Solitons in Nonlocal Media

2.5 Theory of Nonlinear Optical Propagation in NLC
plane wave with the same wavevector 1 . I take k = nek0, being k0 the vacuum wavenum-
ber and ne the linear extraordinary index. Moreover, I define a new reference system
rts, where ˆs and ˆt are parallel to the Poynting and the electric fields, respectively,
and ˆr = ˆt × ˆs. Starting from Maxwell’s equation and considering only extraordinary
components, at the first order the field is polarized along t and obeys the equation (a
complete derivations is reported in appendix A.2) (47; 52)
2ik0ne cos δ ∂Ee
∂s
+ Dt
∂2Ee ∂
+ Dr
∂t2 2Ee ∂r2 + k2 0δǫttEe = 0 (2.6)
where δǫtt = ˆt·ǫ·ˆt is the nonlinear index variation, Dr = 1+ n2 e sin 2 δ
λx
and Dt = n2 e cos 2 δ
λs
are diffraction coefficients (section A.2), different each other due to the anisotropy. I
stress that eq. (2.6) is a scalar NLSE equation [see eq. (1.2)], written along propagation
coordinate s 2 . Therefore, for small perturbations, the nonlinear optical propagation in
anisotropic NLC can be described as in isotropic media.
Now I have to apply eq. (2.6) to the cell geometry sketched in fig. 2.1. Eq. (2.6)
confirms the hypotheses made in section 2.4 to model the soliton trajectory dependence
on the applied bias V, when the soliton was approximated by a plane wave.
Now I have to determine δǫtt, which depends on director reorientation. When V is not
zero, there is a low frequency field parallel to ˆx and an optical field directed like ˆt: I
need to use two angles to describe the director distribution. 3 Moreover, in order to get
a good approximation, I need to take into account the index well induced by V in the
plane xs and, thus, use vectorial equations.
Keeping in mind these considerations, hereafter and for the sake of simplicity, I limit
the investigation to V = 0: in this case the director reorientation takes place only in
the yz plane, being Ee linearly polarized along ˆt, and the transformation from xyz to
rts is a simple rotation around x by an angle δ. Being ˆr = ˆx I call the new reference
system xts. From section 1.4.2 the director profile is governed by
K∇ 2 xtθ + ǫ0ǫa
4 sin [2(θ − δ)] |Ee| 2 = 0 (2.7)
1 see section 1.4.3.
2 Therefore it is written in the paraxial approximation along s.
3 The two directions coincide for high V when all the molecules are parallel to x, but this case is
not interesting because the reorientational nonlinearity goes to zero.
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