## 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.

© 2017 Optical Society of America

## 1. Introduction

Structured optical fields with lattices of intensity [1] coherence [2, 3], phase singularities [4–7] and polarization singularities [8–14] are being studied in recent years. Phase singularities emerge as salient features in interference of three or more non-coplanar plane waves. Polarization singularities on the other hand appear only when the state of polarization (SOP) of interfering beams is modulated judiciously. In the generation of phase singularity lattices by interference, the plane waves themselves do not possess any singularity but their superposition results in the formation of singularity lattices. In this process the charge neutrality is maintained by the formation of dipoles. Similarly in polarization singularity lattices, lemons (monstars) and V-points are accompanied by either stars or V-points of opposite polarity to conserve the net charge. Surprisingly there is no method reported so far to realize lattices consisting of only V-points.

Past few years have seen tremendous increase in research activities involving polarization singularities. They are in general discussed in terms of singularities of a polarization ellipse [15,16]. Circular polarization states have ambiguity in orientation of polarization ellipse, whereas linear polarization states have ambiguity in handedness. C-points and V-points are isolated points in ellipse and vector fields at which ${\oint}_{l}\nabla \gamma .dl\ne 0$ in a closed contour enclosing either of these singular points, where *γ* represents azimuth. V-point singularities are intensity nulls at which all the parameters defining the SOP vanish. They are characterized by Poincaré-Hopf index denoted by *η* [17, 18]. Both radially polarized (RP) and azimuthally polarized (AP) vector beams have a *η* = +1 V-point singularity. The usefulness of polarization singularities have been demonstrated amply for vector field singularities and not for ellipse field singularities. Hence we are motivated to create such lattice fields. The current and earlier reported lattices by us [10, 11] indicate the Wyckoff positions which show that L-lines lie on lines of mirror symmetry, C-points at centers of rotational symmetry and V-points where both occur [19]. These lattices may find applications in structured illumination microscopy in future.

In this paper we demonstrate the generation of V-point lattices by the superposition of uniform amplitude linearly polarized plane waves. Section 2 presents the formulation of superposition of *n* plane waves to generate lattices and a brief methodology to analyze polarization singularities. The plane of polarization of each of these interfering beams is modulated using an S-waveplate to yield V-point lattices. In sections 3 and 4 we explain the generation of lattices consisting of only V-points with translational periodicity that can occur in the interference of symmetrically arranged four and six plane waves respectively. Since the net charge in an unit cell is zero, the polarization singularity lattices are populated with positive and negative index V-points in accordance with the index theorem. Lattice structures obtained by interference of four beams have 4-fold rotation symmetry and that obtained by six beams possess 6-fold rotation symmetry. Concluding remarks are given in section 5.

## 2. Superposition of plane waves

Consider the superposition of even number of plane waves that are linearly polarized. These plane waves are arranged in a symmetric umbrella geometry as shown in [10]. The plane of polarization of each of these beams is arranged in an orderly fashion. In the k-space these *n* plane waves are represented by *n* equidistant points on a projected ring in the transverse plane as depicted by black dots in Fig. 1. Figures 1(a)–1(c) correspond to *n* = 4 plane waves whereas, Figs. 1(d)–1(f) refer to *n* = 6 plane waves. The propagation vector ${\overrightarrow{k}}_{m}$ and the polarization vector ${\widehat{n}}_{m}$ of *m ^{th}* (where

*m*= 1, 2…

*n*) beam can be written as

*k*= (2

_{mx}*π*sin

*θ*

_{m}cos

*ζ*)/

_{m}*λ*,

*k*= (2

_{my}*π*sin

*θ*sin

_{m}*ζ*)/

_{m}*λ*,

*k*= (2

_{mz}*π*cos

*θ*)/

_{m}*λ*. The angle subtended by ${\overrightarrow{k}}_{m}$ with

*k*axis is

_{z}*θ*and

_{m}*ζ*is the angle between projection of ${\overrightarrow{k}}_{m}$ vector onto transverse plane and

_{m}*k*axis.

_{x}*α*is the angle subtended by the plane of polarization of

*m*beam and ${\widehat{\rho}}_{m}$. The polarization vector ${\widehat{n}}_{m}$ of

^{th}*m*beam is radial for

^{th}*α*= 0° and is azimuthal for

*α*= 90° respectively. The plane of polarization of each beam is marked by red arrows in Fig. 1. SOPs are radially arranged in Figs. 1(a) and 1(d), whereas these are azimuthally arranged in Figs. 1(c) and 1(f). Figures 1(b) and 1(e) show SOP arrangement of interfering beams for

*α*= 45°, which can be obtained by linear superposition of radial and azimuthal SOPs. The complex field of the

*m*plane wave can be written as

^{th}*E*

_{0}is a constant. Corresponding spatial frequencies of the

*m*plane wave are given by

^{th}*f*=

_{mx}*k*/2

_{mx}*π*and

*f*=

_{my}*k*/2

_{my}*π*. The resultant field distribution in the overlap region is

*E*and

_{X}*E*are orthogonal and transverse components of resultant field

_{Y}*E*. These components are themselves complex and are given by ${E}_{X}={\displaystyle {\sum}_{m=1}^{n}\left\{{E}_{0}({\widehat{n}}_{m}.\widehat{x})exp\left\{i2\pi ({f}_{mx}x+{f}_{my}y)\right\}\right\}}$ and ${E}_{Y}={\displaystyle {\sum}_{m=1}^{n}\left\{{E}_{0}({\widehat{n}}_{m}.\widehat{y})exp\left\{i2\pi ({f}_{mx}x+{f}_{my}y)\right\}\right\}}$ in the rectangular coordinates. The resultant field

_{R}*E*has spatially varying amplitude and polarization distributions. Transverse intensity distribution corresponding to

_{R}*E*is $I(x,y)={I}_{x}+{I}_{y}={E}_{X}{E}_{X}^{*}+{E}_{Y}{E}_{Y}^{*}$.

_{R}Detailed analysis of polarization distributions can be carried out with the help of normalized Stokes parameters namely *S*_{0}, *S*_{1}, *S*_{2}, and *S*_{3}. Since the resultant intensity *I*(*x*, *y*) is a function of spatial coordinates, the calculated Stokes parameters also have spatial variations. For completely polarized light, C-points are circular polarization points at which *S*_{3} = ±*S*_{0} and *S*_{1} = *S*_{2} = 0. On the other hand, V-points are intensity nulls at which *S*_{1} = *S*_{2} = *S*_{3} = 0. A complex Stokes field *S*_{12} = *S*_{1} + *iS*_{2} = *A*_{12} exp(*iϕ*_{12}) constructed out of otherwise real Stokes parameters *S*_{1} and *S*_{2} is an effective tool to probe polarization singularities in any field distributions. Singularities of *S*_{12} Stokes field are characterized by Stokes singularity index *σ*_{12} which is related to the accumulated Stokes phase Δ*ϕ*_{12} around the singular point, as *σ*_{12} = Δ*ϕ*_{12}/2*π*, where *ϕ*_{12} is Stokes phase. C-point and V-point polarization singularities can be identified using *ϕ*_{12} map. Alternatively, C-points and V-points can be located by the zero crossings of *S*_{1} and *S*_{2}.

## 3. Four beam interference

Consider the interference of four non-coplanar axially symmetric plane waves in which *ζ _{m}* =

*mπ*/2 such that

*ζ*−

_{m}*ζ*

_{m}_{−1}=

*π*/2. The angle

*θ*is same for all the four interfering plane waves and is equal to 0.4°. The SOPs of the four beams are as per Fig. 1(a). The simulated resultant intensity pattern in the overlap region is shown in Fig. 2(a). Stokes Phase and phase contour maps are given in Figs. 2(b) and 2(c) respectively. It is interesting to note that all the singular points in the generated lattice have either

_{m}*σ*

_{12}= +2 or

*σ*

_{12}= −2. Also, equal number of phase contours terminate at each singular point in the lattice structure. The resultant polarization distribution is shown in Fig. 2(d). As the input SOPs (of the interfering plane waves) are changed gradually from radial to azimuthal, the resulting SOP distributions in the lattice also undergo radial to azimuthal variations without affecting the Poincaré-Hopf index of the V-points. The intensity distribution, V-point singularity locations and their signs do not change. The SOP distributions of V-point lattice shown in Figs. 2(e) and 2(f) correspond to that of Figs. 1(b) and 1(c) for the four plane waves respectively. According to sign rule [20], adjacent V-points alternate sign on a constant

*ϕ*

_{12}contour. This can be seen from Fig. 2(c) where positive and negative index V-points are indicated by black and orange dots respectively. Note that each positive index V-point is surrounded by four negative index V-points as immediate neighbours. The generated lattice structures have translational and 4- fold rotational symmetry. Figures 2(g)–2(i) show experimentally obtained polarization distributions for different

*α*values. The experimental setup used to obtain these results is similar to the one given in [10]. But here the spatial light modulator (SLM) is used as a hologram to reconstruct four (six) beams. This is because, the resultant phase variation of four (six) beam interference is binary and do not have any features due to the symmetric arrangement of interfering beams. Such phase patterns are not suitable for SLM to work efficiently as reflective diffractive optical element. S-waveplate in the setup modulates the SOP of each beam and Stokes parameters are captured by using a high resolution Stokes camera.

## 4. Six beam interference

Consider the interference of six non-coplanar axially symmetric plane waves in which *ζ _{m}* =

*mπ*/3 and

*ζ*−

_{m}*ζ*

_{m}_{−1}=

*π*/3. The SOPs of the six beams are as per Fig. 1(d). The resultant intensity pattern for the six beam interference is shown in Fig. 3(a). The corresponding Stokes phase

*ϕ*

_{12}and the phase contour maps are shown in Figs. 3(b) and 3(c) respectively. At the center of each of the hexagonally shaped bright rings, the intensity nulls host a V-point of same charge. As the SOPs of the six plane waves are changed from radial to azimuthal, the polarization distributions in the interference pattern change as shown in Figs. 3(d)–3(f). To invert the sign of all V-points in the lattice simultaneously, a half waveplate (HWP) is inserted in the beam [21] and SOP distributions thus obtained are shown in Figs. 3(g)–3(i). It can be seen from the Fig. 3(c) that a positive (negative) index V-point alternates its sign on constant

*ϕ*

_{12}contours. Lattice structures generated by the interference of six plane waves possess translational and 6-fold rotational symmetry.

Figure 4(a) shows the experimentally obtained intensity distribution for six beam interference. Figures 4(b)–4(d) show lattice structures with a positive V-point at the center of each hexagon, whereas Figs. 4(e) and 4(f) show lattice structures with a negative V-point at the center of each hexagon. Intensity remains invariant for any change in *α* and sign of *η*. V-point singularity locations and their signs also remain invariant for change of *α*. Simulations shown in Figs. 2(a)–2(f) and Figs. 3(a)–3(i) show compliance with theory. But SOP distributions shown in Figs. 2(f) and 3(f) are unstable and revert to spirals in experiments as shown in Figs. 2(i) and 4(d). In Eq. (4), inclusion of *E _{z}* and extending the analysis from 2D to 3D may probably reveal structures such as Möbius strips [22].

## 5. Conclusion

We have shown that interference of four (six) linearly polarized plane waves arranged in predefined fashion results in generation of lattice exclusively embedded with V-points with translational periodicity. As a consequence of periodicity and index theorem, the generated lattices are populated with both positive and negative V-point singularities. The intensity distribution, V-point singularity locations and their signs remains invariant when the SOPs of interfering beams are derived from subset of SOPs having same Poincaré-Hopf index. The lattices generated by interference of four and six plane waves are found to be 4- fold and 6- fold rotationally symmetric. All the first order degenerate states of V-point polarization singularities can be achieved from the same setup by judicious tuning of the S-waveplate.

## Funding

Department of Science and Technology, India, Grant No. SR/S2/LOP-22/2013.

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