Pis’ma v ZhETF, vol. 114, iss. 3, pp. 154 - 155

Enhancement of second-harmonic generation in micropillar resonator

due to the engineered destructive interference

S. A. Kolodny⁺¹⁾, V. K. Kozin^+∗, I. V. Iorsh⁺

⁺ITMO University, 197101 St. Petersburg, Russia

^∗Science Institute, University of Iceland, Dunhagi-3, IS-107 Reykjavik, Iceland

Submitted 10 June 2021

Resubmitted 25 June 2021

Accepted 25

June 2021

DOI: 10.31857/S1234567821150027

It has been recently shown that the engineering of

We investigate the second-harmonic generation

the shape of the dielectric nanoantenna allows to achieve

(SHG) in a micropillar resonator with radius

500

the destructive interference of the low order multipole

nm, consisting of an AlGaAs cylinder, sandwiched

modes [1, 2]. As a result these structures may support

between two DBR mirrors (GaAs-AlGaAs, 30 layers

high quality optical modes, characterized at the same

on top and bottom), as shown on the inset in Fig.1.

time by relatively small mode volumes. Since the effect

The micropillar resonator is tuned to the quasi-BIC

of the emergence of the dark modes due to the destruc-

tive interference is analogous to the bound states in the

continuum (BIC) arising in periodic structures [3], these

modes are usually referred to as quasi-BIC states. It has

been shown experimentally that these states, supported

by the AlGaAs pillars, facilitate substantial increase in

the second harmonic generation efficiency [4].

At the same time, even at the quasi-BIC regime,

individual semiconductor pillars are characterized by

fairly modest quality factors in mid-infrared and op-

tical frequency ranges. The quality factor can be sub-

stantially increased if the pillar is sandwiched between

Bragg reflectors, which suppress the radiation losses

through the top and bottom of the pillar. The result-

ing structures, pillar microcavities, are conventionally

Fig. 1. (Color online) (a) - The sketch of the system under

characterized by large values of Q/V ratio and are rou-

consideration. (b) - Dependence of the wavelength of the

tinely used to enhance the light-matter interactions at

modes on the r/h ratio of the cavity. The wavelength of

the nanoscale [5, 6].

the mode E2 is doubled for better comparison (to demon-

The main source of the radiation losses in pillar mi-

strate that ω2 is close to 2ω1)

crocavities is due to the radiation leakage through the

sidewalls, which increase as the diameter of the cavity is

regime [1, 8]. It means that we consider the same

decreased. We have recently shown, that the at certain

structure where the Q/V ratio enhancement for cavity

ratios of cavity radius to cavity height, the destructive

mode has been observed for micropillar resonator with

interference occurs similar to the one in the quasi-BIC

low-contrast Bragg reflectors [8] and for high-contrast

state, which suppresses the side-wall leakage and reso-

Bragg reflectors [7] due to destructive interference of

nantly increases the quality factor while preserving the

two radiating modes.

effective mode volume [7]. Here, we show that this quasi-

The cavity is placed in the background field (pump),

BIC state occurring in pillar microcavities can be used

which in our case is supposed to be a superposition of

to substantially increase the efficiency of the second har-

two linearly polarized Hermite-Gauss beams [9] that re-

monic generation.

sult in an azimuthally polarized field with the azimuth

number m = 0. The AlGaAs has a non-vanishing tensor

¹⁾e-mail: s.kolodny@metalab.ifmo.ru

of the second-order nonlinear susceptibility χ^(2)ijk. This

154

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2021

Enhancement of second-harmonic generation in micropillar resonator . . .

155

tensor contains only off-diagonal elements in the prin-

increase in the coefficient κ₁, which will lead to a signif-

cipal axis system of the zinc blende crystalline struc-

icant enhancement of the second harmonic power.

ture [10], with the components being non-zero only if

To conclude, in this work we have investigated

= j = k, χ^x

≡ χ^(2)AlGaAs = 290pm/V. As for the

the second harmonic generation in a micropillar

GaAs its tensor of the second-order nonlinear suscepti-

AlGaAs/GaAs resonator, and have shown that it gets

bility χxyz also has non-zero components if i = j = k,

significant enhancement in the quasi-BIC regime. Com-

pared to a single resonator [4], the achieved theoretical

χ^x

≡ χ(2)GaAs = 180 pm/V [11]. The second har-

values are higher by at least an order of magnitude,

monic generation was considered in whole micropillar

despite the fact that the Q-factor is much higher (10⁵

resonator (both in AlGaAs cavity and AlGaAs/GaAs

Bragg reflectors). In our analysis we focus on the two

versus 10²). Thus, we believe that the presented results

can be applied in problems of nonlinear nanophoton-

modes of the pillar: E_1,2 with the real and imaginary

parts of the eigenfrequencies being equal to ω_1,2 and

ics and in the practical implementation of quantum

devices, where high nonlinearity plays an important

γ_1,2, respectively. The pillar is pumped at the frequency

ω close to ω₁, and the frequency ω₂ of the mode E₂ is

role.

The authors would like to thank Kirill Koshelev from

assumed to be close to 2ω. So in our simulations we con-

ITMO University and Sergei Yankin from COMSOL.

sider TE₀₁₂ mode as E₁ since it has a confirmed quasi

This work was supported by the Ministry of

BIC, and the TE₂₁₅ mode as E₂ because of the proxim-

Science and Higher Education of Russian Federation,

ity of ω₂ to 2ω. The cavity radius is fixed in our study

and we vary only its height as well as the Bragg layers

goszadanie #2019-1246.

period to make the center of the bandgap tuned to the

This is an excerpt of the article “Enhancement

mode frequency.

of second-harmonic generation in micropillar resonator

Our main goal is to calculate the nonlinear conver-

due to the engineered destructive interference”. Full

sion coefficient, showing the efficiency of the second-

text of the paper is published in JETP Letters journal.

harmonic generation, and defined as the ratio between

DOI: 10.1134/S0021364021150017

the total SHG power and the pump power squared:

P (2ω)/P₀(ω)². The corresponding expression for the to-

tal SHG power is given by [4]

1. M. V. Rybin, K. L. Koshelev, Z. F. Sadrieva, K. B. Samu-

sev, A. A. Bogdanov, M. F. Limonov, and Y. S. Kivshar,

)₂

8π

(2ω

Phys. Rev. Lett. 119, 243901 (2017).

P (2ω) =

κ₂Q₂L₂(2ω)κ₁₂ ×

2. K. Koshelev, G. Favraud, A. Bogdanov, Y. Kivshar, and

A. Fratalocchi, Nanophotonics 8(5), 725 (2019).

× [Q₁L₁(ω)κ₁(ω)P₀(ω)]² .

(1)

3. C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos,

Here Q_j = ω_j /(2γ_j) is the mode quality factor, L_j is the

and M. Soljačić, Nat. Rev. Mater. 1(9), 1 (2016).

spectral overlap factor, κ₁, κ_1,2 and κ₂ are the so-called

4. K. Koshelev, S. Kruk, E. Melik-Gaykazyan, J.-H. Choi,

coupling, cross-coupling, and decoupling coefficients, re-

A. Bogdanov, H.-G. Park, and Y. Kivshar, Science

spectively, where they are expressed in terms of the spa-

367(6475), 288 (2020).

tial mode profiles E_1,2(r) and the pillar parameters. The

5. G. Lecamp, J.-P. Hugonin, P. Lalanne, R. Braive,

dependence of second harmonic nonlinear coefficient on

S. Varoutsis, S. Laurent, A. Lemaˆıtre, I. Sagnes, G. Pa-

the aspect ratio r/h was studied with a fixed beam

triarche, I. Robert-Philip, and I. Abram, Appl. Phys.

Lett. 90(9), 091120 (2007).

waist radius equal to 1.5 µm (see Fig. 1). As it can be

seen from this plot, the nonlinear coefficient has a pro-

6. Z. Lin, X. Liang, M. Lončar, S. G. Johnson, and

nounced maximum at r/h = 0.745, where the coefficient

A. W. Rodriguez, Optica 3(3), 233 (2016).

is at least an order of magnitude larger in comparison

7. S. Kolodny and I. Iorsh, J. Phys. Conf. Ser. 1461,

with the rest of the aspect ratio area and it is about

012067 (2020).

8 × 10-4 W^-1. At this point the nonlinear coefficient

8. S. Kolodny and I. Iorsh, Opt. Lett. 45(1), 181 (2020).

dependence on the background field frequency in the

9. Q. Zhan, Adv. Opt. Photonics 1(1), 1 (2009).

frequency domain looks like a narrow peak (∼ 1 × 10¹⁰

10. R. W. Boyd, Nonlinear Optics, Academic Press, N.Y.

rad/s) because of the small value of γ₁, which enters the

(2003).

spectral overlap factor L₁(ω). In addition, an important

11. Landolt-Bornstein, Numerical Data and Functional

parameter is the beam waist radius, since its decrease

Relationships in Science and Technology, ed. by

with fixed peak power will lead to a stronger localiza-

O. Madelung, Springer, Berlin (1982), p.17.

tion of the field inside the cavity and, accordingly, an

Письма в ЖЭТФ том 114 вып. 3 - 4

2021