# Graduate School in Nonlinear Science

### Sponsored by the Danish Research Academy

**
MIDIT OFD CATS
Modelling, Nonlinear Dynamics Optics and Fluid Dynamics Chaos and Turbulence Studies
and Irreversible Thermodynamics Risø National Laboratory Niels Bohr Institute and
Technical University of Denmark Building 128 Department of Chemistry
Building 321 P.O. Box 49 University of Copenhagen
DK-2800 Lyngby DK-4000 Roskilde DK-2100 Copenhagen Ø
Denmark Denmark Denmark
**

**HEXAGONAL OPTICAL STRUCTURE IN PHOTOREFRACTIVE CRYSTALS WITH A
FEEDBACK MIRROR**

by** Pavel Lushnikov**

L. D. Landau Inistitute for Theoretical Physics

Moscow University

Moscow, Russia

MIDIT-seminar 431

**Thursday November 26, 1998, 14.00 h**

at MIDIT, IMM Building 305, room 027

**Abstract**: A nonlinear theory is developed to describe the generation of
hexagonal optical structures in a photorefractive medium
with a feedback mirror.
The counterpropagation of light beams in photorefractive crystal results
in transverse instability against the excitation of weak sideband waves
at small angles. The nonlinear three-wave interaction of these weak
waves leads to the explosive amplitude growth of the waves under the
angles ±pi/3 between each other.
It is shown here
that as this instability evolves to its nonlinear stage, the three-
wave interaction between weak sideband beams does not
stabilize it, but rather leads to explosive growth of the
amplitudes of beams whose transverse wave vectors form
angles that are multiples of pi/3. As a result, sidebands beams
at these angles are found to be correlated. A range of
parameters is found in which four-wave interactions saturate
the explosive instability, which explains the appearance of
stable hexagons in the experiment. Outside this region,
nonlinearities of higher order saturate the explosive instability,
and the process of hexagon generation must be studied
numerically. Matrix elements are obtained for the three- and
four-wave interactions as functions of the distance to the
feedback mirror, and an equation for the time evolution of the
sideband wave amplitudes is derived that describes the hexagon
generation. A comparison is made with experimental results
for the photorefractive crystals KNbO_{3} and
BaTiO_{3}.