Wavefront shaping with disorder-engineered metasurfaces
Mooseok Jang
1,†,‡
,
Yu Horie
2,†
,
Atsushi Shibukawa
1,†
,
Joshua Brake
1
,
Yan Liu
1
,
Seyedeh
Mahsa Kamali
2
,
Amir Arbabi
2,§
,
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
Abstract
Recently, wavefront shaping with disordered media has demonstrated optical manipulation
capabilities beyond those of conventional optics, including extended volume, aberration-free
focusing and subwavelength focusing. 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 its exact input-output characteristics
are known
a priori
and can be used with only a few alignment steps. We implement this concept
with a disorder-engineered metasurface, which exhibits additional unique features for wavefront
shaping such as a large optical memory effect range in combination with a wide angular scattering
range, excellent stability, and a tailorable angular scattering profile. Using this designed
metasurface with wavefront shaping, we demonstrate high numerical aperture (NA > 0.5) focusing
and fluorescence imaging with an estimated ~2.2×10
8
addressable points in an ~8 mm field of
view.
Wavefront shaping 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 thousands
1
. This sets
it apart from the regime of wavefront manipulation in adaptive optics where the corrections
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*
Correspondence to: faraon@caltech.edu (A.F.), chyang@caltech.edu (C.Y.).
†
These authors contributed equally to this work.
‡
Present address: Department of Physics, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul 02841, South Korea.
§
Present address: Department of Electrical and Computer Engineering, University of Massachusetts, 151 Holdsworth Way, Amherst,
Massachusetts 01003, USA.
Correspondence and requests for materials
should be addressed to A.F. or C.Y.
Author contributions
M.J. and Y.H. conceived the initial idea. M.J., Y.H., A.S., J.B., Y.L., H.R. and C.Y. expanded and developed the concept. M.J., Y.H.,
and A.S. developed theoretical modeling, designed the experiments, 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 with the help of H.R. 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 contributed to writing the
manuscript. C.Y. and A.F. supervised the project.
Competing financial interests
The authors declare no competing financial interests.
HHS Public Access
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are typically performed for aberrations modeled by a relatively small number of Zernike
orders
2
. As a class of technologies, wavefront shaping is particularly well suited for
applications involving disordered media. These applications can be broadly divided into two
categories. In the first category, wavefront shaping works to overcome intrinsic limitations of
the disordered media. Biological tissue is one such example where scattering is a problem,
with wavefront shaping 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 shaping to unlock an optical space with spatial extent (
x
) and spatial frequency
content (
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
wavefront shaping 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 wavefront shaping to increase the
flexibility of the optical system to, for example, significantly extend the volumetric range in
which aberration-free focusing can be achieved
8
–
10
or generate co-localized, thin light
sheets at several wavelengths
11
.
Unfortunately, this class of methods is stymied by one overriding challenge – the optical
input-output response of the disordered medium needs to be exhaustively characterized
before use
10
,
12
–
15
. Fundamentally, characterizing
P
input-output relationships of a
disordered medium requires
O
(
P
) measurements. 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 shaping. 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
12
. We also note,
although there have been several computational imaging methods proposed to exploit
disordered media for enhanced imaging resolution and field of view (FOV) without such
characterization
16
,
17
, their range of applications is much narrower as they do not enable
active manipulation of the optical field.
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
disordered metasurface provides the optical ‘randomness’ of conventional disordered media,
but in a way that is fully known
a priori
. We note that this approach is conceptually different
from previous techniques aimed at engineering disordered media
11
,
18
–
21
(e.g. Engineered
Diffusers
™
from RPC Photonics) since the disordered metasurface enables the individual
input-output responses rather than the statistical properties of the scattered light pattern to be
engineered. With this approach, we reduce the system characterization to a simple alignment
problem.
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In addition to eliminating the need for extensive characterization measurements, the
disordered metasurface has unique physical properties that make it useful for wavefront
shaping. First, it should be emphasized that the scattering mechanism of the disordered
metasurface is fundamentally different from that of a conventional disordered medium. In
contrast to a conventional disordered medium (e.g. a several micron thick layer of Zinc
Oxide particles or a ground glass diffuser) that has three-dimensional optical inhomogeneity
(i.e. the thickness of the scattering medium is much larger than the optical wavelength), the
disordered metasurface is composed of a two-dimensional array of subwavelength scatterers
of uniform height. These subwavelength scatterers are realized by high refractive index
contrast, dielectric nanoposts which act as truncated multimode waveguides
22
supporting
low quality factor Fabry-Perot resonances. Consequently, the metasurface platform exhibits
useful properties for wavefront shaping such as high transmittance and a very large angular
(tilt/tilt) correlation range due to the low angular sensitivity of the nanoposts’ resonances
23
.
The large transmittance and angular correlation range, which are typically coupled with the
thickness and angular scattering profile of a scattering medium, are independent features in
the metasurface platform. Moreover, the metasurface platform also offers excellent stability
and the capability to tailor the light scattering profile.
Using this disorder-engineered metasurface platform, we demonstrate control over
P
=
1.1×10
13
input-output relationships after a simple alignment procedure. To demonstrate this
new paradigm for controllably exploiting optical ‘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 FOV with a diameter of ~8 mm. In comparison,
for the same FOV, a conventional optical system such as an objective lens can typically at
most access one or two orders of magnitude fewer points.
Principles
The relationship between the input and output optical fields traveling through a disordered
medium
15
can be generally expressed in discretized form as
(1)
where
E
i
is the field vector at the input plane of the medium,
E
o
is the field vector at the
output projection plane behind the medium, and
T
is the transmission matrix connecting the
input field vector
E
i
with the output field vector
E
o
on the other side of the disordered
medium. In the context of addressable focal spots with disordered medium assisted
wavefront shaping,
E
o
is a desired focusing optical field,
E
i
is the linear combination of
independent optical modes controlled by the spatial light modulator (SLM), and
T
is the
transmission matrix, where each element 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 addressable spots on the projection plane, and
T
is a
matrix which connects the input and output fields with
P
elements, where
P
=
M
×
N
(see
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Methods for computation details). 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 wavefront shaping with disordered media is
that it allows access to a broader optical space in both spatial extent and spatial frequency
content than the input optical field can conventionally 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
ν
x
and
ν
y
are spatial
frequency contents along the transverse axes of the SLM and
d
SLM
is the pixel pitch;
typically ~10 μm). As a consequence, the number of addressable 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
, where
λ
is the wavelength of the
light. According to the space-bandwidth product formalism
24
, this means that the number of
addressable focal spots
M
within a given modulation area
S
is maximally improved to
(2)
The scheme for focusing with disordered medium assisted wavefront shaping can be
understood as the process of combining
N
independent optical modes to constructively
interfere at a desired position on the projection plane
4
,
25
,
26
. 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
spots on the output plane are not independent optical modes (see Supplementary Note 1).
Instead, because the transmission through the metasurface can be described as a linear
transformation of
N
input basis vectors, the number of degrees of freedom at the output
plane (i.e. the number of orthogonal patterns at the output plane) remains the same as at the
input plane. More specifically, when the incident wavefronts for
M
different spots are
transmitted through the metasurface, each focal spot exists on top of a background which
contains the correlated contributions from the unoptimised optical modes in the output field,
thus resulting in a maximum of
N
degrees of freedom at the output plane. 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
25
. In
practical situations where, for instance, the addressable spots are used for imaging or photo-
switching, the contrast
η
simply needs to be sufficiently high to ensure the energy leakage
does not harmfully compromise the system performance. In such applications, the
addressable spots can also be described as resolvable spots. Henceforth, we use these terms
interchangeably, depending on whether we are discussing the physics or the specific imaging
applications.
To maximize performance in imaging applications, 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
,
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and in turn
P
, should be as high as possible. Practically, there are two ways to measure the
elements of the transmission matrix – orthogonal input probing and output phase
conjugation (see Supplementary Note 2). In each case, an individual 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 measurement time. As a point of reference, if the fast transmission matrix
characterization method reported in ref.
12
could be extended without complications, it
would still require a measurement 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 is only several hours
25
,
27
,
28
.
In contrast, our disordered 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 steps required to align the disordered
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 optimal 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
that encodes the information for a target
field
is calculated using the concept of phase conjugation (see Methods). This
approach enables us to access the maximum possible number of resolvable spots for
wavefront shaping for a given modulation area
S
with the added benefit of control over the
scattering properties of the metasurface.
Results
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
29
–
35
: subwavelength scatterers or meta-atoms are arranged on a
two-dimensional lattice to purposefully shape optical wavefronts with subwavelength
resolution (Fig. 2a). The disordered metasurface, 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 wavelength of 532 nm. We constrained the design of
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
in free space,
and then chose the width of the individual nanoposts according to the look-up table shown in
Fig. 2b (see Methods). The experimentally measured scattering profile confirms the nearly
isotropic scattering property of the disordered metasurface, presenting a scattering profile
that fully extends to the spatial frequency of 1/
λ
as shown in Fig. 2c. The measured
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transmittance was approximately 50% regardless of the incident pattern. In contrast, it is
known that conventional three-dimensional disordered media in the diffusive regime present
fluctuations in transmittance, depending on the incident field, because the transmittances of
its fundamental transmitting eigenchannels are distributed over a wide range
36
. The
disordered metasurface platform also allows the scattering profile to be tailored, which can
be potentially useful in conjunction with angle-selective optical behaviors such as total
internal reflection. Figure 2c 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 Supplementary Fig. 1 for 2D angular scattering profiles).
In addition to a highly isotropic scattering profile, the disordered metasurface also exhibits a
very large angular (tilt/tilt) correlation range (also known as the optical memory effect
37
).
The correlation is larger than 0.5 even up to a tilting angle of 30 degrees (Fig. 2e). In
conventional disordered media, the volumetric nature of the media makes it very difficult to
achieve a wide angular scattering profile and a large memory effect at the same time due to
the competing influence of the thickness of the scattering medium on both of these
properties. In contrast, due to the low angular sensitivity of the nanoposts’ resonances
23
, the
disordered metasurface can simultaneously achieve a broad angular scattering profile and a
wide memory effect range. For example, conventional scattering 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)
38
.
Although ground glass diffusers present a relatively larger correlation range of ~5 degrees,
their limited angular scattering range makes them less attractive for wavefront shaping (see
Supplementary Fig. 2 for angular tilt/tilt measurement setup and correlation profiles).
Moreover, the disordered metasurface is extraordinarily stable due to its fixed, two-
dimensional fabricated structure
11
. We were able to retain the ability to generate a high
quality optical focus from the same metasurface without observable efficiency loss over a
period of 75 days by making only minor corrections to the system alignment to compensate
for mechanical drift (see Supplementary Fig. 3).
High NA optical focusing over an extended volume
We experimentally tested our wavefront manipulation scheme in the context of disordered
medium assisted focusing and imaging. First, we aligned the disordered metasurface 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 fields
(see Methods for details). Next, to demonstrate the flexibility of this approach, we
reconstructed a converging spherical wave (see Methods for details) for a wide range of
lateral and axial focus positions. Figure 3a presents the simplified schematic for optical
focusing (see also Methods and Supplementary Fig. 4 for more details). Figure 3b1–b3 show
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 corresponding 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
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were reconstructed 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, based on the Nyquist-Shannon
sampling theorem.
Figure 3b4–b6 show the 2D intensity profiles at
x
′
= 0, 1, 4, and 7 mm on the fixed focal
plane of
z
′
= 3.8 mm (corresponding to an 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
estimated 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. This enables high-
NA focusing over a large FOV without the mechanical scanning necessary using
conventional objective lenses, demonstrating that the metasurface-assisted platform, as a
fixed optical system, can access a wider optical space (represented by
x
and
ν
) compared to
conventional optics. For the disordered metasurface, an NA of ~0.5 was maintained in a
lateral FOV with a diameter of ~8 mm, resulting in an estimated 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
39
, an order of magnitude smaller than the number of the spots
demonstrated with the disordered metasurface. We note that although the resolvable spots
for the objective lens have the additional property that they are independent, as long as the
contrast
η
is high enough to enable a sufficient signal-to-noise ratio, the number of
resolvable spots is the appropriate metric of interest, regardless of whether or not they are
independent.
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 can achieve with our
system was 1.1×10
13
which enabled us to address an estimated ~4.3×10
8
focal 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
12
. 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 same number of addressable spots on the output projection
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 Supplementary Fig. 5). 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 wavefront
shaping regardless of the number of degrees of freedom in the input.
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Wide FOV fluorescence imaging
Finally, we implemented a scanning fluorescence microscope for high-resolution, wide FOV
fluorescence imaging (see Methods, Supplementary Figs 4 and 6 for detailed procedure).
Figure 4a presents the wide FOV, low-resolution fluorescence image of
immunofluorescence-labeled parasites (
Giardia lamblia cysts
; see Methods for sample
preparation procedures) captured through the 4× objective lens. As shown in the magnified
view in Fig. 4b3, a typical fluorescence image directly captured with the 4× objective lens
was significantly blurred, so that the shape and number of parasites was not discernible from
the image. Figure 4b1, c, and d present 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.
In addition, the large memory effect range of the metasurface allows us to use galvanometer
mirrors to scan the focus without having to refresh the pattern displayed on the SLM,
improving acquisition speed compared to using the SLM alone (see Methods for details). To
validate the performance of our imaging system, we compared 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.
Discussion
Here we have implemented a disorder-engineered medium using a metasurface platform and
demonstrated the benefit of using it for wavefront shaping. 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 wavefront shaping. 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 spherical wavefronts in this study, our
method is, in principle, generally applicable to produce arbitrary wavefronts for applications
such as beam steering, vector beam generation, multiple foci, or even random pattern
generation (see Supplementary Fig. 7 for experimental demonstrations). We anticipate that
the large gain in the number of addressable optical focal spots (or equivalently angles or
patterns) enabled by our method could potentially be used to improve existing optical
techniques such as fluorescence imaging, optical stimulation/lithography
40
,
41
, free space
coupling among photonic chips/optical networks
42
,
43
, and optical encryption/decryption
44
.
However, because the metasurface does not fundamentally improve the number of degrees of
freedom (
N
<
M
), our method is not well suited for applications in which a large fraction of
the addressable spots are used simultaneously such as holographic displays. In this scenario,
as the contrast
η
decreases in proportion to the number of target modes in the output plane,
the nonzero background would obscure the target pattern.
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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 advantages.
The system is highly scalable and versatile, bypassing the limitations and complexities of
using conventional objective lenses. The scalability of the metasurface could be especially
useful in achieving ultra-long working distances for high NA focusing. The scheme could
also be implemented as a vertically integrated optical device together with electronics
45
(e.g.
a metasurface phase mask on top of a transmissive LCD), providing a compact and robust
solution to render a large number of diffraction-limited spots. Furthermore, the concept is
potentially applicable 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
33
,
46
and
Si for near infrared wavelengths
31
,
47
–
49
), which allows for multiplexing different colors,
useful for multicolor fluorescence microscopy and multiphoton excitation microscopy.
Finally, the planar design provides a platform to potentially achieve ultra-high NA solid-
immersion lenses
50
or total internal reflection fluorescence (TIRF) excitation
51
, suitable for
super-resolution imaging and single-molecule biophysics experiments.
More broadly speaking, we anticipate the ability to customize the design of the disordered
metasurface for a particular application will prove highly useful. For example, we could
potentially tailor the scattering profile of the disordered metasurface to act as an efficient
spatial frequency mixer or to be exploited for novel optical detection strategies
19
,
52
,
53
. The
disordered metasurface could also potentially serve as a collection lens, analogous to the
results obtained for light manipulation, providing an enhanced resolving power and extended
view field. Additionally, while the disordered metasurface in our demonstration was
designed to be insensitive to the polarization of the incident optical field (see Supplementary
Fig. 8), the metasurface platform could be designed independently for orthogonal
polarization states, which provides additional avenues for control in wavefront shaping
54
.
Together, the engineering flexibility provided by these parameters offers unprecedented
control over complex patterned illumination, which offers the potential to directly benefit
emerging imaging methods that rely on complex structured illumination
17
,
55
.
To conclude, we explored the use of a disorder-engineered metasurface in wavefront
shaping, 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 that unlocks the benefits of wavefront shaping, opening new
avenues for the design of optical systems and enabling new techniques for exploring
complex biological systems.
Methods
Design of disordered metasurface
The disordered metasurface consisted 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 was 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
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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 was 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 was randomly chosen from a uniform
distribution between 0 and 2
π
radians. After several iterations, the phase profile converged
such that the far-field pattern had 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 was deposited using plasma enhanced chemical vapor deposition
(PECVD) on a fused silica substrate. The metasurface pattern was first defined in ZEP520A
positive resist using an electron beam lithography system. After developing the resist, the
pattern was transferred onto a 60 nm-thick aluminum oxide (Al
2
O
3
) layer deposited by
electron beam evaporation using the lift-off technique. The patterned Al
2
O
3
served 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 was finally removed by a mixture of ammonium hydroxide and hydrogen
peroxide at 80°C.
Alignment procedure
The alignment procedure consisted of two steps to ensure the proper mapping of the SLM
pixels onto the intended coordinates of the disordered metasurface. Cross-shaped markers
engraved at the four corners of the metasurface were used to guide rough alignment. Then,
the marginal misalignments (e.g. translation and tip-tilt) and aberrations induced by the 4-
f
system were corrected. For this purpose, a collimated laser beam (Spectra-Physics, Excelsior
532) was tuned to be incident on the metasurface and the resulting field was measured with
phase shifting holography. The residual misalignments and aberrations were then calibrated
by comparing the measured complex field with the calculated one and digitally
compensating for the misalignment by adding appropriate correction patterns on the SLM.
Procedure to compute a transmission matrix for optical focusing
The transmission matrix model in our experiments describes the amplitude and phase
relationship between each controllable input mode, given as each SLM pixel, and each
desired focusing optical field. The calculation of
T
was carried out in a row by row manner,
based on the intrinsic phase profile of the disordered metasurface
φ
(
x
,
y
). Setting the
position of the focal spot corresponding to
m
-th row vector as
, the
converging spherical wavefront on the plane of metasurface is given as:
where
z
′
is the focal length. Then, the corresponding input field on the plane of metasurface
was simply given as the product of the spherical wavefront
S
m
(
x
,
y
) and the transmission
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phase profile of the disordered metasurface (see Design of disordered metasurface in
Methods for details):
Next, to calculate the input field on the plane of SLM that corresponds to the input field on
the plane of metasurface
, a low-pass spatial frequency filter
ℒ
was applied to
using a fast Fourier transform algorithm:
Finally,
was sampled at positions corresponding to
N
SLM pixels for
discretization, yielding
N
matrix elements. That is, the discretized complex field composes a
row of the transmission matrix
T
that relates all controllable input modes to a given focal
spot on the projection plane.
The optimal incident field
that reconstructs the target field
(either of a single
spot or a pattern composed of multiple spots) was calculated using
where † represents the conjugate transpose. In the actual experiment, we used an SLM
(Pluto, Holoeye) for phase-only reconstruction of the optimal field
within a circular
aperture with a 4.3 mm radius. In order to measure the focal spot, we used a custom-built
microscope setup consisting of a 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 Supplementary Fig. 5c. For the collection
of the scanned fluorescent signal, an imaging system consisting of a 4× objective lens
(Olympus, 0.1NA, Plan N) and tube lens (Thorlabs, AC508-100-A-ML) was used to cover
most of the FOV of the scanning microscope. We scanned the focal spot created behind the
metasurface across the region of interest with a 10 ms pixel dwell time. A pair of
galvanometric mirrors were used to scan a 2×2 μm patche with a step size of 200 nm, and
the neighboring patches were successively scanned by adding a compensation map on the
SLM to correct coma aberrations, instead of exhaustively calculating and refreshing the
for every spot. Each image in Figs. 4b–4d consists of 15x15 patches. The fluorescence
signal was detected by the sCMOS camera (PCO, PCO.edge 5.5) with an exposure time of 7
ms. The fluorescence signal was extracted from the camera pixels corresponding to the
scanned focus position. The imaging time for a 30×30 μm
2
area was 5 min, which can be
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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 used microscopic parasites,
Giardia lamblia cysts
(Waterborne,
Inc.). Before labeling the Giardia, we first prepared (a) the sample of 10
5
Giardia in 10 μL
phosphate buffered solution (PBS) in a centrifuge tube, (b) 1 μg of Giardia lamblia cysts
antibody (Invitrogen, MA1-7441) in 100 μL PBS, and (c) 2 μg of Goat anti-Mouse IgG (H
+L) Secondary Antibody conjugated with Alexa Fluor® 532 fluorescent dye (Life
Technologies, A-11002) in 100 μL of PBS. The sample (a) was incubated with a blocking
buffer. After the blocking buffer was removed, the sample was again incubated with the
Giardia antibody solution (b). The sample was rinsed twice with PBS to remove the Giardia
antibody solution. The sample was then incubated with the secondary antibody solution with
fluorescent dye (c). Finally, the sample was rinsed twice with PBS to remove the secondary
antibody solution. All incubations were carried out for 30 min at 37°C. The sample in 10 μL
PBS was 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.
Data availability
The data that support the plots within this paper and other findings of this study are available
from the corresponding author upon reasonable request.
Supplementary Material
Refer to Web version on PubMed Central for supplementary material.
Acknowledgments
This work was supported by the National Institutes of Health BRAIN Initiative (U01NS090577), the National
Institute of Allergy and Infectious Diseases (R01AI096226), and a GIST-Caltech Collaborative 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 Foundation 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 Benjamin M. Rosen Bioengineering Center. S.M.K. was supported 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 device nanofabrication was performed at
the Kavli Nanoscience Institute at Caltech.
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Figure 1. Wavefront shaping assisted by a disorder-engineered metasurface
(a)
The system setup consists of two planar components, an SLM and a disorder-engineered
metasurface. (
b
) The disorder-engineered metasurface is implemented by fabricating the
nanoposts with varying sizes, 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.
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Figure 2. Characterization of disorder-engineered metasurfaces
(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.
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Figure 3. Experimental demonstration of diffraction-limited focusing over an extended volume
(
a
) Schematic of optical focusing assisted by the disordered metasurface. The incident light
is polarized along the
x
direction. (
b1
–
6
) Measured 2D intensity profiles for the foci
reconstructed at the positions indicated in (a). b1–b3 are the foci along the optical axis at
z
=
1.4, 2.1, and 3.8, mm, respectively, corresponding to NAs of 0.95, 0.9, and 0.75. b3–b6 are
the foci at
x
= 0, 1, 4, and 7 mm scanned on the fixed focal plane of
z
= 3.8 mm. Scale bar: 1
μm. (
c
) Measured NA (along
x
-axis) of the foci created along the optical axis (red solid line)
compared with theoretical values (black dotted line). When the SLM is used alone, the
maximum accessible NA is 0.033 (orange dotted line), based on the Nyquist-Shannon
sampling theorem. (
d
) Measured NA (along
x
-axis) of the foci created along
x
axis at
z
= 3.8
mm (red solid line) compared with theoretical values (black dotted line). The number of
resolvable focusing points within the 14-mm diameter FOV was estimated to be 4.3×10
8
.
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