Solitons in Nonlocal Media

shape Ee ∝ exp
− (x−a/2)2 +t 2
w 2 S
term in eq. (2.14), the soliton power PS is
2.5 Theory of Nonlinear Optical Propagation in NLC
), I obtain that, given a waist wS and neglecting the Ψ0
16πKDne
PS =
ǫ0ǫ2 aZ0k2 0 sin2
1
[2(θ0 − δ)]
w 2 s
(2.17)
in agreement with Refs. (31; 51; 52). Eq. (2.17) provides the existence curve for
lowest order solitons (in NLC cell as described in fig. 2.10) under the highly nonlocal
approximation.
2.5.2 Numerical Simulations
Resuming former results, the nonlinear optical propagation in the cell depicted in fig.
2.10 and for V = 0 is ruled by the PDE system
2ik0ne cos δ ∂Ee
∂s
+ Dt
∂2Ee ∂
+ Dx
∂t2 2Ee ∂x2 + k2 2 2
0ǫa sin (θ − δ) − sin (θ0 − δ) Ee = 0
K∇ 2 xtθ + ǫ0ǫa
4 sin[2(θ − δ)] |Ee| 2 = 0
(2.18)
The numerical algorithm employed to solve eqs. (2.18) is explained in full details
in appendix B. I now discuss the results.
2.5.2.1 Nonlinear Propagation
In the simulations presented in this section I consider a cell of thickness a = 100µm
and θ0 = π/4 (see figure 2.10), filled up with liquid crystal E7 as in the experiments
previously discussed. Thus, in eqs. (2.18) I use E7 parameters: K = 12 × 10 −12 N and
index dispersion as in fig. 2.11 (79; 82).
I take Gaussian input beam profiles, Ee(x, t, s = 0) =
4Z0P
πnew 2 in
e − x2 +t 2
w 2 in , being Z0
the vacuum impedance, P the power and win the initial waist. I define the transverse
intensity profiles Ix(x, s) = ∞
−∞ |Ee| 2 dt and It(t, s) = a
0 |Ee| 2 dx. In particular, It is
proportional to the scattered light experimentally acquired with the set-up shown in
fig. 2.2. Numerically, It ≈ Ix for beam waists less than 10µm: such property will be
further detailed in section 2.5.2.2.
Fig. 2.12 shows the simulations for the case P = 1mW, win = 2.5µm and λ = 633nm.
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