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Supporting Information: “Highly tunable elastic dielectric
metasurface lenses”
Seyedeh Mahsa Kamali, Ehsan Arbabi, Amir Arbabi, Yu Horie, and Andrei Faraon
1
SUPPLEMENTARY NOTE 1: SAMPLING FREQUENCY OF THE PHASE PROFILE
The lattice constant should be chosen such that the lattice remains non-diffractive and satisfies
the Nyquist sampling criterion. From a signal processing point of view, the locally varying trans-
mission coefficient of a flat microlens can be considered as a spatially band-limited signal with a
2NA
k
0
bandwidth (ignoring the effect of the edges), where
NA
is the microlens numerical aper-
ture, and
k
0
is the vacuum wavenumber. A hypothetical one dimensional band-limited spectrum is
depicted in Fig. S1 (solid blue curve). By sampling the microlens phase profile with sampling fre-
quency of
K
s
, the images (dashed blue curves in Fig. S1) are added to the spectrum. Therefore, for
the perfect reconstruction of the microlens’ transmission coefficient, the Nyquist criterion should
be satisfied:
K
s
>
2NA
k
0
. On the other hand, the lattice should remain subwavelength; the higher
order diffractions (dashed blue curves in Fig. S1) should remain non-propagating. Propagation in
free space can be considered as a low pass filter with
2
nk
0
bandwidth (solid red curve in Fig. S1),
where
n
is the refractive index of the surrounding medium. Therefore, in order to have perfect re-
construction of phase and non-propagating higher order diffractions, the following relation should
be satisfied:
K
s
> nk
0
+ NA
k
0
(1)
Note that the sampling frequency (
K
s
) is a reciprocal lattice vector. For the square lattice
K
s
= 2
π/
Λ
, where
Λ
is the lattice constant. Therefore Eq. (1) would be simplified as follows:
Λ
<
λ
n
+ NA
(2)
Where
λ
is the free space wavelength. Note that the maximum value of numerical aperture is
NA
max
=
n
, which simplifies Eq. (2) to
Λ
< λ/
(2
n
)
. For designing tunable microlenses, Eq.
(2) should be satisfied for all the strains of interest, and
Λ = (1 +

)
a
, where
a
is the unstrained
lattice constant. For the parameters used in the main text, the unstrained lattice constant should be
smaller than 401 nm in order to have tunable microlens up to 50
%
strains. The unstrained lattice
constant was chosen to be 380 nm.
2
SUPPLEMENTARY FIGURES
K
NAk
0
K
s
K
s
-NAk
0
nk
0
-nk
0
-NAk
0
-K
s
Figure S1. Sampling frequency of the phase profile for perfect reconstruction of the wavefront. The locally
varying transmission coefficient spectrum of a flat microlens can be considered as a band-limited signal with
2NA
k
0
bandwidth (solid blue curve). By sampling the transmission coefficient with sampling frequency of
K
s
, displaced copies of the band-limited signal are added to the spectrum (dashed blue curves). In order to
avoid undesirable diffractions, the free space low pass filter (solid red curve) should only filter the zeroth
order diffraction (solid blue curve).
3
0
0.5
1
Intensity (a.u.)
5
μ
m
є
(0%)
Y
X
Z (
P
m)
Y (
P
m)
1600
1400
1200
1000
800
600
400
-20
20
0
є
(0%)
100
200
0
1
Post width (nm)
|t|

‘
t
 S
5
μ
m
є
(10%)
Z (
P
m)
Y (
P
m)
Y
X
1600
1400
1200
1000
800
600
400
-20
20
0
є
(10%)
0
1
Post width (nm)
|t|

‘
t
 S
5
μ
m
є
(20%)
Z (
P
m)
Y (
P
m)
Y
X
1600
1400
1200
1000
800
600
400
-20
20
0
є
(20%)
0
1
Post width (nm)
|t|

‘
t
 S
5
μ
m
є
(30%)
Z (
P
m)
Y (
P
m)
Y
X
1600
1400
1200
1000
800
600
400
-20
20
0
є
(30%)
0
1
Post width (nm)
|t|

‘
t
 S
100
200
5
μ
m
є
(40%)
Z (
P
m)
Y (
P
m)
Y
X
1600
1400
1200
1000
800
600
400
-20
20
0
є
(40%)
0
1
Post width (nm)
|t|

‘
t
 S
100
200
5
μ
m
є
(50%)
Z (
P
m)
Y (
P
m)
Y
X
1600
1400
1200
1000
800
600
400
-20
20
0
є
(50%)
0
1
Post width (nm)
|t|

‘
t
 S
100
200
100
200
100
200
Figure S2. Simulation results of the tunable microlens using the actual nano-posts’ transmission coef-
ficients, extracted from Figs. 2b and 2c in the main text. Intensity profiles of the tunable microlens are
simulated at different strains (

= 0
%
to 50
%
) using the actual transmission coefficients at each strain value.
Intensity and phase of the transmission coefficient at respective strain values are shown in the left, and their
corresponding intensity profiles in the axial plane and in the focal plane are shown in the middle and right,
respectively.
4