of 3
Supplementary information for
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
1
SUPPLEMENTARY NOTE
Here we show that the deflection coefficient of a beam deflector can be estimated using (1).
Consider the aperiodic beam deflector shown in Fig. 1e. Assume that the beam deflector is com-
posed of a large number of extended cells and is illuminated with a normally incident plane wave
with a power amplitude of 1. Depending on the polarization of the incident light, the deflected light
is either TE or TM polarized. The power amplitude of the deflected light
A
along the deflection
angle
θ
is related to the Fourier component of the electric or magnetic fields of the transmitted light
on the output aperture of the device (dashed red line in Fig. 1e) at the spatial angular frequency of
k
0
sin(
θ
)
, and can be found as [1]
A
=
C
L
L
0
F
(
x
)
e
jk
0
sin(
θ
)
x
d
x,
(1)
where
L
is the length of the beam deflector. For TE-polarized incident light
F
=
E
y
and
C
=
cos(
θ
)
2
Z
is a constant that relates the electric field amplitude to the power amplitude, and
Z
is the
wave impedance in the
z >
0
region. For TM-polarized light
F
=
H
y
and
C
=
2
Z
cos(
θ
)
. The
deflection efficiency is the square of the modulus of the power amplitude and is given by
η
=
|
A
|
2
.
Assuming that the beam deflector is composed of
M
extended cells and each extended cell has a
width of
Λ
, we can rewrite (3) as a sum of integrals over extended cells as
A
=
C
L
M
1
p
=0
(
p
+1)Λ
p
Λ
F
(
x
)
e
jk
0
sin(
θ
)
x
d
x.
(2)
Assuming the beam deflector varies slowly from one extended cell to the next, fields over the
p
th
extended cell can be approximated by the fields of a grating created by periodically repeating the
same extended cell. We denote the grating’s transmitted field over its output aperture by
F
g
p
(
x
)
.
Thus,
A
C
L
M
1
p
=0
(
p
+1)Λ
p
Λ
F
g
p
(
x
)
e
jk
0
sin(
θ
)
x
d
x
=
C
L
M
1
p
=0
I
p
,
(3)
where
I
p
=
(
p
+1)Λ
p
Λ
F
g
p
(
x
)
e
jk
0
sin(
θ
)
x
d
x
=
(
p
+1)Λ
p
Λ
F
g
p
(
x
)
e
jk
0
sin(
θ
g
)
x
e
jk
0
(sin(
θ
)
sin(
θ
g
))
x
d
x
(4)
Note that the grating field and
e
jk
0
sin(
θ
g
)
x
are periodic with a period of
Λ
, that is
F
g
p
(
x
+ Λ)
e
jk
0
sin(
θ
g
)(
x
+Λ)
=
F
g
p
(
x
)
e
jk
0
sin(
θ
g
)
x
.
(5)
2
Therefore, we can simplify (6) as
I
p
=
e
jp
φ
g
Λ
0
F
g
p
(
x
)
e
jk
0
sin(
θ
g
)
x
e
j
φ
g
x
Λ
d
x
e
jp
φ
g
Λ
0
F
g
p
(
x
)
e
jk
0
sin(
θ
g
)
x
d
x,
(6)
where we have defined
φ
g
=
k
0
(sin(
θ
)
sin(
θ
g
))Λ
and used the approximation
e
j
φ
g
x
Λ
1
for
φ
g

1
.
φ
g
represents the phase shift from one extended cell to the next.
I
p
can be expressed
in terms of the grating diffraction coefficients as
I
p
e
jp
φ
g
Λ
0
F
g
p
(
x
)
e
jk
0
sin(
θ
g
)
x
d
x
=
Λ
C
e
jp
φ
g
t
n
(
p
φ
g
)
,
(7)
where
t
n
(
p
φ
g
) =
C
Λ
Λ
0
F
g
p
(
x
)
e
jk
0
sin(
θ
g
)
x
d
x,
(8)
represents the diffraction coefficient of the
n
th
diffraction order of an
n
th
-order blazed grating that
is designed with the phase of
p
φ
g
. Plugging
I
p
from (9) into (5) we obtain
A
Λ
D
M
1
p
=0
t
n
(
p
φ
g
)
e
jp
φ
g
=
1
M
M
1
p
=0
t
n
(
p
φ
g
)
e
jp
φ
g
,
(9)
which is the average of
t
n
(
φ
g
)
e
g
over different extended cells. Because
φ
g

1
,
M

1
, and
t
n
(
φ
g
)
e
g
is periodic with a period of
2
π
, its average can also be computed as an integral over its
period as
A
1
2
π
2
π
0
t
n
(
φ
g
)
e
g
d
φ
g
,
(10)
which is the result presented in (1).
arbabi@umass.edu
faraon@caltech.edu
[1] R. F. Harrington,
Time-Harmonic Electromagnetic Fields
(Wiley, 2001).
3