1
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Increasing efficiency of high
numerical aperture metasurfaces
using the grating averaging
technique
Amir
Arbabi
1
✉
, Ehsan
Arbabi
2
, Mahdad
Mansouree
1
, Seunghoon
Han
3
,
Seyedeh Mahsa
Kamali
2
, Yu
Horie
2
& Andrei
Faraon
2
✉
One of the important advantages of optical metasurfaces over conventional diffractive optical elements
is their capability to efficiently deflect light by large angles. However, metasurfaces are conventionally
designed using approaches that are optimal for small deflection angles and their performance for
designing high numerical aperture devices is not well quantified. Here we introduce and apply a
technique for the estimation of the efficiency of high numerical aperture metasurfaces. The technique is
based on a particular coherent averaging of diffraction coefficients of periodic blazed gratings and can
be used to compare the performance of different metasurface designs in implementing high numerical
aperture devices. Unlike optimization-based methods that rely on full-wave simulations and are only
practicable in designing small metasurfaces, the gradient averaging technique allows for the design of
arbitrarily large metasurfaces. Using this technique, we identify an unconventional metasurface design
and experimentally demonstrate a metalens with a numerical aperture of 0.78 and a measured focusing
efficiency of 77%. The grating averaging is a versatile technique applicable to many types of gradient
metasurfaces, thus enabling highly efficient metasurface components and systems.
Flat optical devices based on dielectric metasurfaces have recently attracted significant attention due to their
small size, low weight, and the potential for their low-cost manufacturing using semiconductor fabrication tech-
niques
1
–
8
. The planar form factor of metasurfaces and the high multilayer overlay accuracy of the semiconductor
manufacturing process enable the implementation of low-cost monolithic optical systems composed of cascaded
metasurfaces whose production does not involve post-fabrication assembly and alignment steps
9
,
10
. Used either
as single-layer devices or as integral parts of cascaded metasurface systems, one of the main requirements for
metasurfaces is high efficiency. As a result, increasing the efficiency of metasurfaces has been the subject of recent
studies
11
–
20
, and low numerical aperture (NA) metasurface components (i.e., metasurfaces with small deflection
angles) with efficiencies of more than 97% have been reported
21
. However, the efficiency of metasurfaces is known
to decrease with increasing NA, resulting in a trade-off between the NA and efficiency
11
,
22
–
24
. Several approaches
have been proposed for designing efficient high-NA metasurfaces including adjoint optimization
16
,
18
–
20
,
25
,
26
and
patching together separately designed gratings
14
,
27
. The metasurfaces designed based on iterative adjoint optimi-
zation techniques can be efficient
16
,
18
–
20
; however, their dimensions are inevitably limited by the available com-
putational resources because the optimization process requires full-wave simulation of the entire device in each
iteration, or simulation and optimization of each zone of the device
25
that is applicable to metalenses but cannot
be readily extended to general holograms. Separate optimization of small portions of a gradient metasurface and
then assembling them together to form a large metalens has been proposed
14
,
27
; however, the effect of disconti-
nuities at the boundaries of patches has not been considered and efficient high-NA metasurfaces based on such
approaches have not been demonstrated yet.
Typical metasurfaces are arrays of scatterers (or meta-atoms) arranged on 2D lattices. The conventional
approach of designing metasurfaces involves selection of a set of parameterized meta-atoms and finding a
1
Department of Electrical and Computer Engineering, University of Massachusetts Amherst, 151 Holdsworth Way,
Amherst, MA, 01003, USA.
2
T. J. Watson Laboratory of Applied Physics, California Institute of Technology, 1200 E.
California Blvd., Pasadena, CA, 91125, USA.
3
Samsung Advanced Institute of Technology, Samsung Electronics,
Samsung-ro 130, Suwon-si, Gyeonggi-do, 443-803, South Korea.
✉
e-mail:
arbabi@umass.edu
;
faraon@caltech.edu
open
2
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one-to-one map between the desired optical response (e.g., phase shift) and the meta-atom parameters. The map
typically involves one or two of the meta-atom parameters, and a number of other design parameters such as
the meta-atom height, geometry, and lattice constant are selected by the designer. These parameters are usually
selected for achieving high transmission (or reflection) and a 2
π
phase coverage, but their effects on the perfor
-
mance of high-NA metasurfaces designed using the same design maps cannot be easily evaluated. Here we intro-
duce a novel approach for evaluating the performance of different metasurface designs in implementing high-NA
metasurface components. The approach is based on the adiabatic approximation of aperiodic metasurfaces by
periodic blazed gratings and considers the effect of large deflection angles. Using the proposed approach, we
identify a design for implementing efficient high-NA metasurfaces and experimentally demonstrate a metalens
with an NA of 0.78 and a focusing efficiency of 77% as well as a 50° beam deflector with more than 70% deflection
efficiency for unpolarized light.
We focus our study on the design of gradient metasurfaces composed of meta-atoms that are arranged on
periodic 2D lattices. This category represents a large class of metasurfaces and has been used in the realization of
different types of metasurface optical elements
28
–
35
and systems
9
,
10
. We first discuss the conventional technique
for designing transmissive gradient metasurfaces and then use this discussion to explain the main idea of the
proposed grating averaging technique.
The conventional approach for designing transmissive gradient metasurfaces assumes a local complex trans-
mission coefficient for each meta-atom that only depends on the meta-atom itself (i.e., it is independent of the
neighboring meta-atoms)
11
,
28
–
31
,
33
. For metasurfaces that operate under normal incidence, the local transmission
coefficient for each meta-atom is approximated by the transmission coefficient of a periodic array created by peri-
odically arranging the same meta-atom on the metasurface lattice. A library of meta-atoms is typically formed
that spans the range of required transmission coefficients (e.g., 0 to 2
π
phase shift range). A flat optical component
with a desired spatially varying complex transmission coefficient
t
(
x
,
y
) is realized by an aperiodic meta-atom
array whose local transmission coefficients best approximate
t
(
x
,
y
) at the location of the meta-atoms (i.e., the
lattice sites). Figure
1
shows an example of the conventional approach for designing metasurfaces composed of
nano-post meta-atoms
11
,
34
,
36
. In this case, the meta-atom library comprises square-cross-section nano-posts with
different widths
W
that should provide complex transmission coefficients exp(
−
j
φ
) for
φ
spanning the 0 to 2
π
range. Figure
1a
shows the simulated transmission coefficient of periodic arrays of amorphous silicon nano-posts
(height: 500
nm, width: 60 nm–260
nm, lattice constant: 400
nm, wavelength: 915
nm) that are used as estimates
of local transmission coefficients in designing aperiodic metasurfaces. Refractive indices of 3.65 and 1.45 are
0
Phase (rad)
2
0
100
200
300
W (nm)
0
Phase (rad)
2
0.9
0.95
1
W (nm)
100
150
200
250
0
0.5
1
Phase/(2
)
(a)
500 nm
W
400 nm
(b)
00
.5
11
.5
2
0
20
40
60
80
Efficiecncy (%)
TE
TM
100
g
(rad)
(c)
(d)
(e)
Extended cell
x
y
z
x
y
z
Figure 1.
(
a
) Illustration of an example periodic metasurface composed of amorphous silicon nano-posts with
square cross-sections and its transmission coefficient for normally incident light as a function of the nano-
post’s width (
W
). (
b
) The design curve that maps desired phase shifts to nano-post widths (top plot) and the
corresponding transmittance (bottom plot). The design curve is obtained from the transmission coefficient data
shown in (
a
). (
c
) Diffraction efficiencies of the first transmitted order of blazed gratings with different phases
(
φ
g
). The blazed gratings are designed using the metasurface platform and design curve shown in (
a
) and (
b
).
The gratings’ period is 1200
nm (i.e., three times the metasurface lattice constant) corresponding to a first order
diffraction angle of 49.7°. The efficiency values are computed for light normally incident from the substrate with
transverse electric (TE) and transverse magnetic (TM) polarizations. (
d
) Top view of an aperiodic metasurface
with a deflection angle of 52° that is designed using the metasurface platform and design curves shown in (
a
and
b
). The nano-post dimensions vary rapidly from one metasurface unit cell to the next, but slowly from one
extended cell to the next. (
e
) Side view of the aperiodic beam deflector shown in (
d
). The dashed red line depicts
the output plane of the device. All results are for 915
nm light.
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assumed for the amorphous silicon and the fused silica substrate, respectively. The inverted relation that maps
the desired transmission coefficient exp(
−
j
φ
), which is indexed by
φ
, to the nano-posts width is shown in Fig.
1b
.
A metasurface implementing an arbitrary phase profile can be designed by sampling the desired phase profile at
each lattice site and determining the nano-post’s width at that location using the design curve shown in Fig.
1b
.
The conventional approach provides a simple and scalable technique for designing metasurfaces composed
of a large number of meta-atoms. However, two main approximations are used in this approach. First, it assumes
that the metasurface is locally periodic with a period equal to the lattice constant. In other words, it ignores the
change in the coupling between meta-atoms and their neighbors that is caused by the aperiodicity of the metas
-
urface. Second, it assumes that the radiation pattern of the meta-atoms (i.e., the element factor) is isotropic, thus
the local transmission coefficient of the meta-atoms is assumed to be independent of the incident and scattered
directions. The first approximation is justified if the meta-atom shapes change gradually across the metasurface.
For the validity of the second approximation, the radiation pattern of the meta-atoms should not change signifi-
cantly over the angular range of interest for the incident and transmitted light. As a result, the efficiency of meta-
surfaces designed using the conventional approach depends on the coupling strength among the meta-atoms and
the angular dependence of their radiation patterns.
Low-NA metasurfaces are composed of gradually varying meta-atoms and satisfy the requirement for the first
approximation. They also deflect incident light by small angles, thus the condition for the second approximation
is also satisfied when the incident angle is small. As a result, low-NA metasurfaces designed using the conven-
tional approach can be highly efficient and have achieved experimentally measured efficiency values as high as
97%
21
. These conditions are also satisfied for the low-NA metasurfaces that operate under oblique incidence
provided the design maps are also obtained for the same incident angles
10
. For a more general case where the
incident angle varies across a low-NA metasurface, one may use different design maps for different regions of the
metasurface based on the local incident angle (i.e., phase gradient of the incident field).
As the metasurface NA increases, the approximations involved in the conventional design approach become
less accurate and the metasurface efficiency decreases. Qualitatively, the performance of a metasurface platform
in implementing high-NA devices designed using the conventional approach depends on the coupling among the
meta-atoms and their radiation patterns. However, there is no fast approach to evaluate, compare, and predict the
performance of different designs for the realization of high-NA metasurfaces. The grating averaging technique
discussed in the next section addresses this issue.
Results
Grating averaging technique.
Beam deflectors are basic elements in designing gradient metasurfaces
because such metasurfaces can be considered as beam deflectors with gradually varying deflection angles. As a
result, the deflection efficiency of beam deflectors designed using a metasurface platform can be used to evalu-
ate the performance of the platform in realizing general metasurface components. Metasurface beam deflectors
implement a linear phase ramp (i.e.,
t
=
exp(
−
j
φ
(
x
,
y
)) where
φ
(
x
,
y
) is a linear function of
x
and
y
) and can be
designed for arbitrary deflection angles. A beam deflector in the
z
=
0 plane that deflects normally incident light
propagating along the
z
direction by an angle
θ
toward the
x
axis has a phase profile of
φ
(
x
)
=
2
π
/
λ
sin(
θ
)
x
+
φ
0
,
where
λ
is the light’s wavelength in the
z
>
0 region and
φ
0
is a constant. Now, consider implementing such a beam
deflector by wrapping its phase to 0–2 range and using a metasurface with a square lattice with the lattice constant
of
a
. For specific values of
θ
,
a
sin(
θ
)/
λ
is a rational number (i.e.,
a
sin(
θ
)/
λ
=
n
/
m
, where
n
and
m
are coprime
integers) and the implemented metasurface beam deflector is periodic along the
x
direction. The period is
Λ
=
ma
and the beam deflector may be considered as an
n
th
-order blazed grating. When
a
sin(
θ
)/
λ
is an irrational number
the metasurface is aperiodic and we refer to it as an aperiodic beam deflector. One might consider approximat-
ing the local diffraction efficiency of aperiodic beam deflectors by the diffraction efficiency of a periodic grating
with approximately the same deflection angle. This is particularly interesting because the diffraction efficiency of
gratings can be computed using fast computational methods such as the rigorous coupled mode analysis (RCWA)
technique
37
. However, there are two issues that need to be addressed regarding this approximation.
First, for a given grating period and a design map, there is a family of blazed gratings with the same deflection
angle but different phases and efficiencies. The phase profile of an ideal blazed grating that deflects normally inci-
dent light by an angle
θ
g
is given by
φ
(
x
)
=
2
π
/
λ
sin(
θ
g
)
x
+
φ
g
where
φ
g
is a constant representing the phase of the
diffracted light at
x
=
0. Different values for
φ
g
lead to different blazed grating designs with different efficiencies.
For example, using the metasurface design curve shown in Fig.
1b
, three-post blazed gratings can be designed
by setting the phase delays imparted by the three nano-posts as
φ
g
,
φ
g
+
2
π
/3 and
φ
g
+
4
π
/3. Different three-post
blazed gratings that are obtained for different values of
φ
g
have the same period of 3
a
=
1200 nm and diffraction
angle of
θ
g
=
sin
−
1
(
λ
/(3
a
))
=
49.7° when illuminated with a normally incident 915
nm light from the substrate
side. The simulated diffraction efficiencies of these gratings (for the
+
1 diffraction order) as a function of
φ
g
for
two incident polarizations are shown in Fig.
1c
. As Fig.
1c
shows, the diffraction efficiency of the gratings varies
significantly with
φ
g
; therefore, the deflection efficiency of a periodic metasurface beam deflector is not unique
and depends on its phase.
Now consider an aperiodic beam deflector with a deflection angle
θ
close to the diffraction angle of a blazed
grating
θ
g
. For large deflection angles, the meta-atoms vary significantly from one lattice site (unit cell) to the
next along the direction of the phase gradient. However, the aperiodic beam deflector can be considered as a
slowly varying blazed grating. This can be seen in Fig.
1d
that shows the top view of a portion of an aperiodic
beam deflector with the deflection angle of
θ
=
52° that is designed using the design map shown in Fig.
1b
. The
meta-atoms in the beam deflector shown in Fig.
1d
vary rapidly from one unit cell to the next, but slowly between
extended cells containing three meta-atoms. Therefore, each extended cell of the aperiodic beam deflector may
be approximately considered as a period of a blazed grating with some value of
φ
g
, and from one extended cell to
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the next, the value of
φ
g
varies slowly. As a result, it is reasonable to estimate the deflection efficiency of aperiodic
beam deflectors using the diffraction response of blazed gratings with approximately the same diffraction angle.
Consider an aperiodic beam deflector as schematically shown in Fig.
1e
and assume that the beam deflector is
composed of a large number of extended cells. The beam deflector deflects a normally incident plane wave by an
angle
θ
which is close to the diffraction angle
θ
g
of a family of blazed gratings with different phases
φ
g
. Depending
on the polarization of the incident wave, the deflected light is either TE or TM polarized with respect to
z
. As
shown in the Supplementary Note 1, the deflection coefficient of the aperiodic beam deflector for either TE or
TM polarization is given by
At
e
1
2
()
d,
(1)
n
j
0
2
gg
g
∫
π
φφ
≈
π
φ
where
t
n
(
φ
g
) represents the diffraction coefficient for the same polarization of a blazed grating with the diffraction
angle
θ
g
designed with the phase of
φ
g
. The phase of the diffraction coefficient is the same as the phase of the elec-
tric field of the diffracted wave at
x
=
z
=
0 and its amplitude is given by |
t
n
(
φ
g
)|
=
η
g
, where
η
g
is the diffraction
efficiency of the blazed grating. The deflection efficiency (i.e., the ratio of the power of the deflected beam and the
incident beam power) for the aperiodic grating is given by
η
=
|
A
|
2
.
The deflection efficiency of aperiodic beam deflectors can be obtained according to (1) which is a specially
weighted average of the complex-valued diffraction coefficients of blazed gratings with the same diffraction angle
and different phases. We can compute the deflection efficiency of an aperiodic beam deflector by designing
N
different blazed gratings with different
φ
π
=...
ππ
,,
,2
NN
g
24
i
, finding their diffraction coefficients
t
n
()
g
i
φ
and
approximating the integral in (1) by
φ
∑
φ
=
te
()
N
i
N
n
j
1
1
g
i
i
g
. The main advantage is that the diffraction coefficients of
the blazed gratings can be computed quickly.
In the ideal case, the diffraction coefficient of a blazed grating designed for the phase of
φ
g
is
t
n
(
φ
g
)
=
exp(
−
j
φ
g
) leading to
η
=
|
A
|
2
=
1 (according to (1)) for the ideal beam deflector. In practice, the diffraction efficiency
of the designed gratings (i.e., |
t
g
|
2
) is smaller than unity and there is a difference between their actual and desired
phases
φ
g
. Both of these will lead to the reduction of the efficiency of aperiodic beam deflectors.
In contrast to the periodic beam deflectors (i.e., blazed gratings), the efficiencies of aperiodic beam deflectors
are well-defined and independent of their phases. For example, the deflection efficiency of a beam deflector with
the deflection angle of
θ
g
=
sin
−
1
(
λ
/(3
a
))
=
49.7° which is designed using the metasurface platform of Fig.
1
may
be any of the values shown in Fig.
1c
; however, the deflection efficiency of a large beam deflector with the deflec-
tion angle of 50° which is designed using the same design curve is uniquely obtained from (1) and is ~66% and
~32% for the TE and TM polarizations, respectively.
Low-NA metasurfaces are a special case where the extended cell is the same as the metasurface unit cell, grat-
ing diffraction angle is zero (
θ
g
=
0), and
t
n
is the transmission coefficient of the periodic array of meta-atoms.
Therefore, the efficiency of a low-NA metasurface is given by
te
1
4
()
d,
(2)
j
02
0
2
2
∫
η
π
φφ
=
π
φ
where
t
(
φ
) is the complex-valued transmission coefficient of a periodic array composed of the meta-atom used
for achieving the phase shift
φ
. We note that the effect of infrequent discontinuities violating the adiabatic meta-
surface approximation by gratings, which are caused by wrapping of
φ
g
, is ignored in efficiency estimations using
(1) and (2).
Comparing different metasurface design platforms using the grating averaging tech-
nique.
The efficiency values obtained using the grating averaging technique can be used to evaluate and com-
pare the performance of different designs in implementing metasurfaces with different NAs. To illustrate the
procedure, we consider a second design and compare its performance with the design presented in Fig.
1a,b
.
The second design was selected to offer high efficiency at large deflection angles by exploring different designs
and evaluating their efficiencies using (1). A schematic of the second design is presented in Fig.
2a
. In the sec-
ond design, the nano-posts are 590
nm tall, the lattice constant is 350
nm, and the nano-post widths are varied
between 60
nm and 200
nm. The transmittance and the phase of the transmission coefficient for 915
nm light
normally incident on a periodic array of nano-posts with the parameters of this design are also shown in Fig.
2a
.
Nano-posts with widths larger than 200
nm are excluded in this design and the total phase shift covered by this
design is 1.63
π
which is smaller than its ideal value of 2
π
. The design curve that relates the nano-post width to the
desired phase and the corresponding transmittance values for the second design are shown in Fig.
2b
. Compared
to the first design, the smaller phase shift coverage and the lower average transmittance of the second design indi-
cate its inferior performance when used for implementing low-NA metasurfaces. In fact, the efficiency of low-NA
metasurfaces designed using these designs can be obtained from (2) and the data presented in Figs.
1
a and
2a
, and
are 96% and 92% for the first and second designs, respectively. However, high NA gratings based on the grating
averaging design have higher efficiencies (Fig.
2c
).
The diffraction efficiency of metasurfaces with higher NAs that are designed using these two designs can be
computed using (1) by designing a set of gratings with different periods and phases
φ
g
. We selected different grat-
ing periods for each design and for each grating period we designed 40 gratings with different values of
φ
g
(i.e.,
φ
g
=
0 to 2
π
in 40 steps). The diffraction coefficients of the gratings (
t
n
) were found using RCWA simulations
38
and the deflection efficiencies of aperiodic beam deflectors were obtained by averaging the diffraction coefficients
according to (1), finding the deflection efficiency as
η
=
|
A
|
2
, and averaging
η
values for TE and TM polarizations.
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The unpolarized deflection efficiency versus deflection angle computed for these two design are shown in Fig.
2d
.
As Fig.
2d
shows, for deflection angles larger than ~15° the second design outperforms the first one. As men-
tioned earlier, the second design was found by exploring different designs and comparing their deflection efficien-
cies at large deflection angles using the grating averaging technique. Because the second design was found using
the grating averaging technique, we refer to it as the grating averaging design. We attribute the higher efficiency
of the grating averaging design at large deflection angles to its smaller period and to the exclusion of nano-posts
with large cross-sections. Such nano-posts support high order resonance modes that have off-axis nulls in their
scattering patterns
39
–
41
.
Performance verification: Numerical results.
To demonstrate the effectiveness of the grating averaging
technique in the design of more general metasurfaces, we designed a 50-
μ
m-diameter metalens with a focal
length of
f
=
20
μ
m (NA of 0.78) using the design curve presented in Fig.
2b
. The phase profile of the metalens was
chosen as
Φ
=
−
2
π
/
λ
0
++
xy
f
222
to achieve aberration-free focusing for normally incident light with the
vacuum wavelength of
λ
0
=
915
nm. A schematic of the metalens is shown in Fig.
3a
. The metalens is illuminated
by an
x
-polarized normally incident plane wave and the light impinging outside the clear aperture of the metalens
is blocked by a perfect electric conductor (PEC) as shown in Fig.
3a
. The metalens was simulated using the finite
difference time domain (FDTD) technique
42
. The
x
component of the transmitted electric field on a plane a quar
-
ter of a wavelength above the top of the nano-posts, which is indicated by a dashed red line in Fig.
3a
, is shown in
Fig.
3b
. The transmitted fields in the region above the metalens were computed via the plane wave expansion
method
43
and the electric energy density in the
xz
and
yz
cross-sections are shown in Fig.
3b
.
For comparison, we designed a metalens with similar parameters using the first design (Fig.
1a
) and simulated
it using the same procedure. The corresponding plots for this control metalens are presented in Fig.
3c
. As
E
x
field
distributions presented in Fig.
3b,c
show, the grating averaging metalens has a significantly smaller phase error at
regions close to the circumference of the metalens where the deflection angles are larger. The low efficiency of the
control metalens in deflecting the light toward the focal point means that some of the light is deflected to other
directions. The interference of the light deflected to other directions and the light deflected toward the focus cre-
ates the phase error seen in Fig.
3c
in areas close to the metalens circumference. The smaller phase error leads to
a reduction in the light scattered to other directions as it can be seen in the logarithmic-scale energy distributions
shown in Fig.
3b,c
. The grating averaging metalens also has a higher transmission of 89% compared with 75% for
the control metalens. Figure
3d
shows the focal spots of the two metalenses. The grating averaging metalens has a
more circular and brighter spot than the control metalens.
60
1001
40
180
0
0.5
1
Phase/(2
)
W (nm)
40
120
200
W (nm)
0.85
0.9
0.95
1
0
Phase (rad)
2
0
Phase (rad)
2
)
b
(
)
a
(
First design (Fig. 1)
02
04
06
0
50
60
70
80
90
100
Efficiency (%)
(d)
590 nm
350 nm
W
(c
)
00
.3
0.6
50
75
Efficiecncy (%)
100
g
(rad)
TE
TM
Figure 2.
(
a
) Schematic of a periodic metasurface based on the grating averaging design. Simulated
transmission data for different nano-post widths
W
is also shown. (
b
) Design curve relating the desired phase to
the nano-post width for the metasurface design shown in (
a
) and the corresponding transmittance. The design
curve is obtained using the transmission data shown in (
a
). (
c
) Diffraction efficiencies of the third transmitted
order of blazed gratings with different phases (
φ
g
). The blazed gratings are designed using the metasurface
platform and design curve shown in (
a
and
b
). The gratings’ period is 3500
nm (i.e., ten times the metasurface
lattice constant) corresponding to a third order diffraction angle of 51.7°. The efficiency values are computed
for light normally incident from the substrate with transverse electric (TE) and transverse magnetic (TM)
polarizations. (
d
) Estimated deflection efficiencies of beam deflectors implemented using the metasurface
design shown in Fig.
1
and the grating averaging design shown in (
a
and
b
).
6
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The focusing efficiencies of the grating averaging and the control metalenses are found as 79% and 63%,
respectively. The focusing efficiency is defined as the percentage of the power incident on the metalens aper
-
ture that is focused into and passes through a circle with a radius of 5
μ
m centered around the focal spot of the
metalens. The radius of 5
μ
m for the aperture is selected for a direct comparison with the experimentally meas
-
ured results that are presented in the next section. The on-axis modulation transfer function (MTF) of the two
metalenses are shown in Fig.
3e
. The control metalens MTF along the
x
direction is smaller than the MTF of the
grating averaging metalens at high spatial frequencies. The smaller MTF value is a result of the lower deflection
efficiency of the control metalens along the
x
axis and away from the center of the metalens where phase error is
significant as it can be seen in the
E
x
distribution shown in Fig.
3c
.
Performance verification: Experimental results.
To experimentally confirm the high efficiency of
metasurfaces designed using the grating averaging technique, we designed and fabricated metalenses and meta-
surface beam deflectors. The metasurfaces were designed using the design curve shown in Fig.
2b
and fabricated
by depositing a 590-nm-thick layer of amorphous silicon on a fused silica substrate and pattering it using electron
beam lithography and a dry etching process. Details of the fabrication process were similar to our previous work
and can be found in ref.
11
.
A schematic illustration of one of the fabricated metalenses is shown in Fig.
4a
. The fabricated metalens (diam-
eter: 400
μ
m, focal length: 160
μ
m) is 8 times larger than the simulated metalens shown in Fig.
3a
, but has the
same NA of 0.78. A scanning electron image of the fabricated device is shown in Fig.
4b
. We measured the focal
spot of the metalens using the measurement setup schematically shown in Fig.
4c
. The metalens was illuminated
by a collimated 915
nm laser beam and its focal plane intensity was magnified using the combination of the
objective and tube lenses and captured by the camera (Fig.
4c
). Figure
4d
shows the measured focal spot for an
x
-polarized incident light. We did not observe any noticeable change in the focal spot as we varied the polariza-
tion of the incident beam. The measured focal spot intensity along the
x
-axis (indicated by the dashed black line)
and the simulated focal spot intensity of an ideal metalens with a diameter of 400
μ
m and a focal length of 160
μ
m are also shown in Fig.
4d
. The ideal metalens only modifies the optical wavefront of the incident light while
keeping its local intensity unchanged. As Fig.
4d
shows, the measured focal spot intensity matches well with the
ideal metalens result, thus indicating a negligible wavefront aberration. The measured on-axis MTF of the fabri-
cated metalens for
x
-polarized incident light and the MTF of an ideal metalens with the same NA are presented
in Fig.
4e
. The MTF of the measured device matches well with the ideal MTF at low spatial frequencies but drops
faster at higher frequencies indicating the reduction of the deflection efficiency at larger deflection angles.
We measured the focusing efficiency of the metalens using the setup shown in Fig.
4f
. The metalens was
illuminated by a weakly diverging Gaussian beam that was generated by gently focusing the incident laser beam
before the devices using a lens with a focal length of 5
cm (as shown in Fig.
4f
). The distance of the lens and the
device was adjusted such that the beam radius at the device was ~140
μ
m thus more than 98% of the incident
power impinged on the device aperture (assuming a Gaussian intensity distribution). The intensity distribution
1
0
Intensity (a.u.)
(e)
Cont
rol
(d)
0
-20
-40
-60
Electric ener
gy density (dB)
0
-1
1
E
x
(a.u.
)
Cont
rol
(c)
x
z
x
y
y
z
10
m
0
-20
-40
-60
Electric ener
gy density (dB)
0
-1
1
E
x
(a.u.
)
(b)
x
z
x
y
y
z
10
m
(a)
50
m
20
m
PE
C
x
y
z
E
x
0
500
1000
150
0
0
0.5
1
Contrast
Control, along
x
Control, along
y
1
m
Figure 3.
(
a
) Schematic of metalenses used in numerical simulations. Two metalenses are designed using the
design curves shown Fig.
1b
, referred to as the control metalens, and Fig.
2b
which is referred to as the grating
averaging metalens. (
b
) Full-wave simulation results of the grating averaging, and (
c
) Control metalenses. Top,
x
components of the electric field over the output aperture of the metalenses. Bottom, logarithmic-scale electric
energy density distributions on axial planes of the metalenses. (
d
) Focal plane intensity distributions for the
grating averaging and control metalenses. (
e
) On-axis modulation transfer functions for the grating averaging
and control metalenses. Simulations are performed at 915
nm. PEC: perfect electric conductor.
7
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at the metalens focal plane was magnified by 100
×
, masked by passing through a 1-mm-diameter aperture in the
image plane (corresponding to a 5-
μ
m-radius aperture in the metalens focal plane), and its power was measured.
To measure the incident optical power, we focused the incident beam using a commercial lens (Thorlabs AC254-
030-B-ML with a focal length of 3
cm and a transmission efficiency of 98%) and measured its power in the image
plane (Fig.
4g
). The focusing efficiency of the metalens was found as 77% by dividing the power passed through
the aperture in Fig.
4f
to the incident power.
In addition to the 400-
μ
m-diameter metalens, using the grating averaging design shown in Fig.
2a,b
, we
designed and fabricated 9 beam deflectors with different deflection angles ranging from 7° to 70°, and a metalens
with a diameter of 2
mm and a focal length of 800
μ
m. The beam deflectors had a diameter of 400
μ
m and were
illuminated by a normally incident Gaussian beam with a beam radius of ~100
μ
m on the device (Fig.
5a
). The
deflection efficiencies of the beam deflectors, defined as the ratio of the deflected beam power to the incident
power, were measured for TE and TM polarizations. The average of the measured TE and TM deflection effi
-
ciencies (i.e., the deflection efficiency for unpolarized light) for different beam deflectors are presented in Fig.
5c
.
100X, 0.95 NA
Tube lens
Camera
Device
FC
PC
Laser
(915 nm)
400
m
160
m
)
c
(
)
a
(
(f)
100X, 0.95 NA
Tube lens
Iris (D=1 mm)
Device
f=5 cm
Mirro
r
PD
d>5cm
(g)
100X, 0.95 NA
Tube lens
f=3 cm
PD
Mirro
r
(e)
0
500
1000
1500
0
0.5
1
Contrast
Along y
Along x
Ideal metalens
(d)
1
m
1
0
Intensity (a.u.)
x
y
-2
-1
012
x (
m)
0
0.5
1
Intensity (a.u.)
Ideal met
alens
Measure
d
(b)
1
m
Figure 4.
(
a
) Illustration of the fabricated metalens. (
b
) A scanning electron image of a portion of the metalens.
(
c
) Schematic of the measurement setup used for measuring the focal plane intensity distribution of the
metalens. (
d
) The measured focal plane intensity distribution of the metalens (left) and measured focal plane
intensity along the dashed black line shown in and the corresponding focal plane intensity data for an ideal
metalens. (
e
) Measured on-axis modulation transfer function of the grating averaging and the ideal metalenses.
(
f
) Schematic of the setup used for measuring the optical power focused by the metalens, and (
g
) for measuring
the incident optical power. PC: polarization controller, FC: fiber collimator, PD: photodetector.