110,154 research outputs found
General methods for designing single-mode planar photonic crystal waveguides in hexagonal lattice structures
We systematically investigate and compare general methods of
designing single mode photonic crystal waveguides in a two-dimensional hexagonal lattice of air holes in a dielectric material. We apply the rather general methods to dielectric-core hexagonal lattice photonic crystals since they have not been widely explored before. We show that it is possible to obtain single mode guiding in a limited portion of the photonic bandgap of hexagonal lattice structures. We also compare the potentials of different photonic crystal lattices for designing single-mode waveguides and conclude that triangular lattice structures are the best choice
Quantum duality and Bethe-ansatz for the Hofstadter problem on hexagonal lattice
The Hofstadter problem is studied on hexagonal lattice. We first establish a
relation between the spectra for the hexagonal lattice and for its dual he
triangular lattice. Following the idea of Faddeev and Kashaev, we then obtain
the Bethe-ansatz equations for this system.Comment: 8 pages, latex, revised version for Phys. Lett.
Exact Site Percolation Thresholds Using the Site-to-Bond and Star-Triangle Transformations
I construct a two-dimensional lattice on which the inhomogeneous site
percolation threshold is exactly calculable and use this result to find two
more lattices on which the site thresholds can be determined. The primary
lattice studied here, the ``martini lattice'', is a hexagonal lattice with
every second site transformed into a triangle. The site threshold of this
lattice is found to be , while the others have and
. This last solution suggests a possible approach to establishing
the bound for the hexagonal site threshold, . To derive these
results, I solve a correlated bond problem on the hexagonal lattice by use of
the star-triangle transformation and then, by a particular choice of
correlations, solve the site problem on the martini lattice.Comment: 12 pages, 10 figures. Submitted to Physical Review
On well-rounded sublattices of the hexagonal lattice
We produce an explicit parameterization of well-rounded sublattices of the
hexagonal lattice in the plane, splitting them into similarity classes. We use
this parameterization to study the number, the greatest minimal norm, and the
highest signal-to-noise ratio of well-rounded sublattices of the hexagonal
lattice of a fixed index. This investigation parallels earlier work by
Bernstein, Sloane, and Wright where similar questions were addressed on the
space of all sublattices of the hexagonal lattice. Our restriction is motivated
by the importance of well-rounded lattices for discrete optimization problems.
Finally, we also discuss the existence of a natural combinatorial structure on
the set of similarity classes of well-rounded sublattices of the hexagonal
lattice, induced by the action of a certain matrix monoid.Comment: 21 pages (minor correction to the proof of Lemma 2.1); to appear in
Discrete Mathematic
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