Arbitrary polarization conversion for pure vortex generation with a single metasurface - De Gruyter
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Nanophotonics 2021; 10(1): 727–732
Research article
Marco Piccardo* and Antonio Ambrosio*
Arbitrary polarization conversion for pure vortex
generation with a single metasurface
https://doi.org/10.1515/nanoph-2020-0332 shaped intensity distribution of these beams make them
Received June 17, 2020; accepted July 23, 2020; published online attractive for a number of applications [2], ranging from
August 10, 2020
quantum information [3, 4] to super-resolution microscopy
[5, 6], and have motivated the development of several
Abstract: The purity of an optical vortex beam depends on
techniques of optical vortex generation [7]. However, most
the spread of its energy among different azimuthal and
of these methods, such as spiral phase plates [8],
radial modes, also known as ℓ- and p-modes. The smaller
the spread, the higher the vortex purity and more efficient computer-generated holograms [9, 10], spatial light mod-
its creation and detection. There are several methods to ulators [11], q-plates [12, 13], and J-plates [14, 15], rely on
generate vortex beams with well-defined orbital angular phase-only (PO) transformations. The azimuthal phase
momentum, but only few exist allowing selection of a pure modulation imparted by these optical elements allows
radial mode. These typically consist of many optical ele- creation of an optical vortex, but the lack of amplitude
ments with rather complex arrangements, including active modulation prevents the output beam from being a solu-
cavity resonators. Here, we show that it is possible to tion of the paraxial wave equation. The missing amplitude
generate pure vortex beams using a single metasurface term is compensated by the spreading of the beam energy
plate—called p-plate as it controls radial modes—in com- during propagation on high-order radial modes, thus
bination with a polarizer. We generalize an existing theory leading to impure states consisting of a superposition of
of independent phase and amplitude control with bire- vortex modes [16].
fringent nanopillars considering arbitrary input polariza- There exist a few methods that can provide both the
tion states. The high purity, sizeable creation efficiency, required phase and amplitude (PA) modulation for pure
and impassable compactness make the presented vortex generation, such as mode conversion in active res-
approach a powerful complex amplitude modulation tool onators [17–19], but these require either specific input
for pure vortex generation, even in the case of large topo- beams or rather complex cavity set-ups. Here, we introduce
logical charges. a method that allows conversion of an arbitrary input beam
into a pure vortex mode using a single metasurface plate—
Keywords: dielectric metasurfaces; modal purity; optical
called p-plate as it controls radial modes—which is easy to
vortex generation; polarization conversion.
implement in practical optics experiments and could be of
use in any vortex application, requiring that the beam
power is contained within a specific p-mode.
1 Introduction
The characterizing feature of an optical vortex is a zero of 2 Methods
intensity, which coincides with a phase singularity of the
field. The phase circulates around this point of null in- After presenting our metasurface theory, we will prove the approach
tensity endowing the beam of light with orbital angular by applying it to the design of dielectric metasurfaces operating in the
momentum (OAM) [1]. The OAM and the specific ring- near-infrared region. This spectral range is only chosen for the sake of
illustration, without loss of generality. We consider amorphous silicon
nanopillars with a rectangular cross section lying on a silica substrate.
The pillars have a height of 600 nm and are arranged in a hexagonal
*Corresponding authors: Marco Piccardo and Antonio Ambrosio, close packed lattice (600-nm pillar-to-pillar separation). The wave-
Center for Nano Science and Technology, Fondazione Istituto Italiano length of the source is 1064 nm. A library of the transmission co-
di Tecnologia, Milan 20133, Italy, E-mail: marco.piccardo@iit.it efficients and phases imparted by the pillars as a function of their size
(M. Piccardo); antonio.ambrosio@iit.it (A. Ambrosio). https://orcid. Lx,y (in the 100- to 480-nm range) is constructed using the finite-
org/0000-0001-8851-1638 (M. Piccardo) difference time-domain (FDTD) module of Lumerical and selecting the
Open Access. © 2020 Marco Piccardo and Antonio Ambrosio, published by De Gruyter. This work is licensed under the Creative Commons
Attribution 4.0 International License.
728 M. Piccardo and A. Ambrosio: Single metasurface pure vortex generation
elements with the best performance in terms of amplitude trans- The phase is determined by one of the two dimensions of the
mission and phase accuracy [20]. The average transmittance of the pillar, which sets ψ, while the amplitude can be controlled
pillars chosen from the library to design the metasurfaces considered
via a combination of Δϕ and α, as shown in Figure 1B.
in this work is typically around 95%. The design code sets the sizes and
orientation angles of the pillars according to our theory to produce a Now, we concentrate on the specific problem of finding
desired Laguerre–Gaussian (LG) beam. We validate the method by analytically the exact conditions to obtain the extinction.
carrying out FDTD simulations of p-plates with 30-μm diameter, using This is much more important than determining the condi-
plane wave or Gaussian sources with linear or elliptical polarizations. tions for unity transmission as the extinction is critical to
The beam waist of the designed LG modes is set to ω0 5 μm. The
mask certain parts of the input beam. On the other hand,
simulated near field is propagated to a hemispherical surface in the far
the conditions for unity transmission only influence the
field at 1 m from the metasurface and then projected to a plane for
visual representation using direction cosines. Finally, the propagated efficiency of the device and will be discussed later. The
field is decomposed on an LG basis to evaluate its modal purity. The problem is illustrated in Figure 1C. We consider an incident
basis is truncated at the 20th radial mode. The complex beam elliptical state represented by the Jones vector
parameter of the LG basis is obtained by an optimization algorithm
cosχ
that maximizes the overlap probability of the far-field beam with its
≡ χ, δ⟩ (2)
decomposition over the finite basis [21]. eiδ sinχ
and a linear polarizer with angle θpol π/2 with respect to
the x-axis. Our goal is to find the pillar parameters that
3 Phase and amplitude control convert the input state into a linear state oriented
perpendicularly to the polarizer axis. In the following, the
The operation of a single birefringent nanopillar on an subscript “0” refers to parameters and operators corre-
incident field can be represented in Jones calculus in the sponding to the extinction condition. First of all, we use the
pillar frame of reference as rotation matrix R0 to rotate the field on the pillar basis as
1 0 R0 χ, δ⟩ ≡ χ′, δ′⟩, which turns the ellipse angle from θ to
M eiψ (1) θ − α0 . Then, the new state can be linearized via the pillar
0 eiΔϕ
birefringence operator as M 0 χ′, δ′⟩ with the condition
where the birefringence arises from the form factor of the Δϕ0 −δ′. After linearization, the orientation of the state is
pillar. We assume unity transmission; thus, M is unitary, further rotated to an angle θlin χ′. We impose that after
and the pillar cannot modulate the amplitude of the field. returning back to the original reference frame using
R−1
0 M 0 R0 χ, δ⟩, the output state lies parallel to the x-axis,
Meta-atom geometries capable of controlling directly both
the phase and amplitude of the field have been demon- which gives the condition θlin + α0 0. By combining the
strated [22, 23] but suffer in general from fabrication com- previous conditions, we obtain α0 −χ′. Thus, all that re-
plexities as they rely on resonances, may be limited in mains is to express the rotated state χ′, δ′⟩ in terms of the
efficiency, or may not allow full modulation ranges, i.e. input one χ, δ⟩. This finally leads to the general extinction
from 0 to 1 in amplitude and from 0 to 2π in phase. condition for the orientation angle
However, a simple birefringent pillar represented by the
1
operator M allows conversion of the polarization of the α0 tan−1 tan χ sec δ. (3)
2
incident field, thanks to the phase retardance Δϕ. Thus,
by adding a linear polarizer after the metasurface, one can valid for an arbitrary polarization state. A closed-form
in principle use the projection of the converted polariza- expression can also be deduced for the phase retardance
tion state to modulate the transmitted amplitude of the Δϕ0 , though not as compact (Supplementary Material).
field, as well as control its phase via the global eiψ factor. Figure 1D and E represent the solutions for the
This simple but powerful scheme was demonstrated with extinction condition for all the possible input polarization
metasurfaces designed for the specific cases of linearly states shown on the (χ, δ) plane. In the case of circularly
polarized [24] and circularly polarized [25] light inputs. polarized light (i.e. π/4, π/2⟩), one obtains the intuitive
Here, we generalize this approach to input states of result of Δϕ0 π/2, corresponding to a quarter-waveplate,
arbitrary polarization. and α0 π/4. Clearly, for any polarization state, it is always
The operation principle of the proposed scheme of PA possible to achieve a full dynamic range of amplitude
control is shown in Figure 1A. Given an elliptically polarized modulation by following a suitable path in the parameter
input, the pillar dimensions and orientation angle α are space connecting the zero of amplitude with the global
designed so that the polarization-converted beam after pass- maximum (cf. Figure 1B). However, such trajectory cannot
ing through a polarizer has the desired phase and amplitude. be described with a simple analytical form, and it may be
M. Piccardo and A. Ambrosio: Single metasurface pure vortex generation 729
A nanopillar polarizer B transmitted amplitude
propagation
C D E F
rot. angle, retardance, amplitude,
Figure 1: Mapping polarization to amplitude.
(A) An arbitrary input polarization state is converted by a birefringent nanopillar into a new polarization state that, after passing through a
polarizer, is reduced in amplitude. This approach still allows the phase of the output state to be controlled independently. The nanopillar
orientation angle and polarizer axis are α and θpol , respectively. (B) Contour plot showing the transmitted amplitude of the elliptical state
shown in A after propagation through the nanopillar–polarizer system as a function of the orientation angle α and phase retardance Δϕ of the
pillar. (C) Conditions to achieve amplitude extinction through the nanopillar–polarizer system when the polarizer axis is aligned along the
y-axis. The input is an arbitrary polarization state χ, δ⟩ that is rotated by an angle α 0 in the nanopillar frame of reference (operator R0),
linearized via birefringence (operator M0), and finally brought back to the original frame of reference (operator R −10 ) resulting orthogonal to the
polarizer axis. (D and E) Design parameter maps corresponding to the pillar angle and phase retardance needed to obtain extinction for any
input polarization state, represented in the (χ, δ) plane. (F) Maximum amplitude transmission through the nanopillar–polarization system for
any input state when the phase retardance is fixed to Δϕ0 and only α is adjusted. If instead both parameters are allowed to vary a full
modulation range from 0 to 1 can be achieved, as in B.
convenient in practical metasurface designs to set one of where ℓ and p are indices denoting the OAM and radial
the two parameters, e.g. Δϕ0 , and vary the other one to mode of the LG beam, respectively; r and φ are the radial
modulate the transmitted amplitude according to an and azimuthal coordinates, respectively; ω0 is the beam
analytical formula (Supplementary Material). We show in waist; and Lp|ℓ | (x) is the generalized Laguerre polynomial of
Figure 1F that with this restriction, only a small fraction of argument x. The amplitude mask A(r) is assimilated in the
polarization states remains limited in modulation, while angles of the pillars α(r), which vary along the radial di-
for the majority, it is possible to achieve a large modulation rection of the plate (inset of Figure 2A), while the azimuthal
range, from 0 to almost 1. phase profile ψ(φ) defines the pillar dimension Lx (φ). The
other pillar dimension Ly (φ) is determined by ψ(φ) + Δϕ0 ,
where Δϕ0 is a fixed parameter in the design corresponding
4 Pure vortex generation to the phase retardance for the extinction condition, as
previously described. The output of the metasurface is a
The principle of arbitrary polarization conversion illus- vector vortex beam as it carries OAM and exhibits a
trated above for the single nanopillar is now applied to the nonuniform polarization distribution. After the beam is
design of optical plates for pure vortex generation. The filtered by a linear polarizer, which could be a separate
operation scheme is shown in Figure 2A. The input beam element or directly integrated in the metasurface substrate
can be any wave of known PA distribution, such as a plane as a wire-grid polarizer [24], its amplitude distribution
or Gaussian wave, and arbitrary polarization. In the case of corresponds to that of a pure LG beam.
a plane wave, the p-plate is designed to impart the PA The near-field intensity maps of an LG5,0 vortex
profile of an LG mode [26] generator obtained by FDTD simulations are shown in
√ | ℓ | Figure 2B. The uniform intensity of the input plane wave is
r 2 2r2
L|pℓ | 2 e−r ω0 e−iℓφ ≡ A(r)eiψ(φ) separated by the p-plate into a vertically polarized
2 2
LGℓ, p ∝ (4)
ω0 ω0
component, which is aligned with the polarizer axis and
730 M. Piccardo and A. Ambrosio: Single metasurface pure vortex generation
A polarizer
metasurface
asurf
rface
B metasurface near field
C phase only D phase and amplitude E elliptical polarization F radial mode
intensity
phase
Figure 2: Pure vortex generation from a single metasurface.
(A) Schematic of the system implementation for pure vortex generation. An input wave of known amplitude, phase, and polarization, here
represented as a linearly polarized Gaussian beam, propagates through the metasurface and polarizer and exits as a pure vortex beam. The
angles of the nanopillars along the radial directions α(r) define the amplitude mask of the metasurface, while their rectangular dimensions set
the azimuthal phase profile Lx, y (φ) of the beam. (B) Near-field intensity maps obtained by finite-difference time-domain (FDTD) simulations for
a metasurface generating an LG5,0 mode. Only the vertically polarized component (left) is transmitted through the polarizer. (C–F) Far-field
intensity and phase distributions obtained by FDTD simulations for different structures: (C) a phase-only metasurface generating a
superposition of radial modes; (D–F) phase and amplitude metasurfaces generating pure vortex beams for different input and output states. In
all cases, the bottom row shows the central area (10-μm diameter) of the metasurface structures, the target Laguerre–Gaussian beam, and the
input polarization state (either linear or elliptical).
exhibits the characteristic ring-shaped distribution of a expect for the fundamental radial mode (p = 0). The PO
vortex mode, and a horizontally polarized component, metasurface instead shows multiple rings corresponding to
which contains essentially the complementary intensity the superposition of different p-modes [16]. A modal purity
distribution and will be filtered by the polarizer. (A small analysis shows that 97% of the generated beam power is in
fraction of light goes also into the longitudinal component, the p = 0 mode for the PA metasurface, while this value
which is not shown here.) After propagating the compo- drops to 23% for the PO metasurface. The efficiency in
nent filtered by the polarizer to the far field, we obtain the generating pure beams with the two types of metasurfaces
intensity and phase distributions shown in Figure 2D, will be discussed in Section 5.
which can be compared with those obtained from a PO The purity of the intensity profile generated by the PA
metasurface operating without a polarizer (Figure 2C). metasurface all originates from a proper use of the pillar
While in both cases, the phase shows an azimuthal mod- angle degree of freedom, as highlighted by the comparison
ulation consisting of five sectors (ℓ 5), only the PA met- of the device designs (cf. Figure 2C and D). As described in
asurface produces a single ring of intensity as one would our theory, the PA control works for any input polarization
M. Piccardo and A. Ambrosio: Single metasurface pure vortex generation 731
state. In Figure 2E, we show that also for elliptically from poor phase resolution close to the singularity point
polarized light, where we chose as an example owing to the strong phase gradient of large OAM beams and
χ π/5, δ π/3⟩, one can generate a high-purity vortex finite size of the meta-atoms, in the case of PA meta-
mode with a single intensity ring. While the generation of surfaces, the phase sampling only matters in the ring-
such beams is of interest in OAM applications, where the shaped intensity region of the target LGℓ, 0 mode. Consid-
energy needs to be confined in the fundamental radial ering that the peak intensity of the ring occurs at a radial
mode, the control of higher order p-modes may be position rmax ω0 |ℓ |/2 from the singularity and that the
appealing in other applications, such as super-resolution length of an arc spanning a phase period from 0 to 2π is
microscopy. In Figure 2F, we show that this is possible p 2π rmax /|ℓ |, we can estimate the number of pillars
using a PA metasurface, demonstrating as an example the covering a phase period in a PA metasurface as
generation of an LG1,3 mode. Also, in this case, the purity of
p 2 πω0
the beam is high as a very large fraction of the beam power ∼ (5)
U |ℓ | U
transmitted through the polarizer (92%) is retained in the
target mode. where U is the size of the unit cell of the metasurface. Thus,
for any OAM charge and for a unit cell size fixed by the
phase library, the phase sampling of the PA metasurface
can be chosen arbitrarily large just by adjusting ω0 .
5 Discussion The PA metasurfaces called p-plates introduced here
also present some limitations with respect to the existing
After having demonstrated that PA metasurfaces perform
approaches for optical vortex generation. Differently from
better than PO metasurfaces in terms of pure vortex gen- J-plates [14, 15], which implement PO transformations,
eration, we wish to compare their efficiencies. We define p-plates are not spin–orbit converters. In particular, they
the efficiency as the fraction of power of the beam incident can operate only on a single input polarization state. The
on the metasurface that is converted into the target LGℓ, 0 degree of freedom that is used in J-plates to achieve inde-
mode. For this comparison, it is important to consider an pendent OAM conversion of two orthogonal input polari-
input wave that carries finite power and that can be readily zation states here is used to apply a desired amplitude
available as a practical source in a laboratory; thus, we modulation. In comparison with q-plate lasers [17] and J-
choose a Gaussian beam. In the case of PO metasurfaces, plate lasers [19], which incorporate PO transformation
the efficiency is known to drop rapidly with the OAM
charge [16]. As shown in Figure 3A, the converted beam A phase only B phase and amplitude
1
power spreads towards higher radial modes as |ℓ | grows
relative power
owing to the lack of the amplitude modulation term, and
nearly, no energy is contained in the p = 0 mode for |ℓ | > 10.
In PA metasurfaces instead, all the power transmitted
through the polarizer is already in the p = 0 mode; thus, no 0
modal filtering is needed. What limits the efficiency in this 0 1
case is the fraction of input power that needs to be absor-
bed by the polarizer. To maximize the generation effi- 10-1
radial mode,
ciency, there is an optimum choice for the beam waist ωS of
25
the Gaussian source, which is in general larger than the
10-2
beam waist ω0 of the Gaussian embedded in the target LG
mode (Eq. 4) and is calculated as ωS /ω0 |ℓ | + 1 based on
the overlap of the source intensity profile with the trans- 50 10-3
0 20 40 60 80 100 0 20 40 60 80 100
mittance mask of the metasurface (Supplementary Mate- azimuthal mode, azimuthal mode,
rial). The efficiency of the PA metasurface for the optimum
ωS /ω0 ratio is shown in Figure 3B. It shows a clear benefit Figure 3: Purity efficiency of phase-only and phase-amplitude
for the generation of beams with large OAM charge as it transformations.
Fraction of the input beam power that is contained in the
remains sizeable even for very large values of |ℓ | (e.g. 5%
fundamental radial mode p = 0 (top) and distributed among all
efficiency at |ℓ | 100). Another advantage of PA meta- p-modes (bottom) for different values of orbital angular momentum ℓ
surfaces in this respect lies in the sampling of the imparted by a metasurface plate. Both the case of phase-only
azimuthal phase profile. While PO metasurfaces suffer metasurfaces (A) and phase-amplitude metasurfaces (B) are shown.
732 M. Piccardo and A. Ambrosio: Single metasurface pure vortex generation
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