Modeling of defect modes in photonic crystals using the fictitious source superposition method

Fast and highly accurate boundary element method for scattering calculation of defective gratings

A numerical scattering calculation method for defective gratings is proposed. This method is based on an integral equation method that computes a difference-field distribution, which is the difference between the scattering fields with and without the defect, and it is possible to simulate the arbitrary (finite) size and shape of a defect in the grating without any limitation. A calculation example is also presented to demonstrate the fast convergence and high accuracy of this method.

Analytic theory of defects in periodically structured elastic plates

We consider the problem of localized flexural waves in thin plates that have periodic structure, consisting of a two-dimensional array of pins or point masses. Changing the properties of the structure at a single point results in a localized mode within the band-gap that is confined to the vicinity of the defect, while changing the properties along an entire line of points results in a waveguide mode. We develop here an analytic theory of these modes and provide semi-analytic expressions for the eigenfrequencies and fields of the point defect states, as well as the dispersion curves of the defect waveguide modes. The theory is based on a derivation of Green's function for the structure, which we present here for the first time. We also consider defects in finite arrays of point masses, and demonstrate the connection between the finite and infinite systems.

Convergence analysis and oscillations in the method of fictitious sources applied to dielectric scattering problems

Several wave scattering phenomena in optics are modeled by the method of fictitious sources (MFS). Despite its interesting features, the effectiveness of the MFS and its applicability are restricted by open issues, including the placement of the fictitious sources (FS) and the fields' convergence. Concerning these issues, we investigate here the MFS convergence and study oscillations in its solutions for a representative scattering problem of a dielectric cylinder illuminated by a current filament. It is shown analytically that, when the FS radii lie in the interior and exterior of two disks with certain critical radii, the MFS currents' series diverge while the respective fields converge, as the FS number N tends to infinity. Asymptotic formulas of the divergent currents are established, exhibiting that they increase exponentially with N and oscillate. Numerical simulations are included, demonstrating that (i) the divergent currents oscillate for sufficiently large N, (ii) the oscillating values are fairly approximated by the derived asymptotic expressions, and (iii) these oscillations are inherent in MFS and are not due to ill-conditioning; hence, they cannot be overcome by improving the hardware or software. The possibility of obtaining convergent and correct fields from divergent intermediary currents may lead to a potential significant advance of the applicability of the MFS.

Enhancing the absorption coefficient of a backed rigid frame porous layer by embedding circular periodic inclusions

The acoustic properties of a porous sheet of medium static air flow resistivity (around 10,000 N m s(-4)), in which a periodic set of circular inclusions is embedded and which is backed by a rigid plate, are investigated. The inclusions and porous skeleton are assumed motionless. Such a structure behaves like a multi-component diffraction grating. Numerical results show that this structure presents a quasi-total (close to unity) absorption peak below the quarter-wavelength resonance of the porous sheet in absence of inclusions. This result is explained by the excitation of a complex trapped mode. When more than one inclusion per spatial period is considered, additional quasi-total absorption peaks are observed. The numerical results, as calculated with the help of the mode-matching method described in this paper, agree with those calculated using a finite element method.

Absorption of a rigid frame porous layer with periodic circular inclusions backed by a periodic grating

The acoustic properties of a periodic rigid frame porous layer with multiple irregularities in the rigid backing and embedded rigid circular inclusions are investigated theoretically and numerically. The theoretical representation of the sound field in the structure is obtained using a combination of multipole method that accounts for the periodic inclusions and multi-modal method that accounts for the multiple irregularities of the rigid backing. The theoretical model is validated against a finite element method. The predictions show that the acoustic response of this structure exhibits quasi-total, high absorption peaks at low frequencies which are below the frequency of the quarter-wavelength resonance typical for a flat homogeneous porous layer backed by a rigid plate. This result is explained by excitation of additional modes in the porous layer and by a complex interaction between various acoustic modes. These modes relate to the resonances associated with the presence of a profiled rigid backing and rigid inclusions in the porous layer.

Acoustic response of a rigid-frame porous medium plate with a periodic set of inclusions

The acoustic response of a rigid-frame porous plate with a periodic set of inclusions is investigated by a multipole method. The acoustic properties, in particular, the absorption, of such a structure are then derived and studied. Numerical results together with a modal analysis show that the addition of a periodic set of high-contrast inclusions leads to the excitation of the modes of the plate and to a large increase in the acoustic absorption.

Efficient analysis of photonic crystal devices by Dirichlet-to-Neumann maps

An efficient numerical method based on the Dirichlet-to-Neumann (DtN) maps of the unit cells is developed for accurate simulations of two-dimensional photonic crystal (PhC) devices in the frequency domain. The DtN map of a unit cell is an operator that maps the wave field on the boundary of the cell to its normal derivative and it can be approximated by a small matrix. Using the DtN maps of the regular and defect unit cells, we can avoid computations in the interiors of the unit cells and calculate the wave field only on the edges. This gives rise to a significant reduction in the total number of unknowns. Reasonably accurate solutions can be obtained using 10 to 15 unknowns for each unit cell. In contrast, standard finite element, finite difference or plane wave expansion methods may require a few hundreds unknowns for each unit cell at the same level of accuracy. We illustrate our method by a number of examples, including waveguide bends, branches, microcavities coupled with waveguides, waveguides with stubs, etc.