## Abstract

While a fundamental Gaussian light beam can form stably a spatial soliton in certain self-focusing medium, a single-wave topologically integer-*n*-charge vortex light beam cannot. It breaks up into 2*n* filaments due to symmetry breaking and azimuthal instability, in which every azimuthal section of a $\pi $ phase range from a soliton and repels itself from its azimuthal neighboring soliton. Then what happens to the half-charge vortex light beam, which contains only one section of a $\pi $ phase range? We investigate experimentally and theoretically the propagation and stability of a topologically half-charge vortex light beam in a self-focusing photorefractive medium. We observed that the light beam propagates unstably in a self-focusing medium and breaks up into three filaments. This result is confirmed by numerical simulation and perturbation analysis.

© 2014 Optical Society of America

## 1. Introduction

Vortex [1–4] is a phenomenon that appears in various branches of classical or quantum mechanical physical systems, ranging from fluid mechanics to superconductors and even to Bose-Einstein condensation. In optics, light can carry vortex phase structure and its associated orbital angular momentum [5, 6]. Due to the spatial phase structure and the amplitude properties related to the phase singularity, light beams carrying vortex have applications in optical manipulation [7], optical communication and computation [8] and enhancement of microscopy [9]. Among these, in the scope of the nonlinear optics, the propagation of integer-charged vortex light beams in nonlinear media has also been studied intensively. In general, integral-*n*-charged vortex beams can propagate stably in self-defocusing media [10], while in most cases, they are unstable in self-focusing media such as biased photorefractive (PR) crystals [11], saturable atomic gases [12], quadratic nonlinear media [13] and thermal nonlinear materials [14]. This is due to the so called azimuthal instability, which yields unequal gains for different azimuthal modes and causes the vortex light beam to break up into 2*n* filaments [15]. Nonetheless, some methods are demonstrated to be able to stabilize vortex light beam in such self-focusing media by using partially coherent light beam or adding a small rotating azimuthally modulation to the light beam [16, 17].

Recently, light beams carrying fractional-charge vortices [18–20] in linear medium have invoked research attention by their unique phase discontinuities and geometric-phase properties. These characteristics make their propagation behavior more intriguing than that of the integral ones in both linear [21] and nonlinear [22] media. An single-wave integral-*n-*charge vortex light beam will break up into 2*n* filaments in a self-focusing medium, since every azimuthal section of a $\pi $ phase range forms a soliton and repels itself from its azimuthally neighboring soliton [15]. By this, a single-charge vortex light beam which has a full azimuthal 2$\pi $ phase will break up into two solitons, which are $\pi $ phase different from each other. Similarly, a double-charge vortex light beam which has a full azimuthal 4$\pi $ phase breaks into four. By intuitive induction, one would therefore expect that a half-charge vortex beam, though could not keep its shape, would still keep in one piece because it has only one azimuthal phase range of $\pi $. In this investigation, we want to know if it is really the case and thus study experimentally the propagation of a topologically half-charge vortex light beam in a self-focusing photorefractive medium. We observe that the half-charge light beam do not follow the trivial induction but, to our surprise, breaks up into three filaments. We also confirm this result by numerical simulation and perturbation analysis.

## 2. Experiment

The setup is shown in Fig. 1. A collimated laser beam (of wavelength 532 nm) is focused onto a computer-generated hologram to generate a half-charge vortex light beam [20]. We then use a 4-f image method with lenses of focal lengths of 30 cm and 10 cm, respectively, to relay the vortex beam onto the input face of the SBN60 crystal (a × b × c = 5 × 10 × 5 mm^{3}), of which the electro-optic coefficient ${r}_{33}$is about 250 pm/V. An iris is placed at the Fourier plane of the first lens to filter out the unwanted residues. The extraordinarily polarized vortex light beam with a size about 64 μm (FWHM) on the crystal entrance face is of power 15.3 μW. The light beam propagates along the b-axis. An ordinarily polarized light covers the entire crystal as the background illumination. The intensity ratio between the peak of the vortex light beam and the background is about 1.2 [11,23]. The defect line due to the phase discontinuity on the vortex light beam is parallel to the c-axis. A lens is used to capture the images at the input or output face of the crystal onto the CCD camera and a polarizer filters out the background light.

Shown in Figs. 2(a)-2(b), the half-charge vortex light beam [Fig.2(a)] diffracts naturally from roughly 64 μm at the input face of the crystal to 82 μm at the output face [Fig.2(b)]. Figure 2(c) shows the interference of the light beam with a collinear Gaussian beam from the same laser at the output face without nonlinearity. At the defect line clearly shows a $\pi $ phase difference. When a biasing voltage higher than 1.5 kV is applied on the crystal with polarity against the crystalline c-axis, the light beam does not keep its shape but breaks up into three filaments. Figure 2(d) shows the clearest example when the biasing voltage is 2.1 kV. We interfere the broken light beam with the collinear Gaussian beam to check the phase, shown in Fig. 2(e), in which the positions of the three filaments are marked with yellow dots. It shows that the direct link line between the upper two spots crosses a π phase range and the lower spot is at a position, which is $\pi $ phase different from the upper two. This means that the three bright spots are indeed independent.

## 3. Numerical simulations and perturbation analysis

To understand the mechanism, we adapt the numerical simulations and perturbation analysis. The half charge vortex light beam is composed by a power series as Eq. (1) [18], where α ( = 1/2) is the fractional topological charge. A_{0} is set at 1.2, according to our experimental setting, and ${P}_{n}\left(\rho \right)=\sqrt{\frac{\pi}{8}}{\left(-i\right)}^{\raisebox{1ex}{$\left|n\right|$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\rho \cdot \mathrm{exp}\left(\frac{1}{4}i{\rho}^{2}\right)\times \left[{J}_{\frac{1}{2}\left(\left|n\right|-1\right)}\left(\frac{1}{4}{\rho}^{2}\right)-i{J}_{\frac{1}{2}\left(\left|n\right|-1\right)}\left(\frac{1}{4}{\rho}^{2}\right)\right]$ containing Bessel functions with order of half integer. We take *n* of the summation in Eq. (1) up to ± 80. Then beam propagation method (BPM) is used to solve the wave equation [Eq. (2)] with the photorefractive self-focusing medium [24].

*xy*plane and 400 points in

*z*direction for 10 mm of propagation. The simulation results are shown in Figs. 3(a)–3(h). Notice that though we do observe, in the experiment, transient behavior of the light beam before it comes to steady state, this is not the main concern of this research. As we know the dynamics of the half-charge vortex light beam along the propagation direction depends only on its initial conditions, i.e. its phase and amplitude distribution, as well as the types of the nonlinearity, thus the approach and analysis obtained here can be expanded to other types of nonlinearity. We therefore stick this research to the dynamics along the propagation direction rather than stray from the focus to the temporal dynamics.

The input is shown in Fig. 3(a) and due to phase discontinuity, it has a defect line with zero intensity. Though in the simulation, we can calculate the phase directly, in order to compare with the experiment results, we also show the interfere result with a collinear Gaussian beam. After 10 mm propagation, in Fig. 3(b) the light beam diffracts naturally and reserves its phase property mostly. When a nonlinearity of 3.0 × 10^{−4} (corresponding to the applied voltage of 1.5 kV) is applied, the light beam breaks up into three filaments shown in 2D [Fig. 3(c)] and 3D plots [Fig. 3(d)]. We mark the position of the three peaks at the interference pattern [Fig. 3(g)] and in the phase plot [Fig. 3(h)]. The results are very close to the experimental results.

We use the perturbation analysis for further investigation [15]. We replace $u$ by $u(1+0.03\mathrm{cos}(L\times \theta ))$ as a perturbed vortex beam with *θ* starting at 45 degree above the defect line and let the light beam propagate according to Eq. (2) with a nonlinearity of 5.8 × 10^{−4}. $L$ is the azimuthal periodicity ranging from 1 to 5 and 0.03 is the amplitude of the initial perturbation. The field differences between the original vortex beam and the perturbed vortex beam normalized to the amplitude of the initial perturbation are obtained for different propagation distances [Fig. 4], with the example of $L=3$ at 100 μm of propagation being displayed in the inset. The azimuthal mode of $L=3$ obtains the highest growth among all modes, meaning the azimuthal mode with $L=3$ should dominate and this is just what we observe in the experiment and in the numerical simulation. Notice that, the azimuthal distribution of the perturbed light field is somewhat different from those of the light beam observed at the output face of the crystal for both experiment and simulation. This is due to the nonlinear interaction of the light beam at further propagation after the initial stage (100 μm). Because the growth of all azimuthal modes are positive, vortex light beam cannot propagate stablely in saturable self-focusing medium. Notice that although the light beam here lacks the circular symmetry that is possessed by integral-charged vortex light beam, the perturbational analysis still works consistently with the experiment and simulation. Moreover, if we change the beam size and adjust the nonlinearity accordingly to Ref. 23. then we observe the same result, in which the mode $L=3$ always dominates.

## 4. Filament interactions

We further investigate how the size of the vortex beam affects the propagation and stability. A smaller vortex light beam of 39 μm FWHM [Fig. 5(a)] was launched at the input face and the light beam diffracts naturally at the output face, shown in Fig. 5(b). We also reduce its power to 2.1 μW accordingly to keep roughly the same intensity ratio and increase the nonlinearity (i.e. increasing the biasing voltage) corresponding to the smaller beam size [23]. Shown in Fig. 5(c) at the output face of the crystal, when a biasing voltage of 2.85 kV is applied, we see only two filaments instead of three, different from that of the previous result. When we look into the simulation [Fig. 5(d)–5(g)] with the maximal nonlinear index change of 9.0 × 10^{−4} (corresponding to 4.5 kV in biasing voltage) [Fig. 5(f)] [25], in which the middle section of the propagation can be revealed, we realize that the half-charged vortex light beam actually still breaks up into three filaments [Fig. 5(g)] in 3 mm propagation. However, after some more distance of propagation, since the upper two filaments are much closer in distance and phase, they fused into one due to their nonlinear interaction [26, 27]. That is why the upper filament is brighter than the lower filament in Figs. 5(c) and 5(f).

This result shows that azimuthal instability causes the half-charge vortex light beam to break up into three filaments. Though the half-charge vortex light beam only has a section of phase ranging from 0 to $\pi $, the positions of phase 0 and $\pi $ are on the opposite sides of the defect line and by nonlinear interaction, they repel each other and form two separate filaments. On the opposite side of the defect line where the phase is $\pi /2$ from the defect line, since it is far from those two filaments, stay where it is and form the third filament [see Fig. 5(g)]. After the three filaments are formed, they then interact according to their relative phases and relative distances. It is this particular phase structure that causes the behavior of fractional topologic charge vortex light beams to be much different from integral-charge vortex light beam. Without such phase structure, such phenomenon should not appear. Therefore to prove this idea, we numerically generate an intensity profile exactly the same as that of the half charge vortex light beam but with only plane phase and put that light beam into PR medium in the simulation. The light beam with plane phase blows up much faster, meaning the phase structure plays a very important role. We also suspect with other values of the fractional charge, the vortex light beam should also behave differently.

## 5. Conclusion

In conclusion, azimuthal instability is a well-known phenomenon in nonlinear optics. It is the main reason that integer-charge vortex light beam cannot stably propagate in a self-focusing PR medium. We reveal the instability of the propagation of the topologically half-charge vortex light beam in a biased self-focusing photorefractive medium and confirm that by numerical simulation. With the help of the perturbation analysis, it shows that half-charge vortex light beam is unstable in photorefractive medium due the positive growth gain of different azimuthal modes and the gain to form three filaments is the largest. After the initial splitting, then individual filaments interact. These observed wave phenomena are very general and can exist in other similar nonlinear system, such as light wave in saturable atomic gases or in liquid crystal, or even matter wave in a Bose-Einstein condensate. Since partially spatial coherent light beam can stabilize integral-charge vortex light beam in a self-focusing medium [16], in the future we can further study whether the spatial incoherence can improve the stability of a fractional vortex light beam.

## Acknowledgments

This research is supported by National Science Council, Taiwan.

## References and links

**1. **A. I. Larkin and V. M. Vinokur, “Quantum statistical mechanics of vortices in high-temperature superconductors,” Phys. Rev. B Condens. Matter **50**(14), 10272–10286 (1994). [CrossRef]

**2. **T. F. Fric and A. Roshko, “Vortical structure in the wake of a transverse jet,” J. Fluid Mech. **279**(-1), 1–47 (1994). [CrossRef]

**3. **C. N. Weiler, T. W. Neely, D. R. Scherer, A. S. Bradley, M. J. Davis, and B. P. Anderson, “Spontaneous vortices in the formation of Bose–Einstein condensates,” Nature **455**(7215), 948–951 (2008). [CrossRef]

**4. **A. Bandyopadhyay and R. P. Singh, “Wigner distribution of elliptical quantum optical vortex,” Opt. Commun. **284**(1), 256–261 (2011). [CrossRef]

**5. **J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. **6**, 71 (2004). [CrossRef]

**6. **L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A **45**(11), 8185–8189 (1992). [CrossRef]

**7. **N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: An optical spanner,” Opt. Lett. **22**(1), 52–54 (1997). [CrossRef]

**8. **Z. X. Wang, N. Zhang, and X. C. Yuan, “High-volume optical vortex multiplexing and de-multiplexing for free-space optical communication,” Opt. Express **19**(2), 482–492 (2011). [CrossRef]

**9. **S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral phase contrast imaging in microscopy,” Opt. Express **13**(3), 689–694 (2005). [CrossRef]

**10. **G. A. Swartzlander Jr and C. T. Law, “Optical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. **69**(17), 2503–2506 (1992). [CrossRef]

**11. **Z. G. Chen, M. F. Shih, M. Segev, D. W. Wilson, R. E. Muller, and P. D. Maker, “Steady-state vortex-screening solitons formed in biased photorefractive media,” Opt. Lett. **22**(23), 1751–1753 (1997). [CrossRef]

**12. **V. Tikhonenko, J. Christou, and B. Lutherdaves, “Spiraling bright spatial solitons formed by the breakup of an optical vortex in a saturable self-focusing medium,” J. Opt. Soc. Am. B **12**(11), 2046–2052 (1995). [CrossRef]

**13. **J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov, “Solitary-wave vortices in quadratic nonlinear media,” J. Opt. Soc. Am. B **15**(2), 625–627 (1998). [CrossRef]

**14. **Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Stability of vortex solitons in thermal nonlinear media with cylindrical symmetry,” Opt. Express **15**(15), 9378–9384 (2007). [CrossRef]

**15. **W. J. Firth and D. V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett. **79**(13), 2450–2453 (1997). [CrossRef]

**16. **C. C. Jeng, M. F. Shih, K. Motzek, and Y. Kivshar, “Partially incoherent optical vortices in self-focusing nonlinear media,” Phys. Rev. Lett. **92**(4), 043904 (2004). [CrossRef]

**17. **S. Y. Chen, T. C. Lo, and M. F. Shih, “Stabilization of optical vortices in noninstantaneous self-focusing medium by small rotating intensity modulation,” Opt. Express **15**(21), 13689–13694 (2007). [CrossRef]

**18. **M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. **6**(2), 259–268 (2004). [CrossRef]

**19. **I. V. Basistiy, V. A. Pas ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A, Pure Appl. Opt. **6**(5), S166–S169 (2004). [CrossRef]

**20. **I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wave-front dislocations and their properties,” Opt. Commun. **119**(5-6), 604–612 (1995). [CrossRef]

**21. **J. B. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. **54**(12), 1723–1738 (2007). [CrossRef]

**22. **M. A. Molchan, E. V. Doktorov, and R. A. Vlasov, “Propagation of fractional charge Laguerre-Gaussian light beams in moving defocusing media with thermal nonlinearity,” J. Opt. A: Pure Appl. Opt. **11**(1), 015706 (2009). [CrossRef]

**23. **M. Segev, M. F. Shih, and G. C. Valley, “Photorefractive screening solitons of high and low intensity,” J. Opt. Soc. Am. B **13**(4), 706–718 (1996). [CrossRef]

**24. **B. Crosignani, P. DiPorto, A. Degasperis, M. Segev, and S. Trillo, “Three-dimensional optical beam propagation and solitons in photorefractive crystals,” J. Opt. Soc. Am. B **14**(11), 3078–3090 (1997). [CrossRef]

**25. **Notice there is a small difference between the applied nonlinearity strength in the simulation and in the experiment. This is due to the possible imhomogeneity of electro-optic coefficient of the crystal, or not exactly the same light beam sizes in simulation and in experiment.

**26. **A. V. Buryak, Y. S. Kivshar, M. F. Shih, and M. Segev, “Induced coherence and stable soliton spiraling,” Phys. Rev. Lett. **82**(1), 81–84 (1999). [CrossRef]

**27. **M. F. Shih, M. Segev, and G. Salamo, “Three-dimensional spiraling of interacting spatial solitons,” Phys. Rev. Lett. **78**(13), 2551–2554 (1997). [CrossRef]