![FIG. 1. (Color online) (a) Poincaré sphere representation for the polarizaiton af a light beam with uniform transverse phase distribution. North (south) pole and equator correspond to left (right)-circular and linear polarizations, respectively. The other points have elliptical polarization, and antipodal points are orthogonal to each other. (b) and (c) show the polarization distribution in the transverse plane for m = 1 and m = 2 on the higher-order Poincaré sphere introduced by Milione et al. [27], respectively.](/figures/fig-1-color-online-a-poincare-sphere-representation-for-the-26c44qgc.webp)
Polarization pattern of vector vortex beams generated by q-
plates with different topological charges
Author
Cardano, Filippo, Karimi, Ebrahim, Slussarenko, Sergei, Marrucci, Lorenzo, de Lisio, Corrado,
Santamato, Enrico
Published
2012
Journal Title
Applied Optics
DOI
https://doi.org/10.1364/AO.51.0000C1
Copyright Statement
© 2012 OSA. This paper was published in Applied Optics and is made available as an
electronic reprint with the permission of OSA. The paper can be found at the following URL
on the OSA website: dx.doi.org/10.1364/AO.51.0000C1. Systematic or multiple reproduction
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penalties under law.
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Polarization pattern of vector vortex beams generated by q-plates with different
topological charges
Filippo Cardano,
1
Ebrahim Karimi,
1, ∗
Sergei Slussarenko,
1
Lorenzo
Marrucci,
1, 2
Corrado de Lisio,
1, 2
and Enrico Santamato
1, 3
1
Dipartimento di Scienze Fisiche, Universit`a di Napoli “Federico II”,
Complesso di Monte S. Angelo, 80126 Napoli, Italy
2
CNR-SPIN, Complesso Universitario di Monte S. Angelo, 80126 Napoli, Italy
3
CNISM-Consorzio Nazionale Interuniversitario per le Scienze Fisiche della Materia, Napoli, Italy
We describe the polarization topology of the vector beams emerging from a patterned birefringent
liquid crystal plate with a topological charge q at its center (q-plate). The polarization topological
structures for different q-plates and different input polarization states have been studied experimen-
tally by measuring the Stokes p arameters point-by-point in the beam transverse plane. Furthermore,
we used a tuned q = 1/2-plate to generate cylindrical vector beams with radial or azimuthal polar-
izations, with the p ossibility of switching dynamically between these two cases by simply changing
the linear polarization of the input beam.
OCIS number: 050.4865, 260.6042, 260.5430, 160.3710.
I. INTRODUCTION
The polarization of light is a consequence of the vec-
torial nature of the ele ctromagnetic field and is an im-
portant property in almost every photonic applicatio n,
such as imaging, sp ectroscopy, nonlinear optics, near-
field optics, microscopy, particle tra pping, micromechan-
ics, etc. Most past research dealt with scalar optical
fields, where the polarization was taken uniform in the
beam transverse plane. More recently, the so-called vec-
tor beams were introduced, where the light polarization
in the beam transverse plane is space-variant [1]. As
compared with homogeneously polarized beams, vector
beams have unique featur es. Of particular interest are
the singular vector beams where the polarization distri-
bution in the beam tr ansverse plane has a vectorial sin-
gularity as a C-point or L-line, where the azimuth angle
and o rientation of polarization ellipses are undefined, re-
sp ectively [2, 3]. The p olarization singular points are
often coincident with corresponding singular points in
the optical phase, thus creating what are called vector
vortex beams. Vector vortex beams are strongly corre-
lated to singular optics, where the optical phase at a zero
point of intensity is undetermined [4] and to light beams
carrying definite orbital angular momentum (OAM) [5].
Among vector vortex beams, radially or azimuthally po-
larized vector beams have r eceived particular attention
for their unique behavior under focusing [6–8] and have
been proved to be useful for many applications such as
particle acceleration [9], single molecule imaging [10],
nonlinear optics [11], Coherent anti-Stokes Rama n scat-
tering micro scopy [12], and particle trapping and manip-
ulation [13]. Because of their cylindrical symmetry, the
vector beams with radial and azimuthal polarization are
also named cylindrical vector beams [1].
∗
C
orresponding author: karimi@na.infn.it
The methods to produce vector beams can be divided
into active and passive. Active methods are based on the
output of novel laser sources with specially designed opti-
cal resonators[14–16]. The passive methods use either in-
terferometric schemes [17], or mode-forming holographic
and birefringent elements [6, 18–22]. Light polarization
is usually thought to be independent of other degrees
of freedom of light, but it has been s hown recently that
photon s pin angular momentum due to the polarization
can interac t with the photon or bital a ngular mo mentum
when the light propagate in a homog enous [23] and an in-
homogenous birefringent plate [24, 25]. Such interaction,
indeed, gives the possibility to convert the photon spin
into orbital ang ular momentum and viceversa in both
classical and quantum regimes [26].
In this work, to create optical vector beams we exploit
the spin-to-orbital angular momentum coupling in a bire-
fringent liquid crystal plate with a topological charge q at
its center, named “q-plate” [24, 25]. As it will be shown
later, there is a number of advantages in using q-plates,
mainly because the polarization pattern impressed in the
output b eam can be easily changed by changing the po-
larization of the incident light [28, 29], and q-plates can
be easily tuned to optimal conversion by external tem-
perature [30] or electric fields [31, 32]. Subsequently, the
structure and qua lity of the produced vector field have
been analyzed by point-by-point Stokes parameters to-
mography in the beam transverse plane for different q-
plates and input polarization states. In particular, we
generated and studied in detail the radial and az imuthal
polarizations produced by a q-plate with fractional topo-
logical charge q = 1/2.
2
FIG. 1. (Color on line) (a) Poincar´e sp here representation for the polarizaiton af a light beam with uniform transverse phase
distribution. North (south) pole and equator correspond to left (right)-circular and linear polarizations, respectively. The
other points have elliptical polarization, and antipodal points are orthogonal to each other. (b) and ( c) show the polarization
distribution in the transverse plane for m = 1 and m = 2 on the higher-order Poincar´e sphere introduced by Milione et al. [27],
respectively.
II. POLARIZATION TOPOLOGY
Henry Poincar´e pre sented a nice pictorial way to de-
scribe the light polarization based on the 2 → 1 homo-
morphism of SU(2) and SO(3). In this description, any
polarization state is represented by a po int on the surface
of a unit sphere, named “Poincar´e” or “Bloch” sphere.
The light polarization state is defined by two indepen-
dent real variables (ϑ, ϕ), ranging in the intervals [0 , π]
and [0, 2π], respectively, which fix the colatitude and a z-
imuth angles over the sphere. An alternative algebraic
representation of the light polar ization sta te in terms of
the angles (ϑ, ϕ) is given by
|ϑ, ϕi = cos
ϑ
2
|Li + e
iϕ
sin
ϑ
2
|Ri, (1)
where |Li and |Ri stand for the left and right-circular po-
larizations , respectively. On the Poincar´e sphere , north
and south poles correspond to left and right-circular po-
larization, respectively, while any linear polarization lies
on the equator, as shown in Fig. 1-(a). Special linear po-
larization states are the |Hi, |V i, |Di, |Ai, which denote
horizontal, vertical, diagonal and anti-diagonal polariza-
tions, respectively. In points different from the poles and
the equator the p olarization is elliptical with left (right)-
handed ellipticity in the north (south) hemisphere.
An alternative mathematical description of the light
polarization state, which is based on SU(2) representa-
tion, was given by George Gabriel Stokes in 1852. In this
representation, four parameters S
i
(i = 0, . . . , 3) known
as Stokes parameters nowadays, are exploited to describe
the polarization state. This representation is useful, since
the parameters S
i
are simply related to the light inten-
sities I
p
(p = L, R, H, V, D, A) measured for different po-
larizations , according to
S
0
= I
L
+ I
R
S
1
= I
H
− I
V
S
2
= I
A
− I
D
S
3
= I
L
− I
R
. (2)
The Stokes’ parameters can be used to describe partial
polarization too. In the case of fully polarized light, the
reduced Stokes parameters s
i
= S
i
/S
0
(i = 1, 2, 3) can be
used, instead. We may consider the reduced parameters
s
i
as the Cartesian coordinates on the Poincar´e sphere.
The s
i
are normalized to
P
3
i=1
s
2
i
= 1. The states of
linear polarization, lying on the equator of the Poincar´e
sphere, have s
3
= 0. The two states s
3
= ±1 correspond
to the poles and a re circularly polarized. In singular op-
tics, these two cases may form two different type of po-
larization singularities. For the other states, the sign of
s
3
fixes the polarization helicity; left-handed for s
3
> 0
and right-handed for s
3
< 0. The practical advantage
of using the parameters s
i
is that they are dimens ion-
less and depend on the ratio among intensities. Light
intensities ca n be easily measured by several photodetec-
tors and can be replaced by average photon counts in the
quantum optics experiments. Thus, Stokes’ analys is pro-
vide a very use ful way to p erform the full tomog raphy of
the p olarization state (1) in both classical and quantum
regimes.
The passage of light thro ugh o ptical elements may
change its polarization state. If the optica l element is
fully tr ansparent, the incident power is conserved and
only the light polarization state is affected. The action
of the transparent optical element is then described by
a unitary transformation on the polarization state |ϑ, ϕi
3
in Eq. (1) and corresponds to a continuous path on the
Poincar´e sphere. In most cases, the optical element c an
be considered so thin that the polarization state is seen
to change abruptly from one point P
1
to a differ ent point
P
2
on the sphere. In this case, it can be shown that
the path on the sphere is the geodetic connecting P
1
to
P
2
[33]. Examples o f devices producing a sudden change
of the lig ht polarization in passing through are half-wave
(HW) and quarter-wave (QW) plates . A seq ue nc e of HW
and QW can be used to move the polariz ation state on
the whole Poincar´e sphere, which corresponds to arbi-
trary SU(2) transformation applied to the |ϑ, ϕi state in
Eq. (1). A useful sequence QW-HW-QW-HW (QHQH)
to perfor m arbitrary SU(2) transformation on the light
polarization state is presented in Ref [29].
So far we considered an optical phase that is uniform
in the beam transverse plane. Allowing for a nonuniform
distribution of the optical phas e between different elec-
tric field components gives rise to polarization patterns,
like azimuthal and radial ones, where special topologies
appears in the transverse plane. The topological struc-
ture of the polarizatio n distribution, moreover, remains
unchanged while the beam propa gates. It is worth noting
that most singular polarization patterns in the transverse
plane can still be described by polar angles ϑ, ϕ on the
Poincar´e sphere [27]. The points on the surface of this
higher-order Poincar´e sphere represent polarized light
states where the optical field changes as e
±imφ
, where m
is a positive integer and φ = arctan(y/x) is the a zimuthal
angle in the beam transverse plane. As it is well known,
light beams with optical field proportional to e
imφ
are
vortex beams with topological charge m, which carry a
definite OAM ±m~ per photon along their propagation
axis. Because the beam is polarized, it carries spin an-
gular momentum (SAM) too, so that the photons a re in
what may be calle d a spin-orbit state. Among the spin-
orbit states, only a few states can be described by the
higher-order Poincar´e sphere and, precisely, the states be-
longing to the spin-orbit SU(2)-subspace spanned by the
two base vectors {|Lie
imφ
, |Rie
−imφ
}. In this represen-
tation the north pole, south pole and e quator correspond
to the base sta te {|Lie
imφ
}, the base state {|Rie
−imφ
},
and linear p olarization with rotated topological structure
of charge m, respe ctively. Fig. 1b,c show the polariza-
tion distr ibution for m = 1 and m = 2 spin-orbit sub-
spaces, respectively. The intensity profile for all points on
the higher-order Poincar´e sphere has the same doughnut
shape. The states along the equator are linearly polar-
ized doughnut beams with topological charge m, differing
in their orientation only.
III. THE q-PLATE: PATTERNED LIQUID
CRYSTAL CELL FOR GENERATING AND
MANIPULATING HELICAL BEAM
The q-plate is a liquid cry stal cell patterned in spe-
cific transverse topology, bearing a well-defined integer or
FIG. 2. (Color online) Setup to generate and analyze different
polarization topologies generated by a q-plate. The polariza-
tion state of the input laser beam was prepared by rotating the
two half-wave plates in the QHQH set at angles ϑ/4 and ϕ/4
to p roduce a corresponding rotation of (ϑ, ϕ) on the Poincar´e
sphere as indicated in the inset. The waveplates and polarizer
beyond the q-plate where used to project the beam polariza-
tion on the base states (R, L, H, V, A, D) shown in Fig. 1a.
For each state projection, th e intensity pattern was recorded
by CCD camera and the signals were analyzed pixel-by-pixel
to reconstruct the polarization pattern in the beam transverse
plane. Legend: 4X - microscope objective of 4X used t o clean
the laser mode, f - lens, Q - quarter wave plate, H - half wave
plate, PBS- polarizing beam-splitter.
semi-integer topological charge at its center [24, 25, 28].
The cell gla ss windows are treated so to maintain the
liquid crystal molecular director parallel to the walls (pla-
nar strong anchoring) with non-uniform orientation angle
α = α(ρ, φ) in the cell transverse plane, where ρ and φ
are the polar coordinates. Our q-plates have a singular
orientation with topological charge q, so that α(ρ, φ) is
given by
α(ρ, φ) = α(φ) = qφ + α
0
, (3)
with integer or semi-integer q and real α
0
. T his pat-
tern was obtained with a n azo-dye photo-alignment tech-
nique [32]. Besides its topological charge q, the behavior
of the q-plate depends on its optical birefringent retar-
dation δ. Unlike other LC based o ptical cells [22] used
to produce vector vortex beam, the retardation δ o f our
q-plates can be controlled by temperature control or elec-
tric field [30, 31]. A simple argument based on Jones ma-
trix shows that the unitary action
b
U of the q-plate in the
state (1) is defined by
b
U
|Li
|Ri
= cos
δ
2
|Li
|Ri
+ sin
δ
2
|Rie
+2i(qφ+α
0
)
|Lie
−2i(qφ+α
0
)
.
(4)
The q-plate is said to be tuned when its optical retar-
dation is δ = π . In this case, the first term of Eq. (4)
vanis he s and the optical field gains a helical wavefront
with double of the plate topological charge (2q). More-
over, the handedness of helical wavefront depends on the
helicity of input circular polarization, positive for left-
circular and negative for right-circular polarizatio n.
4
IV. EXPERIMENT
In our experiment, a TEM
00
HeNe laser beam (λ =
632.8nm, 10mW) was spatially cleaned by focusing into
a 50µm pinhole followed by a truncated lens and po-
larizer, in order to have a uniform intensity and a ho-
mogeneous linear polarization. The beam polarization
was then manipulated by a seq ue nc e of wave plates as in
Ref. [29] to reach any point on the Poincar´e sphere. The
beam was then sent into an electrically driven q-plate,
which changed the beam state into an entangled spin-
orbit state as given by Eq. 4. When the voltage on the
q-plate was set for the optical tuning, the trans mitted
beam acquired the characteristic doughnut shape with
a hole at its center. The output beam was analyzed by
point-to-point polarization tomography, by projecting its
polarization into the H, V, A, D, L, R sequence of basis
and measuring the co rresponding intensity a t each pixel
of a 120 ×120 resolution CCD camera (Sony AS-638CL).
Examples of the recorded intensity profiles are shown in
Fig. 3 . A de dic ated software was used to reconstruct the
polarization distribution on the beam transverse plane.
To minimize the error due to small misalignment of the
beam when the polarization was changed, the values of
the measured Stokes parameters were averag e d over a
grid of 20 × 20 squares equally distributed over the im-
age area . No other elaboratio n of the raw data nor best
fit with theory was necessary.
We analyzed the beams generated by two different q-
plates with charges q = 1/2 and q = 1 for two different
input polarization states. The q-plate optical retarda-
tion was optimally tuned for λ = 6 32.8nm by applying
an external voltage of a few volt [3 2]. We considered left-
circular (|Li) and linear-ho rizontal (|Hi) polarized input
beams. These states, after passing through the q-plate
are changed into |R, +2qi and (|R, +2qi + |L, −2qi) /
√
2,
respectively. Figure 3 shows the output intensity pat-
terns for different polarization s elections. Figure 3 (a),
(b) show the results of point-by-point polarization to-
mography of the output fro m q = 1/2-plate for left-
circular and horizontal-linear input polarizations, respec-
tively. Figure 3 (c), (d) show the results o f point-by-
point polariz ation tomography of the output from q = 1-
plate for left-circular and horizontal-linear polarizations,
respectively.
The case of q = 1/2-plate is particularly interest-
ing, because for a linear horizontal input polarization,
it yields to the spin-orbit state (|R, +1i + |L, −1 i) /
√
2
represented by the S
1
-axes over equator of higher-order
Poincar´e sphere (Fig. 2 (b)), which corres ponds to a r a-
dially polarized beam, as shown in Fig. 4 (c). This radial
polarization can be changed into the azimuthal polariza-
tion (corresponding to the antipodal point on S
1
-axes of
the higher-o rder Poincar´e sphere) by just switching the
input linear polarization from horizontal to vertical, as it
is shown in Fig. 4 (b). This provides a very fast and easy
way to switch from radial to azimuthal cylindrical vector
beam. As previously said, cylindrical vector beams have
a number of applications and can be used to generate
uncommon beams such as electric and magnetic needle
beams, where the optical field is confined below diffrac-
tion limits. Such beams have a wide range of applications
in optical lithography, data storag e, material processing,
and optical tweezers [6, 7].
FIG. 3. (Color online) Recorded intensity profiles of the b eam
emerging from the q-plate projected over horizontal (H), ver-
tical (V), anti-diagonal (A), diagonal (D), left-circular (L) and
right-circular (R) polarization base states for different q-plates
and input polarizations. (a) and (b) are for the q = 1/2-plate,
and horizontal-linear (a) and left-circular (b) polarization of
the input beam. (c) and (d) are the same for the q = 1-plate.
The color scale b ar shows the intensity scale (arbitrary units)
in false colors.
Before concluding, it is worth of mention that vortex
vector beams are based on non-separable optical modes,
which is itself an interesting concept in the framework
of classical optics. At the single photon level, however,
the same concept has even more fundamental implica-
tions, because it is at the basis of the so-ca lle d quantum
contextuality [34].