We consider the Laplacian in a domain squeezedbetween two parallel curves in the plane,subject to Dirichlet boundary conditions on one of the curvesand Neumann boundary conditions on the other.We derive two-term asymptotics for eigenvaluesin the limit when the distance between the curves tends to zero.The asymptotics are uniform and local in the sense thatthe coefficients depend only on the extremal points wherethe ratio of the curvature radii of the Neumann boundaryto the Dirichlet one is the biggest.We also show that the asymptotics can be obtainedfrom a form of norm-resolvent convergencewhich takes into account the width-dependenceof the domain of definition of the operators involved.

We consider the Laplace operator in a thin tube of ${\mathbb R}^3$ with a Dirichlet condition on its boundary. We study asymptotically the spectrum ofsuch an operator as the thickness of the tube's cross section goes to zero. In particular weanalyse how the energy levels depend simultaneously on the curvature of the tube's central axisand on the rotation of the cross section with respect to the Frenet frame. The main argument is aΓ-convergence theorem for a suitable sequence of quadratic energies.

Since the development of the light microscope in the 16th century, optical device size and performance have been limited by diffraction. Optoelectronic devices of today are much bigger than the smallest electronic devices for this reason. Achieving control of light—material interactions for photonic device applications at the nanoscale requires structures that guide electromagnetic energy with subwavelength-scale mode confinement. By converting the optical mode into nonradiating surface plasmons, electromagnetic energy can be guided in structures with lateral dimensions of less than 10% of the free-space wavelength. A variety of methods—including electron-beam lithography and self-assembly—have been used to construct both particle and planar plasmon waveguides. Recent experimental studies have confirmed the strongly coupled collective plasmonic modes of metallic nanostructures. In plasmon waveguides consisting of closely spaced silver rods, electromagnetic energy transport over distances of 0.5 m has been observed. Moreover, numerical simulations suggest the possibility of multi-centimeter plasmon propagation in thin metallic stripes. Thus, there appears to be no fundamental scaling limit to the size and density of photonic devices, and ongoing work is aimed at identifying important device performance criteria in the subwavelength size regime. Ultimately, it may be possible to design an entire class of subwavelength-scale optoelectronic components (waveguides, sources, detectors, modulators) that could form the building blocks of an optical device technology—a technology scalable to molecular dimensions, with potential imaging, spectroscopy, and interconnection applications in computing, communications, and chemical/biological detection.

This article reviews the fabrication technologies and optical characteristics of silica-based planar lightwave circuits (PLCs) on Si developed for photonic networks based on wavelength-division multiplexing (WDM). While there have been various planar optical waveguides made with different materials, silica-based PLCs are the most suitable for constructing practical devices because of their excellent design flexibility, stability, and reproducibility. These advantages mainly result from their material characteristics, that is, silica glass is chemically and physically stable. The article also describes the basic characteristics and recent development of arrayed waveguide grating (AWG) multiplexers/demultiplexers as a device application. Since AWGs offer the advantages of low-loss, high-output port counts and mass producibility, they have played a pivotal role in the construction of flexible, large-capacity WDM networks.

We study second order elliptic operators with periodic coefficients in two-dimensional simply connected periodic waveguides with the Dirichlet or Neumann boundary conditions. It is proved that under some mild smoothness restrictions on the coefficients, such operators have purely absolutely continuous spectra. The proof follows a method suggested previously by A. Morame to tackle periodic operators with variable coefficients in dimension 2.

2000 Mathematical Subject Classification: 35J10, 35P05, 35J25.