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Supporting information: Nano-electromechanical spatial light modulator
enabled by asymmetric resonant dielectric metasurfaces
Hyounghan Kwon,
1, 2
Tianzhe Zheng,
1
and Andrei Faraon
1, 2,
1
T. J. Watson Laboratory of Applied Physics and Kavli Nanoscience Institute,
California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125, USA
2
Department of Electrical Engineering, California Institute of Technology,
1200 E. California Blvd., Pasadena, CA 91125, USA
Corresponding author: A.F.: faraon@caltech.edu
1
SUPPLEMENTARY NOTE1: TEMPORAL COUPLED-MODE THEORY FOR ASYMMETRIC
RESONANT DIELECTRIC METASURFACES
For resonant metasurfaces or photonic crystals, it is known that the temporal response of the
resonator can be described by temporal coupled mode theory (TCMT) [1, 2]. As seen in Fig.
1b
in the main text, with normally incident light, the resonant metasurface can be modeled by a
single-mode resonator that is coupled to two ports. The dynamics of the optical resonance can be
generally formulated by:
푑푢
푑푡
=
(
푖푤
0
1
1
1
2
1
푛푟
)
+
(
1
2
)
©
́
+
1
+
2
Æ
̈
,
(
S1
)
©
́
1
2
Æ
̈
=
©
́
+
1
+
2
Æ
̈
+
©
́
1
2
Æ
̈
푢,
(
S2
)
where
and
0
correspond to complex amplitude of the resonance and the central resonance
frequency, respectively;
1
and
2
are the coupling coefficients between the two ports and the reso-
nances; the resonance radiatively decays into port 1 and 2 with decay rates of
1
1
and
1
2
, respectively;
1
푛푟
is the nonradiative decay rate;
+
1
(
1
) and
+
2
(
2
) are amplitudes of the incoming (outgoing)
waves from the ports;
is a direct-transport scattering matrix, written by
=
푖휙
©
́
푟 푖푡
푖푡 푟
Æ
̈
, where
,
, and
are the real reflection coefficient, the real transmission coefficient, and the phase factor,
respectively.
generally describes the direct coupling between the incoming and outgoing waves.
In addition, as
depends on the selection of reference planes in the model,
can be set to 0 for
simplicity. In addition, we here assume that
1
푛푟
is negligible because silicon is almost lossless in the
telecom wavelength range. According to the time-reversal symmetry and the energy conservation,
the coupling coefficients satisfy
1
1
=
2
1
, 푑
2
2
=
2
2
,
(
S3
)
©
́
1
2
Æ
̈
=
©
́
1
2
Æ
̈
,
(
S4
)
2
revealing that the coupling conditions are fundamentally related to the decay rates of the resonances
as well as the direct-transport scattering [2, 3].
According to Refs. [3, 4], we could analytically solve Eqs.
S1
-
S4
to obtain the Eqs.
1
and
2
in
the main text. In detail, when the system is driven by a continuous laser, whose frequency is
, we
can derive
as a function of
from Eq.
S1
=
(
1
2
)
©
́
+
1
+
2
Æ
̈
(
0
)+
1
푡표푡
,
(
S5
)
where
1
푡표푡
=
1
1
+
1
2
. By inserting Eq.
S5
into Eq.
S2
, the outgoing waves can be described by,
©
́
1
2
Æ
̈
=
©
́
+
1
+
2
Æ
̈
+
©
́
1
2
Æ
̈
(
1
2
)
(
0
)+
1
푡표푡
©
́
+
1
+
2
Æ
̈
.
(
S6
)
From Eq.
S6
, we can derive the reflection spectra of port 1 and 2,
1
and
2
:
1
=
1
+
1
+
2
=
0
=
+
1
2
(
0
)+
1
푡표푡
,
(
S7
)
2
=
2
+
2
+
1
=
0
=
+
2
2
(
0
)+
1
푡표푡
.
(
S8
)
Next, Eqs.
S3
and
S4
are employed to eliminate phase ambiguity of
1
and
2
in Eqs.
S7
and
S8
.
Specifically, from Eq.
S3
,
1
and
2
can be described by
1
=
2
1
푖휃
1
, 푑
2
=
2
2
푖휃
2
,
(
S9
)
where
1
and
2
are phases of the coupling coefficients of
1
and
2
, respectively. By inserting
Eq.
S9
into Eq.
S4
,
푐표푠
(
2
1
)
and
푐표푠
(
2
2
)
can be derived by
cos
(
2
1
)
=
1
2
(
2
푡표푡
1
)
,
(
S10
)
cos
(
2
2
)
=
2
2
(
2
푡표푡
+
1
)
,
(
S11
)
3
where
1
=
1
1
1
2
. Then,
푠푖푛
(
2
1
)
and
푠푖푛
(
2
2
)
are expressed as:
sin
(
2
1
)
=
±
1
2
4
2
1
2
4
푡표푡
2
1
2
2
2
푡표푡
,
(
S12
)
sin
(
2
2
)
=
±
2
2
4
2
2
2
4
푡표푡
2
1
2
+
2
2
푡표푡
.
(
S13
)
As
1
and
2
in Eqs.
S7
and
S8
can be described by
cos
(
2
1
)
,
cos
(
2
2
)
,
sin
(
2
1
)
, and
sin
(
2
2
)
, we
can derive the Eqs.
1
and
2
in the main text:
1
=
+
2
1
(
cos
(
2
1
)+
sin
(
2
1
)
)
(
0
)+
1
푡표푡
=
[
(
0
2
2
1
+
2
2
2
2
2
푡표푡
1
2
2
]
1
푟휎
(
0
)+
1
푡표푡
,
(
S14
)
2
=
+
2
2
(
cos
(
2
2
)+
sin
(
2
2
)
)
(
0
)+
1
푡표푡
=
[
(
0
2
2
1
+
2
2
2
2
2
푡표푡
1
2
2
]
+
1
푟휎
(
0
)+
1
푡표푡
.
(
S15
)
The two-port resonator model shown in Fig.
1b
in the main text generally describes any single-
mode resonant metasurfaces under normal incidence. When the light is obliquely incident and
the metasurface is in sub-wavelength regime (i.e. there is no diffraction), the metasurface can be
modeled by a four-port resonator. The general description of the four-port resonator model can be
found in Ref. [4]. Here, we only deal with a fully symmetric case where the resonance equally
decays into the four ports with the decay rate of
1
0
. The reflection spectrum of the symmetric
resonator,
, can be expressed as [4]:
=
[
(
0
2
0
1
2
]
(
0
)+
2
0
.
(
S16
)
We should note that Eq.
S16
becomes identical to Eq.
S14
or
S15
when
1
1
=
1
2
=
1
0
. In other
words, if the structures are symmetric with respect to all available ports and
1
0
0
, the single-mode
resonance is always critically coupled to the excitation.
4
SUPPLEMENTARY NOTE2: NUMERICAL INVESTIGATIONS ON SYMMETRIC RESO-
NANT DIELECTRIC METASURFACES
To physically implement a symmetrical case of the theoretical model in Supplementary Note
1, we simulate the metasurfaces possessing the mirror symmetry in the
-direction. In Fig.
S4a
,
the metasurface grating is composed of Si nanobars and surrounded by air. Specifically, the 841
nm wide and 838nm thick 2D nanostructures are periodically arranged with the lattice constant of
1093 nm. Figure
S4b
shows an electrical field profile of the TE-polarized eigenmode at
Γ
point.
The mode in Fig.
S4b
originates from the Mie mode hosted by individual Si nanostructures (see
Fig.
S1
for details). Reflection and reflected phase spectra are calculated under 0
and 5
tilted
incident lights and plotted in Figs.
S4c
and
S4d
. The resonance is not coupled to the normally
incident light in Figs.
S4c
and
S4d
. That is due to the symmetry mismatch between the excitation
and the eigenmode. In detail, the plane-wave excitation and the eigenmode in Fig.
S4b
are odd
and even under
2
rotation, respectively. In contrast, the 5
tilted light excites the resonant mode
by breaking the odd symmetry of the incident light. However, the critical coupling between the
excitation and the resonant mode occurs, causing negligible reflection at the resonance in Fig.
S4c
and limited phase shifts smaller than 180
in Fig.
S4d
. Finally, the results shown in Figs.
S4c
and
S4d
agree with the previous discussion of the symmetric resonator in Supplementary Note 1.
5
SUPPLEMENTARY NOTE3: NUMERICAL INVESTIGATIONS ON BEAM STEERING OF
THE ASYMMETRIC METASURFACES
Here, we numerically verify the metasurfaces’ capability of beam steering, using a pair of the
nanostructure as a building block of the proposed active metasurfaces. Specifically, the gaps of the
pairs of nanostructure are adjusted by the applied biases such that the metasurface manipulates the
wavefronts of the reflected light. When assuming that the phase is locally determined by the gap of
the two nanostructures, we can exploit the relationship between
0
푡ℎ
and
1
2
2
plotted in Fig.
2d
in
the main text as a lookup table to design the metasurfaces. In other words, once the desired phase
distribution is determined, the gaps of nanostructures can be inversely obtained from Fig.
2d
in the
main text. It is noteworthy to mention that this lookup table approach is widely used in passive
and active metasurfaces. First, we investigate a blazed diffraction grating of which the linear phase
gradient is negative. As shown in Fig.
S6a
, the period of the grating,
, is determined by periodicity
and 2
Λ
, where the periodicity represents the number of the pairs in one period of the blazed grating.
As the blazed grating is designed to have the negative phase gradient of
2
, the metasurfaces
expect to cause the dominant
1st order diffraction at the angle of
=
sin
1
(
)
. We simulate
negative phase gradient gratings having periodicity of 4 and 6. The spectra of reflected power
coefficients for the 0th and
±
1st and order diffractions are plotted in Figs.
S6b
and
S6c
. At 1529
nm, the reflected power coefficients of the
1st order diffraction are 16.7 and 27.0
%
in Figs.
S6b
and
S6c
, respectively. In contrast, the calculated reflected power coefficients of the
+
1st (0th) order
diffraction are 2.53
%
(5.69
%
) and 4.66
%
(7.47
%
) in Figs.
S6b
and
S6c
, respectively. Similarly, the
blazed gratings with positive phase gradients are also investigated. In Fig.
S6d
, the arrangement
of the gap sizes are simply reversed compared to the arrangement of the negative phase gradient
blazed gratings shown in Fig
S6a
. Then, the reversed nanomechanical displacements realize the
positive phase gradient of
2
, expecting to result in the dominant
+
1st order diffraction at the angle
of
. The reflected power coefficient spectra of the positive phase gradient blazed gratings are
plotted in Figs.
S6e
and
S6f
. The dominant
+
1st order diffraction and the suppressed 0th and
1st
order diffractions are observed in Figs
S6e
and
S6f
. At the design wavelength of 1529 nm, the
reflected power coefficients of the
+
1st order diffraction are 10.1
%
and 18.0
%
in Figs.
S6e
and
S6f
,
respectively. At the same wavelength, the calculated reflected power coefficients of the
1st (0th)
diffraction order are 4.26
%
(7.46
%
) and 7.11
%
(8.49
%
) in Figs.
S6e
and
S6f
, respectively. For both
periodicities of 4 and 6, the negative phase-gradient gratings used in Figs.
S6b
and
S6c
perform
6
more efficiently than the positive phase-gradient gratings used in Figs.
S6e
and
S6f
. The same trend
can be also found in the case of periodicity of 1 (see Supplementary Figure
5
for details). We expect
that these differences inherently result from the asymmetry of the structure with respect to
-axis.
Besides, at the design wavelength of 1529 nm, the reflected power coefficients of all available
diffraction orders are plotted in Fig.
S7
, showing that all high-order diffraction components are
suppressed compared to the desired diffraction order. In particular, it is worth noting that the
highest-order diffractions at the angle
±
44
are well suppressed when the periodicity is extended
over 1.
7
|
E
y,max
|
-
|
E
y,max
|
0
(V/m)
Figure S1 Electric field profile of high-order Mie mode resonance in a symmetric Si nanobar.
-components of electric fields are plotted. The eigenmode is found without the periodic boundary
condition. The width and thickness are 841 and 838 nm, respectively. The calculated complex
eigenfrequency is (191 + i0.291) THz, corresponding to the resonant wavelength of 1569 nm and Q-factor
of 328. Scale bar denotes 500 nm.
8
1526
1528
1530
1532
1534
0.75
0.8
0.85
0.9
0.95
1
Reflection
1526
1528
1530
1532
1534
-
3
-
2
-
1
0
1
2
3
Wavelength (nm)
Reflected phase (rad)
1526
1528
1530
1532
1534
0.86
0.864
0.868
0.872
Wavelength (nm)
Reflection
c
a
b
Wavelength (nm)
Figure S2 Analytical fitting of calculated reflection and reflected phase spectra of the asymmetric
metasurface. a
and
b
Numerical data shown in Figs.
1e
and
1f
in the main text is fitted by using Eqs.
1
and
2
in the main text. As shown in Fig.
1c
in the main text,
|
푇퐸
1
|
2
(
|
푇퐸푤
|
2
) and
푇퐸
1
(
푇퐸
2
) represent
reflection and reflected phase for top (bottom) illumination, respectively. From the fitting, we find the
Q-factor and
2
1
of 1004 and 17.52, respectively.
a
: Calculated and fitted spectra of
|
푇퐸
1
|
2
and
|
푇퐸
2
|
2
.
Red asterisks show the calculated spectra of
|
푇퐸
1
|
2
or
|
푇퐸
2
|
2
. The fitted spectrum is plotted by a black
solid line.
b
: Calculated and fitted spectra of
푇퐸
1
and
푇퐸
2
. Red and blue asterisks represent the
calculated spectra of
푇퐸
1
and
푇퐸
2
, respectively. The fitted spectra of
푇퐸
1
and
푇퐸
2
are plotted by solid
and dashed black lines, respectively.
c
Fitted reflection spectra of the direct-transport scattering process.
9
0.5
-0.5
0
(V/m)
Figure S3 Numerical investigation of asymmetric radiation of the proposed metasurface.
Calculated
electric field profiles of the eigenmode at
Γ
point. The
-components of the electric field profiles of the
eigenmode are plotted. Strong radiation toward top direction is observed. In simulation, the power ratio
between the top and bottom radiations and Q-factor of the eigenmode are 17.54 and 1021, respectively.
10
c
E
TE
|
푇퐸
|
E
TE
e
i
푇퐸
a
|
E
y,max
|
-
|
E
y,max
|
b
0
0
0.2
0.4
0.6
0.8
1
1566
1568
1570
1572
1574
0
°
5
°
Reflection
1566
1568
1570
1572
1574
-
1
0
1
2
Wavelength (nm)
0
°
5
°
x
z
Reflected phase (rad)
d
(V/m)
Figure S4 Numerical investigations on resonant reflection behaviors of symmetric metasurfaces. a
Schematic illustration of the suspended symmetric metasurfaces. The metasurface possesses mirror
symmetry in the
-direction with respect to the middle of the nanostructure. The TE polarized light is
incident on the metasurface.
b
Simulated electrical field profile of the eigenmode at
Γ
point. At the
wavelength of 1568 nm, the
-components of the electrical fields are plotted. Scale bar denotes 500 nm.
c
and
d
Calculated reflection and reflected phase spectra of the symmetric metasurfaces. Solid and dashed
curves show the spectra for 0
and 5
tilted TE polarized incident light, respectively.
11
0
20
40
60
80
100
120
1524
1526
1528
1530
1532
1534
0
0.
05
0.
1
(
arb.
u.)
+
1푠푡
2
Wavelength (nm)
0
20
40
60
80
100
120
1524
1526
1528
1530
1532
1534
g
1
g
2
2
(
nm)
1푠푡
2
Wavelength (nm)
0
0.
2
0.
4
(
arb.
u.)
g
1
g
2
2
(
nm)
a
b
Figure S5 Numerical investigations on high diffraction orders. a
and
b
Simulated reflected power
spectra of the
±
1st order diffractions. The reflected power coefficient spectra of the
1
st and
+
1
st order
diffractions,
|
1
푠푡
|
2
and
|
+
1
푠푡
|
2
, are calculated as a function of the nanomechanical tuning,
2
1
2
, and
plotted in
a
and
b
, respectively. It should be noted that the color bars are different in
a
and
b
. The calculated
reflected power coefficient spectra of the
0
th order diffraction is shown in Fig.
2b
in the main text.
12