## Abstract

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 [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.

© 2016 Optical Society of America

## 1. Introduction

Superposition of three or more scalar waves can generate array of optical vortices or phase dislocation points [1–5]. Optical vortex has helical wavefront which ramps spirally around a phase singular point where the amplitude is zero and phase is undefined. Similarly polarization singularity lattice structures, in particular lemon lattice structure can be generated by the interference of three non-coplanar vector beams, each with homogeneous polarization distribution oriented in a designed fashion. Lemons in the lattice correspond to lattice of polarization singular points (C-points) [6–9].

In recent times more attention has been given to beams in which the polarization states are spatially varying across the beam cross section. Cylindrical vector beams such as radially and azimuthally polarized beams are of this nature [10–13]. Such beams are embedded with polarization singularities [14–18] where at least one parameter that defines state of polarization is undefined. C-points (circular polarization), V-points (intensity nulls also called disclinations) and L-lines (linear polarization) are some polarization singularities. In an ellipse field, at C-points the orientation angle of the polarization ellipse is undefined and at V-points in a vector field the orientation angle of the linear polarization is undefined. On L-lines the handedness is not defined. C-points are classified into star, lemon and monstar on the basis of C-point index (${I}_{c}$). For a lower order star it is $-1/2$ and for a lemon and a monstar it is$+1/2$ [19–23]. V-point singularities are characterized by Poincare-Hopf indices.

The C-point and V-point polarization singularities can have positive and negative index values. The sign rule [24] requires that adjacent singularities on a contour have opposite signs. So V-points having negative index values appear along with lemons in the field to maintain the index neutrality. In this paper we demonstrate the creation of lemon fields interlaced in V-point lattice, by interference of three linearly polarized non-coplanar plane waves. The resultant interference pattern consists of regions of ellipse and vector fields. Such lattice structures are envisaged to have applications in optical lattices, optical metrology, photonic crystals and can provide polarization structured illumination in microscopy.

## 2. Three beam interference

We have taken three axially equidistant non-coplanar linearly polarized plane beams whose electric fields are oriented in a predesigned fashion for the formation of lemon polarization lattice structure. The propagation vectors of the three beams have same *z*-component as shown in Fig. 1. The wave vectors corresponding to three interfering beams can be expressed as,

*m*is an index representing the beam number and${k}_{0}=2\pi /\lambda $. As shown in Fig. 1, ${\theta}_{m}$is the angle subtended by these three non-coplanar propagation vectors from the ${k}_{z}$-axis and ${\xi}_{m}$ is the angle between the projection of the propagation vector onto the transverse plane (i.e. ${k}_{x}\text{\hspace{0.17em}}{k}_{y}$plane) and the ${k}_{x}$axis. The value of ${\xi}_{m}$and ${\theta}_{m}$corresponding to these three non-coplanar interfering beams are chosen as$\{{\xi}_{1}=\pi /2\text{\hspace{0.17em}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\xi}_{2}=7\pi /6\text{\hspace{0.17em}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\xi}_{3}=11\pi /6\}$and${\theta}_{1}={\theta}_{2}={\theta}_{3}={0.3}^{0}$respectively.

The electric field vector of the three plane waves are given by

*x*-component of the

*m*beam and so on. The optic axis of the system is along

^{th}*z*-direction. The wave vectors we consider here are limited to the paraxial regime (${\theta}_{m}$is small). Since longitudinal component ${E}_{mz}$of the field vector is negligible as compared to the transverse components ${E}_{mx}$and${E}_{my}$, it can be dropped. The resultant of three-beam interference is${\overrightarrow{E}}_{R}={\displaystyle \sum _{m=1}^{3}{\overrightarrow{E}}_{m}}$and the corresponding intensity is$I={\displaystyle \sum _{p=1}^{3}{\displaystyle \sum _{q=1}^{3}{\overrightarrow{E}}_{p}}}\cdot conj({\overrightarrow{E}}_{q})$.

The *x* and *y* components of the resultant field are complex quantities. Figures 2(a) and 2(b) depict the intensity patterns corresponding to *x* and *y* component of the resultant field respectively. Total transverse intensity along with zero contour lines of real (green and white) and imaginary parts (blue and black) of the *x* and *y* component of the resultant field are shown in Fig. 2(c). In all these figures the numbers along both the axes correspond to pixel numbers. All these four contour lines intersect at the intensity nulls (V-points) which are arranged in a hexagonal lattice.

## 3. Polarization singularities and complex Stokes field

Polarization singularities associated with any polarized field can be completely described by the normalized Stokes parameters [25, 26], which are expressed as

Polarization singularities embedded in an ellipse field and in a vector field can be understood using complex Stokes fields, which are constructed from the Stokes parameters. The complex Stokes fields are expressed as${S}_{pq}(p,q=1,2,3)={A}_{pq}\text{\hspace{0.17em}}{e}^{i{\phi}_{pq}}={S}_{p}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}i{S}_{q}$, where$i=\sqrt{-1}$. C-points appear as phase vortices of topological charge$\pm 1$in${S}_{12}={S}_{1}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}i{S}_{2}$Stokes field. The intensity distribution of the${S}_{12}$Stokes field is given by${I}_{s}={S}_{12}\cdot {S}_{12}^{*}$and is shown in Fig. 3(a). The zero contour lines corresponding to real (white) and imaginary parts (black) of the Stokes field are drawn on Fig. 3(a). These zero crossings are the phase vortices of${S}_{12}$where C-points and disclinations are located. Figure 3(b) shows the phase variation of Stokes field, where vortices with topological charge$+1$and$-2$correspond to C-points (lemons) and disclinations respectively. The C-point index (${I}_{c}$) and the topological charge ($l$) of the phase vortex in the${S}_{12}$field are connected by the following relation

Phase contour lines of the Stokes field${S}_{12}$corresponding to different phase values are shown in Fig. 3(c). It can be seen that the number of contour lines terminating on a V-point (disclination) is double than that terminate on a lemon. These phase contour lines are the *a*-lines in the polarization distributions. According to the sign rule [24] a constant azimuth line (*a*-line) should pass alternatively through charge$+1$and$-1$vortices in a Stokes field. Here the *a*-lines pass through lemons and disclinations alternatively. By comparing Figs. 2(c) and 3(a) one can easily notice that lemons appear at intensity maxima in the three beam intensity pattern. Polarization distribution of the resultant field is given in Figs. 3(d) and 3(e). In Fig. 3(d) red dots and black dots correspond to C-points (lemons) and disclinations respectively. The regions of left handed (black) and right handed (blue) polarization distributions are separated by L-lines, which are marked by green color.

One of the unknown referees observed that “more general than the sign rule (which is a local rule) is the global theorem that the net sum of the Poincare-Hopf indices of all singularities on a manifold (surface) equals the Euler characteristic$\chi $of the surface. In the present case with many ($+1/2$) lemons and ($-1$) V-points, the theorem requires that there be essentially twice as many lemons as V-points.” From Stokes phase distribution of Fig. 3(b), one can observe that for every V-point there are two C-points on either side of it. The unknown referee further observed that “the net index of a unit cell in a periodic array must be zero. For the authors’ lattice one can see this result by looking, say, at a single triangle in Fig. 3(d). At the center of the triangle there is a ($+1/2$) lemon, whereas at each of the three vertices of the triangle there is a ($-1$) V-point. But each V-point is shared by six triangles, so that for a given triangle each vertex contributes ($-1/6$). Adding together the contributions from the three vertices yields a net V-point contribution of ($-1/2$) which exactly cancels the ($+1/2$) lemon contribution.”

## 4. Experiment and results

The experimental setup used to generate lemon lattice structure is shown in Fig. 4(a). A vertically polarized He-Ne laser of wavelength$632.8$nm is spatially filtered and collimated by spatial filter assembly (SF) and lens L_{1} respectively. The collimated light is allowed to pass through a cube beam splitter (BS) and the reflected light coming out of the beam splitter falls on a programmable reflective phase only spatial light modulator (SLM) (Holoeye LC-R2500, Germany). The SLM is used to modify the wavefronts of a beam falling on it. The reflected wave from the SLM carries the phase variation corresponding to the three beam interference of scalar waves. The computed phase variation given to reflective phase SLM is shown in Fig. 4(b). The modified beam is focused to the Fourier plane (FF plane) of the lens L_{2} where undesired Fourier components are filtered out using an amplitude filter shown in Fig. 4(a) inset. The filtered spots are then passed through a polarization converter (PC) (S-waveplate-RPC-632-08-387 Radial-Azimuthal polarization converter, Altechna, Lithuania) placed behind the filter plane, which modifies the polarization state of these spots in azimuthal fashion. A half wave plate (HWP), whose fast axis is shown as red horizontal line, placed after the PC changes the orientation of the plane of polarization of the three beams and the collimating lens L_{3} produces three overlapping plane waves. The SOPs of the three beams after PC (left bottom inset) and after HWP (right bottom inset) are shown in Fig. 4(a).

The intensity pattern and the Stokes parameters are captured by using a high resolution Stokes camera (SC) (Salsa full Stokes polarization imaging camera, 1040x1040 pixels, Bossa Nova, USA). Experimentally recorded three beam intensity pattern and polarization distributions are given in Figs. 5(a)-5(c) respectively, where the black and blue colors correspond to left and right handed polarization ellipses. The polarization distribution corresponding to the region marked by red square in the Fig. 5(b) is expanded and shown in Fig. 5(c).

## 5. Summary

In conclusion, we have proposed and experimentally demonstrated a technique for generating lemon polarization singularity lattice structure by a designed superposition of three linearly polarized plane beams. Lemons and disclinations correspond to phase vortices of topological charge$+1$and$-2$respectively in the Stokes field${S}_{12}$. The fascinating thing is that lemons correspond to intensity maxima in the interference pattern. We envisage that such polarization lattice structure may lead to novel concept of structured polarization illumination methods in super resolution microscopy.

## Funding

Department of Science and Technology, India (SR/S2/LOP-22/2013).

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