University of Southern Denmark
Waveguide metacouplers for in-plane polarimetry
Pors, Anders Lambertus; Bozhevolnyi, Sergey I.
Published in:
Physical Review Applied
DOI:
10.1103/PhysRevApplied.5.064015
Publication date:
2016
Document version:
Final published version
Document license:
CC BY-NC
Citation for pulished version (APA):
Pors, A. L., & Bozhevolnyi, S. I. (2016). Waveguide metacouplers for in-plane polarimetry.
Physical Review
Applied
,
5
(6), 064015-1-064015-9. [064015]. https://doi.org/10.1103/PhysRevApplied.5.064015
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Waveguide Metacouplers for In-Plane Polarimetry
Anders Pors
*
and Sergey I. Bozhevolnyi
SDU Nano Optics, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark
(Received 8 March 2016; revised manuscript received 24 May 2016; published 27 June 2016)
The state of polarization (SOP) is an inherent property of the vectorial nature of light and a crucial
parameter in a wide range of remote sensing applications. Nevertheless, the SOP is rather cumbersome to
probe experimentally, as conventional detectors resp ond only to the intensity of the light, hence losing the
phase information between orthogonal vector components. In this work, we propose a type of polarimeter
that is compact and well suited for in-plane optical circuitry while allowing for immediate determination of
the SOP through simultaneous retrieval of the associated Stokes parameters. The polarimeter is based on
plasmonic phase-gradient birefringent metasurfaces that facilitate normal incident light to launch in-plane
photonic-waveguide modes propagating in six predefined directions with the coupling efficiencies
providing a direct measure of the incident SOP. The functionality and accuracy of the polarimeter, which
essentially is an all-polarization-sensitive waveguide metacoupler, is confirmed through full-wave
simulations at the operation wavelength of 1.55 μm.
DOI: 10.1103/PhysRevApplied.5.064015
I. INTRODUCTION
Despite the fact that the state of polarization (SOP) is the
key characteristic of the vectorial nature of electromagnetic
waves, it is an inherently difficult parameter to experi-
mentally probe owing to the loss of information on the
relative phase between orthogonal vector components in
conventional (intensity) detection schemes. Nevertheless,
the SOP (or the change in SOP) is a parameter often sought
to be measured, since it may carry crucial information
about the composition and structure of the medium that the
wave has been interacting with. As prominent applications,
we mention remote sensing within astronomy [1], biology
[2], and camouflage technology [3] but also more nascent
applications, such as for the fundamental understanding of
processes in laser fusion or within the field of quantum
communication, advantageously exploit the knowledge of
the SOP [4]. Overall, it transpires that polarimetry is of
utmost importance in both fundamental and applied
science.
The SOP evaluation is typically based on the determi-
nation of the so-called Stokes parameters, which are
constructed from six intensity measurements with prop-
erly arranged polarizers placed in front of the detector,
thereb y allowing one to uniquely retr ieve th e SOP [5].
We note that the series of measurements can be automated
(as in commercial polarimeters), though at the expense of
a n on-negligible acquisition time that may induce errors
or limit the possibility to measure transient events.
Alternatively, the Stokes parameters can be measured
simultaneously by splitting the beam into multiple optical
paths and utilizing several polarizers and detectors [6 ].
The downside of this approach, however, amounts to a
bulky, complex, and expensive optical system. Also, we
note that additional realizations of polarimeters do exist,
like using advanced micropolarizers in front of an imaging
detector, but thos e approaches are typically complex
and expensive [3 ]. Overall, it seems that none of the
conventional approaches is ideal with respect to the
simultaneous determination of the Stok es parameters,
compact and inexpensive design, and ease of usage
(e.g., no tedious alignment, etc.).
With the above outline of the current status of polar-
imeters, it is natural to discuss the recent advances within
nanophotonics, particularly the new degrees of freedom
in controlling light using metasurfaces [7,8]. Here, early
approaches in determining the SOP utilize a combination of
a metasurface together with conventional optical elements
[9,10] (like polarizers and wave plates) or the effect of a
polarization-dependent transmission of light through six
carefully designed nanoapertures in metal films [11].
Likewise, different types of metasurfaces are proposed
for the determination of certain aspects of the SOP, like
the degree of linear [12] or circular [13,14] polarization.
Recently, however, metasurface-only polarimeters that
uniquely identify the SOP have been proposed and verified.
For example, a nanometer-thin metadevice consisting of
an array of meticulously designed (rotated and aligned)
metallic nanoantennas features an in-plane scattering pat-
tern that is unique for all SOPs [15]. In a different study, we
propose a reflective metagrating that redirects light into six
diffraction orders, with the pairwise contrast in diffraction
intensities immediately revealing the Stokes parameters of
the incident SOP [16]. Our polarimeter is based on the
optical analog of the reflectarray concept [17], hence
consisting of an optically thick metal film overlaid by a
*
alp@iti.sdu.dk
PHYSICAL REVIEW APPLIED 5, 064015 (2016)
2331-7019=16=5(6)=064015(9) 064015-1 © 2016 American Physical Society
nanometer-thin dielectric spacer and an array of carefully
designed metallic nanobricks. These metasurfaces, also
known as gap-surface-plasmon-based (GSP-based) meta-
surfaces, have the attractive property of enabling the
simultaneous control of either the amplitude and phase
of the reflected light for one polarization or independently
engineering the reflection phases for two orthogonal polar-
izations [18]. These possibilities for light control are
exploited in metasurfaces performing analog computations
on incident light [19], dual-image holograms [20], and
polarization-controlled unidirectional excitation of surface-
plasmon polaritons [21]. Particularly, the last application
inspires us to suggest a type of compact metasurface-based
polarimeter that couples incident light into in-plane wave-
guide modes, with the relative efficiency of excitation
between predefined propagation directions being directly
related to the SOP. We incorporate in our design three
GSP-based metasurfaces that unidirectionally excite the
waveguide modes propagating in six different directions for
the three different polarization bases that are dictated by
the definition of Stokes parameters. By way of an example,
we design the all-polarization-sensitive waveguide meta-
coupler at a wavelength of 1.55 μm and perform full-wave
numerical simulations of a realistic (approximately
100-μm
2
footprint) device that reveals the possibility to
accurately retrieve the Stokes parameters in one shot.
II. STOKES PARAMETERS
Before we begin discussing the realization of the
in-plane polarimeter, it is appropriate to quickly review
the connection between the polarization of a plane wave,
described by the conventional Jones vector, and the Stokes
parameters that are typically measured in experiments. For
a z-propagating monochromatic plane wave, the Jones
vector can be written as
E
0
¼
A
x
A
y
e
iδ
; ð1Þ
where (A
x
, A
y
) are real-valued amplitude coefficients and δ
describes the phase difference between those two compo-
nents. Despite the simplicity in describing the amplitude
and SOP mathematically, the latter parameter is, in contrast,
inherently difficult to probe experimentally, which owes to
the fact that conventional detectors respond to the intensity
of the impinging wave (i.e., I ∝ A
2
x
þ A
2
y
), hence losing
information of the crucial phase relation between the two
orthogonal components. In order to remedy this short-
coming in experiments, the four Stokes parameters are
introduced, which are based on six intensity measurements
and fully describe both the amplitude and SOP of the plane
wave. The Stokes parameters can be written as
s
0
¼ A
2
x
þ A
2
y
; ð2Þ
s
1
¼ A
2
x
− A
2
y
; ð3Þ
s
2
¼ 2A
x
A
y
cos δ ¼ A
2
a
− A
2
b
; ð4Þ
s
3
¼ 2A
x
A
y
sin δ ¼ A
2
r
− A
2
l
; ð5Þ
where it is readily seen that s
0
simply describes the intensity
of the beam, thus retaining the information of the SOP in
s
1
–s
3
. Moreover, s
1
–s
3
can be found by measuring the
intensity of the two orthogonal components of the light in the
three bases (
ˆ
x,
ˆ
y), ð
ˆ
a;
ˆ
bÞ¼ð1=
ffiffiffi
2
p
Þð
ˆ
x þ
ˆ
y; −
ˆ
x þ
ˆ
yÞ, and
ð
ˆ
r;
ˆ
lÞ¼ð1=
ffiffiffi
2
p
Þð
ˆ
x þ i
ˆ
y;
ˆ
x − i
ˆ
yÞ, where the latter two bases
correspond to a rotation of the Cartesian coordinate system
(
ˆ
x,
ˆ
y) by 45° with respect to the x axis and the basis
for circularly polarized light, respectively. It should be
noted that, in describing the SOP of a plane wave, s
1
–s
3
are conventionally normalized by s
0
so that all possible
values lie within 1. Additionally, it is seen that
ðs
2
1
þ s
2
2
þ s
2
3
Þ=s
2
0
¼ 1, which signifies that all polarization
states in the three-dimensional space (s
1
, s
2
, s
3
) represent a
unit sphere, also known as the Poincaré sphere.
Having outlined the connection between the SOP and the
Stokes parameters, it is clear that our waveguide meta-
coupler must respond uniquely to all possible SOPs, with
preferably the most pronounced differences occurring for
the six extreme polarizations jxi, jyi, jai, jbi, jri, and jli,
so that all linear polarizations thereof can be probably
resolved. In order to achieve this property, we base our
design on birefringent GSP-based metasurfaces that can be
used for unidirectional and polarization-controlled inter-
facing of freely propagating waves and waveguide modes
[21]. The in-plane momentum matching to the waveguide
mode is achieved through grating coupling, with an addi-
tional linear phase gradient along the metasurface ensuring
unidirectional excitation. Moreover, and in line with the
previous work in Ref. [16], the metacoupler consists of
three metasurfaces that launch the waveguide modes in
different directions for the orthogonal sets of polarizations
ðjxi; jyiÞ, ðjai; jbiÞ, and ðjri; jliÞ, respectively. In this way,
the contrast between the power carried by the waveguide
mode in the two propagation directions of each metasurface
will mimic the respective dependence of s
1
–s
3
on the SOP.
III. DESIGN OF WAVEGUIDE METACOUPLERS
In the design of any waveguide coupler, the first step
is to specify the properties of the mode to be launched by
the coupler. The waveguide configuration considered here
consists of an optically thick gold film overlaid by a 70-nm-
thick SiO
2
(silicon dioxide) layer and a PMMA [poly
(methylmethacrylate)] layer [see Fig. 1(a)]. Figures 1(c)
and 1(d) show the numerically calculated effective indexes
and propagation lengths of waveguide modes supported by
the configuration as a function of the PMMA thickness at
the telecommunication wavelength of λ ¼ 1550 nm. In the
ANDERS PORS and SERGEY I. BOZHEVOLNYI PHYS. REV. APPLIED 5, 064015 (2016)
064015-2
calculations, performed using the commercially available
finite-element software
C
omsol
M
ultiphysics, the refractive
index of SiO
2
and PMMA is assumed to be 1.45 and
1.49, respectively, while the value for gold is 0.52 þ i10.7
as found from the interpolation of experimental values [22].
It is seen that the first transverse-magnetic (TM) mode
persists for all PMMA thicknesses, but being a surface-
plasmon polariton mode (with the maximum electric field
at the glass-gold interface) it also features a relatively low
propagation length. In order to extend the distance which
information can be carried, we choose to couple light to the
first transverse-electric (TE) mode, which is a photonic
mode with the maximum electric field appearing away from
the metal interface [Fig. 1(b)]. It is clear that one can
achieve propagation lengths of several hundreds of
micrometers by a properly thick PMMA thickness. The
simultaneous increase in the real part of the effective
refractive index, however, signifies the need for an increas-
ingly smaller grating period in order to reach the phase-
matching condition, thus potentially leading to feature sizes
of the metacoupler that prevent the incorporation of a
proper linear phase gradient. In order to avoid such
problems, while still having a waveguide mode that is
reasonably confined to the PMMA layer, we choose a
PMMA thickness of 400 nm corresponding to a TE1 mode
with an effective index of 1.10 and a propagation length of
≃130 μm. The associated metacoupler should then feature
a grating period of Λ
g
¼ λ=1.10 ≃ 1.41 μm in order to
couple normal incident light to the TE1 mode.
Having clarified the waveguide configuration, we next
discuss the procedure of designing the GSP-based meta-
couplers. The basic unit cell is schematically shown in
Fig. 2(a), which is fundamentally the waveguide configu-
ration with a gold nanobrick positioned atop the SiO
2
layer,
thereby ensuring the possibility of controlling the phase of
the scattered light by utilizing nanobrick dimensions in
the neighborhood of the resonant GSP configuration. The
(approximate) linear phase gradient of the metacouplers is
in this work achieved by incorporating three unit cells
within each grating period, with adjacent unit cells featur-
ing a difference in reflection phase of 120°. In order to find
the appropriate nanobrick dimensions, we perform full-
wave numerical calculations of the interaction of normal
incident x- and y-polarized light with the array of unit cells
in Fig. 2(a) when the geometrical parameters take on the
values Λ ¼Λ
g
=3 ¼470 nm, t
s
¼ 70 nm, t
PMMA
¼ 400 nm,
and t ¼ 50 nm. The key parameter is the complex reflec-
tion coefficient as a function of nanobrick widths (L
x
, L
y
),
which is displayed in Fig. 2(b) for x-polarized light, with a
superimposition of the phase contour lines in steps of 120°
for y-polarized light as well. It is seen that the metasurface
is highly reflecting for most configurations. However, near
L
x
¼ 275 nm (keeping L
y
constant), the reflection ampli-
tude features a noticeable dip accompanied by a significant
(a)
(b)
(c)
(d)
FIG. 1. Waveguide configuration. (a) Sketch of the waveguide
configuration that is assumed spatial invariant along the x and z
directions. (b) The electric field of the TE1 mode for propagation
along the z direction and a PMMA thickness of 400 nm. (c) The
real part of the effective index N
eff
¼ β=k
0
, where β is the
propagation constant of the mode and k
0
is the vacuum wave
number, and (d) the propagation length for modes sustained
by the configuration in (a) as a function of the PMMA thickness.
The SiO
2
thickness is fixed at 70 nm, and the wavelength
is λ ¼ 1550 nm.
0.
6
0.
8
1
0.1
0.3
0.4
0.2
L
y
(
µm
)
0.1 0.3 0.4
0.2
L
x
(µm)
|r
xx
|
y-pol.
x-pol.
-165
o
-45
o
75
o
-165
o
-45
o
75
o
x
y
z
t
t
s
t
PMMA
L
x
L
y
Λ
Λ
(a)
(b)
FIG. 2. Design of metacouplers. (a) Sketch of the unit cell of a
GSP-based metacoupler. (b) Calculated reflection coefficient
as a function of nanobrick widths (L
x
, L
y
) for x-polarized
normal incident light and geometrical parameters Λ ¼ 470 nm,
t
s
¼ 70 nm, t
PMMA
¼ 400 nm, t ¼ 50 nm, and wavelength
λ ¼ 800 nm. The color map shows the reflection amplitude,
whereas the solid lines represent contour lines of the reflection
phase for both x - and y-polarized light.
WAVEGUIDE METACOUPLERS FOR IN-PLANE POLARIMETRY PHYS. REV. APPLIED 5, 064015 (2016)
064015-3
change in the reflection phase. This is the signature of the
GSP resonance and, together with the assumption of
negligible coupling between neighboring nanobricks, the
necessary ingredient in designing phase-gradient (i.e.,
inhomogeneous) metasurfaces.
In the design of a unidirectional and polarization-
controlled waveguide coupler for the (
ˆ
x,
ˆ
y) basis, which
we denote metacoupler 1, we follow the previously
developed approach [21], where the Λ
g
× Λ
g
supercell is
populated with nine nanobricks defined by the intersection
of phase contour lines in Fig. 2(b). A top view of the
supercell is displayed in Fig. 3(a), where the nanobricks are
arranged in such a way that xðyÞ-polarized incident light
experiences a phase gradient in the yðxÞ direction, thus
ensuring unidirectional excitation of the TE1 mode. As a
way of probing the functionality of the designed supercell,
we perform full-wave simulations of a coupler consisting of
3 × 3 supercells, with the incident light being a Gaussian
beam with a beam radius of 3 μm. The resulting intensity
distribution in the center of the PMMA layer is shown in
Figs. 3(b) and 3(c) for polarization states jxi and jyi, which
verifies that the TE1 mode is dominantly launched in the
þy- and þx direction, respectively, as a consequence of the
incorporated birefringent phase gradient in the metacou-
pler. Moreover, the coupling efficiency, as defined by the
power carried by the TE1 mode in the desired propagation
direction relative to the incident power, is quite high,
reaching in this numerical example approximately 35%
despite the fact that no attempt is made to reach efficient
coupling.
The second waveguide metacoupler is intended to show
a markedly different directional excitation of the TE1 mode
for the polarization states jai and jbi. As a simple way to
realize this functionality, we reuse the supercell of meta-
coupler 1, though this time the nanobricks are rotated 45°
around their center of mass in the xy plane, followed by an
overall 180° rotation of the supercell [Fig. 3(d)]. The latter
rotation is implemented in order to achieve dominant
|E/E
0
|
2
0
1
2
3
35.1%
%3.
0
3.5%
%2.0
0.3%
%1.53
0.2%
%5.3
1.0%
%4.
0
26.0%
%
7.3
0.6%
%0.
1
2.3%
%0.6
2
0.3%
%2.32
0.3%
%2.3
0.3%
%
9.2
0.3%
%6.
22
x
y
a
b
r
l
(a)
(b)
(c)
(e)
(f)
(h)
(i)
(d)
(g)
x
y
FIG. 3. Performance of the individual metacouplers. (a),(d),(g) Top view of the supercell of coupler 1, 2, and 3, respectively. (b),(c),(e),
(f),(h),(i) Color map of the intensi ty in the center of the PMMA layer for couplers consisting of 3 × 3 superc ells when the incident light
is a Gaussian beam with beam radius 3 μm. Note that the scale bar is chosen to better highlight weak intensity features, while the
numbers (in percent) displayed in the panels correspond to coupling efficiencies through the areas marked by gray lines. (b),(c) Coupler
1 for incident polarization states jxi and jyi; (e),(f) coupler 2 for incident polarization states jai and jbi; (h),(i) coupler 3 for incident
polarization states jri and jli.
ANDERS PORS and SERGEY I. BOZHEVOLNYI PHYS. REV. APPLIED 5, 064015 (2016)
064015-4