Thermo-optic hysteresis with bound states in the continuum D. N. Maksimov123 A. S. Kostyukov1 A. E. Ershov13 M. S. Molokeev12 E. N. Bulgakov23and V. S. Gerasimov13 1IRC SQC Siberian Federal University 660041 Krasnoyarsk Russia

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Thermo-optic hysteresis with bound states in the continuum

D. N. Maksimov1,2,3, A. S. Kostyukov1, A. E. Ershov1,3, M. S. Molokeev1,2, E. N. Bulgakov2,3and V. S. Gerasimov1,3

1IRC SQC, Siberian Federal University, 660041, Krasnoyarsk, Russia

2Kirensky Institute of Physics, Federal Research Center KSC SB RAS, 660036, Krasnoyarsk, Russia and

3Institute of Computational Modelling SB RAS, Krasnoyarsk, 660036, Russia

(Dated: October 6, 2022)

We consider thermo-optic hysteresis in a silicon structure supporting bound state in the contin-

uum. Taking into account radiative heat transfer as a major cooling mechanism we constructed a

non-linear model describing the optical response. It is shown that the thermo-optic hysteresis can

be obtained with low intensities of incident light I0≈1 W/m2at the red edge of the visible under

the critical coupling condition.

I. INTRODUCTION

Recently, we have seen a surge of interest to bound

states in the continuum (BICs) [1–3] that have grown to

an important tool in nanophotonics paving a way to opti-

cal devices with enhanced light-matter interaction. BICs

do not couple the incident light, however, if the sym-

metry of the system is broken the BICs are observed as

narrow Fano resonances in the scattering spectrum [4–

7]. In more detail, the BICs are spectrally surrounded

by a leaky band of high-quality resonances (quasi-BICs)

which can be excited from the far-ﬁeld [8]. The excitation

of the strong resonances results in critical ﬁeld enhance-

ment [9, 10] with the near-ﬁeld amplitude controlled by

the frequency and the angle of incidence of the incom-

ing monochromatic wave. The critical ﬁeld enhancement

triggers nonlinear optical eﬀects even with a low inten-

sity of the incident light. This resonant enhancement

of nonlinear eﬀects can lead to symmetry breaking [11],

channel dropping [12], excitation of non-linear standing

waves [13] as well as self-adaptive robust [14] and tunable

Fabry-Perot [15] BICs.

In the ﬁeld of nonlinear optics the BICs have been

applied for second harmonic generation [16–18]. Quasi-

BICs in subwavelength dielectric resonators [19, 20] are

also shown to demonstrate to enhance second harmonic

generation [21, 22]. In [23] it was found that, otherwise

decoupled, BICs can be excited via second harmonic gen-

eration by illuminating the structure from the far ﬁeld.

At the same time it was shown theoretically that quasi-

BICs allow for optical bistability [8, 24–26] due to the

Kerr eﬀect.

More recently, the research focus shifted towards BICs

in lossy structures. In the presence of material absorption

the BIC are shown to acquire ﬁnite-life, albeit remain

localized and decoupled from the outgoing channels [27].

Quasi-BIC in lossy periodic structures are found to be

instrumental for enhancement of light absorption [28, 29]

in the critical coupling regime even in low loss dielectrics.

This opens novel opportunities for highly eﬃcient light

absorbers [30–34].

Lately, it has been suggested that thermo-optical

eﬀects can be the dominating nonlinear eﬀects in

BIC supporting structures due to heating by absorbed

radiation[35]. In this work we investigate resonantly

enhanced thermo-optical bistability [36–40] in a system

supporting an optical BICs. Taking into account radia-

tive heat transfer as a major cooling mechanism we shall

construct a non-linear model based on the temporal cou-

pled mode theory (TCMT) [41] and theoretically demon-

strate thermo-optic hysteresis.

II. SCATTERING THEORY

We consider an array of identical dielectric rods of ra-

dius R0linearly arranged with period Lin vacuum. The

rods are made of amorphous silicon. The axes of the

rods are collinear and aligned with the z-axis as shown

in Fig. 1 (a). Such a system is known to support an

abundance of BICs as demonstrated in [42, 43]. In this

work we performed numerical simulations with applica-

tion of FDTD Lumerical to examine the properties of the

BIC induced optical response taking into account both

temperature and frequency dependence of the refractive

index. In Fig. 1 (b) we compare the eigenmode proﬁle of

an optical BIC with vacuum wavelength λBIC = 782 nm

against the scattering solution obtained under illumina-

tion by a TE plane wave with λ= 785 nm at the incidence

angle θ= 8.97 deg, and the incident wave vector on the

x0y-plane as shown in Fig. 1 (a). The results are ob-

tained with the numerical values of the refractive index

from [44]. In Fig. 1 (b) one can see a striking similarity

between the two ﬁeld proﬁles.

By deﬁnition a BIC can not couple to incident light.

However, each BIC is a singular point on the dispersion

sheet of a leaky band where the Q-factor diverges to inﬁn-

ity. The BIC is, thus, spectrally surrounded by a family

of high-Qleaky modes (quasi-BICs) with the resonant Q-

factor controlled by variation of the angle of incidence. In

our case the BIC occurs in the Γ-point being symmetri-

cally mismatched from the single radiation channel of the

zeroth diﬀraction order. Therefore, the high-Qresonant

response is triggered by setting oﬀ the angle of incidence

from zero. Thus, the similarity between the BIC and the

scattering solution is explained by both BIC and quasi-

BIC sitting in the same dispersion band.

Our goal in this section is to set up the optimal regime

for enhanced light absorption leading to the most sig-

arXiv:2210.02364v1 [physics.optics] 5 Oct 2022

(a) (b)

2R0

BIC

𝜃=8.97o

-1

-0.5

0.5

𝜃

FIG. 1: (a) Set-up of the array of dielectric rods. (b)

Comparison between the eigenmode proﬁle for a BIC

with λBIC = 782 nm and scattering solution under

illumination by a plane wave with λ= 785 nm. The

solutions are visualized as the z-components of the

electric ﬁeld. The geometric parameters: R0= 128 nm,

L= 428 nm.

niﬁcant change of the refractive index by heating. The

problem of enhanced absorption by quasi-BICs has been

previously considered in the literature [28]. The central

result for upside-down symmetric structures, such as the

one in Fig. 1 (a), is that the maximal absorption of 50%

can be achieved in the critical coupling point where the

radiation and absorption loss rates are equal to one an-

other.

The BIC induced optical response can be understood

in the framework of temporal coupled mode theory

(TCMT) [41]. For the reader’s convenience we list the

most important TCMT formulas below. Let us consider

two-channel scattering for TE-polarized light. The S-

matrix is implicitly deﬁned through the following equa-

tion s(−)

s(−)

2=b

Ss(+)

s(+)

2,(1)

where s(±)

mare the amplitudes of the plane waves in the

far-ﬁeld with subscript m= 1,2 corresponding to the up-

per and lower half-spaces while superscripts (+),(−)stand

for incident and outgoing waves respectively. We assume

that the system is illuminated by a monochromatic wave

of frequency ω. In what follows we introduce the vectors

of incident and outgoing amplitudes |s(±)(t)iwhich oscil-

late in time with the harmonic factor e−iωt. The TCMT

equations [41] take the following form

da(t)

dt =−(iω0+γ+γ0)a(t) + hd∗|s(+)i,

|s(−)i=b

C|s(+)i+a(t)|di,(2)

where b

Cis the matrix of direct (non-resonant) process, ω0

is the resonance center frequency, γis the radiation decay

rate, γ0is material loss decay rate, ais the amplitude of

the resonant eigenmode, and |diis the 2 ×1 vector of

coupling constants satisfying

hd|di= 2γ. (3)

The solution for the S-matrix reads

S=b

C+|dihd∗|

i(ω0−ω) + γ+γ0

.(4)

In the case of the center-plane mirror symmetry we have

C=eiφ ρ iτ

iτ ρ ,(5)

where φ, ρ, τ are real valued parameters such as

ρ2+τ2= 1.(6)

The following equation also holds true [41]

C|d∗i=−|di.(7)

Using Eq. (3) together with Eq. (7) one ﬁnds

|di=eiφ

2rγ

2(1 + ρ)τ−i(1 + ρ)

τ−i(1 + ρ),(8)

for symmetric modes, and

|di=eiφ

2rγ

2(1 + ρ)τ+i(1 + ρ)

−τ−i(1 + ρ),(9)

for anti-symmetric ones. The reﬂectance can be found

from Eq. (4) as

R=ρ2(ω−ω0)2∓2ρτγ(ω−ω0) + τ2γ2+ρ2γ2

(ω−ω0)2+ (γ+γ0)2,(10)

while the transmittance is given by

T=τ2(ω−ω0)2±2ρτγ(ω−ω0) + ρ2γ2+τ2γ2

(ω−ω0)2+ (γ+γ0)2,(11)

where the top sign is used for symmetric modes. Finally,

the absorbance is given by

A=2γγ0

(ω−ω0)2+ (γ+γ0)2.(12)

The maximal absorbance is obtained at the critical cou-

pling condition γ=γ0and ω=ω0. Substituting the

above into Eq. (12) one ﬁnds

Amax =1

2.(13)

Since the system supports its antisymmetric BIC in

the Γ-point, we expect that the resonant frequency ω0

and the radiation decay rate γare given by the following

equations [45]

ω0=ωBIC +κωθ2+O(θ4),

γ=κγθ2+O(θ4),(14)

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Thermo-optichysteresiswithboundstatesinthecontinuumD.N.Maksimov1;2;3,A.S.Kostyukov1,A.E.Ershov1;3,M.S.Molokeev1;2,E.N.Bulgakov2;3andV.S.Gerasimov1;31IRCSQC,SiberianFederalUniversity,660041,Krasnoyarsk,Russia2KirenskyInstituteofPhysics,FederalResearchCenterKSCSBRAS,660036,Krasnoyarsk,Russiaand3Instit...

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Thermo-optic hysteresis with bound states in the continuum D. N. Maksimov123 A. S. Kostyukov1 A. E. Ershov13 M. S. Molokeev12 E. N. Bulgakov23and V. S. Gerasimov13 1IRC SQC Siberian Federal University 660041 Krasnoyarsk Russia.pdf

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Thermo-optic hysteresis with bound states in the continuum D. N. Maksimov123 A. S. Kostyukov1 A. E. Ershov13 M. S. Molokeev12 E. N. Bulgakov23and V. S. Gerasimov13 1IRC SQC Siberian Federal University 660041 Krasnoyarsk Russia

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