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

###### Abstract

We describe the polarization topology of the vector beams emerging from a patterned birefringent liquid crystal plate with a topological charge at its center (-plate). The polarization topological structures for different -plates and different input polarization states have been studied experimentally by measuring the Stokes parameters point-by-point in the beam transverse plane. Furthermore, we used a tuned -plate to generate cylindrical vector beams with radial or azimuthal polarizations, with the possibility 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 vectorial nature of the electromagnetic field and is an important property in almost every photonic application, such as imaging, spectroscopy, nonlinear optics, near-field optics, microscopy, particle trapping, micromechanics, 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 vector beams were introduced, where the light polarization in the beam transverse plane is space-variant zhan09 (). As compared with homogeneously polarized beams, vector beams have unique features. Of particular interest are the singular vector beams where the polarization distribution in the beam transverse plane has a vectorial singularity as a C-point or L-line, where the azimuth angle and orientation of polarization ellipses are undefined, respectively nye99 (); dennis09 (). The polarization 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 correlated to singular optics, where the optical phase at a zero point of intensity is undetermined soskin01 () and to light beams carrying definite orbital angular momentum (OAM) sonja08 (). Among vector vortex beams, radially or azimuthally polarized vector beams have received particular attention for their unique behavior under focusing dorn03 (); wang08 (); zhan02 () and have been proved to be useful for many applications such as particle acceleration varin02 (), single molecule imaging novotny01 (), nonlinear optics bouhelier03 (), Coherent anti-Stokes Raman scattering microscopy lu09 (), and particle trapping and manipulation zhan04 (). Because of their cylindrical symmetry, the vector beams with radial and azimuthal polarization are also named cylindrical vector beams zhan09 ().

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 optical resonatorsoron00 (); kozawa05 (); kawauchi08 (). The passive methods use either interferometric schemes wang07 (), or mode-forming holographic and birefringent elements churin93 (); bomzon02 (); dorn03 (); machavariani08 (); fadeyeva10 (); stalder96 (). Light polarization is usually thought to be independent of other degrees of freedom of light, but it has been shown recently that photon spin angular momentum due to the polarization can interact with the photon orbital angular momentum when the light propagate in a homogenous brasselet09 () and an inhomogenous birefringent plate marrucci06prl (); marrucci11jo (). Such interaction, indeed, gives the possibility to convert the photon spin into orbital angular momentum and viceversa in both classical and quantum regimes nagali09opex ().

In this work, to create optical vector beams we exploit the spin-to-orbital angular momentum coupling in a birefringent liquid crystal plate with a topological charge at its center, named “-plate” marrucci06prl (); marrucci11jo (). As it will be shown later, there is a number of advantages in using -plates, mainly because the polarization pattern impressed in the output beam can be easily changed by changing the polarization of the incident light marrucci06apl (); karimi10praa (), and -plates can be easily tuned to optimal conversion by external temperature karimi09apl () or electric fields piccirillo10apl (); slussarenko11opex (). Subsequently, the structure and quality of the produced vector field have been analyzed by point-by-point Stokes parameters tomography in the beam transverse plane for different -plates and input polarization states. In particular, we generated and studied in detail the radial and azimuthal polarizations produced by a -plate with fractional topological charge .

## Ii Polarization topology

Henry Poincaré presented a nice pictorial way to describe the light polarization based on the homomorphism of SU(2) and SO(3). In this description, any polarization state is represented by a point on the surface of a unit sphere, named “Poincaré” or “Bloch” sphere. The light polarization state is defined by two independent real variables , ranging in the intervals and , respectively, which fix the colatitude and azimuth angles over the sphere. An alternative algebraic representation of the light polarization state in terms of the angles is given by

(1) |

where and stand for the left and right-circular polarizations, respectively. On the Poincaré sphere, north and south poles correspond to left and right-circular polarization, respectively, while any linear polarization lies on the equator, as shown in Fig. 1-(a). Special linear polarization states are the , , , , which denote horizontal, vertical, diagonal and anti-diagonal polarizations, respectively. In points different from the poles and the equator the polarization 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) representation, was given by George Gabriel Stokes in 1852. In this representation, four parameters () known as Stokes parameters nowadays, are exploited to describe the polarization state. This representation is useful, since the parameters are simply related to the light intensities measured for different polarizations, according to

(2) | |||||

(3) | |||||

(4) | |||||

(5) |

The Stokes’ parameters can be used to describe partial polarization too. In the case of fully polarized light, the reduced Stokes parameters can be used, instead. We may consider the reduced parameters as the Cartesian coordinates on the Poincaré sphere. The are normalized to . The states of linear polarization, lying on the equator of the Poincaré sphere, have . The two states correspond to the poles and are circularly polarized. In singular optics, these two cases may form two different type of polarization singularities. For the other states, the sign of fixes the polarization helicity; left-handed for and right-handed for . The practical advantage of using the parameters is that they are dimensionless and depend on the ratio among intensities. Light intensities can be easily measured by several photodetectors and can be replaced by average photon counts in the quantum optics experiments. Thus, Stokes’ analysis provide a very useful way to perform the full tomography of the polarization state (1) in both classical and quantum regimes.

The passage of light through optical elements may change its polarization state. If the optical element is fully transparent, 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 in Eq. (1) and corresponds to a continuous path on the Poincaré sphere. In most cases, the optical element can be considered so thin that the polarization state is seen to change abruptly from one point to a different point on the sphere. In this case, it can be shown that the path on the sphere is the geodetic connecting to bhandari97 (). Examples of devices producing a sudden change of the light polarization in passing through are half-wave (HW) and quarter-wave (QW) plates. A sequence of HW and QW can be used to move the polarization state on the whole Poincaré sphere, which corresponds to arbitrary SU(2) transformation applied to the state in Eq. (1). A useful sequence QW-HW-QW-HW (QHQH) to perform arbitrary SU(2) transformation on the light polarization state is presented in Ref karimi10praa ().

So far we considered an optical phase that is uniform in the beam transverse plane. Allowing for a nonuniform distribution of the optical phase between different electric field components gives rise to polarization patterns, like azimuthal and radial ones, where special topologies appears in the transverse plane. The topological structure of the polarization distribution, moreover, remains unchanged while the beam propagates. It is worth noting that most singular polarization patterns in the transverse plane can still be described by polar angles on the Poincaré sphere milione11prl (). The points on the surface of this higher-order Poincaré sphere represent polarized light states where the optical field changes as , where is a positive integer and is the azimuthal angle in the beam transverse plane. As it is well known, light beams with optical field proportional to are vortex beams with topological charge , which carry a definite OAM per photon along their propagation axis. Because the beam is polarized, it carries spin angular momentum (SAM) too, so that the photons are in what may be called a spin-orbit state. Among the spin-orbit states, only a few states can be described by the higher-order Poincaré sphere and, precisely, the states belonging to the spin-orbit SU(2)-subspace spanned by the two base vectors . In this representation the north pole, south pole and equator correspond to the base state , the base state , and linear polarization with rotated topological structure of charge , respectively. Fig. 1b,c show the polarization distribution for and spin-orbit subspaces, respectively. The intensity profile for all points on the higher-order Poincaré sphere has the same doughnut shape. The states along the equator are linearly polarized doughnut beams with topological charge , differing in their orientation only.

## Iii The -plate: patterned liquid crystal cell for generating and manipulating helical beam

The -plate is a liquid crystal cell patterned in specific transverse topology, bearing a well-defined integer or semi-integer topological charge at its center marrucci11jo (); marrucci06prl (); marrucci06apl ().

The cell glass windows are treated so to maintain the liquid crystal molecular director parallel to the walls (planar strong anchoring) with non-uniform orientation angle in the cell transverse plane, where and are the polar coordinates. Our -plates have a singular orientation with topological charge , so that is given by

(6) |

with integer or semi-integer and real . This pattern was obtained with an azo-dye photo-alignment technique slussarenko11opex (). Besides its topological charge , the behavior of the -plate depends on its optical birefringent retardation . Unlike other LC based optical cells stalder96 () used to produce vector vortex beam, the retardation of our -plates can be controlled by temperature control or electric field karimi09apl (); piccirillo10apl (). A simple argument based on Jones matrix shows that the unitary action of the -plate in the state (1) is defined by

(7) |

The -plate is said to be tuned when its optical retardation is . In this case, the first term of Eq. (7) vanishes and the optical field gains a helical wavefront with double of the plate topological charge (). Moreover, the handedness of helical wavefront depends on the helicity of input circular polarization, positive for left-circular and negative for right-circular polarization.

## Iv Experiment

In our experiment, a TEM HeNe laser beam (nm, mW) was spatially cleaned by focusing into a pinhole followed by a truncated lens and polarizer, in order to have a uniform intensity and a homogeneous linear polarization. The beam polarization was then manipulated by a sequence of wave plates as in Ref. karimi10praa () to reach any point on the Poincaré sphere. The beam was then sent into an electrically driven -plate, which changed the beam state into an entangled spin-orbit state as given by Eq. 7. When the voltage on the -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-point polarization tomography, by projecting its polarization into the H, V, A, D, L, R sequence of basis and measuring the corresponding intensity at each pixel of a resolution CCD camera (Sony AS-638CL). Examples of the recorded intensity profiles are shown in Fig. 3. A dedicated 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 averaged over a grid of squares equally distributed over the image area. No other elaboration of the raw data nor best fit with theory was necessary.

We analyzed the beams generated by two different -plates with charges and for two different input polarization states. The -plate optical retardation was optimally tuned for nm by applying an external voltage of a few volt slussarenko11opex (). We considered left-circular () and linear-horizontal () polarized input beams. These states, after passing through the -plate are changed into and , respectively. Figure 3 shows the output intensity patterns for different polarization selections. Figure 3 (a), (b) show the results of point-by-point polarization tomography of the output from -plate for left-circular and horizontal-linear input polarizations, respectively. Figure 3 (c), (d) show the results of point-by-point polarization tomography of the output from -plate for left-circular and horizontal-linear polarizations, respectively.

The case of -plate is particularly interesting, because for a linear horizontal input polarization, it yields to the spin-orbit state represented by the -axes over equator of higher-order Poincaré sphere (Fig. 2 (b)), which corresponds to a radially polarized beam, as shown in Fig. 4 (c). This radial polarization can be changed into the azimuthal polarization (corresponding to the antipodal point on -axes of the higher-order Poincaré 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 diffraction limits. Such beams have a wide range of applications in optical lithography, data storage, material processing, and optical tweezers dorn03 (); wang08 ().

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 implications, because it is at the basis of the so-called quantum contextuality karimi10prab ().

## V Conclusion

We have generated and analyzed a few vector vortex beams created by a patterned liquid crystal cell with topological charge, named -plate. Radial and azimuthal cylindrical beams have been obtained by acting on the polarization of a traditional laser beam sent through a -plate. In this way, fast switching from the radial to the azimuthal polarization can be easily obtained. Finally, we studied in detail the polarization of a few vector beams generated by different -plates and the polarization distribution patterns have been reconstructed by point-by-point Stokes’ tomography over the entire transverse plane.

In this paper, however, we have investigated the cases and . Future work will address other cases, and in particular the negative ones.

## Vi Acknowledgment

We acknowledge the financial support of the FET-Open program within the 7th Framework Programme of the European Commission under Grant No. 255914, Phorbitech.

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