Origin of degenerate bound states in the continuum in a grating waveguide Parity symmetry breaking due to mode crossing C.B. Reynolds Vl.V. Kocharovsky V.V. Kocharovsky

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Origin of degenerate bound states in the continuum in a grating waveguide:

Parity symmetry breaking due to mode crossing

C.B. Reynolds, Vl.V. Kocharovsky, V.V. Kocharovsky

Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843, USA

(Dated: August 15, 2023)

We explain the origin of bound states in the continuum (BICs) in a planar grating waveguide, in

particular, a mechanism for formation of degenerate BICs, via the analytical theory of the inﬁnite-

grating eigenmodes. Conventional symmetry-protected BICs are formed at normal incidence mainly

by a single inﬁnite-grating eigenmode that has an odd spatial parity on both sides of the BIC

resonance. The odd parity is the reason for a cutoﬀ from the radiation-loss channel and appearance of

such BICs. The mechanism of emergence of a degenerate BIC in a vicinity of a degenerate frequency

of two inﬁnite-grating eigenmodes is diﬀerent. The degenerate BIC is formed by an anti-phased

coherent superposition of two crossing inﬁnite-grating eigenmodes both of which possess a mixed

parity and experience parity symmetry breaking as the frequency scans through the degeneracy

point. In this case a cutoﬀ from the radiation-loss channel and extremely high-Q narrow resonance

is achieved due to the destructive interference of the two crossing eigenmodes. Implementation of

such a mechanism can be instructive for designing BICs in other photonic crystals and structures.

I. INTRODUCTION: A BOUND STATE IN THE

CONTINUUM AS A RESOURCE FOR A

SOLITARY HIGH-Q RESONANCE

By means of the analytical theory of the inﬁnite-

grating eigenmodes [1], we disclose a distinct, explicit

mechanism leading to the formation of the degenerate

bound states in the continuum (BICs) and an entire hier-

archy of BICs in a planar grating waveguide (1D grating

slab with in-plane C2symmetry, Fig. 1).

Namely, far from the crossing points of the inﬁnite-

grating-eigenmode dispersion curves shown in Fig. 2, one

has only symmetry-protected BICs formed at normal in-

cidence mainly by just one relevant single inﬁnite-grating

eigenmode whose in-plane spatial proﬁle is purely odd

with respect to the in-plane mirror symmetry. Such a

BIC waveguide eigenmode involves an admixture of other

odd-parity inﬁnite-grating eigenmodes due to reﬂection

at the boundaries with the substrate and cover, but is

completely disconnected from the zeroth spatial Fourier

harmonic designed to be the only radiation-loss channel

available in the grating waveguide.

On the contrary, a degenerate BIC arises in a vicin-

ity of a degeneracy point where dispersion curves of two

inﬁnite-grating eigenmodes of opposite parity intersect,

as at D0

2,3in Fig. 2. This couple forms an anti-phased

superposition and constitutes a degenerate-BIC eigen-

mode of the grating waveguide due to modes’ mutual

reﬂections at the borders of the grating layer. Although

both inﬁnite-grating eigenmodes possess a mixed parity

and undergo a parity symmetry breaking in a vicinity of

the degeneracy point, their anti-phased superposition re-

mains decoupled from the only radiation channel – emis-

sion of the zeroth spatial Fourier harmonic outside the

grating waveguide into the cover and substrate. All other

Fourier harmonics of higher Bragg diﬀraction orders can-

not be emitted into the cover and substrate because they

are evanescent there. Such a phenomenon of destruc-

tive interference producing the degenerate-BIC waveg-

FIG. 1. The planar grating waveguide. (a) Geometry: the

grating of thickness Land period λgsandwiched between the

substrate and cover with permittivities ε−and ε+, respec-

tively. (b) The wave vectors associated with the propagating

or evanescent eigenmodes inside the grating and the plane

waves (Fourier harmonics) of diﬀerent Bragg diﬀraction or-

ders in the substrate and cover, p= 0,±1, . . ..

uide eigenmode decoupled from the radiation-loss chan-

nel is of general nature and could exist in a number of

other photonic structures and crystals.

The BIC waveguide eigenmode demonstrates an ex-

tremely narrow, solitary high-Q resonance despite the

fact that a continuum of plane waves in a wide range of

frequencies and wave vectors around the resonant ones

arXiv:2210.00628v2 [physics.optics] 13 Aug 2023

FIG. 2. Hierarchy of dispersion curves k2

zn (ω), Eq. (24), and

their intersections (degeneracy points) for the inﬁnite-grating

eigenmodes in the titanium-oxide grating; ε1= 6.25, ε2=

3.9, ρ = 0.39183; kx= 0. The symbol Dl

n,n+1 labels the l-th

intersection of the n-th and (n+1)-th eigenmodes and signiﬁes

the associated BIC.

can freely escape from the waveguide into the cover and

substrate. The latter distinguishes the phenomenon of

BICs from a conventional trapping of ﬁelds inside waveg-

uides or cavities which is based on a complete prevention

(say, by means of reﬂection) of the entire continuum of

waves from propagating outside the waveguide or cav-

ity. The search, analysis, design, and applications of the

BICs constitute a very active ﬁeld of modern research in

a variety of wave systems, especially in photonics, despite

the BIC having ﬁrst been suggested for matter waves in

quantum mechanics [2–4]. The mechanisms of forma-

tion and properties of various BICs in photonic struc-

tures had been widely discussed in the literature (see, for

example, reviews [5–9] and papers [10–21]). In particu-

lar, the BIC in the grating layer sandwiched by the cover

and substrate had been demonstrated via numerical sim-

ulations for a particular set of parameters in [10]. Yet,

despite the fact that the planar lamellar (that is, uniform

along yaxis) grating waveguide is one of the most basic,

simple examples of photonic structures supporting BICs,

a full analytical theory of BICs, in particular, degener-

ate BICs, in the grating waveguide and eigenmode-based

understanding of degenerate-BIC appearance have been

missing until now. This approach, in particular, reveals

exact analytical formulas clarifying universal features of

the BIC formation which, otherwise, would stay hidden.

Apparently, the main reason for that is the fact that

most works on BICs are based on ad hoc numerical sim-

ulations dominating modern analysis of photonic-crystal

structures. They employ various software based on ﬁnite-

element modeling like COMSOL [22], rigorous coupled

wave analysis (RCWA) [24], ﬁnite-diﬀerence time-domain

(FDTD) method [23], codes for layered periodic struc-

tures [25], ﬁnite element method (FEM) [15], etc.

Another reason is that the BIC-formation mechanism

in such a simple, basic photonic structure as the pla-

nar lamellar-grating waveguide turns out to be so special

that it simultaneously merges almost all other known

mechanisms leading to the BIC formation (see reviews

[5–9]), including mechanisms of the symmetry-protected

BIC, accidental BIC, single-resonance parametric BIC,

coupled multiple resonances (Friedrich–Wintgen, but not

Fabry–P´erot) BIC, interference-based BIC through pa-

rameter tuning, and topologically-protected BIC. A more

detailed discussion of this fact requires results presented

in sections II–VI and is postponed to sect. VII.

The present paper is devoted to introducing such an

analytical eigenmode approach per se, not to its thor-

ough application for classiﬁcation of all possible BICs or

discovery of new BICs. We give a new interpretation of

the known symmetry-protected BIC, in particular, de-

generate BICs, but do not attempt, within the scope of

the present paper, to design a BIC of a new type. It

is remarkable how merely splitting the central dielectric

layer of the planar waveguide into two alternating sec-

tions of lengths d1, d2(with a ﬁll factor ρ=d1/(d1+d2))

and diﬀerent permittivities ε1, ε2converts an elementary,

trivially soluble problem of the planar dielectric waveg-

uide [26–28] into a rich, complex problem demonstrating

many generic features of optical crystals.

Optical gratings and related waveguides are also in-

teresting by themselves, irrespective to BICs, since they

have found numerous applications and been studied for

decades (see, for instance, [26,27,29–37] and references

therein). One of the most elegant methods of their stud-

ies is based on the analysis of eigenmodes that is very

fruitful both for the analytical theory and numerical sim-

ulations (see [1,28–31] and references therein). An-

other, more straightforward method is the Fourier modal

method which prevails in the literature and is known as

the rigorous coupled wave theory in the diﬀractive optics

community [11,29,30] or the scattering-matrix approach

in the photonic-crystal community [32–37].

In the present paper, we consider a particular example

of a waveguide eigenmode – a degenerate BIC originating

due to the mechanism outlined in the second paragraph

of the paper. The in-plane x-wavenumber of that BIC

waveguide eigenmode is chosen to be near the Γ-point –

the center of the ﬁrst Brillouin zone of the grating, that

is kx≈0. (We assume that ky= 0 since we consider

only the standard, invariant in the ydirection, grating

problem when ﬁelds are uniform along the yaxis and

propagate in the xz-plane. So, we leave the conical case

of diﬀraction [38] aside.) This choice is predetermined by

strongly enhanced backscattering coupling in this excep-

tional high-symmetry point in the momentum space so

that the corresponding eigenmode looks more like a cav-

ity eigenmode rather than a waveguide eigenmode. In

fact, it has been shown by means of the representation

theory that in the standard photonic crystal slabs the

symmetry-protected BICs exist only at the center of the

Brillouin zone [20,21]. At wavenumbers away from the

Gamma point kx= 0, the waveguide eigenmode radi-

ates and forms a leaky resonance. The latter had been

illustrated in [10] by a numerical example.

We assume that just the central, zeroth diﬀraction or-

der p= 0, spatial Fourier harmonic is emitted out of the

planar grating layer into the cover and substrate, while

all higher-order, p̸= 0, Fourier harmonics are evanescent

and, hence, do not contribute to the radiation losses. Ac-

cording to the plane-wave dispersion relation, the p-th

Fourier harmonic becomes evanescent in the cover (su-

perscript “+”) or substrate (superscript “-”) when its

z-wavenumber squared turns negative:

(k±

zp)2=ε±ω2/c2−(kx+pkg)2<0.(1)

Here ε+or ε−is the permittivity of the cover or sub-

strate, respectively. For modes at the center of the ﬁrst

Brillouin zone, kx= 0, the p-th Fourier harmonic ceases

to provide a radiation-loss channel when

ε±<2πcp

ωλg2.(2)

Below we mainly focus on the degenerate BIC originat-

ing from the degeneracy point D0

2,3(see Fig. 2) when the

condition (2) is satisﬁed for all nonzero diﬀraction orders

p=±1,±2, . . .. The analysis of the degenerate BICs at

other mode-crossing points Dl

n,n+1 is alike.

The content of the paper is as follows. We overview the

genesis and ﬁeld constituents of BICs in a planar grating

waveguide in sect. II. In sect. III we present the nec-

essary analytical formulas describing the inﬁnite-grating

eigenmodes as well as the waveguide eigenmodes. In sect.

IV we disclose the universality of the behavior of the

inﬁnite-grating eigenmodes within the vicinity of a mode

crossing point and a remarkable connection between the

dispersion degeneracy and parity symmetry breaking of

the eigenmode spatial proﬁles changing from odd to even,

or vise versa. We elaborate both on the special case of

normal incidence, kx= 0, and the case of a nonzero in-

plane wavenumber, kx̸= 0. In sect. V we explain how

to design a waveguide manifesting a degenerate BIC at

a given frequency. It is done on the basis of a numerical

example related to a waveguide based on titanium oxide

(TiO2) – an optical material known for many applications

in photonics. The mechanism behind the emergence of

the degenerate BIC is explained and detailed in sect. VI.

It involves a parity symmetry breaking, that occurs in ac-

cord with the above universality, and a decoupling from

a radiation-loss channel due to destructive interference

at mode crossing. Conclusions, discussion of the simul-

taneous manifestation of other known BIC mechanisms

in the formation of the above BIC, and other comments

make up sect. VII.

II. GENESIS AND FIELD CONSTITUENTS OF

BICS IN A PLANAR GRATING WAVEGUIDE

We employ the eigenmode approach which, in the case

of BICs in the planar grating waveguide shown in Fig. 1,

is based on the hierarchy of the following three ﬁeld con-

stituents: (i) the plane waves, (ii) the inﬁnite-grating

eigenmodes, and (iii) the waveguide eigenmodes.

The plane waves are the simplest ﬁeld conﬁgurations

possessing a quasi-harmonic temporal-spatial proﬁle ∝

eikzp z+i(kx+pkg)x−iωt satisfying a standard dispersion law,

zp + (kx+pkg)2=εω2/c2,(3)

in a uniform medium. The latter means that their

phase speed depends on a permittivity εof the dielec-

tric medium in which they propagate. The preﬁx ’quasi’

indicates that the frequency ωand the z-wavenumber

kzp, in general, are complex-valued.

A single plane wave with a wave vector k= (kx, kz)

together with its mirror counterpart with a wave vector

k′= (kx,−kz) constitutes a waveguide eigenmode only

in the trivial case of a slab waveguide with a homoge-

neous central dielectric layer of permittivity ε. A homo-

geneous slab waveguide can be viewed as the zeroth-order

approximation for the grating waveguide if we choose its

core permittivity equal to the average permittivity of two

grating sections, ε= (ε1d1+ε2d2)/λg;λg=d1+d2.

FIG. 3. The ﬁrst three transverse guided TE-eigenmodes of

a homogeneous slab waveguide of thickness L= 0.71587λg:

Dispersion curves kxn(ω) (left) and spatial z-proﬁles E(n)

y(z)

(right). The permittivities of the central layer, substrate and

cover are ε= 4.8208, ε−= 2 and ε+= 1, respectively.

It is immediate to ﬁnd all guided TE-modes of such a

slab waveguide (see, e.g., [26]). The dispersion curves and

spatial z-proﬁles of the ﬁrst three transverse eigenmodes

guided by a slab waveguide are illustrated in Fig. 3. We

enumerate them by the integer nequal to the number of

extrema in the z-proﬁle of the electric ﬁeld E(n)

y(z). All

extrema are localized within the central dielectric layer,

z∈(0, L). The eigenmode amplitude is constant along

the xaxis. For a given frequency ω, the ﬁeld proﬁles

{E(n)

y(z)|n= 1,2, . . .}constitute a series of monochro-

matic transverse eigenmodes associated with a discrete

set of solutions for the eigen x-wavenumber, {kxn(ω)},

to the following dispersion equation

ωL(ε−ε−)

c= (n−1)π+ tan−1rb

1−b+ tan−1rb+a

1−b;

a=ε−−ε+

ε−ε−, b =(ckx/ω)2−ε−

ε−ε−, n = 1,2,3, . . . .

(4)

The series starts from the fundamental eigenmode n= 1.

All those guided modes in the slab have an angle of in-

cident lying inside a sector of the total internal reﬂection.

Hence, they are evanescent, that is, do not radiate out-

side the slab and ideally have an inﬁnite Q factor. How-

ever, they are not BICs since, for a given incident angle,

they have discrete frequencies lying completely outside

the continuous spectrum of the leaky, radiation waves.

The inﬁnite-grating eigenmodes [1] are also closely re-

lated to the plane waves in a uniform dielectric medium.

Yet, they are strongly restructured by multiple Bragg

reﬂections on the 1D lattice of the alternating permittiv-

ities of the grating sections. Such an umklapp scattering

can be visualized via a superposition of the parabolic

uniform-medium dispersion curves k2

zp(kx) of various

diﬀraction orders p= 0,±1,±2, . . . given by Eq. (3) as is

shown in Fig. 4. An eﬀective uniform medium with the

average grating permittivity ε= (ε1d1+ε2d2)/(d1+d2)

yields a good zeroth-order approximation for the disper-

sion curves of the inﬁnite-grating eigenmodes (cf. Fig. 2).

The main nontrivial feature here is opening the gaps in

the k2

zn spectrum near avoiding-crossing points. These

gaps are induced by the Bragg-reﬂection resonances. The

frequency for the plot in Fig. 4is chosen to be the degen-

erate frequency for the second and third inﬁnite-grating

eigenmodes, ω=ωc≡4c

λg. In this case the gap at the

intersection of the dispersion curves k2

z2and k2

z3is absent.

Contrary to the guided modes of the homogeneous

slab, the inﬁnite-grating eigenmodes are the ﬁeld con-

ﬁgurations possessing a trivial quasi-harmonic temporal

and spatial z-proﬁle ∝eikzn z−iωt, but a nontrivial spa-

tial x-proﬁle fn(x) (see Fig. 5and Eq. (6)) such that it

remains invariant in the course of propagation inside the

inﬁnite grating despite a persistent in-plane Bragg scat-

tering. Such an invariance occurs only for a discrete set

of eigen z-wavenumbers kzn enumerated by the integer

n= 1,2, . . . (see Fig. 2). Thus, the inﬁnite-grating eigen-

modes take care of the ﬁeld boundary conditions and ﬁeld

transformation at all yz-plane boundaries between alter-

nating, along the xaxis, sections of the grating, but do

not take into account the grating-waveguide boundaries

FIG. 4. ”Energy bands and gaps” of the 1D grating: An ex-

tended Brillouin zone representation for the dispersion curves

of the inﬁnite-grating eigenmodes, (c/ω)2k2

zn (kx), Eq. (24),

(thick black curves) approximated by a superposition of the

parabolic uniform-medium dispersion curves, (c/ω)2k2

zp (kx),

Eq. (3), (dashed color curves) shifted due to the Bragg reﬂec-

tion resonances of various orders by multiples of the grating

wavenumber pkg= 2πp/λg, p = 0,±1,±2, . . .. The permit-

tivity of the eﬀective uniform medium, ε= (ε1d1+ε2d2)/(d1+

d2)=4.8208, is equal to the weighted sum of the permittivi-

ties ε1= 6.25, ε2= 3.9 of the two grating sections of lengths

d1= 0.39183λg, d2= 0.60817λg, respectively; ω= 4c/λg.

with the substrate and cover. For zero in-plane wavenum-

ber kx= 0, as is illustrated in Fig. 5, the spatial parity of

the x-proﬁles f1, f3, f5, . . . is purely even while the spa-

tial parity of the x-proﬁles f2, f4, . . . is purely odd. Each

inﬁnite-grating eigenmode lives inside the inﬁnite grat-

ing independently of others, is a superposition of many

spatial Fourier harmonics (partial plane waves), and rep-

resents a photon dressed via Bragg, umklapp scattering.

At last, the waveguide eigenmodes are the ﬁeld conﬁgu-

rations which follow a quasi-harmonic evolution ∝e−iωmt

with a complex-valued eigenfrequency ωm=ω′

m−iω′′

enumerated by a composite integer m, while possessing

a stationary (invariant in time) spatial structure. Any,

say the m-th, waveguide eigenmode inside the grating

layer is a superposition of many inﬁnite-grating eigen-

modes, coupled to each other via mutual reﬂections at

the grating-layer borders with the cover and substrate.

Thus, the discrete set of waveguide eigenmodes is de-

termined by and take care of both sets of boundary

planes and corresponding boundary conditions of con-

tinuity of the tangential components of the electric and

magnetic ﬁelds: Two xy-planes, at z= 0 and z=L

FIG. 5. Longitudinal (in-plane) x-proﬁles for the ﬁrst ﬁve

eigenmodes of the inﬁnite grating with two sections of lengths

d1= 0.39183λg, d2= 0.60817λgand permittivities ε1=

6.25, ε2= 3.9, respectively; kx= 0, ω =5c

λg. For simplicity’s

sake, the plotted functions fn(x) diﬀer from those in Eq. (6)

by a phase factor eiφnmaking them real-valued. Note a slight

discontinuity of the derivatives dfn/dx at the boundaries be-

tween the grating sections.

along the zaxis, which determine the series of trans-

verse eigenmodes in the homogeneous slab waveguide as

well as the inﬁnite sequence of yz-planes, at x=pλg

and x=d1+pλg, p = 0,±1,±2, . . ., along the xaxis,

which determine the series of the inﬁnite-grating eigen-

modes. Contrary to the slab’s transverse modes and

inﬁnite-grating eigenmodes, the grating waveguide eigen-

modes possess two-dimensional spatial structure which is

nontrivial in both transverse, z, and longitudinal, x, di-

rections as is shown in Fig. 6.

The waveguide eigenmodes, accordingly, can be enu-

merated by a composite integer m= (mz, mx). Its ﬁrst

component, mz, indicates the number of ﬁeld extrema

in the transverse direction zand can be traced to the

FIG. 6. Two-dimensional (in the xz-plane) ﬁeld patterns,

Re(E(m)

y), of ﬁrst lower frequency grating-waveguide eigen-

modes at a small x-wavenumber, kx≈0. The composite

index (mz, mx) in each row labels the number of transverse

(mz) and longitudinal (mx) ﬁeld extrema per the grating pe-

riod λg. The eigenmodes in each row pair diﬀer by the value

of the binary index, s= 0 or s= 1, labeling modes with the

odd or even parity of the ﬁeld’s x-proﬁle, that is, with the

high or low Q factor. The lengths and permittivities of the

two grating sections are d1= 0.39183λg, d2= 0.60817λgand

ε1= 6.25, ε2= 3.9, respectively. The thickness of the grating

layer is L= 0.71587λg. The permittivities of the substrate

and cover are ε−= 2 and ε+= 1, respectively. Positive and

negative ﬁeld values are marked in red and blue, respectively.

order nof the corresponding transverse mode of the ho-

mogeneous slab in Eq. (4), mz=n. The second compo-

nent, mx, indicates the number of ﬁeld extrema in the

longitudinal direction xper the grating period λgand

depends on the Brillouin zone index, p, of the relevant

inﬁnite-grating eigenmodes. Along with the composite

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Originofdegenerateboundstatesinthecontinuuminagratingwaveguide:ParitysymmetrybreakingduetomodecrossingC.B.Reynolds,Vl.V.Kocharovsky,V.V.KocharovskyDepartmentofPhysicsandAstronomy,TexasA&MUniversity,CollegeStation,TX77843,USA(Dated:August15,2023)Weexplaintheoriginofboundstatesinthecontinuum(BICs)inap...

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Origin of degenerate bound states in the continuum in a grating waveguide Parity symmetry breaking due to mode crossing C.B. Reynolds Vl.V. Kocharovsky V.V. Kocharovsky.pdf

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Origin of degenerate bound states in the continuum in a grating waveguide Parity symmetry breaking due to mode crossing C.B. Reynolds Vl.V. Kocharovsky V.V. Kocharovsky

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