Solitons in Nonlocal Media

2.3 Effect of the Input Interface
Figure 2.2: (a) Experimental set-up. (b) Input field E and its polarization in the plane
xy at z = 0. i.e. at the interface between air and NLC. Sign convention is such that β
shown in the figure is positive.
In order to describe the director orientation, I introduce the two angles ξ and γ, as
shown in fig. 2.3.
As the bulk NLC is preceded by a transition layer
of thickness d following the input interface in z = 0,
I model this transition layer as an anisotropic struc-
ture stratified along z, with optical properties con-
stant in each layer, i.e. with a dielectric tensor which
does not depend on transverse coordinates xy 1 . To
model finite beam behavior I take the director value
in x = a/2, 2 given that the experimental beam width
is much smaller than the cell thickness a. Under such
Figure 2.3: Director ˆn and the
two angles ξ and γ used to describe
its orientation in space.
hypotheses I can apply Berreman’s method (78) to describe the propagation of electro-
magnetic plane waves with k = kˆz, i.e. impinging normally on the sample. I note that
the wavevector k cannot change direction in the transition layer under these hypotheses.
Hence I cast Maxwell’s equations in the form:
dΦ
= iω∆(z)Φ (2.1)
dz
being ω the optical angular frequency and
1 This implies neglecting optical reorientation; such approximation will be justified later.
2 Having neglected the optical reorientation, the profile does not change along y owing to the
symmetry.
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