Scott R. Newman

Fourier–Bessel analysis of localized states and photonic bandgaps in 12-fold photonic quasi-crystals

A Fourier-Bessel (FB) basis is used to solve two-dimensional (2D) cylindrical Maxwells equations for localized states within dielectric structures that possess rotational symmetry. The technique is used to determine the wavelengths and profiles of the stationary states supported by the structure and identify the bandgaps. 12-fold quasi-crystals for the TE and TM polarizations are analyzed. Since the FB approach with 2D photonic crystals in this fashion is new, the accuracy of the results is confirmed using finite-difference time-domain simulations.

FDTD sources for localized state excitation in photonic crystals and photonic quasi-crystals

Numerical methods, such as the finite difference time domain (FDTD) technique, are commonly used to study transmission properties, waveguide modes, and localized states of photonic crystals and photonic quasi-crystals. The degree to which a localized state is excited is dependent on the sources topology. Researchers have proposed a number of different source configurations in order to efficiently excite localized states; dipole sources, random sources, and initial field distributions. The efficient excitation of different localized states in a photonic crystal and quasi-crystal through a general source configuration remains an issue to be addressed. This work re-examines the techniques currently used and determines the most efficient method to excite the modes of a photonic crystal and quasi-crystal without prior knowledge of the localized state profiles.

Modal analysis and device considerations of thin high index dielectric overlay slab waveguides

The effect of adding a thin high index dielectric overlay layer onto a 3-layer slab waveguide demonstrates several interesting features that can be exploited in integrated optical device configurations. A simple modal analysis is employed to examine the behavior of guided light launched from a 3-layer waveguide structure then coupled and propagated in the 4-layer overlay region. Modal properties typically overlooked in conventional slab waveguides are made use of in the design and theoretical analysis of an MMI device and optical index of refraction sensor. The optical structure presented here can form the backdrop waveguide design for more complex and active devices.

Effects of incrementally applied disorder to a triangular lattice creating a 12-fold quasi-crystal

By applying a low level of disorder, in the range of 10 to 20 %, to a translationally symmetric photonic crystal one can obtain the dielectric profile of a rotationally symmetric quasi-crystal. Through the use of a morphing algorithm we study the effects of incrementally applying the disorder to a triangular lattice photonic crystal, converting it to a 12-fold quasi-crystal. Through FDTD simulation, band gap maps and defect states are computed and presented as a function of the morphing process.

Fourier-Bessel expansions of localized light in disorder dielectric lattices

It has been shown that light localization can occur within disordered and random dielectric lattices. The presence and nature of localized light within these dielectric layouts is examined through the rotational order symmetry present within both the field profile and the dielectric. A Fourier-Bessel expansion algorithm using exponentials and Bessel functions as basis functions is employed to decompose the dielectric layout and localized light field profiles. Selecting the coordinate origin for the expansion to coincide with the localized lights field center demonstrates that a relationship exists between the rotational order in the localized light and the dielectric layout.

Photonic quasi-crystals in Fourier and Fourier-Bessel space

Photonic crystals that are aperiodic or quasi-crystalline in nature have been the focus of research due to their complex spatial distributions, resulting in high order rotational symmetries. Recently we proposed aperiodic patterns that were rotationally symmetric while being random in the radial direction. The structures are designed by segmenting the circular design space, randomly populating one segment, and repeating that segment about a center of rotation. Studying the symmetries and geometrical attributes of aperiodic structures is typically performed in reciprocal Fourier space by examining the distribution of the Fourier coefficients. This allows the translational symmetry to be directly extracted and the rotational nature to be interpreted. Instead we propose comparing the typical Fourier analysis with the use of a Fourier-Bessel space. The Fourier-Bessel approach expands the dielectric layout in cylindrical coordinates using exponential and Bessel functions as the angular and radial basis functions. The coefficients obtained in this fashion directly provide the rotational symmetries that are present. This work will examine both the Fourier and Fourier-Bessel distributions of the proposed structures as well as other quasi-crystals in order to explore the strengths and weaknesses of both techniques.

Finite difference simulation of thermally tuned hexagonal photonic crystals

Thermal tuning of hexagonal photonic crystals by absorption of laser energy is examined through finite difference numerical simulation. The photonic crystals are patterned in the device layer of the silicon on insulator (SOI) platform. The thermal equations, which include contributions from laser absorption gain, conduction loss, and radiation loss are combined to obtain a heat balance equation. This governing equation is modeled using a thermodynamic finite difference computation engine. To ensure the stability of the thermal model within the transient regime the velocity of heat propagation is calculated and included as a courant factor controlling the coarseness of the discretization grid and time step interval. The thermal distribution obtained from the numerical simulation, combined with the thermo-optic effect, can be used to alter the initial dielectric distribution of the device layer. The integration of the change in refractive index into the existing dielectric enables the thermal effects to be included into a standard optical finite difference time domain (FDTD) engine. Through the implementation of the optical and thermal simulation tools, the laser thermal tuning of the band gaps and localized states of hexagonal photonic crystals will be explored. The temperature dependence of the central wavelength of the localized states will be calculated.

Low loss off axis photonic crystal waveguides with bends

Photonic crystal waveguide bends are generally designed to follow the crystal symmetry directions. For low angle bends higher propagation losses are typically observed. We present three waveguide bending techniques and the resulting photonic crystal geometries that permit low propagation losses for waveguide directional changes that do not correspond to the symmetry directions.

Quasi-crystal or disordered regular photonic crystal?

Several studies have shown that the incremental introduction of disorder in photonic crystals results in the high frequency band gaps closing followed by the lower frequency band gaps. The level and type of disorder required to pinch off the lower band gap depends on the photonic crystals initial dielectric layout. Our research has shown that a rotationally symmetric 12-fold quasi-crystal structure can be reached by introducing a relatively low level of dielectric disorder to the hexagonal array. A morphing algorithm has been developed that permits the transformation of the hexagonal rod array photonic crystal into a 12-fold quasi-crystal. The intermediate dielectric profiles generated are used to examine the evolution of the band gap and central defect states during the transformation. The resulting FDTD simulations display evidence that the underlying structure of the 12-fold quasi-crystal may be closely related to the hexagonal array.