## Abstract

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.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## Corrections

15 February 2019: A typographical correction was made to Ref. 6.

## 1. Introduction

In recent years, inhomogeneously polarized optical fields have fascinated many researchers [1–13]. Polarization singularities are the points in an inhomogeneously polarized optical field where atleast one of the parameters defining the state of polarization of light is not defined. Homogeneous polarization distribution is represented by a point and inhomogeneous polarization distribution is represented by a region on the Poincare sphere.

Polarization singularities of ellipse fields are C points [5], characterized by a C point index *I _{c}*, and singularities of vector fields are V points [14], characterized by a V point index

*η*, also known as Poincare Hopf index. Generic C points are star, lemon and monstar and generic V points are classified as typeI, typeII, typeIII and typeIV. Hypothetically constructed complex Stokes field

*S*

_{1}+

*iS*

_{2}is useful in the study of polarization singularities. The phase singularities in Stokes field correspond to polarization singularities in the actual optical field. In the Stokes field, a Stokes index is defined as

*σ*= (Δ

*ϕ*)/(2

*π*) where Δ

*ϕ*is the accumulated Stokes phase (

*ϕ*), around the singularity. The Stokes index is related to

*I*and

_{c}*η*by:

*σ*= 2

*I*and

_{c}*σ*= 2

*η*. Around a singularity Δ

*ϕ*can be integral multiple of

*π*. When it is even multiple of

*π*, the distinction between C point and V point can be made from its neighbourhood state of polarization (SOP) distribution. In this paper, we discuss the generation of spatially varying lattice of C points interlaced with V points. The lattice parameters such as periodicity and orientation are smoothly varying across the lattice. We have experimentally shown the generation of lattices with varying a) periodicity, b) orientation and c) both orientation and periodicity. Inverse phase gradient method is used to synthesize phase to achieve such lattices. Spatially varying lattices of C points may find application in polarization dependent structured illumination. The angular momentum content in polarization singularities can be used to rotate particles [15,16], sorting and in communication. These applications have been demonstrated using phase vortices [17,18] but not using polarization singularities so far. In recent years applications of polarization singularities have been demonstrated in robust beam generation [19,20], spatial filtering [10,21] and, for chirality measurement [22].

## 2. Spatially varying lattice

Let us start with three-beam interference method of generating a uniform lattice of C points embedded with V points [7]. The propagation vector for generating this uniform lattice can be written as :

Here *θ* is complement to latitude and Θ* _{j}* is longitude corresponding to interfering beams. Θ

*can be written as (2(*

_{j}*j*− 1)

*π*)/

*n*, where

*n*is the number of interfering beams and

*j*=1,2..

*n*. Periodicity of a lattice depends on the wavelength of light and angle

*θ*that propagation vectors make with the

*z*- axis (axis normal to the interference plane). On increasing angle

*θ*, the frequency of the fringes increases and hence periodicity decreases. Similarly, Θ

*can be varied to change the orientation of the lattice. To have a spatially varying periodicities and orientations in a lattice, local variations of the propagation vectors need to be engineered. This means that the interference of three plane waves each having homogeneous polarization distribution has to be modified to an interference of three non plane waves, each having inhomogeneous polarization distribution. In the non-plane waves, there is a spatial spread of points in momentum space around each of the*

_{j}*k*vectors. A local spread of points in momentum space for three non plane waves can be represented on Bloch sphere as shown in Fig. 1a. Each of the interfering beam has a set of momentum directions in transverse plane. On the Bloch sphere, variation of momentum states along the same latitude lead to orientation change and momentum states along the longitude lead to periodicity change in lattice. The typical spread in

*θ*for periodicity change is between 0.35° to 0.4° with respect to

*k*direction. This can be deduced from the radial spread of the aperture size in the Fourier filter plane of the experimental setup which is described later. These three beams also have spatially varying linear polarization states as shown by three arc segments on the equator of the Poincare sphere (Fig. 1b).

_{z}Lattice orientation and periodicity parameters can be infused in propagation vector equation to generate the required phase mask for spatially varying lattice. The propagation vector for the spatially varying lattice can be written as:

*P*(

*r*) is the periodicity parameter that depends on the variation in angle

*θ*and

*O*(

*r*) is the orientation parameter. As the field distribution is observed at the constant

*z*plane (

*xy*plane), the phase variation does not depend on the

*z*component of the wavevector. Further, the strong

*z*component of the wavevector gives rise to negligible longitudinal electric field components that are neglected in our calculation. For linear variation of periodicity and orientation across the lattice,

*P*(

*r*) and

*O*(

*r*) can be defined as:

*P*(

*r*) =

*C*

_{1}

*x*+

*C*

_{2}

*y*and

*O*(

*r*) =

*C*

_{3}

*x*+

*C*

_{4}

*y*where

*C*

_{1},

*C*

_{2},

*C*

_{3}and

*C*

_{4}are four constants. Such parameters lead to the quadratic variation of phase along the

*x*and

*y*direction of the lattice. In Fig. 2 and 3, for example, we have considered

*C*

_{2}=

*C*

_{4}= 0. The value of

*C*

_{1}decides the rate at which periodicity changes in

*x*direction and

*C*

_{3}decides the rate at which orientation changes in

*x*direction. Say for example, to double the periodicity while traversing the lattice, the periodicity parameter should vary from 1 to 2. For this periodicity variation,

*C*

_{1}= (2 − 1)/(

*N*) where

_{x}δ_{x}*N*is the number of pixels of the SLM in the horizontal direction and

_{x}*δ*is the pixel size. Similarly,

_{x}*C*

_{3}=

*π*/10 − (−

*π*/10)/(

*N*) corresponds to orientation variation from −

_{x}δ_{x}*π*/10 to

*π*/10. Direct construction of spatially varying lattice using this spatially varying propagation vector, will lead to discontinuity in the lattice [23–25]. To avoid the discontinuity in the lattice, phase is calculated using inverse gradient method which uses finite differences, such that: For Θ

_{1}= 0, Θ

_{2}= 2

*π*/3 and Θ

_{3}= 4

*π*/3, the above equation can be solved to find phase functions (

*φ*(

_{j}*r*)) and the transverse components of the resultant fields can be expressed as:

The resultant lattice will have both periodicity and orientation variation. Omitting the term consisting of periodicity parameter or orientation parameter in equation(2) will lead to lattice with only orientation variation or periodicity variation respectively. Resultant lattice consists of both C points and V points in a hexagonal lattice. The polarity of the polarization singularities in the lattice can be changed using half wave plate [9]. Simulated intensity patterns, Stokes phase distributions and polarization patterns with spatially varying periodicity are shown in Figs. 2a, 2d and Fig. 3(a,d) respectively. When there is a variation in orientation, simulated patterns are shown in Figs. 2b, 2e and Fig. 3(b,e). Likewise, for both periodicity and orientation variation, the simulated patterns are shown in Figs. 2c, 2f and Fig. 3(c,f). Stokes phase variations (shown in Fig. 2(d–f)) correspond to star lattice. For a lemon field, polarization distributions (shown in Fig. 3(d–f)) and corresponding Stokes phase distributions can be obtained from Fig. 3(a–c) and Fig. 2(d–f) by using a half wave plate. The Stokes intensity patterns (insets) as well as the total intensity patterns are identical for both lemon and star lattices. Interestingly, in all these lattice structures, not only the unit cell but also the individual SOP distributions rotate across the lattice. Such variations in the lattice only depend on the orientation and periodicity parameters and hence more complex lattice with different spatial variation can also be generated using different spatially varying parameters. A different set of simulation results are shown in Fig. 4 with periodicity variation in radial direction. The periodicity parameter *P*(*r*) chosen here is *P*(*r*) = *r*^{1/3} + 0.4 where *r* is radial distance from the centre of the frame. The size of the frames in all simulations is 4 × 4*μm*^{2} and 400 × 400 pixels are used.

## 3. Experiment and results

The experimental set up for generation of spatially varying lattice is shown in Fig. 5. A He Ne laser beam of wavelength 632.8nm is spatially filtered using spatial filter assembly (SF) and collimated by lens L1 of focal length 20cm. The required phase variation due to three beam interference is imprinted onto a collimated beam using a phase only reflective SLM (Holoeye LC-R2500, Germany). Required beams are extracted using an appropriate filter at the Fourier filter plane (FF). In the Fourier plane of the lens (f=35 cm, for L2) the spots are at a radial distance of approximately 2.5mm from the centre. For experimental convenience, this radial distance can be increased by using a lens (L2) of larger focal length. The polarization distribution in the three beams are modified using an S-wave plate which is placed immediately after FF. A lens L3 is placed at a distance *f* (focal length of lens L3) from FF to produce three beam interference at the camera plane. In this setup, filter FF selects the spatial spread of points in momentum space whereas the S-wave plate imparts spatially varying polarization distributions in the three beams. The required phase mask is generated numerically using MATLAB. High resolution Stokes Camera(SC)(Salsa full Stokes polarization imaging camera, 1040×1040 pixels, BOSSA Nova, USA) is used to record Stokes parameters and intensity distributions for the resultant pattern. Experimentally obtained intensity distributions and the corresponding polarization patterns are shown in Fig. 6. In Fig. 6(b, e, h), the intensity patterns of spatially varying lattices are shown. These three figures depict the variation of (i) periodicity (ii) both periodicity and orientation, (iii) orientation as we move from left to right in each of the figure. Two regions, one on the left and one on the right are selected and magnified from each of these figures and local variation of polarization distributions are shown in Fig. 6(a, c), Fig. 6(d, f) and Fig. 6(g, i) respectively. Like spatially varying polarization distributions, spatially varying polarization lattices are new type of optical fields that can have new features and properties. These fields can be used as new types of structured illumination. If these polarization structures can be recorded onto a polarization sensitive medium, they can have polarization dependence features similar to photonic crystals [26] where intensity dependent structures are engineered.

## 4. Conclusion

In this paper, we have described the synthesis of spatially varying lattices of C points and V points. In a three beam interference, the phase distribution of each beam is synthesized using inverse gradient method which uses finite differences. Lattices of varying parameters such as periodicity and orientation are demonstrated. The momentum and polarization distribution of the interfering beams are explained using Bloch and Poincare spheres. Features of experimentally obtained and theoretically simulated spatially varying polarization singularity lattices are in good agreement with each other.

## Funding

Council of Scientific and Industrial Research (CSIR) [EMR N0-03 (1430)/18/EMR-II].

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