vanishes at
ϕ = 45° (due to symmetry) and at ϕ ≈ 33° (through tuning). The right panel shows the
theoretical results of attenuation in log scale (blue line) and measured resonance
linewidth in linear scale (red crosses). a, periodicity; a.u. arbitrary units; c, vacuum speed
of light; V//, phase velocity; Im(V//), imaginary part of the phase velocity. Part a is from
REF.132, Nature Publishing Group. Part b is adapted with permission from REF.157,
American Institute of Physics, and from REF.159, Elsevier.
REVIEWS
6
|
ADVANCE ONLINE PUBLICATION www.nature.com/natrevmats
Nature Reviews | Materials
a Polarization vector E//
Polarization vector E//
b Topological charges of BICs
Ex = 0Ey = 0BIC
q = 0, no BIC q = 1q = –1 q = –2
ky
kxkxkxkx
s–p plane
x–y plane
Ez
E(k//, kz)
E// = (Ex, Ey)
theoretical hopping rates, the BIC mode intensity, |cn|2,
along with the experimentally measured intensity when
light is launched from the first site are shown in FIG.5b.
Boundary shape engineering. BICs can also arise from
engineering the boundary shape of the structure. This
method was first proposed in water waves, in a system
involving two line sources placed at a certain distance
apart on the water surface such that the propagation
phase shift is π (REF.80). Surface-wave radiations from
the two sources cancel, resulting in a spatially confined
mode profile. Then, the two line sources are replaced
with two obstacles whose boundary shapes correspond
to streamlines of the mode profile that contain the two
sources. In this way, the mode profile in the original
driven system is a BIC in the new undriven system with
obstacles, because it satisfies the Neumann boundary
condition on the obstacle surface, which is a property
of the streamlines.
BICs constructed with two line sources or a ring of
sources typically lead to the Fabry–Pérot type80,82,83,111,112
as described earlier. But this procedure can be extended
to more complex shapes209,210 and to free-floating rather
than fixed obstacles211–213.
Applications of BICs and quasi-BICs
Lasing, sensing and filtering. Structures with BICs are
natural high-Q resonators, because the Qr is, in the ideal
case, infinity. This makes them useful for many optical
and photonic applications. In particular, the macroscopic
size (on the centimetre scale or larger) and ease of fab-
rication make BICs in PhC slabs unique for large-area
high-power applications such as lasers214–219, sensors220,221
and filters222.
Many surface-emitting lasers are based on symmetry-
protected BICs at the Γ point (FIG.2c). This effect was
first observed by the suppression of radiation into the
normal direction in a surface-emitting distributed feed-
back laser with 1D periodicity58,59. This led to PhC sur-
face-emitting lasers (PCSELs) that lase through BICs
with 2D periodicity214,215, followed by the realization
of various lasing patterns216,217, lasing at the blue-violet
wavelengths218 and lasing with organic molecules221.
The suppressed radiation in the normal direction
means that a PCSEL can have a low lasing threshold
but also with a limited output power. Therefore, recent
designs intentionally break the C2 symmetry to allow
some radiation into the normal direction. For example,
the air holes can be intentionally designed as triangular
shapes to break the C2 symmetry (FIG.6a,b); this led to219
continuous-wave lasing at room temperature with 1.5
watt output power and high beam quality (M2 ≤ 1.1),
even though the threshold is still relatively low (FIG.6c).
In addition, PCSELs produce vector beams223,224 with
the order numbers given by the topological charges
of the BICs141 (BOX2), which may find applications in
super-resolution microscopy and in table-top particle
accelerators (see REF.225 for a review on vector beams).
Box 2 | Topological nature of BICs
Perturbations typically turn a bound state in the continuum (BIC) into a leaky resonance. However, some BICs are
protected topologically and cannot be removed except by large variations in the parameters of the system.
The topological nature of BICs can be understood through the robust BICs in photonic crystal (PhC) slabs, which are
fundamentally 2D topological objects141 — vortices. For a general resonance in a PhC slab, the polarization direction of
the far-field radiation is given by a 2D vector, E// = (Ex, Ey) (part a). BICs do not radiate; they exist at the crossing points
between the nodal lines of Ex = 0 and those of Ey = 0. In the k space, the polarization vector forms a vortex around each BIC
with a corresponding ‘topological charge’, q; a few examples (q = 1, −1 and −2) are shown in part b as well as the case with
no BIC (q = 0). Once any crossing (BIC) occurs, large changes in the system parameters are required to remove it.
Topological charges cannot suddenly disappear, because they are conserved and quantized quantities protected by
boundary conditions257; therefore, a BIC of this type cannot be removed unless it is cancelled out with another BIC
carrying the exact opposite topological charge.
The topological properties of BICs were also studied in electron systems with a 2D quantum Hall insulator placed on top
of a 3D bulk normal insulator142. Under low-energy excitation, pure surface modes in the quantum Hall system were found
at isolated k points, embedded in the continuum of the bulk modes of the normal insulator.
Recently, a unified picture of BICs as topological defects has been presented (unpublished observation, H. Zhou). This
study shows that the sum of all topological charges carried by the BICs within the Brilluion zone is governed by a different
topological invariant of the bands — Chern numbers258. The identification and design of BICs in other wave systems, such
as polaritons, magnons and anyons, may be possible using this unified theorem.
REVIEWS
NATURE REVIEWS
|
MATERIALS ADVANCE ONLINE PUBLICATION
|
7
Nature Reviews | Materials
a Potential engineering b Hopping rate engineering
2
1
1 10 20
Lattice index n
Hopping rates
0
1
Mode intensity
Experiment
2κ
–2κ
0
Setup
0
0.4
–0.4
Wave function
0 5 10 15
0 5 10 15
0
1.5
–1.5
Potential
0
100
200
300
400
Radius, r
Radius, r
V(r)
Ψ (r)
E = 4 > Vmax
…
V = 0κ1
κn/κ
κ2κ3κ4
x axis (μm)
z axis
Light
launching
1 10 20 0 4 8
Lattice index, n
|cn|2
z axis (cm)
βz
x axis
Another application lies in chemical and biological
sensing, and particularly in optofluidic setups226. One
sensing mechanism uses the shift of resonance fre-
quency to detect the change of refractive index in the sur-
roundings. Resonators with higher Qs enable narrower
line widths and higher sensitivity, and it is possible to
directly visualize a single monolayer of proteins with the
naked eye using the high-Q resonances close to a BIC220.
Another type of sensing relies on measuring fluorescence
signals. More specifically, the spontaneous emission from
organic molecules can be strongly enhanced and the
angular distribution can be strongly modified near BICs
in a PhC slab, leading to a total enhancement of angular
fluor escence intensity by 6,300 times221. BICs have also
permitted large-area narrow-band filters in the infrared
regime222 as a consequence of their high and tunableQs.
Supersonic surface acoustic wave devices. BICs in
acoustic wave systems, such as the supersonic SAW on
the surface of anisotropic solids (FIG.4b), have enabled
important devices such as supersonic SAW filters. A
schematic setup comprising an interdigital transducer
placed on a piezoelectric substrate is depicted in FIG.6d.
This device converts the input electric signal into an
acoustic wave, which propagates as a SAW to the other
side and reverts to an electric signal on the output side.
In contrast to a regular subsonic SAW — the speed of
which is limited by the speed of the bulk waves — a BIC
allows propagation at a much faster supersonic speed
(FIG.6e), and can therefore be used as a supersonic SAW
filter. The BIC and supersonic SAW are along a fixed
direction (ϕ = 36° in FIG.6f), because other directions are
lossy. The spatial periodicity of the interdigital trans-
ducer determines |k//|, and the central angular frequency
of the SAW filter is given by ω = |k//|V. A characteristic
filtering spectrum176 using a supersonic SAW filter on
Y–X cut LiTaO3 is shown in FIG.6f. Supersonic SAW fil-
ters based on BICs are widely used in mobile phones and
cordless phones, Bluetooth devices and delay lines173–175
because of their low loss, high piezoelectric coupling,
reasonable temperature stability, excellent accuracy and
repeatability, and compatibility with photolithography227.
Guiding photons in gapless PhC fibres. PhC fibres can
guide light in a low-index material through a photonic
bandgap48, but the bandwidth is limited by the width
of the bandgap. A type of hollow-core Kagome-lattice
PhC fibre (FIG.6g) can provide wave guiding inside the
continuum without a bandgap228,229. Its mechanism —
sometimes referred to as inhibited coupling — is a con-
sequence of the dissimilar azimuthal dependence of the
core and cladding modes. More specifically, the core
mode varies slowly with angle, but the cladding mode
is oscillating quickly, as can be seen from the unit-cell
mode (FIG.6h). Although such fibre modes are not true
BICs because there can be residual radiation, they ena-
ble broadband guidance in air and have found many
applications, including multiple-octave frequency comb
generations229 (FIG.6i), all-fibre gas cells230,231 and Raman
sensing232. In addition, by carefully engineering the core
shape, the residual radiation of these quasi-BIC modes is
reduced significantly to 17 dB km−1, which is comparable
to photonic bandgap fibres233.
Outlook
As a general wave phenomenon, BICs arise through several
distinct mechanisms and exist in a wide range of mat-
erial systems. In this Review, we have described the main
mechanisms with examples from atomic and molecular
systems, quantum dots, electromagnetic waves, acoustic
waves in air, water waves and elastic waves insolids.
We have not covered all possible mechanisms. For
example, BICs in systems with chiral symmetry234 are
distinct from the symmetry-protected BICs. In some
two-particle Hubbard models, there are bound states
that can move into and out of the continuum contin-
uously235–238; the confinement requires no parame-
ter tuning and has been credited to integrability236.
Systems with a perfectly flat band can support localized
states239,240. Localization can be induced with strong gain
and/or loss: for example, in a defect site with high loss241
and in the bulk200,202,242,243 or on the surface244 of pari-
ty-time symmetric systems. The latter has been realized
in a synthetic photonic lattice245. There may also be more
constructions not yet discovered.
Even though the very first proposal3 and many subse-
quent studies pointed to the existance of BICs in quantum
systems, there have not been any conclusive observations
of a quantum BIC except for the suppressed linewidth
Figure 5 | A BIC through inverse construction. a|The bound state in the continuum
(BIC) proposed by von Neumann and Wigner3. A potential (left panel) is engineered to
support a localized electron wave function (right panel) with its energy embedded in
the continuous spectrum of extended states. b|Construction of a BIC by engineering
the hopping rates in a semi-infinite lattice system. The hopping rates κn (top left panel)
follow equation 7 to support a bound state (bottom left panel) at βz = 0, embedded in
the continuum of the extended states (−2κ ≤ βz ≤2κ). This BIC is experimentally realized
in an array of coupled optical waveguides (top right panel); light launched at one end
of the array excites the BIC, which propagates along the waveguides. The
corresponding intensity image is shown in the bottom right image. βz, propagation
constant; |cn|2, BIC mode intensity; E, energy; Ψ(r), wave function; V(r), potential
function; Vmax, maximum potential. Part b is adapted with permission from REF.208,
American Physical Society.
REVIEWS
8
|
ADVANCE ONLINE PUBLICATION www.nature.com/natrevmats
Nature Reviews | Materials
PCSELs
c Lasing characterization
b PhC slab resonator
a Experimental setup
x
z
y
x
y
200 nm 200 nm
Output power (W)
0.0
1.0
0.5
1.5
0.0 1.0 2.0 2.50.5 1.5
Current (A)
Supersonic SAW devices
d Supersonic SAW filter setup
Input
Output
SAW
Piezoelectric
Interdigital transducer
f Filtering spectrum
Frequency (MHz)
Loss (dB)
820 900 980
0
–40
–10
–30
–20
860 940
–50
–60
0 30 60 90
Bulk
speed
Velocity (m s–1)
4,500
3,000
4,212
36
3,500
4,000
3,128
e Phase velocity
Guiding photons in gapless PhC fibres
g Hollow-core Kagome-lattice fibre
Unit-cell mode
300
i Frequency comb generation
h Mode profiles
800 6001,500
Output power (dB a.u.)
0
–100
–20
400500
Wavelength (nm)
–40
–60
–80
Core mode Cladding mode
MQWs
Cladding
BICSupersonic
Subsonic
Rotation angle ϕ (degrees)
Continuum
y
x
Photonic crystal
Laser emission
n electrode
Substrate
Cladding
MQWs
p electrode
Cladding
Contact
in Rydberg atoms120. Many researchers mistakenly cite
REF.246 as an experimental realization of a quantum BIC,
but this study concerns a positive-energy defect state
with energy in the bandgap created by a superlattice, not
a BIC. A quantum BIC has been claimed in a study on
multiple quantum wells247; however, data indicate a finite
leakage rate and no evidence for localization. The dif-
ficulty arises from the relatively few control parameters
and the large number of decay pathways in quantum
systems. Therefore, the realization of a quantum BIC
remains a challenge.
Optical systems provide a clean and versatile plat-
form for realizing different types of BICs27,28,49,57,110,132,
208,245, because of nanofabrication technologies that ena-
ble the creation of customized photonic structures. An
optical BIC exhibits an ultrahigh Q — its Qr is technically
Figure 6 | Applications of BICs and quasi-BICs. a–c|Photonic crystal surface-emitting lasers (PCSELs). Schematic
representation of the experimental setup (part a). The lasing mode is a quasi-bound state in the continuum (BIC) because
the 180° rotational symmetry of the photonic crystal (PhC) is broken by the triangular air-hole shapes evident in part b.
The input–output curve of the PCSEL operating under room-temperature continuous-wave condition demonstrates a
low threshold and a high output power (part c). d–f|Supersonic surface acoustic wave (SAW) filters. Schematic
illustration of the setup: two interdigital transducers are placed on a piezoelectric substrate along the direction of the
acoustic BIC (part d). A comparison between the phase velocities of supersonic and subsonic SAWs on Y–X cut LiTaO3
(part e). A BIC appears at ϕ = 36°, which can be used as a supersonic SAW filter with its characteristic transmission
spectrum shown in part f. g–i|Guiding photons without bandgaps. Scanning electron microscope images of a
hollow-core Kagome-lattice PhC fibre are in shown in part g. Photonic guiding in such fibres uses quasi-BICs relying on
the ‘inhibited coupling’ between the core and cladding modes, which can be understood from the dissimilar behaviours
of the core and unit-cell modes along the azimuthal direction (part h). Part i is an image and spectrum showing the
generation and guidance of a three-octave spectral comb using the quasi-BICs in such fibres. a.u., arbitrary units;
MQWs, multiple quantum wells. Parts a–c are from REF.219, Nature Publishing Group. Part e is adapted with permission
from REF.227, Academic Press (Elsevier). Part f is adapted with permission from REF.176, © 2002 IEEE. Parts g–i are
adapted with permission from REF.229, AAAS.
REVIEWS
NATURE REVIEWS
|
MATERIALS ADVANCE ONLINE PUBLICATION
|
9
infinity — which can increase the interaction time
between light and matter by orders of magnitude. In
addition to the high-Q applications described above,
there are many more opportunities in, for example, non-
linearity enhancement and quantum optical applications
that have not been explored. The long-range interactions
in Fabry–Pérot BICs may be useful for nanophotonic cir-
cuits104 and for quantum information processing105,106. It
has also been proposed that the light intensity may act as
a tuning parameter in nonlinear materials, which may
enable robust BICs248, tunable channel dropping249, light
storage and release250,251, and frequency comb genera-
tion252. Finally, it was shown that particle statistics can be
used to modify properties of BICs253.
Considering the many types of BICs, a natural ques-
tion is whether a common concept underlies them
all other than the vanishing of coupling to radiation
through interference. To this end, the topological inter-
pretation of BICs (see BOX2 and REFS141,142) seems
promising. The topological arguments may guide the
discovery of BICs and new ways to trap waves, which
may also exist in quasi-particle systems such as mag-
nons, polaritons, polarons and anyons. Because BICs
defy conventional wisdom and provide new ways to
confine waves, their realization in different material
systems are certain to provide even more surprises and
advances in both fundamental physics and technological
applications.
1. Moiseyev,N. Non-Hermitian Quantum Mechanics
Ch.4 (Cambridge Univ. Press, 2011).
2. Kukulin,V.I., Krasnopol’sky,V.M. & Horáccˇ ek,J.
Theory of Resonances: Principles and Applications
Ch.2 (Springer, 1989).
3. von Neumann,J. & Wigner,E. Über merkwürdige
diskrete Eigenwerte. Phys. Z. 30, 465–467
(in German) (1929).
This paper proposes the possibility of BICs using
an engineered quantum potential as an example.
4. Parker,R. Resonance effects in wake shedding from
parallel plates: some experimental observations.
J.Sound Vib. 4, 62–72 (1966).
This paper reports the observation of
symmetry-protected BICs in acoustic waveguides.
5. Parker,R. Resonance effects in wake shedding from
parallel plates: calculation of resonant frequencies.
J.Sound Vib. 5, 330–343 (1967).
6. Cumpsty,N.A. & Whitehead,D.S. The excitation of
acoustic resonances by vortex shedding. J.Sound Vib.
18, 353–369 (1971).
7. Koch,W. Resonant acoustic frequencies of flat plate
cascades. J.Sound Vib. 88, 233–242 (1983).
8. Parker,R. & Stoneman,S.A.T. The excitation and
consequences of acoustic resonances in enclosed fluid
flow around solid bodies. Proc. Inst. Mech. Eng. C
203, 9–19 (1989).
9. Evans,D.V., Levitin,M. & Vassiliev,D. Existence
theorems for trapped modes. J.Fluid Mech. 261,
21–31 (1994).
10. Evans,D.V., Linton,C.M. & Ursell,F. Trapped mode
frequencies embedded in the continuous spectrum.
Q.J.Mech. Appl. Math. 46, 253–274 (1993).
11. Groves,M.D. Examples of embedded eigenvalues for
problems in acoustic waveguides. Math. Meth. Appl.
Sci. 21, 479–488 (1998).
12. Linton,C. & McIver,M. Trapped modes in cylindrical
waveguides. Q.J.Mech. Appl. Math. 51, 389–412
(1998).
13. Davies,E. & Parnovski,L. Trapped modes in acoustic
waveguides. Q.J.Mech. Appl. Math. 51, 477–492
(1998).
14. Ursell,F. Trapping modes in the theory of surface
waves. Math. Proc. Cambridge Philos. Soc. 47,
347–358 (1951).
15. Jones,D.S. The eigenvalues of ∇2u + λu = 0 when the
boundary conditions are given on semi-infinite
domains. Math. Proc. Cambridge Philos. Soc. 49,
668–684 (1953).
16. Callan,M., Linton,C.M. & Evans,D.V. Trapped
modes in two-dimensional waveguides. J.Fluid Mech.
229, 51–64 (1991).
17. Retzler,C.H. Trapped modes: an experimental
investigation. Appl. Ocean Res. 23, 249–250 (2001).
18. Cobelli,P.J., Pagneux,V., Maurel,A. & Petitjeans,P.
Experimental observation of trapped modes in a water
wave channel. Euro. Phys. Lett. 88, 20006 (2009).
19. Cobelli,P.J., Pagneux,V., Maurel,A. & Petitjeans,P.
Experimental study on water-wave trapped modes.
J.Fluid Mech. 666, 445–476 (2011).
20. Pagneux,V. in Dynamic Localization Phenomena in
Elasticity, Acoustics and Electromagnetism (eds
Craster,R. & Kaplunov,J.) 181–223 (Springer, 2013).
21. Schult,R.L., Ravenhall,D.G. & Wyld,H.W. Quantum
bound states in a classically unbound system of
crossed wires. Phys. Rev. B 39, 5476–5479 (1989).
22. Exner,P., Šeba,P., Tater,M. & Vaneˇk,D. Bound states
and scattering in quantum waveguides coupled
laterally through a boundary window. J.Math. Phys.
37, 4867–4887 (1996).
23. Moiseyev,N. Suppression of Feshbach resonance
widths in two-dimensional waveguides and quantum
dots: a lower bound for the number of bound states in
the continuum. Phys. Rev. Lett. 102, 167404 (2009).
24. Cederbaum,L.S., Friedman,R.S., Ryaboy,V.M. &
Moiseyev,N. Conical intersections and bound
molecular states embedded in the continuum. Phys.
Rev. Lett. 90, 013001 (2003).
25. Longhi,S. Transfer of light waves in optical waveguides
via a continuum. Phys. Rev. A 78, 013815 (2008).
26. Longhi,S. Optical analog of population trapping in the
continuum: classical and quantum interference effects.
Phys. Rev. A 79, 023811 (2009).
27. Dreisow,F. etal. Adiabatic transfer of light via a
continuum in optical waveguides. Opt. Lett. 34,
2405–2407 (2009).
This paper realizes light transfer based on
symmetry-protected and Fabry–Pérot BICs in a
coupled-waveguide array.
28. Plotnik,Y. etal. Experimental observation of optical
bound states in the continuum. Phys. Rev. Lett. 107,
183901 (2011).
This paper realizes an optical symmetry-protected
BIC in a coupled-waveguide array.
29. Shipman,S.P., Ribbeck,J., Smith,K.H. &
Weeks,C.A. Discrete model for resonance near
embedded bound states. IEEE Photonics J. 2,
911–923 (2010).
30. Ptitsyna,N. & Shipman,S.P. A lattice model for
resonance in open periodic waveguides. Discrete
Contin. Dyn. Syst. S 5, 989–1020 (2012).
31. Ladrón de Guevara,M.L., Claro,F. & Orellana,P.A.
Ghost Fano resonance in a double quantum dot
molecule attached to leads. Phys. Rev. B 67, 195335
(2003).
32. Orellana,P.A., Ladrón de Guevara,M.L. & Claro,F.
Controlling Fano and Dicke effects via a magnetic flux
in a two-site Anderson model. Phys. Rev. B 70,
233315 (2004).
33. Ladrón de Guevara,M.L. & Orellana,P.A. Electronic
transport through a parallel-coupled triple quantum
dot molecule: Fano resonances and bound states in
the continuum. Phys. Rev. B 73, 205303 (2006).
34. Voo,K.-K. & Chu,C.S. Localized states in continuum
in low-dimensional systems. Phys. Rev. B 74, 155306
(2006).
35. Solís,B., Ladrón de Guevara,M.L. & Orellan,P.A.
Friedel phase discontinuity and bound states in the
continuum in quantum dot systems. Phys. Lett. A 372,
4736–4739 (2008).
36. Gong,W., Han,Y. & Wei,G. Antiresonance and bound
states in the continuum in electron transport through
parallel-coupled quantum-dot structures. J.Phys.
Condens. Matter 21, 175801 (2009).
37. Han,Y., Gong,W. & Wei,G. Bound states in the
continuum in electronic transport through parallel-
coupled quantum-dot structures. Phys. Status Solidi B
246, 1634–1641 (2009).
38. Vallejo,M.L., Ladrón de Guevara,M.L. &
Orellana,P.A. Triple Rashba dots as a spin filter:
bound states in the continuum and Fano effect. Phys.
Lett. A 374, 4928–4932 (2010).
39. Yan,J.-X. & Fu,H.-H. Bound states in the continuum
and Fano antiresonance in electronic transport
through a four-quantum-dot system. Phys. B 410,
197–200 (2013).
40. Ramos,J.P. & Orellana,P.A. Bound states in the
continuum and spin filter in quantum-dot molecules.
Phys. B 455, 66–70 (2014).
41. Álvarez,C., Domínguez-Adame,F., Orellana,P. A. &
Díaz, E. Impact of electron–vibron interaction on the
bound states in the continuum. Phys. Lett. A 379,
1062–1066 (2015).
42. González,J.W., Pacheco,M., Rosales,L. &
Orellana,P.A. Bound states in the continuum in
graphene quantum dot structures. Euro. Phys. Lett.
91, 66001 (2010).
43. Cortés,N., Chico,L., Pacheco,M., Rosales,L. &
Orellana,P.A. Bound states in the continuum:
localization of Dirac-like fermions. Euro. Phys. Lett.
108, 46008 (2014).
44. Bulgakov,E.N., Pichugin,K.N., Sadreev,A.F. &
Rotter,I. Bound states in the continuum in open
Aharonov–Bohm rings. JETP Lett. 84, 430–435
(2006).
45. Voo,K.-K. Trapped electromagnetic modes in forked
transmission lines. Wave Motion 45, 795–803
(2008).
46. Guessi,L.H. etal. Catching the bound states in the
continuum of a phantom atom in graphene. Phys.
Rev.B 92, 045409 (2015).
47. Guessi,L.H. etal. Quantum phase transition
triggering magnetic bound states in the continuum in
graphene. Phys. Rev. B 92, 245107 (2015).
48. Joannopoulos,J.D., Johnson,S.G., Winn,J.N. &
Meade,R.D. Photonic Crystals: Molding the Flow of
Light Ch. 8,9 (Princeton Univ. Press, 2008).
49. Ulrich,R. in Symposium on optical and acoustical micro-
electronics (ed. Fox,J.) 359–376 (New York,1975).
This work observes a symmetry-protected BIC in a
periodic metal grid.
50. Bonnet-Bendhia,A.-S. & Starling,F. Guided waves by
electromagnetic gratings and non-uniqueness
examples for the diffraction problem. Math. Meth.
Appl. Sci. 17, 305–338 (1994).
51. Paddon,P. & Young,J.F. Two-dimensional vector-
coupled-mode theory for textured planar waveguides.
Phys. Rev. B 61, 2090–2101 (2000).
52. Pacradouni,V. etal. Photonic band structure of
dielectric membranes periodically textured in two
dimensions. Phys. Rev. B 62, 4204–4207 (2000).
53. Ochiai,T. & Sakoda,K. Dispersion relation and optical
transmittance of a hexagonal photonic crystal slab.
Phys. Rev. B 63, 125107 (2001).
54. Fan,S. & Joannopoulos,J.D. Analysis of guided
resonances in photonic crystal slabs. Phys. Rev. B 65,
235112 (2002).
55. Tikhodeev,S.G., Yablonskii,A.L., Muljarov,E.A.,
Gippius,N.A. & Ishihara,T. Quasiguided modes and
optical properties of photonic crystal slabs. Phys.
Rev.B 66, 045102 (2002).
56. Shipman,S.P. & Venakides,S. Resonant transmission
near nonrobust periodic slab modes. Phys. Rev. E 71 ,
026611 (2005).
57. Lee,J. etal. Observation and differentiation of unique
high-Q optical resonances near zero wave vector in
macroscopic photonic crystal slabs. Phys. Rev. Lett.
109, 067401 (2012).
REVIEWS
10
|
ADVANCE ONLINE PUBLICATION www.nature.com/natrevmats
58. Henry,C.H., Kazarinov,R.F., Logan,R.A. & Yen,R.
Observation of destructive interference in the radiation
loss of second-order distributed feedback lasers. IEEE
J.Quantum Electron. 21, 151–154 (1985).
This paper demonstrates lasing through a
symmetry-protected BIC in a distributed feedback
laser with 1D periodicity.
59. Kazarinov,R.F. & Henry,C.H. Second-order
distributed feedback lasers with mode selection
provided by first-order radiation losses. IEEE
J.Quantum Electron. 21, 144–150 (1985).
60. Lim,T.C. & Farnell,G.W. Character of pseudo surface
waves on anisotropic crystals. J.Acoust. Soc. Am. 45,
845–851 (1969).
61. Farnell,G.W. in Physical Acoustics Vol. 6
(edsMason,W.P. & Thurston,R.N.) 109–166
(Academic Press, 1970).
62. Alshits,V.I. & Lothe,J. Comments on the relation
between surface wave theory and the theory of
reflection. Wave Motion 3, 297–310 (1981).
63. Chadwick,P. The behaviour of elastic surface waves
polarized in a plane of material symmetryI. General
analysis. Proc. R.Soc. A 430, 213–240 (1990).
64. Alshits,V.I., Darinskii,A.N. & Shuvalov,A.L. Elastic
waves in infinite and semi-infinite anisotropic media.
Phys. Scr. 1992, 85 (1992).
65. Shipman,S.P. & Welters,A. in Proceedings of the 2012
international conference on mathematical methods in
electromagnetic theory (MMET) 227–232 (Kharkiv,
2012).
66. Robnik,M. A simple separable Hamiltonian having
bound states in the continuum. J.Phys. A 19, 3845
(1986).
This paper proposes BICs in a quantum well based
on separability.
67. Nockel,J.U. Resonances in quantum-dot transport.
Phys. Rev. B 46, 15348 (1992).
68. Duclos,P., Exner,P. & Meller,B. Open quantum dots:
resonances from perturbed symmetry and bound
states in strong magnetic fields. Rep. Math. Phys. 47,
253–267 (2001).
69. Prodanovic´,N., Milanovic´,V., Ikonic´,Z., Indjin,D. &
Harrison,P. Bound states in continuum: quantum dots
in a quantum well. Phys. Lett. A 377, 2177–2181
(2013).
70. Čtyroký,J. Photonic bandgap structures in planar
waveguides. J.Opt. Soc. Am. A 18, 435–441 (2001).
71. Kawakami,S. Analytically solvable model of photonic
crystal structures and novel phenomena. J.Lightwave
Technol. 20, 1644–1650 (2002).
72. Watts,M.R., Johnson,S.G., Haus,H.A. &
Joannopoulos,J.D. Electromagnetic cavity with
arbitrary Q and small modal volume without a complete
photonic bandgap. Opt. Lett. 27, 1785–1787 (2002).
73. Apalkov,V.M. & Raikh,M.E. Strongly localized mode at
the intersection of the phase slips in a photonic crystal
without band gap. Phys. Rev. Lett. 90, 253901 (2003).
74. Rivera,N. etal. Controlling directionality and
dimensionality of radiation by perturbing separable
bound states in the continuum. Preprint at http://
arxiv.org/abs/1507.00923 (2016).
75. Fan,S., Suh,W. & Joannopoulos,J.D. Temporal
coupled-mode theory for the Fano resonance in
optical resonators. J.Opt. Soc. Am. A 20, 569–572
(2003).
76. Haus,H.A. Waves and Fields in Optoelectronics Ch. 7
(Prentice-Hall, 1984).
77. Fan,S. etal. Theoretical analysis of channel drop
tunneling processes. Phys. Rev. B 59, 15882–15892
(1999).
78. Manolatou,C. etal. Coupling of modes analysis of
resonant channel add-drop filters. IEEE J.Quantum
Electron. 35, 1322–1331 (1999).
79. Wang,Z. & Fan,S. Compact all-pass filters in photonic
crystals as the building block for high-capacity optical
delay lines. Phys. Rev. E 68, 066616 (2003).
80. McIver,M. An example of non-uniqueness in the two-
dimensional linear water wave problem. J.Fluid Mech.
315, 257–266 (1996).
This paper proposes a Fabry–Pérot BIC for water
waves by engineering the shapes of two obstacles.
81. Linton,C.M. & Kuznetsov,N.G. Non-uniqueness in
two-dimensional water wave problems: numerical
evidence and geometrical restrictions. Proc. R.Soc. A
453, 2437–2460 (1997).
82. Evans,D.V. & Porter,R. An example of non-
uniqueness in the two-dimensional linear water-wave
problem involving a submerged body. Proc. R.Soc. A
454, 3145–3165 (1998).
83. McIver,M. Trapped modes supported by submerged
obstacles. Proc. R.Soc. A 456, 1851–1860 (2000).
84. Kuznetsov,N., McIver,P. & Linton,C.M. On
uniqueness and trapped modes in the water-wave
problem for vertical barriers. Wave Motion 33,
283–307 (2001).
85. Porter,R. Trapping of water waves by pairs of
submerged cylinders. Proc. R.Soc. A 458, 607–624
(2002).
86. Linton,C.M. & McIver,P. Embedded trapped modes
in water waves and acoustics. Wave Motion 45,
16–29 (2007).
This paper reviews theoretical studies of BICs in
acoustic and water waves.
87. Shahbazyan,T.V. & Raikh,M.E. Two-channel resonant
tunneling. Phys. Rev. B 49, 17123–17129 (1994).
88. Kim,C.S. & Satanin,A.M. Dynamic confinement of
electrons in time-dependent quantum structures.
Phys. Rev. B 58, 15389–15392 (1998).
89. Rotter,I. & Sadreev,A.F. Zeros in single-channel
transmission through double quantum dots. Phys.
Rev. E 71, 046204 (2005).
90. Sadreev,A.F., Bulgakov,E.N. & Rotter,I. Trapping of
an electron in the transmission through two quantum
dots coupled by a wire. JETP Lett. 82, 498–503
(2005).
91. Ordonez,G., Na,K. & Kim,S. Bound states in the
continuum in quantum-dot pairs. Phys. Rev. A 73,
022113 (2006).
92. Tanaka,S., Garmon,S., Ordonez,G. & Petrosky,T.
Electron trapping in a one-dimensional semiconductor
quantum wire with multiple impurities. Phys. Rev. B
76, 153308 (2007).
93. Cattapan,G. & Lotti,P. Bound states in the continuum
in two-dimensional serial structures. Eur. Phys. J.B
66, 517–523 (2008).
94. Díaz-Tendero,S., Borisov,A.G. & Gauyacq,J.-P.
Extraordinary electron propagation length in a
metallic double chain supported on a metal surface.
Phys. Rev. Lett. 102, 166807 (2009).
95. Sadreev,A.F., Maksimov,D.N. & Pilipchuk,A.S. Gate
controlled resonant widths in double-bend
waveguides: bound states in the continuum. J.Phys.
Condens. Matter 27, 295303 (2015).
96. Suh,W., Yanik,M.F., Solgaard,O. & Fan,S.
Displacement-sensitive photonic crystal structures
based on guided resonance in photonic crystal slabs.
Appl. Phys. Lett. 82, 1999–2001 (2003).
97. Suh,W., Solgaard,O. & Fan,S. Displacement sensing
using evanescent tunneling between guided
resonances in photonic crystal slabs. J.Appl. Phys.
98, 033102 (2005).
98. Liu,V., Povinelli,M. & Fan,S. Resonance-enhanced
optical forces between coupled photonic crystal slabs.
Opt. Express 17, 21897–21909 (2009).
99. Marinica,D.C., Borisov,A.G. & Shabanov,S.V.
Bound states in the continuum in photonics. Phys.
Rev. Lett. 100, 183902 (2008).
100. Ndangali,R.F. & Shabanov,S.V. Electromagnetic
bound states in the radiation continuum for periodic
double arrays of subwavelength dielectric cylinders.
J.Math. Phys. 51, 102901 (2010).
101. Bulgakov,E.N. & Sadreev,A.F. Bound states in the
continuum in photonic waveguides inspired by defects.
Phys. Rev. B 78, 075105 (2008).
102. Longhi,S. Optical analogue of coherent population
trapping via a continuum in optical waveguide arrays.
J.Mod. Opt. 56, 729–737 (2009).
103. Hein,S., Koch,W. & Nannen,L. Trapped modes and
Fano resonances in two-dimensional acoustical duct-
cavity systems. J.Fluid Mech. 692, 257–287
(2012).
104. Sato,Y. etal. Strong coupling between distant
photonic nanocavities and its dynamic control. Nat.
Photonics 6, 56–61 (2012).
105. Zheng,H. & Baranger,H.U. Persistent quantum beats
and long-distance entanglement from waveguide-
mediated interactions. Phys. Rev. Lett. 11 0 , 113601
(2013).
106. van Loo,A.F. etal. Photon-mediated interactions
between distant artificial atoms. Science 342,
1494–1496 (2013).
107. Peleg,O., Plotnik,Y., Moiseyev,N., Cohen,O. &
Segev,M. Self-trapped leaky waves and their
interactions. Phys. Rev. A 80, 041801 (2009).
108. Hsu,C.W. etal. Bloch surface eigenstates within the
radiation continuum. Light Sci. Appl. 2, e84 (2013).
109. Longhi,S. Bound states in the continuum in a single-
level Fano–Anderson model. Eur. Phys. J.B 57,
45–51 (2007).
110 . Weimann,S. etal. Compact surface Fano states
embedded in the continuum of waveguide arrays.
Phys. Rev. Lett. 111, 240403 (2013).
This work realizes a Fabry–Pérot BIC in a
coupled-waveguide array.
111. McIver,P. & McIver,M. Trapped modes in an
axisymmetric water-wave problem. Q.J.Mech. Appl.
Math. 50, 165–178 (1997).
112 . Kuznetsov,N. & McIver,P. On uniqueness and trapped
modes in the water-wave problem for a surface-
piercing axisymmetric body. Q.J.Mech. Appl. Math.
50, 565–580 (1997).
113 . Suh,W., Wang,Z. & Fan,S. Temporal coupled-mode
theory and the presence of non-orthogonal modes in
lossless multimode cavities. IEEE J.Quantum Electron.
40, 1511–1518 (2004).
114 . Devdariani,A.Z., Ostrovskii,V.N. & Sebyakin,Y.N.
Crossing of quasistationary levels. Sov. Phys. JETP 44,
477 (1976).
115 . Friedrich,H. & Wintgen,D. Interfering resonances and
bound states in the continuum. Phys. Rev. A 32,
3231–3242 (1985).
This paper proposes that a BIC can arise from two
interfering resonances.
116 . Remacle,F., Munster,M., Pavlov-Verevkin,V.B. &
Desouter-Lecomte,M. Trapping in competitive decay
of degenerate states. Phys. Lett. A 145, 265–268
(1990).
117 . Berkovits,R., von Oppen,F. & Kantelhardt,J.W.
Discrete charging of a quantum dot strongly coupled
to external leads. Euro. Phys. Lett. 68, 699 (2004).
118 . Fonda,L. & Newton,R.G. Theory of resonance
reactions. Ann. Phys. 10, 490–515 (1960).
119 . Friedrich,H. & Wintgen,D. Physical realization of
bound states in the continuum. Phys. Rev. A 31,
3964–3966 (1985).
120. Neukammer,J. etal. Autoionization inhibited by
internal interferences. Phys. Rev. Lett. 55,
1979–1982 (1985).
121. Volya,A. & Zelevinsky,V. Non-Hermitian effective
Hamiltonian and continuum shell model. Phys. Rev. C
67, 054322 (2003).
122. Deb,B. & Agarwal,G.S. Creation and manipulation of
bound states in the continuum with lasers:
applications to cold atoms and molecules. Phys. Rev. A
90, 063417 (2014).
123. Sablikov,V.A. & Sukhanov,A.A. Helical bound states
in the continuum of the edge states in two dimensional
topological insulators. Phys. Lett. A 379, 1775–1779
(2015).
124. Texier,C. Scattering theory on graphs: II. The Friedel
sum rule. J.Phys. A 35, 3389 (2002).
125. Sadreev,A.F., Bulgakov,E.N. & Rotter,I. Bound
states in the continuum in open quantum billiards with
a variable shape. Phys. Rev. B 73, 235342 (2006).
126. Sadreev,A.F. & Babushkina,T.V. Two-electron bound
states in a continuum in quantum dots. JETP Lett. 88,
312–317 (2008).
127. Boretz,Y., Ordonez,G., Tanaka,S. & Petrosky,T.
Optically tunable bound states in the continuum.
Phys. Rev. A 90, 023853 (2014).
128. Lyapina,A.A., Maksimov,D.N., Pilipchuk,A.S. &
Sadreev,A.F. Bound states in the continuum in open
acoustic resonators. J.Fluid Mech. 780, 370–387
(2015).
129. Lepetit,T., Akmansoy,E., Ganne,J.-P. &
Lourtioz,J.-M. Resonance continuum coupling in high-
permittivity dielectric metamaterials. Phys. Rev. B 82,
195307 (2010).
130. Lepetit,T. & Kanté,B. Controlling multipolar radiation
with symmetries for electromagnetic bound states in
the continuum. Phys. Rev. B 90, 241103 (2014).
131. Gentry,C.M. & Popovic´,M.A. Dark state lasers. Opt.
Lett. 39, 4136–4139 (2014).
132. Hsu,C.W. etal. Observation of trapped light within the
radiation continuum. Nature 499, 188–191 (2013).
This work realizes a single-resonance parametric
BIC in a PhC slab.
133. Gansch,R. etal. Measurement of bound states in the
continuum by a detector embedded in a photonic
crystal. Light Sci. Appl. http://dx.doi.org/10.1038/
lsa.2016.147 (2016).
134. Evans,D.V. & Porter,R. On the existence of
embedded surface waves along arrays of parallel
plates. Q.J.Mech. Appl. Math. 55, 481–494 (2002).
135. Porter,R. & Evans,D.V. Embedded Rayleigh–Bloch
surface waves along periodic rectangular arrays. Wave
Motion 43, 29–50 (2005).
136. Bulgakov,E.N. & Sadreev,A.F. Bloch bound states in
the radiation continuum in a periodic array of
dielectric rods. Phys. Rev. A 90, 053801 (2014).
137. Bulgakov,E.N. & Sadreev,A.F. Light trapping above
the light cone in a one-dimensional array of dielectric
spheres. Phys. Rev. A 92, 023816 (2015).
REVIEWS
NATURE REVIEWS
|
MATERIALS ADVANCE ONLINE PUBLICATION
|
11
138. Longhi,S. & Della Valle,G. Floquet bound states in the
continuum. Sci. Rep. 3, 2219 (2013).
139. Yang,Y., Peng,C., Liang,Y., Li,Z. & Noda,S. Analytical
perspective for bound states in the continuum in
photonic crystal slabs. Phys. Rev. Lett. 113, 037401
(2014).
140. Gao,X. etal. Formation mechanism of guided
resonances and bound states in the continuum in
photonic crystal slabs. Preprint at http://arxiv.org/
abs/1603.02815 (2016).
141. Zhen,B., Hsu,C.W., Lu,L., Stone,A.D. &
Soljacˇic´,M. Topological nature of optical bound
states in the continuum. Phys. Rev. Lett. 113 , 257401
(2014).
This paper explains the topological nature of BICs
in PhC slabs.
142. Yang,B.-J., Bahramy,M.S. & Nagaosa,N. Topological
protection of bound states against the hybridization.
Nat. Commun. 4, 1524 (2013).
143. Linton,C.M., McIver,M., McIver,P., Ratcliffe,K. &
Zhang,J. Trapped modes for off-centre structures in
guides. Wave Motion 36, 67–85 (2002).
144. Evans,D. & Porter,R. Trapped modes embedded in
the continuous spectrum. Q.J.Mech. Appl. Math. 51,
263–274 (1998).
145. McIver,M., Linton,C.M., McIver,P., Zhang,J. &
Porter,R. Embedded trapped modes for obstacles in
two-dimensional waveguides. Q.J.Mech. Appl. Math.
54, 273–293 (2001).
146. McIver,M., Linton,C.M. & Zhang,J. The branch
structure of embedded trapped modes in two-
dimensional waveguides. Q.J.Mech. Appl. Math. 55,
313–326 (2002).
147. Koch,W. Acoustic resonances in rectangular open
cavities. AIAA J. 43, 2342–2349 (2005).
148. Duan,Y., Koch,W., Linton,C.M. & McIver,M.
Complex resonances and trapped modes in ducted
domains. J.Fluid Mech. 571, 119–147 (2007).
149. Kim,C.S., Satanin,A.M., Joe,Y.S. & Cosby,R.M.
Resonant tunneling in a quantum waveguide: effect of
a finite-size attractive impurity. Phys. Rev. B 60,
10962–10970 (1999).
150. Linton,C.M. & Ratcliffe,K. Bound states in coupled
guides. I. Two dimensions. J.Math. Phys. 45,
1359–1379 (2004).
151. Cattapan,G. & Lotti,P. Fano resonances in stubbed
quantum waveguides with impurities. Eur. Phys. J.B
60, 51–60 (2007).
152. Olendski,O. & Mikhailovska,L. Bound-state evolution
in curved waveguides and quantum wires. Phys. Rev. B
66, 035331 (2002).
153. Olendski,O. & Mikhailovska,L. Fano resonances of a
curved waveguide with an embedded quantum dot.
Phys. Rev. B 67, 035310 (2003).
154. Chen,Y. etal. Mechanical bound state in the
continuum for optomechanical microresonators. New
J.Phys. 18, 063031 (2016).
155. Zou,C.-L. etal. Guiding light through optical bound
states in the continuum for ultrahigh-Q microresonators.
Laser Photon. Rev. 9, 114–119 (2015).
156. Penunuri,D. & Lakin,K.M. in 1975 IEEE Ultrason.
Symp. 478–483 (IEEE, Los Angeles, 1975).
157. Stegeman,G.I. Normal-mode surface waves in the
pseudobranch on the (001) plane of gallium
arsenide. J.Appl. Phys. 47, 1712–1713 (1976).
158. Aleksandrov,V.V. etal. New data concerning surface
Mandelstamm–Brillouin light scattering from the
basal plane of germanium crystal. Phys. Lett. A 162,
418–422 (1992).
159. Aleksandrov,V.V., Velichkina,T.S., Potapova,J.B. &
Yakovlev,I.A. Mandelstamm–Brillouin studies of
peculiarities of the phonon frequency distribution at
cubic crystal (001) surfaces. Phys. Lett. A 171,
103–106 (1992).
160. Taylor,D.B. Surface waves in anisotropic media:
thesecular equation and its numerical solution. Proc.
R.Soc. A 376, 265–300 (1981).
161. Gundersen,S.A., Wang,L. & Lothe,J. Secluded
supersonic elastic surface waves. Wave Motion 14,
129–143 (1991).
162. Barnett,D.M., Chadwick,P. & Lothe,J. The behaviour
of elastic surface waves polarized in a plane of
material symmetry. I. Addendum. Proc. R.Soc. A 433,
699–710 (1991).
163. Maznev,A.A. & Every,A.G. Secluded supersonic
surface waves in germanium. Phys. Lett. A 197,
423–427 (1995).
164. Darinskii,A.N., Alshits,V.I., Lothe,J., Lyubimov,V.N.
& Shuvalov,A.L. An existence criterion for the branch
of two-component surface waves in anisotropic elastic
media. Wave Motion 28, 241–257 (1998).
165. Xu,Y. & Aizawa,T. Pseudo surface wave on the (1012)
plane of sapphire. J.Appl. Phys. 86, 6507–6511
(1999).
166. Maznev,A.A., Lomonosov,A.M., Hess,P. &
Kolomenskii,A. Anisotropic effects in surface acoustic
wave propagation from a point source in a crystal. Eur.
Phys. J.B 35, 429–439 (2003).
167. Trzupek,D. & Zielin´ski,P. Isolated true surface wave in
a radiative band on a surface of a stressed auxetic.
Phys. Rev. Lett. 103, 075504 (2009).
168. Every,A.G. Supersonic surface acoustic waves on the
001 and 110 surfaces of cubic crystals. J.Acoust. Soc.
Am. 138, 2937–2944 (2015).
169. Every,A.G. Guided elastic waves at a periodic array of
thin coplanar cavities in a solid. Phys. Rev. B 78,
174104 (2008).
170. Maznev,A.A. & Every,A.G. Surface acoustic waves in
a periodically patterned layered structure. J.Appl.
Phys. 106, 113531 (2009).
171. Every,A.G. & Maznev,A.A. Elastic waves at
periodically-structured surfaces and interfaces of
solids. AIP Adv. 4, 124401 (2014).
172. Yamanouchi,K. & Shibayama,K. Propagation and
amplification of rayleigh waves and piezoelectric leaky
surface waves in LiNbO3. J.Appl. Phys. 43, 856–862
(1972).
This work predicts and measures an acoustic
single-resonance parametric BIC on the surface of
a piezoelectric solid.
173. Lewis,M.F. Acoustic wave devices employing surface
skimming bulk waves. US Patent 4159435 (1979).
174. Ueda,M. etal. Surface acoustic wave device using a
leaky surface acoustic wave with an optimized cut angle
of a piezoelectric substrate. US Patent 6037847
(2000).
175. Kawachi,O. etal. Optimal cut for leaky SAW on LiTaO3
for high performance resonators and filters. IEEE
Trans. Ultrason. Ferroelectr. Freq. Control 48,
1442–1448 (2001).
176. Naumenko,N. & Abbot,B. in Proc. 2002 IEEE
Ultrason. Symp. 1, 385–390 (IEEE, 2002).
177. Simon,B. On positive eigenvalues of one-body
Schrödinger operators. Commun. Pure Appl. Math.
22, 531–538 (1969).
178. Stillinger,F.H. & Herrick,D.R. Bound states in the
continuum. Phys. Rev. A 11, 446–454 (1975).
179. Jain,A. & Shastry,C. Bound states in the continuum
for separable nonlocal potentials. Phys. Rev. A 12,
2237 (1975).
180. Molina,M.I., Miroshnichenko,A.E. & Kivshar,Y.S.
Surface bound states in the continuum. Phys. Rev.
Lett. 108, 070401 (2012).
181. Gallo,N. & Molina,M. Bulk and surface bound states
in the continuum. J.Phys. A: Math. Theor. 48,
045302 (2015).
182. Simon,B. Some Schrödinger operators with dense
point spectrum. Proc. Amer. Math. Soc. 125,
203–208 (1997).
183. Moses,H.E. & Tuan,S. Potentials with zero scattering
phase. Il Nuovo Cimento 13, 197–206 (1959).
184. Gazdy,B. On the bound states in the continuum. Phys.
Lett. A 61, 89–90 (1977).
185. Meyer-Vernet,N. Strange bound states in the
Schrödinger wave equation: when usual tunneling
does not occur. Am. J.Phys. 50, 354–356 (1982).
186. Pivovarchik,V.N., Suzko,A.A. & Zakhariev,B.N.
New exactly solved models with bound states above
the scattering threshold. Phys. Scr. 34, 101 (1986).
187. Naboko,S.N. Dense point spectra of Schrödinger and
Dirac operators. Theor. Math. Phys. 68, 646–653
(1986).
188. Darboux,M.G. Sur une proposition relative aux
équations liéaires. C.R.Acad. Sci. 94, 1456–1459
(inFrench) (1882).
189. Svirsky,R. An application of double commutation to
the addition of bound states to the spectrum of a
Schrodinger operator. Inverse Probl. 8, 483 (1992).
190. Pappademos,J., Sukhatme,U. & Pagnamenta,A.
Bound states in the continuum from supersymmetric
quantum mechanics. Phys. Rev. A 48, 3525 (1993).
191. Stahlhofen,A.A. Completely transparent potentials
for the Schrödinger equation. Phys. Rev. A 51,
934–943 (1995).
192. Weber,T.A. & Pursey,D.L. Continuum bound states.
Phys. Rev. A 50, 4478–4487 (1994).
193. Kocˇinac,S.L.S. & Milanovic´ ,V. Bound states in
continuum generated by point interaction and
supersymmetric quantum mechanics. Modern Phys.
Lett. B 26, 1250177 (2012).
194. Ranjani,S.S., Kapoor,A. & Panigrahi,P. Normalizable
states through deformation of Lamé and the
associated Lamé potentials. J.Phys. A Math. Theor.
41, 285302 (2008).
195. Prodanovic´,N., Milanovic´,V. & Radovanovic´,J.
Photonic crystals with bound states in continuum and
their realization by an advanced digital grading
method. J.Phys. A 42, 415304 (2009).
196. Petrovic´,J.S., Milanovic´,V. & Ikonic´,Z. Bound states
in continuum of complex potentials generated by
supersymmetric quantum mechanics. Phys. Lett. A
300, 595–602 (2002).
197. Andrianov,A.A. & Sokolov,A. V. Resolutions of
identity for some non-Hermitian Hamiltonians. I.
Exceptional point in continuous spectrum. SIGMA
http://dx.doi.org/10.3842/SIGMA.2011.111 (2011).
198. Sokolov,A. V. Resolutions of identity for some non-
Hermitian Hamiltonians. II. Proofs. SIGMA http://dx.
doi.org/10.3842/SIGMA.2011.112 (2011).
199. Longhi,S. & Della Valle,G. Optical lattices with
exceptional points in the continuum. Phys. Rev. A 89,
052132 (2014).
200. Longhi,S. Bound states in the continuum in
PT-symmetric optical lattices. Opt. Lett. 39,
1697–1700 (2014).
This paper classifies and compares two types of
BICs in parity-time symmetric systems.
201. Fernández-García,N., Hernández,E., Jáuregui,A. &
Mondragón,A. Bound states at exceptional points in
the continuum. J.Phys. Conf. Ser. 512, 012023 (2014).
202. Garmon,S., Gianfreda,M. & Hatano,N. Bound states,
scattering states, and resonant states in PT-symmetric
open quantum systems. Phys. Rev. A 92, 022125
(2015).
203. Demic´ ,A., Milanovic´,V. & Radovanovic´,J. Bound
states in the continuum generated by supersymmetric
quantum mechanics and phase rigidity of the
corresponding wavefunctions. Phys. Lett. A 379,
2707–2714 (2015).
204. Correa,F., Jakubský,V. & Plyushchay,M.S.
PT-symmetric invisible defects and confluent
Darboux–Crum transformations. Phys. Rev. A 92,
023839 (2015).
205. Pursey,D. & Weber,T. Scattering from a shifted von
Neumann–Wigner potential. Phys. Rev. A 52, 3932
(1995).
206. Weber,T.A. & Pursey,D.L. Scattering from a
truncated von Neumann–Wigner potential. Phys. Rev.
A 57, 3534–3545 (1998).
207. Longhi,S. Non-Hermitian tight-binding network
engineering. Phys. Rev. A 93, 022102 (2016).
208. Corrielli,G., Della Valle,G., Crespi,A., Osellame,R. &
Longhi,S. Observation of surface states with algebraic
localization. Phys. Rev. Lett. 111, 220403 (2013).
This work demonstrates a BIC in a tight-binding
lattice with engineered hopping rates.
209. McIver,M. & Porter,R. Trapping of waves by a
submerged elliptical torus. J.Fluid Mech. 456,
277–293 (2002).
210. McIver,P. & Newman,J.N. Trapping structures in the
three-dimensional water-wave problem. J.Fluid Mech.
484, 283–301 (2003).
211 . McIver,P. & McIver,M. Trapped modes in the water-
wave problem for a freely floating structure. J.Fluid
Mech. 558, 53–67 (2006).
212. McIver,P. & McIver,M. Motion trapping structures in
the three-dimensional water-wave problem. J.Eng.
Math. 58, 67–75 (2007).
213. Porter,R. & Evans,D.V. Water-wave trapping by
floating circular cylinders. J.Fluid Mech. 633,
311–325 (2009).
214. Meier,M. etal. Laser action from two-dimensional
distributed feedback in photonic crystals. Appl. Phys.
Lett. 74, 7–9 (1999).
215. Imada,M. etal. Coherent two-dimensional lasing
action in surface-emitting laser with triangular-lattice
photonic crystal structure. Appl. Phys. Lett. 75,
316–318 (1999).
216. Noda,S., Yokoyama,M., Imada,M., Chutinan,A. &
Mochizuki,M. Polarization mode control of two-
dimensional photonic crystal laser by unit cell
structure design. Science 293, 1123–1125 (2001).
217. Miyai,E. etal. Photonics: lasers producing tailored
beams. Nature 441 , 946–946 (2006).
218. Matsubara,H. etal. GaN photonic-crystal surface-
emitting laser at blue-violet wavelengths. Science 319,
445–447 (2008).
219. Hirose,K. etal. Watt-class high-power, high-beam-
quality photonic-crystal lasers. Nat. Photonics 8,
406–411 (2014).
This paper shows continuous-wave lasing through a
quasi-BIC with a high output-power and a low
threshold at room temperature.
REVIEWS
12
|
ADVANCE ONLINE PUBLICATION www.nature.com/natrevmats
220. Yanik,A.A. etal. Seeing protein monolayers with
naked eye through plasmonic Fano resonances. Proc.
Natl Acad. Sci. USA 108, 11784–11789 (2011).
221. Zhen,B. etal. Enabling enhanced emission and low-
threshold lasing of organic molecules using special
Fano resonances of macroscopic photonic crystals.
Proc. Natl Acad. Sci. USA 110, 13711–13716 (2013).
222. Foley,J.M., Young,S.M. & Phillips,J.D. Symmetry-
protected mode coupling near normal incidence for
narrow-band transmission filtering in a dielectric
grating. Phys. Rev. B 89, 165111 (2014).
223. Iwahashi,S. etal. Higher-order vector beams
produced by photonic-crystal lasers. Opt. Express 19,
11963–11968 (2011).
224. Kitamura,K., Sakai,K., Takayama,N., Nishimoto,M.
& Noda,S. Focusing properties of vector vortex
beams emitted by photonic-crystal lasers. Opt. Lett.
37, 2421–2423 (2012).
225. Zhan,Q. Cylindrical vector beams: from mathematical
concepts to applications. Adv. Opt. Photonics 1, 1–57
(2009).
226. Fan,X. & White,I.M. Optofluidic microsystems for
chemical and biological analysis. Nat. Photonics 5,
591–597 (2011).
227. Morgan,D. Surface Acoustic Wave Filters: With
Applications to Electronic Communications and Signal
Processing Ch. 11 (Academic Press, 2007).
228. Benabid,F., Knight,J.C., Antonopoulos,G. &
Russell,P.S.J. Stimulated raman scattering in
hydrogen-filled hollow-core photonic crystal fiber.
Science 298, 399–402 (2002).
229. Couny,F., Benabid,F., Roberts,P.J., Light,P.S. &
Raymer,M.G. Generation and photonic guidance of
multi-octave optical-frequency combs. Science 318,
1118–1121 (2007).
230. Benabid,F., Couny,F., Knight,J., Birks,T. &
Russell,P.S.J. Compact, stable and efficient all-fibre
gas cells using hollow-core photonic crystal fibres.
Nature 434, 488–491 (2005).
231. Dudley,J.M. & Taylor,J.R. Ten years of nonlinear
optics in photonic crystal fibre. Nat. Photonics 3,
85–90 (2009).
232. Benoît,A. etal. Over-five octaves wide Raman combs
in high-power picosecond-laser pumped H2-filled
inhibited coupling Kagome fiber. Opt. Express 23,
14002–14009 (2015).
233. Debord,B. etal. Hypocycloid-shaped hollow-core
photonic crystal fiber part I: arc curvature effect on
confinement loss. Opt. Express 21, 28597–28608
(2013).
234. Mur-Petit,J. & Molina,R.A. Chiral bound states in
the continuum. Phys. Rev. B 90, 035434
(2014).
235. Zhang,J.M., Braak,D. & Kollar,M. Bound states in
the continuum realized in the one-dimensional two-
particle Hubbard model with an impurity. Phys. Rev.
Lett. 109, 116405 (2012).
236. Zhang,J.M., Braak,D. & Kollar,M. Bound states in
the one-dimensional two-particle Hubbard model with
an impurity. Phys. Rev. A 87, 023613 (2013).
237. Longhi,S. & Della Valle,G. Tamm–Hubbard surface
states in the continuum. J.Phys. Condens. Matter 25,
235601 (2013).
238. Della Valle,G. & Longhi,S. Floquet–Hubbard bound
states in the continuum. Phys. Rev. B 89, 115118
(2014).
239. Vicencio,R.A. etal. Observation of localized states in
Lieb photonic lattices. Phys. Rev. Lett. 11 4 , 245503
(2015).
240. Mukherjee,S. etal. Observation of a localized flat-
band state in a photonic Lieb lattice. Phys. Rev. Lett.
114 , 245504 (2015).
241. Koirala,M. etal. Critical states embedded in the
continuum. New J.Phys. 17, 013003 (2015).
242. Joglekar,Y.N., Scott,D.D. & Saxena,A. PT-symmetry
breaking with divergent potentials: lattice and
continuum cases. Phys. Rev. A 90, 032108 (2014).
243. Molina,M.I. & Kivshar,Y.S. Embedded states in the
continuum for PT-symmetric systems. Studies Appl.
Math. 133, 337–350 (2014).
244. Longhi,S. Invisible surface defects in a tight-binding
lattice. Eur. Phys. J.B 87, 189 (2014).
245. Regensburger,A. etal. Observation of defect states in
PT-Symmetric optical lattices. Phys. Rev. Lett. 11 0 ,
223902 (2013).
This paper demonstrates a BIC in a pair of
parity-time symmetric coupled fibre loops.
246. Capasso,F. etal. Observation of an electronic bound
state above a potential well. Nature 358, 565–567
(1992).
247. Albo,A., Fekete,D. & Bahir,G. Electronic bound
states in the continuum above (Ga, In)(As, N)/(Al, Ga)
As quantum wells. Phys. Rev. B 85, 115307 (2012).
248. Bulgakov,E.N. & Sadreev,A.F. Robust bound state in
the continuum in a nonlinear microcavity embedded in
a photonic crystal waveguide. Opt. Lett. 39,
5212–5215 (2014).
249. Bulgakov,E., Pichugin,K. & Sadreev,A. Channel
dropping via bound states in the continuum in a
system of two nonlinear cavities between two linear
waveguides. J.Phys. Condens. Matter 25, 395304
(2013).
250. Bulgakov,E.N., Pichugin,K.N. & Sadreev,A.F. All-
optical light storage in bound states in the continuum
and release by demand. Opt. Express 23,
22520–22531 (2015).
251. Lannebere,S. & Silveirinha,M.G. Optical meta-atom
for localization of light with quantized energy. Nat.
Commun. 6, 8766 (2015).
252. Pichugin,K.N. & Sadreev,A.F. Frequency comb
generation by symmetry-protected bound state in the
continuum. J.Opt. Soc. Am. B 32, 1630–1636
(2015).
253. Crespi,A. etal. Particle statistics affects quantum
decay and Fano interference. Phys. Rev. Lett. 114 ,
090201 (2015).
254. Silveirinha,M.G. Trapping light in open plasmonic
nanostructures. Phys. Rev. A 89, 023813 (2014).
This paper proposes 3D confinement of light using
an ε = 0 material and provides a non-existence
theorem when ε ≠ 0 (see BOX1).
255. Monticone,F. & Alù,A. Embedded photonic
eigenvalues in 3D nanostructures. Phys. Rev. Lett.
112 , 213903 (2014).
256. Hrebikova,I., Jelinek,L. & Silveirinha,M.G.
Embedded energy state in an open semiconductor
heterostructure. Phys. Rev. B 92, 155303 (2015).
257. Mermin,N.D. The topological theory of defects in
ordered media. Rev. Mod. Phys. 51, 591
(1979).
258. Hasan,M.Z. & Kane,C.L. Colloquium: topological
insulators. Rev. Mod. Phys. 82, 3045
(2010).
Acknowledgements
The authors thank A. Maznev, N. Rivera, F. Wang, H. Zhou,
M.Segev, N. Moiseyev, S. Longhi, P. McIver, M. McIver,
F.Benabid, and S. G. Johnson for discussions. This work was
partially supported by the National Science Foundation through
grant no. DMR-1307632 and by the Army Research Office
through the Institute for Soldier Nanotechnologies under con-
tract no. W911NF-13-D-0001. B.Z., J.D.J. and M.S. were par-
tially supported by S3TEC (analysis and reading of the
manuscript), an Energy Frontier Research Center funded by the
US Department of Energy under grant no. DE-SC0001299. B.Z.
was partially supported by the United States–Israel Binational
Science Foundation (BSF) under award no. 2013508.
Competing interests statement
The authors declare no competing interests.
REVIEWS
NATURE REVIEWS
|
MATERIALS ADVANCE ONLINE PUBLICATION
|
13
