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Ring vortex solitons in nonlocal nonlinear media

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Abstract

We study the formation and propagation of two-dimensional vortex solitons, i.e. solitons with a phase singularity, in optical materials with a nonlocal focusing nonlinearity. We show that nonlocality stabilizes the dynamics of an otherwise unstable vortex beam. This occurs for either single or higher charge fundamental vortices as well as higher order (multiple ring) vortex solitons. Our results pave the way for experimental observation of stable vortex rings in other nonlocal nonlinear systems including Bose-Einstein condensates with pronounced long-range interparticle interaction.

©2005 Optical Society of America

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Supplementary Material (4)

Media 1: MOV (3312 KB)     
Media 2: MOV (1513 KB)     
Media 3: MOV (15520 KB)     
Media 4: MOV (10462 KB)     

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Figures (4)

Fig. 1.
Fig. 1. Examples of ring vortex solitons in a nonlocal medium with charge m=1 (a) and m=3 (b). Solid line - normalized amplitude profile, dotted line - refractive index profile, dashed line - profile of the nonlocal response function. Top row - strongly nonlocal regime (σ0=5, σ=1,Λ=26.0783). Bottom row - weak nonlocality (σ0=0.1, σ=1,Λ=2.0198).
Fig. 2.
Fig. 2. Propagation of nonlocal charge m=1 ring vortex solitons. (a) unstable propagation in the weakly nonlocal case with σ0/σ=0.1 and σ=1, Λ=2.0198, (x,y)∊[-10,10]×[-10,10] and z∊[0,5]. (b) stable propagation in the highly nonlocal case with σ0/σ=10, Λ=101.0199 and σ=1, (x,y)∊[-30,30]×[-30,30] and z∊[0,50].
Fig. 3.
Fig. 3. Transverse structure of the nonlinearity-induced potential nesting the ring vortex soliton from Fig. 2. (a) Movie (3.2MB) depicting spatial evolution of the potential in a weakly nonlocal case (unstable propagation) (http://www.opticsexpress.org/view_media.cfm?umid=11676 for high resolution movie - 15MB); (b) Transverse structure of potential for strong nonlocality (stable propagation of the vortex soliton). Simulation parameters are the same as in Fig. 2.
Fig. 4.
Fig. 4. Stable propagation of a double ring nonlocal vortex soliton with charge m=1, σ0=9, σ=1, (x,y)∊[-25,25]×[-25,25] and z∊[0,25]. (a) Transverse intensity structure. (b) Movie (1.5MB) illustrating dynamics of the vortex nested in the self-induced nonlocal potential. Note the difference in oscillations of the inner and outer rings. (http://www.opticsexpress.org/view_media.cfm?umid=11677 for high resolution movie - 10MB).

Equations (10)

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N ( I ) = R ( r r ) I ( r , z ) d 2 r ,
i z ψ + ( x 2 + y 2 ) ψ + N ( I ) ψ = 0 .
R ( r ) = 1 π σ 0 2 exp ( r 2 σ 0 2 ) .
ψ ( r , z ) = ψ ( r , ϕ , z ) = u ( r ) exp ( im ϕ ) exp ( i Λ z )
r 2 u ( r ) + 1 r r u ( r ) ( m 2 r 2 + Λ ) u ( r ) + u ( r ) 0 2 π 0 R ( r r ) u ( r ) 2 r d r d ϕ = 0
𝓛 = Λ u ( r ) 2 u ( r ) 2 + u ( r ) 2 R ( r r ) u ( r ) 2 d 2 r .
u ( r ) = Ar exp ( r 2 ( 2 σ 2 ) ) .
Λ u n + 1 ( r ) r 2 u n + 1 ( r ) 1 r r u n + 1 ( r ) + m 2 r 2 u n + 1 ( r ) = u n ( r ) N ( u n ( r ) 2 )
ψ ( r , ϕ ) = 2 σ 0 2 σ 4 r m exp ( r 2 ( 2 σ 2 ) ) exp ( im ϕ ) ,
ψ ( r , ϕ ) = 4 2 σ 0 2 σ 4 r ( 1 r 2 ( 2 σ 2 ) ) exp ( r 2 ( 2 σ 2 ) ) exp ( i ϕ ) .
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