Пучки Гельмгольца--Гаусса с квадратичной радиальной зависимостью

A new class of localized solutions of paraxial parabolic equation is introduced. Each solution is a product of some Gaussian-type localized axisymmetric function (different from the fundamental mode) and an amplitude factor. The latter can be expressed via an arbitrary solution of the Helmholtz equation on an auxiliary two-sheet complex surface. The class under consideration contains well known and novel solutions, including those describing optical vortices of various orders.

An Optimal Fourth-Order Finite Difference Scheme for the Helmholtz Equation Based on the Technique of Matched Interface Boundary

In this paper, a new optimal fourth-order 21-point finite difference scheme is proposed to solve the 2D Helmholtz equation numerically, with the technique of matched interface boundary (MIB) utilized to treat boundary problems. For the approximation of Laplacian, two sets of fourth-order difference schemes are derived firstly based on the Taylor formula, with a total of 21 grid points involved. Then, a weighted combination of the two schemes is employed in order to reduce the numerical dispersion, and the weights are determined by minimizing the dispersion. Similarly, for the discretization of the zeroth-order derivative term, a weighted average of all the 21 points is implemented to obtain the fourth-order accuracy. The new scheme is noncompact; hence, it encounters great difficulties in dealing with the boundary conditions, which is crucial to the order of convergence. To tackle this issue, the matched interface boundary (MIB) method is employed and developed, which is originally used to accommodate free edges in the discrete singular convolution analysis. Convergence analysis and dispersion analysis are performed. Numerical examples are given for various boundary conditions, which show that new scheme delivers a fourth order of accuracy and is efficient in reducing the numerical dispersion as well.

The Theory of the Hydrogen Molecule Ion, Scalar Beams, and Scattering by Spheroids

<p>Schrodinger's equation for the hydrogen molecule ion and the Helmholtz equation are separable in prolate and oblate spheroidal coordinates respectively. They share the same form of the angular equation. The first task in deriving the ground state energy of the hydrogen molecule ion, and in obtaining finite solutions of the Helmholtz equation, is to obtain the physically allowed values of the separation of variables parameter. The separation parameter is not known analytically, and since it can only have certain values, it is an important parameter to quantify. Chapter 2 of this thesis investigates an exact method of obtaining the separation parameter. By showing that the angular equation is solvable in terms of confluent Heun functions, a new method to obtain the separation parameter was obtained. We showed that the physically allowed values of the separation of variables parameter are given by the zeros of the Wronskian of two linearly dependent solutions to the angular equation. Since the Heun functions are implemented in Maple, this new method allows the separation parameter to be calculated to unlimited precision. As Schrodinger's equation for the hydrogen molecule ion is related to Helmholtz's equation, this warranted investigation of scalar beams. Tightly focused optical and quantum particle beams are described by exact solutions of the Helmholtz equation. In Chapter 3 of this thesis we investigate the applicability of the separable spheroidal solutions of the scalar Helmholtz equation as physical beam solutions. By requiring a scalar beam solution to satisfy certain physical constraints, we showed that the oblate spheroidal wave functions can only represent nonparaxial scalar beams when the angular function is odd, in terms of the angular variable. This condition ensures the convergence of integrals of physical quantities over a cross-section of the beam and allows for the physically necessary discontinuity in phase at z = 0 on the ellipsoidal surfaces of otherwise constant phase. However, these solutions were shown to have a discontinuous longitudinal derivative. Finally, we investigated the scattering of scalar waves by oblate and prolate spheroids whose symmetry axis is coincident with the direction of the incident plane wave. We developed a phase shift formulation of scattering by oblate and prolate spheroids, in parallel with the partial wave theory of scattering by spherical obstacles. The crucial step was application of a finite Legendre transform to the Helmholtz equation in spheroidal coordinates. Analytical results were readily obtained for scattering of Schrodinger particle waves by impenetrable spheroids and for scattering of sound waves by acoustically soft spheroids. The advantage of this theory is that it enables all that can be done for scattering by spherical obstacles to be carried over to the scattering by spheroids, provided the radial eigenfunctions are known.</p>

The Theory of the Hydrogen Molecule Ion, Scalar Beams, and Scattering by Spheroids

<p>Schrodinger's equation for the hydrogen molecule ion and the Helmholtz equation are separable in prolate and oblate spheroidal coordinates respectively. They share the same form of the angular equation. The first task in deriving the ground state energy of the hydrogen molecule ion, and in obtaining finite solutions of the Helmholtz equation, is to obtain the physically allowed values of the separation of variables parameter. The separation parameter is not known analytically, and since it can only have certain values, it is an important parameter to quantify. Chapter 2 of this thesis investigates an exact method of obtaining the separation parameter. By showing that the angular equation is solvable in terms of confluent Heun functions, a new method to obtain the separation parameter was obtained. We showed that the physically allowed values of the separation of variables parameter are given by the zeros of the Wronskian of two linearly dependent solutions to the angular equation. Since the Heun functions are implemented in Maple, this new method allows the separation parameter to be calculated to unlimited precision. As Schrodinger's equation for the hydrogen molecule ion is related to Helmholtz's equation, this warranted investigation of scalar beams. Tightly focused optical and quantum particle beams are described by exact solutions of the Helmholtz equation. In Chapter 3 of this thesis we investigate the applicability of the separable spheroidal solutions of the scalar Helmholtz equation as physical beam solutions. By requiring a scalar beam solution to satisfy certain physical constraints, we showed that the oblate spheroidal wave functions can only represent nonparaxial scalar beams when the angular function is odd, in terms of the angular variable. This condition ensures the convergence of integrals of physical quantities over a cross-section of the beam and allows for the physically necessary discontinuity in phase at z = 0 on the ellipsoidal surfaces of otherwise constant phase. However, these solutions were shown to have a discontinuous longitudinal derivative. Finally, we investigated the scattering of scalar waves by oblate and prolate spheroids whose symmetry axis is coincident with the direction of the incident plane wave. We developed a phase shift formulation of scattering by oblate and prolate spheroids, in parallel with the partial wave theory of scattering by spherical obstacles. The crucial step was application of a finite Legendre transform to the Helmholtz equation in spheroidal coordinates. Analytical results were readily obtained for scattering of Schrodinger particle waves by impenetrable spheroids and for scattering of sound waves by acoustically soft spheroids. The advantage of this theory is that it enables all that can be done for scattering by spherical obstacles to be carried over to the scattering by spheroids, provided the radial eigenfunctions are known.</p>