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Complex wavefront engineering with disorder-engineered metasurfaces
Mooseok Jang,
1, 3,
Yu Horie,
2,
Atsushi Shibukawa,
1,
Joshua Brake,
1
Yan Liu,
1
Seyedeh
Mahsa Kamali,
2
Amir Arbabi,
2, 4
Haowen Ruan,
1
Andrei Faraon,
2,
and Changhuei Yang
1,
1
Department of Electrical Engineering, California Institute of Technology,
1200 E. California Blvd., Pasadena, California 91125, USA
2
T. J. Watson Laboratory of Applied Physics, California Institute of Technology,
1200 E. California Blvd., Pasadena, California 91125, USA
3
Present address: Department of Physics, Korea University,
145 Anam-ro, Seongbuk-gu, Seoul 02841, South Korea.
4
Present address: Department of Electrical and Computer Engineering,
University of Massachusetts, 151 Holdsworth Way, Amherst, Massachusetts 01003, USA.
Recently, complex wavefront engineering with disordered media has demonstrated optical manipulation
capabilities beyond those of conventional optics. These capabilities include extended volume, aberration-free
focusing and subwavelength focusing via evanescent mode coupling. However, translating these capabilities
to useful applications has remained challenging as the input-output characteristics of the disordered media
(
P
variables) need to be exhaustively determined via
O(
P
)
measurements. Here, we propose a paradigm
shift where the disorder is specifically designed so that its exact characteristics are known, resulting in an
a priori
determined transmission matrix that can be utilized with only a few alignment steps. We implement this
concept with a disorder-engineered metasurface, which exhibits additional unique features for complex wavefront
engineering such as an unprecedented optical memory effect range, excellent stability, and a tailorable angular
scattering profile.
I. INTRODUCTION
Complex wavefront engineering can be best described as a
class of methods that allow control of a very large number of
optical degrees of freedom, ranging up to hundreds of thou-
sands [1]. This sets it apart from the regime of wavefront
manipulation in adaptive optics where the corrections are typ-
ically performed for aberrations modeled by a relatively small
number of Zernike orders [2]. As a class of technologies, com-
plex wavefront engineering is particularly well suited for ap-
plications involving disordered media. These applications can
be broadly divided into two categories. In the first category,
wavefront engineering works to overcome intrinsic limitations
of the disordered media. Biological tissue is one such example
where scattering is a problem, with complex wavefront engi-
neering emerging as a solution to produce a shaped light beam
that counteracts multiple scattering and enables imaging and
focusing deep inside the tissue [3].
In the second category, disordered media are intentionally
introduced in conjunction with wavefront engineering to un-
lock an optical space with spatial extent (
x
) and frequency con-
tent (
ν
) that is inaccessible using conventional optics [4–10].
One of the first demonstrations of this ability was reported by
Vellekoop
et al.
[4], showing that the presence of a disordered
medium (e.g. a scattering white paint layer) between a source
and a desired focal plane can actually help render a sharper
focus. In related efforts, researchers have also shown that
complex wavefront engineering can make use of disordered
media to couple propagating and evanescent modes, in turn
enabling near-field focusing [6, 7]. Recently, there have been
more extensive demonstrations combining disordered media
with complex wavefront engineering to increase the flexibil-
ity of the optical system to, for example, significantly extend
the volumetric range in which aberration-free focusing can be
achieved [8–10].
Unfortunately, this class of methods is stymied by one over-
riding challenge – the optical input-output response of the dis-
ordered medium needs to be exhaustively characterized before
use [10–14]. Fundamentally, characterizing
P
input-output
relationships of a disordered medium requires
O(
P
)
measure-
ments. For most practical applications,
P
greater than
10
12
is
highly desired to enable high fidelity access to the expanded
optical space enabled by the disordered media with wavefront
engineering. Unfortunately, the time-consuming nature of the
measurements and the intrinsic instability of the vast majority
of disordered media have limited the ability to achieve high
values of
P
. To date, the best
P
quantification that has been
achieved is
10
8
with a measurement time of 40 seconds [11].
In this paper, we report the use of a disorder-engineered
metasurface (we call this a disordered metasurface for brevity)
in place of a conventional disordered medium. The disor-
dered metasurface, which is composed of a 2D array of nano-
scatterers that can be freely designed and fabricated, provides
the optical ‘randomness’ of conventional disordered media,
but in a way that is fully known
a priori
. Through this ap-
proach, we reduce the system characterization to a simple
alignment problem. In addition to eliminating the need for ex-
tensive characterization measurements, the disordered meta-
surface platform exhibits a wide optical memory effect range,
excellent stability, and a tailorable angular scattering profile
– properties that are highly desirable for complex wavefront
engineering but that are missing from conventional disordered
media. Using this disorder-engineered metasurface platform,
we demonstrate full control over
P
=
1
.
1
×
10
13
input-output
relationships after a simple alignment procedure. To demon-
strate this new paradigm for controllably exploiting optical
arXiv:1706.08640v1 [physics.optics] 27 Jun 2017
2
‘randomness’, we have implemented a disordered metasurface
assisted focusing and imaging system that is capable of high
NA focusing (
NA
0
.
5
) to
2
.
2
×
10
8
points in a field of
view (FOV) with a diameter of
8 mm
. In comparison, for the
same FOV, a conventional optical system such as an objective
lens can at most access one or two orders of magnitude fewer
points.
II. PRINCIPLES
The relationship between the input and output optical fields
traveling through a disordered medium [14] can be generally
expressed as
E
o
(
x
o
,
y
o
)
=
T
(
x
o
,
y
o
;
x
i
,
y
i
)
E
i
(
x
i
,
y
i
)
d
x
i
d
y
i
,
(1)
where
E
i
is the field at the input plane of the medium,
E
o
is the
field at the output plane of the medium, and
T
is the impulse
response (i.e. Green’s function) connecting
E
i
at a position
(
x
i
,
y
i
) on the input plane with
E
o
at a position (
x
o
,
y
o
) on the
output plane. In the context of addressable focal spots with
disordered medium assisted complex wavefront engineering,
Eq. (1) is discretized such that
E
o
is a desired focusing opti-
cal field,
E
i
is the linear combination of independent optical
modes controlled by the spatial light modulator (SLM), and
T
is a matrix (i.e. the transmission matrix) where each ele-
ment describes the amplitude and phase relationship between
a given input mode and output focal spot. In this scenario,
E
i
has a dimension of
N
, the number of degrees of freedom in the
input field (i.e. the number of SLM pixels),
E
o
has a dimension
of
M
given by the number of resolvable spots on the projection
plane, and
T
is a matrix which connects the input and output
fields with
P
elements, where
P
=
M
×
N
. We note that
the following concepts and results can be generalized to other
applications (e.g. beam steering or optical vortex generation)
simply by switching
E
o
to an appropriate basis set.
One of the unique and most useful aspects of complex wave-
front engineering with disordered media is that it allows ac-
cess to a broader optical space in both spatial extent (
x
) and
frequency content (
ν
) than the input optical field can conven-
tionally access. For example, when an SLM is used alone,
the generated optical field
E
i
contains a limited range of
spatial frequencies due to the large pixel pitch of the SLM
(
ν
x
or
ν
y
1
/(
2
d
SLM
)
where
d
SLM
is the pixel pitch; typi-
cally
10
μ
m). As a consequence, the number of resolvable
spots
M
is identical to the number of controllable degrees of
freedom
N
. In contrast, when a disordered medium is placed
in the optical path, its strongly scattering nature generates an
output field
E
o
with much higher spatial frequencies given by
ν
2
x
+
ν
2
y
1
/
λ
, where
λ
is the wavelength of the light. Ac-
cording to the space-bandwidth product formalism [15], this
means that the number of addressable focal spots
M
within a
given modulation area
S
, is maximally improved to
M
=
S
×
π
λ
2
.
(2)
The scheme for focusing with disordered medium assisted
complex wavefront engineering can be understood as the pro-
cess of combining
N
independent optical modes to construc-
tively interfere at a desired position on the projection plane
[4, 16, 17]. In general, due to the increased spatial frequency
range of the output field, the number of addressable spots
M
is
much larger than the number of degrees of freedom in the input,
N
, and therefore the accessible focal points on the output plane
are not independent optical modes (see supplementary S1).
Instead, each focal spot exists on top of a background which
contains the contributions from the unoptimized optical modes
in the output field. Here the contrast
η
, the ratio between the
intensity transmitted into the focal spot and the surrounding
background, is dictated by the number of controlled optical
modes in the input,
N
[16]. In practical situations where, for
instance, the addressed spots are used for imaging or photo-
switching, the contrast
η
needs to be sufficiently high to ensure
the energy leakage does not harmfully compromise the system
performance.
To maximize performance, we can see it is desirable to have
as many resolvable spots as possible, each with high contrast.
This means that both
M
and
N
, and in turn
P
, should be
as high as possible. Practically, there are two ways to mea-
sure the elements – orthogonal input probing and output phase
conjugation (see supplementary S2). In each case, an indi-
vidual measurement corresponds to a single element in the
transmission matrix and is accomplished by determining the
field relationship between an input mode and a location on the
projection plane. Both still necessitate
O(
P
)
measurements
which, when
P
is large, leads to a prohibitively long measure-
ment time. As a point of reference, if the fast transmission
matrix characterization method reported in Ref. [11] could be
extended without complications, it would still require a mea-
surement time of over 40 days to characterize a transmission
matrix with
P
=
10
13
elements. In comparison, the stability
associated with most conventional disordered media can last
only several hours [16, 18, 19].
In contrast, our disorder-engineered metasurface avoids the
measurement problem altogether since all elements of the
transmission matrix are known
a priori
. This means that
now the procedure to calibrate the system is simplified from
the
O(
P
)
measurements needed to determine the transmission
matrix to the small number of alignment steps for the disorder-
engineered metasurface and the SLM.
A schematic illustration of the technique is presented in
Fig.
1
with the omission of a 4-
f
imaging system optically
conjugating the SLM plane to the disordered metasurface.
An SLM structures a collimated incident beam into an op-
timal wavefront which in turn generates a desired complex
output wavefront through the disordered metasurface. Since
the transmission matrix is known
a priori
, the process to focus
to a desired location is a simple computation. The optimal
incident pattern
E
opt
i
that encodes the information for a target
field
E
target
o
is calculated using the concept of phase conjuga-
tion (see materials and methods). This approach enables us to
access the maximum possible number of resolvable spots for
3
complex wavefront engineering for a given modulation area
S
with the added benefit of control over the scattering properties
of the metasurface.
III. RESULTS
A. The disorder-engineered metasurface
The disordered metasurface platform demonstrated in this
study shares the same design principles as the conventional
metasurfaces that have been previously reported to implement
planar optical components [20–25]: rationally designed sub-
wavelength scatterers or meta-atoms are arranged on a two-
dimensional lattice to purposefully shape optical wavefronts
with subwavelength resolution (Fig.
2A
). The disordered meta-
surface, consisting of Silicon Nitride (SiN
x
) nanoposts sitting
on a fused silica substrate, imparts local and space-variant
phase delays with high transmission for the designed wave-
length of 532 nm. We designed the phase profile
φ
(
x
,
y
)
of the
metasurface in such a way that its angular scattering profile
is isotropically distributed over the maximal possible spatial
bandwidth of
1
/
λ
, and then chose the width of the individual
nanoposts according to the look-up table shown in Fig.
2B
(see
materials and methods for details). The experimentally mea-
sured scattering profile confirms the nearly isotropic scattering
property of the disordered metasurface, presenting a scatter-
ing profile that fully extends to the spatial frequency of
1
/
λ
as
shown in Fig.
2C
. This platform also allows tailoring of the
scattering profile, which can be potentially useful in conjunc-
tion with angle-selective optical behaviors such as total internal
reflection. Figure
2D
presents the measured scattering profiles
of disordered metasurfaces designed to have different angular
scattering ranges, corresponding to NAs of
0
.
3
,
0
.
6
,
and 0
.
9
(see fig.
S1
for 2D angular scattering profiles).
In addition to a highly isotropic scattering profile, the disor-
dered metasurface also exhibits a very large angular (tilt/tilt)
correlation range (also known as the optical memory effect
[26]). The correlation is larger than
0
.
5
even up to a tilting an-
gle of
30
degrees (Fig.
2E
). In comparison, conventional scat-
tering media commonly used for scattering lenses, such as opal
glass and several micron-thick Titanium Dioxide (TiO
2
) white
paint layers, exhibit much narrower correlation ranges of less
than
1
degree (Fig.
2E
) [27]. Although ground glass diffusers
present a relatively wider correlation range of
5
degrees,
their limited angular scattering range makes them less attrac-
tive for complex wavefront engineering (see fig.
S2
for angular
tilt/tilt measurement setup and correlation profiles).
Moreover, the disordered metasurface is extraordinarily sta-
ble. We were able to retain the ability to generate a high
quality optical focus from the same metasurface without ob-
servable efficiency loss over a period of 75 days by making
only minor corrections to the system alignment to compensate
for mechanical drift (see fig.
S3
).
B. High NA optical focusing over an extended volume
We experimentally tested our complex wavefront manipu-
lation scheme in the context of disordered medium assisted
focusing and imaging. First, we aligned the disordered meta-
surface to the SLM by displaying a known pattern on the SLM
and correcting the shift and tilt of the metasurface to ensure
high correlation between the computed and measured output
field. Next, to demonstrate the flexibility of this approach, we
reconstructed a converging spherical wave (see materials and
methods for details) for a wide range of lateral and axial focus
positions. Figure
3A
presents the simplified schematic for op-
tical focusing (see also materials and methods and fig.
S4
for
more details). Figure
3B1-B3
shows the 2D intensity profiles
for the foci reconstructed along the optical axis at
z
=
1
.
4
, 2.1,
and 3.8 mm, measured at their focal planes. The correspond-
ing NAs are 0.95, 0.9, and 0.75, respectively. The full width
at half maximum (FWHM) spot sizes of the reconstructed foci
were 280, 330, 370 nm, which are nearly diffraction-limited as
shown in Fig.
3C
. The intensity profiles are highly symmetric,
implying that the converging spherical wavefronts were recon-
structed with high fidelity through the disordered metasurface.
It is also remarkable that this technique can reliably control
the high transverse wavevector components corresponding to
an NA of
0
.
95
, while the SLM used alone can control only
those transverse wavevectors associated with an NA of
0
.
033
.
Figure
3B4-B6
shows the 2D intensity profiles at
x
=
0
, 1,
4, and 7 mm on the fixed focal plane of
z
=
3
.
8
mm (corre-
sponding to the on axis NA of
0
.
75
). Because the disordered
metasurface based scattering lens is a singlet lens scheme, the
spot size along the
x
-axis increased from 370 to 1500 nm as
the focus was shifted (summarized in Fig.
3D
).
The total number of resolvable spots achievable with the
disordered metasurface,
M
, was experimentally determined
to be
4
.
3
×
10
8
based on the plot in Fig.
3D
, exceeding
the number of controlled degrees of freedom on the SLM
(
N
10
5
) by over
3
orders of magnitude. The NA of
0
.
5
was
also maintained in a lateral FOV with a diameter of
8
mm,
resulting in
2
.
2
×
10
8
resolvable focal spots. For the sake of
comparison, a high-quality objective lens with an NA of
0
.
5
typically has
10
7
resolvable spots, an order of magnitude
smaller than the number of the spots demonstrated with the
disordered metasurface.
With our disordered metasurface platform we control a
transmission matrix with a number of elements
P
given by the
product of the number of resolvable focal spots on the output
plane and the number of controllable modes in the input. The
P
we achieved with our system was
1
.
1
×
10
13
which allowed
us to address
4
.
3
×
10
8
focus spots with a contrast factor
η
of
2
.
5
×
10
4
. This value of
P
is
5
orders of magnitude higher
than what has previously been reported [11]. These findings
testify to the paradigm-shifting advantage that this engineered
‘randomness’ approach brings.
We also experimentally confirmed that even with reduced
control over the number of input modes, we can still access the
4
same number of resolvable spots on the output plane, albeit
with a reduced contrast. By binning pixels on the SLM, we
reduced the number of controlled degrees of freedom on the
SLM by up to three orders of magnitude, from
10
5
to
10
2
,
and verified that the capability of diffraction-limited focusing
over a wide FOV is maintained (see fig.
S5
). Although the
same number of focal spots can be addressed, the contrast
factor
η
is sacrificed when the number of degrees of control
is reduced. Using
10
2
degrees of freedom in the input,
we achieved a contrast factor of
70
. This validates that
the complex wavefront manipulation scheme assisted by the
disordered metasurface can greatly improve the number of
addressable focal spots for complex wavefront engineering
regardless of the number of degrees of freedom in the input.
C. Wide FOV fluorescence imaging
Finally, we implemented a scanning fluorescence micro-
scope for high-resolution wide FOV fluorescence imaging (see
materials and methods, fig.
S4
, and fig.
S6
for detailed pro-
cedure). Figure
4
A presents the wide FOV low-resolution
fluorescence image of immunofluorescence-labeled parasites
(
Giardia lamblia cysts
; see materials and methods for sam-
ple preparation procedures) captured through the
4
×
objective
lens. As shown in the magnified view in Fig.
4B3
, a typical flu-
orescent image directly captured with a
4
×
objective lens was
significantly blurred, so that the shape and number of parasites
was not discernible from the image. Figure
4
,
B1, C, and D
presents the fluorescence images obtained with our scanning
microscope. The scanned images resolve the fine features of
parasites both near the center and the boundary of the
5
-mm
wide FOV (Fig.
4D
). Our platform provides the capability for
high NA focusing (
NA
0
.
5
) within a FOV with a diameter
of
8
mm, as shown in Fig.
3
. To validate the performance of
our imaging system, we compare it to conventional
20
×
and
4
×
objectives. The captured images in Fig.
4
demonstrate that
we can achieve the resolution of the
20
×
objective over the
FOV of the
4
×
objective.
IV. DISCUSSION
Here we have implemented a disorder-engineered medium
using a metasurface platform and demonstrated the benefit of
using it for complex wavefront engineering. Our study is the
first to propose engineering the entire input-output response
of an optical disordered medium, presenting a new approach
to disordered media in optics. Allowing complete control
of the transmission matrix
a priori
, the disorder-engineered
metasurface fundamentally changes the way we can employ
disordered media for complex wavefront engineering. Prior
to this study, to control
P
input-output relationships through
a disordered medium,
O(
P
)
calibration measurements were
required. In contrast, the disorder-engineered metasurface
allows for a transmission matrix with
P
elements to be fully
employed with only a simple alignment procedure.
Although we only demonstrate the reconstruction of spher-
ical wavefronts in this study, our method is generally appli-
cable to produce arbitrary wavefronts for applications such
as beam steering, vector beam generation, multiple foci, or
even random pattern generation (see fig.
S7
for experimental
demonstrations). We anticipate that the large gain in the num-
ber of addressable optical focal spots (or equivalently angles or
patterns) enabled by our method will substantially improve ex-
isting optical techniques such as fluorescence imaging, optical
stimulation/lithography [28, 29], free space coupling among
photonic chips/optical networks [30, 31], and optical encryp-
tion/decryption [32].
In the specific application of focal spot scanning, our basic
system consisting of two planar components, a metasurface
phase mask and a conventional SLM, offers several advan-
tages. The system is highly scalable and versatile, bypassing
the limitations and complexities of using conventional objec-
tive lenses. The scalability of the metasurface can be especially
useful in achieving ultra-long working distances for high NA
focusing. The scheme can also be implemented as a vertically
integrated optical device together with electronics [33] (e.g. a
metasurface phase mask on top of a transmissive LCD), pro-
viding a compact and robust solution to render a large number
of diffraction-limited spots. Furthermore, the concept is ap-
plicable over a wide range of the electromagnetic spectrum
with the proper choice of low-loss materials for the meta-
atoms (e.g. SiN
x
or TiO
2
for entire visible [24, 34] and Si for
near infrared wavelengths [22, 35–37]), which allows for mul-
tiplexing different colors, useful for multicolor fluorescence
microscopy and multiphoton excitation microscopy. Finally,
the planar design provides a platform to achieve ultra-high NA
solid-immersion lenses [38] or total internal reflection fluo-
rescence (TIRF) excitation [39], suitable for super-resolution
imaging and single-molecule biophysics experiments.
More broadly speaking, we anticipate the ability to cus-
tomize the design of the disordered metasurface for a particular
application will prove highly useful. For example, we can tai-
lor the scattering profile of the disordered metasurface to act as
an efficient spatial frequency mixer or to be exploited for novel
optical detection strategies [40–42]. The disordered metasur-
face can serve as a collection lens, analogous to the results ob-
tained for light manipulation, providing an enhanced resolving
power and extended view field. Additionally, the metasurface
platform can be designed independently for orthogonal polar-
ization states, which provides additional avenues for control in
complex wavefront engineering [43]. Together, the engineer-
ing flexibility provided by these parameters offers unprece-
dented control over complex patterned illumination, which
can directly benefit emerging imaging methods that rely on
complex structured illumination [44, 45].
To conclude, we explored the use of a disorder-engineered
metasurface in complex wavefront engineering, challenging a
prevailing view of the ‘randomness’ of disordered media by
programmatically designing its ‘randomness’. The presented
technology has the potential to provide a game-changing shift
5
that unlocks the benefits of complex wavefront engineering,
opening new avenues for the design of optical systems and en-
abling new techniques for exploring complex biological sys-
tems.
MATERIALS AND METHODS
Design of disordered metasurface
The disordered metasurface consists of Silicon Nitride
(SiN
x
) nanoposts arranged on a subwavelength square lattice
with a periodicity of 350 nm as shown in Fig.
2A
. The width of
each SiN
x
nanopost is precisely controlled within a range from
60 nm to 275 nm, correspondingly imparting local and space-
variant phase delays covering a full range of
2
π
with close
to unity transmittance for an incident wavefront at the design
wavelength of 532 nm (Fig.
2B
). The widths of the nanoposts
corresponding to the grayed regions in Fig.
2B
correspond to
high quality factor resonances and are excluded in the design
of the disordered metasurface. The phase profile
φ
(
x
,
y
)
of
the disordered metasurface is designed to yield an isotropic
scattering profile over the desired angular range using the
Gerchberg-Saxton (GS) algorithm. The initial phase profile
of the far-field is randomly chosen from a uniform distribution
between
0
and
2
π
radians. After several iterations, the phase
profile converges such that the far-field pattern has isotropic
scattering over the target angular ranges. This approach helps
to minimize undiffracted light and evenly distribute the input
energy over the whole angular range.
Fabrication of disordered metasurface
A SiN
x
thin film of 630 nm is deposited using plasma en-
hanced chemical vapor deposition (PECVD) on a fused silica
substrate. The metasurface pattern is first defined in ZEP520A
positive resist using an electron beam lithography system.
After developing the resist, the pattern is transferred onto a
60 nm-thick aluminum oxide (Al
2
O
3
) layer deposited by elec-
tron beam evaporation using the lift-off technique. The pat-
terned Al
2
O
3
serves as a hard mask for the dry etching of the
630 nm-thick SiN
x
layer in a mixture of C
4
F
8
and SF
6
plasma
and is finally removed by a mixture of ammonium hydroxide
and hydrogen peroxide at
80
C
.
Alignment procedure
The alignment procedure consists of two steps to ensure the
proper mapping of the SLM pixels onto the intended coordi-
nates of the disordered metasurface. Cross-shaped markers
engraved at the four corners of the metasurface are used to
guide rough alignment. Then, the marginal misalignments
(e.g. translation and tip-tilt) and aberrations induced by the
4-
f
system are corrected. For this purpose, a collimated laser
beam (Spectra-Physics, Excelsior 532) is tuned to be incident
on the metasurface and the resulting field is measured with
phase shifting holography. The residual misalignments and
aberrations are then calibrated by comparing the measured
complex field with the calculated one and digitally compen-
sating for the misalignment by adding appropriate correction
patterns on the SLM.
Procedure for optical focusing
The optimal incident pattern
E
opt
i
that encodes the informa-
tion for a target field
E
target
o
is calculated based on the concept
of phase conjugation using the expression
E
opt
i
(
x
,
y
)
=
L
[
T
(
x
,
y
;
x
o
,
y
o
)
E
target
o
(
x
o
,
y
o
)
d
x
o
d
y
o
]
=
L
[
e
i
φ
(
x
,
y
)
E
target
o
(
x
,
y
)
]
,
where
represents the conjugate transpose, and the function
L
represents the local spatial average of the ideal phase conjuga-
tion field
∫∫
T
E
target
o
d
x
o
d
y
o
within the area corresponding to
each controlled optical mode on the SLM. To produce a focal
spot at
r
=
(
x
,
y
,
z
)
in free space, the target field is set to a
spherical wavefront:
E
target
o
(
x
,
y
)
=
exp
[
i
2
π
λ
(
x
x
)
2
+
(
y
y
)
2
+
z
2
]
,
where
z
is the focal length. To perform the local spatial aver-
age
L
, a low-pass spatial frequency filter is applied using a fast
Fourier transform algorithm so that the SLM can successfully
sample the optimal wavefront
E
opt
i
. Finally, the SLM (Pluto,
Holoeye) is used for phase-only reconstruction of the complex
field
E
opt
i
within a circular aperture with a 4.3 mm radius. In
order to measure the focal spot, we use a custom-built mi-
croscope setup consisting of
100
×
objective lens (Olympus,
UMPlanFl) with an NA of
0
.
95
, a tube lens (Nikon,
2
×
, Plan
Apo), and a CCD camera (Imaging Source, DFK 23UP031).
Procedure for scanning fluorescence imaging
The setup of our scanning microscope is shown in fig.
S5C
.
For the collection of the scanned fluorescent signal, an imag-
ing system consisting of a
4
×
objective lens (Olympus,
0
.
1
NA,
Plan N) and tube lens (Thorlabs, AC508-100-A-ML) is used
to cover most of the FOV of the scanning microscope. We
scan the focal spot created behind the metasurface across the
region of interest with a 10 ms pixel dwell time. A pair of
galvanometric mirrors are used to scan
2
×
2
μ
m
2
patches with
a step size of 200 nm, and the neighboring patches are succes-
sively scanned by adding a compensation map on the SLM to
correct coma aberrations, instead of exhaustively calculating
and refreshing the
E
opt
i
for every spot. The fluorescent signal
is detected by the sCMOS camera (PCO, PCO.edge 5.5) with
6
an exposure time of 7 ms. The fluorescence signal is extracted
from the camera pixels corresponding to the scanned focus
position. The imaging time for a
30
×
30
μ
m
2
area is 5 min,
which can be easily improved by two orders of magnitude
using a high-power laser and resonant scanning mirrors.
Immunofluorescence-labeled sample preparation
As a biological sample, we use microscopic parasites,
Gi-
ardia lamblia cysts
(Waterborne, Inc.). Before labeling the
Giardia, we first prepare (a) the sample of
10
5
Giardia in 10
μ
L
phosphate buffered solution (PBS) in a centrifuge tube, (b) 1
μ
g
of Giardia lamblia cysts antibody (Invitrogen, MA1-7441) in
100
μ
L PBS, and (c) 2
μ
g of Goat anti-Mouse IgG (H+L) Sec-
ondary Antibody conjugated with Alexa Fluor 532 fluorescent
dye (Life Technologies, A-11002) in 100
μ
L of PBS. The sam-
ple (a) is incubated with a blocking buffer. After the blocking
buffer is removed, the sample is again incubated with the Gi-
ardia antibody solution (b). The sample is rinsed twice with
PBS to remove the Giardia antibody solution. The sample
is then incubated with the secondary antibody solution with
fluorescent dye (c). Finally, the sample is rinsed twice with
PBS to remove the secondary antibody solution. All incuba-
tions are carried out for 30 min at
37
C
. The sample in
10
μ
L
PBS is prepared on a slide with Prolong Gold antifade reagent
with DAPI (Life Technologies, P36935) to protect the labeled
sample from fading and covered with a coverslip.
ACKNOWLEDGMENT
This work is supported by the National Institutes of Health
BRAIN Initiative (U01NS090577), and a GIST-Caltech Col-
laborative Research Proposal (CG2012). Y.H. was supported
by a Japan Student Services Organization (JASSO) fellowship.
Y.H. and A.A. were also supported by National Science Foun-
dation Grant 1512266 and Samsung Electronics. A.S. was
supported by JSPS Overseas Research Fellowships. J.B. was
supported by the National Institute of Biomedical Imaging and
Bioengineering (F31EB021153) under a Ruth L. Kirschstein
National Research Service Award and by the Donna and Ben-
jamin M. Rosen Bioengineering Center. S.M.K. was sup-
ported by the DOE “Light-Material Interactions in Energy
Conversion” Energy Frontier Research Center funded by the
US Department of Energy, Office of Science, Office of Basic
Energy Sciences under Award no. DE-SC0001293. The de-
vice nanofabrication was performed at the Kavli Nanoscience
Institute at Caltech.
AUTHOR CONTRIBUTIONS
M.J. and Y.H. conceived the initial idea. M.J., Y.H., A.S.,
J.B., and C.Y. expanded and developed the concept. M.J., Y.H.,
and A.S. developed theoretical modeling, designed the exper-
iments, and analyzed the experimental data. M.J. and A.S.
carried out the optical focusing experiments. Y.H. performed
the full-wave simulation and the design on the metasurface.
A.S. performed the fluorescence imaging experiment. Y.H.,
S.M.K., and A.A. fabricated the metasurface phase mask. Y.L.
performed the measurements on the optical memory effect, the
angular scattering profiles, and the stability. All authors con-
tributed to writing the manuscript. C.Y. and A.F. supervised
the project.
These authors contributed equally to this work.
faraon@caltech.edu
chyang@caltech.edu
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8
A
B
C
D
E
Disorder-engineered metasurface
(calibration
free
transmission
matrix)
0
High NA focusing over a wide FOV
Recongurable control
Wide memory eect range and high stability
Figure 1
.
Complex wavefront engineering assisted by a disorder-engineered metasurface. (A)
The system set-up consists of two planar
components, an SLM and a disorder-engineered metasurface.
(B)
The disorder-engineered metasurface is implemented by varying the size
of nanoposts, which correspond to different phase delays
φ
(
x
,
y
)
on the metasurface.
(C)
The wide angular scattering range enables high
NA focusing over a wide FOV.
(D)
The thin, planar nature of the disordered metasurface yields a large memory effect range and also makes
the transmission matrix of the metasurface extraordinarily stable.
(E)
The SLM enables reconfigurable control of the expanded optical space
available through the disordered metasurface.
9
SiN
x
nanopost
fused silica substrate
1 μm
100
150
200
250
Post width (nm
)
0.0
0.2
0.4
0.6
0.8
1.0
Transm
issi
on
Phase (2π)
0.0
0.3
0.6
0.9
NA
0.0
0.2
0.4
0.6
0.8
1.0
Power (a.u.)
NA=0
.3
NA=0
.6
NA=0
.9
0
10
20
30
40
Angul
ar corre
lation
range
(deg)
0
30
60
90
Maxi
m
um
scatte
ring
angl
e (deg)
Disordered m
etasurfa
ce
Opal glass
White
paint
(TiO
2
)
Ground
glass
D
E
C
B
A
NA = 0.95
0
1
Figure 2
.
Disorder-engineered metasurface. (A)
Photograph and SEM image of a fabricated disorder-engineered metasurface.
(B)
Simulated transmission and phase of the SiN
x
nanoposts as a function of their width at a wavelength of 532 nm. These data are used as a
look-up table for the metasurface design.
(C)
Measured 2D angular scattering profile of the disordered metasurface, normalized to the strongest
scattered field component.
(D)
Measured 1D angular scattering profile of the disordered metasurfaces that were specifically designed to scatter
the incident light to certain angular ranges (
NA
=
0
.
3
,
0
.
6
,
0
.
9
).
(E)
Memory effect range and angular scattering range of the disordered
metasurface compared with conventional random media such as white paint, opal glass, and ground glass diffusers.