of 8
Article
https://doi.org/10.1038/s41467-023-44164-4
High quality factor metasurfaces for two-
dimensional wavefront manipulation
Claudio U. Hail
1
, Morgan Foley
2
,RuzanSokhoyan
1
, Lior Michaeli
1
&
Harry A. Atwater
1
The strong interaction of light with micr
o- and nanostructures plays a critical
role in optical sensing, nonlinear optics
, active optical devices, and quantum
optics. However, for wavefront shaping
, the required local control over light at
a subwavelength scale limits this interact
ion, typically leading to low-quality-
factor optical devices. He
re, we demonstrate an avenue towards high-quality-
factor wavefront shaping in two spatial dimensions based on all-dielectric
higher-order Mie-resonant metasurfaces. We design and experimentally rea-
lize transmissive band stop
fi
lters, beam de
fl
ectors and high numerical aper-
ture radial lenses with measured quality factors in the range of 202
1475 at
near-infrared wavelengths. The excite
d optical mode and resulting wavefront
control are both local, allowing versatile operation with
fi
nite apertures and
oblique illumination. Our results represent an improvement in quality factor
by nearly two orders of magnitude over
previous localized mode designs, and
provide a design approach for a ne
w class of compact optical devices.
The recirculation of light in a con
fi
ned optical mode is a ubiquitous
method to amplify the interaction of light and matter. In this respect,
the ability to con
fi
ne the light to the resonating mode is quanti
fi
ed by
the quality factor,
Q
, as the energy stored per round-trip optical loss
in the resonator. With optical micro- and nanostructures, including
Fabry-Pérot cavities
1
,
2
, whispering gallery mode resonators
3
6
, pho-
tonic crystals
7
,
8
, guided mode structures
9
, and bound states in the
continuum (BIC)
10
13
, quality factors of up to 10
8
have been demon-
strated. The high level of
fi
eld enhancement and con
fi
nement
attained in these structures has led to many advances in sensing
14
,
15
active optical devices
16
,
17
, light sources
11
,
18
, and ampli
fi
cation of
photon
matter coupling
2
,
19
. However, in general, as the mode
volume of an optical resonator decreases and the mode becomes
more localized, more radiative decay channels become available, and
the
fi
eld enhancement and quality factor diminish
20
. As a result, there
is typically a tradeoff between spatial mode localization and the
attainable quality factor.
In optical metasurfaces, a subwavelength-spaced array of loca-
lized resonators is used to abruptly manipulate the phase, amplitude,
polarization, and spectrum of light at an interface
21
. These structures
hold promise to revolutionize many areas of optical imaging,
communication, sensing, and display technology. Numerous optical
components and phenomena have been realized using metasurfaces
such as ef
fi
cient
fl
at lenses
22
24
, on-chip holography
25
,
26
and dynamic
beam steering
27
,
28
. Attaining strong light
matter interaction, and
hence high quality factors, in metasurfaces is particularly desirable, as
it enables the realization of ultra-fast spatial light modulators, non-
linear parametric conversion, highly responsive optical sensing, and
tailored light emission. However, the required subwavelength scale
wavefront control imposes a limit on the resonator size, leading to
signi
fi
cant radiative loss. As a result, most metasurfaces have been
broadband and have relied on dielectric structures with limited light
con
fi
nement and hence quality factor (
Q
<15)
23
,
24
,
29
.Onlyrecently,
advances in wavefront manipulation with increased quality factors
have been made with structures relying on extended guided mode
resonance
30
32
and nonlocal modes based on bound states in the
continuum
33
,
34
. Notably, achieving simultaneous local control over a
wavefront with resonance phase and high quality factor remains an
outstanding challenge
35
,
36
. Here, we demonstrate an avenue towards
high-quality factor metasurfaces by leveraging higher-order Mie
resonances to manipulate the wavefront of light locally in two
dimensions based on the resonance phase.
Received: 25 June 2023
Accepted: 2 December 2023
Check for updates
1
Thomas J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, CA 91125, USA.
2
Department of Physics, California Institute of
Technology, Pasadena, CA 91125, USA.
e-mail:
haa@caltech.edu
Nature Communications
| (2023) 14:8476
1
1234567890():,;
1234567890():,;
Results
Figure
1
a illustrates our high-quality factor optical metasurface for
wavefront manipulation in two dimensions. The surface consists of
subwavelength-spaced amorphous silicon nanoblocks of length
L
and
height
H
arranged in a square array with periodicity
P
on a transparent
glass substrate. The structure dimensions are in the sub-diffractive
regime, both in air and in the substrate, to avoid exciting any lattice
modes. The geometric parameters are chosen to induce and spectrally
overlap an electric dipole (ED) and electric octupole (EO) mode in the
nanoblocks at near-infrared wavelengths. With the ED and EO modes,
the surface operates as a higher-order Mie-resonant metasurface
enabling local control over the transmitted wavefront. Figure
1
bshows
the transmission and re
fl
ection spectrum of the metasurface with a
uniform nanoblock side length
L
= 555 nm, as calculated with
fi
nite
difference time domain (FDTD) simulations (see Methods for details).
A multipole expansion of the modes inside a single nanoblock
embedded in the array shows that the ED and EO modes are in phase
and of similar strength on resonance (see Supplementary Note 1)
37
,
38
.
The destructive interference of the ED/EO mode and the illumination
induces a sharp dip in transmission and near-unity re
fl
ection on
resonance. A quality factor of
Q
= 290 is obtained from a Fano reso-
nance
fi
t
39
. On resonance, the electric
fi
eld in the resonator shows a
strong
fi
eld enhancement of more than 28 times the incident
fi
eld
amplitude (see electric and magnetic
fi
eld pro
fi
les Supplementary
Note 1). Furthermore, the EO mode is clearly visible in the electric and
magnetic
fi
eld pro
fi
les. Notably, interference of multiple extended
11
or
local modes
13
,
40
has been a common strategy to achieve a high-quality
factor optical response. Analyzing the light scattered from the meta-
surface with varying nanoblock aspect ratios suggests that the ED/EO
mode is different from extended supercavity modes
11
or supercavity
modes reported in isolated nanoparticles
13
,
40
(see Supplemen-
tary Note 2).
Due to the near-
fi
eld coupling of the ED and EO among neigh-
boring elements, the resonance is dependent on the array size, i.e., the
number of repetitions of the unit cell, which is the signature of a
partially delocalized mode. This array size dependence is similar to
that seen in low-order Mie-resonant metasurfaces
29
or asymmetry-
induced quasi-BIC structures
41
. Our calculations suggest that for an
array size beyond 10 × 10 unit cells, there is no further signi
fi
cant
change in the modal properties (see Supplementary Fig. 1). For the ED/
EO mode, we attribute this increase in quality factor with increasing
array size to an enhancement of the ED through multipole coupling of
the ED and EO in the array through neighboring particles, as similarly
observed in a different geometry previously
42
.Tofurtherprovide
insight into the ED/EO mode we study its degree of localization.
Exciting the mode in a single nanoblock of the array shows that it is
localized within the nanoblock but extends primarily to its nearest
neighbors along the direction of
polarization (see Supplementary
Note 3). From this calculation, we determine a mode volume of 3.6
times the volume of a unit cell, or 0.86
λ
3
,where
λ
is the wavelength in
free space. The mode localization can also be inferred from its
dependenceontheincidenceangleoftheillumination(seeFig.
1
d, e).
The ED/EO modes can be excited at oblique incidence for both
transverse electric (TE) and transverse magnetic (TM) polarization. We
observe a spectral shift of the resonance of less than 2 nm for a 10°
change in the incident angle. For comparison, a fully delocalized lattice
mode (Wood
s anomaly) or guided mode resonance at the same
resonance wavelength would result in a spectral shift on the order of
130nmforthesame10°change
9
.Theobserved
fl
at angular dispersion
(angular dependent resonance wavelength shift) of our surface indi-
cates that the studied high-Q mode is localized. Furthermore, the
dispersion is of similar order as reported in low-order Mie-resonant
metasurfaces based on the spectral overlap of an electric and magnetic
dipole
43
,
44
.
The local response of our higher-
order Mie-resonant metasurface
enables local wavefront manipulation by controlling the phase of the
transmitted light. Varying the length
L
of the nanoblocks, spectrally
shifts the ED and EO modes and hence the resonance wavelength. The
spectral shift accompanying the change in length
L
allows employing
the resonance phase to impose a phase shift on the transmitted light.
e
d
b
550
555
560
L
(nm)
2


0
Phase (rad)
0
50
100
Transmission (%)
a
c
y
x
z
E
x
P
P
L
L
H


1.26
1.3
1.28
1.32
Wavelength (

m)
0
50
100
(%)
Q
= 290
T
R
0
5
10
15
20
Incidence angle (°)
1260
1280
1300
1320
1340
1360
Wavelength (nm)
0
20
40
60
80
100
Transmission (%)
TE
ED/EO
0
5
10
15
20
Incidence angle (°)
1260
1280
1300
1320
1340
1360
Wavelength (nm)
0
20
40
60
80
100
Transmission (%)
TM
ED/EO
Fig. 1 | High-quality factor metasurfaces for two-dimensional wavefront
manipulation. a
Schematic of the metasurface consisting of amorphous silicon
nanoblocks on a glass substrate. The surface is illuminated at normal incidence
and can de
fl
ect light along the angles
θ
and
φ
in the full upper hemisphere.
b
Calculated transmission (
T
) in blue and re
fl
ection (
R
) spectra in red of the
metasurface with
P
= 736 nm,
H
= 695 nm,
L
= 555 nm.
c
Calculated transmission
intensity (blue) and phase (red) with varying nanoblock side lengths
L
,
P
= 736 nm,
and
H
= 695 nm at a wavelength of
λ
= 1288 nm.
d
,
e
Simulated transmission of
the metasurface in (
a
) with varying angle of incidence for TE (
d
) and TM (
e
)
polarization.
Article
https://doi.org/10.1038/s41467-023-44164-4
Nature Communications
| (2023) 14:8476
2
Figure
1
c shows that varying the nanoblock side length by only ±1%,
allows the phase of the transmitted light to be controlled over almost
the entire 0
2
π
range at a
fi
xed wavelength of
λ
= 1288 nm. The
inherent symmetry of the unit cell and the resonant mode also result in
a polarization-independent response. Most remarkably, a small varia-
tion (
1%) of the length
L
by
dL
of one block in an array of nanoblocks
with uniform length
L
manifests itself in a local resonance shift of the
single nanoblock acting as an effective point source scatterer (see
Supplementary Note 3). Furthermore, the resonant mode remains
intact, and the high quality factor is preserved. These characteristics
render our metasurface, which exhibits four-fold rotational symmetry
and supports partially delocalized ED/EO modes, uniquely suitable for
high-quality factor wavefront manipulation in two dimensions.
We fabricated our high-Q metasurfaces by single-step electron
beam lithography and dry etching of plasma-deposited amorphous
silicon on a glass substrate (see Methods for details). Figure
2
a illus-
trates a scanning electron micrograph of a metasurface with uniform
nanoblock side lengths. To experimentally characterize the surfaces,
we employed linearly polarized, normally incident light through the
substrate from a wavelength-tunable diode laser, collected the trans-
mitted light with an objective lens and imaged it onto an InGaAs IR-
camera, or focused it onto a power meter (see Methods for details).
Figure
2
b shows the measured transmission spectra of metasurfaces
with uniform nanoblock side lengths. A strong dip in transmission is
observed on resonance with a narrow linewidth ranging between 2 and
8 nm. On resonance, the measured minimum transmission ranges
between 6
16%. The corresponding measured quality factors range
between 202 and 668 as determined by
fi
tting a Fano lineshape
function. With uniform nanoblock sizes, this metasurface acts as an
ultra-thin narrowband bandstop
fi
lter. An increase in nanoblock side
length of only a few nanometers signi
fi
cantly redshifts the resonance
by more than the linewidth. Furthermore, increasing the nanoblock
side length beyond
L
= 578 nm leads to a decrease in the quality factor.
The side length is thus a convenient parameter for adjusting the quality
factor by design. The high-frequency oscillation in the transmission
spectrum is due to the interference of light re
fl
ecting at the top and
bottom surfaces of the substrate. The measurements here are taken
with an illumination spot diameter of 30
μ
m. This con
fi
rms that both
the
fi
nite array size and varying incident angles have a negligible effect
on the optical response of the surface. Notably, the measured quality
factors are higher than expected from simulations (see Fig.
1
). How-
ever, when fabrication imperfections, such as non-vertical sidewalls
and a structure undercut, are accounted for in the simulation, the
calculated quality factors increase and good agreement with the
measured transmission spectrum is obtained (see Supplementary
Fig. 2). The measured quality factors of our uniformly-sized meta-
surfaces are higher or on par with measured quality factors of meta-
surfaces based on asymmetry-induced quasi-BIC
12
,
15
,
41
, toroidal modes
45
or electromagnetically induced transparency
46
.However,unlikethese
other designs, our metasurfaces allow for local manipulation of the
resonance phase (see Supplementary Note 4). The measured near-
complete extinction of the transmission on resonance is evidence that
the nanoblocks are fabricated with a high degree of size uniformity. A
comparison with electromagnetic simulations suggests the nano-
blocks are fabricated with a standard deviation in the length
L
of less
than 6 Å (see Supplementary Note 5). For silicon, this represents a near
single atomic layer precision in nanostructuring of the metasurface
and a signi
fi
cant improvement over previous processes for metasur-
face fabrication with standard deviations of 2.1 nm
47
.
To selectively de
fl
ect light over a narrow wavelength range, we
imprinted a linear phase gradient on the transmitted light by varying
the nanoblock side lengths along one of the metasurface in-plane
directions (see Supplementary Table 1 for metasurface design para-
meters). The nanoblock size lengths are set based on the phase look-up
table determined from numerical simulations in Fig.
1
c by applying a
traditional forward design. We performed Fourier plane imaging of the
transmitted light to characterize the light de
fl
ection of the metasur-
face. Figure
3
a shows the measured spectral diffraction ef
fi
ciency and
images of the Fourier plane of the light transmitted through a high-Q
beam de
fl
ector metasurface in the on- and off-resonance case for
de
fl
ecting
x-
polarized light along the
y
direction. This results in a TE
polarization of the de
fl
ected light, referred to as TE de
fl
ection. On
resonance, light is preferentially de
fl
ected to an angle of
θ
=26°
(
φ
= 0°) from the surface normal as determined by the linear phase
gradient of the metasurface along the
y
direction corresponding to 2
π
/
4
P
.Atthedesignwavelength
λ
R
= 1293 nm, a diffraction ef
fi
ciency of
41.2% is attained. The remaining power is coupled to the normal and
opposite direction at
θ
=
26°. Notably, in a representative off-
resonant case, at
λ
O
= 1288 nm, 97.4% of the transmitted light
remains in the surface normal direction. The measured quality factor
of the TE light de
fl
ection is
Q
= 1191. By imprinting a phase gradient
along the orthogonal direction (i.e., along the
x
direction) of the
metasurface, light of the same polarization is de
fl
ected along the
x-
direction to an angle
φ
=26°(
θ
= 0°) with a maximum diffraction ef
fi
-
ciency of 17.1% at
λ
R
= 1293 nm (see Fig.
3
b), due to the polarization-
independent response. The result is a TM polarization of the de
fl
ected
light, referred to as TM de
fl
ection. Here, a larger quality factor of
Q
= 1458 is measured. Off-resonance, at
λ
O
=1288nm, 98.1% of the
transmitted light remains in the normal direction. Two additional
representative measurements of TE and TM de
fl
ection are illustrated
in Supplementary Figs. 3 and 4.
The operating wavelength for beam de
fl
ection can be adjusted by
shifting the average length of nanoblocks. Figure
3
c, d illustrates the
measured spectral diffraction ef
fi
ciency and quality factors for TE and
TM de
fl
ection for surfaces with varying operating wavelength and
fi
xed phase gradient. By modifying the phase gradient imprinted on
the surface, light can be de
fl
ected to different angles. Figure
3
e, f
illustrates the measured spectral diffraction ef
fi
ciency for TE light
de
fl
ection and quality factors for TE and TM de
fl
ection for surfaces
550 600 650
L
(nm)
0
200
400
600
800
Q
(-)
a
b
1260
1280
1300
1320
1340
1360
Wavelength (nm)
0
50
100
Transmission (%)
c
2

m
567 nm 578 nm
604 nm
619 nm
639 nm
Fig. 2 | High-quality factor resonances in silicon nanoblock arrays. a
Scanning
electron micrograph of a metasurface with uniform nanoblock side lengths.
b
Experimentally measured transmission spectra of high-quality factor meta-
surfaces with varying nanoblock side lengths. The measured nanoblock side length
in nm is indicated next to each curve. Here,
P
=736nmand
H
= 695 nm.
c
Experimentally measured quality factors with varying nanoblock side lengths as
determined by a Fano lineshape
fi
t. The error bars illustrate the 95% con
fi
dence
interval of the
fi
t. The metasurface size is 150
μ
m×150
μ
m.
Article
https://doi.org/10.1038/s41467-023-44164-4
Nature Communications
| (2023) 14:8476
3
de
fl
ecting light to different angles (the corresponding TM diffraction
ef
fi
ciency is shown in Supplementary Fig. 5). Overall, the obtained
diffraction ef
fi
ciencies on resonance for the desired de
fl
ection angle
range between 22
76.6% and 4
19.1% for the TE and TM de
fl
ection,
respectively. The lower diffraction ef
fi
ciency obtained for TM polar-
ized de
fl
ection compared to TE is likely due to nearest-neighbor cou-
pling between the nanoblocks. For TM de
fl
ection, the nearest-
neighbor coupling occurs along the direction of the phase gradient,
therefore making this case more prone to local phase errors that arise
from coupling between nanoblocks and fabrication imperfections. The
attained
Q
factors for TE and TM de
fl
ection range between
Q
TE
=298
1191 and
Q
TM
=288
1475. These results illustrate that light
can be de
fl
ected along two dimensions to arbitrary angles
θ
and
φ
with
high quality factors. This contrasts with previous demonstrations of
high-quality factor metasurfaces based on guided mode resonance
structures, which are inherently one-dimensional in their light de
fl
ec-
tion capability, and require large il
lumination apertures and precise
con
fi
guration of the incidence angle
30
,
31
.
To demonstrate the wavefront shaping capabilities of our meta-
surfaces, we realize high-quality factor radial metalenses that focus
light along two dimensions over a narrow wavelength range. The
metalenses are designed by imposing a paraboloidal phase pro
fi
le on
the transmitted light by setting the nanoblock size lengths according
to the look-up table in Fig.
1
c. Figure
4
a, b illustrates the measured
electric
fi
eld intensity in the focal plane and in a cross-section along the
optical axis of a metalens with a numerical aperture (NA) of 0.1 at the
resonance wavelength of
λ
= 1276 nm. On resonance, light is symme-
trically focused to a near-diffraction-limited focal spot. However, at
λ
= 1266 nm, at a wavelength only 10 nm away from the resonance
wavelength, light propagates through the metalens without focusing
(see Fig.
4
c, d). Figure
4
e, shows the measured spectral focusing ef
fi
-
ciency of the metalens, highlighting its wavelength-selective
operation. The maximum focusing ef
fi
ciency is 25.3% on resonance.
From the spectral focusing ef
fi
ciency, a quality factor of
Q
=531 is
determined for the metalens. The measured transmittance of the lens
on resonance is 55.6% (see Supplementary Fig. 6). Figure
4
fillustrates
measured quality factors from different lenses with numerical aper-
tures ranging from 0.1 to 0.8 and varying operating wavelengths. A
similar trend in increased quality factors towards shorter wavelengths,
as in Fig.
2
b, is observed. The highest quality factor
Q
= 880 is obtained
for a lens with 0.18 NA (see Supplementary Fig. 7). To our knowledge,
this represents the highest quality factor radial lens demonstrated to
date and is one order of magnitude higher than previous demonstra-
tions with nonlocal metasurfaces
34
.Figure
4
g
j illustrates the mea-
sured electric
fi
eld intensity in the focal plane for metalenses with
numerical apertures of 0.18, 0.4, 0.6, and 0.8. The observed full width
at half maximum of the focus is in good agreement with the diffraction-
limited values. Additional represen
tative measurements of metalenses
are shown in Supplementary Fig. 7. The Strehl ratios of the char-
acterizedlensesrangebetween0.1and0.57(seeSupplementary
Fig. 8). Both the measured Strehl ratios and focusing ef
fi
ciencies
decrease with increasing numerical aperture, which we attribute to an
increased wavefront distortion due to coupling between neighboring
elements, which increases with larger phase gradients. We note that
the performance of the metalenses demonstrated here is currently
limited by fabrication accuracy. Furthermore, the coupling between
neighboring elements is not optimized in the design but is known to
have a signi
fi
cant effect
43
.
In wavefront shaping with high quality factors, there is an inherent
tradeoff between quality factors and accurate phase sampling due to
fabrication limitations. This is due to the rapid variation of the phase
on resonance. For example, in the structure reported here, the phase
of the transmitted light varies with 9.4 rad/nm within the range of
π
/2
to 3
π
/2 (see Fig.
1
b). As a result, a precision of the nanorod side length

(°)


(°)


(°)


(°)

(°)
60
30
0
0
-30
-60
60
30
0
-30
-60
0


(°)

O

R


(°)
60
30
0
0
-30
-60
60
30
0
-30
-60
0
35.7°
20.4°
16.8°
25.8°
571 nm 577 nm
589 nm
605 nm
615 nm
1280
1290
1300
1310
Wavelength (nm)
0
50
100
Diffraction efficiency (%)
a
f
e


(°)

O

R

R

O
TE
1280
1290
1300
1310
Wavelength (nm)
0
50
100
Diffraction efficiency (%)
b
TM

R

O
Q
= 1458
Q
= 1191
1270
1280
1290
1300
Wavelength (nm)
0
50
100
200
150
Diffraction efficiency (%)
20
30
40
Angle (°)
500
1000
1500
Q
(-)
c
1280
1300
1320
Wavelength (nm)
0
50
100
Diffraction efficiency (%)
d
580
620
600
Length (nm)
500
1500
1000
2000
Q
(-)
TM
TE
TE
TM
TE
TM
+2
+1
0
-1
-2
+2
+1
0
-1
-2
1292
1294
0
10
20
1292
1294
0
20
40
Fig. 3 | High-quality factor beam de
fl
ection along two dimensions.
a
,
b
Experimentally measured diffraction ef
fi
ciencies of the
2 (green),
1(yellow),
0 (blue), +1 (red), and +2 (purple) diffraction orders and Fourier plane images of a
metasurface showing
a
TE de
fl
ection of x-polarized light along the y direction and
b
TM de
fl
ection of x-polarized light along the x direction. The desired diffraction
order is +1, with
θ
= 26° and
φ
= 26°, respectively. The insets show a zoomed-in
region of the plots.
c
Measured diffraction ef
fi
ciency of TE and TM light de
fl
ection
in the +1 diffraction order with varying average nanoblock side lengths. The values
for TM de
fl
ection are shifted vertically by 85% for better illustration.
d
Measured
quality factors of TE (blue circles) and TM (red circles) light de
fl
ection with varying
average nanoblock side lengths.
e
Measured diffraction ef
fi
ciency of TE light
de
fl
ection with varying de
fl
ection angle. The curves are shifted vertically by 50%
from each other for better visibility.
f
Measured quality factors of TE (blue circles)
and TM (red circles) light de
fl
ection with varying de
fl
ection angles. The error bars
in (
d
)and(
f
) illustrate the 95% con
fi
dence interval of the
fi
t. For all metasurfaces
P
=736nmand
H
= 695 nm. The metasurface size is 150
μ
m × 150
μ
m.
Article
https://doi.org/10.1038/s41467-023-44164-4
Nature Communications
| (2023) 14:8476
4
of 0.17 nm is required to sample the phase at a
π
/2 increment in this
range. Furthermore, this required precision generally becomes stricter
when increasing the quality factor. Our fabrication process results in
an accuracy of the nanoblock side length of less than 0.6 nm, which
produces considerable errors in phase sampling, in turn causing the
appearance of stray light in the focal plane of the fabricated meta-
lenses (Fig.
4
j). This shows that fabrication imperfections are currently
a limiting factor in the demonstrated quality factors and diffraction
ef
fi
ciencies. We expect an improved performance with higher fabri-
cation uniformity and lower side wall surface roughness. While there is
still room for improving the electron beam lithography-based process
implemented here, using methods such as scanning probe lithography
or atomic layer etching may allow fabrication with near-atomic layer
accuracy. Another approach is to employ different optical modes and
unit-cell geometries that enable wavefront shaping with similar quality
factors but with more relaxed fabrication requirements. For example, a
con
fi
guration in re
fl
ection with similar quality factors shows a slower
variation of the phase of 2.64 rad/nm requiring only a 0.6 nm precision
to sample the phase at a
π
/2 increment in the
π
/2 to 3
π
/2 range (see
Supplementary Note 6). Furthermore, accounting for fabrication
imperfections in the device design is also expected to result in better
device performance. This could be done by creating a new iteration of
the phase look-up table (Fig.
1
c) by considering the fabrication
imperfections identi
fi
ed in Supplementary Fig. 2. In addition to fabri-
cation constraints, the local approximation used to design the phase
pro
fi
le along the array also limits the maximum ef
fi
ciency of the
devices due to non-negligible long-range coupling between unit cells.
However, this is not a fundamental limitation, and improved perfor-
mance can be attained by accounting for nanoblock coupling in the
design
48
, applying design optimization approaches or by employing
inverse design concepts
49
. Recently, topological optimization has been
employed to realize high-
Q
metagratings for 1D beam de
fl
ection
50
.
Relying on similar concepts, we demonstrate that particle swarm
optimization can be used to optimize the design of our structure for
TM light de
fl
ection resulting in an increase in the diffraction ef
fi
ciency
from 47% to 82% (see Supplementary Fig. 9).
In summary, we have demonstrated the concept of a higher-order
Mie-resonant metasurface as a pathway towards high-quality factor
two-dimensional wavefront manipulation. By spectrally overlapping an
electric dipole and electric octupole mode, a sharp resonance in the
transmission is obtained that enables local control of the wavefront of
light. Using local phase control, we realize beam de
fl
ectors with high
Field int. (arb. units)
-200
-100
0
100
200
z
(

m)
-20
0
20
x
(

m)
-200
-100
0
100
200
z
(

m)
-20
0
20
x
(

m)
-20
0
20
x
(

m)
-20
-10
0
10
20
y
(

m)
-20
0
20
x
(

m)
-20
-10
0
10
20
y
(

m)
a
c
f
e
d
b
1260
1280
1300
Wavelength (nm)
0
10
20
30
Focusing efficiency (%)
Q
= 531
1260
1280
1300
Wavelength (nm)
200
400
600
800
1000
Q
(-)
0.1
0.18
0
0.5
1
Field intensity (arb. units)
0
0.5
1
Field intensity (arb. units)

= 1276 nm
-5
0
5
x
(

m)
0
0.5
1
-5
0
5
y
(

m)
Q
= 391
NA = 0.4
FWHM
2.1

m
h
0
0.5
1
Field int. (arb. units)
Field int. (arb. units)
Field int. (arb. units)
-5
0
5
y
(

m)
-5
0
5
x
(

m)
Q
= 880
NA = 0.18
FWHM
2.9

m
g
0.4
0.6
0.8
-5
0
5
x
(

m)
0
0.5
1
-5
0
5
y
(

m)
Q
= 395
NA = 0.6
FWHM
1.5

m
i
-5
0
5
x
(

m)
0
0.5
1
-5
0
5
y
(

m)
Q
= 429
NA = 0.8
FWHM
1.1

m
j

= 1276 nm

= 1266 nm

= 1266 nm
Fig. 4 | High-quality factor radial metalenses for focusing along two dimen-
sions. a
Measured
fi
eld intensity at the focal plane (
x
y
plane) on resonance at
λ
= 1276 nm of a high-Q metalens with 0.1 NA.
b
Measured
fi
eld intensity along the
optical axis in the
x
z
plane on resonance.
c
Measured
fi
eld intensity at the focal
plane off-resonance at a representative wavelength
λ
= 1266 nm.
d
Measured
fi
eld
intensity along the optical axis in the
x
z
plane off-resonance. The scaling of the
color maps in (
a
d
) is identical.
e
Measured spectral focusing ef
fi
ciency of the
metalens with 0.1 NA and
Q
= 531. The red line shows a Fano
fi
t to the measurement.
f
Measured quality factors of fabricated lenses with different resonance wave-
lengths and numerical apertures of 0.1 (blue disk), 0.18 (red circle), 0.4 (yellow
cross), 0.6 (purple plus), and 0.8 (green asterisk).
g
j
Measured
fi
eld intensity (
fi
eld
int.) at the focal plane (
x
y
plane) on the resonance of four high-
Q
metalenses with
numerical apertures of 0.18, 0.4, 0.6, and 0.8, respectively. The resonance wave-
lengths are 1266 nm, 1291.8 nm, 1285.5 nm, and 1281.4 nm, respectively. The insets
denote the full width at half maximum (FWHM) of the focusing. The metalenses are
100
μ
m in diameter
P
= 736 nm and
H
= 695 nm.
Article
https://doi.org/10.1038/s41467-023-44164-4
Nature Communications
| (2023) 14:8476
5
directivity and quality factors of
Q
=288
1475. We further demon-
strate radial lensing with near-diffraction-limited focusing and quality
factors of
Q
=314
880. Notably, due to the local nature of these
modes, high quality factors are observed even with small illumination
areas, here 30
μ
m in diameter. This is in contrast to recent work with
guided-mode resonance structures, where illumination spots of up to
500
μ
m are required to attain considerable quality factors
30
.Although
the geometric tolerances of the proposed scheme are demanding, we
demonstrate a robust and consistent fabrication of the devices. The
measurements presented here were taken from
fi
ve different samples
fabricated with high yield and high reproducibility. The demonstrated
quality factors are currently limited by fabrication imperfections and
coupling between neighboring structures. Higher quality factors may
be realized by using structures supporting different higher-order Mie
modes. The amplitude variation across the resonant mode can be
reduced by modifying the device design, for example, by operating in
re
fl
ection (see Supplementary Note 6) or by using other types of
higher-order modes such as magnetic octupole, hexapole, or tor-
oidal modes.
The high level of wavefront control and strong light interactions
with our structure, as evidenced by the high quality factor, make our
higher-order Mie-resonant metasurfaces highly suitable for optical
sensing
15
, nonlinear optics
51
, directional lasing
52
and active wavefront
manipulation
53
.Figure
5
illustrates a comparison of the state of the art
of two-dimensional wavefront manipulation in terms of experimen-
tally demonstrated quality factor, numerical aperture, and overall
ef
fi
ciency for radial metalenses
23
,
24
,
30
,
34
,
54
. The concept reported here
enables wavefront control with unprecedented quality factors and
high numerical aperture. As compared to nonlocal metasurfaces
33
,
34
,
our method is polarization independent, robust against incidence
angle variations and shows higher quality factor, ef
fi
ciency, and
numerical aperture (see Supplementary Note 7 for a detailed com-
parison to previous work). Furthermore, in nonlocal metasurfaces the
quality factor for a given numerical aperture is limited by the band-
structure, and a single-layer surface shows an upper bound of 25%
ef
fi
ciency
33
,
34
. The method demonstrated here does not show any of
these limitations. Our design requires no additional polarization optics
and is also advantageous for the implementation of active optical
devices that dynamically modulate the dielectric environment of a
metasurface unit cell. For example, by introducing a refractive index
change in the nanoblocks reported here using either the thermo-optic
or electro-optic effect, our metasurface may be used to dynamically
steer light
55
. The high-
Q
metasurfaces demonstrated here may also be
realized at visible wavelengths, where the narrow spectral response is
advantageous for display and coloring applications
56
,
57
. We envision
that the extraordinary characteristics of higher-order Mie resonances
in high-index nanoblock arrays demonstrated here will lead to
numerous applications, as well as new physics
58
.
Methods
Experiment
The fabricated metasurfaces were characterized on a home-built
optical transmission microscope (schematically illustrated in Supple-
mentary Fig. 10). Coherent light from a wavelength-tunable diode laser
(Santec TSL-510) was loosely focused on the metasurface. The trans-
mitted light was collected with an objective lens (20×, 0.4 NA, Mitu-
toyo) and projected either onto an InGaAs IR camera (Xenics Bobcat
320) or a power meter (Thorlabs S122C). For measuring the trans-
mission spectrum, the laser wavelength was scanned, and the trans-
mitted power was recorded with the power meter. For the power
normalization, the sample was removed, and the illuminated power
was recorded through the same area. For the beam de
fl
ection mea-
surements, a Fourier plane was imaged onto the camera and a 0.9 NA
objective lens was used to capture all diffraction orders. For char-
acterizing the lenses, the focal plane was imaged onto the camera and a
scan along the optical axis was obtained by moving the surface along
the
z
direction.
The transmission of the metasurfaces,
T
=
P
T
/P
I
, is calculated by
recording the power transmitted through the metasurface,
P
T
, nor-
malized by the power incident on the metasurface,
P
I
.Forthebeam
de
fl
ectors, the diffraction ef
fi
ciency is de
fi
ned as the fraction of
transmitted power coupled into a speci
fi
c diffraction order. To this
end, the intensity around each diffraction order is integrated within a
square with a side length of 4 FWHM of the intensity of the diffraction
order. For the metalenses, the focusing ef
fi
ciency is de
fi
ned as the
fraction of transmitted power coupled into a circle around the focal
spot with a radius of 2 times the airy disk radius.
Scanning electron micrographs were acquired on an FEI Nova 200
NanoLab system to measure the sizes of the fabricated structures. For
imaging, the surfaces were covered with a 2 nm thick gold layer by
sputter deposition.
Fabrication
The metasurfaces were fabricated on borosilicate glass substrates
(
n
= 1.503) with a thickness of 220
μ
m. To remove organic residues
from the surface, the substrates were cleaned in an ultrasonic bath in
acetone, isopropyl alcohol, and deionized water each for 15 min, dried
using an N
2
gun, and subsequently cleaned using oxygen plasma.
Amorphous silicon was deposited onto the glass using plasma-
enhanced chemical vapor deposition. In a subsequent step, the
nanoblocks were written in a spin-coated MaN
2403 resist layer by
standard electron beam lithography. The nanoblocks were then
transferred to the amorphous silicon using a SiO
2
hard mask with
chlorine-based inductively coupled reactive ion etching. As a last step,
the residual mask was removed by immersing the samples in buffered
hydro
fl
uoric acid (1:7) for 5 s and subsequent rinsing in deionized
water. The uniform and beam de
fl
ector metasurfaces were fabricated
on an area of 150
μ
m×150
μ
m. The metalenses were fabricated with a
10
0
10
2
10
3
10
4
Q
(-)
10
1
10
0
10
2
10
3
10
4
Q
(-)
0
0.5
1
NA (-)
Khorasaninejad, 2016
Paniagua-
Domínguez, 2018
Arbabi, 2015
Lawrence, 2020
Malek, 2022
This Work
10
1
Local
Nonlocal
GMR 1D
Higher-order
Mie-resonant
0
50
100
Overall efficiency (%)
Nonlocal limit
Nonlocal
Higher-order
Mie-resonant
Local
GMR 1D*
Fig. 5 | Comparison of reported metasurfaces for two-dimensional wavefront
manipulation.
Comparison of our results (light blue crosses) to previously
reported experimental values
23
,
24
,
30
,
34
,
54
for two-dimensional metalenses in terms of
quality factor, numerical aperture, and overall ef
fi
ciency from Arbabi et al.
23
(yellow
cross), Khorasaninejad et al.
24
(blue square), Paniagua-Domínguez et al.
54
(red plus)
and Malek et al.
34
(green triangles). The dotted black line shows the upper bound on
the numerical aperture for nonlocal metasurfaces
33
,
34
. The results for guided mode
resonance (GMR) are shown for comparison
30
(purple circle), but only one-
dimensional wavefront manipulation is possible. *No overall ef
fi
ciency is reported
for GMR
30
.
Article
https://doi.org/10.1038/s41467-023-44164-4
Nature Communications
| (2023) 14:8476
6
diameter of 100
μ
m and a parabolic phase pro
fi
le according to the
equation
φ
x
,
y
ðÞ
=
2
π
λ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x
2
+
y
2
+
f
2
q

f

,
ð
1
Þ
where
λ
isthedesignwavelengthand
f
the focal length. All metalenses
were designed for a wavelength of
λ
= 1280 nm. The variation of the
nanorod side length is set according to Fig.
1
c with a discretization of
0.1 nm. Further metalens design parameters are given in Supplemen-
tary Table 2.
Simulation
The numerical modeling of the nanostructures was carried out using
an FDTD method. Simulations were performed with commercially
available FDTD software (Lumerical FDTD Solutions). A constant
refractive index of
n
= 1.503 was used for the borosilicate glass, and a
constant value of
n
= 3.45 for amorphous silicon, as experimentally
determined by ellipsometry. The simulations were carried out with a
spatially coherent plane wave illumination and periodic boundary
conditions were applied on all sides of the computational domain
unless otherwise noted. The smallest mesh-re
fi
nement of 5 nm was
used. For the oblique illumination simulations, the broadband
fi
xed
angletechniqueisused.
Data availability
All the relevant data of this study are included in the paper and sup-
plementary information
fi
le, and raw data are available from the cor-
responding author upon request.
References
1. Gérard, J. M. et al. Quantum boxes as active probes for photonic
microstructures: the pillar microcavity case.
Appl. Phys. Lett.
69
,
449
451 (1996).
2. Gérard, J. M. et al. Enhanced spontaneous emission by quantum
boxes in a monolithic optical microcavity.
Phys. Rev. Lett.
81
,
1110
1113 (1998).
3. Gorodetsky,M.L.,Savchenkov,A.A.&Ilchenko,V.S.UltimateQof
optical microsphere resonators.
Proc. SPIE
2799
,389
391 (1996).
4. Armani, D. K., Kippenberg, T. J., Spillane, S. M. & Vahala, K. J. Ultra-
high-Q toroid microcavity on a chip.
Nature
421
,925
928 (2003).
5. Pfeiffer,M.H.P.etal.Ultra-smoothsiliconnitridewaveguidesbased
on the Damascene re
fl
ow process: fabrication and loss origins.
Optica
5
, 884 (2018).
6. Ji, X. et al. Ultra-low-loss on-chip resonators with sub-milliwatt
parametric oscillation threshold.
Optica
4
,619(2017).
7. Akahane, Y., Asano, T., Song, B. S. & Noda, S. High-Q photonic
nanocavity in a two-dimensional photonic crystal.
Nature
425
,
944
947 (2003).
8. Deotare,P.B.,McCutcheon,M.W.,Frank,I.W.,Khan,M.&Lon
č
ar,
M. High quality factor photonic crystal nanobeam cavities.
Appl.
Phys. Lett
.
94
, 121106 (2009).
9. Wang, S. S. & Magnusson, R. Theory and applications of guided-
mode resonance
fi
lters.
Appl. Opt.
32
, 2606 (1993).
10. Hsu,C.W.etal.Observationoftrappedlightwithintheradiation
continuum.
Nature
499
,188
191 (2013).
11. Kodigala, A. et al. Lasing action from photonic bound states in
continuum.
Nature
541
,196
199 (2017).
12. Tittl, A. et al. Imaging-based m
olecular barcoding with pixelated
dielectric metasurfaces.
Science
360
, 1105
1109 (2018).
13. Koshelev, K. et al. Subwaveleng
th dielectric resonators for non-
linear nanophotonics.
Science
367
,288
292 (2020).
14. Armani, A. M., Kulkarni, R. P., Fraser, S. E., Flagan, R. C. & Vahala, K. J.
Label-free, single-molecule detection with optical microcavities.
Science
317
,783
787 (2007).
15. Yesilkoy, F. et al. Ultrasensitive hyperspectral imaging and biode-
tection enabled by dielectric metasurfaces.
Nat. Photonics
13
,
390
396 (2019).
16. Xu, Q., Schmidt, B., Pradhan, S. & Lipson, M. Micrometre-scale
silicon electro-optic modulator.
Nature
435
,325
327 (2005).
17. Phare, C. T., Daniel Lee, Y. H., Cardenas, J. & Lipson, M. Graphene
electro-optic modulator with 30 GHz bandwidth.
Nat. Photonics
9
,
511
514 (2015).
18. Spillane, S. M., Kippenberg, T. J. & Vahala, K. J. Ultralow-threshold
Raman laser using a spherical dielectric microcavity.
Nature
415
,
621
623 (2002).
19. Englund, D. et al. Controlling cavity re
fl
ectivity with a single quan-
tum dot.
Nature
450
,857
861 (2007).
20. Tomes, M., Vahala, K. J. & Carmon, T. Direct imaging of tunneling
from a potential well.
Opt. Express
17
, 19160 (2009).
21. Yu, N. et al. Light propagation with phase discontinuities: general-
ized laws of re
fl
ection and refraction.
Science
334
, 333
337 (2011).
22. Lin,D.,Fan,P.,Hasman,E.&Brongersma,M.L.Dielectricgradient
metasurface optical elements.
Science
345
,298
302 (2014).
23. Arbabi,A.,Horie,Y.,Ball,A.J.,Bagheri,M.&Faraon,A.
Subwavelength-thick lenses with high numerical apertures and
large ef
fi
ciency based on high contrast transmitarrays.
Nat. Com-
mun.
6
,1
10 (2015).
24. Khorasaninejad, M. et al. Meta
lenses at visible wavelengths:
diffraction-limited focusing and subwavelength resolution imaging.
Science
352
, 1190
1194 (2016).
25. Ni, X., Kildishev, A. V. & Shalaev, V. M. Metasurface holograms for
visible light.
Nat. Commun.
4
, 2807 (2013).
26. Zheng, G. et al. Metasurface holograms reaching 80% ef
fi
ciency.
Nat. Nanotechnol.
10
,308
312 (2015).
27. Huang, Y. W. et al. Gate-tunable conducting oxide metasurfaces.
Nano Lett.
16
,5319
5325 (2016).
28. Park, J. et al. All-solid-state spatial light modulator with indepen-
dent phase and amplitude control for three-dimensional LiDAR
applications.
Nat. Nanotechnol.
16
,69
76 (2021).
29. Yu, Y. F. et al. High-transmission dielectric metasurface with 2
π
phase control at visible wavelengths.
Laser Photonics Rev.
9
,
412
418 (2015).
30. Lawrence, M. et al. High quality factor phase gradient meta-
surfaces.
Nat. Nanotechnol.
15
,956
961 (2020).
31. Klopfer, E., Lawrence, M., Barton, D. R., Dixon, J. & Dionne, J. A.
Dynamic focusing with high-qu
ality-factor metalenses.
Nano Lett.
20
,5127
5132 (2020).
32. Wu, P. C. et al. Dynamic beam steering with all-dielectric electro-
optic III
V multiple-quantum-well metasurfaces.
Nat. Commun.
10
,
1
9 (2019).
33. Overvig, A. C., Malek, S. C. & Yu, N. Multifunctional Nonlocal
Metasurfaces.
Phys.Rev.Lett.
125
, 17402 (2020).
34. Malek, S. C., Overvig, A. C., Alù, A. & Yu, N. Multifunctional resonant
wavefront-shaping meta-optics based on multilayer and multi-
perturbation nonlocal metasurfaces.
Light Sci. Appl.
11
, 246 (2022).
35. Barton, D. et al. High-Q nanopho
tonics: sculpting wavefronts with
slow light.
Nanophotonics
10
,83
88 (2020).
36. Overvig, A. & Alù, A. Diffractive nonlocal metasurfaces.
Laser Pho-
tonics Rev
.
16
, 2100633 (2022).
37. Alaee,R.,Rockstuhl,C.&Fern
andez-Corbaton, I. An electro-
magnetic multipole expansion
beyond the long-wavelength
approximation.
Opt. Commun.
407
,17
21 (2018).
38. Savinov, V., Fedotov, V. A. & Zheludev, N. I. Toroidal dipolar exci-
tation and macroscopic electromagnetic properties of metama-
terials.
Phys. Rev. B
89
, 205112 (2014).
39. Novotny, L. & Hecht, B.
Principles of Nano-optics
. (John Wiley &
Sons, 2006).
https://doi.org/10.1016/S1748-0132(06)70120-7
.
40. Rybin, M. V. et al. High-Q supercavity modes in subwavelength
dielectric resonators.
Phys.Rev.Lett.
119
,1
5(2017).
Article
https://doi.org/10.1038/s41467-023-44164-4
Nature Communications
| (2023) 14:8476
7
41. Campione, S. et al. Broken symmetry dielectric resonators
for high quality factor fano metasurfaces.
ACS Photonics
3
,
2362
2367 (2016).
42. Prokhorov, A. V. et al. Resonant light trapping via lattice-induced
multipole coupling in symmetrical metasurfaces.
ACS Photonics
9
,
3869
3875 (2022).
43. Gigli, C. et al. Fundamental limitations of Huygens
metasurfaces
for optical beam shaping.
Laser Photonics Rev
.
15
,2000448 (2021).
44. Arslan, D. et al. Angle-selective all-dielectric Huygens
meta-
surfaces.
J. Phys. D Appl. Phys
.
50
,434002(2017).
45. Jeong, J. et al. High quality factor toroidal resonances in dielectric
metasurfaces.
ACS Photonics
7
, 1699
1707 (2020).
46. Yang, Y., Kravchenko, I. I., Briggs,
D. P. & Valentine, J. All-dielectric
metasurface analogue of electromagnetically induced transpar-
ency.
Nat. Commun.
5
,1
7(2014).
47. Kühne, J. et al. Fabrication robustness in BIC metasurfaces.
Nano-
photonics
10
,4305
4312 (2021).
48. Hsu,L.,Dupré,M.,Ndao,A.,Yellowhair,J.&Kanté,B.Localphase
method for designing and optimizing metasurface devices.
Opt.
Express
25
, 24974 (2017).
49. Elsawy, M. M. R., Lanteri, S., Duvigneau, R., Fan, J. A. & Genevet, P.
Numerical optimization methods for metasurfaces.
Laser Photonics
Rev.
14
,1
17 (2020).
50. Zhou, Y., Guo, S., Overvig, A. C. & Alù, A. Multiresonant nonlocal
metasurfaces.
Nano Lett.
23
,6768
6775 (2023).
51. Krasnok, A., Tymchenko, M. & Alù, A. Nonlinear metasurfaces: a
paradigm shift in nonlinear optics.
Mater. Today
21
,8
21 (2018).
52. Ha, S. T. et al. Directional lasing in resonant semiconductor
nanoantenna arrays.
Nat. Nanotechnol.
13
,1042
1047 (2018).
53. Hail, C. U., Michel, A. K. U., Poulikakos, D. & Eghlidi, H. Optical
metasurfaces: evolving from passive to adaptive.
Adv. Opt. Mater.
7
,
1801786 (2019).
54. Paniagua-Domínguez, R. et al. A metalens with a near-unity
numerical aperture.
Nano Lett.
18
,2124
2132 (2018).
55. Sokhoyan,R.,Hail,C.U.,Foley,M.,Grajower,M.&Atwater,H.A.
All-dielectric high-Q dynamica
lly tunable transmissive meta-
surfaces. Preprint at
arXiv
https//arxiv.org/abs/2309.08031 (2023).
56. Hail, C. U., Schnoering, G., Damak, M., Poulikakos, D. & Eghlidi, H. A
plasmonic painter
s method of color mixing for a continuous
red
green
blue palette.
ACS Nano
14
,1783
1791 (2020).
57. Kim,I.etal.Out
fi
tting next generation displays with optical meta-
surfaces.
ACS Photonics
5
, 3876
3895 (2018).
58. Santiago-Cruz, T. et al. Resonant metasurfaces for generating
complex quantum states.
Science
377
, 991
995 (2022).
Acknowledgements
This work was supported by the Air Force Of
fi
ce of Scienti
fi
c Research
under grant FA9550
18-1-0354 (C.U.H., R.S., and H.A.A.) and the Meta-
Imaging MURI grant #FA9550
21
1-0312(M.F.). C.U.H. also acknowl-
edges support from the Swiss National Science Foundation through the
Early Postdoc Mobility Fellowship grant #P2EZP2_191880. L.M.
acknowledges support from the Fulbright Fellowship program and the
Breakthrough Foundation. We gratef
ully acknowledge the critical sup-
port and infrastructure provided f
or this work by The Kavli Nanoscience
Institute at Caltech.
Author contributions
C.U.H, H.A.A., and R.S. conceived the project. C.U.H. performed the
simulations, fabricated the devices, built the experiment, performed the
measurements, and analyzed the results. R.S. and C.U.H. conceived the
metasurface design. M.F. assisted with simulations and fabrication. L. M.
assisted with analyzing the results. C.U.H. wrote the paper with input
from all other authors. H.A.A supervised all aspects of the project.
Competing interests
The Authors declare no competing interests.
Additional information
Supplementary information
The online version contains
supplementary material available at
https://doi.org/10.1038/s41467-023-44164-4
.
Correspondence
and requests for materials should be addressed to
Harry A. Atwater.
Peer review information
Nature Communications
thanks the anon-
ymous, reviewers for their contribution to the peer review of this work. A
peer review
fi
le is available.
Reprints and permissions information
is available at
http://www.nature.com/reprints
Publisher
s note
Springer Nature remains neutral with regard to jur-
isdictional claims in published maps and institutional af
fi
liations.
Open Access
This article is licensed under a Creative Commons
Attribution 4.0 International License, which permits use, sharing,
adaptation, distribution and reproduction in any medium or format, as
long as you give appropriate credit to the original author(s) and the
source, provide a link to the Creati
ve Commons license, and indicate if
changes were made. The images or other third party material in this
article are included in the article
s Creative Commons license, unless
indicated otherwise in a credit line to the material. If material is not
included in the article
s Creative Commons license and your intended
use is not permitted by statutory re
gulation or exceeds the permitted
use, you will need to obtain permission directly from the copyright
holder. To view a copy of this license, visit
http://creativecommons.org/
licenses/by/4.0/
.
© The Author(s) 2023
Article
https://doi.org/10.1038/s41467-023-44164-4
Nature Communications
| (2023) 14:8476
8