Evolution of Bloch-mode envelopes in two-dimensional generalized honeycomb lattices

Journal Article · Thu Jul 15 04:00:00 EDT 2010 · Physical Review. A

DOI:https://doi.org/10.1103/PHYSREVA.82.013840· OSTI ID:21443034

Ablowitz, Mark J; Yi, Zhu ^[1]

---

1. Department of Applied Mathematics, University of Colorado, 526 UCB, Boulder, Colorado 80309-0526 (United States)

Wave propagation in two-dimensional generalized honeycomb lattices is studied. By employing the tight-binding (TB) approximation, the linear dispersion relation and associated discrete envelope equations are derived for the lowest band. In the TB limit, the Bloch modes are localized at the minima of the potential wells and can analytically be constructed in terms of local orbitals. Bloch-mode relations are converted into integrals over orbitals. With this methodology, the linear dispersion relation is derived analytically in the TB limit. The nonlinear envelope dynamics are found to be governed by a unified nonlinear discrete wave system. The lowest Bloch band has two branches that touch at the Dirac points. In the neighborhood of these points, the unified system leads to a coupled nonlinear discrete Dirac system. In the continuous limit, the leading-order evolution is governed by a continuous nonlinear Dirac system. This system exhibits conical diffraction, a phenomenon observed in experiments. Coupled nonlinear Dirac systems are also obtained. Away from the Dirac points, the continuous limit of the discrete equation leads to coupled nonlinear Schroedinger equations when the underlying group velocities are nearly zero. With semiclassical approximations, all the parameters are estimated analytically.

• OSTI ID: 21443034
• Journal Information: Physical Review. A, Journal Name: Physical Review. A Journal Issue: 1 Vol. 82; ISSN 1050-2947; ISSN PLRAAN
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
APPROXIMATIONS
CALCULATION METHODS
COHERENT SCATTERING
DIFFERENTIAL EQUATIONS
DIFFRACTION
DISPERSION RELATIONS
EQUATIONS
INTEGRALS
NONLINEAR PROBLEMS
PARTIAL DIFFERENTIAL EQUATIONS
POTENTIALS
SCATTERING
SCHROEDINGER EQUATION
SEMICLASSICAL APPROXIMATION
TWO-DIMENSIONAL CALCULATIONS
WAVE EQUATIONS
WAVE PROPAGATION