PHYSICAL REVIEW B 100, 115303 (2019)

Multipolar origin of bound states in the continuum

Zarina Sadrieva,1,* Kristina Frizyuk,1,† Mihail Petrov,1 Yuri Kivshar,1,2 and Andrey Bogdanov 1

1ITMO University, Saint Petersburg 197101, Russia2Nonlinear Physics Center, Australian National University, Canberra ACT 2601, Australia

(Received 18 April 2019; revised manuscript received 14 August 2019; published 3 September 2019)

Metasurfaces based on resonant subwavelength photonic structures enable novel methods of wavefrontcontrol and light focusing, underpinning a new generation of flat-optics devices. Recently emerged all-dielectricmetasurfaces exhibit high-quality resonances underpinned by the physics of bound states in the continuumthat drives many interesting concepts in photonics. Here we suggest an approach to explain the physics ofbound photonic states embedded in the radiation continuum. We study dielectric metasurfaces composed ofplanar periodic arrays of Mie-resonant nanoparticles (“meta-atoms”) which support both symmetry protectedand accidental bound states in the continuum, and employ the multipole decomposition approach to reveal thephysical mechanism of the formation of such nonradiating states in terms of multipolar modes generated byisolated meta-atoms. Based on the symmetry of the vector spherical harmonics, we identify the conditions forthe existence of bound states in the continuum originating from the symmetries of both the lattice and the unitcell. Using this formalism we predict that metasurfaces with strongly suppressed spatial dispersion can supportthe bound states in the continuum with the wave vectors forming a line in the reciprocal space. Our resultsprovide a method for designing high-quality resonant photonic systems based on the physics of bound states inthe continuum.

DOI: 10.1103/PhysRevB.100.115303

I. INTRODUCTION

The quest for compact photonic systems with high qual-ity factor (Q factor) modes led to the rapid developmentof optical bound states in the continuum (BICs). BICs arenonradiating states, characterized by the resonant frequenciesembedded to the continuum spectrum of radiating modes ofthe surrounding space [1–3]. The BICs first appeared as amathematical curiosity in quantum mechanics [4]. The dis-covery of BICs in optics immediately attracted broad attention(see, e.g., Refs. [5–7]) due to high potential in applicationsin communications [8,9], lasing [10–13], filtering [14], andsensing [15–17]. Recent achievements in the field of BICs arediscussed in Refs. [18–27].

Decoupling of the resonant mode from the radiative spec-trum, which is the basic idea behind the BIC, can be inter-preted in several equivalent ways. Within the coupled-modetheory, it corresponds to nullifying the coupling coefficientbetween the resonant mode and all radiation channels ofthe surrounding space [28]. Alternatively, the appearance ofBICs is explained as vanishing of Fourier coefficients corre-sponding to open diffraction channels due to the symmetryof the photonic structure. At the particular high-symmetrypoints of the reciprocal space, for example, at the � point,the continuous spectrum is divided into the modes of differentsymmetry classes with respect to the reflectional and rota-tional symmetry of the photonic system. The bound states ofone symmetry class can be found embedded in the continuum

*z.sadrieva@metalab.ifmo.ru†k.frizyuk@metalab.ifmo.ru

of another symmetry class, and their coupling is forbidden aslong as the symmetry is preserved. This kind of BIC is calledsymmetry protected, and it also allows interpretation in termsof topological charges defined by the number of times thepolarization vector winds around the BIC, presented as vortexcenters in the polarization field [29]. In contrast to the sym-metry protected BIC, the so-called accidental BICs [1,29–32]can be observed out of the � point due to an accidentalnullifying of the Fourier (coupling) coefficients via fine tuningof parameters of the photonic system. Such a mechanism isalso known as the Friedrich-Wintgen scenario [33].

There are many methods developed for calculation oftransmission/reflection spectra, eigenmodes, and the complexeigenfrequency spectrum of an open periodic system likephotonic crystal slabs, metasurfaces, and chains of particles.In particular, these methods include guided mode expan-sion method [34], resonance state expansion method [35,36],perturbative methods [37,38], Fourier modal method (alsoknown as rigourous coupled wave analysis) [39–41], quasi-normal-mode expansion [42,43], coupled Bloch mode ap-proach [44,45], methods using multipole scattering theory[46], and Feshbach projection formalism [47–49]. Many ofthese methods are used to explain the appearance of BICsin periodic structures and description of their properties[5,29,50–58].

Despite the number of existing approaches to understand-ing the nature of BICs, there is still room for further de-velopment of the theory. During the previous few years theelectromagnetic multipole theory [59] has been extensivelydeveloped as a natural tool of nanophotonics dealing withthe lowest (fundamental) resonances of the system. The mainadvantage of the multipole decomposition method (MDM)

2469-9950/2019/100(11)/115303(12) 115303-1 ©2019 American Physical Society

ZARINA SADRIEVA et al. PHYSICAL REVIEW B 100, 115303 (2019)

is that it provides a representation of an arbitrary field dis-tribution as a superposition of the fields created by a setof multipoles [60,61]. Namely, the multipole expansion hasbeen widely used to determine the polarization and directivitypatterns of the scattered field of single particles and theirclusters, both plasmonic and dielectric [60–64], for a varietyof applications such as polarization control devices [61],dielectric nanoantennas [65], and light demultiplexing [66].A number of novel optical phenomena have been explainedwithin the MDM such as the anapole effect [67,68], optome-chanical phenomena [69,70], and the Kerker effect [71,72].

In this paper, we extend the MDM approach to explain bothsymmetry protected and accidental BICs. We provide a theoryof BIC origin in terms of MDM for a general case of anyperiodic structure and develop an analytical method, whichdetermines the contribution of the vector spherical harmonics(VSH) to the far field (Sec. II). Working in the VSH basis,we take advantage of the internal symmetry and provide thegroup-symmetry approach to identifying the BIC formationin terms of the unit-cell and lattice symmetries (Sec. III). Weimplement field multipole expansion of the eigenmodes of aperiodic two-dimensional (2D) photonic structure supportingBICs. We illustrate the developed technique by consideringa 2D square array of spheres and extending it to the case ofa photonic crystal slab with a 2D array of cylindrical holes(Sec. IV). The developed approach can be easily extendedeven further to periodic structures with other types of unit-celland lattice symmetries. The proposed method both provides adeeper understanding of the photonic BIC physics and givesa tool for an effective designing of high-Q resonant photonicsystems.

II. MULTIPOLAR APPROACH

In this section we consider the modes of a two-dimensionalperiodic array of dielectric nanoparticles with arbitrary shape(see Fig. 1), and obtain an expression connecting the multi-polar content of the field inside and outside the nanoparticles.We denote the VSH as Mpr mn(k, r) (magnetic) and Npr mn(k, r)(electric), relying on the definition presented in Ref. [73].We introduce an additional notation Wpi pr mn for both typesof VSH, where inversion parity index pi = (−1)n+1 forMpr mn(k, r), and pi = (−1)n for Npr mn(k, r). Index n is themultipole order, m varies from zero to n, and pr = 1 if Wpi pr mn

is even under reflection from the y = 0 plane (ϕ → −ϕ in thespherical system) and pr = −1 if it is odd [74].

In a homogeneous medium with permittivity ε, any solu-tion of the Helmholtz equation with wave vector k = √

ε ωc

can be expanded in terms of electric and magnetic sphericalharmonics [75]. The field inside the medium of the jth cell ofthe array Ein can be written in terms of multipolar decompo-sition:

Ein(r) =∞∑

n=1

n∑m=0pi ,pr

E0[Dpi pr mnW(1)

pi pr mn(k2, r′)ei(kb·r j )], (1)

where k2 = √ε2

ωc is the wave vector in the material, the

superscript (1) stands for spherical Bessel functions in theradial part of the VSH, kb is the Bloch vector, r′ = r − r j ,and r j is the position of a single sphere (see Fig. 1). Dpi pr mn

d

x

y

z

± ± ... +

+

+

+

+

+

+

+

Symmetry-protected BIC:

Accidental BIC:

± ... =

=z

z

+

±±±+

±

(b)

(a)

0rj

r

ε2ε1

FIG. 1. (a) Metasurface with a square unit cell composed ofdielectric meta-atoms. (b) The upper and lower panels show theformation mechanism of symmetry protected (at-�) and accidental(off-�) BIC.

are the coefficients of the multipolar decomposition, whichwill be discussed in Sec. III. In a similar manner, the fieldoutside the array is expressed as follows (see Appendix B):

E(r) = E0

∑K,pi,pr ,m,n

Dpi pr mnVbi−n

2πk1

∫∫ ∞

−∞dk‖δ(kb − k‖ − K)

× eikr

kz

[Ypi pr mn

(k|k|

)]. (2)

Here K is the reciprocal-lattice vector, kz = ±√

k21 − k2

‖ ,|k| = k1 = √

ε1ωc , and Vb is the volume of the first Brillouin

zone. The spherical vector functions Y(n) depend on thespherical coordinates of a unity vector n and they are given inAppendix A. Their symmetry coincides with the symmetry ofthe W function with identical indices. The relation betweenD and D is discussed in Appendix C, and for the case ofan array of spherical particles coefficients D can be derivedanalytically.

The summation in Eq. (2) over the reciprocal vectorscorresponding to open diffraction channels, i.e., the termswith real kz, provides the contribution to the far field. For thefrequencies below the diffraction limit, only the zeroth-orderterm with K = 0 gives a nonzero contribution to the far field:

E(r) = E0Vb

2πk1k1zeik1r

∑pi ,pr ,m,n

i−nDpi pr mn

[Ypi pr mn

(k1

k1

)], (3)

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MULTIPOLAR ORIGIN OF BOUND STATES IN THE … PHYSICAL REVIEW B 100, 115303 (2019)

where k1|| = kb and k1z = ±√

k21 − k2

b . According toEq. (3), the contribution of the multipole with numberspi, pr, m, and n to the far field in the direction defined bythe wave vector k1 is proportional to the multipole expansioncoefficient D and the value of spherical vector function Y inthe given direction. Equation (3) provides the correspondencebetween the radiation pattern of a single unit cell and thefar-field properties of the whole infinite array, allowing forinterpreting the BIC in terms of MDM. In strong contrastto a single nanoparticle, where each multipole contributesto the far field, in the case of a subdiffractive array theremight be a direction in which none of the multipoles givesany contribution, or alternatively the nonzero contributionof different terms may eventually sum up to zero. Theformulated alternative gives a sharp distinction between thesymmetry protected and accidental BIC.

A. At-�-point BIC

The �-point BIC corresponds to the absence of the far-field radiation in the direction along the z axis. Due to thestructure of VSH, it appears that a number of multipoles donot radiate in the vertical direction along the z axis. If thefield inside a single unit cell consists only of such multipoles,there will be no total radiation in the z direction. This simplefact is illustrated in the upper panel in Fig. 1(b). Noticingthat only Ypi pr 1n functions with m = 1 are nonzero in parallelto the z-axis direction, we can conclude that at the � pointin the subdiffractive array all the modes which do not containthe harmonics with m = 1 are symmetry protected BICs. Thisfundamental conclusion lies at the basis of recent experimen-tal demonstration [10] of lasing with BICs in a 2D subdiffrac-tive array of nanoparticles. The particular operational modeconsisted of vertical dipoles oriented along the z axis, thusnot contributing to the only open channel. There exists anapproach [1,29] in which the eigenmodes at the � point canradiate in the normal direction z if their fields are odd underCz

2 rotations and do not have any other rotational symmetryof Cz

n type. In terms of multipole moments, this follows fromthe fact that at the � point any radiative mode should containmultipoles with m = 1. On the other hand, by virtue of thesymmetry, the even modes have zero radiation losses, i.e., aninfinite radiation quality factor, and are known as symmetryprotected BICs.

B. Off-� BIC

Let us now turn to the case of accidental BIC description.In the general case, coefficients D are complex numbers.They define the amplitudes and phase delay between themultipoles. However, in accordance with Ref. [30], if thestructure has time reversal and inversion symmetry, the eigen-modes must satisfy the condition E(r) = E∗(−r). This factimposes strict conditions on the multipoles’ phases; becausesome of them are even under inversion and some of themare odd, the coefficient D−1pr mn before the odd ones must beimaginary.

It follows that every term of the sum in Eq. (3) is purelyreal, and all multipoles are in phase or antiphase. All coeffi-cients depend on k-vector and structural parameters and in the

case of the off-� BICs this sum turns to zero. In other words,for the particular k1 all vector harmonics add up to zero in thedirection of k1, analogously to the anti-Kerker effect, becausethey are already in phase and only amplitudes are modulatedwhile the k vector is changed [Fig. 1(b), lower panel]. Onecan intuitively understand accidental BICs (TM polarized)via a toy dipole model composed of vertical electric andhorizontal magnetic dipoles interfering destructively at some kvector [76].

The expansion coefficients Dpi pr mn depend on shape ofthe nanoparticles, material parameters, and symmetry of thelattice. Obviously, the lattice symmetry imposed restrictionson Dpi pr mn and some coefficients should vanish due to thesymmetry. To explore these selection rules, we employ thegroup-theoretic approach.

III. SYMMETRY APPROACH

The group-theoretic approach is a powerful method whichis widely used for analyzing the properties of periodic pho-tonic systems [31,77–80]. In this section, we apply thismethod to reveal which of the coefficients Dpi pr mn are nonzeroin accordance with the mode symmetry imposed by the sym-metry of the periodic structure [81,82] and to explain theformation of BICs. To make further analysis more illustrativewe will provide the example of a square periodic array ofdielectric spheres shown in the inset in Fig. 2(a). Nevertheless,it is necessary to highlight that further considerations remaintrue for any kind of structures with a square lattice withpoint-group symmetry D4h.

Figure 2(a) shows the dispersion of eigenmodes along the�X direction for a square periodic array of dielectric sphereswith permittivity ε2 = 12 embedded in the air with permit-tivity ε1 = 1. The dispersion is calculated numerically usingthe COMSOL MULTIPHYSICS package. The radius of the spheresis a = 100 nm and the period of the array is d = 600 nm.Analogous to a dielectric slab waveguide, the eigenmodesof the 2D array are split into transverse electric (TE) andtransverse magnetic (TM) modes, which have mainly the zcomponent of magnetic and electric field, correspondingly.Figure 2(b) shows the dependence of the Q factor on theBloch wave number kx for the modes, which are BICs at the� point. One can see that, additionally to at-�-BIC, the TM3

mode turns into off-BIC in the middle of the Brillouin band(accidental BIC).

The eigenmodes of periodic structures have certain sym-metry in accordance with the fact that every mode is trans-formed by an irreducible representation of the structure’ssymmetry group [83,84]. While Bloch functions are alreadythe basis functions of the translation group irreducible repre-sentation labeled by kb, under point-group operations the basisfunctions with different kb transform through each other. Theycarry high-dimensional representations of the space group.We are interested in the multipolar content of the mode withparticular kb, so we look at the point group of kb, i.e., thesubgroup of the whole point group, which transforms the kb

into an equivalent. Note that symmetry of the unit cell shouldbe also taken into account since it alters the point group of thefull structure.

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ZARINA SADRIEVA et al. PHYSICAL REVIEW B 100, 115303 (2019)ω

d / 2

πc

0.4

0.6

1

TE1

TE1

TE2TE3

TE4

TM2

TM1

TM3

TM4

TM4

off-Г BICat-Г BIC

(a)

(b)

TM3

off-Г BIC

at-Г BIC

0 0.2 0.4 0.6 0.8 110

1

103

105

107

XГ kxd / π

0 0.2 0.4 0.6 0.8 1

XГ kxd / π

Qua

lity

fact

or TE4

d

x

yz

FIG. 2. (a) Band diagram for the square periodic array of di-electric spheres at the �X valley. Period d = 600 nm, nanosphereradius a = 100 nm, dielectric permittivity of the spheres ε2 = 12,and medium permittivity ε1 = 1. Modes TE1, TE4, TM3, and TM4

are symmetry protected BICs at the � point. (b) Q factor of the TM3

mode in the �X valley. At the particular wave vector, the Q factortends to infinity and the accidental BIC appears.

At the � point, the group of kb is the whole group D4h,kb = 0 and it is not transformed. At �X valleys the pointgroup of kb is C2v , which consists of π rotation aroundthe x or y axis and two plane reflections at z = 0 and aty = 0 or x = 0. These operations keep vector kb invariant.Analogously, in the �M valley the group is also C2v . Solutionswith particular kb are transformed by one of the kb-groupirreducible representations. Thus, since the solution is trans-formed as a basis function of some particular representation,the multipolar content is strictly limited. Namely, all themultipoles with nonzero contribution must be transformed byan irreducible representation similar to that in the kb group.We will use common notations for irreducible representations,for example, those listed in Refs. [85,86].

FIG. 3. Multipolar content and irreducible representations of TEmodes at the � point and at �X and �M valleys. Spherical harmonicsfor B2 and A1 representations at �X are provided in the right columnof the figure. The multipoles for the same representations at �Mcan be obtained by rotation by π/4. The imaginary unit means thatcoefficients D before these harmonics must be imaginary. The zcomponent of the magnetic field Hz is shown in the insets.

A. � point

For example, we consider the TE1 mode of the squarearray, which is transformed by A2g at the � point. Underthe transformations of the D4h group the only low-ordermultipoles transformed by A2g are magnetic dipole M−101,magnetic octupole M−103, and electric hexadecapole N−144.All of them are invariant under C4 rotations, even underinversion and z = 0-plane reflection, and odd under other D4h

transformations. Higher-order multipoles which behave in thesame way are also presented in the multipolar content of thismode.

Analogously, we classify all possible multipoles at the� point in accordance with their symmetry and provide themultipolar content of the modes (Figs. 3 and 4). Note that TE2

and TE3 modes degenerate, and they transform through eachother as two basis functions of the representation Eu.

B. �X valley

After the symmetry reduction, when we step out the �

point to the �X valley, some symmetry operations remain,e.g., mirror reflections in z = 0 and y = 0 planes and rotationby π around the x axis. Eigenmodes must behave in the sameway under these symmetry operations, as at the � point.

As an example, we consider the TE1 mode, which trans-forms by A2g at the � point, and the TM3 mode (A1u) atthe �X valley. Using the compatibility relations [80], weobtain that the mode which transforms by A2g at the � point

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MULTIPOLAR ORIGIN OF BOUND STATES IN THE … PHYSICAL REVIEW B 100, 115303 (2019)

FIG. 4. Multipolar content and irreducible representations of TMmodes. The z component of the electric field Ez is shown in theinsets. Note that representations can be obtained by replacing for TEmodes A2 ↔ A1, B2 ↔ B1, and u ↔ g, and multipolar content canbe obtained by replacing pr ↔ −pr and M ↔ N.

is transformed under B2 representation of the group C2v inthe �X valley. The TM3 mode, which is A1u at the �, istransformed under B1 in �X . For the A2g mode at the � point,which is odd under reflection in the y = 0 plane and π rotationaround x and even under reflection in the z = 0 plane, theonly possible multipoles in the �X valley should have thesame symmetry properties. For the A1u mode the possiblemultipoles in the �X valley must be odd under reflection inthe z = 0 plane and π rotation and even under reflection inthe x = 0 plane. Low-order possible multipoles in the �Xvalley are listed in the right column of Fig. 3 for TE modes,and for TM modes are easily derived by replacing A2 ↔A1, B2 ↔ B1, u ↔ g, pr ↔ −pr, and M ↔ N. Analogously,for the �M valley we have specific symmetry in the x = yor x = −y direction and possible multipoles are the same asthose which are transformed by the same representation inthe �X valley, but rotated by π/4 with the help of WignerD matrices [87,88].

IV. MULTIPOLAR COMPOSITION OF THE EIGENMODESIN PERIODIC STRUCTURES

A. Multipole analysis of metasurfaces and photonic crystal slabs

Before applying the developed approach to certain struc-tures, we would like to emphasize that the power of the group-theoretic approach and multipole decomposition method isthat these methods can be applied equally for metasurfaces(arrays of meta-atoms) and photonics crystal slabs (slabs withholes). For example, a square array of dielectric spheres and adielectric slab with circular holes arranged in a square latticeare identical from the point of view of group theory. There-fore, these structures have the same classification of eigen-modes and, more importantly, the same multipole content(Mie resonances). The difference will be only in amplitudesof the multipole coefficients.

At first glance it may seem unphysical to characterize theunit cell with a hole by Mie resonances. This contradictioncan be eliminated by means of the proper choice of the unit

D4h D4h

x

zy min

max

min

max

Unit cell 1

bals latsyrc-cinotohPecafrusateM

Unit cell 2 Unit cell 1 Unit cell 2

FIG. 5. Distribution of the Ez component of the electric field ofthe modes corresponding to the same irreducible representation B1u

(see Fig. 4) in the photonic crystal slab and metasurface.

cell. Indeed, Fig. 5 shows the modes of the photonic crystalslab and metasurface corresponding to the same irreduciblerepresentation B1u (see Fig. 4). One can see that the modestructures are completely identical. However, it is more physi-cal to assign Mie resonances not to the unit cell with a hole butto the cross-shaped unit cell (see the lower panel in Fig. 5). Inparticular, it was shown in Ref. [89] that the optical responseof both photonic crystal slab and metasurface is dominatedby the electric and magnetic Mie-type dipole resonances, andboth structures provide 2π phase control of light.

B. Periodic arrays of dielectric spheres

In order to understand how the interference of the mul-tipole moments forms the BICs, we applied a method ofmultipole decomposition by expanding the eigenmode fieldsinside the nanoparticles in terms of spherical harmonics[59,60]. Figure 6 shows the numerical results for the multipoledecomposition for TE1, TE4, and TM4 modes at the � pointand in a point of the �X valley. At the high-symmetry � pointonly a small fraction of all possible multipoles is presented,while after lowering the symmetry extra multipoles appearout of the � point. Figure 7 shows the numerical results forthe multipole decomposition for TM3 modes at the � point, ina point of the �X valley, and at the off-� BIC.

1. At-� BIC

One can see that for the at-� TE1(A2g) BIC mode themagnetic dipole along the z axis (M−101) is mainly con-tributed [Fig. 6(a)]. Its directivity pattern restricts radiationalong the z axis, and the radiation patterns of all remainingmultipoles at the � point are also zero along the z axis. Otherdirections of radiation are forbidden due to the subdiffractionregime. Similarly, the at-� TM3(A1u) BIC is formed by theelectric dipole along the z axis (N101) mostly, prohibiting theradiation in the vertical direction itself [Fig. 7(a)]. Reducingthe unit-cell symmetry to D1h results in the appearance ofthe in-plane dipole moment and higher multipoles radiatingin the vertical direction. Therefore, the BIC turns into aresonant state with a finite Q factor, which could be preciselycontrolled by the asymmetry degree of the unit cell [90]. TheTE4 (B2g) and TM4 (B1u) modes’ lowest multipoles are the

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ZARINA SADRIEVA et al. PHYSICAL REVIEW B 100, 115303 (2019)

10-5

10-4

10-3

10-2

10-1

10 0

TE1

TE4

TM4

x

yz

10-4

10-3

10-2

10-1

10 0

—100d kx=

x

yz

y

x

y

x

M12

2

M12

4

Multipole order Multipole order1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

M12

4+M

144

M12

2

M11

3+M

133

M11

1

N10

1

N11

2

N10

3+N

123

N11

4+N

134

N12

3

M-1

01

M- 1

03

N-1

44

N-1

44+N

-124

N-1

13+N

-133

N-1

22

N-1

11

M-1

14+M

-134

M-1

03+M

-123

M-1

01

M-1

12

ΓX—

Magnetic Electric Magnetic ElectricΓ, kx=0

BIC )b()a(

(e) BIC (f)

10-5

10-4

10-3

10-2

10-1

10 0

x

yz

M-1

23

N-1

24

N-1

44+ N

-124

N- 1

13+N

-133

N-1

22

N-1

11M-1

14+M

-134

M-1

03+M

-123

M-1

01

M-1

12

BIC )d()c( —100d kx=

—8d kx=

y

x

N-1

22

MDMQ

MOMH

EDEQ

EOEH

FIG. 6. Multipolar content of TE1 (a, b), TE4 (c, d), and TM4

(e, f) modes of the metasurface. The vertical axis shows valuesof normalized coefficients D. The horizontal axis shows multipolarorder n. The letters in the legend encode the multipoles as follows:ED (MD), electric (magnetic) dipole (n = 1); EQ (MQ), electric(magnetic) quadrupole (n = 2); EO (MO), electric (magnetic) oc-tupole (n = 3); EH (MH), electric (magnetic) hexadecapole (n = 4).

electric quadrupole N−122 and the magnetic quadrupole M122,respectively. They have m = 2, which also prohibits the radi-ation [Figs. 6(c) and 6(e)]. In contrast to BICs, the radiativemodes (Eg, Eu) are degenerate at the � point since they aretransformed by the two-dimensional representations. From thesymmetry-group approach, we know that TE2,3 and TM1,2

modes contain electric N±111 and magnetic M±111 sphericalharmonics. The numerical multipole expansion shows that de-generated modes contain in-plane electric or magnetic dipolemoments as the main contribution. These numerical resultsvalidate the symmetry-group approach for the system withsquare lattice, confirming that any symmetry protected BICis characterized by multipole moments with m = 1.

TM3

10-4

10-3

10-2

10-1

10 0

—8d kx=

x

yz

y

x

M14

4N

101

N10

3 N10

3+N

123

N10

1

N11

2

N11

4+N

134

M11

3+M

133

M12

2

M11

1

(a) (b)at-Г BIC

M12

4+M

144

—2d kx=

N10

3+N

123N

101

N11

2

N11

4+N

134

M11

3+M

133

M12

2

M11

1

(c)

M12

4+M

144

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4Multipole order Multipole order Multipole order

MDMQ

MOMH

EDEQ

EOEH

off-Г BIC kx=0

FIG. 7. Multipolar content of the TM3 mode of the metasurfacefor three kx Bloch vectors corresponding to the symmetry protectedBIC (a), leaky mode (b), and accidental BIC (c). The vertical axisshows values of normalized D coefficients; the horizontal axis showsmultipolar order n. The letters in the legend encode the multipolesas follows: ED (MD), electric (magnetic) dipole (n = 1); EQ (MQ),electric (magnetic) quadrupole (n = 2); EO (MO), electric (mag-netic) octupole (n = 3); EH (MH), electric (magnetic) hexadecapole(n = 4).

2. Off-� BIC

Away from the � point other multipoles appear in de-composition [Figs. 6(b), 6(d), 6(f), and 7(b)] and the BIC isdestroyed, turning into a resonance state. As mentioned earlierin Sec. II, the accidental off-� BIC is formed due to eitherin-phase or antiphase contributions of different multipoles.We obtain the same result for the TM3 mode with off-�BIC in the �X valley [Fig. 7(c)]. This mode is transformedby B1 representation and consists of the multipoles, whichare odd under reflection in the z = 0 plane and π rotationaround the x axis and even under y = 0 reflection, e.g.,iN101, N112, iN103, iN123, M111, iM122, M113, and M133. Theysum to a TM-polarized wave in the direction given by vectork1 and cancel each other in the off-� point, forming theaccidental BIC, shown in Fig. 2(b). In addition, it is wellknown that the z = 0 plane reflection symmetry of the struc-ture is required to obtain the off-� BIC [30]. Indeed, lackingsuch symmetry, each mode would contain both odd and evenmultipoles under reflection in the z = 0 plane. To restrict theradiation both in the upper and lower half spaces, odd andeven multipoles should be summed to zero independently,while for the symmetric structure only one type of multipolesis presented for each mode, which makes it possible to achievethe BIC by tuning the structure parameters.

C. Photonic crystal slab

We extend our approach beyond the 2D array of spheresand apply it to the photonic crystal slab of the same symmetry.We consider a dielectric slab with a square array of cylindricalholes studied in Ref. [30]. Both at-� (symmetry protected)and off-� (accidental) BICs appear in the lowest TM band,referred to as TMh. This mode has the field profile of the same

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MULTIPOLAR ORIGIN OF BOUND STATES IN THE … PHYSICAL REVIEW B 100, 115303 (2019)

0.2

0.4

0.6

0.8

1kx=

x

yz

d

off-Г BIC

0 0.2 0.4 0.6 0.8

kxd / π

off-Γ BIC

M12

2

M12

4

N12

3

M12

2M

113+

M13

3

N10

1N

112

N10

3+N

123

N11

4+N

134

M11

1 M12

2M

113+

M13

3M

124+

M14

4

M12

4+M

144 N10

1N

112

N10

3+N

123

N11

4+N

134

M11

11 2 3

MDMQ

MOMH

EDEQ

EOEH

0.54

10-2

10-1

100MDMQMOMHEDEQEOEH

(a)

(d)

(b) (c)

0.05

4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

at-Г BIC

Γ X

kx= dkx=0

TMh

Multipole order Multipole order Multipole order

FIG. 8. Multipolar content of the TMh mode of the PhC slabfor three kx Bloch vectors corresponding to the symmetry protectedBIC (a), leaky mode (b), and accidental BIC (c). The vertical axisshows values of normalized D coefficients; the horizontal axis showsmultipolar order n. Panel (d) depicts how the multipolar contributionschange along the �X valley. The letters in the legend encode themultipoles as follows: ED (MD), electric (magnetic) dipole (n = 1);EQ (MQ), electric (magnetic) quadrupole (n = 2); EO (MO), electric(magnetic) octupole (n = 3); EH (MH), electric (magnetic) hexade-capole (n = 4).

symmetry as the TM4 mode of the array of the spheres and istransformed by B1u representation.

The description of the far field defined by Eq. (3) isstill valid, but the coefficients Dpi pr mn in general cannot beexpressed analytically through Dpi pr mn (see Appendix C).However, only the multipoles which are presented in thefield inside the slab contribute to the far field, Dpi pr mn = 0 ifDpi pr mn = 0. The multipolar content for the considered slab isthe same as in the periodic array of spheres because the modesof the slab have the same symmetry as the modes of the array.The multipole decomposition of the TMh mode [Fig. 8(a)]reveals that a magnetic quadrupole M122 and electric octupoleN123 make the major contribution to the at-� BIC, as well asto the TM4 mode of the array of the spheres. However, the

BIC line

at-Г BIС

k

k

x

y

nodal cone

FIG. 9. BIC line formed by the metasurface composed of pointoctupoles with radiation pattern N103.

B1u mode of the photonic crystal slab is the lowest-energyTM mode while for the array of spheres it has the highestenergy among modes under the diffraction limit. Due to thevariational principle [91], for the mode of such symmetry,the electric field is more concentrated inside the high-indexmaterial in the case of a photonic crystal slab, minimizingthe energy of the mode. Although dispersion curves ω(k)of the modes TM4 and TMh behave completely differently,multipole decomposition proves a common origin of them.At the � point, we obtain contributions of multipoles onlywith m = 2, 6, 10, etc., for both modes, and none of thesemultipoles contribute in the far field. At the �X valley, themultipolar content of the TM4 and TMh modes is similar[Figs. 6(e), 6(f), 8(a), and 8(b)]. However, for the TMh mode itis possible to obtain an accidental off-� BIC in the �X valley[Fig. 8(c)]. Away from the � point, multipole contributionschange smoothly, keeping the multipolar content invariable,and at a particular wave vector kx multipoles interfere destruc-tively, forming the accidental BIC.

V. SUMMARY AND OUTLOOK

Importantly, our approach based on the multipole decom-position analysis of individual meta-atoms not only explainsclearly and in simple physical terms the origin of both sym-metry protected and accidental bound states in the continuumbut also has a prediction power and may be employed for bothprediction and engineering of different types of BICs. As anexample, we consider a metasurface consisting of meta-atomspacked in a subwavelength 2D lattice, which are polarizedpurely as octupoles, for example, M−103 (see Fig. 9). Eachoctupole of this type has a nodal cone and, therefore, wecan expect that in a periodic subwavelength array of suchmeta-atoms BICs form a line in the reciprocal space. However,to observe this phenomenon, the effective polarizability ofthe unit cell accounting for the interaction between all meta-atoms should not depend on the Bloch wave number or havevery weak dependence. In other words, observation of BICsin metasurfaces with suppressed spatial dispersion is still achallenging problem. Interestingly, this kind of BIC will be

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ZARINA SADRIEVA et al. PHYSICAL REVIEW B 100, 115303 (2019)

observed for the same directions independently of the latticesymmetry of the metasurface.

We can expand this discussion even further, by consideringthe M−101 dipole. To obtain the Q factor of the at-� BIC, weshould find the dependence of energy-loss rate on the k vectorin Eq. (3), assuming stored energy is almost constant whenθk → 0. The asymptotic behavior for θk → 0 of Ypi pr mn( k

|k| )

functions is defined by dPmn (cos θk )

dθk. So, Y1−103( k

|k| ) ∝ (−6kx +17k3

x /2)eϕ , and Y1−101( k|k| ) ∝ (kx − k3

x /6)eϕ . We can managetheir relative contribution, and at a particular point when thecoefficient before the dipole is six times larger we obtainthat linear terms cancel each other and the field asymptotic isproportional to k3

x , so the quality-factor growth is proportionalto k−6

x . A similar effect was observed in Ref. [22] for thephotonic crystal slab. However, considering realistic situa-tions, we should take into account all possible multipolar con-tributions, including terms with m = 1, where the multipolecontribution growth rate plays a role but not the asymptoticbehavior. Thus, the multipole origin of the BIC is a query formetasurfaces with a suppressed spatial dispersion.

In summary, we have demonstrated that symmetry pro-tected bound states in the continuum in dielectric metasur-faces and photonic crystal slabs at the frequencies belowthe diffraction limit are associated with the multipole mo-ments of the elementary meta-atoms which do not radiate inthe transverse direction. For any type of metasurfaces, thesymmetry protected bound states in the continuum can beobserved only if there exist no multipoles with the azimuthalindex m = 1 in the multipole decomposition. The symmetryapproach allows us to determine what multipolar content thelattice eigenmodes have, and it can be analogously applied tostructures with different symmetries, for example, hexagonallattices or arrays of nanoparticles of an arbitrary shape andin-plane broken symmetry. Similarly, we have revealed thatthe accidental bound state in the continuum correspondingto an off-� point in the reciprocal space is formed due todestructive interference of the multipole fields in the far zone.We have provided general tools for the analysis of boundstates in the continuum based on the irreducible representationof the appropriate photonic band. We believe that our resultswill provide a method for designing high-quality resonantphotonic systems based on the physics of bound states in thecontinuum.

ACKNOWLEDGMENTS

The authors acknowledge useful discussions with I. D.Avdeev and K. L. Koshelev. This work is supported bythe Russian Foundation for Basic Research (Grant No. 19-02-00419), the Ministry of Education and Science of theRussian Federation (Grants No. 3.1668.2017/4.6 and No.3.8891.2017/8.9), and the Grant of the President of the Rus-sian Federation (Grant No. MK-403.2018.2), Russian Foun-dation for Basic Research 18-02-01206. Y.K. acknowledgessupport from the Strategic Fund of the Australian NationalUniversity. Z.S., K.F., M.P., and A.B. acknowledge supportfrom the Foundation for the Advancement of TheoreticalPhysics and Mathematics “BASIS” (Russia).

Z.S. and K.F. contributed equally to this work.

APPENDIX A: BASIC DEFINITIONS

Vector spherical harmonics are defined as [73]

M−11 mn = ∇ × (

rψ 1−1mn

), (A1)

N 1−1mn =

∇ × M−11 mn

k, (A2)

where

ψ1mn = cos mϕPmn (cos θ )zn(kr) , (A3)

ψ−1mn = sin mϕPmn (cos θ )zn(kr). (A4)

Here zn(kr) can be replaced by a spherical Bessel function ofany kind, and Pm

n (cos θ ) are the Legendre polynomials. Theexpansion of the vacuum dyadic Green’s function in terms ofvector spherical harmonics reads as [92,93]

G0(r, r′) = ik1

4π

∞∑n=1

∑pr=±1

n∑m=0

(2 − δ0)2n + 1

n(n + 1)

(n − m)!

(n + m)!

× {[M(3)

pr mn(k1, r) ⊗ M(1)pr mn(k1, r′)

]+ [

N(3)pr mn(k1, r) ⊗ N(1)

pr mn(k1, r′)]}

, for r > r′.

(A5)

Here superscripts (1) and (3) appear, when we replace zn(ρ)by spherical Bessel functions, and the spherical Hankel func-tions of the first kind are δ0 = 1 when m = 0 and δ0 = 0 whenm = 0.

Spherical vectors Ypi pr mn denote two types of functions,Xpr mn and Zpr mn, defined as [94]

X−pr mn

(kk

)= ∇ ×

[kYpr mn

(kk

)], (A6)

Zpr mn

(kk

)= i

kk

× X−pr mn

(kk

), (A7)

where

Y1mn(θ, ϕ) = cos mϕPmn (cos θ ), (A8)

Y−1mn(θ, ϕ) = sin mϕPmn (cos θ ). (A9)

Here pr = (−1)n+1 for X, and pr = (−1)n for Z. Note thatthe transformation behavior is similar for W and Y, X and M,Z and N, and ψ and Y .

APPENDIX B: LATTICE SUMS OF THE SPHERICALHARMONICS

We assume that the multipolar content of the mode isalready known, and coefficients in the formula (1) are given.With the help of vacuum dyadic Green’s function G0, weexpress the field outside the array:

E(r) = k21

4π

∫d3r′′�ε(r′′)G0(r, r′′)Ein(r′′)

= k21

4π(ε2 − ε1)

∑j

∫V

d3r′G0(r, r j + r′)Ein(r j + r′),

(B1)

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MULTIPOLAR ORIGIN OF BOUND STATES IN THE … PHYSICAL REVIEW B 100, 115303 (2019)

where k1 = √ε1

ωc is the vacuum wave vector, and V is the

single nanoparticle’s volume.The Green’s function can be also expressed in terms of

vector spherical harmonics (see Appendix A). Using theproperty G0(r, r j + r′, ω) = G0(r − r j, r′, ω), and substitut-ing (1) into (B1), we obtain

E(r) = E0

∑j

ik31 (ε2 − ε1)

(4π )2

∑pi,pr ,n,m

∑p′

i,p′r ,n

′,m′(2 − δ0)

× 2n + 1

n(n + 1)

(n − m)!

(n + m)!Dp′

i p′r n′m′

× W(3)pi pr mn(k1, r − r j ) · ei(kb·r j )

×∫

Vd3r′[W(1)

pi pr mn(k1, r′) · W(1)p′

i p′r m′n′ (k2, r′)]. (B2)

Here superscript (3) stands for the outgoing spherical Han-kel wave. Now we dwell on the case when the array iscomposed of spherical nanoparticles. Exploiting the VSHorthogonality properties, the integral over the sphere can betaken analytically [75]. It is proportional to Kronecker deltaδpi p′

iδpr p′

rδmm′δnn′ which removes one summation. Combining

all coefficients including the integral into D we obtain that thefield outside the array is expressed with the formula

E(r) = E0

∑pi ,pr ,m,n

j

Dpi pr mnW(3)pi pr mn(k1, r − r j ) · ei(kb·r j ). (B3)

Note that coefficient Dpi pr mn is nonzero only if the harmonicwith such numbers is presented in the field expansion insidethe sphere. If the fields created by each unit cell are alreadyknown, we can also start the considerations from this formula.

To obtain formula (2) we exploit the Weyl identity for VSHexpansion through the plane waves in the case when zn(kr) isreplaced by spherical Hankel functions [95,96]:

M(3)pr mn(k, r) = i−n

2πk

∫∫ ∞

−∞dk||

ei(kxx+kyy±kzz)

kz

[Xpr mn

(kk

)],

(B4)

N(3)pr mn(k, r) = i−n

2πk

∫∫ ∞

−∞dk||

ei(kxx+kyy±kzz)

kz

[Zpr mn

(kk

)].

(B5)

The sign before kz depends on the sign of z. Redefining theharmonics, we have

W(3)pi pr mn(k, r)= i−n

2πk

∫∫ ∞

−∞dk||

ei(kxx+kyy±kzz)

kz

[Ypi pr mn

(kk

)],

(B6)

where kz =√

k2 − k2x − k2

y . Substituting this expansion into

(B3), we get

E(r) = E0

∑j,pi,pr ,n,m

Dpi pr mn · ei(kb·r j ) · i−n

2πk1

×∫∫ ∞

−∞dkxdky

ei(k(r−r j ))

kz

[Ypi pr mn

(kk

)]. (B7)

This expansion helps us to apply the summation formula∑j

eikx j = Vb

∑K

δ(k − K) (B8)

where K is the reciprocal-lattice vector and Vb is the volumeof the Brillouin zone (4π2/d2 for the square lattice), andsubstituting (B8) into (B7) we obtain Eq. (2).

APPENDIX C: RELATION BETWEEN THECOEFFICIENTS D AND D

The coefficients D in (B2) and D in (B3) are connected bythe formula

Dpi pr mn = ik31

(4π )2(2 − δ0)

2n + 1

n(n + 1)

(n − m)!

(n + m)!Dpi pr mn

×∑

p′i,p′

r ,n′,m′

∫V

d3r′(ε2 − ε1)

× [W(1)

pi pr mn(k1, r′) · W(1)p′

i p′r m′n′ (k2, r′)

]. (C1)

This formula describes both the array of nanoparticles andany photonic crystal slab, but in the case of the array ofspheres the integral can be easily taken analytically.

Remark. Here we give the dyadic Green’s functionG0(r, r′) only for the case when r > r′. In the case of aphotonic crystal slab or nonspherical particles, we also needthe part of the Green’s function at which r′ < r [93] to obtainthe near field. This will alter the intermediate calculations,since we have to compute the lattice sum for VSH withspherical Bessel functions. Nevertheless, the answer will havethe same form.

We apply the orthogonality properties of vector sphericalharmonics (see Ref. [75], p. 418), and consider integralsof magnetic and electric harmonics separately. Implement-ing the angular integration, we reduce the integral to theintegral of r-dependent Bessel function products, which alsocan be computed analytically. For magnetic harmonics wehave

DMpi pr mn = iDM

pi pr mna2ε1[k2

1 jn−1(k1a) jn(k2a)

− k1k2 jn−1(k2a) jn(k1a)]

(C2)

where a is the nanoparticle radius, and for electric harmonicswe have

DNpi pr mn = iDN

pi pr mna2ε1

{n + 1

2n + 1

[k2

1 jn−2(k1a) jn−1(k2a)

− k1k2 jn−2(k2a) jn−1(k1a)]

+ n

2n + 1

[k2

1 jn(k1a) jn+1(k2a)

− k1k2 jn(k2a) jn+1(k1a)]}

. (C3)

Note that this expression turns to zero at some frequencies,so we can have zero D when D is nonzero. This refers to theanapoles of the spherical nanoparticles. The frequency wherethe anapole appears is the same as for the single isolatednanoparticle.

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ZARINA SADRIEVA et al. PHYSICAL REVIEW B 100, 115303 (2019)

If we have another type of surface, for example, a pho-tonic crystal slab with holes or an array of cylinders, theorthogonality property cannot be applied and the integral∫

V d3r′�ε[W(1)pi pr mn(k1, r′) · W(1)

p′i p′

r m′n′ (k2, r′)] will mix some

harmonics. However, all the harmonics, which admix, arealready presented in the expansion of the field inside thecell. This will just alter the coefficients before the outgoingmultipoles, but not the multipolar content.

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