Polarization pattern of vector vortex beams generated by $q$-plates with different topological charges

0.43 MB·English

by  Filippo Cardano

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 Santamato1,3
1Dipartimento di Scienze Fisiche, Universita` di Napoli “Federico II”,
Complesso di Monte S. Angelo, 80126 Napoli, Italy
2CNR-SPIN, Complesso Universitario di Monte S. Angelo, 80126 Napoli, Italy
3CNISM-Consorzio Nazionale Interuniversitario per le Scienze Fisiche della Materia, Napoli, Italy
Wedescribethepolarizationtopologyofthevectorbeamsemergingfromapatternedbirefringent
2 liquid crystal plate with a topological charge q at its center (q-plate). The polarization topological
1 structuresfordifferentq-platesanddifferentinputpolarizationstateshavebeenstudiedexperimen-
0 tallybymeasuringtheStokesparameterspoint-by-pointinthebeamtransverseplane. Furthermore,
2 weused a tunedq=1/2-plate togenerate cylindrical vectorbeams with radial or azimuthal polar-

n izations, with the possibility of switching dynamically between these two cases by simply changing
a thelinear polarization of theinput beam.
J OCIS number: 050.4865, 260.6042, 260.5430, 160.3710.

2
1
I. INTRODUCTION The methods to produce vector beams can be divided

] intoactiveandpassive. Activemethodsarebasedonthe
s
outputofnovellasersourceswithspeciallydesignedopti-
c The polarization of light is a consequence of the vec-
i torial nature of the electromagnetic field and is an im- calresonators[14–16]. Thepassivemethodsuseeitherin-
t
p portant property in almost every photonic application, terferometric schemes [17], or mode-forming holographic
o and birefringent elements [6, 18–22]. Light polarization
such as imaging, spectroscopy, nonlinear optics, near-
s. fieldoptics,microscopy,particletrapping,micromechan- is usually thought to be independent of other degrees
c of freedom of light, but it has been shown recently that
ics, etc. Most past research dealt with scalar optical
i
s fields, where the polarization was taken uniform in the photon spin angular momentum due to the polarization
y can interact with the photon orbital angular momentum
beam transverse plane. More recently, the so-called vec-
h whenthelightpropagateinahomogenous[23]andanin-
tor beams were introduced, where the light polarization
p
homogenousbirefringentplate[24,25]. Suchinteraction,
[ in the beam transverse plane is space-variant [1]. As
compared with homogeneously polarized beams, vector indeed, gives the possibility to convert the photon spin
1 into orbital angular momentum and viceversa in both
beams have unique features. Of particular interest are
v classical and quantum regimes [26].
the singular vector beams where the polarization distri-
6
Inthis work,tocreateopticalvectorbeamsweexploit
4 bution in the beam transverse plane has a vectorial sin-
thespin-to-orbitalangularmomentumcouplinginabire-
6 gularity as a C-point or L-line, where the azimuth angle
fringentliquidcrystalplatewithatopologicalchargeqat
2 andorientationof polarizationellipses are undefined, re-
. spectively [2, 3]. The polarization singular points are its center, named “q-plate” [24, 25]. As it will be shown
1
later, there is a number of advantages in using q-plates,
0 often coincident with corresponding singular points in
mainlybecausethepolarizationpatternimpressedinthe
2 the optical phase, thus creating what are called vector
1 vortex beams. Vector vortex beams are strongly corre- output beam can be easily changed by changing the po-
: latedtosingularoptics,wheretheopticalphaseatazero larization of the incident light [28, 29], and q-plates can
v
be easily tuned to optimal conversion by external tem-
i point of intensity is undetermined [4] andto light beams
X perature [30] or electric fields [31, 32]. Subsequently, the
carrying definite orbital angular momentum (OAM) [5].
r Among vector vortex beams, radially or azimuthally po- structure and quality of the produced vector field have
a been analyzed by point-by-point Stokes parameters to-
larized vector beams have received particular attention
mography in the beam transverse plane for different q-
for their unique behavior under focusing [6–8] and have
plates and input polarization states. In particular, we
been proved to be useful for many applications such as
generatedandstudied in detail the radialand azimuthal
particle acceleration [9], single molecule imaging [10],
polarizationsproducedbyaq-platewithfractionaltopo-
nonlinear optics [11], Coherent anti-Stokes Raman scat-
logical charge q =1/2.
teringmicroscopy[12],andparticletrappingandmanip-
ulation [13]. Because of their cylindrical symmetry, the
vector beams with radial and azimuthal polarizationare
also named cylindrical vector beams [1].
∗ Correspondingauthor: [email protected]
2
FIG. 1. (Color online) (a) Poincar´e 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
distributioninthetransverseplaneform=1andm=2onthehigher-orderPoincar´esphereintroducedbyMilioneetal.[27],
respectively.
II. POLARIZATION TOPOLOGY sities I (p = L,R,H,V,D,A) measured for different po-
p
larizations, according to
Henry Poincar´e presented a nice pictorial way to de- S =I +I
0 L R
scribe the light polarization based on the 2 1 homo- S =I I
morphism of SU(2) and SO(3). In this descr→iption, any S1 =IH −IV
2 A D
polarizationstateisrepresentedbyapointonthesurface S =I −I . (2)
3 L R
of a unit sphere, named “Poincar´e” or “Bloch” sphere. −
The Stokes’ parameters can be used to describe partial
The light polarization state is defined by two indepen-
polarization too. In the case of fully polarized light, the
dent real variables (ϑ,ϕ), ranging in the intervals [0,π]
reducedStokes parameters s =S /S (i=1,2,3)can be
and [0,2π], respectively, which fix the colatitude and az- i i 0
used, instead. We may consider the reduced parameters
imuth angles over the sphere. An alternative algebraic
s as the Cartesian coordinates on the Poincar´e sphere.
representation of the light polarization state in terms of i
The s are normalized to 3 s2 = 1. The states of
the angles (ϑ,ϕ) is given by i i=1 i
linear polarization, lying onPthe equator of the Poincar´e
ϑ,ϕ =cos ϑ L +eiϕ sin ϑ R , (1) sphere, have s3 =0. The two states s3 =±1 correspond
to the poles and are circularly polarized. In singular op-
| i (cid:18)2(cid:19)| i (cid:18)2(cid:19)| i
tics, these two cases may form two different type of po-
where L and R standfortheleftandright-circularpo- larization singularities. For the other states, the sign of
| i | i
larizations, respectively. On the Poincar´e sphere, north s fixes the polarization helicity; left-handed for s > 0
3 3
and south poles correspondto left and right-circular po- and right-handed for s < 0. The practical advantage
3
larization, respectively, while any linear polarization lies of using the parameters s is that they are dimension-
i
onthe equator,asshowninFig.1-(a). Speciallinearpo- less and depend on the ratio among intensities. Light
larizationstatesarethe H , V , D , A , whichdenote intensitiescanbeeasilymeasuredbyseveralphotodetec-
| i | i | i | i
horizontal, vertical, diagonal and anti-diagonalpolariza- torsandcanbereplacedbyaveragephotoncountsinthe
tions,respectively. Inpointsdifferentfromthepolesand quantumopticsexperiments. Thus,Stokes’analysispro-
the equatorthe polarizationisellipticalwithleft(right)- vide averyuseful wayto performthe fulltomographyof
handed ellipticity in the north (south) hemisphere. the polarization state (1) in both classical and quantum
An alternative mathematical description of the light regimes.
polarization state, which is based on SU(2) representa- The passage of light through optical elements may
tion,wasgivenbyGeorgeGabrielStokesin1852. Inthis change its polarization state. If the optical element is
representation, four parameters S (i = 0,...,3) known fully transparent, the incident power is conserved and
i
asStokesparametersnowadays,areexploitedtodescribe only the light polarization state is affected. The action
thepolarizationstate. Thisrepresentationisuseful,since of the transparent optical element is then described by
the parameters S are simply related to the light inten- a unitary transformation on the polarizationstate ϑ,ϕ
i
| i
3
in Eq. (1) and corresponds to a continuous path on the
Poincar´e sphere. In most cases, the optical element can
be considered so thin that the polarization state is seen
tochangeabruptlyfromonepointP toadifferentpoint
1
P on the sphere. In this case, it can be shown that
2
the path on the sphere is the geodetic connecting P to
1
P [33]. Examples of devices producing a sudden change
2
ofthe lightpolarizationinpassingthrougharehalf-wave
(HW)andquarter-wave(QW)plates. AsequenceofHW
and QW can be used to move the polarization state on
the whole Poincar´e sphere, which corresponds to arbi- FIG.2. (Coloronline)Setuptogenerateandanalyzedifferent
trary SU(2) transformationapplied to the ϑ,ϕ state in polarization topologies generated by a q-plate. The polariza-
Eq. (1). A useful sequence QW-HW-QW-H| Wi(QHQH) tionstateoftheinputlaserbeamwaspreparedbyrotatingthe
to perform arbitrary SU(2) transformation on the light twohalf-waveplatesintheQHQHset atangles ϑ/4andϕ/4
toproduceacorrespondingrotationof(ϑ,ϕ)onthePoincar´e
polarization state is presented in Ref [29].
sphereasindicatedintheinset. Thewaveplatesandpolarizer
So far we considered an optical phase that is uniform
beyond theq-plate where used to project thebeam polariza-
inthe beamtransverseplane. Allowingforanonuniform
tion on the base states (R, L, H, V, A, D) shown in Fig. 1a.
distribution of the optical phase between different elec- For each state projection, the intensity pattern was recorded
tric field components gives rise to polarization patterns, byCCD camera and thesignals were analyzed pixel-by-pixel
like azimuthal and radial ones, where special topologies toreconstructthepolarizationpatterninthebeamtransverse
appears in the transverse plane. The topological struc- plane. Legend: 4X-microscopeobjectiveof4Xusedtoclean
ture of the polarization distribution, moreover, remains thelasermode,f-lens,Q-quarterwaveplate,H-halfwave
unchangedwhilethebeampropagates. Itisworthnoting plate, PBS-polarizing beam-splitter.
thatmostsingularpolarizationpatternsinthetransverse
plane can still be described by polar angles ϑ,ϕ on the
Poincar´e sphere [27]. The points on the surface of this
semi-integer topological charge at its center [24, 25, 28].
higher-order Poincar´e sphere represent polarized light
states where the opticalfield changesase±imφ, wherem The cell glass windows are treated so to maintain the
isapositiveintegerandφ=arctan(y/x)istheazimuthal liquidcrystalmoleculardirectorparalleltothewalls(pla-
angle in the beam transverse plane. As it is well known, narstronganchoring)withnon-uniformorientationangle
light beams with optical field proportional to eimφ are α = α(ρ,φ) in the cell transverse plane, where ρ and φ
vortex beams with topological charge m, which carry a are the polar coordinates. Our q-plates have a singular
definite OAM m~ per photon along their propagation orientation with topological charge q, so that α(ρ,φ) is
axis. Because ±the beam is polarized, it carries spin an- given by
gular momentum (SAM) too, so that the photons are in
what may be called a spin-orbit state. Among the spin- α(ρ,φ)=α(φ)=qφ+α0, (3)
orbit states, only a few states can be described by the
higher-orderPoincar´esphereand,precisely,thestatesbe- with integer or semi-integer q and real α . This pat-
0
longing to the spin-orbitSU(2)-subspace spannedby the ternwasobtainedwithanazo-dyephoto-alignmenttech-
two base vectors L eimφ, R e−imφ . In this represen- nique [32]. Besides its topologicalchargeq, the behavior
{| i | i }
tationthenorthpole,southpoleandequatorcorrespond of the q-plate depends on its optical birefringent retar-
to the base state L eimφ , the base state R e−imφ , dation δ. Unlike other LC based optical cells [22] used
{| i } {| i }
andlinearpolarizationwithrotatedtopologicalstructure to produce vector vortex beam, the retardation δ of our
of charge m, respectively. Fig. 1b,c show the polariza- q-platescanbecontrolledbytemperaturecontrolorelec-
tion distribution for m = 1 and m = 2 spin-orbit sub- tricfield[30,31]. AsimpleargumentbasedonJonesma-
spaces,respectively. Theintensityprofileforallpointson trixshowsthatthe unitaryactionUofthe q-plateinthe
the higher-orderPoincar´esphere has the same doughnut state (1) is defined by
shape. The states along the equator are linearly polar- b
izeddoughnutbeamswithtopologicalchargem,differing L δ L δ R e+2i(qφ+α0)
in their orientation only. U(cid:18) |Ri (cid:19)=cos2(cid:18) |Ri (cid:19)+sin2(cid:18)|Lie−2i(qφ+α0) (cid:19).
| i | i | i
b (4)
The q-plate is said to be tuned when its optical retar-
III. THE q-PLATE: PATTERNED LIQUID dation is δ = π. In this case, the first term of Eq. (4)
CRYSTAL CELL FOR GENERATING AND vanishes and the optical field gains a helical wavefront
MANIPULATING HELICAL BEAM
with double of the plate topological charge (2q). More-
over,the handednessofhelicalwavefrontdepends onthe
The q-plate is a liquid crystal cell patterned in spe- helicity of input circular polarization, positive for left-
cifictransversetopology,bearingawell-definedintegeror circular and negative for right-circular polarization.
4
IV. EXPERIMENT 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-
In our experiment, a TEM HeNe laser beam (λ =
00
tionlimits. Suchbeamshaveawiderangeofapplications
632.8nm, 10mW) was spatially cleaned by focusing into
in optical lithography,data storage,material processing,
a 50µm pinhole followed by a truncated lens and po-
and optical tweezers [6, 7].
larizer, in order to have a uniform intensity and a ho-
mogeneous linear polarization. The beam polarization
wasthen manipulatedby a sequence ofwaveplates asin
Ref. [29] to reachany point on the Poincar´esphere. 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 transmitted
beam acquired the characteristic doughnut shape with
a hole at its center. The output beam was analyzed by
point-to-pointpolarizationtomography,byprojectingits
polarization into the H, V, A, D, L, R sequence of basis
and measuring the corresponding intensity at each pixel
ofa120 120resolutionCCDcamera(SonyAS-638CL).
×
Examples of the recorded intensity profiles are shown in
Fig.3. A dedicatedsoftwarewasused to reconstructthe
polarization distribution on the beam transverse plane.
FIG.3. (Coloronline)Recordedintensityprofilesofthebeam
To minimize the error due to small misalignment of the
emergingfrom theq-plateprojected overhorizontal(H),ver-
beam when the polarization was changed, the values of tical(V),anti-diagonal(A),diagonal(D),left-circular(L)and
the measured Stokes parameters were averaged over a right-circular(R)polarizationbasestatesfordifferentq-plates
grid of 20 20 squares equally distributed over the im- andinputpolarizations. (a)and(b)arefortheq=1/2-plate,
×
age area. No other elaboration of the raw data nor best and horizontal-linear (a) and left-circular (b) polarization of
fit with theory was necessary. theinputbeam. (c)and(d)arethesamefortheq=1-plate.
Thecolorscalebarshowstheintensityscale(arbitraryunits)
We analyzed the beams generated by two different q-
in false colors.
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 = 632.8nm by applying Before concluding, it is worth of mention that vortex
anexternalvoltageofafewvolt[32]. Weconsideredleft- vector beams are based on non-separable optical modes,
circular(L )andlinear-horizontal(H )polarizedinput which is itself an interesting concept in the framework
| i | i
beams. These states, after passing through the q-plate of classical optics. At the single photon level, however,
arechangedinto R,+2q and(R,+2q + L, 2q )/√2, the same concept has even more fundamental implica-
| i | i | − i
respectively. Figure 3 shows the output intensity pat- tions, because it is at the basis of the so-called quantum
terns for different polarization selections. Figure 3 (a), contextuality [34].
(b) show the results of point-by-point polarization to-
mography of the output from q = 1/2-plate for left-
circularandhorizontal-linearinputpolarizations,respec-
tively. Figure 3 (c), (d) show the results of point-by-
pointpolarizationtomographyofthe outputfromq =1-
plate for left-circular and horizontal-linearpolarizations,
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,+1 + L, 1 )/√2
| i | − i
represented by the S -axes over equator of higher-order
1
Poincar´e sphere (Fig. 2 (b)), which corresponds to a ra-
diallypolarizedbeam,asshowninFig.4(c). Thisradial
polarizationcanbe changedinto the azimuthalpolariza-
tion (corresponding to the antipodal point on S -axes of
1
the higher-order Poincar´e sphere) by just switching the
inputlinearpolarizationfromhorizontaltovertical,asit
isshowninFig.4(b). Thisprovidesaveryfastandeasy
waytoswitchfromradialtoazimuthalcylindricalvector
beam. As previously said, cylindrical vector beams have
5
FIG. 4. (Color online) Highest-row panels: the transverse polarization and intensity distribution in the near field at the
exit face of the q-plate. Lower panels: the polarization and intensity distributions in the far field beyond the q-plate. (a),
(b), and (c): polarization topological structure generated by the q = 1/2-plate for left-circular, V-linear, and H-linear input
polarizations, respectively. (d)-(e): polarizationtopologicalstructuregeneratedbytheq=1-plateforleft-circularandH-linear
input polarizations, respectively. (a) and (d) haveuniform circular polarization distributions.
V. CONCLUSION point-by-pointStokes’tomographyoverthe entiretrans-
verse plane.
In this paper, however, we have investigated the cases
We have generated and analyzed a few vector vortex q =1andq =1/2. Future workwilladdressothercases,
beams created by a patterned liquid crystal cell with and in particular the negative q ones.
topologicalcharge,namedq-plate. Radialandazimuthal
cylindrical beams have been obtained by acting on the
polarization of a traditional laser beam sent through a VI. ACKNOWLEDGMENT
q = 1/2-plate. In this way, fast switching from the ra-
dialtotheazimuthalpolarizationcanbeeasilyobtained. We acknowledge the financial support of the FET-
Finally,westudiedindetailthepolarizationofafewvec- Open program within the 7th Framework Programme
tor beams generated by different q-plates and the polar- of the European Commission under Grant No. 255914,
ization distribution patterns have been reconstructed by Phorbitech.
[1] Q. Zhan, “Cylindrical vector beams: from mathemati- [7] H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard and C.
calconceptstoapplications”, Adv.Opt.Photon.1,1-57 T. Chong, “Creation of a needle of longitudinally polar-
(2009). izedlightinvacuumusingbinaryoptics”,Nat.Photonics
[2] J.F.Nye,Natural focusingand the finestructure oflight 2, 501-505 (2008).
(IOPPublishing, Bristol, UK,1999). [8] Q. Zhan and J. Leger, “Focus shaping using cylindrical
[3] M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Op- vector beams”, Opt.Express 10, 324-331 (2002).
ticalVorticesandPolarization Singularities”,Prog.Opt. [9] C.VarinandM.Pich´e,“Accelerationofultra-relativistic
53, 293-363 (2009). electrons using high-intensity laser beams”, Appl. Phys.
[4] M. S. Soskin, and M. V. Vasnetsov (Ed. E. Wolf), “Sin- B 74, S83-S88 (2002).
gular optics”, Prog. Opt.42, 219-276 (2001). [10] L. Novotny, M. R. Beversluis, K. S. Youngworth, and
[5] S. Franke-Arnold, L. Allen, and M. J. Padgett, “Ad- T.G.Brown,“LongitudinalFieldModesProbedbySin-
vances in optical angular momentum”, Laser Photonics gle Molecules”, Phys. Rev.Lett. 86, 5251- 5254 (2001).
Rev.2, 299313 (2008). [11] A. Bouhelier, M. Beversluis, A. Hartschuh, and
[6] R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for L. Novotny, “Far-field second-harmonic generation in-
a radially polarized light beam”, Phys. Rev. Lett. 91, duced by local field enhancement”, Phys. Rev.Lett. 90,
233901 (2003). 013903 (2003).
6
[12] F. Lu, W. Zheng and Z. Huang, “Coherent anti-Stokes [25] L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo,
Ramanscatteringmicroscopy usingtightlyfocused radi- E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-
ally polarized light”, Opt. Lett.34, 1870-1872 (2009). orbital conversion of the angular momentum of light
[13] Q. Zhan, “Trapping metallic Rayleigh particles with ra- and its classical and quantumapplications”, J. Opt.13,
dial polarization”, Opt. Express12, 3377-3382 (2004). 064001 (2011).
[14] R.Oron,S.Blit,N.Davidson,A.A.Friesen,Z.Bomzon, [26] E. Nagali, F. Sciarrino, F. De Martini, B. Piccirillo,
andE.Hasman,“Theformationoflaserbeamswithpure E.Karimi,L.Marrucci,andE.Santamato,“Polarization
azimuthal or radial polarization”, Appl. Phys. Lett. 77, controlofsinglephotonquantumorbitalangularmomen-
3322-3324 (2000). tum states”, Opt.Express 17 18745-18759 (2009).
[15] Y.Kozawa and S.Sato, “Generation of a radially polar- [27] G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano,
izedlaserbeambyuseofaconicalBrewsterprism”,Opt. “Higher-order Poincar´e sphere, Stokes parameters, and
Lett.30, 3063-3065 (2005). the angular momentum of light”, Phys. Rev. Lett. 107,
[16] H. Kawauchi, Y. Kozawa, and S. Sato, “Generation of 053601 (2011).
radially polarized Ti:sapphire laser beam using a c-cut [28] L.Marrucci,C.Manzo,andD.Paparo,“Pancharatnam-
crystal”, Opt.Lett. 33, 1984-1986 (2008). Berry phase optical elements for wave front shaping in
[17] X. L. Wang, J. Ding, W. J. Ni, C. S. Guo, and thevisibledomain: Switchablehelicalmodegeneration”,
H.T.Wang,“Generationofarbitraryvectorbeamswith Appl. Phys.Lett. 88, 221102 (2006).
a spatial light modulator and a common path interfero- [29] E.Karimi,S.Slussarenko,B.Piccirillo,L.Marrucci,and
metric arrangement”, Opt.Lett. 32, 3549-3551 (2007). E.Santamato,“Polarization-controlled evolutionoflight
[18] E.G.Churin,J.Hoßfeld, T.Tschudi,“Polarization con- transverse modesandassociated Pancharatnam geomet-
figurationswith singular point formed bycomputergen- ric phase in orbital angular momentum”, Phys Rev. A
erated holograms”, Opt.Commun. 99, 13-17 (1993). 81, 053813 (2010).
[19] Z.Bomzon,G.Biener,V.Kleiner,andE.Hasman,“Ra- [30] E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and
dially and azimuthally polarized beams generated by E.Santamato,“Efficientgenerationandsortingoforbital
space-variant dielectric subwavelength gratings”, Opt. angular momentum eigenmodes of light by thermally
Lett.27, 285-287 (2002). tuned q-plates” Appl. Phys.Lett. 94, 231124 (2009).
[20] G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and [31] B.Piccirillo,V.D’Ambrosio,S.Slussarenko,L.Marrucci,
S. Jackel, “Spatially-variable retardation plate for ef- and E. Santamato, “Photon spin-to-orbital angular mo-
ficient generation of radially and azimuthally-polarized mentum conversion via an electrically tunable q-plate”,
beams”, Opt.Commun. 281, 732-738 (2008). Appl. Phys.Lett. 97, 241104 (2010).
[21] T. Fadeyeva, V. Shvedov, N. Shostka, C. Alexeyev, and [32] S. Slussarenko, A. Murauski, T. Du, V. Chigrinov,
A.Volyar,“Naturalshapingofthecylindricallypolarized L. Marrucci, and E. Santamato, “Tunable liquid crystal
beams”, Opt.Lett. 35, 3787-3789 (2010). q-plateswitharbitrarytopologicalcharge”,Opt.Express
[22] M.St`alder,andN.Schadt,“Linearlypolarized lightwith 19, 4085-4090 (2011).
axial symmetry generated by liquid-crystal polarization [33] R. Bhandari, “Polarization of light and topological
converters”, Opt.Lett. 21, 1948-1950 (1996). phases”, Phys. Rep.,281, 1-64 (1997).
[23] E. Brasselet, Y. Izdebskaya, V. Shvedov, A. S. Desyat- [34] E.Karimi,J.Leach,S.Slussarenko,B.Piccirillo,L.Mar-
nikov, W. Krolikowski, and Y. S. Kivshar, “Dynamics rucci,L.Chen,W.She,S.Franke-Arnold,M.J.Padgett,
of optical spin-orbit coupling in uniaxial crystals”, Opt. and E. Santamato, “Spin-orbit hybrid entanglement of
Lett.34 1021-1023 (2009). photons and quantum contextuality” Phys. Rev. A 82,
[24] L.Marrucci,C.Manzo,andD.Paparo,“Opticalspin-to- 022115 (2010).
orbitalangularmomentumconversionininhomogeneous
anisotropic media”, Phys.Rev.Lett. 96, 163905 (2006).