Solitons in Nonlocal Media

2.5 Theory of Nonlinear Optical Propagation in NLC
From eq. (2.13) it is straightforward to compute the coefficient Ψ2 (51)
Ψ2 = − ǫ0ǫa
8K f0
sin[2(θ0 − δ)]
+ Ψ0 cos [2(θ0 − δ)]
2
(2.14)
It is important to remark that eq. (2.14) is obtained without approximations: it is
valid whenever beam and perturbation are radially symmetric. The approximation is
given by the use of the parabolic term in the power expansion of the angle distribution,
justified in the highly nonlocal case (31; 51). In general, the perturbation peak Ψ0
depends on every term of the power expansion, including the effects due to the boundary
conditions.
For small perturbations eq. (2.8) becomes δǫtt ∼ = ǫa sin[2(θ0 − δ)] Ψ; hence, finally I get
δǫtt = ǫa sin[2(θ0 − δ)] 2 2
Ψ0 + Ψ2 (x − a/2) + t
which is the searched parabolic index well.
(2.15)
The term ǫa sin[2(θ0 − δ)] Ψ0 represents a rest energy, which depends on beam shape.
In general, its value changes as light propagates along s, but it is constant for a solitary
wave. Conversely, the term Ψ2 depends on f0, i.e. the peak intensity, owing to the
high nonlocality. Assuming Dx = Dt = D 1 , from quantum harmonic oscillator theory
(31; 49) it stems that solitons of any order are expressed by Hermite-Gauss modes
Ee mn = A0
Ω
π
1
√ 2 m+n n!m! Hm
√ √
Ω(x − a/2) Hn Ωt e −Ω[(x−a/2) 2 +t 2 ]
2 e iβmns
(2.16)
where Ω = − k2 0ǫa sin[2(θ0−δ)]Ψ2
ΩD
D , βmn = (n + m + 1) k0ne cos δ and Hn is the nthdegree
Hermite’s polynomial, whereas A0 is a constant dependent on soliton power.
Given that Ω depends on soliton power through Ψ2, 2 , solitons with a fixed width exists
only for a certain power.
Considering the m = n = 0 case (i.e. the lowest order soliton featuring a Gaussian
1 Actually, this hypothesis is not necessary if I define the new transverse coordinates x ′ =
x/ √ Dx, t ′ = t/ √ Dt.
2 When a specific beam shape is taken as in eq. (2.16) the relationship between f0 and P is a known
linear function depending on parameter Ω.
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