Photonic crystal antireflection coatings, surface modes, and impedances

We present a rigorous definition of a wave impedance for 2D rectangular and triangular lattice photonic crystals (PCs), in the form of a matrix. Reflection and transmission at an interface between PCs can be represented by matrices that relate the Bloch mode (eigenmode) amplitudes in the two PCs; we show that these matrices, which are multi-mode generalisations of reflection and transmission coefficients, may be calculated from the PCs' impedances that we define. Given the impedances and Bloch factors (propagation constants) of a collection of PCs, the reflection and transmission properties of arbitrary stacks of these PCs may be calculated efficiently using a few matrix operations. Therefore our definition enables PC-based antireflection coatings to be designed efficiently: some computationally expensive simulations are required in an initial step to find a range of PCs' impedances, but then the reflectances of every coating that consists of a stack of these PCs can be calculated without any further simulations. We first define the PC impedance from the transfer matrix of a single PC layer (i.e., a grating). Since transfer matrix methods are not especially widespread, we also present a method and associated source code to extract a PC's propagating and evanescent Bloch modes from a scattering calculation that can be performed by any off-the-shelf field solver, and to calculate impedances from the extracted modal fields. Finally, we put our method to use. We apply it to design antireflection coatings, nearly eliminating reflection at a single frequency for one or both polarisations, or lowering it across a larger bandwidth. We use it to find surface modes at interfaces between PCs and air, and their projected band structures. We use the impedance to define effective parameters for PC homogenisation, and we briefly describe how our definition has been used to dispersion engineer a PC waveguide

Rigorous Analysis Of Wave Guiding And Diffractive Integrated Optical Structures

The realization of wavelength scale and sub-wavelength scale fabrication of integrated optical devices has led to a concurrent need for computational design tools that can accurately model electromagnetic phenomena on these length scales. This dissertation describes the physical, analytical, numerical, and software developments utilized for practical implementation of two particular frequency domain design tools: the modal method for multilayer waveguides and one-dimensional lamellar gratings and the Rigorous Coupled Wave Analysis (RCWA) for 1D, 2D, and 3D periodic optical structures and integrated optical devices. These design tools, including some novel numerical and programming extensions developed during the course of this work, were then applied to investigate the design of a few unique integrated waveguide and grating structures and the associated physical phenomena exploited by those structures. The properties and design of a multilayer, multimode waveguide-grating, guided mode resonance (GMR) filter are investigated. The multilayer, multimode GMR filters studied consist of alternating high and low refractive index layers of various thicknesses with a binary grating etched into the top layer. The separation of spectral wavelength resonances supported by a multimode GMR structure with fixed grating parameters is shown to be controllable from coarse to fine through the use of tightly controlled, but realizable, choices for multiple layer thicknesses in a two material waveguide; effectively performing the simultaneous engineering of the wavelength dispersion for multiple waveguide grating modes. This idea of simultaneous dispersion band tailoring is then used to design a multilayer, multimode GMR filter that possesses broadened angular acceptance for multiple wavelengths incident at a single angle of incidence. The effect of a steady-state linear loss or gain on the wavelength response of a GMR filter is studied. A linear loss added to the primary guiding layer of a GMR filter is shown to produce enhanced resonant absorption of light by the GMR structure. Similarly, linear gain added to the guiding layer is shown to produce enhanced resonant reflection and transmission from a GMR structure with decreased spectral line width. A combination of 2D and 3D modeling is utilized to investigate the properties of an embedded waveguide grating structure used in filtering/reflecting an incident guided mode. For the embedded waveguide grating, 2D modeling suggests the possibility of using low index periodic inclusions to create an embedded grating resonant filter, but the results of 3D RCWA modeling suggest that transverse low index periodic inclusions produce a resonant lossy cavity as opposed to a resonant reflecting mirror. A novel concept for an all-dielectric unidirectional dual grating output coupler is proposed and rigorously analyzed. A multilayer, single-mode, high and graded-index, slab waveguide is placed atop a slightly lower index substrate. The properties of the individual gratings etched into the waveguide\u27s cover/air and substrate/air interfaces are then chosen such that no propagating diffracted orders are present in the device superstrate and only a single order is present outside the structure in the substrate. The concept produces a robust output coupler that requires neither phase-matching of the two gratings nor any resonances in the structure, and is very tolerant to potential errors in fabrication. Up to 96% coupling efficiency from the substrate-side grating is obtained over a wide range of grating properties

Sensitivity analysis of 2D photonic band gaps of any rod shape and conductivity using a very fast conical integral equation method

The conical boundary integral equation method has been proposedto calculate the sensitive optical response of 2D photonic band gaps (PBGs),including dielectric, absorbing, and high-conductive rods of various shapes working in any wavelength range. It is possible to determine the diffracted field by computing the scattering matrices separately for any gratingboundary profile. The computation of the matrices is based on the solution of a 2 x 2system of singular integral equations at each interface between two different materials. The advantage of our integral formulation is that the discretization of the integral equations system and the factorization of the discrete matrices, which takes the major computing time, are carried out only oncefor a boundary. It turned out that a small number of collocation points per boundary combined with a high convergence rate can provide adequate description of the dependence on diffracted energy of very different PBGs illuminated at arbitrary incident and polarization angles. Thenumerical results presented describe the significant impact of rod shape on diffraction in PBGs supporting polariton-plasmon excitation, particularly in the vicinity of resonances and at high fillingratios. The diffracted energy response calculated vs. array cell geometry parameters was found to vary from a few percent up to a few hundred percent. The influence of other types of anomalies (i.e. waveguide anomalies, cavity modes, Fabry-Perot and Bragg resonances, Rayleigh orders, etc), conductivity, and polarization states on the optical response has been demonstrated

Propagation of Light in Photonic Crystal Fibre Devices

We describe a semi-analytical approach for three-dimensional analysis of photonic crystal fibre devices. The approach relies on modal transmission-line theory. We offer two examples illustrating the utilization of this approach in photonic crystal fibres: the verification of the coupling action in a photonic crystal fibre coupler and the modal reflectivity in a photonic crystal fibre distributed Bragg reflector.Comment: 15 pages including 7 figures. Accepted for J. Opt. A: Pure Appl. Op

Gratings: Theory and Numeric Applications

International audienceThe book containes 11 chapters written by an international team of specialist in electromagnetic theory, numerical methods for modelling of light diffraction by periodic structures having one-, two-, or three-dimensional periodicity, and aiming numerous applications in many classical domains like optical engineering, spectroscopy, and optical telecommunications, together with newly born fields such as photonics, plasmonics, photovoltaics, metamaterials studies, cloaking, negative refraction, and super-lensing. Each chapter presents in detail a specific theoretical method aiming to a direct numerical application by university and industrial researchers and engineers

Scattering matrix analysis of photonic crystals

Using a scattering matrix approach we analyze and study the scattering and transmission of waves through a two-dimensional photonic crystal which consists of a periodic array of parallel rods with circular cross sections. Without making any assumptions about normal incidence, single mode propagation, and sufficient inter-scatter separation in the direction of propagation, we show how to compute the transmission and reflection coefficients of these periodic structures. The method is based on the computation of a generalized scattering matrix for one column of the periodic structure. We also develop an analytical method to analyze and to study the scattering and transmission of waves through a two-dimensional photonic crystal which consists of a periodic array of parallel metallic rods with rectangular cross sections. The method is based on the computation of generalized scattering matrices for several parts of the periodic entire structure, and their composition to form the scattering matrix for the structure. We derive an explicit formula for the reflection and transmission coefficients when we take into account only one propagating mode in a specific portion of the periodic structure. Finally, we develop an analytical method to analyze and to study Rayleigh-Bloch surface waves propagating along a two-dimensional diffraction grating which again consists of a periodic array of rods with rectangular cross sections. The method is based on mode matching. By taking into account all propagating and only a finite number of evanescent modes in a specific portion of the waveguide we show that the surface waves correspond to the zeros of the determinant of a Hermitian matrix