Stephen P. Shipman

Magnetism and Homogenization of Microresonators

Arrays of cylindrical metal microresonators embedded in a dielectric matrix were proposed by Pendry et al. [IEEE Trans. Microw. Theory Tech., 47 (1999), pp. 2075–2084] as a means of creating a microscopic structure that exhibits strong bulk magnetic behavior at frequencies not realized in nature. This behavior arises for H-polarized fields in the quasi-static regime, in which the scale of the microstructure is much smaller than the free-space wavelength of the fields. We carry out both formal and rigorous two-scale homogenization analyses, paying special attention to the appropriate method of averaging, which does not involve the usual cell averages. We show that the effective magnetic and dielectric coefficients obtained by means of such averaging characterize a bulk medium that, to leading order, produces the same scattering data as the microstructured composite.

RESONANCE AND BOUND STATES IN PHOTONIC CRYSTAL SLABS

Using boundary-integral projections for time-harmonic electromagnetic (EM) fields, and their numerical implementation, we analyze EM resonance in slabs of two-phase dielectric photonic crystal materials. We characterize resonant frequencies by a complex Floquet--Bloch dispersion relation

Boundary-Integral Calculations of Two-Dimensional Electromagnetic Scattering in Infinite Photonic Crystal Slabs: Channel Defects and Resonances

\omega = W(\beta)

Resonant electromagnetic scattering in anisotropic layered media

defined by the existence of a nontrivial nullspace of a pair of boundary-integral projections parameterized by the wave number

A Discrete Model for Resonance Near Embedded Bound States

\beta

Superalgebraically convergent smoothly windowed lattice sums for doubly periodic Green functions in three-dimensional space

and the time-frequency

Sub-wavelength plasmonic crystals: dispersion relations and effective properties

\omega

Convergent Power Series for Fields in Positive or Negative High-Contrast Periodic Media

. At resonant frequencies, the crystal slab supports a source-free EM field. We link complex resonant frequencies, where the imaginary part is small, to resonant scattering behavior of incident source fields at nearby real frequencies and anomalous transmission of energy through the slab. At a real resonant frequency, the source-free field supported by the slab is a bound state. We present numerical examples which demonstrate the effects of structural defects on the resonant properties of a crystal slab and surface waves supported by a die...

Resonance in anisotropic layered media

We compute the transmission of two-dimensional (2D) electromagnetic waves through a square lattice of lossless dielectric rods with a channel defect. The lattice is finite in the direction of propagation of the incident wave and periodic in a transverse direction. We revisit a boundary-integral formulation of 2D electromagnetic scattering [Venakides, Haider, and Papanicolaou, SIAM J. Appl. Math., 60 (2000), pp. 1686--1706] that is Fredholm of the first kind and develop a second-kind formulation. We refine the numerical implementation in the above paper by exploiting separability in the Greens function to evaluate the far-field influence more efficiently. The resulting cost savings in computing and solving the discretized linear system leads to an accelerated method. We use it to analyze E-polarized electromagnetic scattering of normally incident waves on a structure with a periodic channel defect. We find three categories of resonances: waveguide modes in the channel, high-amplitude fields in the crystal...

Total Resonant Transmission and Reflection by Periodic Structures

The resonant excitation of an electromagnetic guided mode of a slab structure by exterior radiation results in anomalous scattering behavior, including sharp energy-transmission anomalies and field amplification around the frequency of the slab mode. In the case of a periodically layered ambient medium, anisotropy serves to couple the slab mode to radiation. Exact expressions for scattering phenomena are proved by analyzing a pole of the full scattering matrix as it moves off the real frequency axis into the lower half complex plane under a detuning of the wavevector parallel to the slab. The real pole is the frequency of a perfect (infinite Q) guided mode, which becomes lossy as the frequency gains an imaginary part. This work extends results of Shipman and Venakides to evanescent source fields and two-dimensional parallel wavevector and demonstrates by example how the latter allows one to control independently the width and central frequency of a resonance by varying the angle of incidence of the source...