Gaussian amplitude functions that are exact solutions to the scalar Helmholtz equation
نویسندگان
چکیده
A new family of exact solutions of the scalar Helmholtz equation is presented. The 0, 0 order of this family represents a new mathematical model for the fundamental mode of a propagating Gaussian beam. The family consists of nonseparable functions in the oblate spheroidal coordinate system and can easily by transformed into a different set of solutions in the prolate spheroidal coordinate system, where the 0, 0 order is a spherical wave. This transformation consists of two substitutions in the coordinate system parameters and represents a more general method of obtaining a Gaussian beam from a spherical wave than assuming a complex point source on axis. Finally, each higher-order member of the family of solutions possesses an amplitude consisting of a finite number of higherorder terms with a zero-order term that is Gaussian.
منابع مشابه
Exact nonparaxial beams of the scalar Helmholtz equation.
It is shown that three-dimensional nonparaxial beams are described by the oblate spheroidal exact solutions of the Helmholtz equation. For what is believed to be the first time, their beam behavior is investigated and their corresponding parameters are defined. Using the fact that the beam width of the family of paraxial Gaussian beams is described by a hyperbola, we formally establish the conn...
متن کاملExplicit multiple singular periodic solutions and singular soliton solutions to KdV equation
Based on some stationary periodic solutions and stationary soliton solutions, one studies the general solution for the relative lax system, and a number of exact solutions to the Korteweg-de Vries (KdV) equation are first constructed by the known Darboux transformation, these solutions include double and triple singular periodic solutions as well as singular soliton solutions whose amplitude d...
متن کاملGaussian beam in oblate spheroidal coordinates
A new geometrical model for the fundamental mode of a Gaussian beam is presented in the oblate spheroidal coordinate system. The model is an interpretation of a Gaussian amplitude wave function, which is an exact solution of the scalar Helmholtz equation. The model uses the skew-line generator of a hyperboloid of one sheet as a raylike element on a contour of constant amplitude. The geometrical...
متن کاملStable Gaussian radial basis function method for solving Helmholtz equations
Radial basis functions (RBFs) are a powerful tool for approximating the solution of high-dimensional problems. They are often referred to as a meshfree method and can be spectrally accurate. In this paper, we analyze a new stable method for evaluating Gaussian radial basis function interpolants based on the eigenfunction expansion. We develop our approach in two-dimensional spaces for so...
متن کاملNumerical solution of second-order stochastic differential equations with Gaussian random parameters
In this paper, we present the numerical solution of ordinary differential equations (or SDEs), from each order especially second-order with time-varying and Gaussian random coefficients. We indicate a complete analysis for second-order equations in special case of scalar linear second-order equations (damped harmonic oscillators with additive or multiplicative noises). Making stochastic differe...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 1987