Topology optimized
multi
-functional
mechanically reconfigurable meta
-optics
studied
at microwave frequencies
C
ONNER
B
ALLEW
,
G
REGORY
R
OBERTS
,
P
HIL
IP
C
AMAYD
-M
UÑOZ
,
M
AXIMILIEN
F
D
EBBAS
,
AND
A
NDREI
F
ARAON
*
1
Kavli Nanoscience Institute and Thomas J. Watson Sr. Laboratory of Applied Physics, California
Institute of Technology, Pasadena, California 91125, USA
*faraon@caltech.edu
Abstract:
Metasurfaces advanced the field of optics by reducing the thickness
of opt
ical
components and
merging multiple functionalities into a single layer
device.
However,
this
generally comes with a reduction
in performance,
especially
for
multifunctional and broadband
applications.
Three
-dimensional
metastructures
can provide the necessary degrees of freedom
for advanced applications, while maintaining minimal thickness. This work explores 3D
mechanically reconfigurable devices that perform focusing, spectral de
multiplexing, and
polarization sorting based on mechanical configuration. As proof of concept, a
rotatable
device,
auxetic device, and a shearing
-based device are designed with adjoint
-based
topology
optimization, 3D
-printed,
and
measured at microwave frequencies (7.6
-11.6 GHz) in an
anechoic chamber.
1. Introduction
Traditional optical design combines independent bulk elements to achieve complex
functionality. Recen
t advances in nano
-fabrication technologies enabled the miniaturization of
bulk components by synthesizing multiple optical functionalities into single, subwavelength-
thick
layers called metasurfaces
[1,2]
. However, the performance of metasurfaces is limited by
the number of optical modes that can be controlled, which scales with the volume of the device
and the maximum refractive index contrast [3,4]
.
Systems of metasurfaces have been used t
o perform functions that are beyond the
capabilities of a single metasurface.
Some novel platforms for this technique include fabricating
two
metasurfaces on oppos
ite sides of a transparent wafer
[5,6]
. A “folded
-optics” platform
uses reflective metasurfaces on both sides of a wafer
to enable many interactions w
ith different
metasurfaces in a small, wafer
-thick
region
[7]
. This technique still uses a modular design
approach,
requir
ing
the different metasurfaces be spatially separated to mitigate the
unaccounted
-for
effects of multiple scattering events between elemen
ts that
effectively
invalidate the local
phase approximation that has made
metasurface design
tractable
[8].
To expand the functionality of metaoptics, adjoint
-based
topology optimization is
capable
of designing
high-
performing
dielectric structures with nonintuitive index distributions and
objective
functions
that are challenging to achieve with traditional methods [9,10]
. While much
of the work has been on 2D platforms such as silicon photonic
waveguides
[11]
, free
-space
2.5D
and
3D
devices
have also been explored
recently
[12
–16]
. In
a previous work, a passive
3D device was designed that functions as an red
-green
-blue (RGB)
color splitter
and polarizer
,
scaled up to microwave frequencies
[13]
. An interesting prospect for this type of
device is to
expand
its functionality while minimizing any corresponding reduction in efficiency. For a
passive device, multiple dependent functio
nalities will necessarily compete for efficiency. For
example, the efficiency of
demultiplexing spectral bins will be reduced if the device
simultaneously sorts polarization. One way to maintain high efficiencies amidst multiple
competing functionalities is to design active device
that can switch functionalities in time to
perform the different tasks.
In
this
work, we use inverse
design techniques and topology optimization to design devices
capable of switching between
optical functions through mechanical reconfiguration. We
explore 3 different methods of reconfigurability: a
rotatable
device
that
switch
es between
three
-
band
spectral
splitting and broadband focusing
when rotated 180
°
; a negative Poisson ratio, or
"auxetic", device
based on rotating rigid squares
that performs the same functions as the
rotatable
device; and a device based on shearing adjacent layers
that can sw
itch between three
-
band
spectral
splitting, broadband focusing, and polarization sorting.
2. Measurement setup and design methods
2.1 Measurement setup
Although 3D devices can be fabricated at the micro
-scale
with methods such as two-
photon
direct laser writing and greyscale lithography, it remains challenging to
fabricate
fully 3D
structures
with the required features sizes and alignment accuracy for optical and near
-infrared
applications
[17]
. For this work, we test our ideas in the mi
crowave range from 7.6 to 11.6 GHz
where
Fused Filament Fabrication 3D
-printing
with
polylactide acid (
PLA
) can generate
subwavelength
feature sizes with minimal material absorption
. The 42% fractional bandwidth
of this frequency range is the same fractional bandwidth as the range of 450 nm to 690 nm light,
which is nearly the full visible spectrum.
Fig.
1. The anechoic chamber used to measure the 3D print
ed devices. (a) A sideview of the
chamber. A TEM mode is injected into the anechoic chamber, exciting the device at
approximately normal incidence. The fields in the cavity, including fields above the device, are
measured with a small antenna probe extendi
ng through the top aluminum plate. The bottom
aluminum plate is scanned in two directions to make a full 2D scan. (b) A picture of the anechoic
chamber with the top aluminum plate removed. The scanning area is surrounded by absorbing
foam. Triangles were cut out of the absorbing foam to reduce reflections. (c) A schematic
representation of the measurement setup. The microwave source (Windfreak SynthHD) is
scanned from 7.6 GHz to 11.6 GHz and is injected through a waveguide adapter. The signal
from the measu
rement probe is first amplified with two low
-noise amplifiers, and the signal is
then passed through a Schottky power detector (Keysight 8470B)
. The output is proportional to
the field amplitude in the cavity.
We use a homemade anechoic microwave chamber t
o measure the electric fields of the
devices
, shown in Fig. 1.
The design and construction of this setup was inspired by the work in
reference
[18]
. The measurement system consists of a parallel plate waveguide supporting
only
the fundamental TEM mode in the frequency range of interest
when propagating through air or
PLA
. A sideview schematic of the system is shown in Fig. 1(a), and the constructed system
with the top plate removed is shown in Fig. 1(b). A circuit view of the system is shown in Fig.
1(c).
The output of a microwave synthesizer (
Windfreak SynthHD
) is injected into the chamber
through a waveguide feed.
This source is scanned f
rom 7.6 GHz to 11.6 GHz,
which is
the
designed bandwidth of operation for all presented devices.
A small probe antenna (SMA
Connector Receptacle) is extended through the top plate of the chamber
to probe the fields
inside the cavity
. The probe antenna signal is amplified with low noise amplifiers
and
propagated through a Schottky diode
detector
(Keysight 8470B)
. The resulting signal is
proportional to the amplitude of the fields in the cavity. The
bottom
plate is moved with stepper
motors to make a 2D scan of the field amplitude inside the chamber. The probe antenna does
not extend into the cavity, so the field intensity in the
small
air gap above the device can be
measur
ed. These fields are approximately proportional to the fields within
the dev
ice material
[18]
.
An important characteristic
of this test setup is it measures 2D devices
under TE
-polarized
illumination
. The respons
e of the device in the microwave chamber is nearly identical to the
response of a device placed in a free space with the same index distribution, assuming the index
distribution is extruded infinitely in the direction separating the waveguide plates
[18]
. This
simplification from 3D to 2D
implies
we can optimize the device in a 2D coordinate system,
thus drastically reducing the overall ti
me of the optimization. Although these devices can be
fully described by a 2D coordinate system, they are analogous to 3D fabricated devices since
patterning occurs in the direction of light propagation.
We
design our
devices assuming a
refractive
index o
f 1.5. While the actual refractive index
of PLA is closer to 1.
65
[19]
, the devices only fill about 80% of the gap in the microwave test
chamber. The effective index is thus approximately
1.5.
There is uncertainty in this value which
likely contributes to discrepancies between simulated and measured device performance. One
source of the uncertainty is predicted to originate from the layered nature of the 3D print which
has previously been report
ed to cause a 7%
anisotropy in PLA
permittivity
, with the
permittivity ranging from 2.75 in the direction of normal to the printing surface and 2.96 in the
direction parallel to the printing surface [19]
. Furthermore, if the permittivity depends on the
layered nature of the print, t
hen it is likely affected by the 3D printer quality and layer height,
which may be different from the reference
s used to estimate the PLA permittivity.
2.2 Topology optimization
The three designs
presented
are
all optimized with the same strategy. The optimization
technique is similar to the techniques
presented
in [13,20]
, with
some
differences to account
for the mechanical reconfiguration of each device, which will be briefly summarized here. The
adjoint variable method is a technique to efficiently compute the gradient
of a figure of merit
with respect
to a design region permittivity by combining the results of just two simulations, a
"forward" simulation and a backward, or "adjoint", simulation. We use a time
-domain
Maxwell's equations solver (Lumerical FDTD), so each simulation contains results for th
e
entire bandwidth of interest.
The optimization begins with each device modelled as a block of material with a
permittivity
betw
een that of air and PLA. T
he gradient of every figure of merit is computed for
each
of its potential configurations
. All
gradients are combined with a weighted average, with
weights chosen according to
Eq. 1 such that all figure of merits seek the same efficiency.
In
this equation,
FoM
represents the current value of a figure of merit,
N
is the total number of
figures of merit, and
w
i
represents the weigh applied to its respective merit function’s gradient.
The maximum operator is used to ensure the weights are never negative, thus ignoring the
gradient of high-
performing figures of merit
rather than forcing the figure of merit to decrease.
The 2/N factor is used to ensure all weights conveniently sum to 1, unless some weights were
negative before the maximum operation.
(1)
푤푤
푖푖
=
max
�
2
푁푁
−
퐹퐹퐹퐹푀푀
푖푖
2
∑퐹퐹퐹퐹푀푀
푗푗
2
, 0
�
The permittivity is stepped in the direction of this weighted gradient, and the process is
repeated until all pixels of the design region have become either PLA or air. It is not guaranteed
that all pixels will ultimately become air or PLA, since they can settle at a local extreme point
at a fictitious permittivity in between PLA and air. Thus, we conclude the optimization by
forcing each pixel to be either PLA or ai
r which has a negligible effect on the performance of
the device. Once the design is finaliz
ed, the resulting
permittivity distribution is exported to an
.STL file and 3D printed in PLA using an Ultimaker S3.
3. Results
Here we describe the results from the three different devices. Measurement results feature
a 2D
scan of the microwave cavity chamber for each configuration, with the red
-green
-blue colors
representing the equivalent hue of visible light when the microwave frequencies
are scaled by
a factor of ×59,618.
The microwave components have varying scatt
ering parameters over the
bandwidth of interest, a scan of the empty cavity is used to normalize device measurements
to
the total power injected into the cavity
. Each device focuses to
one or more of
three separate
pixels, depending on the desired functionality of the configuration. The pixels are depicted as
either red, green, or blue, indicating the assigned color in the spectral
splitting configuration
with red as the lowest frequency bin (7.6
-8.9 GHz), green as the middle frequency bin (8.9-
10.2 GHz), a
nd blue as the high frequency bin (10.2-
11.6 GHz)
. These frequency bins are
referred to simply as red, green, or blue from now on.
The focal plane is analyzed in simulation to determin
e the sorting efficiency, defined as the
fraction of incident power tra
nsmitting through the target pixel.
To quantitatively corroborate
the measured intensity profile with the simulated intensity profile, the measured
devices feature
a series of plots that compare the normalized intensity integrated over the frequency bands of
interest for each functionality.
3.1
Rotatable
device
The first device presented here changes its function by undergoing a 180
°
rotation.
The device
is designed only for TE light.
It focuses broadband light to the center pixel when illuminated
by a norm
ally
-incident planewave from one side, and focuses red, green, and blue light to the
respectively colored
pixels when illuminated from the other side
, as shown in Fig. 2(a
-d). The
footprint of the device
is 6.2 cm × 18.6 cm, which is 2λ × 6λ at the center
wavelength of 3.1
cm
, shown in Fig. 2(e)
. To ensure structural stability
the device
has a thin frame of PLA
, and
a
connectivity constraint is enforced every ten iterations of the optimization
so that the index
distribution converges to a fully connected
structure.
This device is uniquely simple among the devices presented, since a
180°
rotation
does not
alter the scattering matrix
beyond a transposition
due to reciprocity
. A reasonable concern is
that this device will be limited in performance because of this
. Howev
er, reciprocity does not
strongly affect this device, since inputs from both sides are assumed to be normally incident.
Thus, only scattering
components
mapping
a normally incident input to a normally incident
output are coupled by reciprocity, while the d
esired output fields are comprised of many more
uncoupled planewave components.
The sorting efficiency as a function of frequency is shown in Fig. 2(f).
The sorting
efficiency
averaged across the relevant frequencies
for each function is 50.6%.
This
outpe
rforms
a traditional
three
-pixel
absorptive Bayer filter arrays, which have a maximum
theoretical sorting efficiency of 33% for each frequency band.
To quantitatively compare the
intensity profiles of measured and simulated results, normalized intensity pr
ofiles are shown in
Fig. 2(g) for each function.
Fig
. 2. Rotatable
device performing broadband focusing in one configuration and spectral
splitting in the other configuration. (a) Broadband focusing simulation, (b) spectral splitting
simulation, (c) broadband focusing measurement, and (d) spectral splitting measurement. (
e)
Schematic of the device, where blue represents PLA and white represents air. The footprint of
the device is 6.2 cm
× 18.6 cm. (f) Analysis of the simulated fields at the focal plane showing
the sorting efficiency for each function, defined as the fracti
on of incident power reaching the
target pixel. Red, green, and blue light are focused to their respectively colored pixels, and
sorting efficiency is drawn in red, green, and blue, respectively. The broadband focusing
function focuses all light to the middle pixel and is drawn in black. (g) Comparison of simulated
and measurement normalized intensity profiles at the focal plane. Configuration B intensities
are integrated over (left) 7.6 to 8.9 GHz, (center
-left) 8.9 GHz to 10.2 GHz, (center
-right) 10.2
to
11.6 GHz. (right) Configuration A intensity integrated from 7.6 to 11.6 GHz.
3.2
Auxetic device
Auxetic metameterials have a negative Poisson’s ratio –
they have the non
intuitive property of
widening
in the transverse direction
when stretched and narrowing
in the transverse direction
when compressed.
Such metamaterials are useful for tailoring the mechanical properties of
devices, and can increase indentation resistance, shear resistance, energy absorption, hardness,
and fracture toughness
[21]
. They can be fabricated at the macro
-scale through techniques such
as 3D printing and have been studied at the micro-
and nanoscale
[22,23]
. Here we present a
device capable of switching its optical functionality through the well-
studied
auxetic
transformatio
n of rotating rigid squares, yielding devices with
a Poisson’s ratio of
-1 [24]
.
The two functions chosen here are broadband
focusing and
spectral
splitting
for TE
polarization
– the same as the rotatable
device. The first configuration features a 0
°
rotation of
all squares, while the second configuration features a ±90° of each sq
uare, with each square
having an opposite angular rotation as its neighboring squares. The device is fully connected,
with a frame around each square element to ensure structural stability.
The device and its
auxetic transformation are shown in Fig. 3(e).
A total of three devices were fabricated.
The first device demonstrates the auxetic
transformation. To fabricate this, the 3D
-print was paused at half the thickness of the device, a
nylon mesh was
manually
inserted, and the 3D
-print was then
resumed
to com
pletion
. After the
print, the nylon mesh was cut to leave only the required flexible hinges between each square
.
A video of this device being manually actuate
d is available in Visualization
1. The other two
devices are the two different configurations printed directly, which were then measured.
Fig
. 3. Auxetic device performing broadband focusing in one configuration and spectral splitting
in the other configuration. (a) Broadband focusing simulation, (b) spectr
al splitting simulation,
(c) broadband focusing measurement, and (d) spectral splitting measurement. (e) Demonstration
of the auxetic transformation and device footprint, where blue represents PLA and white
represents air. The footprint of the device is 6.
2 cm ×
18.6 cm. (f) Analysis of the focal plane of
this device, showing the sorting efficiency for each function, defined as the fraction of incident
power reaching the target pixel. Red, green, and blue light are focused to their respectively
colored pixe
ls, and sorting efficiency is drawn in red, green, and blue, respectively. The
broadband focusing function focuses all light to the middle pixel and is drawn in black. (g)
Comparison of simulated and measurement normalized intensity profiles at the focal p
lane.
Configuration B intensities are integrated over (left) 7.6 to 8.9 GHz, (center
-left) 8.9 GHz to
10.2 GHz, (center
-right) 10.2 to 11.6 GHz. (right) Configuration A intensity integrated from 7.6
to 11.6 GHz.
The
auxetic
design
and
rotatable
design
have
the same device size and focal length, so
the performance of the two devices can be directly compared.
The simulated sorting efficiency
is shown in Fig. 3(f). Since the auxetic device has a frame around each square element, less of
the design area is avai
lable for optimization.
This detracts from the degrees of freedom of the
device,
which may explain why
the
device has a 48.0% average sorting efficiency
, 2.6% less
than the rotatable
device.
The difference in average efficiency may also arise from the diff
erent
natures of the mechanical reconfiguration.
The simulated and measured intensities within the
full test chamber are shown in Fig 3(a
-d).
To quantitatively compare the intensity profiles of
measured and simulated results, normalized intensity profiles are shown in Fig. 3(g) for each
function.
3.3
Shearing device
The final device features a transformation based
on shearing
alternate
layers
. This
action is
achievable at the microscale through
MEMS electrostatic actuation.
Unlike the previous
dev
ices, this device switches
between three different functionalities: spectral
splitting shown in
Fig. 4(a)
, broadband
focusing shown in Fig. 4(b)
, and polarization splittin
g shown in Fig. 4(c).
All functionalities
are
designed for
both TE and TM polarization
s.
The added polarization control demands
more degrees of freedom
than the rotatable
and
auxetic devices, and it was
found that
the
device
needed to be
nearly 3
×
thic
ker
than
the
rotatable
and auxetic devices
to achieve satisfactory performance
of 59.2% average sorting
efficiency
. This device does
not have a supporting frame and does not
enforce connectivity like
the previous devices.
This device was analyzed only in simulation due to limitations within the measurement
system: the measurement ch
amber only supports a TE
-polarized TEM mode, and the scannable
region is too small to measure this device. A 4-
layer device was designed and tested with similar
agreement between simulation and measurement to the rotatable and auxetic devices, but the
devi
ce could not achieve all objective functions
. Data for this device is available on request.
The
configurations and simulation results for the 8
-layer simulated
device are summarized
in Fig. 4. The device is a stack of eight
8 cm
×
2 cm layers. The mechanical actuation displaces
adjacent layers in equal and opposite directions by 3 cm, which is approximately one
wavelength.
The first configuration
, shown in Fig. 4(a),
shows
spectral
splitting behavior
with
42% average sorting efficiency for TE and 40% average sorting efficiency for TM.
The
crosstalk between the different spectral bins is worse than the rotatable
and auxetic devices
,
with the worst case occurring for the
TM
-polarized green input which focuses only 1.2
×
more
power
to the desired green pixel tha
n the undesired blue and red pixels
.
The neutral position of the device, in which all layers are aligned
as shown in Fig. 4(b)
,
features broadband focusing
with high efficiency
. The aperture size of this
configuration
is
smaller than the other two configurations, and the sorting efficiency is normalized to the power
incident on this smaller aperture when analyzing this configuration.
The sorting efficiency
shown in Fig. 4(e) is expected to be uniformly high across all frequencies and both
polarizations.
The final configuration sorts TE polarization to the leftmost pixel and TM polarization to
the center pixel. The transmission through each pixel
, averaged across the entire spectrum,
is
summarized in matrix form in Fig. 4(
f): a TE input focuses 47% power to the correct
pixel and
28% power to the incorrect pixel, the TM input focuses 62% power to the correct
pixel and
24% power to the incorrect pixel.
More power is coupled to the desir
ed pixel than the undesired
pixel
at all frequencies except for the case of TE input frequencies
below approximately 9 GHz.
In this case more power is directed towards the incorrect green pixel than the correct red pixel.
It is possible that this could be fixed by sacrificing performance in the other functionalities,
such as by tuning the weighting scheme described in Eq. 1,
or by increasing the thickness
or
index contrast
of the device.
Fig.
4. TE and TM fields for a device bas
ed on a net shearing movement of 6 cm. (a) The spectral
splitting configuration splits red, green, and blue light to the red, green, and blue pixels,
respectively. This is analyzed for both TE and TM polarizations. (b) The neutral state of the
device perfo
rms broadband focusing to the center pixel for both TE and TM polarizations. (c)
The polarization sorting configuration sorts broadband TE light to the left (red) pixel and focuses
broadband TM light to the center (green) pixel. (d
-f) Sorting efficiency of
the different
configurations. The line color represents the sorting efficiency to the similarly colored pixel as
depicted in the color plots. Solid lines represent the TE response, and dashed line represent the
TM response. (d) Sorting efficiency in the s
pectral splitting configuration. (e) Sorting efficiency
for the broadband focusing configuration. (f) Sorting efficiency for the polarization splitting
configuration. (Inset) A confusion matrix representation of the sorting efficiency, with true input
on t
he vertical axis and predicted input on the horizontal axis. Each matrix entry is determined
by averaging the relevant trace over the full bandwidth.
4. Discussion
We have studied dielectric scattering elements capable
of
substantially expanding their
functionality through mechanical re
configuration. The work here is a step towards answering
an important question in optics –
how much performance and functionality can
be squeezed
into a certain
volume? The miniaturization of optical systems has led to new applications in
lasers
[25,26]
, biomedical optics
[27,28]
, space instrumentation
[29]
, and generally applications
where strict size requirements exist
. Yet there is still much progress that can be made, since
systems of cascaded metasurfaces still generally
require
some
free
-space propagation to
preserve their independence and the validity
of the techniques
often employed in metasurface
design.
Furthermore,
complex functionality such as combining polarization control, spectral
splitting, and imaging into single elements has been elusive.
Th
e approach shown here showcases the
advantages of
coupl
ing
mechanical
reconfiguration with adjoint
-based electromagnetic optimization.
Miniaturization is achieved
by employing adjoint
-based inverse design to find the optimal shape of a
small
dielectric
volume, while
broad
multifunctional
performance
is achieved through mechanical
reconfiguration.
The designed structures are
difficult to fabricate at the nanoscale for optical
applications, but the design process is flexible enough to incorporate nanoscale fabrication
requirements as shown in other works [30]
. In particular, s
ome areas of fabrication that can
fabricate the required subwavelength features for these devices include: multilayer fabrication
common in CMOS
or MEMS
processes
[31]
; direct
-write two
-photon laser lithography capable
of designing subwavelength 3D elements in the infrared
[17]
; and closely aligned stacks of
Silicon wafers with subwavelength features at terahertz frequencies
[32]
. All of these
techniques have demonstrated active mechanical control that could be useful for multiplexing
functions
like
what was demonstrated in this work.
Adjoint
-optimization is well-
suited for optimizing the optical properties of mechanically
reconfigur
able devices. In this work, the mechanical reconfiguration scheme was
predetermined,
and the material was patterned to enable multifunctionality. However, future
work could combine
existing
optimization techniques for designing mechanical metamaterials
[33,34]
with the techniques presented here
, yielding devices that have co
-optimized
mechanical
and optical performance.
Acknowledgements: This work was funded by Defense Advanced Research Projects
Agency
EXTREME program
(HR00111720035)
.
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