Supporting Information
for
Laser Photonics Rev.
, DOI 10.1002/lpor.202300980
All-Dielectric High-Q Dynamically Tunable Transmissive Metasurfac
es
Ruzan Sokhoyan, Claudio U. Hail, Morgan Foley, Meir Y. Grajower
and Harry A. Atwater*
Supporting Information
All
-
dielectric high
-
Q dynamically tunable transmissive metasurfaces
Ruzan Sokhoyan, Claudio U. Hai
l
, Morgan Foley, Meir Y. Grajower,
and Harry A. Atwater
*
1.
Extraction of the quality factor and
Fano
phase
We extracted the quality factor and Fano phase values by fitting the transmittance spectra to the
Fano formula
:
[1]
푇
=
퐴
|
훾
0
2
−
푒
−
2
푖훥
(
푖
(
휔
−
휔
0
)
−
훾
0
2
)
|
2
(
휔
−
휔
0
)
2
+
(
훾
0
2
)
2
+
푇
푏푔
Here,
푇
푏푔
is a constant offset plus a linear background (
푇
푏푔
=
퐵
+
퐶
(
휔
−
휔
0
)
),
퐴
is the resonance
amplitude,
휔
denotes the frequency of light while
휔
0
is the resonant frequency,
훾
0
is the damping
constant
.
훥
denotes the Fano
phase
,
and
the Fano
asymmetry parameter
q
is related to the Fano
phase Δ as
q
=
-
cot(Δ)
.
[2]
The quality factors are calculated as
푄
=
휔
0
훾
0
.
2.
Field profile of the
high
-
Q
mode in
the
pillar
array
Figure
S1. Spatial distribution of the electric field
amplitude
inside the metasurface unit cell shown
in Figure 1
, which described the
case of an array of a
-
Si pillars on an SiO
2
substrate.
Namely, the
length and width of the pillar are taken to be
l
=
w
= 963 nm. The metasurface period is
P
x
=
P
y
=
1425 nm.
The electric field is plotted in
the
x
-
z
plane
, which goes through the center of the pillar
(the same as
in
Figure
1d
)
.
a)
x
-
component of the electric field, b)
z
-
component of the electric
field.
Field profiles are plotted at a resonant wavelength
of the resonance dip shown in
Figure
1c.
The
E
y
component
is
identically equal to zero
(
E
y
= 0
)
.
Figure
S
2
. Spatial distribution of the electric field
amplitude
inside the metasurface unit cell shown
in Figure 1
, which described the case of an array of a
-
Si pillars on an SiO
2
substrate. The length
and width of the pillar are taken to be
l
=
w
= 963 nm. The metasurface period is
P
x
=
P
y
= 1425
nm.
The electric field is plotted in the
x
-
y
plane, which goes through the center of the pillar. a) An
absolute value of the electric field, b)
x
-
component of the electric field, c)
y
-
component of the
electric field, d)
z
-
component of the electric field.
3.
Optical response of a pillar array
: effect of the pillar height on the array performance
Figure
S3.
Optical
response of a
-
Si pillar array on an SiO
2
substrate.
The metasurface design is the
same as in
Figure
1 and 2. Namely, the length and width of the pillar are taken to be
l
=
w
= 963
nm. The metasurface period is
P
x
=
P
y
= 1425 nm. Transmittance
a)
and
the
phase
of a transmitted
light b) as a function of wavelength and the a
-
Si pillar height
h
.
We observe that modes 2 and 3
coalesce at a pillar height of
h
= 850 nm, and the highest
-
Q
high
-
Q
mode is observed at a pillar
height of
h
= 860 nm.
Figure
S4. Spatial electric field profiles of a lower Q modes supported by the metasurface in the
x
-
z
plane
, which passes through the center of the pillar. In the displayed electric field profiles, the
z
coordinate ranges from
z
=
-
200 nm to
z
= 1000 nm while the top of the SiO
2
substrate
corresponds to the plane
z
=
0. The
x
coordinate ranges from
x
=
-
712.5 to
x
= 712.5 nm. The
displayed false color plot is identical to
Figure
S
3a
. The spatial field profile of the
high
-
Q
mode is
identical to the one shown in
Figure
1, S1, and S2. We observe that
x
-
and
y
-
components of the
electric field of mode 3 have very similar features when varying the pillar height. On the other
hand, mode 2 ‘disappears’ after coalescing with mode 3.
The mod
e
profile of mode 1 is identical
to the one shown in
Figure
1, and S1.
Figure
S5. Spatial electric field profiles of a lower Q modes supported by the metasurface in the
y
-
z
plane
, which passes through the center of the pillar. In the displayed electric field profiles, the
z
coordinate ranges from
z
=
-
200 nm to
z
= 1000 nm while the top of the SiO
2
substrate
corresponds to the plane
z
=
0. The
x
coordinate ranges from
x
=
-
712.5
nm
to
x
= 712.5 nm. The
displayed false color plot is identical to
Figure
S3a. The spatial field profile of the
high
-
Q
mode is
identical to the one shown in
Figure
1, S1, and S2. We observe that
x
-
and
y
-
components of the
electric field of mode 3 have very similar features when varying the pillar height. On the other
hand, mode 2 ‘disappears’ after co
alescing with mode 3. The mod
e
profile of mode 1 is identical
to the one shown in
Figure
1, and S1. In this Figure, we display electric field amplitude |
E
|. On the
considered
y
-
z
plane, the only non
-
zero component is
E
x
.
Figure
S
6
. Optical response of a
-
Si pillar array suspended in air. The assumed geometrical
parameters are the same as in
Figure
1.
Namely, the length and width of the pillar are taken to be
l
=
w
= 963 nm. The metasurface period is
P
x
=
P
y
= 1425 nm.
In a), the transmittance as a
function of wavelength and the a
-
Si pillar height
h
. In b), the phase of the transmitted light as a
function of wavelength and the a
-
Si pillar height
h.
Compared with the case of pillars on an SiO
2
substrate, we observe the abundance of extremely high
-
Q modes, which are marked by a white
circle in a).
At the pillar height of
h
= 870 nm, the Q
-
factor of the observed
high
-
Q
mode is
~4
8
,000.
When considering an a
-
Si pillar array we observe that a
t a pillar height of 870 nm,
we are still able
to couple to the resonant mode, and the resonant spectral feature is still visible from the false color
plots of the transmittance and phase spectra
(
Figure
S6)
.
The extracted Q
-
factor of the resonance
is
~4
8
,000
(
Figure
S7a)
.
In our simulation, we assumed a 5 nm mesh in z direction. Running a
series of simulations with finer mesh could potentially enable us to identify
the parameter values
at which the mode with even higher Q
-
factor can be observed.
We also observe that when take the
meta
surface period as
P
x
= 1520 nm
and
P
y
= 1425 nm, the Q
-
factor of the supported mode is
22
1
,000
(
Figure
S7b)
.
Thus,
increasing the metasurface
period
from
P
x
=
1425 nm to
P
x
=
1520
nm
can strongly affect
the quality factor of the
metasurface.
Figure
S
7
. Transmittance and phase spectrum for an array of a
-
Si pillars
in air
. The assumed width,
length and height of the pillars are
w
=
l
=
963 nm,
h
=
870 nm
(see the schematic in
Figure
1). In
a),
the values for the period are
P
x
=
1425
nm and
P
y
= 1425 nm
. The extracted Q
-
factor of the
resonance is ~
48
,000.
The mesh is 20 nm in the
x
-
and
y
-
directions, and 5 nm in the
z
-
direction. In
b), the values for the period are
as compared with a). In b),
P
x
= 1
520
nm and
P
y
= 1425 nm. To
reduce simulation time, we assumed a mesh of 20 nm in all three directions. The extracted Q
-
factor
of the resonance is ~211,000.
4.
O
p
t
i
c
a
l response of a pillar array: effect of the pillar period on the array performance
We study how the period of the metasurface affects the mode
s
supported by the
metasurface
consisting of an array of a
-
Si pillars on an SiO
2
substrate
(
Figure
1
)
.
For metasurface periods
exceeding 1200 nm, we observe three distinct modes in the transmittance and phase false color
plots, which correspond to the
high
-
Q
mode at wavelength around 1540 nm and two lower
-
Q
modes at wavelengths around 1580 nm and 1590 nm, which correspond to mode 2 and mode 3
from
Figure
S3.
When studying the dependence of the mode
position on the
x
-
period
P
x
, we
observe
that the quality of the
high
-
Q
mode gradually increases with period (
Figure
S8c and
S9c
).
We also observe
that for periods of
P
x
> 1300 nm, the position of the high
-
Q mode is
does not
change significantly with period. We also observe that mode 2 shift
s
stronger with the
x
-
period
P
x
as compared with mode 3.
We also observe, that when we change the
y
-
period
P
y
, the
high
-
Q
mode
shifts stronger as compared with the case
when the
y
-
period is changed (c.f.
Figure
S8c and S9c).
This result is also consistent with the
Figure
S8. Optical response of
an
a
-
Si pillar array on an SiO
2
substrate
when the incoming electric
field is
x
-
polarized (see
Figure
1)
. The metasurface design is the same as in
Figure
1 and 2. Namely,
the length and width of the pillar are taken to be
l
=
w
= 963 nm, the height of the pillar is
h
= 860
nm. The metasurface period in the
y
-
direction is
P
y
= 1425 nm. Transmittance a) and the phase of
a transmitted light b) as a function of wavelength and the
period in the
x
-
direction
P
x
.
c) and d)
take a closer look at the behavior of the
high
-
Q
mode by limiting the range of the
x
-
period to [1300
nm, 1520 nm] in a) and b), respectively. In c) and d)
,
the wavelength range is limited to [
1542.7
nm, 1543.7 nm
]
.
specifics of the spatial mode profiles of the
high
-
Q
mode in the
x
-
z
and
y
-
z
planes (see
Figure
1).
When the
y
-
period
P
y
increase
s
from 1300 nm to 1520 nm, the
high
-
Q
resonance position shift
s
by ~5 nm.
Figure
S9. Optical response of
an
a
-
Si pillar array on an SiO
2
substrate
when the incoming electric
field is
x
-
polarized (see
Figure
1)
. The metasurface design is the same as in
Figure
1 and 2. Namely,
the length and width of the pillar are taken to be
l
=
w
= 963 nm, the height of the pillar is
h
= 860
nm. The metasurface period in the
x
-
direction is
P
x
= 1425 nm. Transmittance a) and the phase of
a transmitted light b) as a function of wavelength and the period in the
y
-
direction
P
y
. c) and d)
take a closer look at the behavior of the
high
-
Q
mode by limiting the range of the
y
-
period to [1300
nm, 1520 nm] in a) and b), respectively. In c) and d), the wavelength range is limited to [1
542
nm, 154
8
nm].
5.
Modes supported by
a
single isolated pillar
So far, in our optical simulations we imposed periodic boundaries in x
-
and y
-
directions implying
that we investigate a periodic array o
f
square a
-
Si pillars.
It is not clear whether the identified
photonic mode is also supported by an isolated a
-
Si pillar.
To address this question, we simulate
scattering cross section of an isolated pillar. In the revised simulation, we use perfectly matched
layer (PML) boundary conditions at all simulation boundaries. First, we consider the case of a
single a
-
Si pillar in air
and calculate its scattering cross section as a function of wavelength and
Figure
S
10
.
a) Scattering cross section of a single a
-
Si pillar
in air
as a function of wavelength and
pillar height. The length and width of the pillar is
l
=
w
=
963 nm. b)
Scattering cross section of a
single a
-
Si pillar
in the air where the pillar height is
h
=
834 nm.
The highest
-
Q mode is observed
at a wavelength of λ = 1519.8 nm.
Figure
S11. a) Scattering cross section of a single a
-
Si pillar on an
SiO
2
substrate
as a function of
wavelength and pillar height. The length and width of the pillar is
l
=
w
= 963 nm. b) Scattering
cross section of a single a
-
Si pillar where the pillar height is
h
= 830 nm. The highest
-
Q mode is
observed at a wavelength of λ = 1519.2 nm.
pillar height (
Figure
S10a). The length and width of the pillar are the same as in
Figure
1 of the
main manuscript (
l
=
w
= 963 nm). As seen in
Figure
S10, in the vicinity of the highest
-
Q mode
we also observe crossing of two other modes. For a pillar height of
h
= 834 nm, the quality factor
of the highest
-
Q mode is ~1000.
Next
,
we study the case of an isolated a
-
Si pillar on an SiO
2
substrate.
Figure S
11
plots the
scattering cross section of an isolated pillar
on an SiO
2
substrate
as a function of wavelength and
pillar height (
Figure
S
11
a
).
We observe an overall broadening of the spectral features as compared
with the case of a single a
-
Si pillar in the air.
We also observe that for a pillar height of 830 nm,
the Q
-
factor of the supported mode is
676
.
Figure
S
1
2
.
Scattering cross section of a single a
-
Si pillar on an SiO
2
substrate as a function of
wavelength and pillar height (the same as
Figure
S
11
a).
We plot the spatial distribution of the
electric field
amplitude
|
E
| in the a
-
Si pillar for the pillar heights of
h
=
720 nm,
h
=
790 nm,
h
=
830 nm,
and
h
=
850 nm.
The electric field is plotted in the
x
-
z
plane
, which passes through the
center of the pillar
.
The solid black lines point towards the scattering peaks a
t which the field
profiles have been simulated.
Figure
S1
3
. Spatial distribution of the
x
-
component
(top panel)
and
y
-
component (bottom panel)
of the electric field
E
in the
x
-
z
plane, which passes through the center of an isolated a
-
Si pillar on
an SiO
2
substrate.
The
x
-
and
z
-
components of the electric field are plotted along the
modal line 3
in
Figure
S10.
Geometrical parameters of the pillar are the same as in
Figure
S10.
The a
-
Si pillar
height and the operating wavelength are marked at the
top of each column
.
We observe that the
x
-
and
z
-
components of the electric field gradually change as we increase the pillar height.
Figure
S1
4
.
Spatial distribution of the
x
-
component of the electric field
E
in the
x
-
z
plane, which
passes through the center of an isolated a
-
Si pillar on an SiO
2
substrate. The
x
-
component of the
electric field is plotted along the
modal line 2
in
Figure
S10. Geometrical parameters of the pillar
are the same as in
Figure
S10. The a
-
Si pillar height and the operating wavelength are marked at
the top of each column. We observe that the
x
-
components of the electric field gradually changes
as we increase the pillar height.
Figure
S1
5
.
Scattering cross section of a single a
-
Si pillar on
an SiO
2
substrate as a function of
wavelength and pillar height (the same as
Figure
S
11
a). We plot the spatial distribution of the
electric field amplitude |E| in the a
-
Si pillar for the pillar heights of
h
= 720 nm,
h
= 790 nm,
h
=
830 nm, and
h
= 850 nm. The electric field is plotted in the
y
-
z
plane, which passes through the
center of the pillar. The solid black lines point towards the scattering peaks at which the field
profiles have been simulated.
To understand the nature of the considered high
-
Q resonance and its relation to the high
-
Q
resonance observed in the case of an array, we display the spatial distribution of
the
electric field
amplitude
in the
y
-
z
cross
-
section of the resonator
(
Figure
S
15
)
.
As seen in
Figure
S
15
,
the spatial
distribution of the electric field
amplitude
of the high
-
Q mode
in the
y
-
z
plane
(for example, at
h
=
830 nm) is identical to the spatial distribution of the electric field observed in the case of the
pillar array (see
Figure
1e).
6.
Controlling the spectral shape of the resonance
Figure
S
1
6
.
Transmittance a) and phase of the transmitted light b) as a
function of the wavelength
and thickness of the SiO
2
spacer
d
(
see the
schematic
in
Figure
3a).
The assumed geometrical
parameters are the same as in
Figure
3.
c) and d) show transmittance and phase spectra for the SiO
2
spacer thickness
for
d
= 510 nm and
d
= 1870 nm.
Figure
S
17
.
Quality factor
and
Fano phase
of the high
-
Q resonance
as a function of the SiO
2
thickness
d
(for schematic see
Figure
3a)
.
Here, the considered structure and geometrical
parameters are identical to the one if
Figure
3. Figure 3c shows that
around the
SiO
2
spacer