## Abstract

We introduce an asymmetry in the core of a high charge optical vortex by using an appropriate computer generated hologram. The splitting of a high charge optical vortex core into unit charge vortices has been found to depend on the extent of the asymmetry. For a second order vortex, the trajectories of the split unit charged vortices and their separation have been recorded as a function of change in the asymmetry of the core. We find a good agreement between the experimentally obtained and numerically calculated results.

©2011 Optical Society of America

## 1. Introduction

Vortices, which are manifestations of phase singularities, are generic to all the wave fields. Therefore, exploring their properties is important to many branches of physics [1]. In optics, these are called optical vortices or light beams having a helical wavefront [2–6]. The helical wavefront is characterized by an azimuthal phase term $\mathrm{exp}\left(jm\theta \right)$ that varies in a corkscrew-like manner along the beam's direction of propagation. Here, the factor *m* is known as the topological charge (can have both positive and negative sign depending on the sense of rotation of the corkscrew) or the order of the vortex. In such a beam of light, each photon acquires an orbital angular momentum of $m\hslash $ [6] and the transverse intensity profile looks like a ring of light with dark core at the centre.

Optical vortices can be found naturally on scattering of light through the rough surfaces. However, they can also be generated in a controlled manner. Diffraction of light through a computer generated hologram (CGH) is one of the most commonly used methods for making optical vortices in the laboratory [7]. Apart from using CGH, vortices can be generated with spiral phase plates, astigmatic mode converters, spatial light modulators [8] as well as interferometric methods [9]. These structures of light find a variety of applications in the fields of optical manipulation [10], optical communication [11], quantum information and computation [12], and astronomy [13].

The stability and the propagation dynamics of a vortex core may be of relevance to the field of optical communication using such beams. Therefore, a great deal of attention has been given to the dynamics and propagation properties of optical vortices in free space and in nonlinear media [14–20]. Ginzburg and Pitaevskii [21] pointed out that a vortex with higher charge (say *m*, where $m>1$) in a superfluid is energetically unfavorable compared to *m* vortices of unit charge. Hence, a vortex with higher charge always has a tendency to break up into unit charge vortices, although the physical mechanism that creates them may be different for different systems. The breaking up of optical vortices can be controlled by several external parameters [22], since propagation and splitting of vortices of high topological charge depends upon the shape of the host beam and its evolution [23–26]. A detailed theoretical description of breaking the degeneracy of an optical vortex core has been given by Freund [27]. The decay of a vortex due to perturbation and anisotropy produced by the nonlinear media has also been explored [28].

The break up of a high charge optical vortex leads to the formation of an array of unit charged vortices lying in the same host beam. However, such a composite structure of vortices has also been formed by the superposition of the two or more phase singular beams [29–33]. By a clever choice of different Laguerre Gaussian (LG) beams, one can generate a linear as well as a circular array of optical vortices. Franke-Arnold et al. [30] have generated a beautiful ring lattice of optical vortices, an optical ferris wheel, which is useful in the trapping of red and blue detuned cold atoms. The superposition of various phase singular beams was also utilized by Berry and Dennis [31] to produce the knots and the links in optical dislocations. The theoretical results [31] were experimentally verified in separate experiments by Leach et al. [32] as well as Dennis et al. [33]. However, in their experiments instead of using two or more separate LG beams, they designed a phase hologram to produce such a superposition by a spatial light modulator (SLM) [32,33].

In all of these experimental as well as theoretical studies, some kind of perturbation has been applied to cylindrically symmetric LG beams. Adding a perturbation breaks the symmetry of such beams and leads to the decay of the LG beams with a high topological strength [34,35]. Dennis [34] has discussed different kinds of perturbations which may cause the breakup of high-order vortices. He differentiates between the elliptically perturbed LG beams and the LG beams perturbed by a small real constant.

We have studied the effect of introducing an asymmetry to a vortex. The asymmetry was introduced through the CGH that produces the vortex. The introduced asymmetry destabilizes a high-order vortex that splits into a number of first order vortices, the number being same as the order of the vortex. We would like to point out that although a perfectly symmetric vortex is used in most of the discussions on vortices, no real system can exhibit a perfect symmetry. Therefore, an elliptically perturbed vortex provides a more realistic and widely applicable vortex model. It should be noted that optical vortices show a deep similarity with the quantized vortices in superfluids and Bose Einstein Condensates. Therefore, the results presented here may also be useful in fields other than optics.

## 2. Theoretical background

Elliptic vortices in an electromagnetic field are one of the solutions of linear as well as nonlinear wave equations [36,37]. Chávez-Cerda et al. [36] have shown the existence of elliptic vortices in scalar wave fields as a solution of Helmholtz and Schrödinger wave equations. The well known LG beams with cylindrical symmetry are also one of the solutions of the Helmholtz equation in the paraxial approximation. For mode indices $l=m$ and $p=0$ in the associated Laguerre polynomial ($L{G}_{p}^{l}$), the LG beam carries an optical vortex of order *m* [6]. The effect of ellipticity in such a cylindrically symmetric beam has been discussed by Dennis [34] and also by Kotlyar et al. [35].

Like all wave fields, propagation of asymmetric vortices in free space can be dealt with using the Huygens-Fresnel principle [38]. We consider our observation plane at $\left(x,y\right)$, which is parallel to the initial vortex plane $\left(\xi ,\eta \right)$ and situated at a normal distance *z* from this plane. On illuminating a diffracting object by a Gaussian laser beam, the resulting expression for the field at $\left(x,y\right)$ can be written as

Under the Fresnel approximation [38], Eq. (1) reduces to

*λ*is the wavelength of light.

## 3. Experiment

The experimental set up is shown in Fig. 1 . An intensity stabilized He-Ne laser (wavelength 632.8 nm, beam waist 0.5 mm and power 1 mW) is used as the light source. The light beam coming from the laser is reflected by a mirror M1 and is sent towards a beam splitter BS1 where it is divided into two parts. The reflected part goes towards the SLM (Holoeye, LC-R 2500) and the transmitted part towards the mirror M2. The SLM is a liquid crystal based device that can modulate light in amplitude as well as in phase; therefore, it can be used as a dynamic diffractive optical element. The diffractive optical elements or the CGH with different fork patterns corresponding to different asymmetry parameters are introduced into the SLM via a computer PC1.

To generate a fork pattern for producing an optical vortex, we imprint an azimuthal phase dependence on a Gaussian beam by calculating the interferogram of the object beam $\mathrm{exp}\left(jm\theta \right)\mathrm{exp}\left(-({\xi}^{2}+{\eta}^{2})/(2{\sigma}^{2})\right)$and the reference beam $\mathrm{exp}\left(jm\xi \right)\mathrm{exp}\left(-({\xi}^{2}+{\eta}^{2})/(2{\sigma}^{2})\right)$. Here, *m* is the charge of the vortex, $\theta =Arg\left[\left(a\xi +j\text{}b\eta \right)\right]$ is the azimuthal angle, *σ* is the beam radius and *k* is the component of the wave vector along the X-axis. The real parameters *a* and *b* define the asymmetry of the vortex core. The interference of these two beams gives the following intensity pattern that is the aperture function in Eq. (2),

Equation (3) provides the required CGH for the production of an optical vortex [7,39,40]. Here, the field amplitudes have been taken as unity.

For a symmetrical vortex the ratio $a/b$ is unity, but in the case of an asymmetrical or an elliptical vortex, $a/b\ne 1$. In our experimental and numerical results we have chosen several values of the asymmetry parameter, $\alpha =a/b$ and recorded the images showing the splitting of the core of the vortices at a fixed distance in free space.

The ellipticity of an optical vortex can be controlled by the azimuthal angle *θ*. The different values of *θ*, yield different diffracting optical elements to produce asymmetric optical vortices [Eq. (3)]. Since *θ* contains *a* and *b*, the same can be achieved by varying the asymmetry parameter *α*. Therefore, to produce asymmetrical vortices, we have made a number of CGH by using different values of *α*. The diffracted light from the SLM contains optical vortices of increasing topological charge with increasing diffraction order. The zero or the central order remains a Gaussian, and vortices on either side of this central order acquire opposite topological charges. To check whether a given beam with a dark core is really a vortex, we use the interferometric method (the Mach-Zehnder Interferometer shown in Fig. 1). If the interference of the beam with a reference beam produces a fork or a spiral pattern (depending on the wavefront curvatures of the interfering beams) then it proves the presence of a vortex.

In our experimental setup, aperture A1 is used to stop the higher diffracted orders. The selected first order diffracted beams along with the central Gaussian are sent to the beam splitter BS1. Therefore, the transmitted part from the BS1 contains the central Gaussian beam and vortices in the first order diffraction. We select one of these vortices by suitably adjusting the aperture A2 and send it to another beam splitter BS2. The Gaussian laser beam, coming from M1 and transimitted through BS1, is reflected by two mirrors M2 and M3. It is finally combined with the selected optical vortex at the beam splitter BS2. In our experiment, first we record the asymmetric vortex by blocking the reference beam. The image is recorded in a computer PC2 using a CCD camera, placed at 50cm from the SLM. After recording the image of the vortex, we let the reference beam interfere with the vortex. To equalize the intensity of the interfering beams, for getting fringes with a better contrast, we have used a neutral density filter NDF1 in the path of the reference beam. The interference fringes are recorded in PC2. Another neutral density filter NDF2, kept in front of the CCD camera, is used to avoid saturation of the CCD.

## 4. Results and discussion

We have explored the splitting of high order optical vortices due to the asymmetry in a vortex core. It has been shown that the asymmetry introduced in the vortex beam during its generation is adequate for the splitting of the dark core. In Fig. 2 , we show the experimental and simulated optical vortices of topological charge two with zero asymmetry and their interference with a reference beam. It is clear from Fig. 2 that a vortex of charge two is located at the centre of the beam, as the corresponding interference fringes show the branching of one line into three lines, the characteristic interferogram of a charge two vortex. However, if we introduce an asymmetry into this vortex beam, the dark core starts breaking (Fig. 3 and Fig. 4 ).

In Fig. 3, we have shown the images of the optical vortices and their corresponding interferograms for different values of the asymmetry parameter. Column (I) shows vortices generated through a hologram with values of the asymmetry parameter $\alpha <1$. It is clear from the images of the vortices and the corresponding interference fringes (column II) that with an increase in the asymmetry, the breaking of the core into vortices of unit charge becomes faster. The same is seen to be true for the values of $\alpha >1$ (columns III and IV), with the only difference being that the singly charged vortices align themselves into a different orientation. The simulated results for the experimental observations are shown in Fig. 4 and found to be in good agreement. These results were obtained by numerically solving the Fresnel integral [Eq. (2)] with the Simpsons’ one third rule. The integral was solved by constructing a grid of 800×800 with a step size of 6.5µm in $\left(\xi ,\eta \right)$ plane. The propagation distance *z* is taken to be 50cm. The program written in C language was run on the PRL 3TFLOP HPC cluster.

To investigate the path followed by singly charged vortices, produced from the splitting of a doubly charged core, we have plotted the trajectories of split vortices in Fig. 5(a)
and Fig. 5(b) for $\alpha <1$ and $\alpha >1$. The centre of a circularly symmetric vortex has been taken as the origin in these plots. In both the graphs, the position of the vortices has been shown in normalized coordinates for different asymmetry parameters (*α*) indicated in the legend. It should be pointed out that the value of asymmetry increases along the trajectory on either side of the origin.

It is clear from the curves that the nature of the trajectories followed by the split vortices is similar in the two cases, $\alpha <1$ and $\alpha >1$, except their orientation. The trajectories followed by the simulated split vortices have been shown along with their experimental counterparts in Fig. 5(a) and Fig. 5(b). Note that all these images have been recorded at $z=50$cm.

It is understood that a perturbation breaks the rotational symmetry of a high-order vortex beam, thereby leading to its decay into single order vortices [34]. In the present case, an elliptic perturbation is given by the substitution $(\xi +j\eta )\to (a\xi +jb\eta )$ into the core of the wavefunction made on the input plane. It is known, when a beam carrying two or more single order vortices propagates in free space, vortices with similar charge rotate in the same direction around the centre of the host beam [3,41,42]. Therefore, for $\alpha <1$, a pair of vortices initially located (at $z=0$) along the X-axis will rotate in the clockwise direction, giving rise to a negative slope as the beam propagates. However, for $\alpha >1$, the vortices which are initially located along the Y-axis rotate clockwise, and give rise to a positive slope.

After propagation, the orientation of split vortices depends on their relative separation at $z=0$. One finds that for $\alpha <1$, contours of the amplitude are elliptical along the X-axis at $z=0$, therefore, the initial split vortices are located along the X-axis only [34]. However, as the beam propagates, the split vortices start rotating in a clockwise direction [3,41]. The vortices with smaller separation (lesser asymmetry) rotate faster than vortices with larger separation (higher asymmetry). This fact is clear in Figs. 5(a) and 5(b). The inset in Fig. 5(a) shows that for a small value of asymmetry ($\alpha =0.90$), the vortex separation is smaller than that for a large asymmetry ($\alpha =0.20$). Also, it is clear from the lines joining the split vortices that the slope for the vortices with smaller asymmetry (line joining rectangles) changes faster than that of larger asymmetries (line joining triangle) as the beam propagates. A similar kind of behavior for the asymmetry parameter values $\alpha >1$ has been shown in the inset of Fig. 5(b).

The effect of asymmetry on the separation between the split vortices has been quantified in Fig. 6 . Both, the experimental as well as the simulated results verify our earlier assertion that the distance between the split vortices increases with increasing the asymmetry.

In order to demonstrate the effect of the asymmetry for high-order vortices, we show the experimental and the simulated results for the third and the fourth order vortices in Fig. 7 and Fig. 8 . The splitting of high charge vortices into a linear array of unit charge vortices (separate isolated dark spots), and their separation with increasing asymmetry is clearly visible in the images as well as in the corresponding interferograms. Circles shown in the interferograms represent the positions of the single fork pattern which is characteristic of the first order optical vortices.

## 5. Conclusion

In conclusion, we have shown the effect of the asymmetry in experimentally produced high-order optical vortices. It is observed that the splitting of high topological charge vortices into vortices of single charge becomes more pronounced with increase in the asymmetry. After the splitting, the single charge vortices align themselves in a straight row. In addition, we have verified our experimental observations with numerical simulations. The experimental and the numerical results presented here confirm results from earlier theoretical studies [34]. We show that the inter-vortex spacing of single charge vortices, which break up from a high charge vortex, can be controlled by the asymmetry parameter. We anticipate that the control over the formation of a stable array of unit charge vortices will find use in a variety of optical trapping experiments.

## Acknowledgments

We acknowledge Profs. P. Janardhan and J. Banerji for a careful reading of the manuscript and suggesting corrections.

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