Solitons in Nonlocal Media

4.3 Highly Nonlocal Limit
(a) Initial position (b) Motion due to the mutual attraction
Figure 4.2: Reciprocal interaction between two beams (red and blue profiles) due to the
nonlocal index perturbation. For the sake of simplicity, one beam (the blue) is fixed in the
space; actually, the interaction is mutual and both beams move along t. In 4.2(a) the blue
beam induces an index well (black line) which exerts a force (proportional to the slope of
black curve) on the red one, consequently moving it from t1 to t2 [fig. 4.2(b)].
d
mj
2 〈xj〉
ds2
= 2 Ψ
j
(1)
2
d
mj
2 〈tj〉
ds2 = 2
j
+ Ψ(2) 2 〈xj〉 (4.16)
Ψ (1)
2 (〈tj〉 − 〈t1〉) + Ψ (2)
2 (〈tj〉 − 〈t2〉)
(j = 1, 2) (4.17)
Being 〈xj〉 = 0 for even intensity profiles, eq. (4.16) shows there is no force acting
on the beams along x, thereby the beam is undeflected in the xs plane. Conversely, in
the xt plane the beams perceive a force proportional to the misplacement between the
two waves. Specifically, I find
d
m1
2 〈t1〉
ds2 = 2Ψ
1
(2)
2 (〈t1〉 − 〈t2〉) (4.18)
d
m1
2 〈t2〉
ds2 = 2Ψ
2
(1)
2 (〈t2〉 − 〈t1〉) (4.19)
Solutions of eqs. (4.18)-(4.19) provide the soliton trajectories in plane tjsj. Note
how every beam is affected only by the potential of the other one and the reciprocal
attraction [Ψ (j)
2 < 0 from (2.14)] increases as the distance, which does not depend on the
reference system, decreases [see figs. 4.2(a)-4.2(b)]. In general, quantities Ψ (j)
2 (j = 1, 2)
depend on sj through the beams’ peak intensity variation, as predicted by (2.14). To
get a closes form for 〈tj〉 I neglect these fluctuations; noteworthy, this condition is
fulfilled if the single beams are solitons.
78