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Supplementary
Information:
High quality factor metasurfaces for two
-dimensional wavefront manipulation
Authors
Claudio U. Hail
1
, Morgan Foley
2
, Ruzan Sokhoyan
1
, Lior Michaeli
1
, Harry A. Atwater
1
*
Affiliations
1
Thomas J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena,
California 91125
2
Department of Physics, California Institute of Technology, Pasadena, California 911125
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Supplementary Figures
Supplementary
Figure
1 | Array
-size dependence. a
, Simulated transmission of a finite array of
N
×
N
nanoblocks with
L =
55 5 nm,
H
= 695 nm, and
P
= 736 nm for varying numbers of repetitions
N
in the array
.
b
,
Maximum electric field enhancement in the central nanoblock of the finite array with varying number of
repetitions.
c
, Quality factor of the transmitted light of the finite array of nanoblocks with varying number of
repetitions. The dashed lines
in (
b
) and (
c
) show the field enhancement and quality factor of the periodic array for
comparison. Beyond
N
= 10 the response of the finite array is similar to the periodic case.
Supplementary
Figure
2 | Comparison of experiment to simulation. a
,
Measured
transmission
(Meas.)
of a
nanoblock array with
L
= 567 nm,
H
= 695 nm
, and
P
= 736 nm
and simulated transmission
(Sim.)
with
modifications in the geometry to account for fabrication
imperfections
. In the simulation, the length of the
nanoblock
is
L
= 516 nm, the side
wall tilt
α
= 12.9°,
the height is
H
= 699 nm, the length and height of the
remaining SiO
2
hard mask are
L
SiO2
=
416 nm and
H
SiO2
= 120 nm, and the undercut is
d
u
=
35 nm deep and 65
nm wide.
b
, Simulated
electric field
amplitude
profile
s in a
n
x-
z and
y-
z cross section
s of a periodic amorphous
silicon nanoblock array on a glass substrate at
λ
= 1267 nm
,
with
modification in the geometry to account for
fabrication imperfections. The similarity to Supplementary Fig.
10
shows that the same ED/EO modes are being
measured. The modeled fabrication imperfections include tilted side
walls, an undercut below the nanoblock and a
residual SiO
2
hard mask as indicated by the white solid lines in (
b
). The
simulated and measured quality factors of
Q
sim
= 660 and
Q
meas
= 668 are in good agreement.
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Supplementary Figure 3
| High quality factor beam deflection to
θ
= 25.8° and
φ
= 25.8°.
a, b,
Experimentally
measured diffraction efficiencies 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 deflection of
x
-polarized light along the
y
direction and
(
b
) TM deflection of
x
-polarized light along the
x
direction.
The desired diffraction order is +1, with
θ
= 25.8° and
φ
= 25.8° respectively. On resonance,
λ
R
= 1280.8
nm, a diffraction efficiency of
55.9% and
17.7
% is attained for
the TE and TM mode,
respectively. Off
-resonance,
λ
O
= 12
73
nm,
99.6% and 99.6% of the transmitted light
remains in the surface normal direction
. The insets show a zoomed in region of the plots.
Supplementary Figure 4
| High quality factor beam deflection to
θ
= 35.7° and
φ
= 35.7°. a, b,
Experimentally
measured diffraction efficiencies of the
-1 (yellow
), 0
(blue)
and
+1
(red)
diffraction
orders
and Fourier plane
images of a metasurface showing (
a
) TE deflection of
x
-polarized light along the
y
direction and (
b
) TM deflection
of
x
-polarized light along the
x
direction. The desired diffraction order is +1, with
θ
= 35.7° and
φ
= 35.7°
respectively. On resonance,
λ
R
= 12
89
nm, a diffraction efficiency of
51
% and
18.1% is attained for the TE and
TM mode,
respectively. Off
-resonance,
λ
O
= 1282
nm,
99.4%
and
50.98% of the transmitted light remains in the
surface normal
direction.
The insets show a zoomed in region of the plots.
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Supplementary
Figure
5 | Spectral diffraction efficiency for TM
deflection at different angles
.
Measured
spectral diffraction efficiency of TM light deflection with varying deflection angle
from the same structures as
shown in Fig. 3e for TE light deflection.
The measured curves are shi
fted by 20% from each other for better
visibility.
Supplementary Figure
6 | Measured transmission of metalenses. a,
Measured transmittance of the metalens
from Fig. 4a
-e. The transmission on resonance is 55.6%.
b
,
Measured transmittance of the metalens from
Supplementary Fig. 7a. The transmission on resonance is 77.8%.
c,
Measured transmittance of the metalens
from Supplementary Fig.
7
b
. The transmission on resonance is 37.
7
%.
The resonance wavelength is the
wavelength where the focusing efficiency is maximized.
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Supplementary Figure 7
| Metalenses
with
high
quality factor. a
-e,
Measured field intensity at the focal plane
on resonance
and off resonance, and the measured
spectral
focusing
efficiency
of
metalens
es with numerical
aperture of (
a
) 0.1 (
b
) 0.18 (
c
) 0.4 (
d
) 0.6 (
e
) 0.8.
The respective resonance wavelengths and determined quality
factors are indicated
on the panels.
The blue line shows
the
meas
ured
focusing efficiency
and the red line shows
a Fano fit to the data
.
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Supplementary Figure 8
| Strehl r
atio
calculations of c
haracterized
metalenses. a-e,
Measured field intensity
profile at the focal plane on resonance
of
high
-Q metalens
es (blue) and an airy disk function (red) with the same
numerical aperture for comparison.
f,
Calculated Strehl ratios for the characterized metalenses
with numerical
apertures of 0.1, 0.18, 0.4, 0.6 and 0.8 and different resonance wavelengths
. The Strehl ratio
(SR)
is calculated
by integrating the intensity in the focal plane around the focal spot within a radius of eight
times
the
diffraction
-
limited airy disk radius.
Supplementary Figure 9 | Numerical optimization of high
-quality factor TM beam deflection to
φ
= 35.
8° .
a,
b,
Simulated
diffraction efficiencies of the
-1 (
yellow
), 0
(blue)
and
+1
( red)
diffraction orders and Fourier plane
images of a metasurface showing TM deflection of
x
-polarized light along the
x
direction
for (
a
) a forward design
structure and (
b
) an optimized structure using a particle swarm optimization. The desired diffraction order is +1,
with
φ
= 35.
8°. On resonance,
λ
R
= 1291.9
nm, a diffraction efficiency of
46.5
% and
81.9% is attained for the
forward design and the optimized design, respectively.
The design includes 3 nanoblocks
per Fresnel zone and
nanoblock side lengths are [553.9, 554.9, 557
] nm and [554.8, 554.9, 558] nm for the forward design and
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optimized design respectively
. For the optimization the rod lengths
L
1
and
L
3
are varied and
L
2
is fixed.
Additionally, the smallest mesh
refinement is set to 10 nm, to reduce the computational cost of the optimization.
Supplementary
Figure
10
| Experimental set
-up
.
The fabricated samples are illuminated in transmission with
loosely focused light from a tunable diode laser. The transmitted light is collected by an imaging objective (
20x,
0.4
NA) and projected onto a InGaAs IR camera through a set of lenses. With a flip mirror the transmitted light can
be either sent to the camera or to a power meter. In the detection path an image plane is formed with an
adjustable iris to limit the area that is projected on the camera or power meter. For Fourier plane imaging a 0.9 NA
objective lens is
used,
and
a Fourier plane is formed on the camera
sensor
by exchanging
the
lens
before the
camera to a different focal length. A linear polarizer (LP) is used to set the incident light polarization.
Supplementary Tables
Figure panel
λ
R
(nm)
θ
(
°
)
Phase gradient
(rad/
μ
m)
Fresnel Zone
s
per surface
Nanoblocks
per
Fresnel Zone
Designed nanoblock
lengths (nm)
3a, b, c, d
1295
26
2.134
51
4
[587.1, 588.6, 588.8, 591.5]
3c, d
127
2.5
25.6
2.134
51
4
[569.1, 570.6, 570.8, 573.5]
3c, d
, e, f
1280.8
25.8
2.134
51
4
[575.1, 576.6, 576.8, 579.5]
3c, d
1306.3
26
.3
2.134
51
4
[603.1, 604.6, 604.8, 607.5]
3c, d
13
19.4
26.6
2.134
51
4
[613.1, 614.6, 614.8, 617.5]
3e, f
1277.7
16.8
1
.423
34
6
[574.2, 576.2, 576.4, 576.6,
577.1, 581.4]
3e, f
1281.2
20.4
1.707
40
5
[
57
4.6, 576.3, 576.5, 576.7,
580.8]
3e, f
1289
35.7
2.846
68
3
[575.6, 576.6, 578.7]
Supplementary Table 1 | Design parameters of beam deflector metasurfaces.
Parameters
are given for all
the beam deflector metasurfaces in Fig
.
3. The
fabricated metasurface size is 150
μ
m x 150
μ
m. The periodicity of
the nanoblocks is
P
= 736 nm.
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Lens
d
iameter
(
μ
m)
Focal
length
(
μ
m)
Numerical
aperture
Fresnel Zones
per surface
Nanoblocks per
Fresnel Zone
100
495
0.1
2
20-48
100
274
0.18
4
9-
36
100
114.6
0.4
4
5-24
100
66.6
0.6
13
3
-18
100
37.5
0.8
39
2-14
Supplementary Table
2 | Design parameters of metalenses.
Design parameters are given for the metalenses
in Fig.
4 and Supplementary Fig. S7
. The periodicity of the nanoblocks is
P
= 736 nm.
The phase distribution is
parabolic and the nanorod lengths are chose
n according to Fig. 1c.
A constant offset is equally added to all
nanoblocks
of a metalens, resulting in the variation of the resonance wavelengths in Figure 4f.
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Supplementary
Note 1:
Mode profile
s and m
ultipole expansion
To gain an understanding of the optical modes supported by
the
metasurface
unit cells
we
perform finite difference time domain simulations. Supplementary Fig. 11
shows the electric and
magnetic field profiles
on resonance in a periodic array of amorphous silicon nanoblocks with the size
corresponding to the
simulation
shown in Fig. 1b. The electric and magnetic field are enhanced by a
factor of 28 and 46, respectively.
Supplementary
Figure
11
| Profiles of electric and magnetic field amplitude in a periodic array
.
Simulated
electric (
a
) and magnetic (
b
) field amplitude in an amorphous silicon nanoblock
in a periodic array on a glass
substrate with
L =
55 5 nm,
H
= 695 nm, and
P
= 736 nm at a wavelength
λ
= 1288 nm. The illumination is incident
along the positive
z
direction with the polarization along the
x
direction.
Cross sections are shown for
y
= 0,
x
= 0
and
z
=
H
/2. The electric and magnetic field are normalized by the incident field amplitude
E
0
and
H
0
, respectively.
The magnetic field profile closely resembles the profile of an electric octupole mode
in an isolated sphere as
shown in Supplementary Fig. 15.
To understand the origin of the
resonance mode and the
high quality factor we perform a
multipole expansion
1
. Supplementary
Fig
. 12
illustrates
the
different resonant components of the
scattering cross section of an individual nanoblock in free space
as calculated from a
multipole
expansion. Clear resonant features are observed such as the magnetic dipole (MD), electric dipole
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(ED), magnetic quadrupole (MQ)
and electric octupole (EO). There is good agreement between the
scattering cross section calculated from the multipole expansion and the corresponding FDTD
calculation. Of all the resonant modes t
he EO at
λ
=
1.16
μ
m
is specifically
in the vicinity of the
operation wavelength of the metasurface. The corresponding electric and magnetic field profiles of the
EO at
λ
= 1.16
μ
m of a single nanoblock in free space are shown in Supplementary Fig. 1
3
as
simulated fr
om FDTD simulations. For these simulations, a total
-field scattered-
field source
and
perfectly matched layer boundary conditions
were used. In the magnetic field profiles the resemblance
to the field profiles of the periodic array in Supplementary Fig. 1
1 is evident. The linewidth of the EO is
approximately
9 nm, suggesting
that in a periodic array the near
-field coupling of the neighboring
nanoblocks further narrows the resonance
, something that is also
observed in low-
order Mie
-resonant
metasurfaces
2
. We expect that for
our metasurface, the influence of the substrate and the neighboring
elements in the array red shift the EO resonance to
λ
= 1.288
μ
m.
Supp
lementary
Figure
12 | Multipole expansion o
f an isolated nanoblock
. a
, Scattering cross section of a
single amorphous silicon nanoblock in free space with
L
= 5
55
nm and
H
= 6
95
nm
calculated using the multipole
expansion method
1
. For comparison,
the sum of the multipoles and the scattering cross section as calculated by
FDTD is shown.
b
,
A zoomed in view of (
a
) over the wavelength range of the electric octupole mode.
c
,
Radiation
pattern of an isolated nanoblock in free space at the EO resonant wavelength
λ
= 1.16
μ
m.
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Supplementary
Figure
13 | Profiles of electric and magnetic field amplitude of an
isolated nanoblock
.
Simulated electric (
a
) and magnetic (
b
) field intensity in an amorphous silicon nanoblock in free space with
L
=
555
nm and
H
= 695 nm
at a wavelength
λ
= 1.16
μ
m. The illumination is along the positive
z
direction with the
polarization along the
x
direction.
Cross sections are shown for
y
= 0,
x
= 0 and
z
=
H
/2.
To account for the
neighboring effect between different nanoblocks,
we adopt the
formalism
by
Savinov et al.
3
to calculate the reflected
field amplitude and phase of a periodic array of nanoblocks
from the multipoles
. Supplementary
Fig
. 14a shows the contributions of the different multipoles towards
the reflected field amplitude. A strong reflectance peak is observed at the resonance wavelength
of our
metasurface
. The main contributions to the reflected field amplitude are due to the
spectrally
overlapped
electric dipole and electric octupole modes
. This spectral overlapping of the ED and EO,
and their interference with the transmitted light, results in a narrow high quality factor resonance with a
vanishing
transmission
on resonance.
The phase of each of the multipole terms in the reflected field i
s
shown in Supplementary
Fig.
1
4b. On resonance the ED and EO are in phase and of equal magnitude
leading to a generalized Kerker effect. We note that although
the formalism by Savinov et al.
3
was
derived for periodic arrays of scatterers without a substrate, a qualitative analysis can still be made in
the presence of a substrate
4
. We further also confirmed the presence of the EO/ED mode without a
substrate and similar behavior in the
reflection amplitude and phase was obtained as for the case with
a substrate shown here.