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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 22 — Oct. 24, 2011
  • pp: 21457–21466
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Dipole azimuthons and vortex charge flipping in nematic liquid crystals

Yana V. Izdebskaya, Anton S. Desyatnikov, Gaetano Assanto, and Yuri S. Kivshar  »View Author Affiliations


Optics Express, Vol. 19, Issue 22, pp. 21457-21466 (2011)
http://dx.doi.org/10.1364/OE.19.021457


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Abstract

We demonstrate self-trapped laser beams carrying phase singularities in nematic liquid crystals. We experimentally observe the astigmatic transformation of vortex beams into spiraling dipole azimuthons accompanied by power-dependent charge-flipping of the on-axis phase singularity. The latter topological reactions involve triplets of vortex lines and resemble pitchfork bifurcations.

© 2011 OSA

1. Introduction

Spatial solitons in nematic liquid crystals, or nematicons [27

27. C. Conti, M. Peccianti, and G. Assanto, “Route to nonlocality and observation of accessible solitons,” Phys. Rev. Lett. 91, 073901 (2003). [CrossRef] [PubMed]

29

29. G. Assanto and M. Karpierz, “Nematicons: self-localized beams in nematic liquid crystals,” Liq. Cryst. 36, 1161–1172 (2009). [CrossRef]

], provide exciting opportunities to test theoretical predictions on the stability of various nonlocal vortex solitons. While spiraling of two nematicons was demonstrated earlier [30

30. A. Fratalocchi, M. Peccianti, C. Conti, and G. Assanto, “Spiraling and cyclic dynamics of nematicons,” Mol. Cryst. Liq. Cryst. 421, 197–207 (2004). [CrossRef]

32

32. A. Fratalocchi, A. Piccardi, M. Peccianti, and G. Assanto, “Nonlinear management of the angular momentum of soliton clusters: Theory and experiment,” Phys. Rev. A 75, 063835 (2007). [CrossRef]

], the dynamics of azimuthons carrying phase singularities and optical vortices remains unexplored. Particularly interesting among them are the most robust lowest-order single-charged vortices [8

8. D. Briedis, D. Petersen, D. Edmundson, W. Krolikowski, and O. Bang, “Ring vortex solitons in nonlocal nonlinear media,” Opt. Express 13, 435–443 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-13-2-435. [CrossRef] [PubMed]

,9

9. A. I. Yakimenko, Yu. A. Zaliznyak, and Yu. S. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” Phys. Rev. E 71, 065603 (2005). [CrossRef]

] and spiraling dipole azimuthons [10

10. S. Lopez-Aguayo, A. S. Desyatnikov, Yu. S. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, “Stable rotating dipole solitons in nonlocal optical media,” Opt. Lett. 31, 1100–1102 (2006). [CrossRef] [PubMed]

16

16. F. Maucher, D. Buccoliero, S. Skupin, M. Grech, A. S. Desyatnikov, and W. Krolikowski, “Tracking azimuthons in nonlocal nonlinear media,” Opt. Quantum Electron. 41, 337–348 (2009). [CrossRef]

]. An important open question is whether radially symmetric or azimuthally modulated vortex nematicons can be supported and stabilized by a reorientational nonlocal nonlinearity.

Here we report on the generation of dipole azimuthons in nematic liquid crystals. We observe experimentally and describe theoretically, for the first time to our knowledge, the formation of dipole azimuthons with nontrivial charge-flipping of on-axis phase dislocations. In these topological reactions the central phase dislocation (vortex line) splits into three lines with alternating topological charges, resembling a pitchfork bifurcation. We argue that these transformations are due to self-induced astigmatic deformation of the vortex beam [22

22. D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Yu. S. Kivshar, “Laguerre and Hermite soliton clusters in nonlocal nonlinear media,” Phys. Rev. Lett. 98, 053901 (2007). [CrossRef] [PubMed]

, 23

23. D. Buccoliero and A. S. Desyatnikov, “Quasi-periodic transformations of nonlocal spatial solitons,” Opt. Express 17, 9608–9613 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-12-9608. [CrossRef] [PubMed]

], which do not rely on the external anisotropy stemming from the boundaries [17

17. Y. V. Kartashov, V. V. Vysloukh, and L. Torner, “Stability of vortex solitons in thermal nonlinear media,” Opt. Express 15, 9378–9384 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-15-9378. [CrossRef] [PubMed]

21

21. Y. Zhang, S. Skupin, F. Maucher, A. Pour, K. Lu, and W. Krolikowski, “Azimuthons in weakly nonlinear waveguides of different symmetries,” Opt. Express 18, 27846–27857 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-18-26-27846. [CrossRef]

].

2. Experimental results

Our experiments are carried out in a planar cell filled with the nematic liquid crystal 6CHBT, with two parallel polycarbonate slides spaced by 110μm and uniformly rubbed at an angle π/4 with respect to the direction z of the input beam wavevector, see Figs. 1(a,b) and Refs. [33

33. Y. V. Izdebskaya, V. G. Shvedov, A. S. Desyatnikov, W. Z. Krolikowski, M. Belic, G. Assanto, and Yu. S. Kivshar, “Counterpropagating nematicons in bias-free liquid crystals,” Opt. Express 18, 3258–3263 (2010), http://www.opticsinfobase.org/abstract.cfm?URI=oe-18-4-3258. [CrossRef] [PubMed]

35

35. Y. V. Izdebskaya, A. S. Desyatnikov, G. Assanto, and Yu. S. Kivshar, “Multimode nematicon waveguides,” Opt. Lett. 36, 184–186 (2011). [CrossRef] [PubMed]

]. Glass plates at the input and output interfaces seal the cell to prevent lens-like effects and avoid light depolarization [see Fig. 1(b)]. A single-charged vortex beam is generated with a fork-type amplitude diffraction hologram using an extraordinarily polarized cw laser beam of wavelength λ = 800 nm and power P. The beam is coupled into the cell with a 10× microscope objective, resulting in an input ring of radius w ≈ 4μm at peak intensity, see Fig. 1(c). The beam dynamics in the medium is monitored by collecting both the light out-scattered through the top plate of the cell (top view, longitudinal dynamics) and the light transmitted through the output interface (output, transverse dynamics), using objectives and high-resolution CCD cameras.

Fig. 1 (a) Perspective and (b) top views of the unbiased planar cell; the ellipses indicate the oriented molecules. MO – microscope objectives; CCD – cameras. (c) Input vortex profile and (d–f) output intensity distributions. (d) Diffracted e-polarized single-charged (m = +1) vortex beam with P = 0.7 mW. (e,f) Vortex breakup into two filaments for P = 2.1 mW, tilted in the opposite direction with respect to the dark stripe, for input topological charges m = +1 in (e) and m = −1 in (f). Dashed lines in (d–f) indicate the edges of the cell.

Figures 1(c–f) present experimental results for vortex beams propagating in the sample. At low input power, P < 0.9 mW, the vortex beam uniformly diffracts without any noticeable self-action, see Fig. 1(d). As the power is increased, 1 < P[mW]< 2.2, the beam experiences self-focusing and the output spot visibly reduces, see Figs. 1(e,f). In this regime, the initial radially symmetric vortex undergoes a drastic transformation: the vortex “doughnut” breaks up into two bell-shaped beams and the dark core becomes a tilted stripe. Such symmetry-breaking can be explained in the context of vortex astigmatic transformations [36

36. E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991). [CrossRef]

, 37

37. V. Denisenko, V. Shvedov, A. S. Desyatnikov, D. Neshev, W. Krolikowski, A. Volyar, M. Soskin, and Y. S. Kivshar, “Determination of topological charges of polychromatic optical vortices,” Opt. Express 17, 23374–23379 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-26-23374. [CrossRef]

] due to the anisotropic structure of the planar cell and the anisotropic character of nematic liquid crystals. Specifically, as the intensity of a vortex beam grows, the (extraordinary) refractive index increases in the middle of the cell but remains constant along x at the cell boundaries where the molecules are anchored, see, e.g. Fig. 1(c) in Ref. [38

38. J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman , “Simulations and experiments on self-focusing conditions in nematic liquid-crystal planar cells,” Opt. Express 12, 1011–1018 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-6-1011. [CrossRef] [PubMed]

]. The index distribution thus becomes cylindrical and the sample behaves as a cylindrical lens. In full agreement with Ref. [37

37. V. Denisenko, V. Shvedov, A. S. Desyatnikov, D. Neshev, W. Krolikowski, A. Volyar, M. Soskin, and Y. S. Kivshar, “Determination of topological charges of polychromatic optical vortices,” Opt. Express 17, 23374–23379 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-26-23374. [CrossRef]

], we observe different transverse tilts of the elongated beam with respect to the dark stripe and the cell boundaries, depending on whether the charge of the input vortex is positive as in Fig. 1(e), or negative as in Fig. 1(f).

A further increase of the input vortex power (P > 2.3 mW) leads to more pronounced narrowing of the output beam and to the formation of self-trapped dipole azimuthons [10

10. S. Lopez-Aguayo, A. S. Desyatnikov, Yu. S. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, “Stable rotating dipole solitons in nonlocal optical media,” Opt. Lett. 31, 1100–1102 (2006). [CrossRef] [PubMed]

], as demonstrated by the results in Fig. 2. First, we discuss the longitudinal dynamics (Figs. 2(a–c)) for three input powers. The whole beam appears tilted and bent owing to walk-off [33

33. Y. V. Izdebskaya, V. G. Shvedov, A. S. Desyatnikov, W. Z. Krolikowski, M. Belic, G. Assanto, and Yu. S. Kivshar, “Counterpropagating nematicons in bias-free liquid crystals,” Opt. Express 18, 3258–3263 (2010), http://www.opticsinfobase.org/abstract.cfm?URI=oe-18-4-3258. [CrossRef] [PubMed]

35

35. Y. V. Izdebskaya, A. S. Desyatnikov, G. Assanto, and Yu. S. Kivshar, “Multimode nematicon waveguides,” Opt. Lett. 36, 184–186 (2011). [CrossRef] [PubMed]

], but its transverse size varies only slightly during propagation (in comparison with the diffraction pattern in Fig. 1(d)) as diffraction is suppressed. Nevertheless, the variation in waist indicates the breathing of the self-trapped beam [28

28. C. Conti, M. Peccianti, and G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92, 113902 (2004). [CrossRef] [PubMed]

, 39

39. A. Snyder and J. Mitchell, “Accessible solitons,” Science 276, 1538–1541 (1997). [CrossRef]

, 40

40. M. Peccianti, A. Fratalocchi, and G. Assanto, “Transverse dynamics of Nematicons,” Opt. Express 12, 6524–6529 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-26-6524. [CrossRef] [PubMed]

], and the excitation of oscillatory modes of a stable dipole soliton [5

5. A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” Prog. Opt. 47, 291–391 (2005). [CrossRef]

], particularly robust and long-lived in nonlocal dielectrics [8

8. D. Briedis, D. Petersen, D. Edmundson, W. Krolikowski, and O. Bang, “Ring vortex solitons in nonlocal nonlinear media,” Opt. Express 13, 435–443 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-13-2-435. [CrossRef] [PubMed]

,9

9. A. I. Yakimenko, Yu. A. Zaliznyak, and Yu. S. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” Phys. Rev. E 71, 065603 (2005). [CrossRef]

].

Fig. 2 Generation and propagation of a dipole azimuthon, with power-driven beam twist and breathing. (a–c) Evolution in the plane (x,z) of a beam with input charge m = +1 for three different input powers. The walk-off angle of about 4.5° is defined by the molecular alignment at the cell boundaries. (d–f) Transverse intensity patterns at the cell output for beams with m = +1; (g–i) as in (d–f) but for m = −1. (j) Beam HWHM versus input power P (see text).

The vortex develops an astigmatic deformation in propagation, splitting into two bright spots as it travels in the first half of the sample (i.e. small z). The longitudinal dark line visible in Figs. 2(a–c) disappears when the two dipole lobes are aligned vertically (i.e. along y), by the output in Fig. 2(b) and well inside the cell in Fig. 2(c). The corresponding output transverse patterns in Figs. 2(d–f) display two well pronounced bright spots, clearly coupled together in a structure similar to that of simulated dipole azimuthons [10

10. S. Lopez-Aguayo, A. S. Desyatnikov, Yu. S. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, “Stable rotating dipole solitons in nonlocal optical media,” Opt. Lett. 31, 1100–1102 (2006). [CrossRef] [PubMed]

]. By varying the power excitation, the output dipole exhibits a strongly varying elliptic shape with tilt at different angles with respect to the boundaries.

Since we observe nematicon deformations in real time, for a better quantitative estimate of soliton parameters we time-average the output images recorded during data acquisition. Figure 2(j) presents our measurements of the output beam half-width at half-maximum (HWHM) wx,y, i.e. the half-size of a rectangle enclosing the contour of the averaged transverse intensity profile at the half-peak. The graph shows that, by increasing the input power from low values up to P ∼ 2.5 mW, the beam width gradually decreases, indicating a transition from self-focusing to solitary regime. The latter corresponds to the interval 2.5 < P[mW]< 6.5 with the formation of stable azimuthons. In this domain the beam can get more elongated in the vertical (y) direction, wy > wx, as emphasized by the shaded regions in Fig. 2(j) and confirming the power-dependent twist. Higher powers, P > 6.8 mW, lead to destabilization of the dipole, in contrast with theoretical predictions [10

10. S. Lopez-Aguayo, A. S. Desyatnikov, Yu. S. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, “Stable rotating dipole solitons in nonlocal optical media,” Opt. Lett. 31, 1100–1102 (2006). [CrossRef] [PubMed]

]. At high excitations, in fact, the temporal dynamics is amplified while the beams develop multi-humped and irregular structures.

Next, we investigate the singular phase structure of dipole azimuthons by employing an interferometric technique: a tilted broad Gaussian beam at an angle with the dipole interferes with it at the output. The interferograms in Fig. 3 allow detecting phase dislocations by the characteristic presence of fork dislocations. Two opposite topological charges, m = ±1, are distinguished by the fork orientation, either down or up.

Fig. 3 (a–h) Acquired intensity profiles of nonlocal dipole azimuthons (left) and corresponding interferograms (right); the circles indicate the positions of vortices with charges +1 (green) and −1 (blue). (i) Topological charge m0 of the on-axis phase singularity; the dashed vertical lines mark the input powers in (a–h).

Figure 3 shows results for an input vortex with m = +1. At small powers, i.e. P = 2.3 mW in Fig. 3(a), the input phase dislocation is preserved, even if the intensity changes substantially, consistently with theoretical predictions [10

10. S. Lopez-Aguayo, A. S. Desyatnikov, Yu. S. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, “Stable rotating dipole solitons in nonlocal optical media,” Opt. Lett. 31, 1100–1102 (2006). [CrossRef] [PubMed]

, 11

11. S. Lopez-Aguayo, A. S. Desyatnikov, and Yu. S. Kivshar, “Azimuthons in nonlocal nonlinear media,” Opt. Express 14, 7903–7908 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-14-17-7903. [CrossRef] [PubMed]

]. However, as the power is increased further, alterations in the interferograms are significant enough that the topological structure can no longer be clearly identified, as in Figs. 3(b,g). This regime corresponds to the flipping of a topological charge in Fig. 3(i), at P ∼ 2.45 mW and P ∼ 3.3 mW, respectively.

Excitations above the critical value corresponding to charge flipping result in stable output interferograms, see Fig. 3(c) for P = 2.6 mW. Remarkably, we observe a triplet of vortices, with the central (on-axis) vortex of charge m0 = −1 and two satellite vortices with an opposite m1,2 = +1, so that the total charge remains unchanged, m = m0 + m1 + m2 = +1. By monitoring the output versus increasing power P [see Figs. 3(c–e)], we observe first the spatial separation of the triplet and then the attraction of vortices in Fig. 3(f), followed by flipping of the central vortex charge from m0 = −1 to m0 = +1 above P ∼ 3.3 mW, see Figs. 3(g–i). With a further increase of power the charge-flipping process repeats again at P ∼ 3.7 mW before a vortex triplet with m0 = −1 remains in the field.

Noteworthy, similar experiments with input vortex beams carrying charge m = −1 lead to an opposite direction of the spatial twist and different profiles, see Fig. 2(g–i). Nevertheless, we reliably measure the same critical power values for charge-flipping as in Fig. 3(i) and observe similar topological reactions involving triplets of vortices. We can therefore conclude that the reported charge-flipping is robust and reproducible.

3. Numerical simulations and discussion

In order to better understand and interpret the surprising complexity of the observed topological structure of the field of nonlocal dipole azimuthons we resort to numerical simulations. Our scope is twofold: verifying the charge-flipping phenomenon and studying its origin while identifying the symmetry-breaking mechanism. In particular, we aim at confirming the hypothesis that neither the boundaries or the medium anisotropy are significant in triggering the topological reactions; hence, the nonlocal self-action of a vortex nematicon itself suffices. To this end, we consider a generic isotropic and unbounded medium governed by the nonlinear Schrödinger equation with a nonlocal nonlinearity [10

10. S. Lopez-Aguayo, A. S. Desyatnikov, Yu. S. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, “Stable rotating dipole solitons in nonlocal optical media,” Opt. Lett. 31, 1100–1102 (2006). [CrossRef] [PubMed]

,11

11. S. Lopez-Aguayo, A. S. Desyatnikov, and Yu. S. Kivshar, “Azimuthons in nonlocal nonlinear media,” Opt. Express 14, 7903–7908 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-14-17-7903. [CrossRef] [PubMed]

],
izE+x2E+y2E+Ee|rρ|2|E(ρ)|2dρ=0.
(1)
Here the propagation variable z is normalized to the diffraction length, x and y are the transverse coordinates. The electric field of a dipole azimuthon is described by the ansatz [10

10. S. Lopez-Aguayo, A. S. Desyatnikov, Yu. S. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, “Stable rotating dipole solitons in nonlocal optical media,” Opt. Lett. 31, 1100–1102 (2006). [CrossRef] [PubMed]

, 11

11. S. Lopez-Aguayo, A. S. Desyatnikov, and Yu. S. Kivshar, “Azimuthons in nonlocal nonlinear media,” Opt. Express 14, 7903–7908 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-14-17-7903. [CrossRef] [PubMed]

], E = A(x + ipy)exp −(x2 + y2)/2w2, with amplitude A and half-width w. The modulation parameter 0 < |p| < 1 (noncanonical vortex strength [44

44. G. Molina-Terriza, E. M. Wright, and L. Torner, “Propagation and control of noncanonical optical vortices,” Opt. Lett. 26, 163–165 (2001). [CrossRef]

, 45

45. G. Molina-Terriza, J. Recolons, J. P. Torres, and L. Torner, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001). [CrossRef]

]) defines a family of dipole azimuthons carrying a vortex with topological charge m = sign(p).

We approximate the experimental conditions by numerically simulating the dynamics of a vortex soliton [8

8. D. Briedis, D. Petersen, D. Edmundson, W. Krolikowski, and O. Bang, “Ring vortex solitons in nonlocal nonlinear media,” Opt. Express 13, 435–443 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-13-2-435. [CrossRef] [PubMed]

,9

9. A. I. Yakimenko, Yu. A. Zaliznyak, and Yu. S. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” Phys. Rev. E 71, 065603 (2005). [CrossRef]

] with p = 1, astigmatically deformed at the input by stretch in one direction, yy/1.2, see Fig. 4. This initial deformation leads to a corresponding elliptic distortion of the optically induced refractive index profile, similar to stressed optical fibers [46

46. D. McGloin, N. B. Simpson, and M. J. Padgett, “Transfer of orbital angular momentum from a stressed fiber-optic waveguide to a light beam,” Appl. Opt. 37, 469–472 (1998). [CrossRef]

48

48. A. V. Carpentier, A. V. Carpentier, H. Michinel, J. R. Salgueiro, S. Doval, and A. Ferrando, “Inversion of a guided optical vortex,” J. Europ. Opt. Soc. 1, 06031 (2006).

] supporting mode-transformations and vortex charge-flipping. However, in our case the trapping potential is nonlinear, i.e. self-induced, and it is nontrivial to aseess whether its initial deformation can propagate and sustain astigmatic mode transformations [36

36. E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991). [CrossRef]

,49

49. J. J. García-Ripoll, G. Molina-Terriza, V. M. Perez-Garcia, and L. Torner, “Structural instability of vortices in Bose-Einstein condensates,” Phys. Rev. Lett. 87, 140403 (2001). [CrossRef] [PubMed]

].

Fig. 4 Numerical dynamics of an initially stretched stable nonlocal vortex soliton. (a) Intensity profiles. (b) Volume half-intensity surface (green) and trajectories of vortices: red for m = +1 and blue for m = −1. (c,d) Magnified fragments with pitchfork topological reactions in (b), including their sequence at z ≃ 29.5. (e) Evolution of the zero-level contours ReE = 0 (red) and ImE = 0 (blue) at the super-critical pitchfork reaction in (c); the circles indicate vortex locations.

At low powers the vortex breaks into two uncoupled filaments [8

8. D. Briedis, D. Petersen, D. Edmundson, W. Krolikowski, and O. Bang, “Ring vortex solitons in nonlocal nonlinear media,” Opt. Express 13, 435–443 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-13-2-435. [CrossRef] [PubMed]

10

10. S. Lopez-Aguayo, A. S. Desyatnikov, Yu. S. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, “Stable rotating dipole solitons in nonlocal optical media,” Opt. Lett. 31, 1100–1102 (2006). [CrossRef] [PubMed]

], while at large powers there is no charge-flipping because of the transition to the quasi-linear limit of high nonlocality [39

39. A. Snyder and J. Mitchell, “Accessible solitons,” Science 276, 1538–1541 (1997). [CrossRef]

]. Closer to the stability border, the dipole azimuthons show strong spatial oscillations in Fig. 4(a,b), similar to the experimental results in Fig. 2. Persistent oscillations indicate the excitation of internal vibrational modes of the dipole azimuthon, which lead to the appearance of additional vortex lines as shown in Fig. 4(b). Several vortices are visible and the central phase dislocation changes its vorticity several times. The two magnified plots in Fig. 4(c,d) confirm the unexpected character of the charge-flipping, with the sudden appearance of a vortex triplet. These topological reactions are reminiscent of super-critical [Fig. 4(c)] and sub-critical [Fig. 4(d) at z ≃ 31.5] pitchfork bifurcations in the theory of nonlinear dynamical systems.

Similar topological reactions were previously observed in interference patterns of two Laguerre-Gaussian vortex beams [50

50. G. Molina-Terriza, J. Recolons, and L. Torner, “The curious arithmetic of optical vortices,” Opt. Lett. 25, 1135–1137 (2000). [CrossRef]

]. In general, perturbations of a perfect pitchfork, as in Figs. 4(c,d) and in Fig. 1(a,b) of Ref. [50

50. G. Molina-Terriza, J. Recolons, and L. Torner, “The curious arithmetic of optical vortices,” Opt. Lett. 25, 1135–1137 (2000). [CrossRef]

], lead to triplet splitting into a perturbed but well defined original vortex line and a vortex-antivortex pair, as in Fig. 1(c,d) of Ref. [50

50. G. Molina-Terriza, J. Recolons, and L. Torner, “The curious arithmetic of optical vortices,” Opt. Lett. 25, 1135–1137 (2000). [CrossRef]

]. Such reaction was previously resported in experiments with photonic lattices [51

51. A. Bezryadina, D. N. Neshev, A. S. Desyatnikov, J. Young, Z. G. Chen, and Yu. S. Kivshar, “Observation of topological transformations of optical vortices in two-dimensional photonic lattices,” Opt. Express 14, 8317–8327 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-18-8317. [CrossRef] [PubMed]

]. The main difference with our experimental data in Fig. 4 is that the spatial separation between phase singularities at the bifurcation points in Figs. 4(b,g) is so small that cannot be resolved. As a result, we cannot trace the original vortex line of charge m0 = +1 in Fig. 4(a) to one of the two side-vortices in Fig. 4(c). Otherwise stated, the experimentally observed topological reactions are indeed very close to the ideal pitchfork and in perfect agreement with numerics in Figs. 4(b,g).

It should be underlined that the analysis is further complicated by the fact that images acquired at the cell output (Fig. 3) cannot be directly compared with propagation dynamics at fixed power (Fig. 4). Nevertheless, since the rotation rate and breathing frequency depend on power in a continuous fashion, the sequence of output images in Fig. 3 can be qualitatively mapped onto the propagation dynamics with pitchfork reactions in Figs. 4(c,d). Furthermore, considering the dynamics of vortex lines and topological reactions versus a control parameter, e.g. either the power in experiments or the propagation variable z in numerics, then the experimental results in Fig. 3 pinpoint the occurrence of pitchfork topological reactions.

4. Conclusions

Finally, we note the relevance of our results in other fields, such as dissipative nonlinear optics, with rich and somewhat unexpected vortex interactions [52

52. P. L. Ramazza, U. Bortolozzo, and L. Pastur, “Phase singularities in triangular dissipative solitons,” J. Opt. A 6S266–S270 (2004).

, 53

53. N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, “Curvilinear motion of multivortex laser-soliton complexes with strong and weak coupling,” Phys. Rev. Lett. 95, 053903 (2005). [CrossRef] [PubMed]

]. Very recently, the nucleation of vortex-antivortex pairs has attracted attention in dissipative optical solitons [54

54. F. Haudin, R. G. Rojas, U. Bortolozzo, M. G. Clerc, and S. Residori, “Vortex emission accompanies the advection of optical localized structures,” Phys. Rev. Lett. 106, 063901 (2011). [CrossRef] [PubMed]

], as well as an evidence of superfluidity in exciton-polariton Bose-Einstein condensates in semiconductor microcavities [55

55. G. Roumpos, M. D. Fraser, A. Löffler, S. Höfling, A. Forchel, and Y. Yamamoto, “Single vortex-antivortex pair in an exciton-polariton condensate,” Nat. Phys. 7, 129 (2011). [CrossRef]

]. It would be rather interesting to extend our observations of complex pitchfork topological reactions of quantized vortices in nematic liquid crystals to dissipative systems, as well.

Acknowledgments

We acknowledge support from the Australian Research Council and thank G. Molina-Terriza, E. Abramochkin, and W. Krolikowski for useful discussions.

References and links

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Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

2.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London A 336, 165–190 (1974). [CrossRef]

3.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001). [CrossRef]

4.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 52, 293–363 (2009). [CrossRef]

5.

A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” Prog. Opt. 47, 291–391 (2005). [CrossRef]

6.

A. S. Desyatnikov and Yu. S. Kivshar, “Rotating optical soliton clusters,” Phys. Rev. Lett. 88, 053901 (2002). [CrossRef] [PubMed]

7.

A. S. Desyatnikov, A. A. Sukhorukov, and Yu. S. Kivshar, “Azimuthons: spatially modulated vortex solitons,” Phys. Rev. Lett. 95, 203904 (2005). [CrossRef] [PubMed]

8.

D. Briedis, D. Petersen, D. Edmundson, W. Krolikowski, and O. Bang, “Ring vortex solitons in nonlocal nonlinear media,” Opt. Express 13, 435–443 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-13-2-435. [CrossRef] [PubMed]

9.

A. I. Yakimenko, Yu. A. Zaliznyak, and Yu. S. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” Phys. Rev. E 71, 065603 (2005). [CrossRef]

10.

S. Lopez-Aguayo, A. S. Desyatnikov, Yu. S. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, “Stable rotating dipole solitons in nonlocal optical media,” Opt. Lett. 31, 1100–1102 (2006). [CrossRef] [PubMed]

11.

S. Lopez-Aguayo, A. S. Desyatnikov, and Yu. S. Kivshar, “Azimuthons in nonlocal nonlinear media,” Opt. Express 14, 7903–7908 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-14-17-7903. [CrossRef] [PubMed]

12.

S. Skupin, O. Bang, D. Edmundson, and W. Krolikowski, “Stability of two-dimensional spatial solitons in nonlocal nonlinear media,” Phys. Rev. E 73, 066603 (2006). [CrossRef]

13.

A. I. Yakimenko, V. M. Lashkin, and O. O. Prikhodko, “Dynamics of two-dimensional coherent structures in nonlocal nonlinear media,” Phys. Rev. E 73, 066605 (2006). [CrossRef]

14.

V. M. Lashkin, A. I. Yakimenko, and O. O. Prikhodko, “Two-dimensional nonlocal multisolitons,” Phys. Rev. A 366, 422–427 (2007).

15.

S. Skupin, M. Saffman, and W. Krolikowski, “Nonlocal stabilization of nonlinear beams in a self-focusing atomic vapor,” Phys. Rev. Lett. 98, 263902 (2007). [CrossRef] [PubMed]

16.

F. Maucher, D. Buccoliero, S. Skupin, M. Grech, A. S. Desyatnikov, and W. Krolikowski, “Tracking azimuthons in nonlocal nonlinear media,” Opt. Quantum Electron. 41, 337–348 (2009). [CrossRef]

17.

Y. V. Kartashov, V. V. Vysloukh, and L. Torner, “Stability of vortex solitons in thermal nonlinear media,” Opt. Express 15, 9378–9384 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-15-9378. [CrossRef] [PubMed]

18.

F. Ye, Y. V. Kartashov, and L. Torner , “Stabilization of dipole solitons in nonlocal nonlinear media,” Phys. Rev. A 77, 043821 (2008). [CrossRef]

19.

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Yu. S. Kivshar, “Boundary effects on the dynamics of higher-order optical spatial solitons in nonlocal thermal media,” J. Opt. A 11, 094014 (2009).

20.

A. A. Minzoni, N. F. Smyth, and Z. Xu, “Stability of an optical vortex in a circular nematic cell,” Phys. Rev. A 81, 033816 (2010). [CrossRef]

21.

Y. Zhang, S. Skupin, F. Maucher, A. Pour, K. Lu, and W. Krolikowski, “Azimuthons in weakly nonlinear waveguides of different symmetries,” Opt. Express 18, 27846–27857 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-18-26-27846. [CrossRef]

22.

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Yu. S. Kivshar, “Laguerre and Hermite soliton clusters in nonlocal nonlinear media,” Phys. Rev. Lett. 98, 053901 (2007). [CrossRef] [PubMed]

23.

D. Buccoliero and A. S. Desyatnikov, “Quasi-periodic transformations of nonlocal spatial solitons,” Opt. Express 17, 9608–9613 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-12-9608. [CrossRef] [PubMed]

24.

C. Rotschild, O. Cohen, O. Manela, Tal Carmon, and M. Segev, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95, 213904 (2005). [CrossRef] [PubMed]

25.

C. Rotschild, M. Segev, Z. Xu, Ya. V. Kartashov, L. Torner, and O. Cohen, “Two-dimensional multipole solitons in nonlocal nonlinear media,” Opt. Lett. 31, 3312–3314 (2006). [CrossRef] [PubMed]

26.

A. Minovich, D. N. Neshev, A. S. Desyatnikov, W. Krolikowski, and Yu. S. Kivshar, “Observation of optical azimuthons,” Opt. Express 17, 23610–23616 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-26-23610. [CrossRef]

27.

C. Conti, M. Peccianti, and G. Assanto, “Route to nonlocality and observation of accessible solitons,” Phys. Rev. Lett. 91, 073901 (2003). [CrossRef] [PubMed]

28.

C. Conti, M. Peccianti, and G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92, 113902 (2004). [CrossRef] [PubMed]

29.

G. Assanto and M. Karpierz, “Nematicons: self-localized beams in nematic liquid crystals,” Liq. Cryst. 36, 1161–1172 (2009). [CrossRef]

30.

A. Fratalocchi, M. Peccianti, C. Conti, and G. Assanto, “Spiraling and cyclic dynamics of nematicons,” Mol. Cryst. Liq. Cryst. 421, 197–207 (2004). [CrossRef]

31.

A. Fratalocchi, A. Piccardi, M. Peccianti, and G. Assanto, “Nonlinearly controlled angular momentum of soliton clusters,” Opt. Lett. 32, 1447–1449 (2007). [CrossRef] [PubMed]

32.

A. Fratalocchi, A. Piccardi, M. Peccianti, and G. Assanto, “Nonlinear management of the angular momentum of soliton clusters: Theory and experiment,” Phys. Rev. A 75, 063835 (2007). [CrossRef]

33.

Y. V. Izdebskaya, V. G. Shvedov, A. S. Desyatnikov, W. Z. Krolikowski, M. Belic, G. Assanto, and Yu. S. Kivshar, “Counterpropagating nematicons in bias-free liquid crystals,” Opt. Express 18, 3258–3263 (2010), http://www.opticsinfobase.org/abstract.cfm?URI=oe-18-4-3258. [CrossRef] [PubMed]

34.

Y. V. Izdebskaya, V. G. Shvedov, A. S. Desyatnikov, W. Krolikowski, and Yu. S. Kivshar, “Soliton bending and routing induced by interaction with curved surfaces in nematic liquid crystals,” Opt. Lett. 35, 1692–1694 (2010). [CrossRef] [PubMed]

35.

Y. V. Izdebskaya, A. S. Desyatnikov, G. Assanto, and Yu. S. Kivshar, “Multimode nematicon waveguides,” Opt. Lett. 36, 184–186 (2011). [CrossRef] [PubMed]

36.

E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991). [CrossRef]

37.

V. Denisenko, V. Shvedov, A. S. Desyatnikov, D. Neshev, W. Krolikowski, A. Volyar, M. Soskin, and Y. S. Kivshar, “Determination of topological charges of polychromatic optical vortices,” Opt. Express 17, 23374–23379 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-26-23374. [CrossRef]

38.

J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman , “Simulations and experiments on self-focusing conditions in nematic liquid-crystal planar cells,” Opt. Express 12, 1011–1018 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-6-1011. [CrossRef] [PubMed]

39.

A. Snyder and J. Mitchell, “Accessible solitons,” Science 276, 1538–1541 (1997). [CrossRef]

40.

M. Peccianti, A. Fratalocchi, and G. Assanto, “Transverse dynamics of Nematicons,” Opt. Express 12, 6524–6529 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-26-6524. [CrossRef] [PubMed]

41.

W. Krolikowski, E. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Y. S. Kivshar, C. Denz, and B. Luther-Davies, “Observation of dipole-mode vector solitons,” Phys. Rev. Lett. 85, 1424 (2000). [CrossRef] [PubMed]

42.

T. Carmon, R. Uzdin, C. Pigier, Z. H. Musslimani, M. Segev, and A. Nepomnyashchy, “Rotating propeller solitons,” Phys. Rev. Lett. 87, 143901 (2001). [CrossRef] [PubMed]

43.

D. Neshev, G. McCarthy, W. Krolikowski, E. A. Ostrovskaya, Yu. S. Kivshar, G. F. Calvo, and F. Agullo-Lopez, “Dipole-mode vector solitons in anisotropic nonlocal self-focusing media,” Opt. Lett. 26, 1185–1187 (2001). [CrossRef]

44.

G. Molina-Terriza, E. M. Wright, and L. Torner, “Propagation and control of noncanonical optical vortices,” Opt. Lett. 26, 163–165 (2001). [CrossRef]

45.

G. Molina-Terriza, J. Recolons, J. P. Torres, and L. Torner, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001). [CrossRef]

46.

D. McGloin, N. B. Simpson, and M. J. Padgett, “Transfer of orbital angular momentum from a stressed fiber-optic waveguide to a light beam,” Appl. Opt. 37, 469–472 (1998). [CrossRef]

47.

A. Ya. Bekshaev, M. V. Vasnetsov, V. G. Denisenko, and M. S. Soskin, “Transformation of the orbital angular momentum of a beam with optical vortex in an astigmatic optical system,” JETP Lett. 75, 127–130 (2002). [CrossRef]

48.

A. V. Carpentier, A. V. Carpentier, H. Michinel, J. R. Salgueiro, S. Doval, and A. Ferrando, “Inversion of a guided optical vortex,” J. Europ. Opt. Soc. 1, 06031 (2006).

49.

J. J. García-Ripoll, G. Molina-Terriza, V. M. Perez-Garcia, and L. Torner, “Structural instability of vortices in Bose-Einstein condensates,” Phys. Rev. Lett. 87, 140403 (2001). [CrossRef] [PubMed]

50.

G. Molina-Terriza, J. Recolons, and L. Torner, “The curious arithmetic of optical vortices,” Opt. Lett. 25, 1135–1137 (2000). [CrossRef]

51.

A. Bezryadina, D. N. Neshev, A. S. Desyatnikov, J. Young, Z. G. Chen, and Yu. S. Kivshar, “Observation of topological transformations of optical vortices in two-dimensional photonic lattices,” Opt. Express 14, 8317–8327 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-18-8317. [CrossRef] [PubMed]

52.

P. L. Ramazza, U. Bortolozzo, and L. Pastur, “Phase singularities in triangular dissipative solitons,” J. Opt. A 6S266–S270 (2004).

53.

N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, “Curvilinear motion of multivortex laser-soliton complexes with strong and weak coupling,” Phys. Rev. Lett. 95, 053903 (2005). [CrossRef] [PubMed]

54.

F. Haudin, R. G. Rojas, U. Bortolozzo, M. G. Clerc, and S. Residori, “Vortex emission accompanies the advection of optical localized structures,” Phys. Rev. Lett. 106, 063901 (2011). [CrossRef] [PubMed]

55.

G. Roumpos, M. D. Fraser, A. Löffler, S. Höfling, A. Forchel, and Y. Yamamoto, “Single vortex-antivortex pair in an exciton-polariton condensate,” Nat. Phys. 7, 129 (2011). [CrossRef]

OCIS Codes
(160.3710) Materials : Liquid crystals
(050.4865) Diffraction and gratings : Optical vortices
(190.6135) Nonlinear optics : Spatial solitons

ToC Category:
Nonlinear Optics

History
Original Manuscript: August 5, 2011
Revised Manuscript: October 4, 2011
Manuscript Accepted: October 5, 2011
Published: October 17, 2011

Citation
Yana V. Izdebskaya, Anton S. Desyatnikov, Gaetano Assanto, and Yuri S. Kivshar, "Dipole azimuthons and vortex charge flipping in nematic liquid crystals," Opt. Express 19, 21457-21466 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-22-21457


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References

  1. Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).
  2. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London A336, 165–190 (1974). [CrossRef]
  3. M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt.42, 219–276 (2001). [CrossRef]
  4. M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt.52, 293–363 (2009). [CrossRef]
  5. A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” Prog. Opt.47, 291–391 (2005). [CrossRef]
  6. A. S. Desyatnikov and Yu. S. Kivshar, “Rotating optical soliton clusters,” Phys. Rev. Lett.88, 053901 (2002). [CrossRef] [PubMed]
  7. A. S. Desyatnikov, A. A. Sukhorukov, and Yu. S. Kivshar, “Azimuthons: spatially modulated vortex solitons,” Phys. Rev. Lett.95, 203904 (2005). [CrossRef] [PubMed]
  8. D. Briedis, D. Petersen, D. Edmundson, W. Krolikowski, and O. Bang, “Ring vortex solitons in nonlocal nonlinear media,” Opt. Express13, 435–443 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-13-2-435 . [CrossRef] [PubMed]
  9. A. I. Yakimenko, Yu. A. Zaliznyak, and Yu. S. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” Phys. Rev. E71, 065603 (2005). [CrossRef]
  10. S. Lopez-Aguayo, A. S. Desyatnikov, Yu. S. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, “Stable rotating dipole solitons in nonlocal optical media,” Opt. Lett.31, 1100–1102 (2006). [CrossRef] [PubMed]
  11. S. Lopez-Aguayo, A. S. Desyatnikov, and Yu. S. Kivshar, “Azimuthons in nonlocal nonlinear media,” Opt. Express14, 7903–7908 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-14-17-7903 . [CrossRef] [PubMed]
  12. S. Skupin, O. Bang, D. Edmundson, and W. Krolikowski, “Stability of two-dimensional spatial solitons in nonlocal nonlinear media,” Phys. Rev. E73, 066603 (2006). [CrossRef]
  13. A. I. Yakimenko, V. M. Lashkin, and O. O. Prikhodko, “Dynamics of two-dimensional coherent structures in nonlocal nonlinear media,” Phys. Rev. E73, 066605 (2006). [CrossRef]
  14. V. M. Lashkin, A. I. Yakimenko, and O. O. Prikhodko, “Two-dimensional nonlocal multisolitons,” Phys. Rev. A366, 422–427 (2007).
  15. S. Skupin, M. Saffman, and W. Krolikowski, “Nonlocal stabilization of nonlinear beams in a self-focusing atomic vapor,” Phys. Rev. Lett.98, 263902 (2007). [CrossRef] [PubMed]
  16. F. Maucher, D. Buccoliero, S. Skupin, M. Grech, A. S. Desyatnikov, and W. Krolikowski, “Tracking azimuthons in nonlocal nonlinear media,” Opt. Quantum Electron.41, 337–348 (2009). [CrossRef]
  17. Y. V. Kartashov, V. V. Vysloukh, and L. Torner, “Stability of vortex solitons in thermal nonlinear media,” Opt. Express15, 9378–9384 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-15-9378 . [CrossRef] [PubMed]
  18. F. Ye, Y. V. Kartashov, and L. Torner , “Stabilization of dipole solitons in nonlocal nonlinear media,” Phys. Rev. A77, 043821 (2008). [CrossRef]
  19. D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Yu. S. Kivshar, “Boundary effects on the dynamics of higher-order optical spatial solitons in nonlocal thermal media,” J. Opt. A11, 094014 (2009).
  20. A. A. Minzoni, N. F. Smyth, and Z. Xu, “Stability of an optical vortex in a circular nematic cell,” Phys. Rev. A81, 033816 (2010). [CrossRef]
  21. Y. Zhang, S. Skupin, F. Maucher, A. Pour, K. Lu, and W. Krolikowski, “Azimuthons in weakly nonlinear waveguides of different symmetries,” Opt. Express18, 27846–27857 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-18-26-27846 . [CrossRef]
  22. D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Yu. S. Kivshar, “Laguerre and Hermite soliton clusters in nonlocal nonlinear media,” Phys. Rev. Lett.98, 053901 (2007). [CrossRef] [PubMed]
  23. D. Buccoliero and A. S. Desyatnikov, “Quasi-periodic transformations of nonlocal spatial solitons,” Opt. Express17, 9608–9613 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-12-9608 . [CrossRef] [PubMed]
  24. C. Rotschild, O. Cohen, O. Manela, Tal Carmon, and M. Segev, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett.95, 213904 (2005). [CrossRef] [PubMed]
  25. C. Rotschild, M. Segev, Z. Xu, Ya. V. Kartashov, L. Torner, and O. Cohen, “Two-dimensional multipole solitons in nonlocal nonlinear media,” Opt. Lett.31, 3312–3314 (2006). [CrossRef] [PubMed]
  26. A. Minovich, D. N. Neshev, A. S. Desyatnikov, W. Krolikowski, and Yu. S. Kivshar, “Observation of optical azimuthons,” Opt. Express17, 23610–23616 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-26-23610 . [CrossRef]
  27. C. Conti, M. Peccianti, and G. Assanto, “Route to nonlocality and observation of accessible solitons,” Phys. Rev. Lett.91, 073901 (2003). [CrossRef] [PubMed]
  28. C. Conti, M. Peccianti, and G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett.92, 113902 (2004). [CrossRef] [PubMed]
  29. G. Assanto and M. Karpierz, “Nematicons: self-localized beams in nematic liquid crystals,” Liq. Cryst.36, 1161–1172 (2009). [CrossRef]
  30. A. Fratalocchi, M. Peccianti, C. Conti, and G. Assanto, “Spiraling and cyclic dynamics of nematicons,” Mol. Cryst. Liq. Cryst.421, 197–207 (2004). [CrossRef]
  31. A. Fratalocchi, A. Piccardi, M. Peccianti, and G. Assanto, “Nonlinearly controlled angular momentum of soliton clusters,” Opt. Lett.32, 1447–1449 (2007). [CrossRef] [PubMed]
  32. A. Fratalocchi, A. Piccardi, M. Peccianti, and G. Assanto, “Nonlinear management of the angular momentum of soliton clusters: Theory and experiment,” Phys. Rev. A75, 063835 (2007). [CrossRef]
  33. Y. V. Izdebskaya, V. G. Shvedov, A. S. Desyatnikov, W. Z. Krolikowski, M. Belic, G. Assanto, and Yu. S. Kivshar, “Counterpropagating nematicons in bias-free liquid crystals,” Opt. Express18, 3258–3263 (2010), http://www.opticsinfobase.org/abstract.cfm?URI=oe-18-4-3258 . [CrossRef] [PubMed]
  34. Y. V. Izdebskaya, V. G. Shvedov, A. S. Desyatnikov, W. Krolikowski, and Yu. S. Kivshar, “Soliton bending and routing induced by interaction with curved surfaces in nematic liquid crystals,” Opt. Lett.35, 1692–1694 (2010). [CrossRef] [PubMed]
  35. Y. V. Izdebskaya, A. S. Desyatnikov, G. Assanto, and Yu. S. Kivshar, “Multimode nematicon waveguides,” Opt. Lett.36, 184–186 (2011). [CrossRef] [PubMed]
  36. E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun.83, 123–135 (1991). [CrossRef]
  37. V. Denisenko, V. Shvedov, A. S. Desyatnikov, D. Neshev, W. Krolikowski, A. Volyar, M. Soskin, and Y. S. Kivshar, “Determination of topological charges of polychromatic optical vortices,” Opt. Express17, 23374–23379 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-26-23374 . [CrossRef]
  38. J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman , “Simulations and experiments on self-focusing conditions in nematic liquid-crystal planar cells,” Opt. Express12, 1011–1018 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-6-1011 . [CrossRef] [PubMed]
  39. A. Snyder and J. Mitchell, “Accessible solitons,” Science276, 1538–1541 (1997). [CrossRef]
  40. M. Peccianti, A. Fratalocchi, and G. Assanto, “Transverse dynamics of Nematicons,” Opt. Express12, 6524–6529 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-26-6524 . [CrossRef] [PubMed]
  41. W. Krolikowski, E. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Y. S. Kivshar, C. Denz, and B. Luther-Davies, “Observation of dipole-mode vector solitons,” Phys. Rev. Lett.85, 1424 (2000). [CrossRef] [PubMed]
  42. T. Carmon, R. Uzdin, C. Pigier, Z. H. Musslimani, M. Segev, and A. Nepomnyashchy, “Rotating propeller solitons,” Phys. Rev. Lett.87, 143901 (2001). [CrossRef] [PubMed]
  43. D. Neshev, G. McCarthy, W. Krolikowski, E. A. Ostrovskaya, Yu. S. Kivshar, G. F. Calvo, and F. Agullo-Lopez, “Dipole-mode vector solitons in anisotropic nonlocal self-focusing media,” Opt. Lett.26, 1185–1187 (2001). [CrossRef]
  44. G. Molina-Terriza, E. M. Wright, and L. Torner, “Propagation and control of noncanonical optical vortices,” Opt. Lett.26, 163–165 (2001). [CrossRef]
  45. G. Molina-Terriza, J. Recolons, J. P. Torres, and L. Torner, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett.87, 023902 (2001). [CrossRef]
  46. D. McGloin, N. B. Simpson, and M. J. Padgett, “Transfer of orbital angular momentum from a stressed fiber-optic waveguide to a light beam,” Appl. Opt.37, 469–472 (1998). [CrossRef]
  47. A. Ya. Bekshaev, M. V. Vasnetsov, V. G. Denisenko, and M. S. Soskin, “Transformation of the orbital angular momentum of a beam with optical vortex in an astigmatic optical system,” JETP Lett.75, 127–130 (2002). [CrossRef]
  48. A. V. Carpentier, A. V. Carpentier, H. Michinel, J. R. Salgueiro, S. Doval, and A. Ferrando, “Inversion of a guided optical vortex,” J. Europ. Opt. Soc.1, 06031 (2006).
  49. J. J. García-Ripoll, G. Molina-Terriza, V. M. Perez-Garcia, and L. Torner, “Structural instability of vortices in Bose-Einstein condensates,” Phys. Rev. Lett.87, 140403 (2001). [CrossRef] [PubMed]
  50. G. Molina-Terriza, J. Recolons, and L. Torner, “The curious arithmetic of optical vortices,” Opt. Lett.25, 1135–1137 (2000). [CrossRef]
  51. A. Bezryadina, D. N. Neshev, A. S. Desyatnikov, J. Young, Z. G. Chen, and Yu. S. Kivshar, “Observation of topological transformations of optical vortices in two-dimensional photonic lattices,” Opt. Express14, 8317–8327 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-18-8317 . [CrossRef] [PubMed]
  52. P. L. Ramazza, U. Bortolozzo, and L. Pastur, “Phase singularities in triangular dissipative solitons,” J. Opt. A6S266–S270 (2004).
  53. N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, “Curvilinear motion of multivortex laser-soliton complexes with strong and weak coupling,” Phys. Rev. Lett.95, 053903 (2005). [CrossRef] [PubMed]
  54. F. Haudin, R. G. Rojas, U. Bortolozzo, M. G. Clerc, and S. Residori, “Vortex emission accompanies the advection of optical localized structures,” Phys. Rev. Lett.106, 063901 (2011). [CrossRef] [PubMed]
  55. G. Roumpos, M. D. Fraser, A. Löffler, S. Höfling, A. Forchel, and Y. Yamamoto, “Single vortex-antivortex pair in an exciton-polariton condensate,” Nat. Phys.7, 129 (2011). [CrossRef]

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