Discrete Hankel Transform Research Articles

Correction to: Fractionalization of a Discrete Hankel Transform Based on an Involutory Symmetric Kernel Matrix

Correction to: Fractionalization of a Discrete Hankel Transform Based on an Involutory Symmetric Kernel Matrix

Discrete fractional Hankel transform based on a nonsymmetric kernel matrix

A discrete fractional Hankel transform (DFRHT) is developed based on the eigen decomposition of a nonsymmetric involutory kernel matrix Y of a discrete Hankel transform (DHT). The two eigen spaces pertaining to the two distinct eigenvalues of Y are not orthogonal because Y is not a normal matrix. Explicit expressions are derived for the oblique projection matrices of Y on its eigen spaces by applying a contributed generalized spectral theorem of linear algebra. Expressions are derived for the dimensions of the two eigen spaces in terms of the trace of matrix Y. An analytic form is derived for the inverse of a modal matrix of Y in order to circumvent the need for numerically evaluating it when computing the sophisticated fractional transform.The fact that the Hankel transform (HT) is self-reciprocating implies that taking the HT once can be viewed as a rotation by an angle π radians in the space – Hankel frequency plane. Being a generalization of the HT, the fractional Hankel transform (FRHT) of fractional order a corresponds to a rotation by an arbitrary angle α where α=πa.Initial orthonormal bases are individually generated for the two eigen spaces by the singular value decomposition of the oblique projection matrices on those spaces. In order for the DFRHT to approximate its continuous counterpart, namely the FRHT, it is desirable to define it in terms of eigenvectors which are as close as possible to samples of the eigen functions of the FRHT. After deriving a three term recursion formula for the eigen functions, a sampling scheme is proposed such that a finite number of samples will be representative of the behavior of those functions. Final optimal orthonormal eigenvectors are individually generated for the two eigen spaces.

Fractionalization of a Discrete Hankel Transform Based on an Involutory Symmetric Kernel Matrix

Fractionalization of a Discrete Hankel Transform Based on an Involutory Symmetric Kernel Matrix

Full-Vectorial Fiber Mode Solver Based on a Discrete Hankel Transform

It is crucial to be time and resource-efficient when enabling and optimizing novel applications and functionalities of optical fibers, as well as accurate computation of the vectorial field components and the corresponding propagation constants of the guided modes in optical fibers. To address these needs, a novel full-vectorial fiber mode solver based on a discrete Hankel transform is introduced and validated here for the first time for rotationally symmetric fiber designs. It is shown that the effective refractive indices of the guided modes are computed with an absolute error of less than 10−4 with respect to analytical solutions of step-index and graded-index fiber designs. Computational speeds in the order of a few seconds allow to efficiently compute the relevant parameters, e.g., propagation constants and corresponding dispersion profiles, and to optimize fiber designs.

Modal Network Formulation for the Analysis and Design of Mode-Converting Metasurfaces in Cylindrical Waveguides

Modal network theory has proven to be a useful tool in analyzing waveguide discontinuities. Ports of a modal network correspond to waveguide eigenmodes on either side of the discontinuity. In this work, modal network theory is extended beyond conventional/geometrical waveguide discontinuities. We use the modal network formulation to analyze discontinuities resulting from metasurfaces. Specifically, we analyze azimuthally invariant cascaded sheet metasurfaces placed in a cylindrical waveguide excited by TM <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0n</sub> modes. The metasurface comprises multiple radially varying electric admittance sheets separated by dielectric spacers. We derive the modal representation of a single inhomogeneous, electric sheet admittance using two approaches: a conventional numerical integration approach and a discrete Hankel transform (DHT) approach. We show that the DHT approach is more efficient. Subsequently, the modal matrices of cascaded sheet metasurfaces are derived using the DHT. Finally, we propose an optimization-based procedure to synthesize transparent metasurface-based mode converters. The mode converters transform a set of incident modes on one side to another set of desired modes on the opposite side of the metasurface. Two designs are synthesized at 10 GHz: a single-mode converter, and a mode splitter. The designs are verified using the commercial finite element electromagnetic solvers.

Discrete two dimensional Fourier transform in polar coordinates part II: numerical computation and approximation of the continuous transform.

The theory of the continuous two-dimensional (2D) Fourier Transform in polar coordinates has been recently developed but no discrete counterpart exists to date. In the first part of this two-paper series, we proposed and evaluated the theory of the 2D Discrete Fourier Transform (DFT) in polar coordinates. The theory of the actual manipulated quantities was shown, including the standard set of shift, modulation, multiplication, and convolution rules. In this second part of the series, we address the computational aspects of the 2D DFT in polar coordinates. Specifically, we demonstrate how the decomposition of the 2D DFT as a DFT, Discrete Hankel Transform and inverse DFT sequence can be exploited for coding. We also demonstrate how the proposed 2D DFT can be used to approximate the continuous forward and inverse Fourier transform in polar coordinates in the same manner that the 1D DFT can be used to approximate its continuous counterpart.

Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules

The theory of the continuous two-dimensional (2D) Fourier transform in polar coordinates has been recently developed but no discrete counterpart exists to date. In this paper, we propose and evaluate the theory of the 2D discrete Fourier transform (DFT) in polar coordinates. This discrete theory is shown to arise from discretization schemes that have been previously employed with the 1D DFT and the discrete Hankel transform (DHT). The proposed transform possesses orthogonality properties, which leads to invertibility of the transform. In the first part of this two-part paper, the theory of the actual manipulated quantities is shown, including the standard set of shift, modulation, multiplication, and convolution rules. Parseval and modified Parseval relationships are shown, depending on which choice of kernel is used. Similar to its continuous counterpart, the 2D DFT in polar coordinates is shown to consist of a 1D DFT, DHT and 1D inverse DFT.

Matlab Code for the Discrete Hankel Transform

Previous definitions of a Discrete Hankel Transform (DHT) have focused on methods to approximate the continuous Hankel integral transform without regard for the properties of the DHT itself. Recently, the theory of a Discrete Hankel Transform was proposed that follows the same path as the Discrete Fourier/Continuous Fourier transform. This DHT possesses orthogonality properties which lead to invertibility and also possesses the standard set of discrete shift, modulation, multiplication and convolution rules. The proposed DHT can be used to approximate the continuous forward and inverse Hankel transform. This paper describes the matlab code developed for the numerical calculation of this DHT.

Theory and operational rules for the discrete Hankel transform.

Previous definitions of a discrete Hankel transform (DHT) have focused on methods to approximate the continuous Hankel integral transform. In this paper, we propose and evaluate the theory of a DHT that is shown to arise from a discretization scheme based on the theory of Fourier-Bessel expansions. The proposed transform also possesses requisite orthogonality properties which lead to invertibility of the transform. The standard set of shift, modulation, multiplication, and convolution rules are derived. In addition to the theory of the actual manipulated quantities which stand in their own right, this DHT can be used to approximate the continuous forward and inverse Hankel transform in the same manner that the discrete Fourier transform is known to be able to approximate the continuous Fourier transform.

Numerical solution of the modified Bessel equation

A Green’s function based solver for the modified Bessel equation has been developed with the primary motivation of solving the Poisson and biharmonic equations in cylindrical geometries. The method is implemented using a Discrete Hankel Transform and a Green’s function based on the modified Bessel functions of the first and second kind. The computation of these Bessel functions has been implemented to avoid scaling problems due to their exponential and singular behaviour, allowing the method to be used for large order problems, as would arise in solving the Poisson equation with a dense azimuthal grid. The method has been tested on monotonically decaying and oscillatory inputs, checking for errors due to interpolation and/or aliasing. The error has been found to reach machine precision and to have computational time linearly proportional to the number of nodes.

Reconstruction of optical fields with the Quasi-discrete Hankel transform

Unlike the FFT, the Quasi Discrete Hankel Transform (QDHT) is not sampled on a uniform grid; in particular the field may no longer be sampled on axis. We demonstrate how the generalised sampling theorem may be applied to optical problems, evaluated with the QDHT, to efficiently and accurately reconstruct the optical field at any point. Without sacrificing numerical accuracy this is demonstrated to be typically 50x faster than using an equivalent 2D FFT.