## Abstract

Polarization singularity lattices exhibit richer features and varieties than their scalar counterparts, namely phase vortex lattices. Lattices consisting of only *C*-points or only *V*-points of different Stokes indices are possible by phase and polarization engineering. In this article we show the generation of one generic and three non-generic lattices—all abiding the enlarged sign principle in six-beam interference. Two of them are vector fields and two of them are ellipse fields. In the vector fields, one lattice consists of *V*-points of the same magnitude and the other consists of *V*-points of different magnitude of the Poincare-Hopf index. Similarly in the two ellipse fields, the same and different *C*-point index lattices are there. Interestingly all the *C*-points are of the same handedness. In one particular case the *C*-point lattice is interlaced with saddle points, in which formation of *C*-lines is also noticed. These are saddles in both azimuth and ellipticity distributions. The governing rules for realizing these lattices are given and these lattices are experimentally demonstrated.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

In scalar optics the two important parameters that are related to an optical field are amplitude and phase. In real valued functions features such as saddles and extrema are the only possibilities, whereas in complex valued functions additional topological features namely phase vortices can be found. In inhomogeneous polarization distributions the number of topological features increases by many-fold due to the increase in the number of parameters.

The generic polarization singularities in two dimensional ellipse fields are star, lemon and monstar [1–7]. In the spatially varying polarization distributions, these are isolated singular points where the orientation of the azimuth is indeterminate. The *C*-points can have positive or negative, integral or half integral index values. They can be of left or right handed and can be dark or bright. The half integral *C*-points are more common and familiar, whereas the integral *C*-points are uncommon. There are many reports on generation of generic polarization singularity lattices [8–18]. The polarization distributions of these lattice fields can be mapped all over the Poincare sphere and hence qualify to be called as full Poincare beams.

The lattice patterns that are discussed in this article are formed by the interference of six phase engineered radially polarized plane waves. In the absence of phase engineering, normally this would yield *V*-point lattice [18]. Lattices consisting of only *C*-points or only *V*-points of different Stokes indices are experimentally produced. The enlarged sign rule is found to hold good in one of the lattices where *C*-points of same sign are accompanied by saddles. The wave vectors of these six beams are symmetric and lie on the surface of a cone such that, ${\overrightarrow{k}}_{j}=\frac{2\pi}{\lambda}\{\mathrm{cos}({\xi}_{j})\mathrm{sin}({\xi}_{j})\mathrm{sin}(\theta ),\mathrm{cos}(\theta )\}$, where *j* is the beam number and *j* = 1 − 6. The angle between any of these wave vectors and *k*_{z}-axis is *θ* = 0.15° and *ξ _{j}* is the angle between the projection of a wave vector onto the transverse plane (i.e.

*k*plane) and the

_{x}k_{y}*k*axis. These six non-coplanar linearly polarized uniform amplitude plane beams are chosen such that the angle between any two adjacent beams is 2

_{x}*π*/6. The transverse components of these

*k*-vectors are shown as dots on rings, viewed in a

*k*plane. In the transverse plane the electric field vectors of these six interfering beams are oriented in a radial fashion and are shown in Figs. 1(a)–1(d). The resultant interference field of six phase engineered vector beams can be written as

_{x}k_{y}*ϕ*=2

_{l}*lπ*represents total initial constant phase offset shared between all the six interfering beams. The value of

*l*is 0≤

*l*≤(

*n*/2), where

*n*is the number of interfering beams. For

*n*= 6, the range for

*l*is given by 0≤

*l*≤3. We are limiting ourselves only to the integral values of

*l*. For

*n*= 6, the possible four integral

*l*values indicate four possible phase engineered lattices, that are presented in this article. The phase offsets associated with

*j*interfering beam can be written as

^{th}*jϕ*/6.

_{l}The constant initial phase offset that is given to each interfering beam for generating these four lattices (*E*_{60}, *E*_{61}, *E*_{62}, and *E*_{63}) are shown in Table 1. The lattice wave fields *E*_{60} and *E*_{63} correspond to vector fields, which are embedded with only *V*-points. The wave fields *E*_{61} and *E*_{62} are very interesting as they contain only *C*-points. The wave field *E*_{61} is embedded with both integral and half integral *C*-points, whereas in wave field *E*_{62} the half integral *C*-points are interlaced with vacancies, which are intensity zero points with missing polarization singularities. In three of the fields namely *E*_{60}, *E*_{61}, and *E*_{63} both positive and negative index polarization singularities are present, whereas in the field *E*_{62} the negative (or positive) index *C*-points are embedded with saddle points.

Normalized Stokes parameters (*S*_{0}, *S*_{1}, *S*_{2}, *S*_{3}) are widely used to describe spatial distribution of state of polarizations (SOPs) in an optical field [19, 20]. Complex Stokes field *S*_{12} = *S*_{1} + *iS*_{2} can be used to locate the positions of the polarization singularities in the spatially varying polarization distribution. The irradiance profile of six-beam interference is given by $I={\displaystyle {\sum}_{p=1}^{6}{\displaystyle {\sum}_{q=1}^{6}{\overrightarrow{E}}_{p}\cdot conj({\overrightarrow{E}}_{q})}}$. Figures 2(a)–2(d) correspond to normalized irradiance profiles of four different lattice wave fields and the corresponding normalized *S*_{12} Stokes intensity in each case is shown as inset. Note for pure vector lattice fields the normalized Stokes intensity is constant.

The polarization distributions in the six-beam interference due to phase engineering are shown in Fig. 3, with *S*_{12} Stokes phases as insets. Figure 3(a) shows the polarization distribution of the resultant field *E*_{60}, without phase engineering. In this case, the resultant field *E*_{60} is embedded with positive and negative *V*-points of polarization singularity index 1. When the phase offset to the *j ^{th}* interfering beam is (

*j*2

*π*/6), the polarization distribution of the resultant field

*E*

_{61}happens to be embedded with only

*C*-points [Fig. 3(b)]. In this lattice an integral

*C*-point (

*I*= +1) is surrounded by six half integral

_{c}*C*-points (

*I*= −1/2). This is very interesting as the lattice pattern contains both integral (Type I) and half integral

_{c}*C*-points (stars). In the

*S*

_{12}Stokes phase distribution the integral and half integral

*C*-points appear as phase vortices of topological charge (+2) and (−1) respectively. Figure 3(c) shows the polarization distribution of the resultant field

*E*

_{62}, when the total initial phase offset between the interfering beams is 4

*π*(

*l*= 2). Surprisingly the polarization distribution is embedded with saddles of azimuth and ellipticity.

*C*-points and their orthogonal polarization distributions have been introduced recently [21]. These orthogonal

*C*-points are seen in the lattice formations reported earlier [15–17]. One interesting point is that polarization distributions of Fig. 3(b) and Fig. 3(c) are unique as they are made up of non-orthogonal

*C*-points. Another unique feature observable in Fig. 3(b) and Fig. 3(c) is that all the

*C*-points are of same handedness and each

*C*-point is enclosed by

*L*-lines (not shown). In Fig. 3(c) the formation of

*C*-lines (green lines) can also be observed. Polarization distribution of the resultant field

*E*

_{63}, when initial phase offsets among the interfering beams is 6

*π*(

*l*= 3), is shown in Fig. 3(d). Here polarization distribution is embedded with two different types of

*V*-points of unequal topological charges and is different from the polarization distributions as shown in Fig. 3(a). In Fig. 3(d) a higher order

*V*-point of Poincare Hopf index (−2) is interlaced with

*V*-points of Poincare Hopf index (+1). All the four lattices satisfy the enlarged sign rule [5, 22, 23].

A half wave plate (HWP) can be used to invert the polarity of *C*-points and *V*-points in a lattice field [24]. The index inversed polarization distributions corresponding to the fields *E*_{60}, *E*_{61}, *E*_{62}, and *E*_{63} are shown in Fig. 4(a)–4(d) respectively. The respective Stokes phase distributions are shown as inset. The positions of polarization singularities (*C*-points and *V*-points) remain invariant during the process of index inversion.

The *V*-points always occur at intensity nulls, whereas *C*-points can occur at any intensity level. It is interesting to note that in Fig. 3(b) and Fig. 4(b) the integral index *C*-points occur at intensity maxima and half integral index *C*-points occur at intermediate intensity level. When only half integral index *C*-points are there [Fig. 3(c) and Fig. 4(c)], they lie at intensity. In the *S*_{12} Stokes phase distribution though the topological charge associated to *V*-points (in *E*_{60}) and integral *C*-points (in *E*_{61}) are same, the *C* and *V*-points can be distinguished by observing the SOP distributions in their respective neighborhoods.

At the saddle point zero crossing of *S*_{1} can happen (Fig. 5, location 3) by adding constant phase to *E*_{62}. Similarly crossing of *S*_{2} = 0 can be achieved at the saddle point. When contour of *S*_{1} = 0 (or *S*_{2} = 0) goes through a saddle, the *C*-points on either side of it have same polarity. Lattice consisting of *C*-points of same polarity is possible with the help of saddles. Under sufficient perturbation zero crossings of *S*_{1} (or *S*_{2}) separate and no longer cross each other at saddles as shown in locations 1, 2, 3 and 5 of Fig. 5. But at a *C*-point (Fig. 5, location 4) the zero contours of *S*_{1} and *S*_{2} intersect each other at all times irrespective of perturbation.

In the experimental setup as shown in Fig. 6, a spatially filtered collimated laser light of 633*nm* wavelength falls on a SLM and the light reflected from the SLM when focused produces six bright spots at the Fourier plane. Depending on which of the four phase distributions among Fig. 6(a)–6(d) is displayed on the SLM, these six bright spots on the Fourier plane are capable of producing six plane waves with appropriate phase shifts, upon collimation by lens *L*3. These six spots are filtered and then passed through an S-waveplate, which is placed just after the filter plane. The S-waveplate modulates polarization of the individual spots as radial. Lattice intensity patterns as captured by Stokes camera are shown in Fig. 6(e)–6(h). Polarization distributions for the experimentally recorded fields *E*_{60}, *E*_{61}, *E*_{62} and *E*_{63} are depicted in Fig. 7(a)–7(d). A HWP is used just behind the S-waveplate to obtain the index inversed fields and are shown in Fig. 8(a)–8(d). The experimental and theoretical results are nicely matching.

In conclusion we have demonstrated the effect of phase and polarization engineering to realize four novel lattice structures in six-beam interference. Lattices populated with (a) only *V*-points, (b) integral and half integral index *C*-points, (c) saddles interlaced with half integral *C*-points of same polarity and (d) *V*-points of different Stokes indices are realized from the same six-beam interference setup through phase engineering. The salient features of these lattices are presented.

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