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.
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 light's field center demonstrates that a relationship
exists between the rotational order in the localized light and the dielectric layout.
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.
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.
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 source's 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.
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 crystal's 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
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.