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We report a method to determine the sign as well as the magnitude of the Poincaré Hopf index (PHI) of electromagnetic Gaussian Schell model (EGSM) polarization singular vector beams. This method is based on the analysis of the orientation angle (OA) of polarization. OA distribution is found to form petal like structure around the singularity. The sign and magnitude of PHI are related to the number and the orientation direction of petals around the singularity.

In this work, the far-field properties of non-isotropic partially coherent vector vortex beams (PCVVBs) are investigated both theoretically and experimentally. The term non-isotropic signifies that the spatial correlations between the parallel and orthogonal electric field components are distinguishable. It is found that self-orientation and shaping of intensity profile, correlation-induced polarization and depolarization are highly dependent on both the non-isotropic correlation parameters and Poincaré-Hopf index (PHI) of the beam. The simultaneous depolarization and polarization effects are due to the difference in the input correlation parameters that alter the state of polarization (SOP) and degree of polarization (DOP) distributions. The experimental results are in good agreement with the theoretical predictions. The distinguishability of correlation parameters at the source plane leads to significant changes on its intensity profile, DOP, and SOP distributions on far-field propagation, which may found potential applications in beam shaping, detecting and imaging atmospheric lidar, optical imaging and directional transportation where the self-rotation characteristic of beam plays an important role.

Vector vortex beams play an important role to tailor tightly focused ﬁelds by creating an additional longitudinal component at the focal plane. Till now, mainly focusing properties of fundamental, higher order vector ﬁelds and negative indexed ellipse ﬁelds have been investigated. In this paper we numerically analyze tight focusing behaviour of ellipse ﬁelds embedding C-point singularities of index IC = 12, IC = 1, IC = 32and IC = 2. We show that both sign and absolute value of C-point index (IC) of the ellipse ﬁelds play an important role in tailoring the focal intensity landscapes. For negative index C-points the intensity distribution of the longitudinal components show symmetries that correspond to the number of separatrices present in the polarization distribution. At the focal plane both transverse and longitudinal components of the ellipse ﬁelds are found to be embedded with phase singularities. These ellipse ﬁelds can be useful in advanced microscopy for shaping the focus of light ﬁelds for various applications.

The Young’s double-slit interference of partially coherent polarization singular vector beams (PC-PSVBs) is investigated both theoretically and experimentally. Unlike the case of scalar or linearly polarized light beams, the far-field interference pattern of PC-PSVB’s have spatial dependence both in horizontal and vertical directions. The visibility of these patterns is governed by the input spatial correlation length. The number of minima along the vertical direction (y-axis) depicts the magnitude of Poincaré-Hopf index (PHI) of the incident beam. Our results reveal that the far-field statistical properties namely, intensity pattern, degree of polarization (DOP) distribution and Stokes-parameters distributions of the PC-PSVBs diffracted through a double-slit are strongly influenced by the spatial correlation length of the incident beam. It is observed that the DOP distribution of the diffracted PC-PSVB does not follow the azimuthally symmetric DOP profile of the input PC-PSVB. Interestingly, the DOP distributions contain the information of the magnitude of PHI of the incident PC-PSVB even for a low value of spatial correlation length. However, the maximum value of DOP deteriorates with the decrease in the input spatial correlation length. Furthermore, the Stokes field (S12(ρ,z)) of the diffracted PC-PSVB contains the information of both the magnitude and polarity of the PHI of the input beam. The experimental results are in good agreement with the theoretical predictions, and may find applications in the areas where the statistical properties of a light field are important.

In this Letter, Young's double-slit experiment with vector vortex beams is investigated. We present the results for various Poincaré-Hopf index beams of this class considering all four major types. Polarization associated morphological changes in the far-field interference pattern are studied both theoretically and experimentally. The Fraunhofer pattern consists of lattices of polarization singularities of the generic type, located on a line, in a direction perpendicular to the slit. The number of linear lattices varies as a function of Poincaré-Hopf index η of the beam that is diffracted, and the number of intensity nulls occurring along the vertical line is equal to |η|.

We investigate the statistical properties of partially coherent polarization singular beams embedded with a V-point polarization singularity. An analytical formula for the cross-spectral density matrix is derived for the family of partially coherent polarization singular vector beams (PSVBs) propagating through a paraxial ABCD optical system. It is observed that the far-field intensity profiles and the coherence-induced depolarization effect in partially coherent PSVBs depend on both the input spatial coherence length and the Poincaré-Hopf index (PHI) of the beam. Interestingly, it is found that in this process of coherence degradation, the polarization (Stokes, S12) vortices are preserved. The depolarization is due to an enhanced unpolarized light field that in turn modulates the beam profile, the transverse distribution of the degree of polarization (DOP) and the degree of coherence (DOC). Furthermore, the Gaussian distribution of the DOC evolves into a non-Gaussian profile in the far-field with the number of ring dislocations equal to the magnitude of PHI of the beam. The degeneracy associated with the intensity profile, the Stokes intensity distribution, the DOP, and DOC profiles of these partially coherent PSVBs carrying opposite polarity of PHI are also discussed to complete this study. Subsequently, all of these findings are experimentally verified by generating a family of partially coherent PSVBs with controllable spatial coherence. The modulation of the spatial coherence length in the source plane leads to efficient control of its intensity, the DOC and DOP profiles on propagation, which are of importance in particle trapping, material thermal processing, free-space optical communications, and detection of a phase object.

We have demonstrated, theoretically and experimentally, a method based on Stokes polarimetry to detect the polarization singularity index (PSI) and the state-of-polarization (SOP) distribution of a partially coherent polarization singular vector beam. It is observed that on reducing the spatial correlation in polarization singular vector beams, the information of the phase vortex and polarization vortex is lost, yielding identical intensity profiles tending toward Gaussian distribution. However, the residual correlated field that comprises the polarized part of irradiance distribution still preserves these vortex structures. The PSI of a partially coherent vector beam can be readily obtained from the flower patterned image of the Stokes parameter's distribution. A uniform deterioration in the magnitude of Stokes intensity all across the beam cross section is observed as a signature of reduced two-point correlation. The respective Stokes phase maps are also invariant and can be used to determine the SOP distribution. The present study foresees potential application in free-space optical communication, optical trapping, and imaging, where the coherence properties of the polarization singular vector beams are of considerable importance.

We investigate the peculiar diffraction behavior of the vector fields on passing through a diamond shaped aperture. The higher-order vector fields embedded with V-point singularity are generated using a rectangular path Sagnac interferometer. It is witnessed that the order (magnitude of Poincare-Hopf index) of the vector field singularity can be directly predicted by counting the number of intensity nulls of the diffraction pattern. The method of diffraction and polarization transformations reported here can be used to generate variety of on-axis and off-axis polarization singular beams, beams with multiple polarization singularities, full Poincare’ beams etc.

Stokes phase is the phase difference between orthogonal component states in the decomposition of any polarization state. Phase singularities in the Stokes phase distribution are Stokes singularities of an inhomogeneous polarization distribution. Under circular decomposition, Stokes phase distribution (ϕ12) represents polarization azimuth (γ) distribution and the singularities present in it are polarization singularities. Therefore, the charge of the Stokes vortices depicted as Stokes index σ12 is an important parameter associated with the polarization singularity. The Hybrid order Poincaré sphere (HyOPS)/Higher order Poincaré sphere (HOPS) beams, all having same Stokes index, contain a Stokes singularity at the center of the beam as these beams are constructed by vortex superposition. These beams, being superposition of orthogonal orbital angular momentum (OAM) states in orthogonal spin angular momentum (SAM) states can offer great multiplexing capabilities in communication. In this article, we identify these degenerate Stokes index states and discuss the ways and means of lifting this degeneracy. Otherwise, there are limitations on intensity based detection techniques, where demultiplexing or segregation of different HOPS/HyOPS beams is warranted. The method adduced here uses the diffraction of these beams through an equilateral triangular aperture in combination with polarization transformation as a probe to lift the Stokes index/Stokes phase degeneracy. Successively, the novelty of the detection scheme is discussed in the context of beams with alike polarization distributions where even the technique of Stokes polarimetry fails to predict the OAM and SAM content of the beam.

In optical testing, the well-known peak valley detection ambiguity exhibited by degenerate interference intensity patterns is due to phase. The interplay between phase and polarization is evident in coherence theory. So the theme of intensity degeneracy arising due to polarization is taken up in this article for discussion. Fringes with high contrast (visibility) occur when the interfering beams are in same state of polarization (SOP). But when multiple beams are involved in interference, high contrast fringes are possible even if the SOP of each of the interfering beams is different. We show the superposition of multiple beams in different SOPs form lattice patterns consisting of polarization singularities and the intensity distribution in the interference patterns exhibit high contrast. By changing the SOPs of the individual beams, same intensity distributions can be produced. These intensity patterns are termed as degenerate intensity patterns, but have different polarization distributions. The SOP changes must follow certain rules to achieve degenerate intensity patterns. We also demonstrate intensity degeneracies in Fraunhoffer diffraction patterns of apertures illuminated by beams having polarization singularities. This study therefore illustrates the limitations on intensity based measurements in identifying polarization singularities as these singularities are expected to play a major role in future in diverse areas of optics.

In this Letter, we present a recipe for the generation of full Poincaré beams that contain all Stokes vortices (SVs), namely ϕ12, ϕ23, and ϕ31 vortices. Superposition of two scalar vortex beams with charges l1 and l2 (where |l1|≠|l2|) in orthogonal states of polarization (SOP) generates all three types of SVs, out of which two types of them are generic and always lie in a ring, with the third type at the center of the ring with index value (l2−l1). Thus, generation of hitherto unknown dark SVs is shown. The number of SVs in a ring is 4|l2−l1|. Index sign inversion for all SVs can be achieved by swapping l1 and l2. By changing the orthogonal pairs of SOPs of the interfering beams, the SV at the center of the ring can be changed from one to another type such that the other two types take part in the formation of the ring of generic SVs. We have also deduced the expressions for the location of all the SVs in the beam. Experimental results are presented.

In this article we show that diffraction segregates the polarization singularities according to their handedness. Polarization singularities are superpositions of left and right handed circular polarization vortex states. In the superposition, the component states possess different orbital angular momenta depending on the type of the singularity. A fork grating that can generate different orbital angular momentum (OAM) states in different diffraction orders is shown to segregate right and left handed polarization singularities. A V-point polarization singularity that corresponds to one combination of OAM states incident on the fork grating is found to diffract in such a way that the same OAM combination does not occur in all the nonzero diffraction orders. As a result, each of the diffraction orders will have different polarization singularities. This OAM transfer by the fork grating segregates the right and left handed polarization singularities thereby, making the diffraction helicity dependent.

In singular beams, topological charge is conserved during diffraction. Like scalar field diffraction, in vector field diffraction also, there are conserved quantities. A diffracting V-point disintegrates into a number of C-points of the same polarity in which the polarization singularity index is conserved. In this Letter, we show for the first time, to the best of our knowledge, that apart from the index, the helicity (handedness) is also conserved in V-point diffraction. Since V-point is devoid of any handedness, the helicity conservation entails that there is an equal number of opposite handed C-points in the diffracted field, which are interestingly also found to be orthogonal pairs. Further, coexistence of C-points of opposite handedness in the diffraction demands the presence of L-line, which is also shown. We experimentally demonstrate these by studying the diffraction phenomenon through two different types of apertures.

We report investigations on propagation of converging vector beams containing C-point and V-point polarization singularities through atmospheric turbulence. The C-point singularity is generated by superposition of the 𝑙=0 and 𝑙=1 orbital angular momentum (OAM) states, whereas the V-point singularity is generated by a superposition of the 𝑙=−1 and 𝑙=1 OAM states in orthogonal polarizations. The propagation of these beams through extended atmosphere is modeled by placing random phase screens along a 2 km propagation path. The random phase screens were generated using the FFT method with von Karman spectrum and 𝐶2𝑛=10−14 m−2/3. The quality of intensity profile of the focused vector beams after propagation through turbulence is assessed using the instantaneous signal-to-noise ratio and the on-axis scintillation index measurements. Our simulation results show that although both the C-point and V-point beams perform better than their scalar OAM components, C-point beams are seen to maintain much better beam intensity profile compared to the V-point beams. This observation is explained in terms of the OAM diversity of the individual polarization states and the correlation of their associated speckle patterns. The results presented here are important for engineering laser beams that can maintain a robust intensity profile on propagation through long-range atmospheric turbulence.

Fields containing polarization singularities normally do not host all the Stokes vortices. For example, fields with S12 Stokes vortices (C-points, V -points) in general do not contain S23 Stokes vortices (Poincare vortices). In this paper, we demonstrate for the first time, to the best of our knowledge, synthesis of a structured field that hosts all three Stokes vortices, namely, S12, S23, and so far unexplored S31 Stokes vortices. Only S12 Stokes fields that host C-points and V -points have been experimentally realized by many research groups. Generation of S23 Stokes fields that contain Poincare vortices have been proposed earlier but not experimentally demonstrated so far. There are no reports on the generation of S31 Stokes field containing the third set of Stokes vortices.

We present a method to generate a spatially varying lattice of polarization singularities. The periodicity and orientation of the lattice can be varied spatially by engineering phase and polarization gradients in the interfering beams. A spatial light modulator and an S-wave plate are used to control the phase and polarization gradients, respectively, in the interfering beams. A filter in the Fourier space selects the required spatial frequency components of the interfering beams. Experimentally realized lattices are presented. These spatially varying lattices may find applications in polarization dependent structured illumination, particle sorting, and optical trapping.

In this paper we present a novel experimental method to find the Poincare-Hopf index (η), also called, V-point index of the polarization singular beams by diffraction. We have recently reported the diffraction of V-point singularities through triangular apertures. For a given V-point index, four types of polarization distributions are possible namely type I to IV and they all have similar intensity distribution. It is observed that the diffraction patterns for all these four types of beams have same intensity distribution. This phenomenon is termed as intensity degeneracy. As a result the information about the sign of the Poincare-Hopf index cannot be ascertained. This is similar to the peak valley degeneracy exhibited by interferograms. That means, in interferograms convex and concave surfaces, for example produce similar fringe patterns. Optical vortices (phase singularities) are known to lift this degeneracy. Here the degeneracy itself is due to the polarization singularity. In this presentation, we show a method to find the polarization singularity index magnitude and the sign by diffraction. The diffraction patterns are subjected to polarization transformations using appropriate elements. The resultant diffraction pattern now clearly distinguishes between the positive and negative V-point index and its magnitude. Experimentally higher order V-points are generated using an interferometer. Experimental results supported by theoretical simulations will be presented.

Beams carrying C-point polarization singularity (lemon and star) are experimentally shown to maintain robust intensity profile on passing through a random medium compared to beams carrying V-points polarization singularity (radially and azimuthally polarized).

We report experiments on propagation of scalar and vector optical beams through random phase screens mimicking turbulence and show that the intensity profile of the beam containing a C-point polarization singularity shows maximally robust behavior. This observation is explained in terms of the polarization and orbital angular momentum (OAM) diversity in the beam. The l=0 and l=1 OAM states whose vector superposition leads to the C-point singularity are seen to produce complementary speckle intensity patterns with significant negative correlation on propagation through a random phase screen. This unique property of C-point singularity makes it superior to other inhomogeneous polarization states as demonstrated in our simulations and experiments. The results provide an important generic guideline for designing beams that can maintain superior beam intensity profile on passing through random phase fluctuations and are expected to have a number of applications

We report experiments on propagation of scalar and vector optical beams through random phase screens mimicking turbulence and show that the intensity profile of the beam containing a C-point polarization singularity shows maximally robust behavior. This observation is explained in terms of the polarization and orbital angular momentum (OAM) diversity in the beam. The $l = 0$ and $l = 1$ OAM states whose vector combination leads to the C-point singularity are seen to produce complementary speckle intensity patterns with significant negative correlation on propagation through a random phase screen. This unique property of C-point singularity makes it superior to other inhomogeneous polarization states as demonstrated in our experiments. The results provide an important generic guideline for designing beams that can maintain optimally robust beam intensity profile on passing through random phase fluctuations and are expected to have a number of applications.

Cylindrical vector beams with azimuthal and radial polarization distributions are studied for singularities. It is shown experimentally that these beams have screw dislocation as well as edge dislocation at the same time. The relation between phase and polarization of light beam is the key to understand this fact. We envisage that this has potential application in phase synthesis using polarization engineering. Further, the polarization singularities in these inhomogeneously polarized beams are examined by measuring Stokes parameters across the cross-section of these beams.

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 S 12 , 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 [24]. We envisage that such polarization lattice structure may lead to novel concept of structured polarization illumination methods in super resolution microscopy.

V-points are polarization singularities in spatially varying linearly polarized optical fields and are characterized by the Poincare-Hopf index η. Each V-point singularity is a superposition of two oppositely signed orbital angular momentum states in two orthogonal spin angular momentum states. Hence, a V-point singularity has zero net angular momentum. V-points with given |η| have the same (amplitude) intensity distribution but have four degenerate polarization distributions. Each of these four degenerate states also produce identical diffraction patterns. Hence to distinguish these degenerate states experimentally, we present in this Letter a method involving a combination of polarization transformation and diffraction. This method also shows the possibility of using polarization singularities in place of phase singularities in optical communication and quantum information processing.

Abstract: 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.
⃝c 2017 Optical Society of America

In this paper we present experimental studies on diffraction of V-point singularities through equilateral and isosceles right triangular apertures. When V-point index, also called Poincare-Hopf index (η), of the optical field is +1, the diffraction disintegrates it into two monstars/lemons. When V-point index η is -1, diffraction produces two stars. The diffraction pattern, unlike phase singularity, is insensitive to polarity of the polarization singularity and the intensity pattern remains invariant. Higher order V-point singularities are generated using Sagnac interferometer and it is observed that the diffraction disintegrates them into lower order C-points.

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 chang- ing the polarity of the polarization singularity index by a HWP holds good both for ellipse fields as well as for vector fields. © 2017 Optical Society of America

The effect on the Stokes parameters of a Gaussian Schell model beam on propagation in free space is studied experimentally and results are matched with the theory [X. H. Zhao, et al. Opt. Express 17, 17888 (2009)] that in general the degree of polarization of a Gaussian Schell model beam doesn't change on propagation if the three spectral correlation widths δ<sub>xx</sub>, δ<sub>yy</sub>, δ<sub>xy</sub> are equal and the beam width parameters σ<sub>x</sub>=σ<sub>y</sub>. It is experimentally shown that all the four Stokes parameters at the center of the beam decrease on propagation while the magnitudes of the normalized Stokes parameters and the spectral degree of polarization at the center of the beam remain constant for different propagation distances.

In this paper, generation of the Dammann grating by using the spatial light modulator (SLM) is discussed, and quality of the grating is examined by using the polarization interferometer and Fourier fringe analysis technique.

Controlled generation of the periodic polarization structure by interferometer is investigated. Periodic polarization structure is generated by interference of two spatially separated orthogonally polarized beams. Advantage of this technique is discussed and results are presented.

Abstract: We describe a laser beam engineered to carry l = 0 and l = 1 orbital angular momentum (OAM) states in orthogonal polarizations. It is observed that on collinear transmission through a random phase screen, the far field diffraction intensity patterns for the individual polarization states are complementary with significant negative correlation. As a result the combined intensity profile for the beam remains robust against time varying phase fluctuations. In our simulation and experiment the SNR for the central lobe of the combined far-field diffraction pattern defined as mean divided by the standard deviation of intensity values shows significant improvement over that for individual polarizations. The concept of polarization and OAM diversity as demonstrated here can be considered valuable for robust laser beam engineering without the requirement of any active real-time correction methods.

Using polarization as an additional parameter apart from amplitude and phase in spatial filtering experiments offers additional advantages and possibilities. An S-waveplate that can convert a linearly polarized light into radially or azimuthally polarized light can also be used for isotropic edge enhancement. For anisotropic edge enhancement, introduction of a polarizer at the output was recommended and edge selection was done by orientation of the polarizer. But the full potential of the S-waveplate as a spatial filter has not been exploited so far. Unlike the standard amplitude and phase-based Fourier filters, which are independent to the state of polarization of the illuminating beam, the S-waveplate acts in a different way depending on the state of polarization. The edge selection does not need to be carried out by changing the orientation of the polarizer. With a fixed polarizer at the output, we show that either isotropic or anisotropic edge enhancement in any desired orientation can be performed by operating the same spatial filter setup in different illuminating polarization states.

Interferometric methods for array generation offer various advantages over the diffractive methods, but the efficiency of such interferometric methods so far reported are very low, making its applicability very low. In view of this, an experimental setup using polarizing elements for an eight-fold increase in the efficiency of an interferometric array generator—the Michelson interferometers in tandem setup (P. Senthilkumaran and R. S. Sirohi, Optics Communications, 105 (1994) 158)—is presented in this paper.

We present an experimental method based on a modified multiple beam interference approach to generate an optical vortex array arranged in a spatially varying lattice. This method involves two steps which are: numerical synthesis of a consistent phase mask by using two-dimensional integrated phase gradient calculations and experimental implementation of produced phase mask by utilizing a phase only spatial light modulator in an optical 4f Fourier filtering setup. This method enables an independent variation of the orientation and period of the vortex lattice. As working examples, we provide the experimental demonstration of various spatially variant optical vortex lattices. We further confirm the existence of optical vortices by formation of fork fringes. Such lattices may find applications in size dependent trapping, sorting, manipulation and photonic crystals.