Three-dimensional model for light-induced chaotic rotations in liquid crystals under spin
and orbital angular momentum transfer processes
Etienne Brasselet
Centre de Physique Moléculaire Optique et Hertzienne, Université Bordeaux 1,
CNRS, 351 Cours de la Libération, 33405 Talence Cedex, France
Bruno Piccirillo
*
and Enrico Santamato
*
Dipartimento di Scienze Fisiche, Università di Napoli “Federico II,” via Cintia, 80126 Napoli, Italy
Received 9 April 2008; revised manuscript received 22 August 2008; published 3 September 2008
Liquid crystals interacting with light represent a unique class of soft-matter systems that exhibit various
generic nonlinear behaviors, including chaotic rotational dynamics. Despite several experimental observations,
complex nematic liquid crystal director rotations in presence of spin and orbital angular momentum transfer
processes were left unexplained. We present a self-consistent three-dimensional model able to describe the
previous experimental observations, accounting for the dependence on the incident beam intensity, polariza-
tion, finite size and shape. More generally, our model is able to describe quantitatively the dynamics of, and
beyond, the optical Fréedericksz transition under realistic experimental conditions almost three decades after
its experimental discovery.
DOI: 10.1103/PhysRevE.78.031703 PACS numbers: 42.70.Df, 05.45.a, 42.65.Sf
I. INTRODUCTION
Dynamical systems that are described by a set of coupled
first-order autonomous differential equations are usually
characterized in terms of rotational motions, which is a ge-
neric feature of many processes in physics, chemistry, or
biology. This is especially the case when such nonlinear sys-
tems exhibit chaotic dynamics. Then, the system trajectory
can be considered as going through an infinite number of
rotationlike motions characterized by a suitable definition of
the phase 1, which is of special interest in the framework of
phase synchronization 2,3. Due to the modeling difficulties
and/or achievable computational handling of real dynamical
systems exhibiting chaotic rotations, their description is usu-
ally reduced to standard nonlinear models, such as the Lor-
entz 4 or Rössler 5 systems. Exact modeling of real sys-
tems exhibiting chaotic rotations is much less widespread.
Quite recently, chaotic rotations generated by a light field
in nematic liquid crystals NLCs have been reported 6,7.
In these experiments, the light was circularly polarized and
sent at normal incidence onto a homeotropically aligned
nematic film. In Ref. 6 an elliptically shaped light beam
was used, while in Ref. 7 the incident beam shape was
cylindrically symmetric. In both cases the observations were
left unexplained despite some attempt to describe the NLC
continuum medium as a discrete set of coupled nonlinear
rotators driven by light, which led to a generic Kuramoto
model 7,8. The major difficulty in modeling this kind of
experiment is that the dynamics of the NLC director is
strongly affected by the finite size and shape of the incoming
light beam so that the plane-wave approach, where all the
fields depend on one coordinate only, is inapplicable. All the
three space coordinates and time must be retained. A previ-
ous attempt to model complex rotational dynamics account-
ing for the finite size of the light beam, but neglecting higher
order reorientational modes and nonadiabatic propagation of
light in the distorted liquid crystal, failed to retrieve experi-
mental observations 9 and to give the expected sequence of
bifurcations in the plane-wave limit 10. Therefore a suit-
able model is still lacking.
In this paper, we present the derivation of a set of coupled
ordinary differential equations ODEs for the motion of the
local averaged molecular orientation defined by a unit vector
n, called director, from the fundamental equations for LCs
and electromagnetic waves. The dependence of the liquid
crystal dynamics on the incident beam intensity, polarization,
finite size and shape are obtained, which naturally accounts
for spin angular momentum SAM and orbital angular mo-
mentum OAM transfer processes between light and matter.
This is another interesting yet intriguing feature of this class
of experiments where both the photon SAM and OAM come
into play and couple with the NLC anisotropic fluid. The
model capabilities are illustrated in the case where SAM and
OAM transfer processes are simultaneously present and a
successful description of the complex sequence of director
rotations observed in Ref. 6 is obtained. More generally,
our model is able to describe quantitatively the dynamics of,
and beyond, the optical Fréedericksz transition OFT under
realistic experimental conditions almost three decades after
its experimental discovery.
From a general point of view, the angular momentum
balance in NLCs can be written in the form 11 r v˙
= div L
ˆ
+ w and In n˙ = div S
ˆ
- w where an upper dot stands
for partial time derivative, , I, and v are, respectively, the
density, inertial momentum, and velocity field of the fluid, L
ˆ
and S
ˆ
are the orbital and spin total matter + light angular
momentum flux tensors and, finally, w is the internal torque
density due to the anisotropy of the elastic constants of the
material. Note that L
ˆ
and S
ˆ
can be uniquely split into a mat-
*
Also at Consorzio Nazionale Interuniversitario per la Struttura
della Materia CNISM, Sezione di Napoli, Italy.
PHYSICAL REVIEW E 78, 031703 2008
1539-3755/2008/783/0317034 ©2008 The American Physical Society 031703-1