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We provide a brief overview on our recent experimental work on linear and nonlinear localization of singly charged vortices (SCVs) and doubly charged vortices (DCVs) in two-dimensional optically induced photonic lattices. In the nonlinear case, vortex propagation at the lattice surface as well as inside the uniform square-shaped photonic lattices is considered. It is shown that, apart from the fundamental (semi-infinite gap) discrete vortex solitons demonstrated earlier, the SCVs can self-trap into stable gap vortex solitons under the normal four-site excitation with a self-defocusing nonlinearity, while the DCVs can be stable only under an eight-site excitation inside the photonic lattices. Moreover, the SCVs can also turn into stable surface vortex solitons under the four-site excitation at the surface of a semi-infinite photonics lattice with a self-focusing nonlinearity. In the linear case, bandgap guidance of both SCVs and DCVs in photonic lattices with a tunable negative defect is investigated. It is found that the SCVs can be guided at the negative defect as linear vortex defect modes, while the DCVs tend to turn into quadrupole-like defect modes provided that the defect strength is not too strong.

Vortices and vortex solitons are ubiquitous in many branches of sciences such as hydrodynamics, superfluid, high-energy physics, laser and optical systems, and Bose-Einstein condensates [

Several different physical mechanisms have been proposed for suppression of the azimuthally instability [

In this paper, we present a brief overview of our recent work on both linear and nonlinear propagation of SCV and DCV beams in 2D photonic lattices in bulk photorefractive material by optical induction. In the nonlinear cases, vortex beam propagation both inside and at the surface of photonic lattices is considered. We show that under appropriate nonlinear conditions both SCV and DCV beams can stably self-trapped into a gap vortex soliton in photonic lattices with self-defocusing nonlinearity. While a SCV beam can turn into a stable gap vortex soliton under four-site excitation, a DCV beam tends to turn into a quadrupole gap soliton which bifurcates from the band edge. However, under geometrically extended eight-site excitation, the DCV can evolve into a stable gap vortex soliton. At the interface between photonics lattice and a homogenous media, under appropriate conditions, the SCV beam can turn into a stable surface discrete vortex soliton under four-site excitation with self-focusing nonlinearity, while it self-traps into a quasi-localized surface state under single site excitation. The stability of such nonlinear localized states is monitored by the numerical simulation to long propagation distance. In the linear case, the SCV beam can also evolve into a linear localized vortex defect mode in photonic lattice with a negative defect, while the DCV beam turns again into a quadrupole-like defect mode, provided that the defect strength is not too deep. Our results may prove to be relevant to the studies of similar phenomena in superfluids and Bose-Einstein condensates.

The experimental setup for our study is similar to those we used earlier for observation of discrete solitons [^{3}) photorefractive crystal illuminated by a laser beam with wavelength 488 nm from the Argon ion lasers. The biased crystal provides a self-focusing or defocusing noninstantaneous nonlinearity by simply reversing the external biased field. To generate a 2D-waveguide lattice, we use an amplitude mask to spatially modulate the otherwise uniform incoherent beam after the diffuser. The rotating diffuser turns the laser beam into a partially spatially incoherent beam with a controllable degree of spatial coherence. The mask is then imaged onto the input face of the crystal; the periodic intensity pattern is a nearly nondiffracting pattern throughout the crystal after proper spatial filtering. The lattice beam is ordinarily polarized, so it will induce a nearly linear waveguide array that remains invariant during propagation. We can create not only uniform lattices but also lattices with negative defects and lattices with sharp surfaces by simply changing the specially designed amplitude mask. The extraordinarily polarized beam passes through a computer generated hologram to create a ring vortex beam, and then it is focused by a circular lens. The vortex beam is then used as our probe beam or

Experimental setup for optical induction of photonic lattices in a biased photorefractive crystal by amplitude modulation of a partially coherent beam. PBS: polarizing beam splitter; SBN: strontium barium niobate. Top path is the lattice-inducing beam, the middle path is the probing vortex beam, and the bottom path is the reference beam. The top right insert illustrates the scheme of induced waveguide arrays in the otherwise uniform SBN nonlinear crystal.

A piezotransduced (PZT) mirror is added to the optical path of the reference beam which can be used to actively vary the relative phase between the plane wave and the vortex beam. In addition, a white-light background beam illuminating from the top of the crystal is typically used for fine-tuning the photorefractive nonlinearity [

First, we study the SCV and DCV beams propagating inside optically induced photonic lattices under the self-defocusing nonlinearity. We find that a donut-shaped SCV beam can self-trap into a stable gap vortex soliton, while the DCV beam evolves into a quadruple soliton under the four-site excitation. Spectrum measurement and numerical analysis suggest that the gap vortex soliton does not bifurcate from the edge of the Bloch band, quite different from previously observed spatial gap solitons.

In our experiment, the lattice beam induces a “backbone” lattice with a self-defocusing nonlinearity where an intensity minimum corresponds to an index maximum [

Experimental results of self-trapping of SCVs (top) and DCVs (bottom) in a defocusing photonic lattice. (a) Interferograms showing the phase of the input vortex beams, (b) intensity patterns of self-trapped vortex beams at lattice output, (c, d) interferograms between (b) and a tilted plane wave (c) and an on-axis Gaussian beam (d), respectively, and (e) the

We also measure the spatial spectrum of self-trapped vortices by using the Brillouin Zone (BZ) Spectroscopy technique [

The experimental observations are also compared with the numerical results obtained using beam propagation method with initial conditions similar to that for the experiment. The numerical model is a nonlinear wave equation with a 2D square lattice potential under saturate self-defocusing nonlinearity. Excellent agreement can be found for the propagation distance of 10 mm (i.e., our crystal length) for both

Simulation results of singly charged (a) and doubly charged (b) vortex beams propagating to a longer distance of 40 mm. Shown are the output transverse (

It is interesting to explore whether stable doubly charged gap vortex soliton exists in

Experimental observation of an extended

Discrete surface solitons form an important family of discrete solitons that exist at interface between the semi-infinite photonic lattices and a homogenous media [

First, we use the single-site excitation so the vortex ring covers only one lattice site at the surface (Figure

Experimental results of vortex self-trapping under single-site surface excitation (top) and four-site surface excitation (bottom). (a) Lattice beam superimposed with the vortex beam, (b) linear and (c) nonlinear output of the vortex beam, where blue dashed line indicates the surface location, (d) zoom-in interferogram of (c) with an inclined plane wave, and (e)

Next, we use the four-site excitation so the vortex ring is expanded to cover four nearest lattice sites at the surface (the vortex core locates in the index minimum as off-site excitation, as shown in Figure

Due to that the induced surface is slightly deformed, the four-spot does not match a perfect square pattern. To monitor the phase structure of the self-trapped vortex, an inclined plane wave is again introduced for interference. The vortex singularity persists clearly in the nonlinear output (Figure

Experimentally, it is difficult to test the stability of the self-trapped surface vortex structures due to the limited crystal length (i.e., 1 cm). Thus, we study the stability of the self-trapped vortex states by numerical simulation with parameters similar to the experimental conditions, but to much longer propagation distance (4 cm). For the four-site excitation vortex in the semi-infinite gap is stable, but for the single-site vortex in the Bragg reflection gap is unstable as shown in Figure

Numerical simulation of single-site (top) and four-site (bottom) surface excitation of a SCV after 4-cm of nonlinear propagation. Shown are (a) vortex output intensity pattern and (b) its corresponding interferogram, and (c) side-view of the vortex propagation to

In the previous experiments, the localization of the vortex beam inside and at the surface of the photonic lattice is related to the nonlinearity-induced defect by the vortex beam itself. These self-trapped nonlinear localized topological states can be considered as self-induced

Different from light guided in higher refractive-index region due to the total internal reflection, light guidance in negative defects results from the Bragg reflection in periodic structures. Here, we show that both a SCV and DCV beams can be guided in the negative defects in 2D photonic lattices provided the defect strength is not too deep. The experimental setup is similar to Figure

Experimental results of bandgap guidance of a vortex beam in a tunable negative defect. (a) Vortex at input. (b)–(d) Induced lattices with nonzero-intensity defect, no defect, and zero-intensity defect, respectively. (e), (f) Vortex output from the defect in (b) and its zoom-in interferogram. The circles in (f) mark the location of the vortex pairs. (g) Interferogram when the vortex is excited at nondefect site. (h) Vortex diffraction output when lattice is absent.

Typical experimental results are presented in the bottom panels of Figure

The formation of the vortex DMs depends on the defect strength. A series of experiments is performed to illustrate the influence of the defect strength. To do so, we keep the bias field and

Experimental results of SCV beam output after propagating through (a) a half-intensity (

Finally, a DCV beam is used as a probe beam sent into the tunable negative defect to check whether a high-order optical vortex can be guided in it. We find the DCV cannot be guided and it tends to break up again into a quadruple-like structure when the defect strength is not too deep. Typical experimental results are shown in Figure

Experimental results of bandgap guidance f a DCV beam in a tunable negative defect. (a) interference pattern of input vortex with an inclined plane wave (zoomed in), (b) lattice beam with a nonzero-intensity defect, (c) linear output of the doubly charged vortex beam exiting the defect channel in the lattice (b), (d) lattice beam with a zero-intensity defect, and (e) linear output of the DCV through the defect in lattice (d).

We have studied experimentally the dynamical propagation of optical vortices in optically induced square photonic lattices under different settings. Inside the uniform photonic lattices, we find that the SCV beams self-trap into stable gap vortex solitons under four-site excitation with a defocusing nonlinearity. However, the DCV beams tend to break up and turn into quadrupole gap solitons. With geometrically extension eight-site excitation, the DCV beams can evolve into stable gap vortex solitons. Both the SCV and DCV gap solitons do not bifurcate from modes at the band edges, which is different from all the previously observed gap solitons. At the lattice surface, the SCV beams turn into stable surface vortex solitons in the semi-infinite gap under the four-site excitation with a self-focusing nonlinearity. However, they evolve into unstable quasivortex solitons with propagation constants in the first Bragg gap under the single-site excitation. In photonic lattices with negative defects, the SCV beams can be guided as linear vortex defect modes under appropriate defect conditions due to the bandgap guidance, while the DCV beams are unstable and evolve into quadrupole-like defect modes. Our results about such linear and nonlinear localized vortex states may prove to be relevant to similar vortex phenomena in other discrete systems beyond optics.

This work was supported in part by the 973 Program (2007CB613203) and NSFC (10904078) in China and by US National Science Foundation (NSF) and Air Force Office of Scientific Research (AFOSR). The authors thank J. Yang, P. G. Kevrekidis, K. Law, and X. Wang for helpful discussions and assistance.