1,648 research outputs found
Lorenz-Mie theory for 2D scattering and resonance calculations
This PhD tutorial is concerned with a description of the two-dimensional
generalized Lorenz-Mie theory (2D-GLMT), a well-established numerical method
used to compute the interaction of light with arrays of cylindrical scatterers.
This theory is based on the method of separation of variables and the
application of an addition theorem for cylindrical functions. The purpose of
this tutorial is to assemble the practical tools necessary to implement the
2D-GLMT method for the computation of scattering by passive scatterers or of
resonances in optically active media. The first part contains a derivation of
the vector and scalar Helmholtz equations for 2D geometries, starting from
Maxwell's equations. Optically active media are included in 2D-GLMT using a
recent stationary formulation of the Maxwell-Bloch equations called
steady-state ab initio laser theory (SALT), which introduces new classes of
solutions useful for resonance computations. Following these preliminaries, a
detailed description of 2D-GLMT is presented. The emphasis is placed on the
derivation of beam-shape coefficients for scattering computations, as well as
the computation of resonant modes using a combination of 2D-GLMT and SALT. The
final section contains several numerical examples illustrating the full
potential of 2D-GLMT for scattering and resonance computations. These examples,
drawn from the literature, include the design of integrated polarization
filters and the computation of optical modes of photonic crystal cavities and
random lasers.Comment: This is an author-created, un-copyedited version of an article
published in Journal of Optics. IOP Publishing Ltd is not responsible for any
errors or omissions in this version of the manuscript or any version derived
from i
Spectral dimension reduction of complex dynamical networks
Dynamical networks are powerful tools for modeling a broad range of complex
systems, including financial markets, brains, and ecosystems. They encode how
the basic elements (nodes) of these systems interact altogether (via links) and
evolve (nodes' dynamics). Despite substantial progress, little is known about
why some subtle changes in the network structure, at the so-called critical
points, can provoke drastic shifts in its dynamics. We tackle this challenging
problem by introducing a method that reduces any network to a simplified
low-dimensional version. It can then be used to describe the collective
dynamics of the original system. This dimension reduction method relies on
spectral graph theory and, more specifically, on the dominant eigenvalues and
eigenvectors of the network adjacency matrix. Contrary to previous approaches,
our method is able to predict the multiple activation of modular networks as
well as the critical points of random networks with arbitrary degree
distributions. Our results are of both fundamental and practical interest, as
they offer a novel framework to relate the structure of networks to their
dynamics and to study the resilience of complex systems.Comment: 16 pages, 8 figure
Adding SALT to Coupled Microcavities: the making of active photonic molecule lasers
A large body of work has accumulated over the years in the study of the
optical properties of single and coupled microcavities for a variety of
applications, ranging from filters to sensors and lasers. The focus has been
mostly on the geometry of individual resonators and/or on their combination in
arrangements often referred to as photonic molecules (PMs).
Our primary concern will be the lasing properties of PMs as ideal candidates
for the fabrication of integrated microlasers, photonic molecule lasers.
Whereas most calculations on PM lasers have been based on cold-cavity (passive)
modes, i.e. quasi-bound states, a recently formulated steady-state ab initio
laser theory (SALT) offers the possibility to take into account the spectral
properties of the underlying gain transition, its position and linewidth, as
well as incorporating an arbitrary pump profile. We will combine two
theoretical approaches to characterize the lasing properties of PM lasers: for
two-dimensional systems, the generalized Lorenz-Mie theory will obtain the
resonant modes of the coupled molecules in an active medium described by SALT.
Not only is then the theoretical description more complete, the use of an
active medium provides additional parameters to control, engineer and harness
the lasing properties of PM lasers for ultra-low threshold and directional
single-mode emission.Comment: 16th International Conference on Transparent Optical Networks (2014
Optimization of integrated polarization filters
This study reports on the design of small footprint, integrated polarization
filters based on engineered photonic lattices. Using a rods-in-air lattice as a
basis for a TE filter and a holes-in-slab lattice for the analogous TM filter,
we are able to maximize the degree of polarization of the output beams up to 98
% with a transmission efficiency greater than 75 %. The proposed designs allow
not only for logical polarization filtering, but can also be tailored to output
an arbitrary transverse beam profile. The lattice configurations are found
using a recently proposed parallel tabu search algorithm for combinatorial
optimization problems in integrated photonics
Phase Space Engineering in Optical Microcavities I: Preserving near-field uniformity while inducing far-field directionality
Optical microcavities have received much attention over the last decade from
different research fields ranging from fundamental issues of cavity QED to
specific applications such as microlasers and bio-sensors. A major issue in the
latter applications is the difficulty to obtain directional emission of light
in the far-field while keeping high energy densities inside the cavity (i.e.
high quality factor). To improve our understanding of these systems, we have
studied the annular cavity (a dielectric disk with a circular hole), where the
distance cavity-hole centers, d, is used as a parameter to alter the properties
of cavity resonances. We present results showing how one can affect the
directionality of the far-field while preserving the uniformity (hence the
quality factor) of the near-field simply by increasing the value of d.
Interestingly, the transition between a uniform near- and far-field to a
uniform near- and directional far-field is rather abrupt. We can explain this
behavior quite nicely with a simple model, supported by full numerical
calculations, and we predict that the effect will also be found in a large
class of eigenmodes of the cavity.Comment: 12th International Conference on Transparent Optical Network
S and Q Matrices Reloaded: applications to open, inhomogeneous, and complex cavities
We present a versatile numerical algorithm for computing resonances of open
dielectric cavities. The emphasis is on the generality of the system's
configuration, i.e. the geometry of the (main) cavity (and possible inclusions)
and the internal and external dielectric media (homogeneous and inhomogeneous).
The method is based on a scattering formalism to obtain the position and width
of the (quasi)-eigenmodes. The core of the method lies in the scattering
S-matrix and its associated delay Q-matrix which contain all the relevant
information of the corresponding scattering experiment. For instance, the
electromagnetic near- and far-fields are readily extracted. The flexibility of
the propagation method is displayed for a selected system.Comment: 15th International Conference on Transparent Optical Networks (2013
Ab initio investigation of lasing thresholds in photonic molecules
We investigate lasing thresholds in a representative photonic molecule
composed of two coupled active cylinders of slightly different radii.
Specifically, we use the recently formulated steady-state ab initio laser
theory (SALT) to assess the effect of the underlying gain transition on lasing
frequencies and thresholds. We find that the order in which modes lase can be
modified by choosing suitable combinations of the gain center frequency and
linewidth, a result that cannot be obtained using the conventional approach of
quasi-bound modes. The impact of the gain transition center on the lasing
frequencies, the frequency pulling effect, is also quantified
Percolation on random networks with arbitrary k-core structure
The k-core decomposition of a network has thus far mainly served as a
powerful tool for the empirical study of complex networks. We now propose its
explicit integration in a theoretical model. We introduce a Hard-core Random
Network model that generates maximally random networks with arbitrary degree
distribution and arbitrary k-core structure. We then solve exactly the bond
percolation problem on the HRN model and produce fast and precise analytical
estimates for the corresponding real networks. Extensive comparison with
selected databases reveals that our approach performs better than existing
models, while requiring less input information.Comment: 9 pages, 5 figure
Growing networks of overlapping communities with internal structure
We introduce an intuitive model that describes both the emergence of
community structure and the evolution of the internal structure of communities
in growing social networks. The model comprises two complementary mechanisms:
One mechanism accounts for the evolution of the internal link structure of a
single community, and the second mechanism coordinates the growth of multiple
overlapping communities. The first mechanism is based on the assumption that
each node establishes links with its neighbors and introduces new nodes to the
community at different rates. We demonstrate that this simple mechanism gives
rise to an effective maximal degree within communities. This observation is
related to the anthropological theory known as Dunbar's number, i.e., the
empirical observation of a maximal number of ties which an average individual
can sustain within its social groups. The second mechanism is based on a
recently proposed generalization of preferential attachment to community
structure, appropriately called structural preferential attachment (SPA). The
combination of these two mechanisms into a single model (SPA+) allows us to
reproduce a number of the global statistics of real networks: The distribution
of community sizes, of node memberships and of degrees. The SPA+ model also
predicts (a) three qualitative regimes for the degree distribution within
overlapping communities and (b) strong correlations between the number of
communities to which a node belongs and its number of connections within each
community. We present empirical evidence that support our findings in real
complex networks.Comment: 14 pages, 8 figures, 2 table
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