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Letter Optics Letters 1
Freestanding dielectric nanohole array metasurface for
mid-infrared wavelength applications
JUN RONG ON G1,+, HONG SON CHU1, VALERIAN HONGJIE CHEN1,2, ALEXANDER YUTONG ZHU3,A ND
PATRICE GENEVET4,*
1Institute of High Performance Computing, 1 Fusionopolis Way, 16-16 Connexis, 138632, Singapore
2Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford, UK
3John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, United States
4Universite Cote d’Azur, CNRS, CRHEA, rue Bernard Gregory, Sophia Antipolis 06560 Valbonne, France
*Corresponding author: patrice.genevet@crhea.cnrs.fr
+ongjr@ihpc.a-star.edu.sg
Compiled June 11, 2017
We designed and simulated freestanding dielectric op-
tical metasurfaces based on arrays of etched nanoholes
in a silicon membrane. We showed
2
πphase control
and high forward transmission at mid-infrared wave-
lengths around
4.2
µm by tuning the dimensions of the
holes. We also identified the mechanisms responsible
for high forward scattering efficiency and showed that
these conditions are connected with the well-known
Kerker conditions already proposed for isolated scat-
terers. A beam deflector was designed and optimized
through sequential particle swarm and gradient de-
scent optimization to maximize transmission efficiency
and reduce unwanted grating orders. Such freestand-
ing silicon nanohole array metasurfaces are promising
for the realization of silicon based mid-infrared optical
elements. © 2017 Optical Society of America
OCIS codes:
(050.6624) Subwavelength structures; (160.3918)
Metamaterials.
http://dx.doi.org/10.1364/ol.XX.XXXXXX
Metasurfaces are two-dimensional planar metamaterials
formed through suitable arrangement of etched subwavelength
structures. They have the inherent advantage of being thin,
lightweight compared to traditional optics and yet straightfor-
ward to fabricate compared to three-dimensional metamaterials
[
1
–
6
]. Inheriting concepts proposed in pioneering works on high
contrast gratings and microwave reflect and transmit-arrays,
metasurface technology is now suitable for applications at opti-
cal wavelengths [
7
–
14
]. Engineered optical metasurfaces have
the ability to impart an abrupt phase change to the incident
wavefront over a subwavelength thickness by utilizing struc-
tural resonances, enabling many potential applications in beam
deflection, lensing and wavefront control.
Mid-IR photonics at wavelengths of 2 to 20
µ
m has wide
ranging applications in spectroscopy, chemical and biomolecu-
lar sensing, and detection [
15
,
16
]. Metasurface optics at mid-IR
b
a
ૃൌǤૄܕ
ʹ
Ʌ
ࢠ
࢞
ʹɎ
࢟
࢞ࢊ
Fig. 1.
(a) Schematic of freestanding silicon nanohole array
metasurface acting as a beam deflector with deflect angle
θ
.
(b) Top-down view of silicon nanohole array beam deflector
metasurface. The metasurface is constructed using repeating
supercells. Within each supercell, the holes are separated by
a center-to-center distance
d
. From first hole to last hole there
is a linear phase gradient and the end-to-end the phase shift
difference is 2π.
wavelengths could potentially be simultaneously highly trans-
parent, easy to manufacture and of low-cost. However, fabri-
cating a metasurface on a supporting substrate material may
impose an additional undesirable material constraint on the
intended application, e.g. high absorption of silicon dioxide
substrate in the mid-IR wavelengths. A freestanding and trans-
mitting silicon based metasurface is thus a highly attractive
optical element at mid-IR wavelengths. In this article, we report
the design of a freestanding and transmitting beam deflector
made using an arrangement of subwavelength holes in a silicon
membrane (see Fig. 1). Similarly configured silicon based free-
standing nanohole arrays have previously been reported in the
literature[
17
,
18
]. However, these nanohole arrays were either
designed as reflectors or were unintentionally strongly reflecting.
In our deflector, after a traditional parametric study, we further
Letter Optics Letters 2
performed optimization of the hole dimensions and positions us-
ing particle swarm and gradient descent techniques to maximize
forward transmission and suppress unwanted grating orders.
We first consider the phase change and transmission through
a periodic square lattice of circular holes in a freestanding sili-
con membrane. We performed 3D finite difference time domain
(FDTD) simulations [
19
] while changing the hole radius
r
and
hole period
d
and fixing the slab thickness at
t=
1.118
µ
m. The
silicon nanohole array is surrounded by air (
n=
1). The exci-
tation source is chosen to be plane waves polarized along the
x-axis (parallel to the periodicity) and normally incident on one
side of the silicon. Figure 2shows the simulation results for
transmission and phase at
r=
770 to 1150 nm and
d=
2650 to
2900 nm. For many potential applications of metasurfaces, phase
control over a full 2
π
is needed. From Fig. 2(a), we can identify
the region of interest where 2
π
phase control is achievable, as de-
marcated by the solid white line borders. Correspondingly, Fig.
2(b) shows the transmission levels for the dimensions within
this region of interest. Based on these simulation results, we
choose a hole period
d
of 2800 nm (dotted line)for our subse-
quent calculations so as to have
T>
0.6 for all hole radii
r
within
the region.
0.8 0.9 11.1
2.65
2.7
2.75
2.8
2.85
r (
μ
m)
Phase (
π
)
a (
μ
m)
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0.8 0.9 11.1
2.65
2.7
2.75
2.8
2.85
r (
μ
m)
T
a (
μ
m)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0-2π
region
a
d
d
b
Fig. 2.
3D FDTD simulation results for (a) phase and (b) trans-
mission
T
of nanohole arrays at
r=
770 to 1150 nm and
d=
2650 to 2900 nm. The solid white line demarcates the
region with full 2
π
range of phase shift. The dotted white line
indicates a hole period
d
of 2800 nm which was used in subse-
quent simulations.
It has been shown both theoretically and experimentally that
in-phase interference of electric and magnetic dipole modes, i.e.
Kerker conditions, in high index dielectric nanoparticles can
produce high forward light scattering [
20
–
23
]. To show that the
high forward transmission
T
of the nanohole array is due to
the fulfillment of Kerker conditions, we performed a multipole
decomposition of the fields being excited within the structure
by normally incident plane waves [
24
–
26
]. The electromagnetic
fields were extracted from our FDTD simulations and decom-
posed into the electric and magnetic dipole contributions to the
total scattering cross section. We consider nanoholes in a peri-
odic square lattice where the lattice interactions are taken into
account by the periodic boundary conditions of our simulations.
In Fig. 3we plot the normalized forward transmission spectrum
for holes of different radii
r
. In Fig. 3(a), we overlay the peaks in
the electric and magnetic dipole contributions. We found good
correlations between coincident dipole contributions and high
forward transmission, particularly along the diagonal running
from
λ=
4.2
µ
m and
r=
850 nm to
λ=
3.8
µ
m and
r=
1040
nm. This allows us to infer that there is fulfillment of Kerker
conditions within these regions.
As the structure is a periodic array of nanoholes etched in a
membrane, it can also be analyzed as a photonic crystal slab. In
Fig. 3(b), we overlay the TE and TM bandstructure of the square
lattice hole array at the
Γ
-point (
kk=
0). We again see correlation
between coincident bands and high forward transmission. The
bandstructure was calculated using FDTD simulations. Note
that these bands are leaky modes above the light line and hence
can radiate out of the slab and contribute to the forward trans-
mission. We surmise that the coincidence of these
Γ
-point modes
is also an indication of the fulfillment of Kerker conditions.
850 900 950 1000 1040
3.8
4
4.2
4.4
4.6
4.8
5x 10
-6
850 900 950 1000 1040
3.8
4
4.2
4.4
4.6
4.8
5x 10
-6
0
0.2
0.4
0.6
0.8
1
T
r (nm)
= TE bands
= TM bands
T
Wavelength (µm)
r (nm)
= Magnetic dipole
= Electric dipole
ab
Fig. 3.
3D FDTD simulation results for transmission
T
vs.
wavelength for
r=
850 to 1040 nm. In (a),the peaks of the
electric and magnetic dipolar contributions to the scattering
cross section are overlayed showing good correlation with
high forward transmission when they are coincident. In (b),
the bandstructure of the nanohole array at the
Γ
-point in k-
space is overlayed.
We designed a mid-IR beam deflector at 4.2
µ
m as a demon-
stration of the functionality of such freestanding nanohole array
metasurfaces. To form a beam deflector, we chose the spatial
variation of the hole radii such as to impart a phase shift onto
the incoming wave which varies linearly with distance in the x-
direction, as in Fig. 1. The deflection angle
θ
of the wavefront is
given by the linear phase gradient of the deflector metasurface,
θ=sin−12π
kNd (1)
where a supercell of
N
number of holes is formed when the
total phase shift difference is 2
π
,
d
is the hole spacing and
k
is
the free space wave number. A supercell can then be repeated
to provide the linear phase gradient over the entire structured
metasurface. As a start, we allow the hole spacing
d
to remain
constant over the entire design. For the current deflector design,
we have chosen a repeating supercell of 10 nanoholes. To form
the supercell, we have 5 different chosen hole radii
r(nm) =
[
849, 863, 890, 1021, 1044
]
, i.e. adjacent pairs of holes will be of
the same radii. The holes are arranged in order of increasing
hole radii going from left to right. The deflection angle obtained
is
θ=sin−10.4π
k(2d)=
8.63
◦
. The hole radiii w chosen purely by
the phase gradient requirement and not all radii will have high
transmission according to Fig. 2and 3. We therefore further
optimized the positions and radii of the holes as described in
later section.
In order to verify the beam deflecting property of the de-
signed freestanding metasurface, we calculated the far field scat-
tering pattern from the 3D FDTD results (see Fig. 4(a)). The 3D
FDTD simulation was performed with a plane wave source lin-
early polarized along the long axis of the supercell (x-axis). The
strong peak at
θ=
8.63
◦
shows the metasurface can efficiently
Letter Optics Letters 3
deflect the incident light with good transmission and directivity.
The far field pattern also reveals the presence of grating order
peaks at angles given by the diffraction grating equation
θ=sin−1mλ
Nd (2)
where
N
is the number of holes in the supercell, the order
m=
0,
±
1,
±
2..., and
mλ
Nd 6
1. These peaks reduce the over-
all efficiency of the beam deflector since not all scattered energy
is directed towards a single desired peak. Moreover, we find
there is some residual reflection which reduces overall trans-
mission. In the following section, we describe how the beam
deflector efficiency was optimized.
0 5 10 15 20
0
0.5
1
Angle (deg)
Normalized |E|
2
Before opt.
After opt.
a
b
-2 -1 0 1 2
Before opt.
After opt.
Fig. 4.
(a) Normalized far field radiation pattern of nanohole
array beam deflector, before and after optimization of hole
sizes and positions. The numbers indicate the grating or-
ders
m
. (b) Normalized far field radiation pattern, showing
stronger scattering into the main lobe and suppression of un-
wanted grating lobes after optimization.
To improve the maximum beam deflector efficiency at 4.2
µ
m, we propose to tune the supercell hole radii and also their
positions along the left-right axis in order to maximize a figure-
of-merit (FOM) [
10
,
11
]. Similar methods are also commonly
used to optimize linear antenna arrays and in designs of metasur-
faces [
27
–
29
]. In total, we have 10 radius parameters
r1
to
r10
and
10 position parameters
∆d1
to
∆d10
. The position parameters
∆di
denote the hole positions relative to perfect periodicity, i.e. the
i
-th shifted hole position will be
(i−
1
)×d+∆di
. We adopted
a sequential particle swarm optimization (PSO) and gradient
descent (GD) optimization coupled with FDTD simulations to
tune the 20 parameters. The PSO is used to initially search a
large number of candidate solutions [
30
]. Once a pre-determined
number of iterations is completed, a GD is performed using the
solution with the best FOM as a starting position. As such we
are able to arrive at the local optimum around the best position
as determined by the PSO. We choose a FOM such as to maxi-
mize the forward transmission into the main lobe at the desired
deflection angle. Explicitly, the FOM is calculated as
FOM =T×Rθ2
θ1|E(θ,φ0)|2dθ
Rπ/2
−π/2 |E(θ,φ0)|2dθ
(3)
which is the product of the normalized forward transmission
and the fraction that is directed at the main lobe of the deflector.
E(θ
,
φ)
is the far field on the hemisphere,
θ1
and
θ2
are chosen to
be at the midpoints to the adjacent grating orders and
φ0=
0
◦
is
aligned parallel to the x-axis.
The minimum FDTD mesh size was set to 20 nm within the
volume of the silicon slab. Symmetric boundary conditions were
used along the long edge of the supercell, periodic boundary
conditions were used in the short edge of the supercell and in
the z-direction PML boundary conditions were used. We set the
number of generations in the PSO algorithm to 50 and the num-
ber of particles to 20. The PSO was then repeated 3 times, each
time re-centering and shrinking the parameter range around the
best parameters found from the previous PSO run. Subsequently,
we used the best parameters obtained from the final PSO as the
starting point for the GD optimization which was done for 30
iterations, each iteration consisting of 21 simulations. We moni-
tored the transmission and reflection of the metasurface and also
calculated the near to far field projection. For the optimization,
the FOM was calculated at a single wavelength of 4.2 µm.
Figure 4(a) compares the normalized far field pattern before
and after the sequential PSO and GD optimization. We can see
that the unwanted grating orders are reduced after optimization.
In Fig. 4(b) we have normalized the far fields to the maximum
of the peak obtained after optimization so as to make relative
comparisons. The main lobe is shown to have doubled and the
adjacent grating orders are reduced. After optimization, the
FOM was increased from 0.365 to 0.795, as shown in Fig. 5(a).
Also, the normalized forward transmission
T
increased from
0.64 to 0.92. This compares very well to an unpatterned silicon
membrane in which
T
is about 0.6. The final parameters used
for the solution with the best FOM are listed in Table 1.
44.05 4.1 4.15 4.2 4.25 4.3 4.35 4.4
0
0.2
0.4
0.6
0.8
1
Wavelength (
μ
m)
FOM
Before opt.
After opt.
1.08 1.1 1.12 1.14 1.16
0
0.2
0.4
0.6
0.8
1
Thickness (
μ
m)
FOM
150 100 150 180
0.4
0.5
0.6
0.7
0.8
Iteration
FOM
a
bc
Fig. 5.
(a) FOM vs. wavelength, before and after optimization.
(b) FOM vs. silicon membrane thickness. (c) FOM vs. opti-
mization iteration number.
We studied the spectral bandwidth of operation of our
nanohole array metasurface deflector. The FWHM of the FOM
after optimization is 80 nm (see Fig. 5(a)), which is sufficiently
broad for applications in integrated laser collimators and lenses.
Prior to optimization, the actual peak FOM is 0.447 located at
4.23
µ
m. After optimization, the peak was raised and shifted to
4.2
µ
m. We also studied the sensitivity of the optimized FOM to
Letter Optics Letters 4
Radii (nm) Shift (nm)
r1949 ∆d11
r2807 ∆d226
r3854 ∆d3-16
r4894 ∆d4-7
r5851 ∆d5-3
r6847 ∆d6-76
r7916 ∆d7-77
r81033 ∆d874
r9987 ∆d926
r10 1120 ∆d10 29
Table 1. Optimized parameters.
silicon membrane thickness as shown in Fig. 5(b). At approxi-
mately
±
25 nm, the FOM falls to half the maximum. For future
work, we can also take into consideration the design sensitivity
to thickness through robust optimization methods [
31
]. We have
plotted the change in FOM with optimization iteration number
in Fig. 5(c) for reference.
A practical concern of such photonic device optimization al-
gorithms that utilise full 3D FDTD simulations to calculate the
FOM is the computational cost of the large number of iterations
required. For the optimization procedure described in the above
section, each simulation took on average 14 minutes to complete.
The total time needed to complete the sequential PSO and GD
optimization is close to 850 hours. This large computational time
becomes prohibitive when designing multiple complex optical
elements. A possible method to reduce the total time required
would be to perform an initial coarse mesh optimization fol-
lowed by a refined mesh optimization. By halving the mesh
size on all three dimensions in the FDTD simulation domain,
the total simulation time can be reduced by a factor of 8. We
verified that our coarse mesh simulation was indeed completed
in 1.75 minutes. Subsequently, we completed the PSO optimiza-
tion with a coarse mesh arriving at a best FOM of 0.791. After
mesh refinement, the best FOM was reduced to 0.774 which
verifies that a coarse mesh optimization could be used to reduce
computation time.
In this work, we have designed a freestanding silicon
nanohole array beam deflector for the mid-IR wavelength of
4.2
µ
m. Our silicon metasurface can be fabricated with con-
ventional techniques and has high transmission due to the lack
of substrate, fulfillment of Kerker conditions and subsequent
further optimization. Similar principles can be used to design
various other optical metasurfaces, e.g. flat lenses. Rectangu-
lar, elliptical, or other asymmetrical hole shapes can be used to
impart birefringence, as required in waveplates or polarisation
beam splitters. Hole shapes with handedness can give chiral
optical properties, such as in optical rotation. In the future, we
expect our nanohole array design to be able to form useful meta-
surface optical elements at wavelengths spanning from visible
to infrared.
FUNDING INFORMATION
P. G. gratefully acknowledges financial support from the Euro-
pean Research Council (ERC) under the European Union’s Hori-
zon 2020 research and innovation programme (grant agreement
FLATLIGHT No. 639109). J.R.O., H.S.C. and V.H.C. acknowl-
edge the funding support from Agency for Science, Technology
and Research - Science and Engineering Research Council for
Pharos grant award No. 152-73-00025.
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