FFTW is an implementation of the discrete

We describe a fully-vectorial, three-dimensional algorithm to compute the definite-frequency eigenstates of Maxwell's equations in arbitrary periodic dielectric structures, including systems with anisotropy (birefringence) or magnetic materials, using preconditioned block-iterative eigensolvers in a planewave basis. Favorable scaling with the system size and the number of computed bands is exhibited. We propose a new effective dielectric tensor for anisotropic structures, and demonstrate that O Delta x;2 convergence can be achieved even in systems with sharp material discontinuities. We show how it is possible to solve for interior eigenvalues, such as localized defect modes, without computing the many underlying eigenstates. Preconditioned conjugate-gradient Rayleigh-quotient minimization is compared with the Davidson method for eigensolution, and a number of iteration variants and preconditioners are characterized. Our implementation is freely available on the Web.

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This paper describes Meep, a popular free implementation of the finite-difference time-domain (FDTD) method for simulating electromagnetism. In particular, we focus on aspects of implementing a full-featured FDTD package that go beyond standard textbook descriptions of the algorithm, or ways in which Meep differs from typical FDTD implementations. These include pervasive interpolation and accurate modeling of subpixel features, advanced signal processing, support for nonlinear materials via Padé approximants, and flexible scripting capabilities. PACS

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⇤ These authors contributed equally to this work.The ability to confine light is important both scientifically and technologically. Many light confinement methods exist, but they all achieve confinement with materials or systems that forbid outgoing waves. Such systems can be implemented by metallic mirrors, by photonic band-gap materials 1 , by highly disordered media (Anderson localization 2 ) and, for a subset of outgoing waves, by translational symmetry (total internal reflection 1 ) or rotation/reflection symmetry 3, 4 . Exceptions to these examples exist only in theoretical proposals [5][6][7][8] . Here we predict and experimentally demonstrate that light can be perfectly confined in a patterned dielectric slab, even though outgoing waves are allowed in the surrounding medium. Technically, this is an observation of an "embedded eigenvalue" 9 -namely a bound state in a continuum of radiation modes-that is not due to symmetry incompatibility [5][6][7][8][10][11][12][13][14][15][16] . Such a bound 1 state can exist stably in a general class of geometries where all of its radiation amplitudes vanish simultaneously due to destructive interference. This method to trap electromagnetic waves is also applicable to electronic 12 and mechanical waves 14,15 .The propagation of waves can be easily understood from the wave equation, but the localization of waves (creation of bound states) is more complex. Typically, wave localization can only be achieved when suitable outgoing waves either do not exist or are forbidden due to symmetry incompatibility. For electromagnetic waves, this is commonly implemented with metals, photonic bandgaps, or total internal reflections; for electron waves, this is commonly achieved with potential barriers. In 1929, von Neumann and Wigner proposed the first counterexample 10 , in which they designed a quantum potential to trap an electron whose energy would normally allow coupling to outgoing waves. However, such artificially designed potential does not exist in reality. Furthermore, the trapping is destroyed by any generic perturbation to the potential. More recently, other counterexamples have been proposed theoretically in quantum systems 11-13 , photonics 5-8 , acoustic and water waves 14,15 , and mathematics 16 ; the proposed systems in refs. 6 and 14 are most closely related to what is demonstrated here. While no general explanation exists, some cases have been interpreted as two interfering resonances that leaves one resonance with zero width 6,11,12 . Among these many proposals, most cannot be readily realized due to their inherent fragility. A different form of embedded eigenvalue has been realized in symmetry-protected systems 3, 4 , where no outgoing wave exists for modes of a particular symmetry.To show that an optical bound state is feasible even when it is surrounded by symmetrycompatible radiation modes, we consider a practical structure: a dielectric slab with a square array 2 of cylindrical holes (Fig. 1a), an example of photonic crystal (PhC) slab 1 . The periodic geomet...

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We analyze the properties of two-dimensionally periodic dielectric structures that have a band gap for propagation in a plane and that use index guiding to confine light in the third dimension. Such structures are more amenable to fabrication than photonic crystals with full three-dimensional band gaps, but retain or approximate many of the latter's desirable properties. We show how traditional band-structure analysis can be adapted to slab systems in the context of several representative structures, and describe the unique features that arise in this framework compared to ordinary photonic crystals. ͓S0163-1829͑99͒00832-2͔