A micro-optic bottle resonators constructed by azimuthal sculpting a pair of rings along the perimeter of a glass whisker is proposed. The structures are examined using a numerical solver optimized for the cylindrical symmetry of such resonators. The modal space and field profiles are computed as a function of ring spacing and demonstrates that multiple glass region confined states are available and operate similar to the well-known Whispering-Gallery-Modes. Additional computation results are presented when the structure is configured as an index of refraction sensor.
The design of optical resonator structures usually processed by proposing the structure and then through some numerical or experimental process, determining the supported states. This is often a hit and miss approach as the desired properties of the state do not match the application requirements. A numerical design approach is presented in which the input is the desired optical resonator state and the output is the geometrical and material properties of the resonator structure that will support the state. The technique is presented for cylindrically symmetric structures using the Fourier-Bessel numerical mode solver.
For spherically symmetric dielectric structures, a basis set composed of Bessel, Legendre and Fourier functions, BLF, are
used to cast Maxwell's wave equations into an eigenvalue problem from which the localized modes can be determined.
The steps leading to the eigenmatrix are reviewed and techniques used to reduce the order of matrix and tune the
computations for particular mode types are detailed. The BLF basis functions are used to expand the electric and
magnetic fields as well as the inverse relative dielectric profile. Similar to the common plane wave expansion technique,
the BLF matrix returns the eigen-frequencies and eigenvectors, but in BLF only steady states, non-propagated, are
obtained. The technique is first applied to a air filled spherical structure with perfectly conducting outer surface and then
to a spherical microsphere located in air. Results are compared published values were possible.
Maxwell’s wave equations can be solved using different techniques in order to extract optical properties of a variety
of dielectric structures. For structures that contain an extended axis which serve for the reference for cylindrical
symmetry, we have shown that an expansion of the fields and inverse of the relative dielectric profile using a
simplified and complete set of basis functions of Fourier-Bessel terms provide access to an eigenvalue formulation
from which the eigen-states can be computed. We review the steps used to convert Maxwell’s equation into an
eigenvalue formulation, and then proceed to discuss several applications of the technique. For cylindrically
symmetric structures, the computational technique provides a significantly reduced matrix order to be populated.
New target structure for the presentation consists of cylindrical space slot channel waveguide in which the channel
extends in azimuthal (ϕ) direction. The channel is provided by considering the etching of external side walls of
“Bragg fiber”. The configuration is similar to a structure that can support whispering-gallery modes, except that the
modes highest field locations are within the ambient medium of the channel. Optical properties of this structure can
be best examined through field component which is discontinuous by ratio of relative dielectric constants when
passing air–Bragg interfaces. The ability to select Bragg dielectric properties and to introduce non-uniformities in
Bragg plane spacing provides access to tuning slot channel waveguide properties and design several novel
configurations.
The plane wave expansion (PWM) technique applied to Maxwell’s wave equations provides researchers with a
supply of information regarding the optical properties of dielectric structures. The technique is well suited for
structures that display a linear periodicity. When the focus is directed towards optical resonators and structures that
lack linear periodicity the eigen-process can easily exceed computational resources and time constraints. In the case
of dielectric structures which display cylindrical or spherical symmetry, a coordinate system specific set of basis
functions have been employed to cast Maxwell’s wave equations into an eigen-matrix formulation from which the
resonator states associated with the dielectric profile can be obtained. As for PWM, the inverse of the dielectric and
field components are expanded in the basis functions (Fourier-Fourier-Bessel, FFB, in cylindrical and Fourier-
Bessel-Legendre, BLF, in spherical) and orthogonality is employed to form the matrix expressions. The theoretical
development details will be presented indicating how certain mathematical complications in the process have been
overcome and how the eigen-matrix can be tuned to a specific mode type. The similarities and differences in PWM,
FFB and BLF are presented. In the case of structures possessing axial cylindrical symmetry, the inclusion of the z
axis component of propagation constant makes the technique applicable to photonic crystal fibers and other
waveguide structures. Computational results will be presented for a number of different dielectric geometries
including Bragg ring resonators, cylindrical space slot channel waveguides and bottle resonators. Steps to further
enhance the computation process will be reported.
The whispering-gallery mode space of a cylindrically symmetric silicon micro-optic bottle resonator with an axial Gaussian dielectric profile is theoretically examined. Maxwell’s wave equation for the magnetic field, expressed in cylindrical coordinates, is converted into an eigenvalue problem using a Fourier–Bessel basis space to expand the inverse dielectric and field components. The order of the eigen-matrix is reduced using symmetry arguments and azimuthal order segmentation such that the matrix can be solved on a desktop PC. The eigen-space reveals that several states can be confined in the bottle region, whereas another set of states extends the axial length of the structure similar to ridge waveguide modes. The hollow core bottle is also presented as a design modification. The micro-optic bottle resonator structure size makes it directly compatible with silicon-based photonics increasing its potential as a key component in sensing, communication, and other applications.
The insertion of high dielectric rods in the low dielectric region of photonics crystal enables the optical properties to be
reconfigurable. We show that for a square array of holes, the inserted rods define the waveguide region, wavelength
of operation and functionality of the photonic crystal device (directional coupler presented). Also are examined the
modification of the resonator state’s wavelength and field profile when rods are introduced in the central region of two
types of quasi-crystals. Based on these results more elaborate reconfigurable devices can be derived.
The availability of low loss waveguide bends in photonic crystal structures makes possible numerous integrated optic
devices. The method proposed in this presentation consists of introducing a dielectric shift equivalent to a sheer
displacement along a segment of the photonic crystal waveguide and results in a double bent waveguide. The degree of
sheer determines the waveguide deflection angle. Theoretical analysis is performed using FDTD and PWM and predicts
low loss bends. Experiments are performed in the microwave regime using an array of alumina rods in air over the 2 to
14 GHz range and confirm theoretical predictions.
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.
We theoretically examine the optical guiding properties of typical 3-layer oxide waveguides with an additional thin
high dielectric cladding applied on one side. Modal and FDTD simulations techniques are performed to determine
the field evolution and coupling coefficients for light transitioning between the un-clad and clad regions. Results
indicate that such waveguide structures lead to novel integrated optic device designs. We also address the issue of
efficiently coupling light from a low dielectric value slab waveguide, into the lowest order mode supported in the
clad layer by using waveguide tapers. In particular a waveguide structure suitable for efficiently interfacing glass
based and silicon based waveguide structures is presented for the SOI fabrication platform.
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.
The modifications to the optical properties of a slab waveguide coated with a patterned thin high dielectric overlay layer
are examined. The asymmetric nature of the entire waveguide configuration makes it possible to keep the waveguide
mode highly confined in the slab waveguide while enhancing the evanescent wave-overlay interaction. The sensitivity
of the guided light to variations in the refractive index, thickness, period, and length of patterned overlay are examined
using Finite Difference Time Domain (FDTD) simulations. For various waveguide geometries, transmission and
reflection spectrums are obtained from which active and passive optical device configurations are explored.
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.
A charge moving faster than the phase velocity in a medium can produces conical radiation known as Cherenkov radiation. The FDTD numerical technique is used to model this radiation process and the generated light spectrum is used to explore the optical properties of photonic crystal and quasi-crystals. It is shown that the radiation from the fast moving charge is able to provide information on the band gap and transmission spectrums, defect state wavelengths and waveguide propagation.
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
hexagonal array.
A silicon-on-insulator (SOI) rib waveguide integrated with a Bragg grating and nickel chromium heating elements has
been fabricated. The heaters enable tuning of the Bragg wavelength through a thermo-optic effect. As the temperature
is increased, the Bragg wavelength also increases due to the increasing silicon refractive index and effective mode index
of the guide. The device is designed so that an additional processing step can remove the buried oxide beneath the
grating to form a suspended waveguide. This structure would further decrease the power input required to tune the filter,
and would allow the waveguide to buckle at a critical temperature. In this work the un-released structure is fabricated
and mounted in a standard dual in-line pin (DIP) package to allow optical and electrical characterization. Test results
demonstrating thermo-optic tuning show a 1 nm shift in Bragg wavelength with a power input of 58 mW.
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.
The new term "Perfectly Periodic Photonic Quasi-Crystals" (P3QC) applies to 2-D and 3-D dielectric arrangements that
posses a high rotational order about a central pivot point (standard photonic quasi-crystal) and in the same pattern posses
a radial periodicity as viewed from the same central pivot point. These structures display no translational symmetry as
associated with standard photonic crystals. In a 2-D structure, P3QC periodicity is observed for the polar coordinates (r,
φ) and a unit cell of surface dS =rdrd θ serves as the building block of the pattern at each of the radial "Lattice Points".
A generating algorithm based on orthogonal functions is used to produce many different types of P3QC patterns for latter
analysis through FDTD simulations. The presence of bandgaps in the transmission spectrum for these structures is
observed when the dielectric fill factor, rotational order and dielectric contrast are carefully selected. Central localized
light states are commonly observed in these structures.
The FDTD technique is employed to examine the optical properties of 12-fold rotational order quasi-crystal structures. Bandgap and defect states are shown to exits in the quasi-crystal patterns for in-plane propagating light. Out-of-plane propagation is examined by infinitely extending the planar quasi-crystal in the third direction. The resulting micro-structured optical fiber guides show supported modes confined by index guiding. A stack and draw technique is presented making quasi-crystal core designs possible.
The key feature that gives photonic crystals (PhCs) their ability to form a photonic band structure (PBS) is their translational symmetry. Structures that do not have translational symmetries also have a PBS. One explanation as to how these structures generate a PBS involves long-range interactions resulting in fractal dispersion relations (e.g. 1-D quasi-crystals, Fibonacci layers, etc.) However, long-range interactions do not fully explain why 2-D quasi-crystals structures also exhibit a PBS. This paper proposes an explanation for such results: by applying rotation operators from the SO(2) group to 1-D quasi-crystal dispersion relations. This process mimics the way electronic state amplitudes are calculated when such states have an angular dependence. Simulations results are presented in this paper.
The key feature that gives photonic crystals (PhCs) their ability to form photonic band gaps (PBGs) analogous to electronic band gaps of semiconductors is their translation symmetries. In recent years, however, it has been found that structures that possess only rotational symmetries can also have PBGs. In addition, these structures, known as Photonic Quasicrystals (PhQs), have other interesting qualities that set them apart of their translational cousins. One interesting feature is how defect states can be created in PhQs. If the rotational symmetry is disturbed, defect states analogous to defects states that are created in PhCs can be obtained. Simulation results of these defect states and other propagation properties of planar 12-fold photonic quasicrystal patterns, and its physical implementations in Silicon-On-Insulator (SOI) are presented. The main mechanisms required to make any optical multiplexing system is propagation; stop bands and add/drop ports. With the rotationally symmetry of the PhQ causing the stop bands, line defects facilitating propagation and now these specially design defect states acting as add/drop ports, a physical implementation of an OADM can be presented. Theoretical, practical and manufacturing benefits of PhQs are discussed. Simulated transmission plots are shown for various fill factors, dielectric contrast and propagation direction. It is shown that low index waveguides can be produced using the quasi-crystal photonic crystal pattern. Fabrication steps and results are shown.
The effect of thermal tuning on the optical properties of an SOI based suspended waveguide is analysed. This analysis is based on the model that a fixed-fixed suspended beam, which forms the optical waveguide, will buckle when thermal expansion causes an axial stress that exceeds the critical buckling pressure of the beam. The analysis of the waveguide response will be broken up into the pre- and post-buckle stages of thermal actuation. Each stage of actuation will have a separate relationship for the shift in optical response as a function of temperature, which will include a combination of the thermo-optic, photo-elastic, and thermo-elastic effects. Given a corrugated, or "Bragg grating" version of the waveguide, it will be shown that thermally tuning the Bragg wavelength involves a change in index via a change in temperature and stress, and a change in grating pitch via a change in temperature. Particular attention will be paid to the evolving stress field over the length of the waveguide and its relationship to the stress-optic effect. It will also be shown that the pre- and post-buckle temperatures are path-dependant. Finally, examples of device implementation will be explored.
We present a dual beam multiple exposure technique that can generate complex 2-D and 3-D bandgap template structures in a photosensitive material. The system parameters related to the planar interference pattern produced by the two laser beams and reorientation effect of the sample relative to these planes is discussed. Photonic crystal structures such as the 2-D square and hexagonal arrays of dielectric "rods" and "holes" and the 3-D Yablonovite and other profiles are given. We show band gap calculations for these structures and discuss a technique for increasing the band gap through laser sculpting the dielectric template profile. In addition the paper discusses how the dual beam multiple exposure technique can be used to design 2-D quasicrystal structures of low to high rotational symmetry.
The power coupling expressions for any pair of end separated waveguides are derived in general form for TE modes. The expressions are shown to agree with the accepted mode coupling expressions when the separation approaches zero. Furthermore, the expressions are used to discuss a particular type of integrated-optic device which makes use of the electro- optic effect and end separated waveguides.
A digital fiber optic temperature sensor based on the abrupt change in the absorption coefficient observed when a two constituent mixture undergoes a solid/liquid phase transition is presented. Varying the weight ratio of the two constituent mixture, the phase transition temperature can be tuned over several degrees Celsius.
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