Solitons in Nonlocal Media

Appendix B
Numerical Algorithm
B.1 Simulations of Nonlinear Optical Propagation in NLC
My purpose is to numerically simulate the PDE system given by (2.18), here rewritten
2ik0ne cos δ ∂Ee
∂s
+ Dt
∂2Ee ∂
+ Dx
∂t2 2Ee ∂x2 + k2 2 2
0ǫa sin (θ − δ) − sin (θ0 − δ) Ee = 0
(B.1)
K∇ 2 xtθ + ǫ0ǫa
4 sin [2(θ − δ)] |Ee| 2 = 0 (B.2)
Eqs. (B.1) and (B.2) rule optical nonlinear propagation and director reorientation
in NLC, respectively. My integration scheme works as follows: I compute the θ distribu-
tion at the input plane through eq. (B.2), with Ee(x, t, s = 0) known because its profile
is determined by the specific input beam. From the knowledge of θ(x, t, s = 0), I can
2 2 easily compute the nonlinear refractive index, given by ǫa sin (θ − δ) − sin (θ0 − δ) ,
in the same plane. Afterwards, I can use eq. (B.1) to find how the optical field Ee
propagates until s = ∆s, being ∆s the integration step along s (in this way I neglect
reflections along s, see sections B.1.1 for further details.). I can repeat the same set of
operations for the plane s = ∆s and, iterating the procedure, it is straightforward to
find the beam profile in a zone as long as I wish. Now I discuss the single algorithm
implemented in C++ to solve the single equations.
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