Solitons in Nonlocal Media

3.4 Soliton Oscillations in a Finite-Size Geometry
behavior (its slope increases in proximity of the boundaries) and is stronger in the
Poisson case being the nonlocality higher. Moreover, in the Poisson case the force is
almost independent from the beam waist (for ω < 0.1 and 〈ξ〉 < 0.9) as in the 1D case,
while in the screened case force it varies with the waist due to the lower nonlocality, the
latter stronger for smaller beam widths. Fig. 3.11(c) shows the soliton trajectories in
the plane ξs for beams at a fixed power, impinging normally on the cell (i.e. with a null
initial velocity) and computed through eq. (3.56): the soliton oscillates sinusoidally,
with a period Λ [shown in fig. 3.11(d)] decreasing as the beam is launched closer to a
boundary, due to the anharmonicity of the potential V m X
(see section 3.3.2). Finally,
given the linear relationship between the nonlinear index perturbation and the intensity
profile, the period decreases with power as P −1/2 .
W2 can be computed from [see eqs. (3.69) and (3.61)]:
W2(〈ξ〉) = −C
∞
(πm) 2 V ξ mV υ m cos(πm 〈ξ〉) (3.72)
m=1
Fig. 3.11(e) shows the first two terms of the force F m X [see eq. (3.62)], W0 and
W2 〈y〉 2 , respectively. The first order is dominant, being typically about three magni-
tude orders larger than the other ones.
Finally, in the highly nonlocal approximation and in the Poisson case, the potential
V m X
is given by (3.67), with [see appendix C.5 for details]:
c 1 0 = 2C
∞
πmV υ m(−1) m
m=1
0.5
0
t2
−
e w2 cos(πmt)dt (3.73)
A comparison between the complete form of W0 and its linear approximation
c1 0 (〈ξ〉 − 0.5) is reported in fig. 3.11(e): good accuracy is obtained for beams with
oscillation amplitudes less than 0.15.
3.4.3 Liquid Crystals
3.4.3.1 Model
In the case of liquid crystals, where optical propagation is governed by the first of
eqs. (2.18), eq. (3.54) is valid with the positions V = −
1 For the sake of simplicity I assumed Dx = Dt.
65
k 2 0
2m
Dx∆n 1 and ∆n =