Solitons in Nonlocal Media

ǫ sin 2 ξbulk + ǫ⊥ cos 2 d
ξbulk
2V 2.3 Effect of the Input Interface
dx ′2 + ∆ǫLF sin(2ξbulk) dξbulk
dx ′ dx
2 dξbulk
K3 sin 2 ξbulk + K1 cos 2 d
ξbulk
2ξbulk dx ′2 + K3 − K1
sin(2ξbulk)
2
+ ∆ǫLF
2
dV
sin(2ξbulk)
dx ′
dV
= 0 (2.4)
′
dx ′
2
= 0
(2.5)
being V the electrostatic potential. I introduced a new reference system x ′ y ′ z ′
defined by ˆ y ′ = ˆn(V = 0), ˆ x ′ = ˆx and ˆ z ′ = ˆ x ′ × ˆ y ′ . In eqs. (2.4)-(2.5) V is the root
mean square (RMS) of the voltage, K1 and K3 are the Frank’s elastic constants for
splay and bend, respectively, and ∆ǫLF is the dielectric anisotropy at low frequencies.
The boundary conditions are V (x ′ = 0) = 0, V (x ′ = a) = V and ξbulk(x ′ = 0) =
ξbulk(x ′ = a) = 2π/180, the last one stemming from the pre-tilt (see section 2.1). The
system composed by eqs. (2.4) and (2.5) has to be solved numerically: results for
various V are shown in fig. 2.4.
As stated above, I need to know ξ in x = a/2 in bulk NLC. From fig. 2.4(a) this
corresponds to the maximum ξ for a fixed V, i.e. ξmax(V).
Taking a linear input polarization at the first slice (see section 2.2), I can split the elec-
tric field vector into ordinary (o) and extraordinary (e) components; the latter matches
the director orientation in z = 0. From eq. (2.3) and taking d = 20µm, I find that
the input e component transfers nearly all its power to the e-wave in the bulk NLC
for every V; similarly, o-wave components in z = 0 remain o-waves in bulk. This is
represented by the two straight horizontal lines in fig. 2.5(b) at β = 45 ◦ (e input)
and β = 135 ◦ (o input): as the bias increases, for every z > d the director elevation
increases as well [see fig. 2.4(c)], and the NLC principal axes rotate. e- and o-wave
input components, however, remain decoupled as they evolve through the transition
layer. Otherwise stated, if the rotation of the NLC dielectric tensor with z is slow
enough that the index change is adiabatic, the e-wave displacement field De rotates
with z, remaining parallel to the director projection onto xy (58). It is clear how this
result maintains its validity also for larger d; hence, given that transition layers d are
about 100µm, the numerical results retain their validity also for the cell in fig. 2.1.
To validate these predictions, I varied V and experimentally studied the input polar-
ization maximizing energy coupling on the e (o) component. For low V, it is easy
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