D.E. Pelinovsky and A. Scheel
Stability analysis of stationary light transmission in
nonlinear photonic structures
J. Nonlin. Sci. 13, 347-396 (2003)
We study optical bistability of stationary light transmission
in nonlinear periodic structures of finite and semi-infinite
length. For finite-length structures, the system exhibits
instability mechanisms typical for dissipative dynamical
systems. We construct a Leray-Schauder stability index and show
that it equals the sign of the Evans function in $\lambda=0$. As
a consequence, stationary solutions with negative-slope
transmission function are always unstable.
In semi-infinite structures, the system may have stationary
localized solutions with non-monotonically decreasing amplitudes.
We show that the localized solution with a positive-slope amplitude
at the input is always unstable. We also derive expansions
for finite size effects and show that the bifurcation
diagram stabilizes in the limit of the infinite domain size.
OPTICAL BISTABILITY, EVANS FUNCTION, BRAGG RESONANCE, PHOTONIC GRATINGS