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OPTICS
Subwavelength dielectric resonators for
nonlinear nanophotonics
Kirill Koshelev
1,2
, Sergey Kruk
1
, Elizaveta Melik-Gaykazyan
1,3
, Jae-Hyuck Choi
4
, Andrey Bogdanov
2
,
Hong-Gyu Park
4
*, Yuri Kivshar
1,2
*
Subwavelength optical resonators made of high-index dielectric materials provide efficient ways to
manipulate light at the nanoscale through mode interferences and enhancement of both electric and
magnetic fields. Such Mie-resonant dielectric structures have low absorption, and their functionalities
are limited predominantly by radiative losses. We implement a new physical mechanism for suppressing
radiative losses of individual nanoscale resonators to engineer special modes with high quality factors:
optical bound states in the continuum (BICs). We demonstrate that an individual subwavelength
dielectric resonator hosting a BIC mode can boost nonlinear effects increasing second-harmonic
generation efficiency. Our work suggests a route to use subwavelength high-index dielectric resonators
for a strong enhancement of light–matter interactions with applications to nonlinear optics, nanoscale
lasers, quantum photonics, and sensors.
High-index resonant dielectric nanostruc-
tures emerged recently as a new plat-
form for nano-optics and photonics
to complement plasmonic structures
in a range of functionalities (1,2). All-
dielectric nanoresonators benefit from low
material losses and allow the engineering of
artificial magnetic responses. Progress in all-
dielectric nanophotonics led to the develop-
ment of efficient flat-optics devices that reached
and even outperformed the capabilities of con-
ventional bulk components (3). These advances
motivated further diversification of applica-
tions of dielectric nanostructures (4), especially
toward nonlinear optics (5–7). Efficiencies of
nonlinear optical processes in all-dielectric
nanostructures have exceeded by several or-
ders of magnitude the efficiencies demonstrated
in metallic nanoparticles with plasmonic reso-
nances (8,9).
One of the main limiting factors for high
efficiencies of all-dielectric nanostructures as
functional devices is the quality (Q) factor of
their resonant modes. Traditionally, response
of dielectric nanoparticles is governed by low-
order geometrical resonances, resulting in
low Q factors. An elegant solution for Q-factor
control and its increase is provided by the
physics of bound states in the continuum
(BICs). BICs were first proposed in quantum
mechanics as localized electron waves with the
energies embedded within the continuous
spectrum of propagating waves (10). Recently,
BICs have attracted considerable attention in
photonics (11,12). Mathematical bound states
have infinitely large Q factors and vanishing
resonant linewidth. In practice, BICs are lim-
ited by a finite sample size, material absorp-
tion, and structural imperfections (13), but
they manifest themselves as resonant states
with large Q factors, also known as quasi-BICs
or supercavity modes. Until now, optical BICs
have been observed only for extended systems
(12,14–16) and used for various applications
including lasing (17) and sensing (18). For
individual isolated dielectric resonators, gen-
uine nonradiative states require extreme mate-
rial parameters diverging toward infinity or zero
(19,20). In realistic individual resonators, there
are an infinite number of possible paths for
radiation to escape (12), which limits the Q
factor substantially. However, the concept of
quasi-BICs allows us to come close to reaching
nonradiative states for individual dielectric
resonators (21–23). The modes forming a quasi-
BIC belong to the same resonator, which allows
the system footprint to be kept very small.
Except for specific composite structures (24),
suchasmallfootprintischallengingtoachieve
for resonators relying on alternative mecha-
nisms of localization, including whispering
gallery mode resonators and cavities in pho-
tonic bandgap structures.
Here, we studied individual dielectric nano-
resonators hosting a quasi-BIC resonance at
telecommunication wavelengths and demon-
strated its capability for second-harmonic gen-
eration (SHG). Our subwavelength resonator
exploits mutual interference of several Mie
modes, which results in a quasi-BIC regime.
We designed a 635-nm-tall nonlinear nano-
resonator of cylindrical shape made of AlGaAs
(aluminum gallium arsenide) placed on an
engineered three-layer substrate (SiO
2
/ITO/
SiO
2
) (Fig. 1A). For a cylindrical particle, Mie
resonances are classified with an azimuthal
order and can be loosely sorted into two groups
distinguished by the number of oscillations in
the radial and axial directions [see part 1 of the
supplementary text (25)]. We selected a pair of
modes from different groups with uniform
azimuthal field distribution (Fig. 1B), both of
which demonstrated a magnetic dipolar behav-
ior [see parts 1 and 2 of the supplementary text
(25)]. By changing the resonator’sdiameter,the
spectral mismatch of dipolar modes can be
decreased, which induces their strong cou-
pling in the parametric space and produces
the characteristic avoided resonance cross-
ing of frequency curves (Fig. 1C). In the strong
coupling regime, the modes are hybrid with
a combination of radial or axial oscillations
and thus do not belong to any of the defined
groups [see part 1 of the supplementary text
(25)]. Open boundaries of the nanoresonator
enable constructive and destructive mode in-
terference in the far field (26), which results
in modification of the mode Q factors because
of their identical dipolar nature (Fig. 1D). The
quasi-BIC regime with suppressed dipolar ra-
diation (Fig. 1E) and thus an increased Q
factor is reached for a particle of a specific
diameter of ~930 nm.
Wefurthercompensatedforthedecreaseof
the Q factor induced by energy leakage into
the substrate (27)byaddingalayerofITO
(indium tin oxide) exhibiting an epsilon-near -
zero transition acting as a conductor above
a 1200-nm wavelength (e.g., at the quasi-BIC
wavelength) and as an insulator below this
wavelength [e.g., at the second harmonic (SH)
wavelength]. The ITO layer is separated from
the resonator by a SiO
2
spacer. The thickness of
the SiO
2
spacer layer provides control over the
phase of reflection, further enhancing the de-
structive interference of the two magnetic
dipoles in the far field and thus increasing
theQfactor(Fig.1F).Fortheoptimalspacer
thickness between 300 and 400 nm, the Q
factor reaches the maximal predicted value
of 235.
We fabricated a set of individual AlGaAs
nanoresonators with diameters varying from
890 to 980 nm from epitaxially grown AlGaAs
(crystal axes orientation [100], 20% Al) by
means of electron-beam lithography and a
dry-etching process. The nanoparticles were
subsequently transferred to a substrate made
of a commercial film of 300-nm ITO on glass
with an added SiO
2
spacer 350-nm thick [see
materials and methods and part 7 of the sup-
plementary text (25)]. We measured scatter-
ing spectra from individual nanoparticles with
a laser tunable within the wavelength range of
1500 to 1700 nm. To maximize light coupling
to the quasi-BIC mode, we illuminated each
nanoresonator with a tightly focused, azi-
muthally polarized light [see the materials
and methods and part 8 of the supplementary
text (25)]. The scattering spectra are evaluated
as the difference between the bare substrate
RESEARCH
Koshelev et al., Science 367, 288–292 (2020) 17 January 2020 1of5
1
Nonlinear Physics Center, Australian National University,
Canberra ACT 2601, Australia.
2
Department of Physics and
Engineering, ITMO University, St. Petersburg 197101, Russia.
3
Faculty of Physics, Lomonosov Moscow State University,
Moscow 119991, Russia.
4
Department of Physics, Korea
University, Seoul 02841, Republic of Korea.
*Corresponding author. Email: hgpark@korea.ac.kr (H.-G.P.);
ysk@internode.on.net (Y.K.)
on January 16, 2020 http://science.sciencemag.org/Downloaded from
reflectivity and the normalized measured back-
ward scattering of the nanoresonator. We ob-
served a symmetric peak with the extracted
Q factor of 188 ± 5 for the particle diameter
of ~930 nm, which corresponds to the quasi-
BIC condition [see Fig. 1G and the materials
and methods (25) for details on the Q-factor
extraction procedure]. We further measured
the dependence of the Q factor on the nano-
resonator diameter (dots in Fig. 1D), which
showed good agreement with numerical
simulations.
Next, we exploited the designed quasi-BIC
resonator as a nonlinear nanoantenna for SH
generation (Fig. 2A). At the SH wavelength,
the nanoresonator supports a high-order Mie
mode with a Q factor of 65 [see part 1 of the
supplementary text (25)]. For SH wavelengths,
the material properties of ITO are similar to
glass, so the spacer and ITO thickness are in-
essential. To increase the nonlinear conver-
sion efficiency, we developed the consistent
theory of SHG for nanoscale resonators using
the eigenmode expansion method [see parts 4
and 5 of the supplementary text (25)], which
goes beyond the phase-matching approach
used for nonlinear optics of macroscopic
structures (28).
The optical response of designed nonlinear
nanoantenna is driven by the quasi-BIC with
complex frequency w
1
–ig
1
and the SH Mie
mode with frequency w
2
–ig
2
. The total SH
power radiated by the nanoresonator [see
part 5 of the supplementary text (25)] is:
P2w¼ak
2Q2L2k12 ½Q1L1k1Pw2ð1Þ
This expression allows a step-by-step expla-
nation of the SHG process (Fig. 2B). The inci-
dent power P
w
is coupled to the quasi-BIC
depending on the spatial overlap k
1
between
the pump and the mode. The coupled power
is resonantly enhanced depending on the
quasi-BIC Q factor Q
1
anddampedbythespec-
tral overlap factor L1ðwÞ¼g2
1=½ðww1Þ2þ
g2
1, which is the unity at the resonance. The
efficiency of upconversion of the total ac-
cumulated power is determined by the cross-
coupling coefficient k
12
,whichdependsonthe
symmetry of the nonlinear susceptibility tensor
of AlGaAs and the spatial overlap between
the generated nonlinear polarization current
and SH mode [see part 5 of the supplemen-
tary text (25)]. The converted SH power is
increased by a high Q factor of the SH mode
but at the same time is decreased because of
the spectral mismatch with the quasi-BIC,
L2ð2w1Þ¼g2
2=½ð2w1w2Þ2þg2
2.Theout-
coupling factor k
2
(2w)determinesafraction
of the radiated SH power and is the unity in
Koshelev et al., Science 367, 288–292 (2020) 17 January 2020 2of5
Fig. 1. Optical quasi-BIC mode in an individual dielectric nanoresonator.
(A) Scanning electron micrograph (top) and schematic (bottom) of an individual
dielectric nanoresonator. (B) Simulated near-field patterns of the two modes
for different diameters. (C) Calculated mode wavelengths versus resonator
diameter. (D) Calculated (lines) and measured (dots) Q factors of modes versus
resonator diameter. Calculations in (C) and (D) are done for a 350-nm SiO
2
spacer. (E) Simulated far-field patterns of the high-Q mode for disks of different
diameters shown schematically. For calculations, |E|
2
is normalized to the full
mode energy. (F) Calculated Q factor of the quasi-BIC versus SiO
2
spacer
thickness compared with the Q factors of a nanoresonator in air and on a bulk
SiO
2
substrate (dashed lines). (G) Measured scattering spectrum and retrieved
Q factor of the observed resonance for a disk with a diameter of ~930 nm.
RESEARCH |REPORT
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the vicinity of w
2
. The exact expressions for
coupling coefficients k
1
,k
12
,andk
2
and the
constant aare given in part 5 of the supple-
mentary text (25). Note that the effective mode
volume does not appear in Eq. 1 because k
12
takes into account the explicit spatial dis-
tributions of the electric field of the modes.
With this theoretical analysis, we can specify
the optimal conditions to maximize the SHG
efficiency from an individual dielectric nano-
resonator. First, the spatial profile of the pump
must be structured to match the distribution
of the excited mode; therefore, we used the
cylindrical vector beam with azimuthal po-
larization. We estimated k
1
as 33% for the
experimental conditions using a model of a
free-standing resonator in air (Fig. 2C) [see
parts 5 and 10 of the supplementary text (25)].
Next, the optimal structure must be resonant
simultaneously at pump and SH wavelengths
(9). Maximization of Q
1
is critical compared
with maximization of Q
2
because of the quad-
ratic over linear dependence of P
2w
[see Eq. (1)
and part 6 of the supplementary text (25)].
For the designed nanoresonator with a diam-
eter of ~930 nm, the factor of spectral overlap
reaches50%(Fig.2D).Finally,thecollec-
tion efficiency must be increased, which
can be achieved by engineering the substrate
properties. The epsilon-near-zero transition
of ITO makes it effectively “invisible”to the
SH radiation, allowing it to propagate in
both the forward and backward directions
(Fig. 2E).
To perform systematic experimental anal-
ysis of the SHG enhancement in quasi-BIC
resonators, we excited the fabricated set of
nanoparticles with laser pulses of 2-ps dura-
tion in the wavelength range from 1500 to
1700 nm [see the materials and methods and
part 9 of the supplementary text (25)]. Figure 3,
AtoD,showsthemapsoftheSHGintensity
versus the pump wavelength and resonator
diameter for the nanoresonators pumped by
the azimuthal, radial, and linearly polarized
beams, respectively. The experimental data
reveal a sharp enhancement of the nonlinear
signal in the quasi-BIC regime selectively for
the azimuthally polarized pump. We measured
directionality diagrams of the SH signal in the
backward and forward directions within the
numerical apertures of a pair of confocal ob-
jectivelenses[seeFig.3,EandF,andpart
12 of the supplementary text (25)]. The dia-
gram in the backward direction features dis-
tinct maxima in fo ur directions that ar e
qualitatively similar to the theoretical SHG
directionality shown in Fig. 2A and the far-field
pattern of the mode excited at the SH wave-
length [see part 1 of the supplementary text (25)].
Figure 4, A and B, shows a wavelength cut
(at the quasi-BIC diameter of ~930 nm) and a
size cut (at the quasi-BIC wavelength of 1570 nm)
of the measured 2D SHG maps (see Fig. 3, B to
D). Both plots demonstrate that the observed
SH intensity for the azimuthal pump sur-
passes the SH intensity for the other polar-
izations by several orders of magnitude, which
confirms high spatial selectivity of the quasi-
BIC [see also part 3 of the supplementary
text (25)]. With these experiments, we
reached beyond the predictions of the theo-
retical model (see Fig. 2C) and measured an
observable SH signal for radial and linear
polarizations caused by off-resonant excita-
tion of other nanoparticle modes (Fig. 4B).
However, this signal remains several orders
of magnitude lower compared with azimuthal
polarization. We further experimentally mea-
sured the SHG conversion efficiency. The nu-
merical analysis of quasi-BICs in a nonlinear
nanoresonator [see Fig. 2 and (23)] does not
account for the trade-off between pulse dura-
tion and laser damage threshold. The high Q
factor of the quasi-BIC requires relatively long
pu lses to pump the mode effectively. At the same
time,apeakpoweroflongerpulsesbecomes
limited by the material laser damage thresh-
old. From this point of view, theoret ica l or
numerical analysis does not answer the ques-
tion of whether a nanoresonator made of com-
mon dielectric materials can indeed function
as an efficient nonlinear nanoantenna.
We conducted an experimental verification
of this by detecting the peak pump power Pw
p
incident onto the sample and the peak SH
power P2w
pcaptured by the two objective lenses
in the forward and backward directions (Fig.
4C). The directly measured conversion efficien-
cy P2w
p=ðPw
pÞ2was 1.3 × 10
−6
W
−1
[see part 11 of
the supplementary text (25)]. The observed
SHG efficiency at the quasi-BIC was more than
two orders of magnitude higher than that
demonstrated with earlier implementations
using other approaches (5–7,9). We further
estimate the total SHG efficiency as 4.8 × 10
−5
W
−1
using the common approach by taking
into account only the coupled part of Pw
p,theo-
retically estimated as 33%, and the total SH
power, estimated using the calculated collec-
tion efficiency of 24%. A detailed list of the
Koshelev et al., Science 367, 288–292 (2020) 17 January 2020 3of5
Fig. 2. Second-harmonic generation with a dielectric nanoantenna.
(A) Diagram of the SHG in a nanoresonator under azimuthally polarized vector beam
excitation. (B) Schematic of the SHG process in a nonlinear dielectric nanoantenna.
Each term of the formula describes one step of the process. (C) Percentage of pump
power coupled to the quasi-BIC for different polarizations of pump depending on the
ratio between the beam waist radius w
0
and the pump wavelength. The calculation is
done for a free-standing nanoresonator in air. The diffraction limit is 0.61. (D)Spectral
overlap L
2
(2w
1
) between the high-Q mode at the pump frequency and the high-order
Mie mode at the SH frequency versus the disk diameter. The inset shows the near-field
profiles of both modes. (E) Experimental ellipsometry data for the permittivity of the
ITO layer. Wavelength ranges of the excitation and collection are marked with red and
blue shading, respectively.
RESEARCH |REPORT
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experimental parameters and an elaborated
comparison with the earlier results for in-
dividual nanoresonators is presented in part
13 of the supplementary text (25). The SHG
efficiency of an individual nanoantenna dem-
onstrated here is qualitatively comparable
to the best-to-date efficiencies of nonlinear
metasurfaces (29,30) based on hybrid multiple-
quantum-well structures, whereas a quantitative
comparison cannot be done without some
ambiguity. Although high nonlinear coeffi-
cients P2w
p=ðPw
pÞ2were demonstrated in such
systems in the far- to mid-infrared spectral
ranges, the reported conversion efficiencies
P2w
p=Pw
pof 2 × 10
–4
%(29)and7.5×10
–2
%
(30), respectively, remain low and are lim-
ited by a peak pump power of 100 mW that
they can sustain, compared with 10 W for our
nanoresonator.
Our results illustrate, for the first time to our
knowledge, manifestation of high Q-factor op-
tical modes in individual nanoresonators in
the linear and nonlinear regimes governed by
the physics of bound states in the continuum.
Koshelev et al., Science 367, 288–292 (2020) 17 January 2020 4of5
Fig. 3. Experimental characterization of the SHG enhancement. (A) 3D map of SH intensity measured as a function of the pump wavelength and particle diameter
for an azimuthally polarized beam. The SH intensity is normalized on the square of the pump power. (Bto D) Top views of the maps of SHG with the azimuthal,
radial, and linear pump, respectively. (Eand F) Experimentally measured directionality diagrams of SHG for a nanoresonator with a diameter of ~930 nm in the
(E) backward and (F) forward direction.
Fig. 4. Experimental nonlinear conversion efficiency. (Aand B) Measured SH intensity as a function of the pump wavelength for the nanoresonator with
the diameter of ~930 nm (A) and as a function of the nanoresonator diameter at a 1570-nm pump wavelength (B) for different pump polarizations. The SH intensity
is normalized on the square of the pump power. (C) Measured peak SH power versus the peak pump power for a nanoresonator with a diameter of ~930 nm
(log
10
–log
10
scale). Line shows the fit with a quadratic dependence with the nonlinear conversion coefficient 1.3 × 10
−6
W
−1
.
RESEARCH |REPORT
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Our experiments demonstrate that quasi-BIC
engineering for individual nanoparticles in
the optical frequency range is feasible despite
fabrication tolerances and material absorp-
tion. Individual high-Q nanoresonators with
a subwavelength footprint promise specific
applications as nonlinear nanoantennas, low-
threshold nanolasers, and compact quantum
sources.
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ACKNOW LEDGM ENTS
We thank K. Ladutenko, B. Luther-Davies, D. Smirnova, and
L. Wang for their valuable inputs into this project at various
stages of its development. Funding: This work was supported
bytheAustralianResearchCouncil, the Strategic Fund of the
Australian National University, the National Research Foundation
of Korea (NRF) under grant no. 2018R1A3A3000666 funded
by the Korean Government (MSIT), and the Russian Science
Foundation under grant no. 18-72-10140. K.K. and A.B.
acknowledge support from the Foundation for the Advancement
of Theoretical Physics and Mathematics “BASIS.”Author
contributions: K.K., S.K., and Y.K. conceived the research;
K.K. and A.B. performed theoretical analysis, numerical
simulations, and data analysis; J.-H.C. and H.-G.P. fabricated
the samples; S.K. and E.M.-G. conducted experimental studies;
K.K., S.K., and Y.K. wrote the manuscript based on input
from all authors. Competing interests: The authors declare
no competing interests. Data and materials availability:
All data needed to evaluate the conclusions in this paper are
available in the main text or the supplementary materials.
SUPPLEMENTARY MATERIALS
science.sciencemag.org/content/367/6475/288/suppl/DC1
Materials and Methods
Supplementary Text
Figs. S1 to S15
Table S1
References (31–43)
5 September 2019; accepted 27 November 2019
10.1126/science.aaz3985
Koshelev et al., Science 367, 288–292 (2020) 17 January 2020 5of5
RESEARCH |REPORT
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science.sciencemag.org/content/367/6475/288/suppl/DC1
Supplementary Materials for
Subwavelength dielectric resonators for nonlinear nanophotonics
Kirill Koshelev, Sergey Kruk, Elizaveta Melik-Gaykazyan, Jae-Hyuck Choi, Andrey Bogdanov,
Hong-Gyu Park*, Yuri Kivshar*
*Corresponding author. Email: hgpark@korea.ac.kr (H.-G.P.); ysk@internode.on.net (Y.K.)
Published 17 January 2020, Science 367, 288 (2020)
DOI: 10.1126/science.aaz3985
This PDF file includes:
Materials and Methods
Supplementary Text
Figs. S1 to S15
Table S1
References
CONTENTS
Materials and Methods 3
Supplementary Text 6
Part 1. Mode analysis for an isolated dielectric cylindrical nanoresonator 6
Part 2. Linear simulations and multipolar decomposition 8
Part 3. Nonlinear simulations and multipolar decomposition 9
Part 4. Eigenmode expansion method for open optical resonators 9
Part 5. Derivation of Equation 1 for the second harmonic power 10
Part 6. Comparison of theory and experiment 12
Part 7. Sample fabrication 13
Part 8. Linear spectroscopy 13
Part 9. Experimental setup for nonlinear spectroscopy 13
Part 10. Knife-edge experiment 14
Part 11. Peak power dependence of the measured second-harmonic signal 14
Part 12. Directivity diagram of the second harmonic 14
Part 13. Comparison of SHG efficiencies from nanoscale dielectric and plasmonic resonators 14
Figures 16
Tables 31
3
MATERIALS AND METHODS
Numerical calculations. For numerical simulations, shown in Figs. 1 and 2 of the main text,
we use the finite-element-method eigenvalue solver in COMSOL Multiphysics. All calculations
are realized for a single nanoresonator of a specific size (height of 635 nm) on a semi-infinite
structured substrate surrounded by a perfectly matched layer mimicking an infinite region. All
material properties including losses are imported from the tabulated data for AlGaAs (20% Al) [for
the visible range see (31) and for the near-IR range see (32)] and SiO2[see (33)] and extracted from
the experimental ellipsometry data for the ITO layer. The difference in simulated and measured
resonant peak positions can be explained by fabrication imperfections such as inclination of the
disk walls, which are not taken into account in the simulations. Since the actual percentage of
Al in the deposited AlGaAs compound is determined with a certain accuracy, the refractive index
of the sample can be slightly different from the tabulated data used for the simulations. The
measured Q factor is extracted from the experimental scattering spectrum using the single-peak
fitting to the generalized Fano lineshape (22) implemented into Python programming language via
the LevenbergMarquardt algorithm with custom multi-step re-evaluation of initial conditions and
extraction of resonance related data from the background. The error is directly extracted from the
fitting procedure and explained by the mismatch between the shape of the peaks in the measured
spectrum and the generalized Fano lineshape. More details on the numerical calculations can be
found in Part 2 and Part 3 of the Supplementary text.
For quantitative characterization of the disk eigenmodes, we employ the mode decomposition
method over the irreducible spherical multipoles (34), characterized by the orbital (l= 1,2, . . .)
and azimuthal (m= 0,±1,...,±|l|) indices defined with respect to the disk axis. The decompo-
sition is realized as a custom built-in routine for the eigenmode solver in COMSOL Multiphysics.
For multipole classification, we use the following notations: electric/magnetic dipole (ED/MD,
l= 1), electric/magnetic quadrupole (EQ/MQ, l= 2), electric/magnetic octupole (EO/MO, l= 3),
electric/magnetic hexadecapole (EH/MH, l= 4). The multipolar decomposition for eigenmodes
of open resonators is approximate due to divergence of field amplitudes in the far-field zone. De-
spite that, it still gives good quantitative results for the modes with a Q factor more than 10. The
presence of a substrate breaks the spherical symmetry and induces interference between different
multipoles, which leads to some error in the results of the standard multipolar decomposition (34).
If the optical dielectric contrast between the nanoresonator and the substrate is high (>2), the
4
error of the multipolar decomposition procedure is of order of a dozen percent for low-order Mie
modes (35).
Theoretical methods. A detailed description of the derivation of Eq. (1) of the main text is
provided in Part 4 and Part 5 of the Supplementary text.
Sample fabrication. We epitaxially grow AlGaAs (20% Al) film on a buffer AlInP layer on
top of a [100] GaAs substrate. The thickness of the AlGaAs film is about 635 nm. We de-
posit a SiO2spacer on the ITO/SiO2substrate by plasma-enhanced chemical vapour deposition.
The thickness of the SiO2spacer is about 350 nm. The thickness of ITO layer is 300 nm. The
AlGaAs disk resonators are fabricated with an electron-beam lithography with chemically as-
sisted ion-beam etching. The AlInP buffer layer is wet etched by diluted HCl solution. Then,
the AlGaAs disks are attached to a polypropylene carbonate (PPC)-coated polydimethylsiloxane
(PDMS) stamp. The AlGaAs resonators are transferred to the SiO2/ITO/SiO2substrate from the
GaAs wafer. Finally, the AlGaAs disks are detached from the PDMS stamp by applying heat to
the thermal adhesive PPC layer. More details on the fabrication procedure can be found in Part
7of the Supplementary text. The dimensions of the fabricated nanoresonators are determined by
scanning electron microscopy (FEI Verios) images. The horizontal error bars in Fig. 1D of the
main text is due to the error of order of 5 nm in determination of disk diameters.
Optical experiments. In order to conduct both linear and nonlinear optical measurements, a
spectroscopy experimental setup based on the tunable optical parametric amplifier (MIROPA-fs-M
from Hotlight Systems) that generates 2ps duration pulses at a repetition rate of 5.144 MHz and
pumped by a pulsed laser 1030 nm (FemtoLux3 by Ekspla) is constructed. The considered spectral
range of the pump is from 1500 to 1700 nm. To create vector beams (36) over the pump spectral
range, we use two custom q-plates optimized for 1570 nm and 1640 nm wavelengths (Thorlabs).
A train of wavelength tunable laser pulses is focused from the air side of the sample mounted on
a three-dimensional stage by a Mitutoyo MPlanApo NIR objective lens, ×100 infrared, 0.70 NA.
For linear optical characterization of the resonators, we measure the ratio between the spectra of
the power reflected from a single nanoparticle and from the multilayered substrate. The reflected
signal at the fundamental wavelength is collected by the same objective lens with 0.70 NA, and,
after passing through a non-polarizing beam splitter cube BS015 by Thorlabs, it is detected by
an Ophir power meter head PD300-IR. Part 8 of the Supplementary contains the corresponding
experimental setup schematic.
For nonlinear optical measurements, we pump the resonators by azimuthally, radially, and lin-
5
early polarized laser beams (36) to experimentally verify the selective spatial coupling to the mode
at the fundamental wavelength. The pump spectrum is cleared up by a long-pass infrared filter
(Thorlabs FELH1300). The pump beam is observed by a near-infrared InGaAs camera Xenics
Bobcat-320 with a 150 mm focal distance achromatic doublet (AC508-150-CML). The polariza-
tion type of a pump beam is checked by full polarimetry performed with a combination of an
achromatic quarter wave-plate and broadband wire-grid polarizer Thorlabs AQWP05M-1600 and
WP25M-UB, respectively. For the linearly polarized excitation, the polarization angle with respect
to the AlGaAs crystalline axes is chosen to provide the highest second-harmonic (SH) signal. The
forward SH signal is collected by an Olympus objective lens MPlanFL N (×100 visible, 0.90 NA)
and detected by a visible cooled CCD camera (Starlight Xpress Ltd, Trius-SX694) with a 150
mm focal distance achromatic doublet. The detected signal is filtered out by a set of filters (a
coloured glass bandpass filter Thorlabs FGB25 and UV fused silica filter with dielectric coating
FELH0650). The focusing objective lens with 0.70 NA also collects the nonlinear optical signal
in the backward direction. The incident pump and the backward generated SH are separated by
a45◦dichroic mirror DMLP950R by Thorlabs. Detailed experimental setup schematics can be
found in Part 9 of the Supplementary. We measure the diameter of the azimuthally, radially, and
linearly polarized focused pump laser beam by performing knife-edge experiments which results
are presented in Part 10. The beam waist radius is 1.8µm for the azimuthal pump while the fabri-
cated nanoparticles are isolated from each other by spacing of 10 µm. The SH signal is normalized
over a spectral function of the setup which includes filter transmittance, laser power, and detector
sensitivity spectra. The origin of the SH signal is verified by the direct measurement of its spec-
trum (by a visible spectrometer Ocean Optics QE Pro) and its power dependence which is in a
quadratic manner what Part 11 demonstrates. The directionality diagrams of the SH are measured
by recording the back-focal plane images of the two objective lenses with an additional lens (75
mm focal distance achromatic doublet). Polarization states of the SH over the directionality dia-
gram are retrieved with Stokes vector formalism by employing an achromatic quarter-wave plate
and a polarizer, see Part 12 of the Supplementary text.
6
SUPPLEMENTARY TEXT
Part 1. MODE ANALYSIS FOR AN ISOLATED DIELECTRIC CYLINDRICAL
NANORESONATOR
In this section we explain the mechanism of formation of quasi-BICs in an individual cylin-
drical nanoresonator. The eigenmodes of a disk resonator in the free space can be rigorously
classified according to their azimuthal order (m= 0,±1,±2, . . .) with respect to the disk axis
and their parity with respect to the up-down inversion along the disk axis (p= 0,1). Away from
the avoided resonance crossings (ARCs) of the frequency curves, each mode of the cylindrical
nanoresonator can be also sorted in one of two groups distinguished by the number of oscillations
in the radial and axial directions. This sorting is not rigorous, but it is qualitatively justified and
is useful to categorize the modes. The justification can be done by considering the extreme case
of an infinitely long cylinder, where radial modes represent the well-known Mie modes (37) while
axial modes appear only for a finite length of the cylinder.
We select a pair of modes from different groups (one radial and one axial) with the same m
and p. With the change of the cylinder aspect ratio their frequencies change differently because
of their distinct nature. In the vicinity of the specific aspect ratios, the mode dispersion curves
tend to cross. However, the nanoresonator has open boundaries which makes it a non-Hermitian
electromagnetic system and allow so-called internal and external coupling between its modes [see
Section VI in (38)]. In the strong coupling regime, the mode dispersion curves exhibit the ARC
behavior and the mode radiative losses change due to the external coupling. Near the ARC the
modes are hybrid with a combination of radial or axial oscillations and thus do not belong to any
of the defined groups. Here, we also note that modes with different mor pdo not interact and their
frequency curves always cross.
The proper quantitative description of the mode hybridization can be achieved by analyzing the
eigenfunction evolution in the vicinity of the ARC of two modes [see Fig. S1A]. We denote the
hybrid eigenfunctions of the cylinder resonator as ϕ1,2, they are given by the linear combination
of the eigenfunctions of the uncoupled radial and axial modes ϕa,b, respectively
ϕ1,2=Ca
1,2ϕa+Cb
1,2ϕb.(S1)
7
The complex coefficients Ca,b
1,2can be determined by solving the 2x2 matrix equation (22)
ωa−iγa0
0ωb−iγb
Ca
1,2
Cb
1,2
=ω1,2
1 + Vaa Vab
Vba 1 + Vbb
Ca
1,2
Cb
1,2
.(S2)
Here, Vij describes the coupling between radial and axial modes in the parameter space, which
is due to the change of the resonator aspect ratio (the disk diameter for the fixed height). If
|Ca
1,2||Cb
1,2|or |Ca
1,2| |Cb
1,2|, then the modes can be considered as dominantly radial or axial,
otherwise, for |Ca
1,2|'|Cb
1,2|both modes are hybrid. The values Vij can be extracted from the
calculated dispersion ω1,2, by using the two-level model approximation (22).
To simplify the analysis of mode hybridization, it is also possible to compare the mode polar-
ization ratio. For nanoresonator modes with fixed m6= 0 and fixed p, the radial modes possess one
dominant polarization (TE or TM), while the axial modes always have another polarization. Then,
the criterion of mode hybridization is the degree of TE or TM polarization. For nanoresonator
modes with fixed m= 0, such as the modes studied in the main text, the polarization is pure. For
pure TE modes, as in the paper, which have only three nonzero field components Eϕ,Hz,Hrin
the cylindrical coordinate frame, the criterion of mode hybridization is the ratio of two nonzero
magnetic field components, e.g. |Hz|/|Hr|, as shown in Fig. S1B. At the same time, the absolute
value of the electric field component |Eϕ|is almost constant in the vicinity of the ARC.
To compare, how quasi-BIC modes form with and without a substrate, we consider m= 0 and
two different cases: (i) the experimental design – a disk of 635 nm height on top of a three-layer
substrate (SiO2/ITO/SiO2) with a 300 nm ITO layer and a 350 nm SiO2spacer and (ii) the same
disk suspended in air without a substrate. As shown in Figs. S2A-D, for both cases strong coupling
between the modes produces the characteristic ARC of frequency curves and modification of mode
radiative Q factors. Figure S2G shows the transformation of the far-field patterns while passing the
ARC. Away from the ARC each mode represents a MD aligned with the disk axis. At the quasi-
BIC regime the dipolar contribution to the radiation of the high-Q mode is suppressed because of
destructive interference between two MDs, and the radiation pattern changes to a MO. Figures S2E
and F show the multipolar decomposition of the radiated power for the high-Q mode. It can be
seen, that in the vicinity of the quasi-BIC the MD contribution is strongly suppressed and the MO
contribution dominates which is in agreement with the evolution of the far-field patterns shown in
8
Fig. S2G. We also note that in the presence of the substrate, pis not a mode index any more and
can be used only approximately, while the modes with different pcan be hybridized in the vicinity
of ARCs.
Due to the structure of the second-order susceptibility tensor of AlGaAs, the nonlinear polar-
ization induced by the modes with m= 0 can excite only the resonator modes with |m|= 2 (for
details see Part 5). At the second-harmonic wavelengths, in the vicinity of the doubled frequency
of the quasi-BIC the nanoresonator supports a single high-order Mie mode with |m|= 2 which
is a linear combination of two degenerate modes with m= 2 and m=−2. Figure S3 shows the
mode dispersion (panel A), the Q factor evolution (panel B) and the electric field profiles for the
second-harmonic (SH) mode (panel C) in the near- and far-field zones. At the quasi-BIC regime,
the SH mode has a Q factor of about 65. The mode far-field profile is highly-symmetric which
corresponds to the dominant MH component.
Part 2. LINEAR SIMULATIONS AND MULTIPOLAR DECOMPOSITION
For numerical simulations of the linear spectrum we use the finite-element-method solver in
COMSOL Multiphysics in the frequency domain. To compare the simulations with the measured
spectra, we consider an AlGaAs disk on top of a three-layer substrate (SiO2/ITO/SiO2) with a
350 nm SiO2spacer, 300 nm ITO layer and the azimuthally polarized pump (36). To obtain the
exact expression for the background field for scattering simulations, we derive the angular spec-
trum representation (39) for the azimuthal cylindrical vector beam in each layer of the multilayered
structure and match the solutions at the boundaries between the layers.
We calculate the reflected power coming through 0.7NA aperture in the backward direction and
normalise it on the reflectance of the multilayered substrate collected in the same aperture. The
comparison between the measured and simulated reflectance is shown in Fig. S4. We observe that
the dip in the simulated spectrum is red-shifted by 28 nm with respect to the dip observed in the
experiment. To understand the nature of the resonant dip in the reflectivity spectrum, we perform
the multipolar decomposition of the total scattered power in the full solid angle shown in Fig. S5.
It demonstrates that the quasi-BIC is determined by the magnetic dipolar and octupolar patterns in
accordance to the results of the eigenmode simulations (see Fig. S2E).
9
Part 3. NONLINEAR SIMULATIONS AND MULTIPOLAR DECOMPOSITION
For numerical simulations of the nonlinear response at the second-harmonic wavelength, we
employ the approach based on the undepleted pump approximation (40) which is justified because
the pump power is below the saturation threshold of the resonator. We use two steps to calculate
the intensity of the radiated nonlinear signal. Using the simulated field amplitudes at the pump
wavelength, we obtain the nonlinear polarization induced inside the disk. Then, we employ the
polarization as a source for the electromagnetic simulation at the harmonic wavelength to ob-
tain the generated second-harmonic field. The nonlinear susceptibility tensor corresponds to the
zincblende crystalline structure with χ(2)
xyz = 290 pm/V (40).
We calculate the sum of the radiated second harmonic intensity coming through 0.7NA aperture
in the backward direction and 0.9NA aperture in the forward direction. A comparison between the
measured and simulated second harmonic intensity is shown in Fig. S6.
Part 4. EIGENMODE EXPANSION METHOD FOR OPEN OPTICAL RESONATORS
In this section we present the basics of the eigenmode expansion method for non-Hermitian
optical systems recently developed for confined optical resonators of arbitrary shape (41). We
study an open optical resonator, which possesses the properties of a non-Hermitian system and its
modes are inherently leaky. The main idea is to expand the resonator’s Green’s function ˆ
Gof the
resonator into a series of its eigenfunctions Ej(r)
ˆ
G(ω, r,r0) = X
j
c2Ej(r)⊗Ej(r0)
2Njω(ω−ωj+iγj).(S3)
Here, ωjand γjare the real and imaginary parts of the eigenfrequencies, respectively, and Njis
the normalisation constant for Ej. The expansion holds only for r,r0inside the resonator, where
the permittivity and permeability are different from the background values. The eigenfunctions
are diverging in space for large arguments |r| λ, which is the result of their leaky nature. The
main difficulty of such approach is the accurate and correct normalisation of eigenmodes, which
was proposed only recently (41)
Nj=ZV
dV εEj·Ej+c2
2(ωj−iγj)2IS
dS "Ej·∂
∂r r∂Ej
∂r −r∂Ej
∂r 2#,(S4)
where both the volume and surface integration goes over the spherical shell Slocated in the far
field zone.
10
Using Eq. (S3) for the Green’s function it is possible to find the solution of the Maxwell’s
equations for any given source. The eigenfunctions can be found numerically using full-wave
computational packages, or semi-analytically, using the resonant state expansion method (41).
Part 5. DERIVATION OF EQUATION 1 FOR THE SECOND HARMONIC POWER
We consider a subwavelength AlGaAs disk on a substrate. We neglect the absorption losses,
since the material Q factor Re[ε]/Im[ε]>103within the range of wavelengths from 780 to
1670 nm. Also we neglect the losses due to surface roughness are low because of high quality
of fabrication. The disk is excited by the pump with a given distribution of electric field Ebg.
The total electric field E(r)in each point of the space can be divided into the background and the
scattered as
E(ω, r) = Esc(ω, r) + Ebg (ω, r).(S5)
The scattered field can be found using the Green’s function (42)
Esc(ω, r) = −ω2
c2Zdr0∆(ω, r0)ˆ
G(ω, r,r0)·Ebg (ω, r0),(S6)
where ∆(ω, r) = (ω, r)−bg(ω, r).
The scattered field can be rigorously expanded into a series of the resonator’s eigenmodes.
We consider the case when only one mode E1is resonantly excited in the vicinity of the pump
frequency
Esc(ω, r) = a(ω)E1(r).(S7)
Using the expansion in Eq. (S3) we find the resonant amplitude aat the pump wavelength
a(ω) = −ω
2N1(ω−ω1+iγ1)Zdr0∆(ω, r0)E1(r0)·Ebg (ω, r0).(S8)
The energy accumulated inside the resonator W(ω)is proportional to |a|2, which can be re-written
as
W(ω)∝ |a(ω)|2=c
N1ω1
Q1L1(ω)κ1(ω)P0
0(ω).(S9)
Here, Qj=ωj/2γjis the mode quality factor, the spectral overlap factor Lj(ω)is
Lj(ω) = γ2
j
(ω−ωj)2+γ2
j
,(S10)
11
and the coupling coefficient κ1is
κ1(ω) = (ω/c)Rdr0∆(ω, r0)E1(r0)·Ebg(ω, r0)
2
(2γ1/c)N1P0(ω).(S11)
The coefficient P0(ω)is proportional to the total incident power P(ω)
P0(ω) = 8π
cP(ω).(S12)
To analyse the resonator response at the SH wavelength we calculate the nonlinear SH polarization
PNL
PNL
i(2ω) = X
j,k
χ(2)
ijk Ej(ω)Ek(ω),(S13)
where χ(2)
ijk is the second-order susceptibility tensor. For AlGaAs χ(2) has the symmetry of the
zincblende crystalline structure. We assume the resonant conditions when the amplitude a(ω)is
large, so E(ω, r)'Esc(ω, r)and
PNL
i(2ω)=[a(ω)]2X
j,k
χ(2)
ijk E1,j E1,k .(S14)
The induced field E(2ω)at the SH wavelength can be found using the resonator Green’s function
similar to Eq. (S6)
E(2ω, r) = −(2ω)2
c2Zdr0ˆ
G(2ω, r,r0)·PNL(2ω, r0).(S15)
We assume that E(2ω)is dominated by a single resonant state E2with frequency ω2lying in the
vicinity of 2ω
E(2ω, r) = b(2ω)E2(r).(S16)
Thus, the amplitude bcan be found as
b(2ω) = −2ω
2N2(2ω−ω2+iγ2)Zdr E2(r)·PNL(2ω, r) =
=−2ω[a(ω)]2
2N2(2ω−ω2+iγ2)X
i,j,k
χ(2)
ijk ZdrE2,iE1,j E1,k.(S17)
The energy of the SH field is proportional to |b(2ω)|2
|b(2ω)|2=(2ω/c)2
(2γ2/c)N2
Q2L2(2ω)κ12 hω1
cN1|a(ω)|2i2,(S18)
12
where the cross-coupling coefficient κ12 is
κ12 =Pi,j,k χ(2)
ijk RdrE2,i(r)E1,j (r)E1,k(r)
2
(N2ω2/c)(N1ω1/c)2.(S19)
The total SH power
P(2ω) = c
8πIS
dS·Re [E(2ω)×H∗(2ω)] .(S20)
Here, the integral is evaluated at the disk surface, where Eq. (S16) is valid.
Finally, combining Equations (S9), (S16), (S18) and (S20), we get the expression for the total
SH power
P(2ω) = α(2ω)κ2Q2L2(2ω)κ12 [Q1L1(ω)κ1(ω)P(ω)]2.(S21)
Here, the decoupling coefficient κ2is
κ2=HSdS·Re [E2×H∗
2]
(2γ2/c)N2
,(S22)
and the smooth envelope coefficient αis
α(2ω) = 8π
c2ω
c2
.(S23)
We note, that the effective mode volume does not appear in the resulting expression for the SH
power Eq. (S21). The reason is that the effective mode volume approach is used to approximate
a mode with some nontrivial spatial distribution as a mode with uniform pattern which takes the
effective mode volume. Here, we use the explicit expressions for the mode overlap integrals [see
Eq. (S19)], which take into account the actual spatial distribution of the mode electric fields.
We also note that for the modes of the cylindrical AlGaAs nanoresonator with m= 0 excited
by the azimuthally polarized pump, Eq. (S19) shows that only eigenmodes with |m|= 2 can be
excited at SH wavelengths. This simplifies the search of SH eigenmodes presented in Part 1.
Part 6. COMPARISON OF THEORY AND EXPERIMENT
In this section we compare the measured SH power spectrum with the evaluated dependence
via Eq. (S21). The main parameters of Eq. (S21) are the spectral overlap coefficient L2(2ω1)(see
Fig. S7A) and the combination of the mode Q factors Q2Q2
1(see Fig. S7B). Figures S7C and D
show the comparison between calculated L2Q2Q2
1and the measured SH power P(2ω)/[P(ω)]2,
respectively. The mismatch for the high values of the disk diameter is because the objective col-
lection efficiency and the dependence of other coefficients in Eq. (S21) on the disk diameter were
neglected.
13
Part 7. SAMPLE FABRICATION
The fabrication procedure is shown in Fig. S8. The polymethyl methacrylate (PMMA) mask is
defined on top of the AlGaAs/AlInP/GaAs wafer structure using electron-beam lithography. The
vertical pillar structure is fabricated by chemically-assisted ion beam etching. The PMMA mask is
removed by O2plasma. The AlInP buffer layer is selectively wet-etched by diluted HCl solution.
Then, the AlGaAs nanodisks are put on the GaAs substrate. The AlGaAs nanodisks are picked
up from the GaAs substrate using a polypropylene carbonate (PPC)-coated polydimenthylsilox-
ane (PDMS) stamp. The nanodisks are dropped down to a target substrate such as glass or ITO
substrates by applying heat (up to 90◦C) to the thermal adhesive PPC layer. The PPC layer is then
removed thoroughly by acetone.
Part 8. LINEAR SPECTROSCOPY
The schematic of the experimental setup for linear spectroscopy by using pulsed laser radiation
generated by a tunable optical parametric amplifier (MIROPA-fs-M from Hotlight Systems) is
presented in Fig. S9.
Representative raw data of the reflected power spectra are shown in Fig. S10.
Part 9. EXPERIMENTAL SETUP FOR NONLINEAR SPECTROSCOPY
The schematics of the experimental setups for SH generation spectroscopy in the forward and
backward directions are presented in Fig. S11.
The representative raw data detected by the camera for the SH signal from a subwavelength
resonator pumped by an azimuthally polarized vector laser beam is shown in Fig. S12. These
presented raw camera counts are not normalized over the spectra of optical elements transmission,
detector sensitivity, or pump power values. A spectrum of average pump power values measured
after a focusing objective lens is also shown in Fig. S12. The pump power variations are taken into
account during post-processing by dividing the integrated number of camera counts over the square
of a pump power value. In order to recalculate the detector counts to the second harmonic optical
signal power values the data of quantum efficiency for the Trius-SX694 CCD camera (Starlight
Xpress Ltd) is used. The calibration coefficient is also estimated by the use of a power meter and
CW-laser.
14
Part 10. KNIFE-EDGE EXPERIMENT
The geometrical parameters of cylindrical vector pump beams are measured using the knife-
edge method (43). The experimental results at the pump wavelength of 1640 nm for both linearly
and azimuthally polarized pump beams are shown in Fig. S13. The considered parameter of the
linearly polarized beam is the radius windicated by the level of 1/e2of the peak value from the
data derivative approximated by a Gaussian profile. The parameter w0defining an azimuthally
polarized beam is estimated as the distance between two peaks of a derivative of beam profiling
data because the derivative as a function of knife-edge position xis proportional to the expression
(1 + 4x2/w2
0)e−2x2/w2
0.
Part 11. PEAK POWER DEPENDENCE OF THE MEASURED SECOND-HARMONIC SIGNAL
Using an attenuator (a set of polarizing optical elements) we are able to conduct measurements
of the SH response with respect to the pump power. The experimental results are presented in
Fig. S14 for the dependence of the SH peak power P2ω
pon the pump peak power Pω
p(see Table S1
for details). As the considered particle is pumped, the resonant nonlinear response is blue-shifted.
Part 12. DIRECTIVITY DIAGRAM OF THE SECOND HARMONIC
By adding an additional lens we obtain back-focal plane images of the SH signal. An achro-
matic quarter-wave plate and a wire-grid polarizer are implemented into the experimental setup to
retrieve polarization states of the nonlinear response on the base of Stokes vector formalism. The
obtained directivity diagrams are presented in Fig. S15.
Part 13. COMPARISON OF SHG EFFICIENCIES FROM NANOSCALE DIELECTRIC AND
PLASMONIC RESONATORS
To compare the observed SH generation conversion efficiency for our structure with previously
demonstrated results in Refs. (5-7, 9) of the main text, we evaluate and combine all the relevant
parameters taking them directly from the references. They are summarized in Table S1 which
contains the data on the experimental setup properties. We note that in our experiments we mea-
sure the spot radius using the knife-edge method (see Part 10), while for other references it was
15
estimated using the diffraction limit criterion of the first null of the Airy disk.
Table S1 summarizes the directly measured pump and second harmonic powers. The SH power
does not account for collection efficiency of the receiving objective and the pump power does not
account for the coupling (insertion) efficiency to the mode at the pump wavelength. The peak
values of the powers are evaluated considering the Gaussian pulse shape, which gives the factor
of 0.94. We assume the second harmonic pulse is √2longer than the pump pulse, which is the
approximation valid for a non-resonant bulk material.
For direct comparison of our results with the earlier works we use the directly measured value
of the dimensional conversion efficiency, defined as
ηexp =P2ω
p
Pω
p2.(S24)
To evaluate and compare the intrinsic performance of the mode independently on the pump
coupling and the objective collection efficiency we estimate the total SH generation conversion
efficiency. Its value is given by the ratio of the total radiated SH power estimated using the col-
lection efficiency of the objective βand the amount of the pump power coupled to the resonator
proportional to the coupling coefficient κ1
ηest =ηexp
βκ2
1
.(S25)
To calculate the coupling coefficient for this work κ1we use the exact expression [see
Eq. (S11)]. However, for other references the numerical data for field profiles is not available, thus
we use the simplified definition for κ1, evaluating it as the doubled percentage of pump power
coming through the top surface of the resonator, which is the common approach used in the litera-
ture. For the linearly polarized fundamental Gaussian beam focused on the top surface of the disk
resonator with radius r0the coupling efficiency can be calculated analytically
κ1= 2 [1 −exp (−2r2
0/w2
0)].(S26)
The collection efficiency βis calculated as a percentage of the second harmonic power coming
through the collecting objective, which is estimated by using the explicit radiation pattern of the
eigenmode for a disk on a structured substrate.
16
FIGURES
Figure S1. (A) Calculated mode wavelengths vs. resonator diameter for the disk on a structured substrate.
The dashed lines qualitatively show the dispersion of uncoupled radial and axial modes. (B) Calculated
ratio |Hz|/|Hr|as a function of the disk diameter. The regime of hybrid modes is qualitatively shown by
the blurred blue area.
17
Figure S2. Behavior of eigenmodes in an individual disk resonator in the strong mode coupling regime (A,
C, E) on a three-layer (SiO2/ITO/SiO2) substrate with a 300 nm ITO layer and a 350 nm SiO2spacer and (B,
D, F) in the free space. (A,B) Dispersion of the low-Q and high-Q modes vs. resonator diameter. (C,D) Q
factor evolution for the low-Q and high-Q modes vs. resonator diameter. (E,F) Multipolar decomposition
of the radiated power for the high-Q mode vs. resonator diameter. MD and MO denote the magnetic dipole
and magnetic octupole contributions. (G) Transformation of the radiation pattern (far-field distribution) for
the high-Q and low-Q mode while passing the avoided resonance crossing. For comparison of calculations
for different disk diameters, |E|is normalized on the square root of the full mode energy. Points A1,2,3,
B1,2,3correspond to the disk diameters of 750 nm, 900 nm, 1250 nm, respectively.
18
Figure S3. Mode dispersion at the second harmonic wavelength. (A) Dispersion of the high-Q mode at the
pump wavelength (red) and the mode at the second harmonic wavelength (green) vs. resonator diameter.
The wavelength of the SH mode is doubled for a direct comparison. (B) Q factor evolution for the mode at
the second harmonic wavelength vs. resonator diameter. (C) Near-field and far-field patterns for mode at
the second harmonic wavelength. The far-field pattern is calculated for a disk without a substrate.
19
Figure S4. Reflectance spectrum and retrieved Q factor of the observed resonance for a 930 nm disk on a
three-layer substrate excited with an azimuthal pump. The spectra are normalised on the reflectivity of the
structured substrate in 0.7NA aperture. (A) Measurements. (B) Simulations. The inset shows the near-field
pattern of the excited quasi-BIC resonance at the dip wavelength.
20
Figure S5. Multipolar decomposition of the scattered power at the pump wavelength in the full solid angle
for a 930 nm disk on a three-layer substrate excited with an azimuthal pump.
21
Figure S6. Experimental and simulated spectra of the second-harmonic intensity for a 930 nm disk on a
three-layer substrate excited with an azimuthal pump. The simulated spectrum is artificially blue-shifted by
34 nm. The simulated and experimental data are normalized independently. The inset shows the near-field
pattern at the resonant peak wavelength.
22
Figure S7. Comparison between theory and experiment for SH power. (A) Spectral overlap between the
high-Q mode at the pump frequency and the high-order Mie mode at the SH frequency vs. the disk diameter.
(B) Q factor combination vs. the disk diameter. (C)L2Q2Q2
1vs. the disk diameter. (D) The SH signal
evaluated along the quasi-BIC dispersion measured experimentally. The data in (C) and (D) is normalised
independently.
23
Figure S8. Schematics of the fabrication procedure. (A) Mask formation with electron-beam lithography.
(B) Chemically-assisted ion beam etching. (C) The PMMA mask removement and AlInP buffer layer wet-
etching. (D) The AlGaAs nanodisks transfer. (E) The drop down of nanodisks and PPC layer removement.
24
Figure S9. Experimental setup for linear spectroscopy in reflection. MIROPA-fs-M is the optical parametric
amplifier, M1 and M2 are the mirrors, GP1 and GP2 are the Glan prisms, P is the wire-grid polarizer, IFG
is the infrared glass filter, QP is the commercial liquid crystal q-plate, NPBS is the non-polarizing 50/50
beamsplitter, O1 is the objective, S is the sample on a three-dimensional stage, PM is the power meter head.
25
Figure S10. Spectra of the power reflected by the AlGaAs nanoresonator with 930 nm diameter (blue dots)
and by the SiO2/ITO/SiO2substrate with the 350 nm SiO2spacer and the 300 nm ITO layer (black dots).
Radiation is collected in the backward direction by the objective lens with 0.70 NA.
26
Figure S11. Second-harmonic generation experimental setup in (A) transmission and (B) reflection.
MIROPA-fs-M is the optical parametric amplifier, M1 and M2 are the mirrors, GP1 and GP2 are the Glan
prisms, P is the wire-grid polarizer, IFG is the infrared glass filter, QP is the commercial liquid crystal q-
plate, O1 and O2 are the objectives, S is the sample on a three-dimensional stage, GFG is the set of filters
which transmit the SH signal, L1 is the lens, CCD is the detecting visible camera, DM is the dichroic mirror.
27
Figure S12. Representative spectra of the raw data for the SH optical signal (black dotes) collected by the
visible CCD-camera and average pump power values (gray dotes) measured after the focusing objective
lens. The presented data corresponds to the nanoresonator which diameter is 925 ±5nm.
28
Figure S13. Knife-edge experimental results for linearly (A, B, C) and azimuthally (D, E, F) polarized
focused beams. Power dependence on the knife-edge position (A,D) and corresponding derivatives of
beam profiling data (B,E). Beam parameters (C)wand (F)w0in dependence on the knife-edge position
on the optical axis.
29
Figure S14. Peak power dependence of the total SH signal generated by an AlGaAs nanoresonator (height
is 635 nm, diameter – 930 nm). Dot colors correspond to different central fundamental wavelengths of
1562,1564,1565,1566,1567 nm. The polarization of the pump beam is azimuthal. The brown line depicts
a quadratic fit P2ω
p= 1.3·10−6Pω
p2
30
Figure S15. Experimentally measured directionality diagrams of the SH in the backward (A) and forward
(B) directions. Arrows visualize the polarization states. The insets (C) show retrieved spatially-resolved
ellipticity (top) and polarization inclination (bottom) angles for the SH signal.
Parameter Pump λ
(nm)
Laser
repetition
rate
(MHz)
Laser
pulse
duration
(ps)
Excitation
objective
NA
Pump
beam
waist
radius
(µm)
Resonator
radius
(nm)
Pump
average
power
(mW)
Pump
peak
power
(W)
SH
average
power
(nW)
SH peak
power
(mW)
Coupling
coeffi-
cient
(%)
Collection
efficiency
(%)
Measured
conversion
efficiency
(W−1)
Estimated
total con-
version
efficiency
(W−1)
Symbol λ ν τ NA w0r0PωPω
pP2ωP2ω
pκ1β ηexp ηest
Formula 0.61λ/NA 0.94 Pω
ντ 0.94 P2ω
√2ντ Eq. (S26) Eq. (S24) Eq. (S25)
This work∗1570 5.144 2 0.7 1.8∗∗ 465 7.9·10−27.2 0.93 6·10−233∗∗∗ 24 (1.3·
10−6)∗∗∗∗
4.8·10−5
Ref. (5)∗1556 5 0.5 0.85 1.1 245 1.0 376 8 2.1 19 30 1.5·10−81.4·10−6
Ref. (6)∗1556 80 0.17 0.85 1.1 193 0.75 52 0.7 3.4·10−212 10 1.3·10−88.7·10−6
Ref. (7)∗910 0.1 0.18 0.9 0.62 200 5·10−3261 1·10−20.42 38 28 6.2·10−91.5·10−7
Ref. (9)∗1550 80 0.12 1.35 0.7 N/A 0.12 11.8 7.6·10−45.3·10−5N/A N/A 3.8·10−10 N/A
Table S1: Experimental parameters for the comparison of SHG efficiencies from nanoscale dielectric and plasmonic resonators.
∗Material system, resonator geometry, mode type at pump wavelength:
This work - AlGaAs disk on SiO2/ITO/SiO2, quasi-BIC;
Ref. (5) - AlGaAs disk covered in BCB on SiO2, electric and magnetic dipole;
Ref. (6) - AlGaAs disk on AlOx/GaAs, magnetic dipole;
Ref. (7) - GaP disk on GaP, magnetic quadrupole;
Ref. (9) - gold nanorod dimer on fused silica, electric dipole;
∗∗Knife-edge measurements (see Part 10).
∗∗∗ Evaluated using Eq. (S11) for a disk suspended in air.
∗∗∗∗ Evaluated from the quadratic fit to the data shown in Fig. 4C of the main text
31
32
32
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