Felix J. Lawrence

Antireflection coatings for two-dimensional photonic crystals using a rigorous impedance definition

We show that no consistent scalar definition of impedance is generally possible for photonic crystals. Instead, we present a rigorous semianalytic matrix definition of impedance for square lattice photonic crystals, defined in terms of Bloch modes. We then apply our definition to design a range of multilayer photonic crystal antireflection coatings efficiently.

Modeling photonic crystal interfaces and stacks: impedance-based approaches

In many research areas, the reflective properties of a bulk medium are characterized by its impedance or an impedance-like quantity. Such a quantity is essential for the efficient design of stacked structures such as antireflection coatings and thin-film filters. For 2D photonic crystals and metamaterials, the literature contains multiple definitions of impedance, not all of which are consistent. We review these proposed definitions, evaluate their regions of applicability, and numerically test their accuracy in a variety of salient photonic crystal examples.

Coupled waveguide modes in hexagonal photonic crystals

We investigate the modes of coupled waveguides in a hexagonal photonic crystal. We find that for a substantial parameter range the coupled waveguide modes have dispersion relations exhibiting multiple intersections, which we explain both intuitively and using a rigorous tight-binding argument.

EMUstack: An open source route to insightful electromagnetic computation via the Bloch mode scattering matrix method

Abstract We describe EMUstack, an open-source implementation of the Scattering Matrix Method (SMM) for solving field problems in layered media. The fields inside nanostructured layers are described in terms of Bloch modes that are found using the Finite Element Method (FEM). Direct access to these modes allows the physical intuition of thin film optics to be extended to complex structures. The combination of the SMM and the FEM makes EMUstack ideally suited for studying lossy, high-index contrast structures, which challenge conventional SMMs. Program summary Program title: EMUstack Catalogue identifier: AEZI_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEZI_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: GNU General Public License, version 3 No. of lines in distributed program, including test data, etc.: 154301 No. of bytes in distributed program, including test data, etc.: 5308635 Distribution format: tar.gz Programming language: Python, Fortran. Computer: Any computer with a Unix-like system with Python, a Fortran compiler and F2Py [1]. Also required are the following free libraries LAPACK and BLAS [2], UMFPACK [3]. Developed on 1.6 GHz Intel Core i7. Operating system: Any Unix-like system; developed on Ubuntu 14.04 (using Linux kernel 3.16). RAM: Problem dependent; specifically on the resolution of the FEM mesh and the number of modes included. The given example uses approximately 100 MB. Classification: 10. External routines: Required are the following free libraries LAPACK and BLAS [2], UMFPACK [3]. Optionally exploits additional commercial software packages: Intel MKL [4], Gmsh [5]. Nature of problem: Time-harmonic electrodynamics in layered media. Solution method: Finite element method and the scattering matrix method. Running time: Problem dependent (typically about 3 s per wavelength including plane wave orders ≤ 3 ). References: [1] P. Peterson, F2PY: A tool for connecting Fortran and Python programs, International Journal of Computational Science and Engineering 4 (4) (2009) 296. [2] LAPACK, http://www.netlib.org/lapack [3] T.A. Davis, Algorithm 832: UMFPACK V4.3 - An Unsymmetric-Pattern Multifrontal Method, ACM Transactions on Mathematical Software 30 (2) (2004) 165–195. [4] Intel MKL, http://www.software.intel.com/intel-mkl [5] C. Geuzaine, J.-F. Remacle, Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities, International Journal for Numerical Methods in Engineering 79 (2009) 1309–1331.

Semi-analytic method for slow light photonic crystal waveguide design

Abstract We present a semi-analytic method to calculate the dispersion curves and the group velocity of photonic crystal waveguide modes in two-dimensional geometries. We model the waveguide as a homogenous strip, surrounded by photonic crystal acting as diffracting mirrors. Following conventional guided-wave optics, the properties of the photonic crystal waveguide may be calculated from the phase upon propagation over the strip and the phase upon reflection. The cases of interest require a theory including the specular order and one other diffracted reflected order. The computational advantages let us scan a large parameter space, allowing us to find novel types of solutions.

A flexible Bloch mode method for computing complex band structures and impedances of two-dimensional photonic crystals

We present a flexible method that can calculate Bloch modes, complex band structures, and impedances of two-dimensional photonic crystals from scattering data produced by widely available numerical tools. The method generalizes previous work which relied on specialized multipole and finite element method (FEM) techniques underpinning transfer matrix methods. We describe the numerical technique for mode extraction, and apply it to calculate a complex band structure and to design two photonic crystal antireflection coatings. We do this for frequencies at which other methods fail, but which nevertheless are of significant practical interest.

Supermodes of hexagonal lattice waveguide arrays

We present a semianalytical formulation for calculating the supermodes and corresponding Bloch factors of light in hexagonal lattice photonic crystal waveguide arrays. We then use this formulation to easily calculate dispersion curves and predict propagation in systems too large to calculate using standard numerical methods.

Diffraction engineering with braided modes in photonic crystal waveguide arrays

We consider discrete diffraction in coupled photonic crystal waveguides in a hexagonal lattice. We show that in these structures the (discrete) diffraction coefficient depends strongly on frequency and can even change sign. This behavior does not occur in photonic crystal waveguides in square lattices. This behavior is interesting in its own right and has intriguing consequences for the propagation of discrete spatial solitons.

Bloch-mode based homogenisation of photonic crystals

We propose a method for photonic crystal (PC) homogenisation based on the PCs Bloch modes. The resulting quantities may be used in Snells law; to calculate reflections, transmissions, and propagation; and to locate surface modes.

Impedance of photonic crystals

We show how the concept of impedance can be defined rigorously in terms of Bloch modes for photonic crystal applications and then exploit this to accurately and efficiently design multilayer anti-reflection coatings for photonic crystals.