Solitons in Nonlocal Media

2.5 Theory of Nonlinear Optical Propagation in NLC
having neglected the derivative along s, as already discussed in section 1.3. The
nonlinear index perturbation is
2.5.1.1 The Highly Nonlocal Case
2 2
δǫtt = ǫa sin (θ − δ) − sin (θ0 − δ)
(2.8)
Let me define the nonlinear perturbation of the director angle Ψ = θ − θ0. For small
Ψ, eq. (2.7) becomes
K∇ 2 xtΨ + ǫ0ǫa
4 |Ee| 2 sin [2(θ0 − δ)] + ǫ0ǫ1
2 |Ee| 2 cos [2(θ0 − δ)]Ψ = 0 (2.9)
I assume the optical field Ee is cylindrically symmetric, which means Ψ has the
same property if asymmetric boundary conditions are neglected (actually, this is true
also for asymmetric boundary conditions for the zone close to the beam peak if the
beam waist is negligible compared to the cell size: see sections 2.5.2.1 and 2.5.2.2). I
can write the field and the perturbation using a Taylor series around x = a/2, t = 0
2 2
Ψ = Ψ0 + Ψ2 (x − a/2) + t + o (x − a/2) 2 + t 2
|Ee(x, t)| 2 2 2
= f0 + f2 (x − a/2) + t + o (x − a/2) 2 + t 2
(2.10)
(2.11)
being Ψ0 = Ψ| x=a/2,t=0 and f0 = |Ee| 2 | x=a/2,t=0 the
maxima of the induced pertur-
and f2 =
x=a/2,t=0 1
.
x=a/2,t=0
bation and the field, respectively, whereas Ψ2 = 1 ∂
2
2Ψ ∂x2 Substituting eqs. (2.10) and (2.11) into (2.9) I get:
ǫ0ǫa
4
4KΨ2 + o [(x − a/2) + t]+
2 2
f0 + f2 (x − a/2) + t + o (x − a/2) 2 + t 2 sin[2(θ0 − δ)]+
ǫ0ǫa 2 2
f0 + f2 (x − a/2) + t
2
+ o (x − a/2) 2 + t 2
2 2
Ψ0 + Ψ2 (x − a/2) + t + o (x − a/2) 2 + t 2 cos [2(θ0 − δ)] = 0
2
∂ 2 |Ee| 2
∂x 2
(2.12)
From eq. (2.12) all the coefficients in front of every power of x or t must be equal
to 0. For the zero-th order power this gives
4KΨ2 + ǫ0ǫa
4 f0 sin [2(θ0 − δ)] + ǫ0ǫa
2 f0Ψ0 cos [2(θ0 − δ)] = 0 (2.13)
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