Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part I. Method

Summation of a Schlömilch type series

The ability to accurately and efficiently characterize multiple scattering of waves of different nature attracts substantial interest in physics. The advent of photonic crystals has created additional impetus in this direction. An efficient approach in the study of multiple scattering originates from the Rayleigh method, which often requires the summation of conditionally converging series. Here summation formulae have been derived for conditionally convergent Schlömilch type series ∑ s = − ∞ ∞ Z n ( | s D − x | ) × e − i n arg ⁡ ( s D − x )   e i s D sin ⁡ θ 0 , where Z n ( z ) stands for any of the following cylindrical functions of integer order: Bessel functions J n ( z ), Neumann functions Y n ( z ) or Hankel functions of the first kind H n ( 1 ) ( z ) = J n ( z ) + i Y n ( z ) . These series arise in two-dimensional scattering problems on diffraction gratings with multiple inclusions per unit cell. It is shown that the Schlömilch series involving Hankel functions or Neumann functions can be expressed as an absolutely converging series of elementary functions and a finite sum of Lerch transcendent functions, while the Schlömilch series of Bessel functions can be transformed into a finite sum of elementary functions. The closed-form expressions for the Coates's integrals of integer order have also been found. The derived equations have been verified numerically and their accuracy and efficiency has been demonstrated.

Roundtrip matrix method for calculating the leaky resonant modes of open nanophotonic structures

We present a numerical method for calculating quasi-normal modes of open nanophotonic structures. The method is based on scattering matrices and a unity eigenvalue of the roundtrip matrix of an internal cavity, and we develop it in detail with electromagnetic fields expanded on Bloch modes of periodic structures. This procedure is simpler to implement numerically and more intuitive than previous scattering matrix methods, and any routine based on scattering matrices can benefit from the method. We demonstrate the calculation of quasi-normal modes for two-dimensional photonic crystals where cavities are side-coupled and in-line-coupled to an infinite W1 waveguide, and we show that the scattering spectrum of these types of cavities can be reconstructed from the complex quasi-normal mode frequency.

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.

Matrix Fabry-Perot resonance mechanism in high-contrast gratings

We present a simple analytic formalism to explain the unique resonance phenomenon in subwavelength high-contrast gratings (HCG). We show that the resonances are due to strong coupling between two surface-normal waveguide array modes resulting from abrupt and large index contrast. Simple expression for HCG quality factor is derived that agrees with spectral-fitting approaches reported in literature.

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.

Weaving the invisible thread: design of an optically invisible metamaterial fibre

We present the design of an invisible metamaterial fibre operating at optical frequencies, which could be fabricated by adapting existing fibre drawing techniques. The invisibility is realised by matching the refractive index of the metamaterial fibre with the surroundings. We present a general recipe for the fabrication of such fibres, and numerically characterise a specific example using hexagonally arranged silver nanowires in a silica background. We find that invisibility is highly sensitive to details of the metamaterial boundary, a problem that is likely to affect most invisibility and cloaking schemes.

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.

Evidence of a mobility edge for photons in two dimensions

A scaling analysis of conductance for photons in two dimensions is carried out and, contrary to widely held belief, we find strong evidence of a mobility edge. Such behavior is compatible with the existence of an Anderson transition for electronic systems under symplectic symmetry, and indeed we show that the transfer matrix in the photonic system we have modelled has such a symmetry. We verify single parameter scaling of the conductance and demonstrate the transition from the metallic phase to localization. Key parameters, including the critical disorder, the conductance, and the critical exponent of the localization length are calculated, and it is shown that the value of the critical exponent is similar to that for electronic systems with symplectic symmetry.

Analysis of scattering by a linear chain of spherical inclusions in an optical fiber

The scattering by a linear chain of spherical dielectric inclusions, embedded along the axis of an optical fiber, is analyzed using a rigorous integral equation formulation, based on the dyadic Green's function theory. The coupled electric field integral equations are solved by applying the Galerkin technique with Mie-type expansion of the field inside the spheres in terms of spherical waves. The analysis extends the previously studied case of a single spherical inhomogeneity inside a fiber to the multisphere-scattering case, by utilizing the classic translational addition theorems for spherical waves in order to analytically extract the direct-intersphere-coupling coefficients. Results for the transmitted and reflected power, on incidence of the fundamental HE(11) mode, are presented for several cases.

Transformation of radially traveling cylindrical waves between two skew cylindrical coordinate systems

The transformation of radially traveling cylindrical waves between two cylindrical coordinate systems with skew (nonparallel) axes is derived for the first time to our knowledge. The analytical procedure is based on the complex integral representation of the Hankel function and appropriate contour deformation and change of variables to obtain a final Fourier transform expression of the cylindrical wave in the new system. Scalar and vector waves are considered. This new result is a powerful tool for the rigorous analysis of scattering and coupling in nonparallel optical fiber configurations.

Wide-angle coupling into rod-type photonic crystals with ultralow reflectance

We describe the surprising phenomenon of near-perfect coupling from free space into uniform two-dimensional rod-type photonic crystals over a wide range of incident angles. This behavior is shown to be a generic feature of many rod-type photonic crystal structures that is related to strong forward scattering resonances of the individual cylinders. We explain these results using both semianalytic analysis and two-dimensional numerical calculations and identify the conditions under which efficient, wide-angle coupling can occur. The results may lead to more efficient designs for in-band photonic crystal devices such as superprisms and self-collimation based photonic circuits.

Diamagnetic response of metallic photonic crystals at infrared and visible frequencies

We show analytically and numerically that diamagnetic response (effective magnetic permeability mue<1) at infrared and visible frequencies can be achieved in photonic crystals composed of metallic nanowires or nanospheres when the wavelength is much larger than the lattice constant a (lambda approximately 2000a). When lambda approximately100a, the metallic photonic crystals will exhibit strong diamagnetic response (mue<0.8), leading to many interesting phenomena such as the unusual Brewster angle for s waves and incident-angle-and-polarization-independent reflection and transmission.

Two-dimensional treatment of the level shift and decay rate in photonic crystals

We present a comprehensive treatment of the level shift and decay rate of a model line source in a two-dimensional photonic crystal (2D PC) composed of circular cylinders. The quantities in this strictly two-dimensional system are determined by the two-dimensional local density of states (2D LDOS), which we compute using Rayleigh-multipole methods. We extend the critical point analysis that is traditionally applied to the 2D DOS (or decay rate) to the level shift. With this, we unify the crucial quantity for experiment--the 2D LDOS in a finite PC--with the band structure and the 2D DOS, 2D LDOS, and level shift in infinite PC's. Consistent with critical point analysis, large variations in the level shift are associated with large variations in the 2D DOS (and 2D LDOS), corroborating a giant anomalous Lamb shift. The boundary of a finite 2D PC can produce resonances that cause the 2D LDOS in a finite 2D PC to differ markedly from the 2D LDOS in an infinite 2D PC.

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

We present an exact theory for modeling defect modes in two-dimensional photonic crystals having an infinite cladding. The method is based on three key concepts, namely, the use of fictitious sources to modify response fields that allow defects to be introduced, the representation of the defect mode field as a superposition of solutions of quasiperiodic field problems, and the simplification of the two-dimensional superposition to a more efficient, one-dimensional average using Bloch mode methods. We demonstrate the accuracy and efficiency of the method, comparing results obtained using alternative techniques, and then concentrate on its strengths, particularly in handling difficult problems, such as where a mode is highly extended near cutoff, that cannot be dealt with in other ways.

Conductance of photons in disordered photonic crystals

The conductance of photons in two-dimensional disordered photonic crystals is calculated using an exact multipole-plane wave method that includes all multiple scattering processes. Conductance fluctuations, the universal nature of which has been established for electrons in the diffusive regime, are studied for photons, in both principal polarizations and for varying disorder. Our simulations show that universal conductance fluctuations can be observed in H(||) (TE) polarization for weak and intermediate disorder while, for E(||) (TM) polarization, we show that the conductance variance is essentially independent of sample size but strongly dependent on disorder. The probability distribution of the conductance is also calculated in the diffusive and localized regimes, and also at their transition, for which the distributions for both polarizations are seen to be very similar.

Wide-angle beam splitting by use of positive-negative refraction in photonic crystals

We present a positive-negative refraction effect in which, under certain conditions, an incident plane wave launched into a photonic crystal excites a positive-refracted Bloch wave and a negative-refracted Bloch wave simultaneously, both of which maintain the polarization. By utilizing this phenomenon, wide-angle beam splitting can be realized at the microscale level. Numerical simulations are employed to demonstrate this anomalous refraction behavior.

Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory

We present a rigorous Bloch mode scattering matrix method for modeling two-dimensional photonic crystal structures and discuss the formal properties of the formulation. Reciprocity and energy conservation considerations lead to modal orthogonality relations and normalization, both of which are required for mode calculations in inhomogeneous media. Relations are derived for studying the propagation of Bloch modes through photonic crystal structures, and for the reflection and transmission of these modes at interfaces with other photonic crystal structures.

Effects of disorder in two-dimensional photonic crystal waveguides

The effects of randomness on the guiding properties of waveguides embedded in disordered two-dimensional photonic crystals composed of a finite cluster of circular cylinders of infinite length are investigated for TM-polarized radiation. Different degrees of disorder in the radius, filling fraction, refractive index, and position are considered for both straight and 90 degrees bent guides. The crystals exhibit similar sensitivity to refractive index and radius disorder, with a degree of disorder from 15%-20% yielding little substantial change in the guiding properties. A smaller range of position disorder is also considered. For strong disorder in radius and refractive index, the guide effectively closes. These results were obtained by a Monte Carlo simulation method, and the performance of this method is analyzed. The method requires at least ten realizations in some cases for convergence to commence; substantially more realizations are required for moderate and strong disorder to achieve accurate results.

Semianalytic treatment for propagation in finite photonic crystal waveguides

We present a semianalytic theory for the properties of two-dimensional photonic crystal waveguides of finite length. For single-mode guides, the transmission spectrum and field intensity can be accurately described by a simple two-parameter model. Analogies are drawn with Fabry-Perot interferometers, and generalized Fresnel coefficients for the interfaces are calculated.

Cylinder gratings in conical incidence with applications to woodpile structures

We use our previous formulation for cylinder gratings in conical incidence to discuss the photonic band gap properties of woodpile structures. We study scattering matrices and Bloch modes of the woodpile, and use these to investigate the dependence of the optical properties on the number of layers. We give data on reflectance, transmittance and absorptance of metallic woodpiles as a function of wavelength and number of layers, using both the measured optical constants of tungsten and using a perfect conductivity idealization to characterize the metal. For semi-infinite metallic woodpiles, we show that polarization of the incident field is important, highlighting the role played by surface effects as opposed to lattice effects.

Diffusion and anomalous diffusion of light in two-dimensional photonic crystals

The transport properties of electromagnetic waves in disordered, finite, two-dimensional photonic crystals composed of circular cylinders are considered. Transport parameters such as the transport and scattering mean free paths and the transport velocity are calculated, for the case where the electromagnetic radiation has its electric field along the cylinder axes. The range of the parameters in which the diffusion process can take place is specified. It is shown that the transport velocity upsilon(E) can be as much as 10(8) times less than its free space value, while just outside the cluster upsilon(E) can be 0.3c. The effects of weak and strong disorders on the transport velocity are investigated. Different regimes of the wave transport-ordered propagation, diffusion, and anomalous diffusion-are demonstrated, and it is inferred that Anderson localization is incipient in the latter regime. Exact numerical calculations from the Helmholtz equation are shown to be in good agreement with the diffusion approximation.

Novel electromagnetic approach to photonic crystals with use of the C method

We introduce a new method allowing rigorous electromagnetic analysis of scattering through photonic crystals comprising polygonal or round rods. For this purpose, we reformulate the C method with adaptive spatial resolution by utilizing the hybrid-spectrum connection method, permitting the use of nonidentical trapezoidal profiles. Considering polygonal rods as gratings consisting of different piecewise-differentiable surfaces, we are able to analyze the reflection and the transmittance of crystals by means of the C method. To enhance computational efficiency, we apply the recursive S-matrix approach with Redheffer's star product to solve the transfer matrix for structures of numerous successive layers of rods.

Cylinder gratings in conical incidence with applications to modes of air-cored photonic crystal fibers

We develop a formulation for cylinder gratings in conical incidence, using a multipole method. The theory, and its numerical implementation, is applied to two-dimensional photonic crystals consisting of a stack of one-dimensional gratings, each characterized by its plane wave scattering matrix. These matrices are used in combination with Bloch's theorem to determine the band structure of the photonic crystal from the solution of an eigenvalue problem. We show that the theory is well adapted to the difficult task of locating the complete band gaps needed to support air-guided modes in microstructured optical fibers, that is, optical fibers in which the confinement of light in a central air hole is achieved by photonic band-gap effects in a periodic cladding comprising a lattice of air holes in a glass matrix.

Photonic band structure calculations using scattering matrices

We consider band structure calculations of two-dimensional photonic crystals treated as stacks of one-dimensional gratings. The gratings are characterized by their plane wave scattering matrices, the calculation of which is well established. These matrices are then used in combination with Bloch's theorem to determine the band structure of a photonic crystal from the solution of an eigenvalue problem. Computationally beneficial simplifications of the eigenproblem for symmetric lattices are derived, the structure of eigenvalue spectrum is classified, and, at long wavelengths, simple expressions for the positions of the band gaps are deduced. Closed form expressions for the reflection and transmission scattering matrices of finite stacks of gratings are established. A new, fundamental quantity, the reflection scattering matrix, in the limit in which the stack fills a half space, is derived and is used to deduce the effective dielectric constant of the crystal in the long wavelength limit.

Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part II. Properties and implementation

A numerical implementation and generalized conservation properties of a formulation for calculating wave propagation through stacked gratings comprising metallic and dielectric cylinders are presented. The basic formulation of the method was given in a companion paper [J. Opt. Soc. Am. A. 17, 2165 (2000)]. Here, details of the numerical implementation of the method are discussed and are illustrated for the ensemble average of a strongly scattering structure with refractive index and radius disorder. Also presented are a comprehensive treatment of energy conservation and generalized phase relations, as well as reciprocity.