S
1
Supplementary Material
Miniaturization of a
-
Si guided mode resonance filter arrays for near
-
IR
multi
-
spectral filtering
Ryan C. Ng
,
1
,a)
Juan C. Garcia
,
2
Julia R. Greer,
3
and
Katherine T
.
Fountaine
2
1
Division of Chemistry and
Chemical Engineering
,
California Institute of Technology
,
Pasadena
,
California
91125
,
USA
2
NG Next, Northrop Grumman Corporation, One Space Park, Redondo Beach, California 90278
,
USA
3
Division of Engineering and Applied Sciences
,
California
Institute of Technology
,
Pasadena
,
California
91125
, USA
2D nanopillar array
While
the design and discussion in this work is restricted to a 1D design, all the ideas here are fully
applicable to a 2D design should a polarization
-
independen
t
response be desired at normal
incidence using an array of pillars or cubes with mirrors on all fo
ur sides. To illustrate this idea, in
Figure S
1
we present a proof
-
of
-
concept design in 2D that has not been optimized, but still exhibits
the GMR rapid spectral variation. Figure S
1
(
a
)
presents a schematic of this design, with Al mirrors
on only two sides
, and Figure S
1
(
b
)
shows the associated reflection spectrum. This array has
geometric parameters
h
= 150 nm,
a
= 800 nm,
f
= 0.5, and
s
= 300 nm for a 7x7 array of a
-
Si
nanopillars embedded in SiO
2
with Al mirrors. The spectral characteristics can be signi
ficantly
improved through optimization of the geometric parameters. While mirrors are required on all four
sides for polarization
-
independence, the ability to miniaturize this design and visualize the GMR
with mirrors only along one direction further suppo
rt the GMR mechanism.
S
2
F
IG.
S1
.
GMR in compact finite 2D design incorporating mirrors. (a) Schematic of a 2D design
incorporating mirrors only along two boundaries for an array with geometric parameters
h
= 150
nm,
a
= 800 nm,
f
= 0.5, and
s
= 300 nm for a 7x7 array of a
-
Si nanopillars e
mbedded in SiO
2
with
Al mirrors. (b) Associated reflection spectrum calculated in FDTD demonstrating a non
-
optimized
proof
-
of
-
concept of the compact finite array design and ability to observe the GMR utilizing a 2D
array should polarization
-
independence be
desired.
Number of periods
required to obtain desired spectral characteristics
For comparison
,
to
quantify
the advantage of incorporating a mirror
in terms of filter footprint
,
FDTD simulations were done to determine the number of periods
required in a finite design that
does not incorporate any mirrors to match the amplitude of other higher
-
performance designs
(Figure S
2
). Figure S
2
(
a
)
compares the performance of a 17
-
period finite mirrorless design with
the 7
-
period finite design incorpor
ating mirrors proposed in this work. Figure S
2
(
b
)
compares the
performance of a 141
-
period finite design that does not incorporate mirrors with an infinite design.
While it is possible to obtain high spectral resolution and high signal
-
to
-
noise ratio spect
ral peaks
using GMR filter designs that do not incorporate any kind of reflective boundaries, this comes at
the cost of extremely
large
lateral footprints.
S
3
F
IG.
S2
.
FDTD simulations comparing spectra between designs that incorporate mirrors and
designs that do not incorporate mirrors, indicating the number of periods required in a mirrorless
design to match the performance of other designs in terms of
peak
amplitude.
(a) Comparison
between the finite 7
-
period design that incorporates mirrors in this work (green) with a finite 17
-
period design that does not incorporate mirrors (pink). (b) Comparison between an infinite design
(yellow) and a finite 141
-
period design tha
t does not incorporate mirrors (purple).
Effect of varying
period
in finite GMR design
The effect of varying the number of period
s
in the finite design incorporating mirrors is shown in
FDTD calculated spectra in Figure S
3
. With a lower number of periods
, the amplitude of the
peak
decreases
,
the
FWHM
broadens
,
and
background noise
increases
. However, depending on the
specific signal
-
to
-
noise that may be desired in the final imaging device, a filter incorporating only
3 periods still exhibits a GMR
that may be useful if the lower signal
-
to
-
noise ratio and lower
spectral resolution is acceptable in a particular application.
S
4
F
IG.
S
3
.
FDTD generated reflection spectra demonstrating the effect of
varying the number of
periods
and varying filter footprint in the finite design incorporating mirrors. As the number of
periods decreases, the amplitude decreases and the bandwidths broaden wit
h increased background
noise.
Spectral characteristics of arrays
Table S1 shows the spectral characteristics of the fabricated arrays measured experimentally
compared to
experiment.
We observe FWHM values from 140.5 to 205.1 nm in experiment, and
from
102.7 to 120.6 nm in simulation (overestimation of actual FWHM and underestimation of
their performance due to defining the FWHM as half of the peak reflection
due to the asymmetric
resonance
, defined
in the main text
).
For example, to calculate the FWHM f
or the 731 nm period
array in simulation, we find the bandwidth at half of the reflection amplitude (82.6/2), which results
in a bandwidth of 102.7 nm. This is to avoid ambiguity since the baseline has a slightly different
amplitude on either side of the a
symmetric resonance.
Our
definition of FWHM causes a
significant overestimation of the bandwidth of the 881 nm array in experiment.
The transmission
dips are as low as 58.1% experimentally, compared to 17.4% in simulation. We attribute this
efficiency loss
to fabrication imperfections, experimental normalization, and alignment errors in
fabrication and measurement (angle and polarization).
S
5
Table S1
.
Spectral characteristics for each array in {experiment / simulation} for variable
periodicity
with 7 periods
. The color scheme is consistent with that in Figure 3.
Period (nm)
Peak Position (nm)
Reflection Amplitude
FWHM (nm)
731
1180 / 1176
41.9 / 8
3.0
140.
5
/ 10
3
.
6
781
1230 /
1244
39.0 /
7
9.4
157.5
/ 10
6
.
2
831
1300 /
131
2
34.0 /
7
5.4
143.9
/
110.4
881
1350 /
138
1
32.5 /
7
1
.
1
205.1
/ 11
6
.
0
931
1400 /
14
49
23.0
/ 66.
7
188.6
/
122.7
Top down schematic of filter array and surrounding reflective frame
Rather
than patterning and depositing two rectangular blocks for the mirrors as suggested by
Figure 1
(
a
)
, we surround the filter with an Al frame.
This Al frame now serves two purposes: 1)
to allow the GMR to reflect back to approximate infinit
e periodicity
and
2) to enable normalization
during measurements through the array.
T
ransmission through mirror frames matching the area of
the sample but with no patterned area was measured
to properly normalize the power transmitted
through the patterned sample area, thou
gh only reflective mirrors on two sides of the array are
required to observe the
laterally propagating
GMR in the finite array
.
An aerial schematic of this
layout is shown in Figure S4 and the dimensions of the frame and distances from the filter are
indic
ated. The a
-
Si array is indicated in red and the surrounding Al frame in dark blue. This frame
is
discontinuous with slits in the frame for ease of experimental lift
-
off. Frames of the same lateral
dimensions were fabricated without gratings for normalizat
ion of the transmission measurements.
S
6
F
IG.
S
4
.
Top
-
down schematic of the
7 period a
-
Si
filter array and surrounding
Al mirror/
frame.
The frame is introduced for ease of normalization during measurements. Slits are included in the
frame for ease of lift
-
off during fabrication.
The array of a
-
Si slab
s has variable
periodicity,
a
, fill
fraction,
f
,
and
spacer region,
s
.
Dimensions for the frame, slits, distance from the array, and lateral
length of the array slabs are indicated in the figure.
This schematic is not to scale.
Ellipsometry
Ellipsometry data
was
obtained
(Figure S
5
)
for
a
100 nm thick
film of
a
-
Si
deposited
by PECVD
at 200
o
C, 800 m
T
orr, and 10 W with 250 sccm of 5% SiH
4
diluted in Ar
. This n,k
data was
used
in
in the FDTD simulations
to model a
-
Si.
S
7
F
IG.
S
5
.
Raw n and k data for a
-
Si determined from ellipsometry.
Fresnel Correction
This calculation
allows for
the
additional
top and bottom
interfaces
of the glass substrate
to be
accounted for, that are not otherwise accounted for in FDTD simulations. The calculation can be
done for a fixed number of layers, and then generalized to an arbitrary number of layers. First, we
assume we have a homogeneous film of a single material
in air (i.e. air
-
film
-
air with each layer
numbered as layers 0, 1, and 2, respectively).
This stack is schematically shown in Figure S6.
R
A
is the amplitude of the wave reflected directly from the top surface of the film,
R
B
is the amplitude
of the wave r
eflected from the bottom surface of the film that transmits back through the top
surface,
R
C
is the wave that reflects from the bottom surface, then internally reflects again off the
top surface then bottom surface, before transmitting through the top surf
ace, and so on. Then,
R
ij
,
T
ij
,
r
ij
are the reflectance, transmittance, and reflection coefficients at each interface between each
layer, respectively. We define:
푅
01
=
푟
01
2
At normal incidence
:
푅
01
=
(
푛
0
−
푛
1
푛
0
+
푛
1
)
2
S
8
As
R
A
is the reflectance off the top
interface
,
푅
A
=
푅
01
Then,
푅
B
=
푇
01
푅
12
푇
10
푅
C
=
푇
01
푅
12
푅
10
푅
12
푇
10
More generally, to describe the reflectance out of the top surface after
m
internal reflections off
the back
-
side of the film for
α
≥
1:
푅
m
=
푇
01
푇
10
푅
12
훼
푅
10
훼
−
1
The total reflectance out of the top surface is the sum of all of these
R
m
terms that transmit from
the film to the top surface
:
푅
tot
=
∑
푅
m
=
푅
01
+
푇
01
푇
10
푅
12
∑
푅
12
훼
푅
10
훼
∞
훼
=
0
This is a geometric series:
푅
tot
=
푅
01
+
푇
01
푇
10
푅
12
(
1
1
−
푅
12
푅
10
)
For normal incidence, TE and TM polarization are the same, i.e.
T
= 1
–
R
for TE, but
R
TE
=
R
TM
.
We also utilize
R
ij
=
-
R
ji
. For a 3
-
layer air
-
film
-
air stack,
R
02
describes the total reflectance in this
system considering all interfaces. Simplifying:
푅
to
t
=
푅
02
=
푅
01
+
푅
12
1
+
푅
12
푅
01
This result can similarly be generalized to a different number of layers. For a stack of 4 layers:
S
9
푅
tot
=
푅
03
=
푅
01
+
(
푅
12
+
푅
23
1
+
푅
23
푅
12
)
1
+
(
푅
12
+
푅
23
1
+
푅
23
푅
12
)
푅
01
This is the expression utilized in this work for a 4
-
lay
er film of air
-
glass
-
glass
-
air, where the FDTD
simulated spectrum represents
R
12
, the reflection at the interface between the glass
-
glass layers.
F
IG.
S6.
Schematic of Fresnel reflection through the interfaces of a single slab of homogeneous
material. Here, layer 0 and layer 2 are air, though this can be generalized to any cover or substrate.
R
A
is the amplitude of the wave reflected directly from the top su
rface of the film,
R
B
is the
amplitude of the wave reflected from the bottom surface of the film that transmits back through
the top surface,
R
C
is the wave that reflects from the bottom surface, then internally reflects again
off the top surface then bott
om surface, before transmitting through the top surface, and so on.
Methods
Fabrication
:
The filter arrays were fabricated using a top
-
down methodology on a glass substrate.
Approximately 64 nm of a
-
Si was deposited onto the substrate by PECVD at 200
o
C, 800 mTorr,
and 10 W with 250 sccm of 5% SiH
4
diluted in Ar for 2 minutes and 18 seconds. The thickness
deposited in PECVD dictates the height of the a
-
Si slabs in the array. Prior to all electron beam
lithography writes in the fabrication process, a sac
rificial layer (a solution of poly(4
-
styrenesulfonic acid) mixed with 1% by volume of Triton X
-
100 surfactant) was spin
-
coated above
the resist at 3000 rpm and baked at 90
o
C for 3 minutes, followed by 10 nm of electron beam
S
10
evaporated Au for charge dissipa
tion. The sacrificial layer gold was removed following EBL in
water and developed as normal. Ti alignment markers (
h
= 200 nm) were created by first patterning
in polymethyl methacrylate (PMMA) 950 A8 positive
-
tone electron beam resist with a Raith
5000+ e
lectron beam writer at 100 kV, developing in methyl isobutyl ketone:isopropanol
(MIBK:IPA) in a 1:3 ratio for 90 seconds, and then depositing Ti in electron beam evaporation
and lifting off. The next EBL write utilized MaN
-
2403 negative
-
tone electron beam
resist applied
onto the a
-
Si. The grating slabs were then exposed in an aligned write to the Ti markers and are 21
m
each in length. Following electron beam exposure, the pattern was developed in MF
-
319 for
40 seconds and the pattern was transferred into
the a
-
Si layer with a pseudo
-
Bosch SF
6
/C
4
F
8
etch
with ICP
-
RIE at 15
o
C with 40 W ICP power, 1500 W forward power, 26 sccm of SF
6
and 35 sccm
of C
4
F
8
.
The MaN
-
2403 resist mask was removed by cleaning in an oxygen plasma for 10 minutes
(10 mtorr and 80 W wi
th 20 sccm O
2).
Finally, the mirrors were aligned and patterned
with PMMA
950 A4 and developed in 1:3 MIBK:IPA for 90 seconds, and a 65 nm thick layer of Al that dictated
the mirror height was deposited in electron beam evaporation and subsequently lifted
-
off. Rather
than patterning and depositing two rectangular blocks for the mirrors as suggested by Figure 1
(
a
)
,
we surround the filter with an Al frame. This frame is the length of the filter and spacer regions in
one lateral direction
perpendicular to the grating beams, and 24
m in the other lateral direction
parallel to the grating beams (Figure S4). It is discontinuous with slits in the frame for ease of
experimental lift
-
off. Frames of the same lateral dimensions were fabricated wit
hout gratings for
normalization of the transmission measurements. The resulting filter was in
-
filled with 500 nm of
methylsiloxane based spin
-
on glass solution (Filmtronics 500F).
Measurement
:
Measurements were made with a Fianium white light source coupl
ed to a near
-
IR
monochromator with Ge photodetectors and a 20X Mitutuyo objective with a range from 480
-
S
11
1800 nm. This objective allowed focusing of the spot size down to ~10
m. Transmission
measurements were taken
from 1100 to 1600 nm spaced evenly 10 nm
apart from one another. To
properly normalize to the power transmitted through the patterned sample area, transmission
through mirror frames matching the area of the sample but with no patterned area was measured.
To account for any power fluctuations betw
een the measurement of the sample and the frame, a
pair of Ge photodetectors simultaneously recorded the intensity of the beam transmitting through
the sample, I
ref
, and the intensity of the incident beam, I
inc
, using a beamsplitter prior to the
objective.
The transmissivity of the sample, T, was then calculated by:
푇
=
(
퐼
ref
퐼
inc
)
sample
(
퐼
inc
퐼
ref
)
frame
Simulatio
n: Full
-
field,
2D
simulations were computed with Lumerical
FDTD, a commercial
electromagnetics software package.
For infinite simulations (Figures 1
(
b
)
and 1
(
c
)
) and crosstalk
simulations (Figure 4)
,
periodic
boundary conditions were used on the lateral boundaries to
reduce
the simulation region and
emulate infin
ite periodicity.
For finite simulations (Figures 1
(
b
)
, 1
(
d
)
,
1
(
e
)
, 2, and 3)
,
perfectly matched layers were used for the lateral boundaries as an artificial
absorbing region to emulate infinite space
.
In all cases, perfectly matched layers were used for th
e
axial boundarie
s
.
For the polarization, the E
-
field is parallel to the direction of the length of the
grating slabs.
For the materials in the simulation, Palik data was used for
Al and
SiO
2
and
ellipsometric data was used for a
-
Si (Figure S
5
). A finer ov
erride mesh was applied over the a
-
Si
slabs
with mesh sizes <5%
of
the height and
width
of the
slabs
. The spectra in this work utilized
single broadband (
800
-
20
00 nm) simulations.
A Fresnel correction was applied to all simulations
to account for interface
s that were not included in simulation (Figure S6).