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
and Andrei Faraon
2,
†
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
Abstract
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 de-
signed 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 partic-
ular 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. Un-
like 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 averag-
ing is a versatile technique applicable to many types of gradient metasurfaces, thus enabling highly efficient
metasurface components and systems.
∗
arbabi@umass.edu
†
faraon@caltech.edu
1
arXiv:2004.06182v1 [physics.optics] 13 Apr 2020
INTRODUCTION
Flat optical devices based on dielectric metasurfaces have recently attracted significant atten-
tion due to their small size, low weight, and the potential for their low-cost manufacturing using
semiconductor fabrication techniques [1–8]. The planar form factor of metasurfaces and the high
multilayer overlay accuracy of the semiconductor manufacturing process enable the implementa-
tion 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 optimization techniques can be efficient [16, 18–20]; however, their di-
mensions are inevitably limited by the available computational resources because the optimization
process requires full-wave simulation of the entire device in each iteration, or simulation and op-
timization 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 discontinuities 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 con-
ventional approach of designing metasurfaces involves selection of a set of parameterized meta-
atoms and finding a 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 performance of high-NA metasur-
faces designed using the same design maps cannot be easily evaluated. Here we introduce a novel
2
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 sys-
tems [9, 10]. We first discuss the conventional technique for designing transmissive gradient meta-
surfaces 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 transmission 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 approx-
imated by the transmission coefficient of a periodic array created by periodically 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 conven-
tional 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(
−
jφ
)
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 meta-
surfaces. Refractive indices of 3.65 and 1.45 are assumed for the amorphous silicon and the fused
silica substrate, respectively. The inverted relation that maps the desired transmission coefficient
exp(
−
jφ
)
, 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
3
0
Phase (rad)
2
0
100
200
300
W (nm)
0
Phase (rad)
2
0.9
0.95
1
Transmi�ance
W (nm)
100
150
200
250
0
0.5
1
Transmi�ance
Phase/(2
)
(a)
500 nm
W
400 nm
(b)
0
0.5
1
1.5
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.
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 metasur-
4
faces 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 metasurface. 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 metasur-
face. For the validity of the second approximation, the radiation pattern of the meta-atoms should
not change significantly over the angular range of interest for the incident and transmitted light. As
a result, the efficiency of metasurfaces 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 require-
ment for the first approximation. They also deflect incident light by small angles, thus the condi-
tion for the second approximation is also satisfied when the incident angle is small. As a result,
low-NA metasurfaces designed using the conventional 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 de-
sign 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 differ-
ent 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 ap-
proach become less accurate and the metasurface efficiency decreases. Qualitatively, the per-
formance of a metasurface platform in implementing high-NA devices designed using the con-
ventional 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.
5
RESULTS
Grating averaging technique
Beam deflectors are basic elements in designing gradient metasurfaces because such metasur-
faces 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
evaluate the performance of the platform in realizing general metasurface components. Metasur-
face beam deflectors implement a linear phase ramp (i.e.,
t
= exp(
−
jφ
(
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 wave-
length 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
π
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 peri-
odic 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 ape-
riodic and we refer to it as an aperiodic beam deflector. One might consider approximating 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 incident 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
6
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 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
A
≈
1
2
π
∫
2
π
0
t
n
(
φ
g
)
e
jφ
g
d
φ
g
,
(1)
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 electric 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
7
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
g
i
=
2
π
N
,
4
π
N
,...,
2
π
, finding their diffraction coefficients
t
n
(
φ
g
i
)
and approximating the integral in (1) by
1
N
∑
N
i
=1
t
n
(
φ
g
i
)
e
jφ
g
i
. 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(
−
jφ
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 deflection 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, grating 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
η
0
=
1
4
π
2
∣
∣
∣
∣
∫
2
π
0
t
(
φ
)
e
jφ
d
φ
∣
∣
∣
∣
2
,
(2)
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 metasurface 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 technique
The efficiency values obtained using the grating averaging technique can be used to evaluate
and compare 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
8
60
100
140
180
0
0.5
1
Transmi�ance
Phase/(2
)
W (nm)
40
120
200
W (nm)
0.85
0.9
0.95
1
Transmi�ance
0
Phase (rad)
2
0
Phase (rad)
2
(a)
(b)
First design (Fig. 1)
Gra�ng avg. design
0
20
40
60
Deflec�on angle (degree)
50
60
70
80
90
100
Efficiency (%)
(d)
590 nm
350 nm
W
(c)
0
0.3
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 trans-
mission 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 trans-
verse 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).
the design presented in Figs. 1a and 1b. 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 second 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
9
1
0
Intensity (a.u.)
(e)
Control
Gra�ng averaging
(d)
0
-20
-40
-60
Electric energy density (dB)
0
-1
1
E
x
(a.u.)
Control
(c)
x
z
x
y
y
z
10
μ
m
0
-20
-40
-60
Electric energy density (dB)
0
-1
1
E
x
(a.u.)
Gra�ng avera�ng
(b)
x
z
x
y
y
z
10
μ
m
(a)
50
μ
m
20
μ
m
PEC
x
y
z
E
x
0
500
1000
1500
Spa�al frequency (1/mm)
0
0.5
1
Contrast
Control, along x
Control, along y
Gra�ng avg, along x
Gra�ng avg, along y
1
μ
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.
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 indicate 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.
1a 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
10