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Circularly polarized spatial solitons in Kerr media beyond paraxial approximation

Hongcheng Wang and Weilong She

Open Access Open Access

Abstract

We show that, in (2+1)-D case, both bright and dark solitons can exist in optical Kerr media when the optical beams are cylindrically symmetric and almost circularly polarized. We characterize the dependence of the properties associated with these solitons, such as their spatial width and intensity profiles, on their normalized intensity and the non-paraxial degree.

©2005 Optical Society of America

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Figures (4)
Fig. 1.
Fig. 1. Normalized soliton profile u(ρ)/u 0 of a non-paraxial bright soliton for various values of u 0 at a 2=0.015.

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Fig. 2.
Fig. 2. Existence curve of nonpraxial bright solitons for a 2=0.015.

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Fig. 3.
Fig. 3. Non-paraxial vortex soliton solutions for various values of u ∞: (a) m=1; (b) m=2.

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Fig 4.
Fig 4. Existence curves of 2-D non-paraxial vortex solitons for m=1 (dashed curve) and m=2 (solid curve) when a 2=0.015.

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Equations (18)

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∇ × E = iω B , ∇ × B = − i ω c 2 n 0 2 E − iω μ 0 P nl .
P nl = 4 3 ε 0 n 0 n 2 [ ∣ E ∣ 2 E + 1 2 ( E · E ) E * ] ,
∇ × ∇ × E = k 2 E + 4 3 k 2 n 2 n 0 [ ∣ E ∣ 2 E + 1 2 ( E · E ) E * ] ,
∇ · E = − 4 3 n 2 n 0 ∇ · [ ∣ E ∣ 2 E + 1 2 ( E · E ) E * ] .
s = x r 0 , t = y r 0 , ξ = z ( k x 0 2 ) , U = k r 0 ∣ n 2 ∣ n 0 E ,
a 2 ∂ 2 U + ∂ ξ 2 + 2 i ∂ U + ∂ ξ + Δ ⊥ U + + 4 γ 3 ∣ U + ∣ 2 U +
= − 2 γ 3 a 2 [ 4 ( ∣ ∂ U + ∂ s ∣ 2 + ∣ ∂ U + ∂ t ∣ 2 ) U + + [ ( ∂ U + ∂ s ) 2 + ( ∂ U + ∂ t ) 2 ] U + * + ∣ U + ∣ 2 ∇ ⊥ U + + U + 2 ∇ ⊥ U + * ] ,
2 i ∂ U − ∂ ξ + Δ ⊥ U − + 8 γ 3 ∣ U + ∣ 2 U − = − 2 γ 3 a 2 ( ∂ ∂ s − i ∂ ∂ t ) [ U + ( ∂ ∂ s − i ∂ ∂ t ) ∣ U + ∣ 2 ] ,
U 3 = a 2 ( i ∂ U + ∂ s + ∂ U + ∂ t ) ,
U + ( ρ , φ , ξ ) = u ( ρ ) exp ⁡ ( iβξ ) ,
− ( 2 β + β 2 a 2 ) u + u ″ + u ′ ρ + 4 u 3 3 + 2 a 2 3 [ 5 a 2 u u ′ 2 + 2 u 2 ρ u ′ + 2 u 2 u ″ ] = 0 ,
u ( ρ ) = u 0 + 3 ρ 2 u 0 2 ( 3 + 4 a 2 u 0 2 ) ( 2 β + β 2 a 2 − 4 3 u 0 2 ) , as ρ → 0 .
U + ( ρ , φ , ξ ) = u ( ρ ) exp ⁡ ( imφ ) exp ⁡ ( iβξ ) ,
− ( 2 β + β 2 a 2 ) u + u ″ + u ′ ρ − m 2 ρ 2 u − 4 u 3 3 − 2 a 2 3 [ 5 u u ′ 2 + 2 u 2 ρ u ′ + 2 m 2 u 2 ρ 2 + 2 u 2 u ″ ] = 0 ,
u ( 0 ) = 0 , u ( ∞ ) = u ∞ , u ′ ( ∞ ) = 0 , u ″ ( ∞ ) = 0 .
u ( ρ ) = δ ρ m + O ( ρ m + 2 ) .
β = a − 2 ( − 1 ± 1 − 4 a 2 u ∞ 2 3 ) ,
u ∞ < 3 ( 4 a 2 ) ,
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