Stabilization and breakup of optical vortices in presence of hybrid nonlinearity

doi:10.1364/OE.17.023130

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Stabilization and breakup of optical vortices in presence of hybrid nonlinearity

Xuetao Gan, Peng Zhang, Sheng Liu, Youming Zheng, Jianlin Zhao, and Zhigang Chen »View Author Affiliations

Optics Express, Vol. 17, Issue 25, pp. 23130-23136 (2009)
http://dx.doi.org/10.1364/OE.17.023130

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Abstract

We study nonlinear propagation of optical vortices in presence of hybrid nonlinearity as established in a nonconventional biased photorefractive crystal. Our results indicate that under hybrid nonlinearity the breakup of a singly-charged vortex along with the loss of its angular momentum is suppressed considerably as compared with that under conventional self-focusing or self-defocusing nonlinearity. Disintegration of a doubly-charged vortex under hybrid nonlinearity is also presented. Our experimental results are in good agreement with the numerical simulations.

1. Introduction

Vortices are universal in nature and share many common properties in different physical systems [1

L. M. Pismen, Vortices in nonlinear fields: from liquid crystals to superfluids, from non-equilibrium patterns to cosmic strings. (Oxford University Press, New York, 1999).

]. In optics, vortices with orbital angular momentum (OAM) are promising for applications in optical manipulations [2

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 ( 2003). [CrossRef] [PubMed]

] and optical communications [3

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 ( 2004). [CrossRef] [PubMed]

]. In the past few decades, the interplay between natural diffraction of helical-phase structures of optical vortices (OVs) and nonlinearities has been investigated extensively in various nonlinear media [4

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

–15

A. V. Gorbach, D. V. Skryabin, and C. N. Harvey, “Vortex solitons in an off-resonant Raman medium,” Phys. Rev. A 77(6), 063810 ( 2008). [CrossRef]

]. Photorefractive media have provided an ideal platform for studying the nonlinear dynamics of OVs [7

Z. 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] [PubMed]

–10

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, and A. Yariv, “Dark photorefractive spatial solitons and photorefractive vortex solitons,” Phys. Rev. Lett. 74(11), 1978–1981 ( 1995). [CrossRef] [PubMed]

]. The intrinsic anisotropy and nonlocality mediated by photorefractive nonlinear materials led to unusual features of dynamical vortex propagation [8

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Decay of high order optical vortices in anisotropic nonlinear optical media,” Phys. Rev. Lett. 78(11), 2108–2111 ( 1997). [CrossRef]

–10

]. Specially, it was found that the anisotropy of the nonlinearity might be responsible for the nonlinear decay of higher-order vortices [8

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Decay of high order optical vortices in anisotropic nonlinear optical media,” Phys. Rev. Lett. 78(11), 2108–2111 ( 1997). [CrossRef]

], whereas the nonlocality could dramatically suppress the azimuthal modulation instability of the vortex beam for formation of bright ring vortex solitons [10

]. Actually, suppression the azimuthal instability and stabilization of nonlinear optical vortices (OVs) have been studies extensively in the last decade [4

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

], including for example self-trapping of OVs in saturable self-defocusing [7

,11

V. Tikhonenko, Y. S. Kivshar, V. V. Steblina, and A. A. Zozulya, “Vortex solitons in a saturable optical medium,” J. Opt. Soc. Am. B 15(1), 79 ( 1998). [CrossRef]

] and nonlocal self-focusing media [12

D. Briedis, D. E. Petersen, D. Edmundson, W. Krolikowski, and O. Bang, “Ring vortex solitons in nonlocal nonlinear media,” Opt. Express 13(2), 435–443 ( 2005). [CrossRef] [PubMed]

], in models with competing nonlinearities [13

T. J. Alexander, Y. S. Kivshar, A. V. Buryak, and R. A. Sammut, “Optical vortex solitons in parametric wave mixing,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 61(2), 2042–2049 ( 2000). [CrossRef] [PubMed]

,14

I. Towers, A. V. Buryak, R. A. Sammut, B. A. Malomed, L. Crasovan, and D. Mihalache, “Stability of spinning ring solitons of the cubic-quintic nonlinear Schrödinger equation,” Phys. Lett. A 288(5-6), 292–298 ( 2001). [CrossRef]

], and also for multi-component vortices in presence of parametric wave mixing [13

,15

A. V. Gorbach, D. V. Skryabin, and C. N. Harvey, “Vortex solitons in an off-resonant Raman medium,” Phys. Rev. A 77(6), 063810 ( 2008). [CrossRef]

]. Recently, we have demonstrated that a nonconventionally biased (NCB) photorefractive crystal, whose bias field has an arbitrary angle with respect to the crystalline c-axis, can support hybrid nonlinearity, i.e., co-existence of self-focusing and -defocusing nonlinearity [16

P. Zhang, S. Liu, C. Lou, Y. Gao, J. Zhao, J. Xu, and Z. Chen, “Hybrid nonlinearity supported by nonconventionally biased photorefractive crystals,” Appl. Phys. B 95(3), 559–563 ( 2009). [CrossRef]

]. Such hybrid nonlinearity along with the enhanced anisotropy and nonlocality brings about novel beam dynamics in both continuous and discrete regimes, including for example the generation of “saddle solitons” [17

Y. Hu, C. Lou, P. Zhang, J. Xu, J. Yang, and Z. Chen, “Saddle solitons: a balance between bi-diffraction and hybrid nonlinearity,” Opt. Lett. 34(21), 3259–3261 ( 2009). [CrossRef] [PubMed]

]. In this paper, we focus on numerical and experimental studies of the nonlinear propagation of OVs in the presence of hybrid nonlinearity. We found that the enhanced nonlocality and anisotropy play a nontrivial role on the stabilization and breakup of OVs. In particular, our results indicate that, under the hybrid nonlinearity, the breakup of a singly-charged vortex along with loss of its angular momentum is suppressed considerably as compared with that under conventional self-focusing or -defocusing nonlinearity, while there seems to be an aggravation of the breakup of doubly-charged higher-order vortices.

2. Theoretical model

We study the nonlinear propagation dynamics of OVs in a NCB photorefractive crystal with an input light field taken to be

B (r) = \sqrt{I_{0}} r \exp (- \frac{{| r |}^{2}}{σ^{2}}) \exp (i m θ),

(1)

where r=(x ²+y ²)^1/2 and θ is the azimuth angle, σ is the diameter of the Gaussian beam in which the vortex embeds, m is a signed integer called topological charge, I ₀ is the characteristic intensity of the vortex. The normalized OAM L_z of the vortex beam can be calculated by L_z=i∫∫[x(B*∂B/∂y- B∂B*/∂y)-y(B*∂B/∂x-B∂B*/∂x)]dxdy, where B* is the conjugation of B [18

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] [PubMed]

]. We take the same coordinate system as that constructed in Ref. [19

P. Zhang, J. Zhao, C. Lou, X. Tan, Y. Gao, Q. Liu, D. Yang, J. Xu, and Z. Chen, “Elliptical solitons in nonconventionally biased photorefractive crystals,” Opt. Express 15(2), 536–544 ( 2007). [CrossRef] [PubMed]

], where the external biased field E is oriented along the y axis, and the crystal c-axis possesses an arbitrary angle α with respect to it. Here, we will focus on three cases at α=0, α=π, and α=π/2, corresponding to the biased field is parallel (E//+c), antiparallel (E//-c), and perpendicular (E⊥+c) to c-axis, respectively. An extraordinary polarized input vortex beam B( r ) propagates along the z axis, which is perpendicular to the c-axis, and its nonlinear propagation dynamics under different biased conditions can be determined by [19

]

(\frac{\partial}{\partial z} - \frac{i}{2} \nabla^{2}) B (r) = i (\frac{\partial ϕ}{\partial x} \sin α + \frac{\partial ϕ}{\partial y} \cos α) B (r),

(2a)

\nabla^{2} ϕ + \nabla ϕ \cdot \nabla \ln [1 + {| B (r) |}^{2}] = \frac{\partial}{\partial y} \ln [1 + {| B (r) |}^{2}],

(2b)

where

\nabla = \hat{x} (\partial / \partial x) + \hat{y} (\partial / \partial y)

, and φ is the light-induced electrostatic potential.

3. Vortex induced index changes

By choosing the vortex beam depicted in Eq. (1) with I ₀ =0.55 and σ=4 as the input beam [Fig. 1(a) ] of Eq. (2), the vortex-induced refractive index changes Δn in the conventional biased (CB) cases of α=0 and π, and the NCB case of α=π/2 are calculated as shown in Figs. 1(b)-1(d), respectively. In local self-focusing (-defocusing) media, such as Kerr material, index change can only be induced at the locations receiving light-illumination and its value is positive (negative) alone. In contrast, due to the nonlocality of photorefractive response, Δn shown in Fig. 1 are no longer localized at the bright-ring of the vortex beam, and possess both positive and negative values simultaneously. Specifically, at α=0 (π), the vortex circular ring induces Δn possessing inhomogeneous pseudoelliptical rings with two highest (lowest) lobes, which implies that the crystal mainly exhibits self-focusing (-defocusing) nonlinearity with somewhat anisotropy. In addition, the two peaks of the Δn coincide with the intensity maximum of the vortex beam, reflecting relatively weak nonlocality. However, at α=π/2, an interesting structure of Δn arises, i.e. symmetric profile but with opposite signs along two diagonal directions and each diagonal direction involves higher and lower lobes simultaneously, representing a more complex anisotropic nonlinearity. What’s more, Δn is almost zero at points where the intensity of the vortex beam is maximum, exemplifying an enhanced nonlocality. These features manifest the enhanced nonlocal and anisotropic nature of the hybrid nonlinearity.

Fig. 1 Numerical simulations of the refractive index change in a biased photorefractive crystal induced by a vortex beam. (a) Input vortex beam; (b)-(d) Index changes at α=0, π, and π/2, respectively.

4. Nonlinear propagation of singly-charged vortices

By solving Eq. (2) with the beam propagation method, the nonlinear evolutions of the singly-charged (m=1) vortex beam depicted in Fig. 1(a) are numerically simulated under different bias conditions. Figures 2(a) –2(c) display the evolution results for α=0, π, and π/2, respectively, and the corresponding display areas are 40×40, 256×256, and 256×256, respectively. From left to right they correspond to the output intensity patterns at different normalized propagation lengths z=20, 40, 60, 80, and 100, respectively. Acting like the anisotropic boundary condition as mentioned in Ref. [6

V. Tikhonenko, J. Christou, and B. Luther-Daves, “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 ( 1995). [CrossRef]

], the original anisotropic index change depicted in Fig. 1(b) expedites the breakup of the vortex beam at α=0 with the self-focusing nonlinearity, as shown in Fig. 2(a). The nontrivial interplay between the inherent angular momentum of the vortex beam and the anisotropic nonlinearity makes the two collapsed fragments present clockwise and anticlockwise damped rotations in turn during the propagation [Figs. 2(a1)–2(a5)]. While in the case of α=π, as shown in Fig. 2(b), the self-defocusing nonlinearity stretches the vortex beam into an ellipse with its major axis along the c-axis and leads to the formation of two lobes. Meanwhile, the two lobes flow along the elliptical ring clockwise gradually, driven by the helical-phase structure. However, the energy flows will be retarded by the anisotropic nonlinearities, as demonstrated in Ref. [9

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Vortex evolution and bound pair formation in anisotropic nonlinear optical media,” Phys. Rev. Lett. 77(22), 4544–4547 ( 1996). [CrossRef] [PubMed]

]. Figures 2(c1)–2(c5) present the evolution results in presence of the hybrid nonlinearity at α=π/2. Comparing with the above two CB cases, we can find that the vortex beam experiences a more stable nonlinear evolution process. Notwithstanding the coexistence of self-focusing and -defocusing nonlinearities, it neither collapses into two separated lobes as that happens at α=0 nor expands and stretches as much as that does at α=π. This characteristic nonlinear dynamics manifests the suppression of the azimuthal modulation instability exerted by the enhanced nonlocality under the NCB condition. In addition, the results reveal that the weak stretching along the direction that has an angle of 45° with respect to the c-axis, which is also determined by the hybrid nonlinearity as mentioned in Ref. [16

]. Furthermore, the dynamics of OAMs during the vortex evolutions are monitored and given in Fig. 2(d). It can be seen that, accompanying with the breakup of the vortex beam at α=0, its OAM drops sharply at the beginning, and then performs damped oscillations corresponding to the alternative rotations of the two fragments. In contrast, in both cases of α=π/2 and π, the OAMs decay slowly and remain positive even after long distance propagation. Remarkably, the loss of OAM is dramatically reduced in the NCB case as compared to other cases, indicating the suppression of azimuthal modulation instability with enhanced nonlocality in presence of hybrid nonlinearity.

Fig. 2 Nonlinear evolution of the singly-charged vortex beam at α=0 (a), π (b), and π/2 (c). From left to right are the output intensity profiles at different normalized propagation distances z=20, 40, 60, 80, and 100, respectively; (d) OAM versus z.

We use the experimental setup shown in Fig. 3 to observe the nonlinear propagation of OVs in a biased photorefractive crystal. A thin light beam emitted from a cw He-Ne laser (@633nm) is split into two beams. One of them is expanded and collimated by a reversed telescope and is used as a reference beam. The other beam passes through a computer-generated holographic mask and a spatial filter to generate a vortex beam [20

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221 ( 1992). [CrossRef] [PubMed]

], which is focused onto the input facet of the SBN:60:Ce crystal with dimensions of 6.5mm×3.5mm×6.7(c)mm and 0.025wt.% dopant. The input and output facets of the crystal are monitored by a CCD camera with an imaging lens. The biased field can be applied onto the crystal either parallel or perpendicular to the crystalline c-axis. A cold white light source is used to provide an incoherent background illumination to finely tune the nonlinearity.

Fig. 3 Experimental setup for observation of vortex propagation in a biased photorefractive crystal. BS: Beam splitter; RT: Reversed telescope; M: Mirror; L: Lens; F: Spatial filter; WLS: White light source, Mask: Computer-generated holographic mask.

Typical experimental results are shown in Fig. 4 . Where, Figs. 4(a) and 4(b) display the intensity pattern and interferogram of the input vortex beam, respectively. The output intensity profile in absence of applied voltage is shown in Fig. 4(c) and exhibits obvious linear diffraction. Figures 4(d) and 4(e) depict the nonlinear outputs with the external biased field parallel and antiparallel to the c-axis, respectively, in the presence of gradually increased voltages [(1)-(3) are corresponding to 0.3kV, 0.5kV, and 0.7kV, respectively]. According to Fig. 4(d), under self-focusing nonlinearity (α=0), the singly-charged vortex beam as expected will break up into two filaments, which rotate gradually counterclockwise. Unfortunately, the oscillatory rotations are not observed here due to the short length of our crystal. In Fig. 4(e), we can find that the anisotropic self-defocusing nonlinearity (α=π) causes the vortex beam stretch along the c-axis, and the ellipticity of the beam profile increases gradually with the enhancement of the external biased field. Figure 4(f) displays the nonlinear output of NCB case with the external field perpendicular to the c-axis. Although the nonlinear evolution is similar with that in Fig. 4(e), the orientation of the induced elliptical output possesses an angle of about 45° with respect to the c-axis, and the output intensity patterns are much smaller. Note that to obtain clear evolution results and struggle against the enhanced nonlocality for NCB case, much higher voltages are applied [(1)-(3) are corresponding to 1kV, 2kV, and 3kV, respectively]. In comparison with Fig. 2, our experimental observations agree with the numerical expectations.

Fig. 4 Experimental results of a singly-charged vortex under different bias conditions. (a) Intensity distribution and (b) interferogram of the input vortex beam; (c) Linearly diffracted output beam; (d)-(f) Nonlinear output at α=0, π, and π/2, respectively; where results from top to bottom are obtained at gradually increased voltages.

5. Breakup of doubly-charged vortices

It has been demonstrated that a higher-order vortex will break up into an array of vortices with unit charge under the anisotropic boundary conditions or anisotropic nonlinearities [8

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Decay of high order optical vortices in anisotropic nonlinear optical media,” Phys. Rev. Lett. 78(11), 2108–2111 ( 1997). [CrossRef]

]. In presence of hybrid nonlinearity, the enhanced anisotropy is expected to influence the breakup of higher-order vortices. In self-focusing medium, the azimuthal modulation instability of higher-order vortices will be much severer than that of singly-charged vortices, leading to a quick collapse of the input vortex. Therefore, here we use the self-defocusing case at α=π to compare with the breakup of higher-order vortices under hybrid nonlinearity. Figures 5(a) and 5(b) display the simulation results of a doubly-charged vortex beam at α=π/2 and π, respectively, obtained at z=45, I ₀=0.64, σ=3, and m=2. It is obvious that, in both cases, the doubly-charged vortex decays into two separate singly-charged vortices, which can be seen from the fork fringes in the interferograms as described in the bottom row in Fig. 5. However, although the nonlinear expansion of input beam under the NCB condition is smaller than that at α=π, the separation of the two decay-generated dark cores under hybrid nonlinearity is more distinct. This aggravation of the nonlinear decay can be readily interpreted by the enhancement of the anisotropic nonlinearity driven by the NCB filed. In addition, the orientations of the singly-charged vortex pairs for the two cases are obviously distinguishing, which are mainly determined by the anisotropy of the nonlinearity [16

]. With the hybrid nonlinearity, the bright stripe separating the two dark cores [see Fig. 5(a)] will experience self-focusing during propagation [16

], which will result in further separation of the two charge-1 vortices. However, for the conventional case with self-defocusing, the bright stripe is almost parallel to the c-axis, which means it will experience less expansion during propagation to make sure the dark cores can be separated. Our simulation results also get confirmed in experiment. By launching a doubly-charged vortex beam [see Fig. 5(c)] into the crystal, we observe the nonlinear decay of the input vortex with the biased field perpendicular (2kV) and antiparallel (1kV) to the c-axis, as shown in Figs. 5(d) and 5(e), respectively.

Fig. 5 Numerical (a, b) and experimental (c-e) results showing breakup of double-charged vortices. Top and bottom correspond to intensity patterns and interferograms, respectively. (a) and (b) Simulations at z=45 for α=π/2 and α=π, respectively; (c) Input vortex; (d) and (e) Experimental results corresponding to (a) and (b).

6. Conclusions

In conclusion, we have studied the nonlinear dynamics of OVs propagating under the hybrid nonlinearity in a NCB photorefractive crystal. We found that the hybrid nonlinearity under NCB condition prevents the singly-charged OVs from rapid breakup as occurred in conventional nonlinear self-focusing media, while there seems to be an aggravation of the breakup of doubly-charged OVs. Future work on vortex propagation under the hybrid nonlinearity in discrete regimes may lead to new features of nonlinear vortex phenomena investigated previously, including discrete vortex solitons [21

B. A. Malomed and P. G. Kevrekidis, “Discrete vortex solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(2), 026601 ( 2001). [CrossRef] [PubMed]

–24

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, “Observation of vortex-ring “discrete” solitons in 2D photonic lattices,” Phys. Rev. Lett. 92(12), 123904 ( 2004). [CrossRef] [PubMed]

], charge-flipping [25

T. J. Alexander, A. A. Sukhorukov, and Y. S. Kivshar, “Asymmetric vortex solitons in nonlinear periodic lattices,” Phys. Rev. Lett. 93(6), 063901 ( 2004). [CrossRef] [PubMed]

,26

A. Bezryadina, E. Eugenieva, and Z. Chen, “Self-trapping and flipping of double-charged vortices in optically induced photonic lattices,” Opt. Lett. 31(16), 2456–2458 ( 2006). [CrossRef] [PubMed]

], and topological transformations [27

A. Ferrando, M. Zacarés, M. A. García-March, J. A. Monsoriu, and P. F. de Córdoba, “Vortex transmutation,” Phys. Rev. Lett. 95(12), 123901 ( 2005). [CrossRef] [PubMed]

,28

A. Bezryadina, D. N. Neshev, A. S. Desyatnikov, J. Young, Z. Chen, and Y. S. Kivshar, “Observation of topological transformations of optical vortices in two-dimensional photonic lattices,” Opt. Express 14(18), 8317 ( 2006). [CrossRef] [PubMed]

Acknowledgments

This work was supported by the NPU Foundation for Fundamental Research, the Doctorate Foundation of NPU, the 973 program, and by NSF and AFOSR.

References and links

1.	L. M. Pismen, Vortices in nonlinear fields: from liquid crystals to superfluids, from non-equilibrium patterns to cosmic strings. (Oxford University Press, New York, 1999).
2.	D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 ( 2003). [CrossRef] [PubMed]
3.	G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 ( 2004). [CrossRef] [PubMed]
4.	A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” Prog. Opt. 47, 291 ( 2005). [CrossRef]
5.	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] [PubMed]
6.	V. Tikhonenko, J. Christou, and B. Luther-Daves, “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 ( 1995). [CrossRef]
7.	Z. 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] [PubMed]
8.	A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Decay of high order optical vortices in anisotropic nonlinear optical media,” Phys. Rev. Lett. 78(11), 2108–2111 ( 1997). [CrossRef]
9.	A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Vortex evolution and bound pair formation in anisotropic nonlinear optical media,” Phys. Rev. Lett. 77(22), 4544–4547 ( 1996). [CrossRef] [PubMed]
10.	G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, and A. Yariv, “Dark photorefractive spatial solitons and photorefractive vortex solitons,” Phys. Rev. Lett. 74(11), 1978–1981 ( 1995). [CrossRef] [PubMed]
11.	V. Tikhonenko, Y. S. Kivshar, V. V. Steblina, and A. A. Zozulya, “Vortex solitons in a saturable optical medium,” J. Opt. Soc. Am. B 15(1), 79 ( 1998). [CrossRef]
12.	D. Briedis, D. E. Petersen, D. Edmundson, W. Krolikowski, and O. Bang, “Ring vortex solitons in nonlocal nonlinear media,” Opt. Express 13(2), 435–443 ( 2005). [CrossRef] [PubMed]
13.	T. J. Alexander, Y. S. Kivshar, A. V. Buryak, and R. A. Sammut, “Optical vortex solitons in parametric wave mixing,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 61(2), 2042–2049 ( 2000). [CrossRef] [PubMed]
14.	I. Towers, A. V. Buryak, R. A. Sammut, B. A. Malomed, L. Crasovan, and D. Mihalache, “Stability of spinning ring solitons of the cubic-quintic nonlinear Schrödinger equation,” Phys. Lett. A 288(5-6), 292–298 ( 2001). [CrossRef]
15.	A. V. Gorbach, D. V. Skryabin, and C. N. Harvey, “Vortex solitons in an off-resonant Raman medium,” Phys. Rev. A 77(6), 063810 ( 2008). [CrossRef]
16.	P. Zhang, S. Liu, C. Lou, Y. Gao, J. Zhao, J. Xu, and Z. Chen, “Hybrid nonlinearity supported by nonconventionally biased photorefractive crystals,” Appl. Phys. B 95(3), 559–563 ( 2009). [CrossRef]
17.	Y. Hu, C. Lou, P. Zhang, J. Xu, J. Yang, and Z. Chen, “Saddle solitons: a balance between bi-diffraction and hybrid nonlinearity,” Opt. Lett. 34(21), 3259–3261 ( 2009). [CrossRef] [PubMed]
18.	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] [PubMed]
19.	P. Zhang, J. Zhao, C. Lou, X. Tan, Y. Gao, Q. Liu, D. Yang, J. Xu, and Z. Chen, “Elliptical solitons in nonconventionally biased photorefractive crystals,” Opt. Express 15(2), 536–544 ( 2007). [CrossRef] [PubMed]
20.	N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221 ( 1992). [CrossRef] [PubMed]
21.	B. A. Malomed and P. G. Kevrekidis, “Discrete vortex solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(2), 026601 ( 2001). [CrossRef] [PubMed]
22.	J. Yang and Z. H. Musslimani, “Fundamental and vortex solitons in a two-dimensional optical lattice,” Opt. Lett. 28(21), 2094–2096 ( 2003). [CrossRef] [PubMed]
23.	D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92(12), 123903 ( 2004). [CrossRef] [PubMed]
24.	J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, “Observation of vortex-ring “discrete” solitons in 2D photonic lattices,” Phys. Rev. Lett. 92(12), 123904 ( 2004). [CrossRef] [PubMed]
25.	T. J. Alexander, A. A. Sukhorukov, and Y. S. Kivshar, “Asymmetric vortex solitons in nonlinear periodic lattices,” Phys. Rev. Lett. 93(6), 063901 ( 2004). [CrossRef] [PubMed]
26.	A. Bezryadina, E. Eugenieva, and Z. Chen, “Self-trapping and flipping of double-charged vortices in optically induced photonic lattices,” Opt. Lett. 31(16), 2456–2458 ( 2006). [CrossRef] [PubMed]
27.	A. Ferrando, M. Zacarés, M. A. García-March, J. A. Monsoriu, and P. F. de Córdoba, “Vortex transmutation,” Phys. Rev. Lett. 95(12), 123901 ( 2005). [CrossRef] [PubMed]
28.	A. Bezryadina, D. N. Neshev, A. S. Desyatnikov, J. Young, Z. Chen, and Y. S. Kivshar, “Observation of topological transformations of optical vortices in two-dimensional photonic lattices,” Opt. Express 14(18), 8317 ( 2006). [CrossRef] [PubMed]

OCIS Codes
(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in
(190.5330) Nonlinear optics : Photorefractive optics
(050.4865) Diffraction and gratings : Optical vortices

ToC Category:
Nonlinear Optics

History
Original Manuscript: October 20, 2009
Revised Manuscript: November 14, 2009
Manuscript Accepted: November 17, 2009
Published: December 2, 2009

Citation
Xuetao Gan, Peng Zhang, Sheng Liu, Youming Zheng, Jianlin Zhao, and Zhigang Chen, "Stabilization and breakup of optical vortices in presence of hybrid nonlinearity," Opt. Express 17, 23130-23136 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-25-23130

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References

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