Elsevier

Physics Letters A

Volume 341, Issues 1–4, 20 June 2005, Pages 145-155
Physics Letters A

Ring-shaped solitons in a “dartboard” photonic lattice

https://doi.org/10.1016/j.physleta.2005.04.074Get rights and content

Abstract

We introduce a two-dimensional model of photorefractive media, with the usual saturable nonlinearity, and a cylindrical photonic lattice periodic in the radial direction. The saturable nonlinearity may be either self-focusing or self-defocusing, and the cylindrical lattice is introduced so that, in the former case, the potential has a maximum at the center. Ring-shaped solitons are constructed numerically. In the case of self-focusing, they are typically unstable against azimuthal perturbations, which is shown by means of numerical computations, and is also explained with the help of an analytical approximation. On the contrary, solitons of the same type may be stable if the saturable nonlinearity is self-defocusing.

Introduction

It is well known that photonic lattices induced by properly patterned light beams launched into a photorefractive medium in the ordinary polarization (in which the medium is nearly linear), provide a versatile tool for creation of various solitons in the extraordinary polarization, where the light feels strong saturable nonlinearity. After the prediction of this possibility [1], many experimental results have been reported, including the first creation of quasi-discrete two-dimensional (2D) solitons [2] (1D solitons [3] and soliton trains [4] were also created in this setting) and vortex solitons of the same type [5], vortices in the second lattice bandgap and composite inter-band solitons [6] and necklace-shaped solitons [7] among others. Stability of 2D solitons in the corresponding model was also studied in some detail numerically [8].

Another geometry, in which 2D photorefractive solitons can be studied, is one imposed by a cylindrical photonic lattice, in the form of a system of concentric rings (which, in principle, can be readily created, passing the cylindrical ordinarily polarized beam through an appropriate phase mask). The objective of the present Letter is to introduce solitons of this type in the photorefractive model and investigate their existence and stability conditions. It should be said that 2D (spot-like) solitons supported by a similar Bessel lattice (with the radial shape of the lattice following the Bessel function), but in the model with the usual cubic nonlinearity (without saturation, which is a distinguishing feature of the photorefractive medium) were introduced in Ref. [9], and further theoretically investigated in this setting, including the study of dipole-mode solitons [10], ones in an azimuthally modulated lattice [11], and vorticity-bearing azimuthally uniform patterns in the model with the self-defocusing cubic nonlinearity [12].

In this Letter, we assume the simplest periodic modulation of the intensity of the ordinarily polarized beam along the radial variable, V(r)=I0sin2(2πr/ρ), that can be created by means of phase mask with the radial period ρ. The evolution of the signal field u along the propagation coordinate z obeys the following equation: iuz=Δu+E0u1+I0sin2(2πr/ρ)+|u|2, where Δ is the 2D Laplacian. Note that Eq. (2) is a special case of the well-known photorefractive model with the saturable self-focusing nonlinearity [1], if the radial photonic lattice is introduced as per Eq. (1). Eq. (2) is expressed in the normalized form, with the constant E0 proportional to the bias dc voltage, which induces the photorefractive nonlinearity for the signal field. The remaining scaling invariance of the equation makes it possible to fix the value of ρ or E0 (an invariant combination is E02/ρ2). We will use this fact, fixing E07.5 throughout the paper (see [8]), while ρ will be varied. The lattice's strength I0 is another nontrivial parameter. Numerical results demonstrate that the generic situation is quite adequately represented by fixing I0=1, which will be assumed below.

It is relevant to note that the linearized version of Eq. (2), obtained by dropping the term |u|2 in the denominator, is the 2D linear Schrödinger equation, with the “dartboard” effective potential Veff(r)=E0/(1+V(r)). Due to the shape of the 2D potential, we will call nonlinear radially localized solutions, to be found in it, “dart solitons”.

As E0 and I0 in Eq. (2) are positive, the effective potential has the first local maximum at the central point, r=0, which is surrounded by the first ring-shaped minimum at r=ρ/4; subsequent maxima and minima are located at the rings with rmin=nρ/2 and rmax=(2n+1)ρ/4, respectively, where n=1,2,. An essential difference from the situation for the model with the Bessel radial lattice and cubic self-focusing nonlinearity, introduced in Ref. [9], is in the fact that the model considered in that work had a (deepest) minimum of the potential at r=0. In fact, the radial effective potential with a minimum at r=0 can also be considered in the present model. To this end, one can either use the phase mask which generated the intensity distribution in the form of V(r)=I0cos2(2πr/ρ), instead of the one corresponding to Eq. (1), or simply reverse the sign of the bias dc voltage, setting E0<0 in Eq. (2). In that case, one expects to find stable solitons trapped at the central potential well.

The Letter is organized as follows. In Section 2, we construct, in a numerical form, a family of axially uniform dart soliton solutions, both fundamental and higher-order ones, and investigate their stability—by computation of the linearization eigenvalues for small perturbations, and in direct simulations. The result is that, typically, the dart solitons, fundamental ones and their higher-order counterparts alike, are unstable (for the focusing case with E0>0) against azimuthal perturbations breaking the axial symmetry. The instability evolution is shown to “break” the original ring-shaped soliton into a few highly localized spots, which are solitons themselves (the so-called “rotary” ones, since they may perform circular motion in the ring-shaped potential well [9]).

The modulational instability (MI) of the axially uniform state against symmetry-breaking azimuthal perturbations is considered in Section 3. In a simple approximation, it is reduced to the MI in the usual 1D nonlinear Schrödinger (NLS) equation [14]. It is also shown that, with the E0<0, when the saturable nonlinearity in Eq. (2) is self-defocusing, all the dart solitons are stable. Section 4 concludes the Letter.

Section snippets

Axially uniform solitons and their instability

Eq. (2) may support vortex-ring soliton solutions in the form u(r,ϕ)=w(r)exp(iμz)exp(imϕ), where m0 is integer vorticity (m=0 corresponds to the fundamental soliton), and −μ is the (real) propagation constant. The substitution of ansatz (3) in Eq. (2) leads to a stationary equation for the real radial function w(r), d2wdr2+1rdwdrm2r2w+μw(r)E0w1+I0sin2(2πr/ρ)+w2=0. Relevant solutions must decay exponentially at r and satisfy the following boundary conditions at r0, limr0(rmw(r))=const,

Stability analysis

The above numerical results clearly demonstrate that typical dart solitons are unstable for focusing nonlinearities (in the model with the potential maximum at r=0), and ensuing dynamics is dominated by the modulational instability (MI) of the solitons in the azimuthal direction. To illustrate this property more explicitly, we consider in this section the model with the usual cubic nonlinearity, instead of the saturable one of Eq. (2), and a radial potential V(r), i.e., the NLS equation of the

Conclusion

In this Letter, we have introduced a 2D model of the photorefractive medium with the saturable nonlinearity and a radially periodic cylindrical photonic lattice. The lattice is introduced in such a way that, in the case of self-focusing, the effective potential has a maximum at the central point. The main result is that, in the latter case, the ring-shaped (dart) solitons are typically unstable against azimuthal perturbations due to a quasi-one-dimensional, effective modulational instability

Acknowledgments

P.G.K. gratefully acknowledges support from the Eppley Foundation for Research, NSF-DMS-0204585 and NSF-CAREER, as well as numerous insightful discussions with Zhigang Chen about the experimental feasibility of the dartboard lattices.

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