Polarization singularities are superpositions of orbital angular momentum (OAM) states in orthogonal circular polarization basis. The intrinsic OAM of light beams arises due to the helical wavefronts of phase singularities. In phase singularities, circulating phase gradients and, in polarization singularities, circulating Stokes phase gradients are present. At the phase and polarization singularities, undefined quantities are the phase and Stokes phase, respectively. Conversion of circulating phase gradient into circulating Stokes phase gradient reveals the connection between phase (scalar) and polarization (vector) singularities. We demonstrate this by theoretically and experimentally generating polarization singularities using phase singularities. Furthermore, the relation between scalar fields and Stokes fields and the singularities in each of them is discussed. This paper is written as a tutorial-cum-review-type article keeping in mind the beginners and researchers in other areas, yet many of the concepts are given novel explanations by adopting different approaches from the available literature on this subject.

V-points normally do not occur in generic light fields as compared to C-points and L-lines. In structured optical fields, simultaneous existence of C-points, V-points and L-lines can be engineered in lattice forms. But lattices consisting only of V-points have not been realized so far. In this paper we demonstrate creation of lattices of V-point polarization singularities with translational periodicity. These lattice structures are obtained by the interference of four (six) linearly polarized plane waves arranged in symmetric umbrella geometry. The state of polarization of each beam is controlled by an S-waveplate. Since in a periodic lattice of polarization singularities the net charge in a unit cell is zero, the lattices are populated with positive and negative index V-point singularities. All the first order degenerate states of V-point singularities can be realized in the same setup by selective excitation of the S-waveplate.

Inhomogeneous polarization distributions can host polarization singularities such as lemons, monstars, stars, flowers, spider webs, and higher-order C-points in optical beams. Singularities in ellipse fields are characterized by a C-point index and singularities in vector fields by the Poincare–Hopf index. These singularities can be generated by diffractive or interference methods. In this paper, we show that a half-wave plate (HWP) can be used for polarization singularity index sign inversion. The result presented here is powerful, and it shows the importance of a HWP in the study of polarization singularities. The HWP affects the entire state of polarization (SOP) distribution in the index sign inversion process. The concomitant global change of the SOP distribution happens in an orderly fashion to change the polarity of the polarization singularity index. This method of changing the polarity of the polarization singularity index by a HWP holds good both for ellipse fields as well as for vector fields.

The sign rule requires that adjacent singularities on a contour have opposite signs and hence cultivation of lemon only fields poses problem as all lemons have positive index. In this paper we show that the interference of three linearly polarized plane waves can create regions of ellipse and vector fields in 2-dimensions (2D) in which a lemon lattice is interlaced in a V-point lattice. The lemons appear at intensity maxima of the lattice structure while the V-points take care of index conservation by sitting at intensity minima. In the Stokes field S₁₂, lemons and disclinations (V-points) appear as phase vortices of topological charge + 1 and −2 respectively. In the polarization distribution the constant azimuth lines (a-lines) are seen running through lemon and disclination alternatively obeying the sign rule [Opt. Lett. 27, 995 (2002) [CrossRef] ]. We envisage that such polarization lattice structure may lead to novel concept of structured polarization illumination methods in super resolution microscopy.

In this paper, we present a simple and flexible non-interferometric method to generate various polarization singularity lattice fields. The proposed method is based on a double modulation technique that uses a single reflective spatial light modulator to generate different lattice structures consisting of V-point and C-point polarization singularities. The present technique is compact with respect to previous experimental realization techniques. Different structures having star and lemon fields are generated without altering the experimental setup. In addition, the same setup can be used to obtain different types of inhomogeneous fields embedded with isolated polarization singularities even of higher orders. The Stokes polarimetry method has been used to obtain the polarization distributions of generated fields, which are in good agreement with simulated results.