Pierre R. Villeneuve

Photonic bandgaps in periodic dielectric structures

Abstract Photonic bandgaps are defined as frequency intervals for which propagation of electromagnetic waves is forbidden in all 4π steradians within a dielectric structure with a periodic index of refraction. Such structures consist, for example, of dielectric spheres in suspension or air holes in a dielectric material, with a spatial period comparable to the electromagnetic wavelength. The principal feature of periodic structures is their ability to perturb the density of electromagnetic states within the structures. Since photonic bandgap materials can essentially suppress all states, the radiative dynamics within the materials can be strongly modified. By changing the atom-field radiative coupling, photonic bandgap materials could lead to the inhibition of spontaneous emission; if a local defect is introduced within the structure, it will behave like a high-Q microcavity. The existence of bandgaps can be predicted from a classical treatment of the vector wave equation. The use of the plane-wave expansion method can lead to accurate results but introduces two problems related to the dielectric discontinuities and the plane-wave cutoff. Experimental investigations at microwave frequencies have demonstrated many of the properties of photonic bandgap structures.

Photonic band gaps of transverse-electric modes in two-dimensionally periodic media

A rigorous analysis of the propagation of transverse-electric modes in two-dimensionally periodic media is presented, based on an exact solution of the wave equation. The modulation functions studied lead to Mathieu and Hill equations. It is shown that the complete inhibition of the propagation of transverse-electric modes in a modulated plane is possible. Several modulation functions of the permittivity are studied in order to find the one for which the modulation depth necessary for the observation of a frequency band gap is minimized. In particular, thin grids and closely packed arrays of square rods are considered. The allowed angles of propagation of waves whose frequency is not contained within a forbidden band are also presented.

Photonic Band Gaps in Two-Dimensional Square and Triangular Lattices

While three-dimensional lattices can generate complete band gaps between the first and second bands for both orthogonal polarizations p and s, the overlap of the p and s gaps in two-dimensional lattices occurs between low s bands and higher p bands. This leads to significant differences between two- and three-dimensional lattices. Dielectric grids with cylindrical air holes located at the corners of a square or triangle can yield photonic band gaps common to p and s polarizations. The cross-sectional geometry of the cylindrical holes plays a vital role in determining the conditions to open a gap. Dielectric grids with holes of circular cross section located at the corners of a triangle or square require a similar index contrast to generate a band gap. Dielectric grids with holes of square cross section located at the corners of a square require a much larger index contrast to generate a band gap. Any defect in the cross-sectional symmetry of the cylindrical air holes can significantly affect the gap size in a square lattice with square holes, while the consequences are moderate in a triangular lattice with circular holes.

Confinement and inhibition of transverse-electric modes in a one-dimensionally periodic medium

Abstract The paper presents a rigorous analysis of the propagation of transverse-electric modes in a planar one-dimensionally periodic medium. By changing the modulation depth, the propagation of some optical frequencies can be: (i) confined strictly along the non-modulated axis, (ii) confined within an angle θ max from the axis of modulation or (iii) inhibited along the non-modulated axis. Inhibition along the non-modulated axis is interpreted using a parallel-plate waveguide analogy.