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|Title: ||New methods for computing transmission and reflection spectra of photonic crystals and diffraction gratings|
|Other Titles: ||Ji suan guang zi jing ti he yan she guang shan zhe she he fan she pu de ji zhong xin fang fa|
|Authors: ||Wu, Yumao (吴语茂)|
|Department: ||Department of Mathematics|
|Degree: ||Doctor of Philosophy|
|Issue Date: ||2010|
|Publisher: ||City University of Hong Kong|
|Subjects: ||Photonic crystals.|
|Notes: ||CityU Call Number: QD924 .W8 2010|
vii, 120 leaves : ill. 30 cm.
Thesis (Ph.D.)--City University of Hong Kong, 2010.
Includes bibliographical references (leaves 112-119)
|Abstract: ||Due to their unusual optical properties and significant potentials in applications, photonic
crystals (PhCs) have been extensively studied both theoretically and experimentally.
Many PhCs devices, such as waveguide bends, branches, frequency filters and
waveguide couplers, are important building blocks of integrated optical circuits. Threedimensional
(3D) PhCs with a complete bandgap in the optical wavelength region
have potential applications, such as ultrahigh quality-factor cavities and zero-threshold
lasers. The woodpile structures composed of alternating layers of rods have attracted
much attention due to their relatively simple fabrication process compared with other
3D PhCs. Diffraction gratings have been extensively studied for many years, and they
are important in practical applications, such as monochromators, spectrometers, lasers,
wavelength division multiplexing devices, optical pulse compressing devices, etc.
Numerical methods are essential in the design, analysis and optimization of photonic
crystals and diffraction gratings. Some of these methods, such as the finitedifference
time-domain (FDTD) method, are general methods that can be used to study
various aspects of PhCs and diffraction gratings, but their accuracy and efficiency are
often limited. FDTD requires a small mesh size to resolve curved material interfaces
and often has difficulties truncating periodic structures that extend to infinity. The
Fourier modal method (FMM) is suitable for diffraction gratings with uniform layers,
but they are not so efficient and may have convergence problems when a general
grating with sloping interfaces or a photonic crystal composed of cylinders must be approximated.
The boundary integral equation (BIE) method is somewhat complicated
to implement, since the integral operators are related to the quasiperiodic Green’s function
which requires sophisticated lattice sums techniques to evaluate. The finite element method (FEM) is very general, but it gives rise to large, complex and indefinite
linear systems that are expensive to solve.
In this thesis, efficient numerical methods based on the Dirichlet-to-Neumann
(DtN) or Neumann-to-Dirichlet (NtD) maps (of unit cells or homogeneous sub-domains)
are developed for accurate simulations of two- and three-dimensional photonic crystals
and diffraction gratings. For photonic crystals composed of interpenetrating cylinders,
i.e. when the radius of the cylinders is larger than √3/4 of the lattice constant, an
efficient DtN map method is developed for computing the reflection and transmission
spectra. Our method manipulates a pair of operators defined on a set of curves. It
is efficient since the wave field in the interiors of the unit cells are never calculated.
This is achieved by using the DtN maps which map the wave field on the boundaries
of the unit cells to its normal derivative. The DtN map method is also developed for
two-dimensional photonic crystals with oblique incident waves. In that case, the DtN
operator maps the two longitudinal field components to their normal derivatives on the
boundary of the unit cell. For three-dimensional photonic crystals composed of crossed
arrays of circular cylinders, including woodpile structures as special cases, we develop
an efficient and accurate computational method. The method relies on marching a few
operators from one side of the structure to another. The marching step makes use of the
DtN maps for two-dimensional unit cells in each layer where the structure is invariant
in the direction of the cylinder axes. A further simplification is developed based on
the so-called Tangential-to-Tangential (T2T) operator which maps two transverse field
components to two different transverse field components on the boundary of a 2D unit
cell. These operators are approximated by matrices based on expansions in cylindrical
For analyzing diffraction gratings, a new method is developed based on dividing
one period of the grating into homogeneous sub-domains and computing the NtD maps
for these sub-domains by boundary integral equations. For a sub-domain, the NtD
operator maps the normal derivative of the wave field to the wave field on its boundary.
The method retains the advantages of existing boundary integral equation methods for
diffraction gratings, but avoids the quasi-periodic Green's functions that are expensive to evaluate. For diffraction problems in conical mounting, the NtD operator maps
the normal derivatives of two longitudinal components of the electromagnetic field to
these two components on the boundary of the sub-domain. A differentiation operator
along the boundary is also needed to impose proper interface conditions. The method
performs equally well for dielectric or metallic gratings.
Keywords: Photonic crystal, Diffraction grating, Dirichlet-to-Neumann map, Operator
marching, Neumann-to-Dirichlet map, Boundary integral equation|
|Online Catalog Link: ||http://lib.cityu.edu.hk/record=b3947532|
|Appears in Collections:||MA - Doctor of Philosophy |
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