Third harmonic
generation
enhancement and wavefront control using
a local high
-Q metasurfa
ce
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Supplementary Information: Third harmonic generation
enhancement
and
wavefront control using
a local high-
Q metasurface
Claudio U. Hail
1
, Lior Michaeli
1
, Harry A. Atwater
1
*
1
Thomas J. Watson Laboratory of Applied Physics, California Institute of Technology,
Pasadena, California 91125, United States
* Correspondence and requests for materials should be addressed to H.A.A (email:
haa@caltech.edu
).
Third harmonic
generation
enhancement and wavefront control using
a local high
-Q metasurfa
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Supplementary Figure
S1
| Multipole expansion of the reflected field amplitude
.
Calculated
contributions of the electric dipole (ED), magnetic dipole (MD), magnetic quadrupole (MQ), electric
quadrupole (EQ), and electric octupole (EO) to the reflected field amplitude determined from a multipole
expansion of a nanoblock within a periodic array
1
. The metasurface dimensions are
P
= 736 nm,
H
= 695
nm
and
L
= 555 nm.
Supplementary Figure
S2 | Simulated angular dispersion for the metasurface in TE and TM
polarization
.
Simulated linear transmission (
T
) of the metasurface in Fig. 1b with
P
= 736 nm,
H
= 695 nm,
L
= 555 nm
for varying incident angles
for TE (
a
) and TM (
b
) polarization.
Supplementary Figure
S3 | NIR
pump spectr
a.
Experimentally measured pump spectra corresponding
to the resonant pump excitation of the metasurfaces illustrated in Fig. 2d.
Third harmonic
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enhancement and wavefront control using
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Supplementary Figure
S4 | TH
phase look up table.
Simulated TH phase with varying nanoblock side
length
L,
P
= 736 nm, and
H
= 695 nm at a wavelength of
λ
= 1260
nm
, with the same geometrical
parameters as in Fig. 1b.
Note that the characterized TH
metalens
in Fig. 4 is spectrally shifted with respect
to this design wavelength to maximize THG due to critical coupling.
Hence, w
e assume that the phase look
up table does not significantly change with respect to the resonant wavelength.
Supplementary Figure S
5 | Experimental set
-up.
The fabricated samples are illuminated in transmission
with loosely focused light from
either a white light supercontinuum laser or a fs pulsed laser. The transmitted
light
is collected by an imaging objective (
50x, 0.
95
NA) and projected on the respective sensor for
detection.
With
a set of flip
mirrors,
the transmitted light can be either sent to
a visible camera,
power meter
or spectrometer, or then to a NIR camera or spectrometer
. For
the measurements presented in Fig. 3, the
focusing lens is replaced with an objective lens (10x, 0.25 NA) and
the numerical aperture of the illumination
is set with an adjustable iris
that
is placed in front of the objective lens
.
Third harmonic
generation
enhancement and wavefront control using
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-Q metasurfa
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Supplementary Figure S
6 | Measured refractive index of
amorphous silicon.
Real and imaginary
refractive index of amorphous silicon as fitted to ellipsometry measurements using a Tauc
-Lorentz model
.
Third harmonic
generation
enhancement and wavefront control using
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-Q metasurfa
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Method
Maximum Q
factor
TH
Efficiency
η
(×10
-6
)
Normalized TH
efficiency
η
/
I
p
2
(cm
4
/GW
2
)
Peak pump
intensity
I
p
(GW/cm
2
)
Wavefront
manipulation
Ref. 34
Guided mode
~20
0.176
0.65
0.52
No
Ref. 35
EIT
-analogue
466
1.2
0.117
3.2
No
Ref. 44
Higher order
Mie resonances
NA
0.28
0.194
1.2
No
Ref. 38
q- BIC
18511
10.9
90
0.1
No
Ref. 40
q- BIC
~160
~1
~0.149
2.59
No
Ref. 41
q- BIC
600
1.8
11.25
0.4
No
Ref. 28
Non
-
resonant
geometric
phase
NA
10
-5
0.0015
0.082
Yes
Ref. 24
Low order Mie
resonances
NA
~1
NA
NA
Yes
Ref. 25
Low order Mie
resonances
NA
~1
~0.44
1.5
Yes
Ref. 27
Low order Mie
resonances
~40
1.1
0.001
33
Yes
This
work
Higher order
Mie resonances
470
26.5
0.25
10.3
Yes
Supplementary Table
S1
| Comparison to the state of the art of THG with silicon based
metasurfaces.
Comparison of the current state of the art of THG with silicon based metasurfaces.
Experimentally reported values are compared.
Lens diameter
(
μ
m)
Focal
length (
μ
m)
Numerical
aperture
Fresnel Zones
per surface
Nanoblocks per
Fresnel Zone
100
329.5
0.1
5
6
6–
29
Supplementary Table
S2
| Metalens design parameter
s.
Design parameters for the metalens in Fig.
4 .
The periodicity of the nanoblocks is
P
= 736 nm
and
H
= 695 nm. The phase distribution is parabolic and
the nanoblock side
lengths are set
according to Supplementary
Fig.
4.
Third harmonic
generation
enhancement and wavefront control using
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Supplementary Note 1: Methods
Experiment
The fabricated metasurfaces were characterized on a home
-built optical
transmission microscope (schematically illustrated in Fig. S5). For the linear optical
characterization, coherent light from a supercontinuum laser (NKT, Super K Extreme) was loosely
focused on the metasurface with a lens of
f
=
50 mm. The transmitted light was collected with an
objective lens (50x, 0.95 NA, Zeiss) and projected on to a NIR grating spectrometer (Princeton
Instruments, Acton 2500, PylonNIR). For the nonlinear optical characterization, a Ti:Sapphire
laser with 100 fs pulse length and 10 kHz repetition rate (Coherent Libra) was used to drive an
optical parametric amplifier (Coherent, OPerA Solo) to generate a NIR pump beam within the
wavelength range 1.2–
1.6
μ
m. The pump beam was loosely focused on the metasurface with a
lens of
f
=
50 mm, the transmitted light was filtered with two short pass filters (Thorlabs FGS900),
and the remaining TH beam was projected onto a visible grating spectrometer (Princeton
Instruments, Acton 2500, Pixis). The pump spot size was 100
μ
m unless noted otherwise. The
pump power was measured by placing a power meter (Thorlabs S122C) at the location of the
metasurface. The third harmonic power was measured by integrating the intensity vs wavelength
obtained by the grating spectrometer camera. For this the spectrometer counts were converted
to optical power by performing a calibration with a power meter (Thorlabs S130C). For the
metalens characterization, the focal plane was imaged on a CMOS camera (Thorlabs
DCC1645C
-HQ) and a scan along the optical axis was performed by moving the surface along
the
z
direction. The pump intensity and beam profile were measured by projecting the collected
pump beam on to an InGaAs camera without the low pass filters in place.
By imaging the sample
plane on the InGaAs camera we monitor the metasurface position and the position, size and
angular content of the pump beam. For the experiments reported in Fig. 3 an objective lens (10x,
0.25 NA, Olympus) was used to focus the pump on the sample. With an overfilled aperture, the
NA was set with an iris placed directly before the objective lens.
T he dominant sources of
Third harmonic
generation
enhancement and wavefront control using
a local high
-Q metasurfa
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measurement
errors are the measurement and calibration of the TH power with the spectrometer,
and the measurement of the incident pump power with the photodiode power meter. For the
spectrometer, these errors are
±
0.3% noise,
±
1% in linearity error, as per manufacturer
specification, and
±
8% in power calibration error. For the power meter there is a
±
5%
measurement uncertainty and a
±
0.5% linearity error. In addition to that, there can be errors
arising from thermal drift in the optical components of the microscope over time.
Scanning electron micrographs were acquired on an FEI Nova 600 NanoLab system to
measure the sizes of the fabricated structures. For imaging, the surfaces were covered with a
4 nm thick carbon 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 a N
2
gun. 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 an SiO
2
hard mask with chlorine-
based inductively
coupled reactive ion etching. The uniform metasurfaces were fabricated on an area of 150
μ
m ×
150
μ
m. The metalens was fabricated with a diameter of 100
μ
m and a parabolic phase profile
according to the equation
휑휑
(
푥푥
,
푦푦
)
=
2
휋휋
휆휆
�
�
푥푥
2
+
푦푦
2
+
푓푓
2
−푓푓�
,
(1)
where
λ
is the design wavelength (
λ
= 442
nm) and
f
the focal length. The variation of the
nanoblock side length is set with a discretization of 0.1 nm according to Fig. S4. Further metalens
design parameters are given in Table S2.
Third harmonic
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enhancement and wavefront control using
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Simulation
The numerical modelling of the nanostructures was carried out using an FDTD
method. Simulations were performed with a commercially available FDTD software (Lumerical
FDTD Solutions). In the NIR spectral range a constant refractive index of
n
= 1.503 was used for
the borosilicate glass and
n
= 1.453 for the SiO
2
. In the TH spectral range
n
= 1.53 and
n
= 1.47
are used, respectively. For amorphous silicon experimentally measured values were used as
determined by ellipsometry (see Fig. S6). The simul
ations
were
carried out with a spatially
coherent plane wave illumination, a perfectly matched layer boundary condition along the
z
direction, and periodic boundary conditions were applied along the
x
and
y
direction. For the
nanoblock a smallest mesh-
refinement of 5 nm was used.
The third harmonic phase was determined from finite element simulations by solving
Maxwell’s equations in the frequency domain using the commercially available software COMSOL
Multiphysics. A first simulation is performed over the spectral range of the pum
p to obtain the
electric field in the silicon nanoblocks, and thereby the nonlinear polarization
P
NL
(3
ω
) =
ε
0
χ
(3)
E
3
(
ω
).
A second simulation is then performed in the TH spectral range introducing a current source
generated by
P
NL
. For the simulations we adopt the undepleted pump approximation
2
, we
approximate the susceptibility tensor for amorphous silicon to a scalar non-
dispersive value of
χ
(3)
= 2.45 ×
10
-19
m
2
/V
2
,
3,4
and we use periodic boundary conditions along the
x
and
y
direction,
and a perfectly matched layer along the
z
direction.
Supplementary Note
2: Numerical aperture dependent TH power
To provide further insight into the incident angle dependence of TH power from a
metasurface we calculate the TH power emitted when illuminat
ing
with a beam with varying
focusing numerical aperture. For this we assume a thin metasurface (
d
<<
λ
) and an illumination
with a
monochromatic
gaussian beam
of width
w,
an illumination cone angle
θ
, and total
incident
power
P
ω
. Additionally,
we adopt the undepleted pump approximation. Furthermore, w
e adopt the
hypothesis that
the effective third order nonlinear susceptibility of the metasurface
χ
eff
(3)
is angle
Third harmonic
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independent
, which we then later test against the experimental data
( see
Fig. 3b)
. The electric
field
amplitude
generated by the metasurface at the TH is approximated
by
2,5
퐸퐸
3휔휔
(
푟푟
) =
푖푖
3
휔휔휔휔
8
푛푛
3휔휔
푐푐
휒휒
푒푒푒푒푒푒
(
3
)
퐸퐸
휔휔
3
(
푟푟
),
(1)
where
E
ω
is the field from the gaussian illumination at the metasurface,
d
is the thickness of the
metasurface
, and
n
3
ω
is the refractive index at the
TH
. Equation (1) is used as an approximation
as it assumes collimated illumination. The TH power
,
P
3ω
,
is obtained by integrating the third
harmonic intensity over the entire area of the illuminating beam
푃푃
3휔휔
=
�
1
2
푛푛
3휔휔
휀휀
0
푐푐
|
퐸퐸
3휔휔
(
푟푟
)
|
2
휔휔
푑푑
=
�
9
휀휀
0
휔휔
2
휔휔
2
128
푛푛
3휔휔
푐푐
�휒휒
푒푒푒푒푒푒
(
3
)
퐸퐸
휔휔
3
(
푟푟
)
�
2
휔휔
푑푑
,
(
2
)
where
n
ω
is the refractive index at the
pump wavelength,
c
represents the speed of light and
ε
0
the vacuum permittivity. This can be rewritten to
푃푃
3휔휔
=
�
9
휀휀
0
휔휔
2
휔휔
2
128
푛푛
3휔휔
푐푐
�휒휒
푒푒푒푒푒푒
(
3
)
�
2
�
2
푛푛
휔휔
휀휀
0
푐푐
퐼퐼
(
푟푟
)
�
3
휔휔
푑푑
,
(
3
)
where
퐼퐼
(
푟푟
)
=
2푃푃
휔휔
휋휋푤푤
2
exp
�
−2푟푟
2
푤푤
2
�
i s the intensity of the gaussian beam. Inserted into the integral
, the
following expression is obtained
푃푃
3휔휔
=
9
휔휔
2
휔휔
2
16
푛푛
3휔휔
푛푛
휔휔
3
휀휀
0
2
푐푐
4
�휒휒
푒푒푒푒푒푒
(
3
)
�
2
8
푃푃
휔휔
3
휋휋
3
푤푤
6
� ��푒푒
−
2푟푟
2
푤푤
2
�
3
푟푟
휔휔
푟푟
휔휔
푟푟
∞
0
2
휋휋
0
.
(
4
)
This is then evaluated to
푃푃
3휔휔
=
3
휔휔
2
휔휔
2
푃푃
휔휔
3
4
푛푛
3휔휔
푛푛
휔휔
3
휀휀
0
2
푐푐
4
휋휋
2
푤푤
4
�휒휒
푒푒푒푒푒푒
(
3
)
�
2
~
푁푁
푑푑
4
.
(
5
)
Supplementary Note 3
: Effective third order nonlinear susceptibility
To determine the effective third order nonlinear susceptibility of the metasurface we follow
a similar analysis as in the theoretical calculation
described
in Supplementary Note 2
. The
pump
intensity
I(r)
of the illuminating beam, the pump power
P
ω
,
and the TH power
P
3ω
are now known
quantities from the measurement.
However, s
ince the camera only provides relative intensity
Third harmonic
generation
enhancement and wavefront control using
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-Q metasurfa
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data, the measured intensity of the camera,
I
cam
, needs
to be renormalized to the total pump power
by integration
over the entire beam size
푃푃
휔휔
=
�
1
2
푛푛
휔휔
휀휀
0
푐푐
|
퐸퐸
휔휔
|
2
휔휔
푑푑
=
�휂휂퐼퐼
푐푐푐푐푐푐
휔휔
푑푑
.
(
6
)
This allows
to
determine the scaling factor
η
, which then allows obtaining the
squared
electric
field amplitude
|
퐸퐸
휔휔
|
2
=
2
휂휂
푛푛
휔휔
휀휀
0
푐푐
퐼퐼
푐푐푐푐푐푐
.
(
7
)
With this
adapted intensity
we can rewrite Eq. (3) to
푃푃
3휔휔
=
�
9
휀휀
0
휔휔
2
휔휔
2
128
푛푛
3휔휔
푐푐
�휒휒
푒푒푒푒푒푒
(
3
)
�
2
�
2
휂휂
푛푛
휔휔
휀휀
0
푐푐
퐼퐼
푐푐푐푐푐푐
�
3
휔휔
푑푑
=
�
9
휔휔
2
휔휔
2
휂휂
3
16
푛푛
3휔휔
푛푛
휔휔
3
휀휀
0
2
푐푐
4
�휒휒
푒푒푒푒푒푒
(
3
)
�
2
퐼퐼
푐푐푐푐푐푐
3
휔휔
푑푑
.
(
8
)
Solving this for
χ
eff
(3)
yields
휒휒
푒푒푒푒푒푒
(
3
)
=
�
푃푃
3휔휔
∫
9
휔휔
2
휔휔
2
휂휂
3
16
푛푛
3휔휔
푛푛
휔휔
3
휀휀
0
2
푐푐
4
퐼퐼
푐푐푐푐푐푐
3
휔휔
푑푑
,
(
9
)
where the integral is evaluated numerically.
In the
experiment
, the pump power is measured before the focusing objective lens. As a
result
, w
e perform a separate measurement of the transmission of the focusing objective lens for
varying numerical apertures
to account for this effect
when obtaining the pump power incident on
the metasurface. The measured transmission of the focusing objective lens is illustrated in
Supplementary Fig. 7.
Third harmonic
generation
enhancement and wavefront control using
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-Q metasurfa
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Supplementary Figure
S7
| Measured objective transmission vs NA at
λ
= 1327 nm.
For the
measurement we use two identical colinearly aligned objective lenses (10x, 0.25 NA, Olympus) and
measure the power before and after the two lenses for varying beam sizes, corresponding to the respective
numerical aperture. This avoids errors induced by the incident angle dependent absorption usually
encountered in photodiode power meters.
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Savinov, V., Fedotov, V. A. & Zheludev, N. I. Toroidal dipolar excitation and macroscopic
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, (2014).
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Boyd, R. W.
Nonlinear Optics
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(Academic Press, 2008).
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Wang, L.
et al.
Nonlinear Wavefront Control with All
-Dielectric Metasurfaces.
Nano Lett.
18
, 3978–
3984 (2018).
4.
Koshelev, K.
et al.
Nonlinear Metasurfaces Governed by Bound States in the Continuum.
ACS Photonics
6
, 1639–
1644 (2019).
5.
Kumar, N.
et al.
Third harmonic generation in graphene and few
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Phys. Rev. B
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, 121406 (2013).