Solitons in Nonlocal Media

4.3 Highly Nonlocal Limit
Figure 4.3: Plot of vector soliton trajectory. Angle between vector soliton direction and z
is given by β + ρ, where β is the angle between axis s and z (see note 2) and ρ = arctan k3
from eq. (4.23). Single solitons oscillate sinusoidally around this direction, keeping a
phase shift equal to π. s1 and s2 represent single beam energy direction when other beam
is lacking. In this plot beams are launched at the same point, i.e. their positions are
identical in z = 0.
d〈t2〉
ds
= tan ∆β, where ∆β =
s=0 δ2−δ1 1 (I suppose that for each beam the derivative in
2
s = 0 is unchanged with respect to the single beam case). After some simple algebra I
find k1 = k4 = 0, k3 = tan∆β 1 + and k2 = tan∆β
in eq. (4.23) the beams trajectories are
⎧
1
⎪⎨
〈t1〉 = −
α
⎪⎩
〈t2〉 = − 1
α
2 + m2
2γ2Ψ (1)
2
(1)
2 tan ∆β4γ2Ψ 2
αm2
4γ2Ψ (1)
2
α 2 m2
tan ∆β
4γ2Ψ (1)
2
αm2
sin(αs) + tan ∆β
sin(αs) + tan∆β
1 +
(1)
4γ2Ψ 2 . Replacing these
αm2
(1)
4γ2Ψ 2
α2
s
m2
1 +
(1)
4γ2Ψ 2
α2
s
m2
(4.24)
Therefore, the vector soliton propagates along a direction at an angle ∆ with z,
given by:
∆ = β + ρ = β + arctan
tan ∆β
1 In my case ∆β > 0 because I have λ1 > λ2.
1 −
80
m1γ2Ψ (1)
2
2m1γ2Ψ (1)
2
+ m2γ1Ψ (2)
2
(4.25)