What contributions have the authors mentioned in the paper "Asymmetric metasurfaces with high-q resonances governed by bound states in the continuum" ?

The authors reveal that metasurfaces created by seemingly different lattices of ( dielectric or metallic ) meta-atoms with broken in-plane symmetry can support sharp high-Q resonances arising from a distortion of symmetry-protected bound states in the continuum. The authors develop a rigorous theory of such asymmetric periodic structures and demonstrate a link between the bound states in the continuum and Fano resonances. Their results suggest the way for smart engineering of resonances in metasurfaces for many applications in nanophotonics and metaoptics.

(Open Access) Asymmetric Metasurfaces with High-Q Resonances Governed by Bound States in the Continuum. (2018) | Kirill Koshelev

Asymmetric Metasurfaces with High-Q Resonances Governed by Bound States

in the Continuum

Kirill Koshelev,

1,2

Sergey Lepeshov,

Mingkai Liu,

Andrey Bogdanov,

and Yuri Kivshar

1,2

Nonlinear Physics Centre, Australian National University, Canberra ACT 2601, Australia

ITMO University, St. Petersburg 197101, Russia

(Received 28 June 2018; publis hed 9 November 2018)

We reveal that metasurfaces created by seemingly different lattices of (dielectric or metallic) meta-atoms

with broken in-plane symmetry can support sharp high-Q resonances arising from a distortion of

symmetry-protected bound states in the continuum. We develop a rigorous theory of such asymmetric

periodic structures and demonstrate a link between the bound states in the continuum and Fano resonances.

Our results suggest the way for smart engineering of resonances in metasurfaces for many applications in

nanophotonics and metaoptics.

DOI: 10.1103/PhysRevLett.121.193903

Metasurfaces have attracted a lot of attention in recent

years due to novel ways they provide for wave front

control, advanced light focusing, and ultrathin optical

elements [1]. Recently, metasurfaces based on high-index

resonant dielectric nanoparticles [2] have emerged as

essential building blocks for various functional metaoptics

devices [3] due to their low intrinsic losses and unique

capabilities for controlling the propagation and localization

of light. A key concept underlying the specific function-

alities of many metasurfaces is the use of constituent

elements with spatially varying optical properties and

optical response characterized by high quality factors (Q

factors) of the resonances.

Many interesting phenomena have been demonstrated

for metasurfaces composed of arrays of meta-atoms with

broken in-plane inversion symmetry (see Fig. 1), which all

demonstrate the excitation of high-Q resonances for the

normal incidence of light. Examples include imaging-based

molecular bar coding with pixelated dielectric metasurfaces

[4] and polarization-induced chirality in metamaterials [5],

which both involve asymmetric pairs of tilted bars [see

Fig. 1(a)], trapped modes in arrays of dielectric nanodisks

with asymmetric holes [6] [see Fig. 1(b)], sharp trapped-

mode resonances in plasmonic and dielectric split-ring

structures [7,8] [see Fig. 1(c)], broken-symmetry Fano

metasurfaces for enhanced nonlinear effects [9,10] [see

Fig. 1(d)], tunable high-Q Fano resonances in plasmonic

and dielectric metasurfaces [11–13] [see Fig. 1(e)], trapped

light and metamaterial-induced transparency in arrays of

square split-ring resonators [14,15] shown in Fig. 1(f).

Here, we reveal that all such seemingly different structures

can be unified by a general concept of bound states in the

continuum, and we prove rigorously their link to the Fano

resonances.

Bound states in the continuum (BIC) represent a general

wave phenomenon observed in acoustics, hydrodynamics,

and optics [16–19]. Originally, this concept appeared in

quantum mechanics [20], but later it was explained in terms

of destructive interference when the coupling constants

with all radiating waves vanish accidentally via continuous

(d)

(c)

(a) (b)

(e)

(f)

FIG. 1. Top: Schematic for the scattering of light by a metasur-

face. Bottom: Designs of the unit cells of asymmetric meta-

surfaces with a broken in-plane inversion symmetry of

constituting meta-atoms supporting sharp resonances, as consid-

ered in Refs. [4–15].

PHYSICAL REVIEW LETTERS 121, 193903 (2018)

tuning of parameters [21]. Such a mechanism is known as

the Fridrich-Wintgen scenario. If the coupling constants

vanish due to symmetry, such BIC are called symmetry

protected.

A true BIC is a mathematical object with an infinite value

of the Q factor and vanishing resonance width, and it can

exist only in ideal lossless infinite structures or for extreme

values of parameters [22–24]. In practice, BIC can be

realized as a quasi-BIC, also known as a supercavity mode

[25], when both the Q factor and resonance width become

finite [26,27]. Nevertheless, the BIC-inspired mechanism

of light localization makes possible to realize high-Q quasi-

BICs in optical cavities and photonic crystal slabs

[19,23,25,28], coupled optical waveguides [29–32], and

even isolated subwavelength dielectric particles [33].

Here, we reveal that sharp spectral resonances recently

reported and even observed for various types of seemingly

different (plasmonic and dielectric) metasurfaces originate

from the powerful physics of BIC as a result of distortion of

the symmetry-protected bound state in the continuum. We

demonstrate that true BICs transform into quasi-BICs when

the in-plane inversion symmetry of a unit cell becomes

broken, and we derive the universal formula for the Q

factor as a function of the asymmetry parameter. We

develop an analytical approach to describe light scattering

by arrays of meta-atoms based on the explicit expansion of

the Green’s function of open systems into eigenmode

contributions, and demonstrate rigorously that reflection

and transmission coefficients are linked to the conventional

Fano formula. We prove that the Fano parameter becomes

ill-defined at the BIC condition which corresponds to a

collapse of the Fano resonance [34,35].

To gain a deeper insight into the physics of BICs in

metasurfaces with in-plane asymmetry, we focus on one of

the examples recently suggested for biosensing [4], a square

array of tilted silicon-bar pairs shown in Fig. 2(a). For our

analysis, we consider a homogenous background with

permittivity 1, and calculate both the eigenmodes and

transmission spectra, as shown in Fig. 2(b). An asymmetry

parameter here is the angle θ between the y axis and the long

axis of the bar. The ideal (lossless and infinite) structure

supports a symmetry-protected BIC [36] at θ ¼ 0°. Such an

ideal BIC is unstable against perturbations that break the in-

plane inversion symmetry ðx; yÞ → ð−x; −yÞ, and it trans-

forms into a quasi-BIC with a finite Q factor.

The eigenmode and transmission spectra are shown in

Fig. 2(b) as functions of θ, where t is the amplitude of the

transmitted wave, and T ¼jtj

is the transmittance. We

observe that BIC with infinite Q factor at θ ¼ 0° transforms

into high-Q quasi-BIC whose radiation loss grows with θ.

The detailed transmission spectra shown in Fig. 2(c) exhibit a

narrow dip that vanishes when the pair becomes symmetric,

Transmission

1563 1565 1567 1569 1571

Wavelength, nm

BIC

θ=10˚

0˚

0.5˚

1˚

2˚

3˚

4˚

5˚

6˚

8˚

Wavelength, nm

30 40

1600

1560

1520

1480

1440

Wavelength, nm

θ, deg

|H|

(c)

(b)

(d)

Eigenmode analysis

quasi-BIC

BIC quasi-BIC

1600

1560

1520

1480

1440

Transmission

(a)

FIG. 2. (a) A square lattice of tilted silicon-bar pairs with a design of the unit cell. Parameters: the period is 1320 nm, bar semiaxes are

330 and 110 nm, height is 200 nm, distance between bars is 660 nm. (b) Eigenmode spectra and transmission spectra with respect to

pump wavelength and angle θ . Error bars show the magnitude of the mode inverse radiation lifetime. (c) Evolution of the transmission

spectra vs angle θ. Spectra are relatively shifted by 1.5 units. (d) Distribution of the electric and magnetic fields for both BIC and

quasi-BIC.

PHYSICAL REVIEW LETTERS 121, 193903 (2018)

193903-2

which confirms the results of the eigenmode analysis.

Figure 2(d) demonstrates the similarity of the electric and

magnetic fields in BIC and quasi-BIC within a unit cell. Our

analysis shows that BIC is characterized by a polarization

vortex in the reciprocal space with a unity topological charge

(see details in the Supplemental Material [37]).

We analyze the transmission spectra of the metasurfaces

and prove rigorously that they can be described by the

classical Fano formula, and the observed peak positions and

linewidths correspond exactly to the real and imaginary parts

of the eigenmode frequencies. While the analytical solution

of Maxwell’s equations does not exist, the description of the

transmission T with the Fano formula is still widely used by

introducing the Fano parameter phenomenologically [42].

Here, we demonstrate the explicit correspondence between

the Fano line shape of the transmission spectra and properties

of the eigenmode spectra, discussed previously only for

special cases [18,23,43].

To derive an analytical expression for light transmission,

we expand the transmitted field amplitude into a sum of

independent terms where each term corresponds to an

eigenmode of the photonic structure. This becomes pos-

sible by applying the recently developed procedure

allowing for rigorous characterization of the Green’s

function and, therefore, the eigenmode spectrum of open

optical resonators [44]. The eigenmodes of a metasurface

are treated as self-standing electromagnetic excitations with

a complex spectrum describing both the resonant frequen-

cies ω

and inverse lifetimes γ. Straightforward but rather

cumbersome calculations (see Supplemental Material [37])

reveal that the frequency dependence of the transmission T

is described rigorously by the Fano formula, and the Fano

parameters are expressed explicitly through the material

and geometrical parameters of the metasurface and dimen-

sionless frequency Ω ¼ 2ðω − ω

Þ=γ,

TðωÞ¼

1 þ q

ðq þ ΩÞ

1 þ Ω

þ T

ðωÞ: ð1Þ

Here q is the Fano asymmetry parameter, T

and T

describe

the background contribution of nonresonant modes to the

resonant peak amplitude and the offset, respectively (see the

explicit expressions in the Supplemental Material [37]).

Importantly, the frequency dependence of q, T

, and T

is smooth, and it can be neglected, that is valid when the

neighboring resonances do not overlap spectrally with the

resonance under study. Remarkably, the exact formulashows

that the parameter q in Eq. (1) becomes ill-defined for a true

BIC thus corresponding to a collapse of the Fano resonance

when any features in the transmission spectra disappear, and

the resonant mode is transformed into a “dark mode,” which

does not manifest itself in scattering spectra.

Next, we describe analytically the behavior of the

radiative Q factor of the quasi-BIC as a function of θ

shown in Figs. 2(b) and 2(c). We consider the radiation

losses as a perturbation which is a natural approximation

valid when θ remains relatively small. Then, the inverse

radiation lifetime γ

rad

can be calculated as a sum of

radiation losses into all open radiation channels. We focus

on quasi-BICs with the frequencies below the diffraction

limit for which only open radiation channels represent the

zeroth-order diffraction. Then γ

rad

of quasi-BIC takes the

form (see Supplemental Material [37]),

rad

¼jD

þjD

; ð2aÞ

x;y

¼ −

ﬃﬃﬃﬃﬃﬃﬃ



x;y

∓

y;x

xz;yz



: ð2bÞ

Here, k

¼ ω

=c, S

is the unit cell area, D

x;y

are the

coupling amplitudes between quasi-BIC and zero-order

diffraction channels of two orthogonal polarizations, and p,

m, and

Q are the normalized electric dipole, magnetic

dipole, and electric quadrupole moments per unit cell.

Equations 2(a) and 2(b) show that a symmetry mismatch

for a true BIC leads to zero D

x;y

and vanishing radiation

losses [45]. In other words, the electric field components E

and E

of BIC are odd with respect to the inversion of

coordinates ðx; yÞ → ð−x; −yÞ, so that γ

rad

¼ 0. For quasi-

BICs, we perform straightforward transformations of

Eqs. 2(a) and 2(b) (see Supplemental Material [37])to

show that the radiative quality factor Q

rad

¼ ω

=γ

rad

depends on θ as

rad

ðθÞ¼Q

½αðθÞ

−2

; ð3Þ

where α ¼ sin θ and Q

is a constant determined by the

metasurface design, being independent on θ. In general,

Eq. (3) remains valid for metasurfaces placed on a substrate

as long as the quasi-BIC frequency is below the diffraction

limit of the substrate [27].

Next, we demonstrate that the quadratic dependence

defined by Eq. (3) represents a universal behavior for the Q

factor of a quasi-BIC as a function of the asymmetry

parameter for all studied metasurfaces with broken-

symmetry meta-atoms. We introduce the generalized asym-

metry parameter α, which has distinct definitions for

different structures but takes values between 0 and 1.

We derive Eq. (3) using the second-order perturbation

theory for open electromagnetic systems and confirm the

result by independent calculations of the eigenmode and

reflectance spectra for all designs presented in Figs. 1(b)–1(f)

(see Supplemental Material [37]). Since plasmonic meta-

surfaces possess significant absorption, we extract the bare

radiative Q factor by evaluating the inverse radiative lifetime

rad

being a difference between the total inverse lifetime γ

and nonradiative damping rate evaluated at α ¼ 0.

Figure 3(a) shows a direct comparison of the values

of the radiative Q factor of quasi-BICs as functions of

the asymmetry parameter α for dielectric and plasmonic

PHYSICAL REVIEW LETTERS 121, 193903 (2018)

193903-3

metasurfaces with various broken-symmetry meta-atoms in

the unit cell. All curves represent the results of our direct

numerical simulations being shifted relatively in the ver-

tical direction to originate from the same point. As can be

noticed from Fig. 3(a), for small values of α the behavior of

rad

for all metasurfaces is clearly inverse quadratic.

Importantly, for most of the structures the law α

−2

is valid

beyond the applicability limits of the perturbation theory.

Figure 3(b) defines the asymmetry parameter α for different

metasurface designs.

The quadratic scalability of the Q factor of quasi-BICs,

combined with linear scalability of Maxwell’s equations,

suggests a straightforward way of smart engineering of

photonic structures with the properties on demand. As an

example, we focus on a design of tilted silicon-bar pairs and

suggest a very simple algorithm for a design of metasur-

faces with a wide range of Q factors and operating

wavelengths ranging from visible to THz. First, since

the refractive index dispersion for silicon is relatively

weak, we can tune the operating wavelength from 0.5 to

300 μm by a linear geometric scaling of the structure in all

dimensions. Second, we can control the mode radiative Q

factor in a wide range of parameters by changing the

asymmetry parameter α ¼ sin θ according to Eq. (3). The

total Q factor of quasi-BIC is limited by absorption, which

can be estimated as ReðεÞ=ImðεÞ. For silicon in the

frequency range from near-IR to THz, the Q factor can

exceed 10

Using this approach, we calculate the dependence of the

total Q factor of quasi-BIC vs the operating wavelength and

asymmetry parameter for a square lattice of tilted silicon-

bar pairs, and summarize our results in Fig. 4. For each data

point of this map, we perform only one numerical simu-

lation with fixed material parameters to obtain the value of

required for Eq. (3), and then employ the advantages

of the geometrical scaling combined with rotation of bars.

Asymmetry parameter α

0.001

Δs

α = sinθ

α = Δs / S

(b)

(a)

add remove

Perturbation

ΔL

α = ΔL / L

Radiative Q factor

0.01 0.1

FIG. 3. Effect of the in-plane asymmetry on the radiative Q

factor of quasi-BICs. (a) Dependence of the Q factor on the

asymmetry parameter α for all designs of symmetry-broken met a-

atoms shown in Figs. 1(a)–1(f) (log-log scale). (b) Definitions of

the asymmetry parameter α for different metasurfaces.

Campione, 2016

Vabishchevich, 2018

Tittl, 2018 Lim, 2018

Singh, 2011

0.17

0.34

0.50

0.64

0.5 1 1.5 5 10 15 100

200

300

Asymmetry parameter α

Wavelength, μm

LATEMCIRTCELEID

Forouzmand, 2017

1015.00025 100

Scaling factor

FIG. 4. Map of operating wavelengths and quality factors for

silicon metasurfaces with tilted-bar pairs for varying orientation

of the bars (α ¼ sin θ) and scaling factor. The radiative part of the

total Q factor is evaluated using Eq. (3), the nonradiative part is

taken equal 10

. All calculations are confirmed by direct

numerical simulations with realistic dispersion of silicon (see

Supplemental Material [37]). The geometric scaling factor is

shown in the upper horizontal axis. Colored rectangles corre-

spond to the structures considered in the previous studies,

see Fig. 1.

PHYSICAL REVIEW LETTERS 121, 193903 (2018)

193903-4

We observe that a variety of the required Q factors can be

achieved in a broad range from 10 up to 10

for each

wavelength domain, by using the same material and

design. The colored rectangles show the range of the Q

factors and operating wavelengths related to the previous

studies of metasurfaces with symmetry-broken meta-atoms

[4,8–10,13,15]. We verify the applicability of the proposed

analytical approach by three-dimensional electromagnetic

simulations taking into account the silicon dispersion and

by using the

COMSOL

finite-element method, the results

are summarized in the Supplemental Material [37].We

observe a good agreement, and thus justify the validity of

our analytical scaling method.

We believe that our approach based on the BIC concept

can describe many other cases of symmetry-broken meta-

surfaces and photonic-crystal slabs studied earlier by other

approaches and different applications in mind [46–50].

Also, our approach can be helpful to get a deeper physical

insight into many other problems in optics, including the

existence of dark states in dielectric inclusions coupled to

the external waves by small nonresonant metallic antennas

[51] and electromagnetically induced transparency [52,53].

We argue that almost any problem involving the so-called

“dark states” may find its rigorous formulation in the

framework of the theory of BIC resonances.

In conclusion, we have revealed that high-Q resonances

recently observed in metasurfaces composed of dissimilar

meta-atoms with broken in-plane symmetry are closely

associated with the physics of bound states in the continuum.

We have proven a direct link between peculiarities in the

transmission spectra and Fano resonances, and demonstrated

analytically a large variation of the Q factors that can be

realized with changing the unit-cell asymmetry, paving the

way towards smart engineering of resonances in metaoptics

for nanolasers, light-emitting metasurfaces, nonlinear nano-

photonics, optical sensors, and ultrafast active metadevices.

The authors acknowledge financial support from the

Australian Research Council, the Alexander von

Humboldt Stiftung, and the Russian Science Foundation

(18-72-10140), and also thank H. Altug, H. Atwater, F.

Capasso, A. Krasnok, S. Kruk, W. Liu, Th. Pertsch, M.

Rybin, R. Singh, I. Staude, V. Tuz, and N. Zheludev for

useful discussionsand suggestions.K. K. andA. B. acknowl-

edge a support from the Foundation for the Advancement of

Theoretical Physics and Mathematics “BASIS” (Russia).

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193903-5

Asymmetric Metasurfaces with High-Q Resonances Governed by Bound States in the Continuum.

Figures

Citations

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High-Q Quasibound States in the Continuum for Nonlinear Metasurfaces.

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Angle-multiplexed all-dielectric metasurfaces for broadband molecular fingerprint retrieval

References

Flat Optics With Designer Metasurfaces

Optically resonant dielectric nanostructures

Fano resonances in photonics

Sharp trapped-mode resonances in planar metamaterials with a broken structural symmetry

Observation of trapped light within the radiation continuum

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Frequently Asked Questions (1)

Q1. What contributions have the authors mentioned in the paper "Asymmetric metasurfaces with high-q resonances governed by bound states in the continuum" ?