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

Absorption and scattering by structured interfaces in X-rays

Promising achievements of resonance inelastic X-ray scattering and other spectroscopy studies in the range from hard X-ray to extreme ultraviolet require the development of exact tools for modeling energy characteristics of state-of-the-art optical instruments for bright coherent X-ray sources, space science, and plasma and superconductor physics. Accurate computations of the absorption and scattering intensity by structured interfaces in short wavelength ranges, i.e. realistic gratings, zone plates and mirrors, including multilayer-coated, are not widely explored by the existing methods and codes, due to some limitations connected, primarily, with solving difficult problems at very small wavelength-to-period (or to correlation length) ratios and accounting for random roughness statistics. In this work, absorption integrals and scattering factors are derived from a rigorous solution of the vector Helmholtz equations based on the boundary integral equations and the Monte Carlo method. Then, using explicit formulae (in quadratures), the author finds the absorption and scattering intensity of one- and bi-periodic gratings and mirrors, which may have random roughnesses. Examples of space and spectral power distributions for gratings and mirrors working in X-rays are compared with those derived using the usual indirect approach and well known approximations.

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.

Semi-analytic impedance modeling of three-dimensional photonic and metamaterial structures

We define the concept of an impedance matrix for three-dimensional (3D) photonic and metamaterial structures relative to a reference medium and show that it satisfies a matrix generalization of the basic algebraic properties of the wave impedance between homogeneous media. This definition of the impedance matrix is motivated by the structure of the Fresnel reflection and transmission matrices at the interface between the media. In the derivation of the Fresnel scattering matrices, the field in each medium is expressed by a Bloch mode expansion, with field matching at the interface being undertaken in a least-squares manner by exploiting a biorthogonality relation between primal and adjoint Bloch modes. A semi-analytic technique, based on the impedance matrix, is developed for modeling the scattering of light by 3D periodic photonic and metamaterial structures. The advantages (in design and intuition) of the formalism are demonstrated through two applications.

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.

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.

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.

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.

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 I. Method

We present a formulation for wave propagation and scattering through stacked gratings comprising metallic and dielectric cylinders. By modeling a photonic crystal as a grating stack of this type, we thus formulate an efficient and accurate method for photonic crystal calculations that allows us to calculate reflection and transmission matrices. The stack may contain an arbitrary number of gratings, provided that each has a common period. The formulation uses a Green's function approach based on lattice sums to obtain the scattering matrices of each layer, and it couples these layers through recurrence relations. In a companion paper [J. Opt Soc. Am. A 17, 2177 (2000)] we discuss the numerical implementation of the method and give a comprehensive treatment of its conservation properties.