of 27
Articles
https://doi.org/10.1038/s41566-019-0536-x
Single-shot quantitative phase gradient
microscopy using a system of multifunctional
metasurfaces
Hyounghan Kwon
1,2
, Ehsan Arbabi
1,2
, Seyedeh Mahsa Kamali
1,2
, MohammadSadegh Faraji-Dana
1,2
and Andrei Faraon
1,2
*
1
T. J. Watson Laboratory of Applied Physics and Kavli Nanoscience Institute, California Institute of Technology, Pasadena, CA, USA.
2
Department of
Electrical Engineering, California Institute of Technology, Pasadena, CA, USA. *e-mail: faraon@caltech.edu
SUPPLEMENTARY INFORMATION
In the format provided by the authors and unedited.
NA
turE PHotoNicS
|
www.nature.com/naturephotonics
Supplementary information: Single Shot Quantitative Phase Gradient
Microscopy Using a System of Multifunctional Metasurfaces
Hyounghan Kwon,
1, 2
Ehsan Arbabi,
1, 2
Seyedeh Mahsa Kamali,
1, 2
MohammadSadegh Faraji-Dana,
1, 2
and Andrei Faraon
1, 2,
1
T. J. Watson Laboratory of Applied Physics and Kavli Nanoscience Institute,
California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125, USA
2
Department of Electrical Engineering, California Institute of Technology,
1200 E. California Blvd., Pasadena, CA 91125, USA
Corresponding author: A.F.: faraon@caltech.edu
1
SUPPLEMENTARY NOTE 1: WAVE PROPAGATION SIMULATION AND PHASE MAPS OF
THE METASURFACES
In this section we present numerical analysis results of the QPGM. First, it is worth explaining
why the QPGM consists of two metasurface layers, and a single layer is not capable of doing so.
Then, we verify that the QPGM based on the two metasurface layers addresses the issues faced
in the system using a single metasurface. We used wave-propagation simulations to analyze the
systems in Fig.
S1
. In the simulations, the metasurfaces and the conventional lens are modeled as
ideal phase plates. Moreover, the thickness of all fused silica substrates is fixed at 1 mm. For all
metasurfaces, the distance between the optical axes for TE and TM polarizations,
s
, is fixed at
1.5
μ
m along the
y
-axis. Finally, the phase sample shown in Fig. 1
c
is used for the all simulations.
Figure
S1a
shows a schematic of a DIC system based on a single birefringent metasurface lens
and a regular refractive lens, forming a 4-
f
imaging system. To implement the single metasurface
layer system, we use the thin lens equation for the phase profiles of the metasurface lens (ML
1
,
a
)
and the conventional lens (Lens
2
,
a
). More specifically, ML
1
,
a
has two different phase profiles for
TE and TM polarizations, (
φ
ML
1
,
a
,
TE
and
φ
ML
1
,
a
,
TM
), given by:
φ
ML
1
,
a
,
TE
=
π
λ
f
1
(
x
2
+
(
y
+
s
2
)
2
)
(
S1
)
φ
ML
1
,
a
,
TM
=
π
λ
f
1
(
x
2
+
(
y
s
2
)
2
)
,
(
S2
)
where
x
and
y
are Cartesian coordinates from the center of ML
1
,
a
and
λ
is the operating wavelength
in vacuum. Moreover, the polarization-insensitive phase profile of Lens
2
,
a
is written as,
φ
Lens
2
,
a
=
π
λ
f
2
(
x
2
+
y
2
)
. While one might expect that the configuration in Fig.
S1a
works similar to a
conventional DIC microscope, an additional spurious intensity gradient shows up in the formed
interference pattern,
I
sin
g
le
, as seen in Fig.
S1b
. The reason is that despite the 4-
f
imaging
system, the two slightly separated optical axes for the TE and TM polarizations cause the intensity
gradient not present in the original target (see Supplementary Note 2 and Supplementary Fig.
S2
for experimental results of the single metasurface system and theoretical analysis about the
degradation). Another clear issue with the system shown in in Fig.
S1a
which undermines
its miniature size is that it would require a variable phase retarder to retrieve quantitative phase
gradient information.
2
To resolve the spurious intensity gradient issue first, we replace the refractive lens with a second
birefringent metasurface lens in Fig.
S1c
. For the system based on the two bifocal metasurface
lenses shown in Fig.
S1c
, the metasurface lens 1,
ML
1
,
c
, is the same as
ML
1
,
a
shown in Fig.
S1a
.
A second birefringent metasurface lens,
ML
2
,
c
, is used in the system shown in Fig.
S1c
instead of
Lens
2
,
a
in Fig.
S1a
. The two phase profiles of
ML
2
,
c
for TE and TM polarizations, (
φ
ML
2
,
c
,
TE
and
φ
ML
2
,
c
,
TM
), are given by:
φ
ML
2
,
c
,
TE
=
π
λ
f
2
(
x
2
+
(
y
+
s
2
)
2
)
(
S3
)
φ
ML
2
,
c
,
TM
=
π
λ
f
2
(
x
2
+
(
y
s
2
)
2
)
+
φ
o
,
(
S4
)
where
φ
o
is the phase offset between the two orthogonal polarizations.
φ
o
is fixed at
3
π
4
for the
simulations. As a result, the simulated interference intensity map shown in Fig.
S1d
,
I
double
, is
a clear DIC image of the transparent object with no intensity artifacts (see Supplementary Note
2 for theoretical explanation about the system of the two birefringent metasurface lenses). In the
experimental implementation, the values of
φ
o
are
3
π
4
and
π
4
for the QPGM using two separate
substrates and the double-sided QPGM, respectively.
In addition to birefringence, the capability of metasurfaces to simultaneously perform multiple
independent functions allows us to eliminate the requirement of a variable retarder, since several
images with different phase offsets can be captured simultaneously as shown schematically in Fig.
S1e
. Moreover, we employed the ray tracing method instead of the thin lens equation to mitigate
geometric aberrations. In other words, the phase profiles are optimized for the metasurface layers
1 and 2 in Fig.
S1e
,
Layer
1
,
e
and
Layer
2
,
e
, to minimize the off-axis aberrations and increase the
field of view. Specifically, the phase profiles of the metasurfaces are defined by two terms. One
is a sum of even-order polynomials of radial coordinates (mostly implementing the focusing), and
the other is a linear phase gradient term associated with the three-directional splitting of light. The
phase profiles of
Layer
1
,
e
for TE and TM polarizations,
φ
Layer
1
,
e
,
TE
and
φ
Layer
1
,
e
,
TM
, are given by:
φ
Layer
1
,
e
,
TE
=
5
n
=
1
a
n
R
2
n
(
x
2
+
(
y
+
s
2
)
2
)
n
k
g
r at
,
1
y
(
S5
)
φ
Layer
1
,
e
,
TM
=
5
n
=
1
a
n
R
2
n
(
x
2
+
(
y
s
2
)
2
)
n
k
g
r at
,
1
y
(
S6
)
3
φ
Layer
1
,
e
,
TE
=
5
n
=
1
a
n
R
2
n
(
x
2
+
(
y
+
s
2
)
2
)
n
k
g
r at
,
1
x
(
S7
)
φ
Layer
1
,
e
,
TM
=
5
n
=
1
a
n
R
2
n
(
x
2
+
(
y
s
2
)
2
)
n
k
g
r at
,
1
x
(
S8
)
φ
Layer
1
,
e
,
TE
=
5
n
=
1
a
n
R
2
n
(
x
2
+
(
y
+
s
2
)
2
)
n
+
k
g
r at
,
1
y
(
S9
)
φ
Layer
1
,
e
,
TM
=
5
n
=
1
a
n
R
2
n
(
x
2
+
(
y
s
2
)
2
)
n
+
k
g
r at
,
1
y
,
(
S10
)
where
a
n
are the optimized coefficients of the even-order polynomials in the shifted radial coordi-
nates,
k
g
r at
,
1
is the linear phase gradient, and
R
denotes the radius of the metasurfaces. Detailed
information about
a
n
,
k
g
r at
,
1
, and
R
is given in Table
S1
. Since a single set of rectangular nano-
posts can only implement one pair of the birefringent phase maps, three different sets of rectangular
nano-posts are designed to achieve the three pairs of phase maps in (Eqs.
S5
and
S6
), (Eqs.
S7
and
S8
), and (Eqs.
S9
and
S10
). Then, the three maps of the rectangular nano-posts are interleaved
along the
x
-axis using the spatial multiplexing method [1–3]. With the phase profiles in Eqs.
S5
-
S10
, the Layer
1
,
e
in Fig.
S1e
plays the role of a lens and a three directional beam-splitter at the
same time, at the cost of a drop in efficiency.
Layer
2
,
e
has three different birefringent metasurfaces which are identically displaced from the
center of Layer
2
,
e
. The distance from the center of the Layer
2
,
e
to the center of each lens,
D
,
is 660
μ
m. The three coordinates of the centers of the lenses measured from the center of the
Layer
2
,
e
are (0,-
D
) (-
D
,0) and (0,
D
). The six phase profiles of the three lenses for TE and TM
polarizations are written as:
φ
Layer
2
,
e
,
TE
=
5
n
=
1
b
n
R
2
n
(
x
2
+
(
y
+
D
+
s
2
)
2
)
n
+
k
g
r at
,
2
y
(
S11
)
φ
Layer
2
,
e
,
TM
=
5
n
=
1
b
n
R
2
n
(
x
2
+
(
y
+
D
s
2
)
2
)
n
+
k
g
r at
,
2
y
+
φ
o
(
S12
)
φ
Layer
2
,
e
,
TE
=
5
n
=
1
b
n
R
2
n
((
x
+
D
)
2
+
(
y
+
s
2
)
2
)
n
+
k
g
r at
,
2
x
(
S13
)
4
φ
Layer
2
,
e
,
TM
=
5
n
=
1
b
n
R
2
n
((
x
+
D
)
2
+
(
y
s
2
)
2
)
n
+
k
g
r at
,
2
x
+
φ
o
+
2
π
3
(
S14
)
φ
Layer
2
,
e
,
TE
=
5
n
=
1
b
n
R
2
n
(
x
2
+
(
y
D
+
s
2
)
2
)
n
k
g
r at
,
2
y
(
S15
)
φ
Layer
2
,
e
,
TM
=
5
n
=
1
b
n
R
2
n
(
x
2
+
(
y
D
s
2
)
2
)
n
k
g
r at
,
2
y
+
φ
o
+
4
π
3
,
(
S16
)
where
b
n
are the optimized coefficients of the even-order polynomials of the shifted radial coor-
dinates and
k
g
r at
,
2
is the linear phase gradient. The detailed information about
b
n
and
k
g
r at
,
2
is
given in Table
S1
. As a result, the two layers form the three different DIC images at the image
plane. Specifically, combinations of (Eqs.
S5
,
S6
,
S11
, and
S12
), (Eqs.
S7
,
S8
,
S13
, and
S14
), and
(Eqs.
S9
,
S10
,
S15
, and
S16
) result in the three phase-shifted DIC images in Fig.
S1f
,
I
1
,
I
2
and
I
3
,
respectively. To be specific, the desired phase offsets for the three-step phase shifting are achieved
by the phase maps of the second metasurface layer in Eqs.
S11
-
S16
. Moreover, we should point
out that
I
1
,
I
2
and
I
3
in Fig.
S1f
are comparable to the ideal results shown in Fig.
1d
. Figure
S1g
shows the PGI calculated from the three DIC images in Fig.
S1f
by using Eq.
2
in the main text,
and is in good agreement with the ideal PGI shown in Fig.
1e
.
As shown in Figs.
4a
and
4b
in the main text, the QPGM is also implemented using double-sided
metasurfaces on a 1-mm thick fused silica wafer. The double-sided metasurface QPGM is designed
through an identical process used for the design of the QPGM based on two metasurface layers on
two separate substrates. The phase profiles of the double-sided metasurfaces are determined by
Eqs.
S5
-
S16
with the optimized phase profile parameters such as
a
n
,
b
n
,
R
,
D
,
s
,
k
g
r at
,
1
, and
k
g
r at
,
2
given in Table
S2
. The distances from the object plane to the metasurface layer 1 and from
the image plane to the metasurface layer 2 are 317
μ
m and 397
μ
m, respectively.
5
SUPPLEMENTARY NOTE 2: EXPERIMENTAL AND THEORETICAL RESULTS WITH A
SINGLE BIFOCAL METASURFACE LENS
We fabricated a single bifocal metasurface and measured its performance. The phase map of
the device is determined by Eqs.
S1
and
S2
. For the fabricated single bifocal metasurface lens,
the diameter, focal length, and separation of the focal points are 900
μ
m, 1.2 mm, and 2
μ
m,
respectively. The two focal points are characterized through the optical setup shown in Fig.
S2a
.
An 850-nm semiconductor laser (Thorlabs, L850P010) is coupled to a single mode fiber and a laser
collimator for illumination. The linear polarizer in front of the laser collimator is aligned to 0
(for TE), 90
(for TM), and 45
for characterization of the bifocal metasurface lens. The measured
intensity maps shown in Supplementary Fig.
S2b
clearly show the polarization dependent bifocal
property. The intensity profiles on the black dashed lines in Fig.
S2b
are shown in Fig.
S2c
. In Fig.
S2c
, the FWHM of the two focal points and the distance between the two focal points are 1.21-1.35
μ
m and 2
μ
m, respectively. Moreover, the measured focusing efficiencies through a pin-hole with
a diameter of about 4
×
FWHM at the focal plane are
78
%
for both TE and TM polarizations.
We performed imaging experiments with the single bifocal metasurface lens using the optical
setup schematically shown in Fig.
S2d
. We employed a variable retarder to capture three different
DIC images. Moreover, oblique illumination from an LED was used to avoid saturation at the
central pixels of the camera resulting from undiffracted light. To limit the bandwidth, we employed
a band-pass filter with a center wavelength and bandwidth of 850 nm and 10 nm, respectively. The
three captured DIC images,
I
1
,
I
2
, and
I
3
, are shown in Fig.
S2e
. The results clearly show that the
spurious graded intensity patterns degrade the DIC images as expected from Fig.
S1b
. In Fig.
S2f
, the PGI is calculated from the three DIC images in Fig.
S2e
through the three-step phase
shifting method [4]. The graded phase gradient in the background in Fig.
S2f
also indicates the
imperfect imaging performance of the single bifocal metasurface lens.
The degradation exists both in the simulation results shown in Fig.
S1b
and the measurement
in Figs.
S2e
and
S2f
. The degradation can be explained using Fourier optics. If we consider the
optical system consisting of one metasurface bi-focal lens and a normal thin lens as in Fig.
S1a
such that the z-axis passes through the center of the second lens (i.e. the optical axes of the first
lens for TE and TM polarizations are
s/2 distant from the z-axis), we can write down the relations
between the field at the object plane (
U
) and the fields at the Fourier plane (
U
F
,
T E
and
U
F
,
T M
)
through the Fresnel diffraction. For simplicity, let’s assume that the focal lengths of the two lenses
6
in Fig.
S1a
are identical. In this case, the two optical fields formed by the first metasurface lens,
U
F
,
T E
and
U
F
,
T M
are no longer simple Fourier transforms of
U
, but would instead be written as:
U
F
,
T E
(
x
,
y
)
=
∫∫ ∫∫
U
(
x
′′
,
y
′′
)
ex p
[
i
π
λ
f
((
x
x
′′
)
2
+
(
y
y
′′
)
2
)]
dx
′′
d
y
′′
×
ex p
[−
i
π
λ
f
(
x
2
+
(
y
+
s
2
)
2
))])
ex p
[
i
π
λ
f
((
x
x
)
2
+
(
y
y
)
2
)]
dx
d
y
=
A
1
∫∫
U
(
x
′′
,
y
′′
)
ex p
[
i
π
λ
f
(
x
2
+
x
′′
2
+
y
2
+
y
′′
2
)]
×
∫∫
ex p
[
i
π
λ
f
(
x
2
+
y
2
2
(
x
+
x
′′
)
x
2
(
y
+
y
′′
+
s
2
)
y
]
dx
d
y
dx
′′
d
y
′′
=
A
2
ex p
(−
i
2
π
λ
f
s
2
y
)
∫∫
U
(
x
′′
,
y
′′
)
ex p
[−
i
2
π
λ
f
(
x x
′′
+
(
y
+
s
2
)
y
′′
)]
dx
′′
d
y
′′
=
A
2
ex p
(−
i
2
π
λ
f
s
2
y
)
̃
U
(
x
λ
f
,
y
+
s
2
λ
f
)
U
F
,
T M
(
x
,
y
)
=
A
2
ex p
(
i
2
π
λ
f
s
2
y
)
̃
U
(
x
λ
f
,
y
s
2
λ
f
)
.
where
λ
, f, and
s are the wavelength, focal length of the two lenses, and the separation between
the optical axes of the metasurface lens, respectively. Also,
̃
U
represents the 2D Fourier transform
of
U
.
A
1
and
A
2
are complex constants. Then, the fields at the image plane (
U
i
,
T E
and
U
i
,
T M
) can
be written through the 2D Fourier transform of
U
F
,
T E
and
U
F
,
T M
;
U
i
,
T E
(
x
,
y
)
=
∫∫
U
F
,
T E
(
x
,
y
)
ex p
[−
i
2
π
λ
f
(
x x
+
y y
)]
dx
d
y
=
B
1
ex p
(
+
i
2
π
λ
f
s
2
y
)
U
(−
x
,
−(
y
+
s
2
))
U
i
,
T M
(
x
,
y
)
=
∫∫
U
F
,
T M
(
x
,
y
)
ex p
[−
i
2
π
λ
f
(
x x
+
y y
)]
dx
d
y
=
B
1
ex p
(−
i
2
π
λ
f
s
2
y
)
U
(−
x
,
−(
y
s
2
))
,
where
B
1
is a complex constant. We should point out that the fields at the image plane are not
only shifted laterally by the separation
s, but also accompanied with a different phase gradient
for TE and TM polarizations. The different phase gradients for TE and TM polarizations cause
the spurious degradation shown in Figs.
S1b
,
S2e
, and
S2f
. Considering that
s is close to the
diffraction-limit scale, the simulation result in Fig.
S1b
and the measurement in Figs.
S2e
and
S2f
clearly reveal that the system is very sensitive to even very small values of
s. Furthermore,
the inevitable axial misalignment between the double axes of the metasurface lens and the single
axis of the normal lens results in the phase gradient difference at the image plane. In other words,
lens-based DIC microscopy typically necessitates at least three optical elements which are two
lenses and a birefringent crystal such as a Wollaston prism between the two lenses. Although one
might suppose that the different linear phase gradients can be employed instead of the lateral shifts
along the
y
-axis in Eqs.
S1
and
S2
to remove the spurious degradation, it is worth pointing out
that both schemes are mathematically equivalent with a different complex constant and result in a
similar degradation.
In contrast, the two metasurface layers can avoid the unwanted degradation because the two
lenses have independent optical axes for both polarizations. For the system in Fig.
S1c
, the fields
at the image plane can be written as the Fourier transform with respect to the optical axis of each
7
polarization. As a result, the fields at the image plane
U
i
,
T E
and
U
i
,
T M
can be written as;
U
i
,
T E
(
x
,
y
)
=
U
(−
M x
,
M
(
y
+
s
2
)−
s
2
)
U
i
,
T M
(
x
,
y
)
=
U
(−
M x
,
M
(
y
s
2
)
+
s
2
)
,
where M is the magnification of the system and is mainly determined by the phase maps of the
two bi-axial lenses. These equations clearly show that the two birefringent metasurface lenses are
able to capture the conventional DIC images with two polarizers. More interestingly, the simulation
results based on the wave propagation in Fig.
S13
reveal that the proposed bi-axial system becomes
very robust against other kinds of misalignments in the fabrication or optical alignment procedures
once the separations of the optical axes for the both lenses are identical. As explained in the
main manuscript, it is worth noting that the metasurfaces uniquely admit arbitrary phase maps
in subwavelength scale for both polarizations. Thus, the separation is one of the most accurately
controllable variables in the design process of the metasurfaces. This is one of the reasons why
our system performs adequately in the experiments. Furthermore, we remark that bi-focal lenses
having an accurate separation close to the diffraction-limit are difficult to implement using any
conventional birefringent optics.
8
SUPPLEMENTARY NOTE 3: THREE-STEP PHASE SHIFTING METHOD FOR UNIDIREC-
TIONAL QUANTITATIVE PHASE GRADIENT IMAGING
Phase-shifting is a widely known technique for phase retrieval in common interferometers.
Moreover, it has been also used for quantitative phase gradient imaging by modifying the classical
DIC microscope [5]. The technique utilizes multiple phase-shifted interference patterns to retrieve
the phase information. To be specific, a captured differential interference contrast (DIC) image at
the object plane is effectively written as;
I
j
=
|
U
(
x
,
y
)−
U
(
x
,
y
y
)
e
i
φ
j
|
2
=
B
(
x
,
y
)−
C
(
x
,
y
)
cos
(
θ
(
x
,
y
)−
φ
j
)
(
S17
)
, where
U
(
x
,
y
)
=
A
(
x
,
y
)
e
i
φ
(
x
,
y
)
,
B
(
x
,
y
)
=
|
A
(
x
,
y
)|
2
+
|
A
(
x
,
y
)|
2
,
C
(
x
,
y
)
=
2
|
A
(
x
,
y
)
A
(
x
,
y
)|
,
θ
(
x
,
y
)
=
φ
(
x
,
y
)−
φ
(
x
,
y
y
)
, and
φ
j
is a phase offset. Since
B
(
x
,
y
)
,
C
(
x
,
y
)
, and
θ
(
x
,
y
)
are unknown, it can be seen that a minimum of three independent measurements are required
for unambiguous retrieval of all unknowns. This is why the second metasurface layer in the
current work consists of three different polarization sensitive off-axis lenses. Moreover, it is worth
noting here that extensive investigations have previously been done to develop two-step phase
shifting algorithms with additional assumption, some form of a priori knowledge, or complicated
computations [6–8]. In the three-step phase shifting techniques,
φ
1
,
φ
2
, and
φ
3
are usually set to
0,
2
π
3
, and
4
π
3
. Using the Eq.
S17
,
I
1
,
I
2
, and
I
3
are written as;
I
1
=
B
(
x
,
y
)−
C
(
x
,
y
)
cos
(
θ
(
x
,
y
))
,
I
2
=
B
(
x
,
y
)−
C
(
x
,
y
)
cos
(
θ
(
x
,
y
)−
2
π
3
)
, and
I
3
=
B
(
x
,
y
)−
C
(
x
,
y
)
cos
(
θ
(
x
,
y
)−
4
π
3
)
.
With further calculations, one can see that
θ
(
x
,
y
)
can be expressed in terms of
I
1
,
I
2
, and
I
3
;
θ
(
x
,
y
)
=
ar ctan
(
3
I
2
I
3
2
I
1
I
2
I
3
)
Considering that
θ
(
x
,
y
)
=
φ
(
x
,
y
)−
φ
(
x
,
y
y
)≈∇
y
φ
×
y
and
y
is small compared to the
sample feature sizes,
y
φ
can be calculated as;
y
φ
1
y
ar ctan
(
3
I
2
I
3
2
I
1
I
2
I
3
)
In Eq. 2 in the main manuscript, a calibration term,
y
φ
cali
, is added to remove any kind of
unwanted background coming from imperfect experimental conditions. Specifically,
y
φ
cali
is the
background signal measured without any sample. The calibration process also allows for arbitrarily
choosing three phase offsets with the same difference of
2
π
3
. For example,
φ
1
is set to
3
π
4
and
π
4
9
for the device using the two separate metasurface layers and the device based on the double-sided
metasurfaces, respectively.
10
I. REFERENCES
[1] Maguid, E.
et al.
Photonic spin-controlled multifunctional shared-aperture antenna array.
Science
352
,
1202-1206 (2016).
[2] Arbabi, E., Arbabi, A., Kamali, S. M., Horie, Y. & Faraon, A. Multiwavelength polarization-insensitive
lenses based on dielectric metasurfaces with meta-molecules.
Optica
3
, 628–633 (2016).
[3] Lin, D.
et al.
Photonic multitasking interleaved Si nanoantenna phased array.
Nano Lett.
16
, 7671–7676
(2016).
[4] Huang, P. S. & Zhang, S. Fast three-step phase-shifting algorithm.
Appl. Opt.
45
, 5086–5091 (2006).
[5] Cogswell, C. J., Smith, N. I., Larkin, K. G. & Hariharan, P. Quantitative DIC microscopy using a
geometric phase shifter.
Proc. SPIE
2984
, 72–81 (1997).
[6] Meng, X. F.
et al.
Two-step phase-shifting interferometry and its application in image encryption.
Opt.
Lett.
31
, 1414–1416 (2006).
[7] Liu, J.-P. & Poon, T.-C. Two-step-only quadrature phase-shifting digital holography.
Opt. Lett.
34
,
250–252 (2009).
[8] Zhang, Y, Tian, X & Liang, R. Random two-step phase shifting interferometry based on Lissajous
ellipse fitting and least squares technologies.
Opt. Express
26
, 15059–15071 (2018).
11
Metasurface
R
(
μ
m)
D
(
μ
m)
s
(
μ
m)
a
1
a
2
a
3
a
4
a
5
k
g
r at
,
1
(rad/
μ
m)
Layer1
300
660
1.5 -4.70
×
10
2
2.05
×
10
1
-3.88
×
10
0
7.69
×
10
1
-7.30
×
10
2
2.23
Metasurface
R
(
μ
m)
D
(
μ
m)
s
(
μ
m)
b
1
b
2
b
3
b
4
b
5
k
g
r at
,
2
(rad/
μ
m)
Layer2
300
660
1.5 -2.37
×
10
2
3.58
×
10
1
1.07
×
10
1
-7.58
×
10
0
1.92
×
10
1
2.22
Table S1
Phase profile parameters for the two separate metasurface layers
12
Metasurface
R
(
μ
m)
D
(
μ
m)
s
(
μ
m)
a
1
a
2
a
3
a
4
a
5
k
g
r at
,
1
(rad/
μ
m)
Layer1
100
210
1.5 -1.29
×
10
2
7.96
×
10
0
-1.17
×
10
1
6.98
×
10
0
-1.34
×
10
0
2.21
Metasurface
R
(
μ
m)
D
(
μ
m)
s
(
μ
m)
b
1
b
2
b
3
b
4
b
5
k
g
r at
,
2
(rad/
μ
m)
Layer2
100
210
1.5 -7.88
×
10
1
-5.19
×
10
0
4.53
×
10
0
-1.43
×
10
0
1.61
×
10
1
2.19
Table S2
Phase profile parameters for the double-sided metasurface QPGM
13
Figure S1 System-level design and numerical analysis of the QPGM. a
Schematic of an optical system
consisting of a single metasurface and a conventional thin lens. The metasurface works as a bifocal lens
with two focal points for TE and TM polarizations separated along the
y
axis.
b
Simulated intensity map at
the image plane formed by the system shown in
a
.
c
Schematic of an optical system consisting of two
birefringent metasurfaces. Each metasurface acts as a bifocal lens with two focal points for TE and TM
polarizations separated along the
y
axis.
d
Simulated intensity map formed in the image plane by the
system shown in
c
.
e
Schematic of the optical system composed of one multi-functional metasurface and
three bifocal lenses. As mentioned in Fig.
1b
, the first metasurface collimates light from two focal points
for the TE and TM polarizations that are separated in the
y
direction and splits it in three directions towards
the three metasurfaces on the second layer. Similar to the system in
c
, the first metasurface forms a separate
imaging system with each of the metasurface lenses on layer 2.
f
Three simulated DIC images at the image
plane using the system shown in
e
.
g
The PGI calculated from the three DIC images in
f
. Pol.: linear
polarizer; ML: metasurface lens;
d
a
: 2.20 mm;
d
c
: 2.81 mm;
d
e
: 2.70 mm ;
f
1
: 687
μ
m; and
f
2
: 1.51 mm.
14
Figure S2 Phase gradient imaging with a single bifocal metasurface lens. a
Schematic illustration of
the optical setup used for capturing focuses of the single bifocal metasurface lens and measuring focusing
efficiencies. The linear polarizer (L-Pol.) in front of the laser collimator is accordingly adjusted to confirm
the input polarization states.
b
Measured focal points for different input polarization states. Left, center,
and right images are the measured intensity maps at the focal plane with the linear polarizer aligned to 0
(TE), 90
(TM), and 45
, respectively. Scale bars: 2
μ
m.
c
Normalized intensity profiles at the focal plane
on the three black dashed lines in
b
. Green and blue dashed lines are the intensity profiles for TE and TM
polarized input light, respectively. The black line is the intensity profile with the 45
linear polarized light.
d
Schematic illustration of the optical setup capturing the three interference intensity patterns with a single
bifocal metasurface and a variable retarder. An LED is used for the oblique illumination. BP: bandpass
filter.
e
Measured interference intensity patterns,
I
1
,
I
2
, and
I
3
that have phase offsets of 0,
2
π
3
, and
4
π
3
between the TE and TM polarizations, respectively.
f
PGI calculated through the three-step phase shifting
method from the three interference intensity maps in
e
. Scale bars: 30
μ
m.
15
Figure S3 Schematic illustration showing the relation between
s
and
y
.
s
is the separation
between the two optical axes for TE and TM polarizations. The green and blue dashed lines denote the
optical axes for TE and TM polarizations, respectively.
y
is the effective shearing distance at the object
plane. In other words, two points that are
y
apart along the
y
-axis in the object plane, are imaged to the
same points at the image plane for the two different polarizations. The green and blue solid lines represent
the rays coming from the green and blue points at the object plane, respectively. While the blue dot in the
object plane is imaged along the blue dashed line representing the optical axis of TM polarization, the
green dot is actually off-axis imaged at the position of the blue dot with respect to the green dashed line
showing the optical axis of TE polarization. The black dashed box shows the relation between
s
and
y
given by
y
=
s
(
1
+
1
M
)
, where M is the magnification of the optical system.
16
Figure S4 Simulation results of the nano-posts at the wavelength of 850 nm. a
-
d
Simulated
transmittance and transmitted phase of TE and TM polarized light for periodic arrays of meta-atoms as
functions of
D
x
and
D
y
. The amorphous silicon layer is 664 nm thick, and the lattice constant is 380 nm.
a
and
b
: transmittance,
c
and
d
: transmitted phase
e
-
h
The optimized simulation results calculated from
a
-
d
for complete polarization and phase control. Calculated optimal
D
x
and
D
y
as functions of the required
φ
T E
and
φ
T M
are shown in
e
and
f
, respectively. Simulated transmittance of TE and TM polarized light as
functions of
φ
T E
and
φ
T M
are shown in
g
and
h
, respectively.
i-l
The optimized simulation results of the
nano-posts composed of 664-nm thick amorphous silicon and 60-nm thick Al
2
O
3
on the fused silica
substrate. The nano-posts are arranged on a square lattice with a 390-nm lattice constant and cladded by an
8-
μ
m-thick SU-8 layer. Calculated optimal
D
x
and
D
y
as functions of the required
φ
T E
and
φ
T M
are
shown in
i
and
j
, respectively. Simulated transmittance of TE and TM polarized light as functions of
φ
T E
and
φ
T M
are shown in
k
and
l
, respectively.
17
Figure S5 Simulated point spread functions of the QPGM. a
Schematic showing the locations of point
sources used at the object plane for characterizing the point spread functions (PSFs). Cartesian coordinates
of each point are given under it. The coordinates are calculated from the center of the metasurface layer 1.
b
The normalized PSFs at the image plane for TE polarization of the optical system shown in Fig.
S1e
.
The coordinates under the PSFs are the coordinates of the corresponding point sources. It is assumed that
the PSFs are identical for TE and TM polarizations.
c
The magnified intensity maps of the normalized
PSFs in
b
. The coordinates of the corresponding point sources are shown above the intensity maps. The
dashed black circles are airy disks whose radii are 2.54
μ
m at the image plane. All coordinates in the
figures are given in microns.
18
Figure S6 Schematic of the measurement setup.
Schematics of the custom-built microscope setup used
to measure the three DIC images captured by the QPGMs. L-Pol.: linear polarizer; BP: band-pass filter.
19
Figure S7 Magnified PGIs of the phase resolution targets having different thicknesses.
The
thicknesses of the targets are as follows;
a
: 54 nm;
b
: 159 nm;
c
: 261 nm; and
d
: 371 nm. Note that the
color bar scale in
a
is different from the other panels. Scale bars: 15
μ
m.
20