Photonic Gap Research Articles

Existence of a photonic gap in periodic dielectric structures.

Using a plane-wave expansion method, we have solved Maxwell's equations for the propagation of electromagnetic waves in a periodic lattice of dielectric spheres (dielectric constant ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{a}}$) in a uniform dielectric background (${\mathrm{\ensuremath{\epsilon}}}_{\mathit{b}}$). Contrary to experiment, we find that fcc dielectric structures do not have a ``photonic band gap'' that extends throughout the Brillouin zone. However, we have determined that dielectric spheres arranged in the diamond structure do possess a full photonic band gap. This gap exists for refractive-index contrasts as low as 2.

Chaotic trajectories in the standard map. The concept of anti-integrability

A rigorous proof is given in the standard map (associated with a Frenkel-Kontorowa model) for the existence of chaotic trajectories with unbounded momenta for large enough coupling constant k > k 0. These chaotic trajectories (with finite entropy per site) are coded by integer sequences { m i } such that the sequence b i = |m i+1 + m i−1−2m i| be bounded by some integer b. The bound k 0 in k depends on b and can be lowered for coding sequences { m i } fulfilling more restrictive conditions. The obtained chaotic trajectories correspond to stationary configurations of the Frenkel-Kontorowa model with a finite (non-zero) photon gap (called gap parameter in dimensionless units). This property implies that the trajectory (or the configuration { u i }) can be uniquely continued as a uniformly continuous function of the model parameter k in some neighborhood of the initial configuration. A non-zero gap parameter implies that the Lyapunov coefficient is strictly positive (when it is defined). In addition, the existence of dilating and contracting manifolds is proven for these chaotic trajectories. “Exotic” trajectories such as ballistic trajectories are also proven to exist as a consequence of these theorems. The concept of anti-integrability emerges from these theorems. In the anti-integrable limit which can be only defined for a discrete time dynamical system, the coordinates of the trajectory at time i do not depend on the coordinates at time i - 1. Thus, at this singular limit, the existence of chaotic trajectories is trivial and the dynamical system reduces to a Bernoulli shift. It is well known that the KAM tori of symplectic dynamical originates by continuity from the invariant tori which exists in the integrible limit (under certain conditions). In a similar way, it appears that the chaotic trajectories of dynamical systems originate by continuity from those which exists at the anti-integrable limits (also under certain conditions).

Self-localized light: launching of low-velocity solitons in corrugated nonlinear waveguides

We show that stationary gap solitons are just a limiting case of a wider set of (slowly) moving solitons associated with the photonic stop gap of a periodic nonlinear structure. We give an analytic description of these solitons and provide the prescription for launching such solitons in a realistic waveguide geometry.

Spontaneous emission of atoms coupled to frequency-dependent reservoirs.

We study the spontaneous emission of atoms coupled to frequency-dependent photon-mode reservoirs, and find atomic emission spectra qualitatively different from those of free space. Frequency-dependent photon-mode reservoirs are found in a variety of physical situations (e.g., in waveguides, microcavities, dielectrics, systems with photonic spectral gaps, etc.), and are shown to lead to nonexponential decay and concomitant complicated non-Lorentzian emission line shapes. By studying Autler-Townes spectra in the presence of a frequency-dependent photon-mode density, we present simple examples of dynamical modifications of spontaneous decay properties.