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 λ = 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 nonmonotonically 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.
Bibliographical noteFunding Information:
D. P. was partially supported by the NSERC grant RGP-238931-01. A. S. was partially supported by the NSF grant DMS-0203301.
- Bragg resonance
- Evans function
- optical bistability
- photonic gratings