Journal ArticleDOI

# Waveguide metacouplers for in-plane polarimetry

07 Jul 2016-arXiv: Optics-

AbstractThe 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 only respond to the intensity of the light, hence loosing the phase information between orthogonal vector components. In this work, we propose a new 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$\mu$m.

Topics: Polarimeter (57%), Polarimetry (53%), Stokes parameters (53%), Ray (52%), Photonics (52%)

### Introduction

• You may not further distribute the material or use it for any profit-making activity or commercial gain .
• The SOP evaluation is typically based on the determination of the so-called Stokes parameters, which are constructed from six intensity measurements with properly arranged polarizers placed in front of the detector, thereby allowing one to uniquely retrieve the SOP [5].
• Also, the authors note that additional realizations of polarimeters do exist, like using advanced micropolarizers in front of an imaging detector, but those approaches are typically complex and expensive [3].
• Particularly, the last application inspires us to suggest a type of compact metasurface-based polarimeter that couples incident light into in-plane waveguide modes, with the relative efficiency of excitation between predefined propagation directions being directly related to the SOP.

### II. STOKES PARAMETERS

• Before the authors 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.
• 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 ∝ A2x þ A2y), hence losing information of the crucial phase relation between the two orthogonal components.
• Þðx̂þ iŷ; x̂ − iŷÞ, where the latter two bases correspond to a rotation of the Cartesian coordinate system (x̂, ŷ) by 45° with respect to the x axis and the basis for circularly polarized light, respectively.
• 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.

### III. DESIGN OF WAVEGUIDE METACOUPLERS

• The waveguide configuration considered here consists of an optically thick gold film overlaid by a 70-nmthick SiO2 (silicon dioxide) layer and a PMMA [poly ] 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.
• The linear phase gradient of the metacouplers is in this work achieved by incorporating three unit cells within each grating period, with adjacent unit cells featuring a difference in reflection phase of 120°.
• 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.
• This fact is evidenced in Figs. 3(e) and 3(f), where approximately 26% of the incident power is coupled to the TE1 mode in the desired direction, hence verifying the unidirectional and birefringent response of this waveguide metacoupler.

### IV. PERFORMANCE OF THE IN-PLANE POLARIMETER

• The previous section outlines the design of three waveguide metacouplers that each launch the TE1 modes traveling primarily along two directions, with the maximum contrast occurring for the polarization states ðjxi; jyiÞ, ðjai; jbiÞ, and ðjri; jliÞ, respectively.
• The exact size of the waveguide metacoupler is not a critical parameter, but in order to avoid too-divergent TE1 beams the authors ensure that each side of the hexagon is considerably larger than the wavelength.
• The authors emphasize that, unlike related work [16], there is no mathematical equivalence between D1–D3 and s1=s0–s3=s0, nor is it even possible to find a linear relation (i.e., device matrix) between those quantities that is valid for all SOPs.
• The authors note that these small errors, 064015-6 corresponding to determining the Stokes parameters with an accuracy of six decimals, are obtained using the numerically calculated coupling efficiencies with full precision [i.e., not the rounded-off data presented in Table I and Eq. (7)], thereby highlighting the perfect linear relationship between coupling efficiencies and the Stokes parameters.
• As a final comment to the above discussion, it should be noted that most polarimeter designs utilize the linear relation C¼BS4, where the four-vector S4¼½s0;s1;s2;s3.

### V. CONCLUSION

• In summary, the authors design a compact in-plane polarimeter that couples incident light into waveguide modes propagating along six different directions, with the coupling efficiencies being dictated by the SOP.
• This allows one to realize simultaneous detection of the Stokes parameters.
• Regarding the spectral bandwidth of the proposed design, it should be noted that phase matching with the TE1 mode is achieved through grating coupling, which makes the polarimeter inherently narrow band, since the period of the grating must be close to the wavelength of the mode.
• Also, it is worth noting that conventional polarimeters typically measure the SOP in a destructive (i.e., strongly modifying or extinction of the incident beam) or perturbative way.
• Moreover, the authors foresee the possibility of a compact circuitry with built-in plasmonic detectors that are integrated into spatially confined waveguides [33,34].

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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
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=
ﬃﬃ
2
p
Þð
ˆ
x þ
ˆ
y;
ˆ
x þ
ˆ
yÞ, and
ð
ˆ
r;
ˆ
lÞ¼ð1=
ﬃﬃ
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; jy, ðjai; jb, and ðjri; jl, 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

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Abstract: The optical constants $n$ and $k$ were obtained for the noble metals (copper, silver, and gold) from reflection and transmission measurements on vacuum-evaporated thin films at room temperature, in the spectral range 0.5-6.5 eV. The film-thickness range was 185-500 \AA{}. Three optical measurements were inverted to obtain the film thickness $d$ as well as $n$ and $k$. The estimated error in $d$ was \ifmmode\pm\else\textpm\fi{} 2 \AA{}, and that in $n$, $k$ was less than 0.02 over most of the spectral range. The results in the film-thickness range 250-500 \AA{} were independent of thickness, and were unchanged after vacuum annealing or aging in air. The free-electron optical effective masses and relaxation times derived from the results in the near infrared agree satisfactorily with previous values. The interband contribution to the imaginary part of the dielectric constant was obtained by subtracting the free-electron contribution. Some recent theoretical calculations are compared with the results for copper and gold. In addition, some other recent experiments are critically compared with our results.

15,901 citations

Journal ArticleDOI
Abstract: A quantal system in an eigenstate, slowly transported round a circuit C by varying parameters R in its Hamiltonian Ĥ(R), will acquire a geometrical phase factor exp{iγ(C)} in addition to the familiar dynamical phase factor. An explicit general formula for γ(C) is derived in terms of the spectrum and eigenstates of Ĥ(R) over a surface spanning C. If C lies near a degeneracy of Ĥ, γ(C) takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy. As an illustration γ(C) is calculated for spinning particles in slowly-changing magnetic fields; although the sign reversal of spinors on rotation is a special case, the effect is predicted to occur for bosons as well as fermions, and a method for observing it is proposed. It is shown that the Aharonov-Bohm effect can be interpreted as a geometrical phase factor.

6,612 citations

Journal ArticleDOI
TL;DR: This Review focuses on recent developments on flat, ultrathin optical components dubbed 'metasurfaces' that produce abrupt changes over the scale of the free-space wavelength in the phase, amplitude and/or polarization of a light beam.
Abstract: Metamaterials are artificially fabricated materials that allow for the control of light and acoustic waves in a manner that is not possible in nature. This Review covers the recent developments in the study of so-called metasurfaces, which offer the possibility of controlling light with ultrathin, planar optical components. Conventional optical components such as lenses, waveplates and holograms rely on light propagation over distances much larger than the wavelength to shape wavefronts. In this way substantial changes of the amplitude, phase or polarization of light waves are gradually accumulated along the optical path. This Review focuses on recent developments on flat, ultrathin optical components dubbed 'metasurfaces' that produce abrupt changes over the scale of the free-space wavelength in the phase, amplitude and/or polarization of a light beam. Metasurfaces are generally created by assembling arrays of miniature, anisotropic light scatterers (that is, resonators such as optical antennas). The spacing between antennas and their dimensions are much smaller than the wavelength. As a result the metasurfaces, on account of Huygens principle, are able to mould optical wavefronts into arbitrary shapes with subwavelength resolution by introducing spatial variations in the optical response of the light scatterers. Such gradient metasurfaces go beyond the well-established technology of frequency selective surfaces made of periodic structures and are extending to new spectral regions the functionalities of conventional microwave and millimetre-wave transmit-arrays and reflect-arrays. Metasurfaces can also be created by using ultrathin films of materials with large optical losses. By using the controllable abrupt phase shifts associated with reflection or transmission of light waves at the interface between lossy materials, such metasurfaces operate like optically thin cavities that strongly modify the light spectrum. Technology opportunities in various spectral regions and their potential advantages in replacing existing optical components are discussed.

3,712 citations

Journal ArticleDOI
, Qiong He1, Qin Xu1, Xin Li1, Lei Zhou1
TL;DR: It is demonstrated theoretically and experimentally that a specific gradient-index meta-surface can convert a PW to a SW with nearly 100% efficiency, and may pave the way for many applications, including high-efficiency surface plasmon couplers, anti-reflection surfaces, light absorbers, and so on.
Abstract: The arbitrary control of electromagnetic waves is a key aim of photonic research. Although, for example, the control of freely propagating waves (PWs) and surface waves (SWs) has separately become possible using transformation optics and metamaterials, a bridge linking both propagation types has not yet been found. Such a device has particular relevance given the many schemes of controlling electromagnetic waves at surfaces and interfaces, leading to trapped rainbows, lensing, beam bending, deflection, and even anomalous reflection/refraction. Here, we demonstrate theoretically and experimentally that a specific gradient-index meta-surface can convert a PW to a SW with nearly 100% efficiency. Distinct from conventional devices such as prism or grating couplers, the momentum mismatch between PW and SW is compensated by the reflection-phase gradient of the meta-surface, and a nearly perfect PW-SW conversion can happen for any incidence angle larger than a critical value. Experiments in the microwave region, including both far-field and near-field characterizations, are in excellent agreement with full-wave simulations. Our findings may pave the way for many applications, including high-efficiency surface plasmon couplers, anti-reflection surfaces, light absorbers, and so on.

1,253 citations

Journal ArticleDOI
J. Scott Tyo
TL;DR: The foundations of passive imaging polarimetry, the phenomenological reasons for designing a polarimetric sensor, and the primary architectures that have been exploited for developing imaging polarimeters are discussed.
Abstract: Imaging polarimetry has emerged over the past three decades as a powerful tool to enhance the information available in a variety of remote sensing applications. We discuss the foundations of passive imaging polarimetry, the phenomenological reasons for designing a polarimetric sensor, and the primary architectures that have been exploited for developing imaging polarimeters. Considerations on imaging polarimeters such as calibration, optimization, and error performance are also discussed. We review many important sources and examples from the scientific literature.

1,163 citations