#### Abstract

Vortices of different charges are embedded in a wavefront that has quadratic phase variation, and the intensity distribution near the focal plane is studied. This method may be useful in realizing complicated beam profiles. We have experimentally demonstrated the generation of vortex arrays having integer as well as fractional topological charges that produce different intensity profiles at the focal plane. The phase variation realized on a spatial light modulator (SLM) acts as a Fresnel lens with embedded vortices.

#### 1. Introduction

A modulo 2 operation carried out on a spherical wavefront results in concentric zones called Fresnel zones. A Fresnel lens is formed by segmenting the continuous surface of a conventional lens into a set of surfaces of the same curvature. This process allows fabrication of a large diameter lens with very short focal length thereby ensuring substantial increase in its numerical aperture. Further, there is a significant reduction in the thickness and weight of the lens. Thus, a plane wave incident on a Fresnel lens is split into multiple segments (zones), and the combination of multiple successive wavefront zones reconstructs the spherical wavefront.

An optical beam with phase singularity also called optical vortex or a screw dislocation is a helical phase ramp which focuses to a ring-like intensity pattern [1]. An optical vortex is characterized by azimuthal phase dependence of the type where is the topological charge. The magnitude of topological charge represents the total phase accumulated by the helical wave in one complete revolution around the vortex point. The sign of topological charge represents left- or right-handed helicity. When the topological charge has a noninteger value, it results in the formation of a fractional vortex. Wavefront containing fractional charge vortex exhibits a half-cut (low-intensity radial opening in the doughnut structure), corresponding to a mixed edge-screw dislocation [2]. Such vortices, on propagation, transform into several opposite-sign screw dislocations which disappear on the beam periphery [3].

Optical vortices find applications in particle trapping [4], data encryption [5], phase contrast microscopy [6], image processing [7], collimation testing [8], vortex coronagraphy, [9] and in various other areas. Recently, optical vortex arrays which are regular nets of vortices have generated an enormous interest among many research groups [10, 11]. Optical vortex arrays, also called vortex lattices, find applications in multiple trap optical tweezers [12], spatial solitons [13], microfluidic sorting [14], metrology [15], micromechanics [16], and atom optics [17]. Vortex arrays are generated using interferometry [18–20], Talbot imaging [21], and spatial filtering [22]. In this paper, we present a simple and effective method of generating array of vortices by embedding vortex phase distribution in the Fresnel lens. Specially designed phase masks are produced which are encoded onto the SLM to achieve array of vortices near the focal plane. A vortex array of integer as well as fractional topological charge can be produced by this method. As this method uses a single lens function, there is no cross-talk and overlapping of different beams. The method offers flexibility in the selection of location, number density, and topological charge of vortices in the beam.

#### 2. Theory

A convex lens brings a collimated incident beam to a focus at a distance from the lens. The phase delay suffered by the beam due to lens is where is the focal length, , and is the wavelength of light used. The polar coordinates in the element plane , are related to Cartesian coordinates by . A thin lens can be considered equivalent to a transparency with a complex amplitude transmittance given by Fresnel zone sectioning of this transmittance function leads to a Fresnel lens. The radius of the nth zone is . Figure 1(a) shows the profile of a conventional Fresnel lens.

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A vortex or phase singularity in the wave field can be implanted by using a phase mask or by a spiral phase plate. The transmission function of such a phase mask is given by where is the topological charge of vortex. A vortex lens [23] has the transmission function given by A phase mask having this transmittance function transforms incident plane wave into converging helicoidal wave . Presence of Fresnel lens factor in a wave field gives desired curvature to the beam. When the topological charge is a fraction, it is called fractional vortex lens [24]. This specific transmittance imparts an on-axis vortex on the incident optical field. Phase variation corresponding to multiple vortices can be embedded into the wavefront by using the expression where is the topological charge of th vortex, and is helical phase variation corresponding to the th vortex. The position of the core of the th vortex is , where where , are the Cartesian coordinates of the th vortex core. Similar method of embedding vortices at desired locations is used in the iterative computational phase synthesis algorithms for diffractive elements, in wave optical engineering [25, 26]. This process is used to annihilate stagnation causing vortices, in the iterative algorithms.

When the extremum of a quadratic phase variation of a lens coincides with the position of the core of the vortex, spiral zones are formed. Figure 1(b) shows the profile of a vortex Fresnel lens. When the position of the vortex does not coincide with the extremum of , branching of the zone boundary occurs as can be seen from Figure 1(c). The number of branches is equal to the magnitude of the charge of the vortex. The sign of the vortex decides the sense (clockwise or anticlockwise) in which the branching occurs. Figure 1(d) shows profile of a vortex Fresnel lens in which opposite vortices of unequal fractional charges are embedded at two different locations.

The complex amplitude at the focal plane can be evaluated by using the Fresnel-Kirchhoff diffraction integral which can be written [27] as where and correspond to object and diffracted field planes expressed in cylindrical coordinates. is the wavelength of light used, and is the focal length of the lens. The multiplicative factor is ignored since the amplitude is only a scale factor, and the phase has no effect on the intensity. The field at the focal plane is proportional to the Fourier transform of the transmittance function given in (7). Intensity distribution at the observation plane is then given by Here, the intensity distribution refers to the far-field image of the object. We have carried our simulation studies to find the distribution of intensity in the focal plane.

#### 3. Experimental Setup

A phase-only SLM can impose an azimuthal phase term directly on the incident beam converting it into a helical beam. The method of generating optical vortices using phase mask is more versatile because it allows precise and dynamic control over vortex parameters and possibility of generating specific patterns of optical vortices. The experimental arrangement is shown in Figure 2. A linearly polarized light from a He-Ne laser ( nm) is spatially filtered, expanded, and collimated to form a plane wave. The collimated beam of light is directed on the SLM (Holoeye LC-2002 with pixels, *μ*m). By using properly oriented linear polarizer or by generating appropriate elliptically polarized light, one can modulate the phase while keeping the amplitude constant. We have used elliptically polarized light to achieve phase-only modulation for the SLM using a combination of two polarizers and two quarter wave plates, as shown in Figure 2. This architecture consists of a combination of an input polarizer P and a quarter wave plate QW_{1} in front of the SLM and an output quarter wave plates QW_{2} and polarizer P_{2} behind the SLM. We adjust the polarizers and wave plates such that the SLM modulates only the phase, leaving the amplitude constant. A gray-scale phase mask corresponding to the transmittance of a lens, embedded with required vortex phase distributions, is displayed on SLM, and the corresponding intensity distribution is recorded on a CCD.

#### 4. Results and Discussion

The Airy pattern at the focal plane of a lens is replaced by a doughnut pattern in a vortex lens. In a Fresnel vortex lens, the vortex phase is embedded at the extremum of the lens phase distribution that can produce a ring focus. By embedding multiple vortices in different locations in a Fresnel lens, vortex lattices can be realized at the focal plane [28]. Figure 3(a) shows the phase distribution of the modified Fresnel lens that can generate a vortex lattice of four vortices with positive topological charge. Figures 3(b) and 3(c) depict the simulation and experimental results of the intensity pattern in the focal plane of the Fresnel lens embedded with vortex lattice, as described above. The position and the charge of each of the vortices can be modified. Figure 4 shows one such configuration where two positive charge and two negative charge vortices are formed. The phase distributions are similar to that of a conventional Fresnel lens with branching of zones occurring at off-axis locations [29]. For positive (negative) vortex, the branch or fork occurs in anticlockwise (clockwise) sense as we move along the zone boundary. For a diverging Fresnel lens, the sense of fork formation is opposite to that of a positive lens. Similarly, vortex phase of fractional charge can also be embedded in a Fresnel lens. Figure 5 shows two positively charge fractional vortices of magnitude 3.5, embedded at off-axis locations. In this case, apart from branching of zones at vortex points, there is an additional edge dislocation connecting the two vortices. In the intensity distribution near the focal plane, the two dark cores are connected by a curved dark line. The two dark cores and the connecting dark curved line are surrounded by bright intensity pattern. In Figure 6, oppositely charged fractional vortices are embedded in the phase distribution of Fresnel lens. The dark cores in this case are connected by the dark line that has different geometry. These intensity patterns are unique. Two fractional vortices are connected to each other, irrespective of the charge of the individual vortex. Figures 7 and 8 depict arrays of vortices with fractional topological charges. The dark line that connects two vortices is decided by the manner, the edge dislocation connecting the two fractional vortices, appears in the phase distribution. If the edge dislocation line starting from one fractional vortex is not connected to another fractional vortex, it runs towards the boundary of the intensity pattern and terminates. Simulations are carried out to see the focal plane intensity distributions of such lenses. In all the Figures 3–8, it can be clearly seen that the experimental and simulation results match exactly. Figure 9 depicts vortex lattices of arrays of the vortices with fractional topological charges. The charge distributions of the vortices in each of the lattices are different. Simulated intensity distributions at the focal plane corresponding to these lattices reveal that all the vortices are not connected by dark lines to form pairs. Only three pairs are connected, and in the remaining three fractional vortices, the dark lines from their cores are diverging outward.

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#### 5. Conclusion

Array of vortices of different charges are embedded in a wavefront with quadratic phase variation and the intensity distribution near the focal plane is studied. The intensity patterns, when fractional vortices are embedded in the Fresnel lens, possess unique interesting features. The intensity patterns produced by any two fractional charge vortices are connected by dark line, irrespective of the sign of the vortices. The geometry of this dark line is decided by the manner in which the edge dislocation line appears in the phase distribution of the Fresnel lens.