of 7
1
Supplementary Information for
2
Metasurface generated complex three dimensional optical fields for interference lithography
3
Seyedeh Mahsa Kamali, Ehsan Arbabi, Hyounghan Kwon, and Andrei Faraon
4
Andrei Faraon
5
E-mail: faraon@caltech.edu
6
This PDF file includes:
7
Supplementary text
8
Figs. S1 to S6
9
Seyedeh Mahsa Kamali, Ehsan Arbabi, Hyounghan Kwon, and Andrei Faraon
1 of
7
www.pnas.org/cgi/doi/10.1073/pnas.1908382116
Supporting Information Text
10
S1. Fully periodic 3D structures.
According to the grating equation, to have a fully periodic structure, the gyroid and diamond
11
patterns should have square in-plane lattices with lattice constants of 479 nm and 500 nm, which result in in-depth periods of
12
1184 nm and 500 nm, respectively. The rotated cubic pattern has a triangular in-plane lattice with a lattice constant of 408 nm,
13
resulting in an in-depth period of 500 nm. Supplementary Fig. 1 demonstrates in-plane and in-depth periods of interference
14
patterns of different diffraction orders for square and triangular lattices.
15
Triangular lattice
In-depth period (nm)
0
1000
0
408
500
3000
In-plane period (nm)
0
th
order
1
st
order
2
nd
order
3
rd
order
Rotated cubic lattice
0
th
order
1
st
order
2
nd
order
3
rd
order
4
th
order
0
900
0
3000
500
In-plane period (nm)
500
1184
Diamond lattice
Gyroid lattice
Rectangular lattice
In-depth period (nm)
A
B
Fig. S1. Square and triangular lattice diffraction orders and their corresponding in-plane and in-depth periods.
(A) Different excited diffraction orders and the
corresponding in-depth periods in a triangular lattice as a function of in-plane periods. The designed rotated cubic lattice is shown with black star (B) Same as (A), but for
rectangular lattice. The designed gyroid and diamond lattices are shown with black hexagon and black star, respectively. Here the lattices are two dimensional and the number
of diffraction orders are only referring to the order of their excitations.
S2. The level set representation of 3D structures.
The gyroid (
F
G
), rotated cubic (
F
C
), and diamond (
F
D
) lattices are defined
16
with the level set approximations as follows:
17
F
G
(
x,y,z
) = sin 2
π
(
x
P
x
) cos 2
π
(
y
P
y
) + sin 2
π
(
y
P
y
) cos 2
π
(
z
P
z
)
+ sin 2
π
(
z
P
z
) cos 2
π
(
x
P
x
)
t
[1]
18
where,
P
x
= 479
nm,
P
y
= 479
nm,
P
z
= 1184
nm,
19
F
C
(
x,y,z
) = cos 2
π
(
x
P
x
) + cos 2
π
(
y
P
y
) + cos 2
π
(
z
P
z
)
t
[2]
20
where,
P
x
=
P
y
=
P
z
= 408
nm, and
21
x
=
0
.
71
x
0
.
41
y
0
.
58
z,
y
= 0
.
71
x
0
.
41
y
0
.
58
z,
z
= 0
.
81
y
0
.
58
z,
[3]
22
23
F
D
(
x,y,z
) = sin 2
π
(
x
P
x
+
y
P
y
+
z
P
z
) + sin 2
π
(
x
P
x
y
P
y
+
z
P
z
)
+ sin 2
π
(
x
P
x
+
y
P
y
z
P
z
) + sin 2
π
(
x
P
x
+
y
P
y
+
z
P
z
)
t
[4]
24
where,
P
x
=
P
y
=
P
z
= 500
nm.
25
2 of 7
Seyedeh Mahsa Kamali, Ehsan Arbabi, Hyounghan Kwon, and Andrei Faraon
408 nm
√3×
408 nm
500 nm
500 nm
500 nm
500 nm
479 nm
1184 nm
479 nm
b
1
b
2
b
1
b
2
b
1
b
2
b
1
b
2
x-polarization
diraction orders
y-polarization
diraction orders
Amplitude
rad/(2
π
)
1
0
Amplitude (a. u.)
1
0
Amplitude
Phase
Phase
b
1
b
2
b
1
b
2
b
1
b
2
b
1
b
2
x-polarization
diraction orders
y-polarization
diraction orders
Amplitude
rad/(2
π
)
1
0
Amplitude (a. u.)
1
0
Amplitude
Phase
Phase
b
1
b
2
b
1
b
2
b
1
b
2
b
1
b
2
x-polarization
diraction orders
y-polarization
diraction orders
Amplitude
rad/(2
π
)
1
0
Amplitude (a. u.)
1
0
Amplitude
Phase
Phase
Gyriod pattern
Rotated cubic pattern
Diamond pattern
A
B
C
Fig. S2.
Target 3D structures and their corresponding optimized diffraction order coefficients. (A) Target gyroid pattern defined with the level set approximation with
t
=
0.25
(top). The optimized diffraction order coefficients for the x and y polarizations for gyroid structure with square lattice (bottom). (B) Same as (A), but for rotated cubic pattern with
triangular lattice and
t
=
0.15. (C) Same as (A), but for diamond pattern with square lattice and
t
=
0.3.
Seyedeh Mahsa Kamali, Ehsan Arbabi, Hyounghan Kwon, and Andrei Faraon
3 of 7
|
t
x
|
2
|
t
y
|
2
60
180
120
120
180
D
x
(nm)
D
y
(nm)
60
60
180
120
120
180
D
x
(nm)
D
y
(nm)
60
Transmission (%)
0
100
x
z
|
t
x
|E
x
e
i
t
x
E
x
E
y
|
t
y
|E
y
e
i
t
y
t
y
500 nm
t
x
rad/(2
π
)
1
0
Sampling points
A
B
Fig. S3. Diamond meta-mask sampling points and transmission powers of uniform nanoposts.
(A) The desired diamond phase-masks are sampled at four points with
with a 250-nm period. Position of the sampling points are shown with black squares. (B) Transmission powers of the x- and y-polarized light at 532 nm for the uniform array of
nanoposts as functions of the nano-post widths.
4 of 7
Seyedeh Mahsa Kamali, Ehsan Arbabi, Hyounghan Kwon, and Andrei Faraon
t
x
500 nm
|t
y
|
t
y
|t
x
|
500 nm
500 nm
500 nm
rad/(2
π
)
1
0
Transmission
1
0
Initial design
Optimized design
t
x
500 nm
|t
y
|
t
y
|t
x
|
500 nm
500 nm
500 nm
rad/(2
π
)
1
0
Transmission
1
0
A
B
Fig. S4.
Transmission amplitudes and phases of the initial and optimized diamond meta-masks. (A) x- and y-polarized transmission phases (top row) and amplitudes (bottom
row) of the initial design. The input polarization is chosen to be linearly polarized with
|
E
x
|
/
|
E
y
|
= 0
.
84
. (B) Same as (A), but for the optimized design. The input polarization
is chosen to be linearly polarized with
|
E
x
|
/
|
E
y
|
= 0
.
5
.
Seyedeh Mahsa Kamali, Ehsan Arbabi, Hyounghan Kwon, and Andrei Faraon
5 of 7
-6
0
6
-6
6
0
1
0
Intensity (a. u.)
x (μm)
y (μm)
Fig. S5.
Larger area of measured in-plane intensity profile of the diamond meta-mask.
6 of 7
Seyedeh Mahsa Kamali, Ehsan Arbabi, Hyounghan Kwon, and Andrei Faraon