8,296 research outputs found
Efficient computation of TM- and TE-polarized leaky modes in multilayered circular waveguides
In combination with the perfectly matched layer (PML)-paradigm, eigenmode expansion techniques have become increasingly important in the analysis and design of cylindrical and planar waveguides for photonics applications. To achieve high accuracy, these techniques rely on the determination of many modes of the modal spectrum of the waveguide under consideration. In this paper, we focus on the efficient computation of TM- and TE-polarized leaky modes for multilayered cylindrical waveguides. First, quasi-static estimates are derived for the propagation constants of these modes. Second, these estimates are used as a starting point in an advanced Newton iteration scheme after they have been subjected to an adaptive linear error correction. To prove the validity of the computation technique, it is applied to technologically important cases: vertical-cavity surface-emitting lasers and a monomode fiber
Rotation method for accelerating multiple-spherical Bessel function integrals against a numerical source function
A common problem in cosmology is to integrate the product of two or more
spherical Bessel functions (sBFs) with different configuration-space arguments
against the power spectrum or its square, weighted by powers of wavenumber.
Naively computing them scales as with the number of
configuration space arguments and the grid size, and they cannot be
done with Fast Fourier Transforms (FFTs). Here we show that by rewriting the
sBFs as sums of products of sine and cosine and then using the product to sum
identities, these integrals can then be performed using 1-D FFTs with scaling. This "rotation" method has the potential to
accelerate significantly a number of calculations in cosmology, such as
perturbation theory predictions of loop integrals, higher order correlation
functions, and analytic templates for correlation function covariance matrices.
We implement this approach numerically both in a free-standing,
publicly-available \textsc{Python} code and within the larger,
publicly-available package \texttt{mcfit}. The rotation method evaluated with
direct integrations already offers a factor of 6-10 speed-up over the
naive approach in our test cases. Using FFTs, which the rotation method
enables, then further improves this to a speed-up of
over the naive approach. The rotation method should be useful in light of
upcoming large datasets such as DESI or LSST. In analysing these datasets
recomputation of these integrals a substantial number of times, for instance to
update perturbation theory predictions or covariance matrices as the input
linear power spectrum is changed, will be one piece in a Monte Carlo Markov
Chain cosmological parameter search: thus the overall savings from our method
should be significant
A fast analysis-based discrete Hankel transform using asymptotic expansions
A fast and numerically stable algorithm is described for computing the
discrete Hankel transform of order as well as evaluating Schl\"{o}milch and
Fourier--Bessel expansions in
operations. The algorithm is based on an asymptotic expansion for Bessel
functions of large arguments, the fast Fourier transform, and the Neumann
addition formula. All the algorithmic parameters are selected from error bounds
to achieve a near-optimal computational cost for any accuracy goal. Numerical
results demonstrate the efficiency of the resulting algorithm.Comment: 22 page
An algorithm for the rapid numerical evaluation of Bessel functions of real orders and arguments
We describe a method for the rapid numerical evaluation of the Bessel
functions of the first and second kinds of nonnegative real orders and positive
arguments. Our algorithm makes use of the well-known observation that although
the Bessel functions themselves are expensive to represent via piecewise
polynomial expansions, the logarithms of certain solutions of Bessel's equation
are not. We exploit this observation by numerically precomputing the logarithms
of carefully chosen Bessel functions and representing them with piecewise
bivariate Chebyshev expansions. Our scheme is able to evaluate Bessel functions
of orders between and 1\sep,000\sep,000\sep,000 at essentially any
positive real argument. In that regime, it is competitive with existing methods
for the rapid evaluation of Bessel functions and has several advantages over
them. First, our approach is quite general and can be readily applied to many
other special functions which satisfy second order ordinary differential
equations. Second, by calculating the logarithms of the Bessel functions rather
than the Bessel functions themselves, we avoid many issues which arise from
numerical overflow and underflow. Third, in the oscillatory regime, our
algorithm calculates the values of a nonoscillatory phase function for Bessel's
differential equation and its derivative. These quantities are useful for
computing the zeros of Bessel functions, as well as for rapidly applying the
Fourier-Bessel transform. The results of extensive numerical experiments
demonstrating the efficacy of our algorithm are presented. A Fortran package
which includes our code for evaluating the Bessel functions as well as our code
for all of the numerical experiments described here is publically available
Dispersion modeling and analysis for multilayered open coaxial waveguides
This paper presents a detailed modeling and analysis regarding the dispersion
characteristics of multilayered open coaxial waveguides. The study is motivated
by the need of improved modeling and an increased physical understanding about
the wave propagation phenomena on very long power cables which has a potential
industrial application with fault localization and monitoring. The
electromagnetic model is based on a layer recursive computation of
axial-symmetric fields in connection with a magnetic frill generator excitation
that can be calibrated to the current measured at the input of the cable. The
layer recursive formulation enables a stable and efficient numerical
computation of the related dispersion functions as well as a detailed analysis
regarding the analytic and asymptotic properties of the associated
determinants. Modal contributions as well as the contribution from the
associated branch-cut (non-discrete radiating modes) are defined and analyzed.
Measurements and modeling of pulse propagation on an 82 km long HVDC power
cable are presented as a concrete example. In this example, it is concluded
that the contribution from the second TM mode as well as from the branch-cut is
negligible for all practical purposes. However, it is also shown that for
extremely long power cables the contribution from the branch-cut can in fact
dominate over the quasi-TEM mode for some frequency intervals. The main
contribution of this paper is to provide the necessary analysis tools for a
quantitative study of these phenomena
Beyond the traditional Line-of-Sight approach of cosmological angular statistics
We present a new efficient method to compute the angular power spectra of
large-scale structure observables that circumvents the numerical integration
over Bessel functions, expanding on a recently proposed algorithm based on
FFTlog. This new approach has better convergence properties. The method is
explicitly implemented in the CLASS code for the case of number count
's (including redshift-space distortions, weak lensing, and all other
relativistic corrections) and cosmic shear 's. In both cases our
approach speeds up the calculation of the exact 's (without the Limber
approximation) by a factor of order 400 at a fixed precision target of 0.1%.Comment: 40 pages, 6 figures; v2: one reference adde
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