Solitons in Nonlocal Media

B.1.1 Optical Equation
B.1 Simulations of Nonlinear Optical Propagation in NLC
To solve the optical equation I implemented a beam propagation method (BPM) that
allows to compute the field distribution from an input field, neglecting reflections in
propagation 1 . A splitting method (113) was applied to solve the initial value problem
of equation (B.1). Rewriting eq. (B.1) in order to isolate the operator governing the
evolution along s, I get
where I defined the operators Lt = i
(diffraction along x), L∆n = i
∂Ee
∂s = LtEe + LxEe + L∆nEe = LEe
(B.3)
Dt ∂
2k0ne cos δ
2
∂t2 Dx ∂ (diffraction along t), Lx = i2k0ne
cos δ
2
∂x2 k0
2ne cos δδǫtt (index-well action) and L = Lt + Lx + L∆n.
Formally, the solutions of (B.3) in the interval [s s+∆s] can be written as Ee(s+∆s) =
e i∆sL Ee(s) = e i∆sLt e i∆sLx e i∆sL∆nEe(s). Let me consider the three equations
∂Ee
= LtEe
(B.4)
∂s
∂Ee
= LxEe
(B.5)
∂s
∂Ee
= L∆nEe
(B.6)
∂s
and assume that an exact or approximated method is available to solve each equa-
tion in the interval [s s+∆s], i.e., there are three discretized operators Uj (j = t, s,∆n)
such that
Ee(s + ∆s) = Uj(s + ∆s, s)Ee(s) (B.7)
For a small enough propagation step 2 I get that a correct numerical solutions of eq.
(B.3) in the interval [s s + ∆s] is (113)
Ee(s + ∆s) = U∆n(s + ∆s, s)Ux(s + ∆s, s)Ut(s + ∆s, s)Ee(s) (B.8)
1 The nonlinear index variations I study are small, thereby this condition is satisfied.
2 The step must be chosen empirically; in particular, I have run numerical simulations for several
∆s: the accuracy is sufficient when the solutions become independent from the employed propagation
step.
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