# Extended Stokes parameters for cylindrically polarized beams

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## Abstract

The extended Stokes parameters for arbitrary cylindrically polarized beams are newly introduced to evaluate their quality. A set of the parameters, expressing a cylindrically polarized state, permits the definition of the degree of polarization that indicates the purity of the spatial symmetry of polarization of a light beam. In addition, the Pancharatnam–Berry phase related to the cylindrically polarized states is described by the new Stokes parameters.

### Keywords

Extended Stokes parameters Cylindrically polarized beam Optical vortex Poincaré sphere Pancharatnam–Berry phaseCylindrically polarized (CP) beams [1] are laser mode beams whose polarization distributions have symmetry of rotation about the beam axis. These beams, including axisymmetrically polarized beams (such as a radially polarized beam) and anti-vortex beams, are promising mode for vector-vortex-mode-division multiplexing in optical communications [2, 3]. They have been consequently generated in a lot of studies [4, 5] so far. For establishing vector-vortex-mode-division multiplexing technology, quantitative evaluation of the CP beams is significantly important. Earlier researches have introduced the high-order [6, 7] and the hybrid [8] Stokes parameters to characterize the CP beams. These parameters, however, cannot be responsible for the local deviation in the symmetry of CP states, which often appears in experimental measurements. To overcome this issue, we have introduced the extended Stokes parameters (ESPs) for the polarization states with \(C_\infty\) symmetry in consideration of the local deviation of the symmetry [9]; on the other hand, the CP states other than with \(C_\infty\) symmetry have not yet been covered.

In the present paper, we extend the coverage of the ESPs so that they describe the CP beams without \(C_\infty\) symmetry. As a natural extension of the degree of polarization (DOP) from the ordinary Stokes parameters, we define the spatial DOP from the introduced ESPs, which represents a measure in the purity of the symmetry of a CP beam. Furthermore, we discuss the Pancharatnam–Berry geometric phase [10] for the new ESPs.

Notation of various Stokes parameters (\(i=0-3\))

\(s_i^0(r,\phi ,z)\) | The ordinary Stokes parameters at \((r,\phi ,z)\) |

\(s_i^l(r,\phi ,z)\) | The Stokes parameters (SPs) for CP beams at \((r,\phi ,z)\) |

\(S^\mathrm {E}_{i,l}(z)\) | The extended Stokes parameters (ESPs) |

Comparison of DOP

DOP \(\mathcal {P}\) | The purity of polarization with respect to time |

DOP from ESPs \(\mathcal {P}_l\) | The purity of polarization with respect to time and space |

DOP-SD from ESPs \(\mathcal {P}_l^\mathrm {space}\) | The purity of polarization with respect to space |

DOP from higher order [6, 7] or hybrid [8] Stokes parameters | Undefinable |

Our ESPs define the spatially averaged DOP that takes into account of the purity of the cylindrical symmetry, whereas the high-order [6, 7] and the hybrid [8] Stokes parameters are for the perfectly symmetrical polarization states. In this part, we describe the coherency matrix consisting of time and space averages of electric field components. This coherency matrix gives the new DOP from the ESPs in the same way that the ordinary coherency matrix provides the definition of DOP \(\mathcal {P}\) [12, 13].

The DOP defined from the ESPs is an important value for both generating and using CP modes, providing the purity or the quality factor of the polarization distribution.

The Pancharatnam–Berry phase [10, 16, 17] is an additional phase factor when a quantum (such as a photon and an electron) traveled along a closed loop in a state space, which is first discussed by Pancharatnam [10]. Reference 18 has experimentally shown that the Pancharatnam–Berry phase shift occurs in optical systems using waveplates and polarizers. A light beam undergoes a phase shift which is half of the solid angle \(\Omega\) subtended to a closed path on the Poincaré sphere [10, 18].

On the assumption that the electric field is monochromatic as \({\mathbf E}(r,\phi ,z,t) = \tilde{\mathbf E}(r,\phi ,z) \exp (\mathrm{i}(kz-\omega t))\), where \(\omega\) and \(k\) are respectively the angular frequency and the propagation constant at \(\omega\), the DOP \(\mathcal {P}_{l_\mathrm {WP}}\) is conserved through the propagation of \(z_\mathrm {in}\le z\le z_\mathrm {out}\) for the light of which \(\mathcal {P}_{l_\mathrm {WP}}\) is 1 at \(z=z_\mathrm {in}\).

This means that the symmetry of the polarization distribution is conserved and the point \((\tilde{S}^{\mathrm {E}}_{1,l},\tilde{S}^{\mathrm {E}}_{2,l},\tilde{S}^{\mathrm {E}}_{3,l})\) draws a trajectory on the surface.

Figure 2 shows a typical example of the Pancharatnam–Berry phase for CP beams. An \(s\!=\!+\hbar , l\!=\!+\hbar\) optical vortex beam [20] passes through two \(q(=\!-1)\) half-wave plates; the one for \(q\!=\!-1, \alpha _0\!=\!\alpha _1\), and the other for \(q\!=\!-1, \alpha _0\!=\!\alpha _2\) (Fig. 2(a); \(q\) and \(\alpha _0\) are defined in ref. 19). Conserving the \(C_2\) symmetry of the polarization distribution, the beam experiences the Pancharatnam–Berry phase \(\phi _\mathrm {PBP}^{l_\mathrm {WP}}=2(\alpha _1-\alpha _2)\), which corresponds to the half of the solid angle subtended to the trajectory of the beam on the Poincaré sphere for \(l=-1\) ESPs (Fig. 2(b)).

We introduced the ESPs for arbitrary CP beams with \(C_{|l-1|}\) symmetry in addition to the previous report for those with \(C_\infty\) symmetry. Using the introduced parameters, we defined the spatial DOP and discussed the Pancharatnam–Berry phase in the analogy to the conventional Stokes parameters.

The ESPs are of use for evaluating the properties of CP laser beams. While spatial polarization properties were so far qualitatively estimated by monitoring the intensity distribution after passing through a polarizer, measuring our ESPs and thereby obtaining the spatial DOP, which are related to the purity of the spatial symmetry of the CP beams, give the qualitative information of the polarization distribution. Our ESPs and DOP are crucial for applications such as laser processing [21] and spectroscopy [22] using CP modes.

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