Articles
https://doi.org/10.1038/s41566-017-0078-z
Wavefront shaping 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, Pasadena, CA, USA.
2
T. J. Watson Laboratory of Applied Physics, California
Institute of Technology, Pasadena, CA, USA. Present addresses:
3
Department of Physics, Korea University, Seoul, South Korea.
4
Department of Electrical
and Computer Engineering, University of Massachusetts, Amherst, MA, USA. Mooseok Jang, Yu Horie and Atsushi Shibukawa contributed equally
to this work. *e-mail: faraon@caltech.edu; chyang@caltech.edu
© 2018 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
SUPPLEMENTARY INFORMATION
In the format provided by the authors and unedited.
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Supplementary Notes
Supplementary Note 1. Number of addressable output modes and th
e number of degrees of
freedom in the disordered metasurface assisted wavefront engine
ering system
In this supplementary section, we describe the disorder-enginee
red metasurface and phase-only
SLM optical system from the main text in a general mathematical
framework to clearly explain
why we can address a number of output modes (
M
) which is larger than the controlled number of
independent degrees of freedom (
N
). We show that the linear operator connecting the input and
output optical modes always has a rank ≤
N
(the number of pixels in the SLM). Therefore, the
number of degrees of freedom for the output modes is equal to t
he rank of the transmission
matrix. However, even though we are limited to at most
N
degrees of freedom for the output
modes, it is still possible to ha
ve a large numbe
r of resolvabl
e focal spots within a field of view.
Any linear optical device can be described by a linear operator
T
which takes an input
function
E
i
and generates a linear combination of modes
E
o
, given as
ܧ
ൌ
ܶ
ܧ
.
In the scenario of disordered media assisted wavefront shaping,
T
is an
M
×
N
matrix where
N
is
the number of independent modes controlled on the SLM (the numb
er of elements in
E
i
) and
M
is the number of addressable fo
cal spots within an area of inte
rest on the other side of the
disordered medium (the number of elements in
E
o
). In general,
M
is larger than
N
to take
advantage of the disordered medium’s ability to access an exten
ded optical space. This
transmission matrix is of rank ≤
N
. This means that we cannot exercise complete control over all
M
target output modes to achieve, for example, a perfect focus w
ith no background. However, as
previous work in the field of wavefront shaping has shown
1,2
, it is possible to combine the
fraction of the
N
independent input modes (given by the rank of
T
) that are transmitted through
open channels of the disordered medium to optimize the light in
tensity delivered into a desired
output mode such as a focusing field.
When a disordered medium is used in this way, it is called a “s
cattering lens.” If each
resolvable focal spot in the output space was treated as one mo
de (the total number of which is
defined as
M
according to the space-bandwidth product formalism in the main
text), we would
seemingly be able to achieve a number of degrees of freedom lar
ger than the rank of our linear
system. However, it is not valid to count each resolvable focal
spot as an independent mode,
because the focal spots created by the scattering lens have cor
related, speckle-
like backgrounds.
Although the number of resolvable focal spots is not equivalent
to the number of degrees of
freedom, it is an important and useful parameter in many applic
ations. In our focus-scanning
scattering lens microscope, since the intensity of an achieved
focal spot is significantly higher
(>10
4
) than the background intensity, we can count the addressed foc
al spots as resolvable focal
spots.
Supplementary Note 2. Conventional measurement of the transmiss
ion matrix using
O(P)
measurements
In previous reports, measurements of the transmission matrix ha
ve been performed in one of two
ways. The first method can be implemented by displaying
N
orthogonal patterns on the SLM and
recording the output field for each pattern
3,4
. This approach can be understood as measuring the
transmission matrix one column at a time, where each column cor
responds to one SLM pattern,
and each element in the column r
epresents the output field cont
ribution at a unique focal point on
the projection plane. To focus to a given point on the projecti
on plane, the pattern displayed on
the SLM is selected as a linear combination of the SLM patterns
such that the output field
constructively interferes at the desired focal point. In the co
ntext of phase-only modulation, this
means that the phase of each fie
ld vector, controlled by their
respective pixels on the SLM, is
aligned so as to maximize the su
m over all the field vectors at
that location. In order to enable
focusing at all
M
focal spots, the output field for each SLM pattern must be mea
sured at each of
the
M
focal spot locations.
An alternate way to measure the transmission matrix is using op
tical phase conjugation
5
.
This scheme is typically implemented by creating a calibration
light focus from an external lens
positioned at the desired focus location and recording the opti
cal field transmitted in the reverse
direction through the disordered medium toward the SLM. Then th
is procedure is repeated by
scanning the focus to all
M
desired focal spots on the output plane. Mathematically, this
approach can be interpreted as measuring the transmission matri
x one row at a time, where the
elements in each row describe the phase and amplitude relations
hip between a pixel on the SLM
and the desired focal point.
While both of these approaches provide a way to characterize th
e transmission matrix of a
disordered medium, they each suffer from practical limitations
that prevent them from being
practically useful for achieving control over large transmissio
n matrices (
P
> 10
12
). These stem
from the sheer number of measurements and time required to char
acterize the transmission
matrix. The first method i
s infeasible for large
M
due to the lack of commercially available
camera sensors with the required number of pixels. Thus far, to
the best of our knowledge, the
largest reported transmission matrix measured using this method
contained
P
= 10
8
elements.
While the second method is not limited by the availability of t
he requisite technology, it requires
mechanically scanning the focus to each spot. Assuming the rele
vant measurement technology
existed for both cases, with
a measurement speed of 10
8
measurements (i.e. transmission matrix
elements) per second (equivalent to 5 megapixels at 100 frames
per second), the measurement
for all
P
= 10
13
elements in our demonstrated transmission matrix would require
a measurement
time of over 24 hours. To make matters worse, conventional diso
rdered media used with
wavefront engineering such as white paint made of TiO
2
or ZnO nanoparticles have a stability of
only several hours
1,6,7
, so the measured transmission matrix would be invalid by the t
ime the
measurement was complete.
Supplementary Figures
Supplementary Figure 1
Supplementary Figure 1. Measured angular scattering profiles of
disordered metasurfaces
as well as those of conventional disordered media.
A collimated laser beam illuminated the
scattering media and a 4
f
system imaged the back focal plane of an objective lens (NA =
0.95) to
a camera. (
a
to
c
) Angular scattering profiles of disordered metasurfaces with d
ifferent designs,
normalized to strongest scattered field component. The disorder
ed metasurfaces were
specifically designed such that they scatter the incident light
to certain angular ranges of (
a
) NA
= 0.3, (
b
) 0.6, (
c
) 0.9, which are denoted with red dotted lines. See also Fig. 2
c in the main text
for the scattering profiles of the disordered metasurface used
in the experiment. (
d
to
f
) Angular
scattering profiles of conventional scattering media. (
d
) The 20-μm-thick white paint (made of
TiO
2
nanoparticles) and (
e
) opal glass diffuser (10DIFF-VI
S, Newport) show isotropic scat
tering
over the wide angular ranges, while (
f
) the ground glass diffuser (DG10-120, Thorlabs) has a
very limited angular range for scattering. The black dotted lin
es correspond to the cutoff
frequencies of the objective le
ns (NA = 0.95), which is the lim
it in our measurement set-up.
Supplementary Figure 2
Supplementary Figure 2. Optical memory effect measurement.
(
a
) Schematic of the optical
set-up to measure the angular correlation range of different sc
attering media. The output of a
long coherence length, 532-nm, continuous-wave laser was attenu
ated by a variable attenuator
composed of a half-wave plate (HWP) and a polarizing beam split
ter (PBS) where the unwanted
power was sent into a beam dump (BD). After it was expanded to
a beam diameter of 8 mm by
lenses L1 and L2, the laser beam
illuminated the scattering med
ium to be tested, and the speckle
pattern was detected by a camera. The camera and a camera lens
L3 were positioned 7.4 degrees
from the optical axis, to avoid collecting any undiffracted lig
ht. The series of speckle patterns
were recorded as we rotated the scattering medium, and we compu
ted the correlation coefficient
between the first frame and each of the ensuing frames. (
b
) The measured memory effect ranges
for the disordered metasurface, ground glass (DG10-120, Thorlab
s), opal glass (10DIFF-VIS,
Newport), and 20-μm-thick white paint (made of TiO
2
nanoparticles). See also Fig. 2e in the
main text. Error bars indicate the standard deviation of three
measurements.
Supplementary Figure 3
Supplementary Figure 3. Extraordi
nary stability of a disordered
metasurface
. Over a period
of 75 days, a high quality optical focus was obtained from the
same metasurface without
observable efficiency loss by small system alignments to compen
sate for mechanical drift. (
a
)
Reconstructed focus on the 1st day. The measured contrast was 1
9,800. (
b
) Reconstructed focus
on the 75th day. The measured contrast was 21,500. Scale bar: 1
μm.
Supplementary Figure 4
Supplementary Figure 4. E
xperimental set-up.
See Methods for detailed procedures for
different experiments. (
a
) Phase-shifting holography set-up used for calibrating the ali
gnment for
the disordered metasurface and the SLM. Zeroth-order block betw
een L7 and L6 was used to
block an undiffracted light from the disordered metasurface, wh
ich was experimentally measured
to be 1.5% with respect to the incident intensity. (
b
) Custom-built microscope set-up used for
characterizing high-NA fo
cusing over a wide FOV. (
c
) Focus-scanning fluorescence imaging set-
up. M: mirror, L: lens, HWP: half-wave plate, PBS: polarizing
beam splitter, S: shutter, EOM:
electro-optic modulator, GM: galvanometric mirror, BS: beam spl
itter, sCMOS: scientific
CMOS camera, CCD: CCD camera, SLM: spatial light modulator, ZB:
zeroth-order block, DM:
disordered metasurface, FM: flip
mirror, PSM: polarization-main
taining single-mode fiber, FL:
fluorescence filter.
Supplementary Figure 5
Supplementary Figure 5. Demonstra
tion of ultra-high number of r
esolvable spots
M
(~4.5
10
8
) even with a handful of physically controlled degrees of freed
om (~2.5
10
2
) as
inputs.
(
a1-2, b1-2
) Cropped phase images displayed on the SLM (
a1
,
b1
) as well as the
corresponding 2D intensity profiles (
a2
,
b2
) of the foci reconstructed at
z
́ = 3.8 mm on axis (NA
= 0.75). The controlled number of input optical modes displayed
SLM was (
a1
) 1.0×10
5
and (
b1
)
2.5×10
2
, respectively. Scale bars for the phase images and the 2D inte
nsity profiles are 500 μm
and 1 μm, respectively. (
c
) Measured NA of the foci created along the
x
-axis. The measured NA
shows good agreement with theory, regardless of the number of i
nput modes controlled on the
SLM. (
d
) Measured number of resolvable spots
M
as a function of the number of optical modes
N
controlled on the SLM. (
e
) Dependence of contrast factor
ߟ
on the number of optical modes
controlled on the SLM.
Supplementary Figure 6
Supplementary Figure 6. Electrical signal flow diagram for scan
ning fluorescence imaging.
(
a
) The system control diagram. (
b
) A data acquisition card (DAQ) outputted voltage stepping
signals to a pair of galvanometric mirrors (GM1 and GM2) to per
form bi-directional raster
scanning with a pixel dwell time of 10 ms. At the same time, th
e DAQ outputted a synchronized
trigger signal with a 7 ms durat
ion (corresponding to the expos
ure time) to a camera for detecting
fluorescent signals. After one patch of 11
11 spots were scanned by the galvanometric mirrors,
the galvanometric mirrors returned to the original position. Du
ring a 100 ms period, the phase
map for correcting coma aberration was updated on a spatial lig
ht modulator (SLM). Then, the
raster scanning by the galvanometric mirrors was resumed again
to constitute another patch.
Supplementary Figure 7
Supplementary Figure 7. Demonstration of arbitrary complex wave
front modulation with
a disordered metasurface.
(
a, b
)
Simultaneous generation of multiple foci. Scale bars: 1 μm.
(a) Four foci on a 4 μm pitch grid were reconstructed simultane
ously along the lateral axes. (b)
Two foci separated by 10 μm were reconstructed simultaneously a
long the optical axis. (
c, d
)
Optical vortex focusing with topology charges of (
c
)
m
=1 and (
d
)
m
=2. Scale bars: 1 μm. (
e
to
h
)
3D display using letters of ‘C’, ‘I’, and ‘T’ placed at (
f
)
z
= -10 μm, (
g
) 0 μm, and (
h
) 10 μm.
Scale bars: 2 μm.
Supplementary Figure 8. Demonstration of polarization insensiti
vity of current disordered
metasurface design.
Due to the symmetry of the lateral size of the nanoposts, the
current
disordered metasurface design is insensitive to the incident po
larization state. Foci with (
a
)
horizontal and (
b
) vertical linear polarizations.