of 9
Dielectric Metasurfaces for Complete Control of Phase and
Polarization with Subwavelength Spatial Resolution and High
Transmission
Amir Arbabi, Yu Horie, Mahmood Bagheri, and Andrei Faraon
1
Dielectric metasurfaces for complete control
of phase and polarization with subwavelength
spatial resolution and high transmission
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DOI: 10.1038/NNANO.2015.186
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S1. ARBITRARY POLARIZATION AND PHASE TRANSFORMATION USING SYMMETRIC
AND UNITARY JONES MATRICES
Here we show that any arbitrary polarization and phase transformation can always be per-
formed using a unitary and symmetric Jones matrix. We prove this by determining the unitary
and symmetric Jones matrix
T
that maps a given input electric field
E
in
to a desired output elec-
tric field
E
out
. For polarization and phase transformations (i.e. no amplitude modification), the
Jones matrix should be unitary since the transmitted power is equal to the incident power and
|
E
out
|
=
|
E
in
|
. The general relation between the electric fields of input and output optical waves
for normal incidence is expressed as
E
out
=
TE
in
. For a symmetric and unitary Jones matrix we
have
T
xx
E
in
x
+
T
yx
E
in
y
=
E
out
x
,
(1a)
T
yx
E
in
x
T
yx
T
yx
T
xx
E
in
y
=
E
out
y
,
(1b)
where
E
in
x
and
E
in
y
are the
x
and
y
components of the electric field of the input light,
E
out
x
and
E
out
y
are the
x
and
y
components of the electric field of the output light,
T
ij
(
i,j
=
x,y
) are the elements
of the 2
×
2 Jones matrix, and * represents complex conjugation. In deriving Eqs. 1a and 1b, we
have used the symmetric properties
T
xy
=
T
yx
, and the unitary condition
T
xx
T
xy
+
T
yx
T
yy
=0
.
By multiplying Eq. 1a by
T
xx
and Eq. 1b by
T
yx
we obtain
|
T
xx
|
2
E
in
x
+
T
yx
T
xx
E
in
y
=
T
xx
E
out
x
,
(2a)
|
T
yx
|
2
E
in
x
T
yx
T
xx
E
in
y
=
T
yx
E
out
y
.
(2b)
By adding Eqs. 2a and 2b, using the unitary condition
|
T
xx
|
2
+
|
T
yx
|
2
=1
, and taking the complex
conjugate of the resultant relation, we find
T
xx
E
out
x
+
T
yx
E
out
y
=
E
in
x
.
(3)
Finally, by expressing Eqs. 3 and 1a in the matrix form, we obtain
E
out
x
E
out
y
E
in
x
E
in
y
T
xx
T
yx
=
E
in
x
E
out
x
.
(4)
2
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Therefore, for any given
E
in
and
E
out
, we can find
T
xx
and
T
yx
from Eq. 4, and
T
xy
and
T
yy
from
the symmetry and unitary conditions as
T
xy
=
T
yx
,
(5a)
T
yy
=
exp(2
i
T
yx
)
T
xx
.
(5b)
Thus, we can always find a unitary and symmetric Jones matrix that transforms any input optical
wave
E
in
to any output optical wave
E
out
.
S2. REALIZATION OF ANY SYMMETRIC AND UNITARY JONES MATRICES USING A UNI-
FORM BIREFRINGENT METASURFACE
Here we show that any symmetric and unitary Jones matrix can be realized using a uniform
birefringent metasurface shown in Fig. 2a in the main text, if
φ
x
,
φ
y
, and the angle between one
of the principal axis of the metasurface and the
x
axis (
θ
) could be chosen freely. Any symmetric
and unitary matrix is decomposable in terms of its eigenvectors and eigenvalue matrix (
) as
T
=
V
e
x
0
0e
y
V
T
=
R
(
θ
)
∆R
(
θ
)
,
(6)
where superscript
T
represents the matrix transpose operation.
V
is a real unitary matrix; there-
fore, it corresponds to an in-plane geometrical rotation
R
by an angle that we refer to as
θ
, and
since
V
T
=
V
1
,
V
T
represents a rotation by
θ
. According to Eq. 6, the operation of a meta-
surface that realizes the Jones matrix
T
can be considered as rotating the electric field of the input
wave (
E
in
) by
θ
, phase shifting the
x
and
y
components of the rotated
E
in
respectively by
φ
x
and
φ
y
, and rotating back the rotated and phase shifted vector by angle
θ
. Equivalently,
T
can be
implemented using a metasurface that imposes phase shifts
φ
x
and
φ
y
to the components of
E
in
along angles
θ
and
90
+
θ
, respectively. Such a metasurface is realized by starting with a metasur-
face whose principal axis are along
x
and
y
directions and imparts
φ
x
and
φ
y
phase shifts to
x
and
y
-polarized waves, and rotating it anticlockwise by angle
θ
. Therefore, any symmetric and unitary
Jones matrix can be realized using a metasurface if its
φ
x
,
φ
y
, and in-plane rotation angle (
θ
) could
be chosen freely.
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S3. INDEPENDENT WAVEFRONT CONTROL FOR TWO ORTHOGONAL POLARIZATIONS
In this section, we derive the necessary condition for the design of a device that imposes two
independent phase profiles to two optical waves with orthogonal polarizations. The four elements
of the Jones matrix
T
are found uniquely using Eqs. 4 and 5, when the determinant of the matrix
on the left hand side of Eq. 4 is nonzero. Therefore, a devices that is designed to map
E
in
to
E
out
,
converts an optical wave whose polarization is orthogonal to
E
in
to an optical wave polarized
orthogonal to
E
out
. For example, an optical element designed to generate radially polarized light
from
x
polarized input light, will also generate azimuthally polarized light from
y
polarized input
light.
In the special case that the determinant of the matrix on the left had side of Eq. 4 is zero we
have
E
out
x
E
in
y
E
out
y
E
in
x
=0
,
(7)
and because
T
is unitary we have
|
E
in
|
=
|
E
out
|
; therefore we find
E
out
=exp(
)
E
in
where
φ
is an arbitrary phase. This special case corresponds to a device that preserves the polarization
ellipse of the input light, switches its handedness (helicity), and imposes a phase shift on it. In
this case, the
T
matrix is not uniquely determined from Eq. 4, and an additional condition, such
as the phase profile for the orthogonal polarization, can be imposed on the operation of the device.
Therefore, the device can be designed to realize two different phase profiles for two orthogonal
input polarizations.
SUPPLEMENTARY VIDEO LEGENDS
Supplementary Video 1
|
Polarization switchable phase hologram
. Movie showing the evo-
lution of the image generated by a polarization switchable hologram as the polarization direction
(shown by an arrow on the bottom left) of the illumination light is changed.
4
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SUPPLEMENTARY FIGURES
-10
0
10
dB
1
μ
m
1
μ
m
Glass substrate
z
x
y
y
z
x
x
z
y
a-Si post
k
E
i
Scatterd light
Plane wave incident
Supplementary Fig.
1.
Large forward scattering by a single amorphous silicon post
. Schematic illus-
tration and finite element simulation results of light scattering by a single 715 nm tall circular amorphous
silicon post with a diameter of 150 nm. The simulation results show the logarithmic scale energy density of
the light scattered by the single amorphous silicon post over the
xz
and
yz
planes. The energy densities are
normalized to the energy density of the 915 nm
x
-polarized incident plane wave.
5
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φ
x
/(2
π
)
φ
y
/(2
π
)
|t
x
|
2
|t
y
|
2
0.25
0.5
0.75
1
0
D
x
(nm)
D
y
(nm)
100
200
300
400
100
200
300
400
D
x
(nm)
D
y
(nm)
100
200
300
400
100
200
300
400
D
x
(nm)
D
y
(nm)
100
200
300
400
100
200
300
400
D
x
(nm)
D
y
(nm)
100
200
300
400
100
200
300
400
Supplementary Fig.
2.
Phase shifts and intensity transmission coefficients as a function of elliptical
post diameters, used to derive data in Fig. 2b-e of the main text
. Intensity transmission coefficients
(
|
t
x
|
2
and
|
t
y
|
2
) and the phase of transmission coefficients (
φ
x
and
φ
y
) of
x
and
y
-polarized optical waves
for the periodic array of elliptical posts shown in Fig. 2a of the main text as functions of the post diameters.
6
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Diffraction limited spot (NA=0.6)
0
Intensity (a.u.)
1
2
μ
m
2
μ
m
0
Intensity (a.u.)
1
Measured focal spot
x (
μ
m)
-2
-1
012
Intensity (a.u.)
0
0.5
1
x (
μ
m)
-2
-1
012
Intensity (a.u.)
0
0.5
1
x (
μ
m)
-2
-1
012
Intensity (a.u.)
0
0.2
0.4
0.6
0.8
1
Measured intensity
Airy pattern
(NA=0.58)
a
c
b
Supplementary Fig.
3.
Diffraction limited focusing by device shown in Fig. 5c
.
a
, Theoretical diffraction
limited focal spot (Airy disk) for a lens with numerical aperture (NA) of 0.6 at the operation wavelength
of 915 nm. Inset shows the intensity along the dashed line.
b
, Measured focal spot for the device shown in
Fig. 5c when the device is uniformly illuminated with right handed circularly polarized 915 nm light. Inset
shows the intensity along the dashed line.
c
, Measured intensity along the dashed line shown in (
b
) and its
least squares Airy pattern fit which has an NA of 0.58.
7
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850
900
950
1000
0
0.5
1
1.5
Wavelength (nm)
850
900
950
1000
0
0.5
1
Wavelength (nm)
850
900
950
1000
0
0.5
1
1.5
Wavelength (nm)
850
900
950
1000
0
0.5
1
Wavelength (nm)
850
900
950
1000
0
0.5
1
1.5
Wavelength (nm)
850
900
950
1000
0
0.5
1
Wavelength (nm)
850
900
950
1000
0
0.5
1
1.5
Wavelength (nm)
850
900
950
1000
0
0.5
1
Wavelength (nm)
D
x
=100 nm
D
y
=200 nm
D
x
=180 nm
D
y
=200 nm
D
x
=185 nm
D
y
=230 nm
D
x
=150 nm
D
y
=300 nm
65
150
250
350
450
D
y
D
x
|t
x
|
2
|t
y
|
2
φ
x
/(2
π
)
φ
y
/(2
π
)
Diameter (nm)
φ
x
/(2
π
)
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
φ
y
/(2
π
)
φ
x
/(2
π
)
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
φ
y
/(2
π
)
Supplementary Fig.
4.
Transmission spectra of periodic arrays of elliptical posts showing that the op-
eration wavelength does not overlap with resonances
. Wavelength dependence of the intensity transmis-
sion coefficients (
|
t
x
|
2
and
|
t
y
|
2
) and the phase of transmission coefficients (
φ
x
and
φ
y
) of
x
and
y
-polarized
optical waves for the periodic arrays schematically shown in Fig. 2a of the main text. The spectra are shown
for a few arrays with different (
D
x
,
D
y
) combinations: (100 nm, 200 nm), (180 nm, 200 nm), (150 nm, 300
nm), (185 nm, 230 nm). The corresponding phase shift values and post diameters for these arrays are shown
on the
D
x
and
D
y
graphs on the right with black symbols. For brevity, only the spectra for the arrays with
D
y
>D
x
are shown. The transmission and phase spectra for the arrays with
D
x
>D
y
(which are shown
with red symbols on the
D
x
and
D
y
graphs) can be obtained by swapping
x
and
y
in the spectra graphs.
The desired operation wavelength (
λ
=
915 nm) is shown with dashed red vertical lines in the spectra plots,
and it does not overlap with any of the resonances of the periodic arrays.
8
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Objective lens
Tube lens
Camera
Polarizer
Device
lens 1
Fiber
collimator
Pinhole
Laser
Polarization
controller
Device
lens 1
Optical power meter
a
b
Fiber
collimator
Laser
Polarization
controller
Supplementary Fig.
5.
Measurement setup
.
a
, Schematic illustration of the measurement setup used for
characterization of devices modifying polarization and phase of light. The linear polarizer was inserted into
the setup only during the polarization measurements.
b
, Schematic drawing of the experimental setup used
for efficiency characterization of the device shown in Fig. 4b.
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