of 8
Supporting Information for “Experimental Demonstrat
ion of >230° Phase
Modulation in Gate-Tunable Graphene-Gold Reconfigur
able Mid-Infrared
Metasurfaces”
Michelle C. Sherrott
1,2ǂ
, Philip W.C. Hon
2,3ǂ
, Katherine T. Fountaine
2,3
, Juan C. Garcia
3
,
Samuel M. Ponti
3
, Victor W. Brar
1,4
, Luke A. Sweatlock
2,3
, Harry A. Atwater
1,2
*
1. Thomas J. Watson Laboratory of Applied Physics,
California Institute of Technology,
Pasadena, CA 91125, USA
2. Resnick Sustainability Institute, California Ins
titute of Technology, Pasadena, CA
91125, USA
3. Northrop Grumman Corporation, NG Next Nanophoton
ics & Plasmonics Laboratory,
Redondo Beach, CA 90278, USA
4. Department of Physics, University of Wisconsin;M
adison, Madison, WI 53706, USA
ǂ
Equal contributors
*Corresponding author: Harry A. Atwater (
haa@caltech.edu
)
I.
Fermi energy-dependent Field Profiles (8.5 μm)
As the Fermi energy of the graphene is changed from
0 to 0.4 eV, the graphene
permittivity shifts the resonance to shorter wavele
ngths, therefore decreasing the field
confinement in the gap between gold dipoles at a wa
velength of 8.5 Cm. This transition
happens quickly between 0.2 and 0.4 eV, whereas les
s change is apparent between E
F
= 0
and 0.2 eV, consistent with absorption and phase mo
dulation results presented in the
main text, Figure 1d and 1e.
Figure S1
: Electric field profiles in the gap between gold d
ipoles at a wavelength of 8.5
Cm. Fermi energies of 0, 0.2, and 0.4 eV are presen
ted.
II.
Angle-dependent absorption spectra
To address the angle;dependent response of our stru
cture, we present absorption spectra
of our structure at angles of 0 – 30° for Fermi ene
rgies of 0, 0.25, and 0.5 eV. It is clear
that at larger angles of incidence, representative
of those present in an FTIR
measurement, a shoulder in the data emerges at 9 Cm
. This explains qualitatively the
experimental results presented in Figure 2c of the
main text, where a clear shoulder is
observed. Additionally, the spectra blue shifts wit
h increasing angle of incidence,
explaining the blue shift and broadening observed i
n the FTIR measurements with respect
to the simulations.
Figure S2
: Angle;dependent absorption spectra of resonant ge
ometry at different Fermi
energies. a) E
F
= 0 eV, b) E
F
= 0.25 eV, c) E
F
= 0.5 eV.
III.
Absorption Contributions
To analyze the contributions to absorption in our d
evice and address ways to improve the
efficiency, we performed simulations of our structu
re and quantitatively separated out the
absorption in each component. This is done by spati
ally integrating the power absorbed
as given by
ε
”|
E
|
2
, where
ε
” is the imaginary permittivity and
E
the complex electric
field, for each frequency in a 2D COMSOL full;wave
simulation. Results are presented
in Figure S3. The majority of absorption is localiz
ed to the silicon nitride and graphene,
suggesting that by utilizing a lossless substrate a
nd higher mobility graphene the
reflection efficiency of the structure could be imp
roved.
Figure S3
:
Simulated absorption in different components of res
onant geometry, a) Spectroscopic
absorption at E
F
= 0.3eV. b) Absorption as a function of Fermi ener
gy at 8.62 Cm
(wavelength of maximum phase modulation in 2D simul
ations).
IV.
Extraction of Reflectance from Interferograms
To assess the reflection efficiency of our device,
we extract the reflectance data from the
QCL interferograms. Given that the inteferogram’s m
aximum and minimum reflectance
is associated with constructive and destructive int
erference from the two legs (reference
leg and the sample leg), respectively, one can calc
ulate the intensity from the sample. The
reflectance of the sample is calculated after norma
lization with the intensity from a
reference mirror, which replaced the sample. We est
imate that there is a +/; 10%
deviation in the reflectance calculated in this man
ner due to a laser spot size that is
slightly larger than the sample size – estimate for
the power distribution within a
Gaussian beam was taken into account.
V.
Reflectance Data at Additional Wavelengths
To assess the reflection efficiency of our device,
we re;plot here the simulated and
experimental reflectance data as a function of E
F
at three representative wavelengths: 8.2,
8.5, and 8.7

m, Figure S4. This may be used to approximate the r
eflection measured in
interferometry measurements. Agreement between simu
lation and experiment is good.
Figure S4:
a) FTIR reflectance results as a function of Fermi
energy at 8.2, 8.5, and 8.7

m and b) Corresponding simulated reflectance.
VI.
Interferometry Data at Additional Wavelengths
We present simulation and experimental phase data f
or all measured wavelengths as a
function of Fermi energy. This data is summarized i
n Figure 4b of the main text.
Figure S5
: Reflected phase as a function of Fermi energy for
remaining wavelengths,
simulation (line) and experiment (dot).
λ=8.15 m
λ=8.25 m
λ=8.35 m
λ=8.40 m
λ=8.45 m
λ=8.55 m
λ=8.60 m
λ=8.65 m
λ=8.75 m
∆φ(°)
∆φ(°)
∆φ(°)
∆φ(°)
∆φ(°)
∆φ(°)
∆φ(°)
0 0.1 0.2 0.3 0.4
E
F
(eV)
0 0.1 0.2 0.3 0.4
E
F
(eV)
0 0.1 0.2 0.3 0.4
E
F
(eV)
0 0.1 0.2 0.3 0.4
E
F
(eV)
0 0.1 0.2 0.3 0.4
E
F
(eV)
0 0.1 0.2 0.3 0.4
E
F
(eV)
0 0.1 0.2 0.3 0.4
E
F
(eV)
0 0.1 0.2 0.3 0.4
E
F
(eV)
0 0.1 0.2 0.3 0.4
E
F
(eV)
∆φ(°)
50
0
50
100
150
200
250
∆φ(°)
50
0
50
100
150
200
250
∆φ(°)
50
0
50
100
150
200
250
∆φ(°)
50
0
50
100
150
200
250
∆φ(°)
50
0
50
100
150
200
250
∆φ(°)
50
0
50
100
150
200
250
∆φ(°)
150
100
50
0
50
100
150
200
250
∆φ(°)
50
0
50
100
150
200
250
∆φ(°)
150
100
50
0
50
100
150
200
250
VII.
Phase Modulation Range for Different Scattering Rat
es
Graphene quality is an important factor in determin
ing the achievable phase modulation
range at different wavelengths. For short scatterin
g rates, the mode becomes over;
damped and therefore minimal phase modulation can b
e achieved over a wide range of
wavelengths. As the scattering rate is increased, t
he mode is well;defined over a wider
spectral band, and therefore a larger spectral rang
e of phase modulation is achieved.
Interestingly, for higher quality graphene, a smoot
h phase trend is observed consistently
for a wide range of wavelengths, but the maximum ac
hievable phase modulation is
smaller than for 10 – 30 fs scattering times. This
trend occurs because the spectral phase
curves for lower quality graphene are steeper and t
herefore, result in a larger phase range
at wavelengths near the resonance.
Figure S6
: Phase modulation range achievable for a Fermi ene
rgy range of 0 to 0.44 eV
for scattering times of 10 – 40 fs.
VIII.
Antenna Array Theory Calculated Phase Gradients
a
b
c
Figure S7
: Assuming a desired steered beam direction of five
degrees, a 69 element 1;D
metasurface, and an element tunable reflection phas
e range of 215° with a hypothetical
unity reflection amplitude, antenna array theory pr
edicts the a) desired phase gradient, the
error in phase due to the limited phase range and b
) their respective 1;D far;field beam
patterns. c) Resulting beam pattern after including
the proposed device’s non;unity
reflection amplitude into the same set of calculati
ons.
Antenna array theory is a powerful and computationa
lly non;intensive method
that can capture the far;field response of a metasu
rface. In the example illustrated here,
the limited reflection phase tuning range results i
n less than 145° degrees of maximum
phase error (Fig. S7a).Traditionally, microwave ant
enna arrays have been lauded for their
robustness in maintaining desired beam shapes and d
irections even when hampered by
poorly functioning or dead elements. Likewise, the
69 element metasurface with its
associated phase errors suggests minimal pointing e
rror with the cost of side lobe levels
higher than that of the ideal case (Fig. S7b). A mo
re realistic illustration that accounts for
the non;unity reflection amplitude and limited phas
e range of the specific device
proposed in the text is shown in Fig. S7c. The resu
lting predicted far;field beam pattern
shows multiple lobes in addition to the steered mai
n lobe. The analysis does not use a
“smart” algorithm to avoid reflection minimums. One
could realize a practical device
with a beam pattern closer to that of Fig S7b by us
ing a “smart” algorithm (at the expense
of using a less than ideal phase) or choosing a des
ign with less variability in its reflection
amplitude across the desired phase range as referen
ced in [6] in the main text.